author | nipkow |
Tue, 05 Nov 2019 13:56:34 +0100 | |
changeset 71032 | acedd04c1a42 |
parent 71029 | 934e0044e94b |
parent 71031 | 66c025383422 |
child 71172 | 575b3a818de5 |
permissions | -rw-r--r-- |
69517 | 1 |
section \<open>Complex Path Integrals and Cauchy's Integral Theorem\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3 |
text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2015)\<close> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
4 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63589
diff
changeset
|
5 |
theory Cauchy_Integral_Theorem |
71031 | 6 |
imports |
7 |
Complex_Transcendental |
|
8 |
Henstock_Kurzweil_Integration |
|
9 |
Weierstrass_Theorems |
|
10 |
Retracts |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
11 |
begin |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
12 |
|
70196
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
13 |
lemma leibniz_rule_holomorphic: |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
14 |
fixes f::"complex \<Rightarrow> 'b::euclidean_space \<Rightarrow> complex" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
15 |
assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
16 |
assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
17 |
assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
18 |
assumes "convex U" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
19 |
shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
20 |
using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)] |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
21 |
by (auto simp: holomorphic_on_def) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
22 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
23 |
lemma Ln_measurable [measurable]: "Ln \<in> measurable borel borel" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
24 |
proof - |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
25 |
have *: "Ln (-of_real x) = of_real (ln x) + \<i> * pi" if "x > 0" for x |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
26 |
using that by (subst Ln_minus) (auto simp: Ln_of_real) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
27 |
have **: "Ln (of_real x) = of_real (ln (-x)) + \<i> * pi" if "x < 0" for x |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
28 |
using *[of "-x"] that by simp |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
29 |
have cont: "(\<lambda>x. indicat_real (- \<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel_measurable borel" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
30 |
by (intro borel_measurable_continuous_on_indicator continuous_intros) auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
31 |
have "(\<lambda>x. if x \<in> \<real>\<^sub>\<le>\<^sub>0 then ln (-Re x) + \<i> * pi else indicator (-\<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel \<rightarrow>\<^sub>M borel" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
32 |
(is "?f \<in> _") by (rule measurable_If_set[OF _ cont]) auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
33 |
hence "(\<lambda>x. if x = 0 then Ln 0 else ?f x) \<in> borel \<rightarrow>\<^sub>M borel" by measurable |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
34 |
also have "(\<lambda>x. if x = 0 then Ln 0 else ?f x) = Ln" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
35 |
by (auto simp: fun_eq_iff ** nonpos_Reals_def) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
36 |
finally show ?thesis . |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
37 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
38 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
39 |
lemma powr_complex_measurable [measurable]: |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
40 |
assumes [measurable]: "f \<in> measurable M borel" "g \<in> measurable M borel" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
41 |
shows "(\<lambda>x. f x powr g x :: complex) \<in> measurable M borel" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
42 |
using assms by (simp add: powr_def) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
43 |
|
70136 | 44 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Homeomorphisms of arc images\<close> |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
45 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
46 |
lemma homeomorphism_arc: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
47 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
48 |
assumes "arc g" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
49 |
obtains h where "homeomorphism {0..1} (path_image g) g h" |
68339 | 50 |
using assms by (force simp: arc_def homeomorphism_compact path_def path_image_def) |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
51 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
52 |
lemma homeomorphic_arc_image_interval: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
53 |
fixes g :: "real \<Rightarrow> 'a::t2_space" and a::real |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
54 |
assumes "arc g" "a < b" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
55 |
shows "(path_image g) homeomorphic {a..b}" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
56 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
57 |
have "(path_image g) homeomorphic {0..1::real}" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
58 |
by (meson assms(1) homeomorphic_def homeomorphic_sym homeomorphism_arc) |
68339 | 59 |
also have "\<dots> homeomorphic {a..b}" |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
60 |
using assms by (force intro: homeomorphic_closed_intervals_real) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
61 |
finally show ?thesis . |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
62 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
63 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
64 |
lemma homeomorphic_arc_images: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
65 |
fixes g :: "real \<Rightarrow> 'a::t2_space" and h :: "real \<Rightarrow> 'b::t2_space" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
66 |
assumes "arc g" "arc h" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
67 |
shows "(path_image g) homeomorphic (path_image h)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
68 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
69 |
have "(path_image g) homeomorphic {0..1::real}" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
70 |
by (meson assms homeomorphic_def homeomorphic_sym homeomorphism_arc) |
68339 | 71 |
also have "\<dots> homeomorphic (path_image h)" |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
72 |
by (meson assms homeomorphic_def homeomorphism_arc) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
73 |
finally show ?thesis . |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
74 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
75 |
|
65037
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
76 |
lemma path_connected_arc_complement: |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
77 |
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
78 |
assumes "arc \<gamma>" "2 \<le> DIM('a)" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
79 |
shows "path_connected(- path_image \<gamma>)" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
80 |
proof - |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
81 |
have "path_image \<gamma> homeomorphic {0..1::real}" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
82 |
by (simp add: assms homeomorphic_arc_image_interval) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
83 |
then |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
84 |
show ?thesis |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
85 |
apply (rule path_connected_complement_homeomorphic_convex_compact) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
86 |
apply (auto simp: assms) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
87 |
done |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
88 |
qed |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
89 |
|
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
90 |
lemma connected_arc_complement: |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
91 |
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
92 |
assumes "arc \<gamma>" "2 \<le> DIM('a)" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
93 |
shows "connected(- path_image \<gamma>)" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
94 |
by (simp add: assms path_connected_arc_complement path_connected_imp_connected) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
95 |
|
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
96 |
lemma inside_arc_empty: |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
97 |
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
98 |
assumes "arc \<gamma>" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
99 |
shows "inside(path_image \<gamma>) = {}" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
100 |
proof (cases "DIM('a) = 1") |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
101 |
case True |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
102 |
then show ?thesis |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
103 |
using assms connected_arc_image connected_convex_1_gen inside_convex by blast |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
104 |
next |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
105 |
case False |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
106 |
show ?thesis |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
107 |
proof (rule inside_bounded_complement_connected_empty) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
108 |
show "connected (- path_image \<gamma>)" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
109 |
apply (rule connected_arc_complement [OF assms]) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
110 |
using False |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
111 |
by (metis DIM_ge_Suc0 One_nat_def Suc_1 not_less_eq_eq order_class.order.antisym) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
112 |
show "bounded (path_image \<gamma>)" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
113 |
by (simp add: assms bounded_arc_image) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
114 |
qed |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
115 |
qed |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
116 |
|
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
117 |
lemma inside_simple_curve_imp_closed: |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
118 |
fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
119 |
shows "\<lbrakk>simple_path \<gamma>; x \<in> inside(path_image \<gamma>)\<rbrakk> \<Longrightarrow> pathfinish \<gamma> = pathstart \<gamma>" |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
120 |
using arc_simple_path inside_arc_empty by blast |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
121 |
|
68493 | 122 |
|
70136 | 123 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Piecewise differentiable functions\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
124 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
125 |
definition piecewise_differentiable_on |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
126 |
(infixr "piecewise'_differentiable'_on" 50) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
127 |
where "f piecewise_differentiable_on i \<equiv> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
128 |
continuous_on i f \<and> |
68284 | 129 |
(\<exists>S. finite S \<and> (\<forall>x \<in> i - S. f differentiable (at x within i)))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
130 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
131 |
lemma piecewise_differentiable_on_imp_continuous_on: |
68284 | 132 |
"f piecewise_differentiable_on S \<Longrightarrow> continuous_on S f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
133 |
by (simp add: piecewise_differentiable_on_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
134 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
135 |
lemma piecewise_differentiable_on_subset: |
68284 | 136 |
"f piecewise_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_differentiable_on T" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
137 |
using continuous_on_subset |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
138 |
unfolding piecewise_differentiable_on_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
139 |
apply safe |
68339 | 140 |
apply (blast elim: continuous_on_subset) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
141 |
by (meson Diff_iff differentiable_within_subset subsetCE) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
142 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
143 |
lemma differentiable_on_imp_piecewise_differentiable: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
144 |
fixes a:: "'a::{linorder_topology,real_normed_vector}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
145 |
shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
146 |
apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
147 |
apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
148 |
done |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
149 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
150 |
lemma differentiable_imp_piecewise_differentiable: |
68284 | 151 |
"(\<And>x. x \<in> S \<Longrightarrow> f differentiable (at x within S)) |
152 |
\<Longrightarrow> f piecewise_differentiable_on S" |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
153 |
by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
154 |
intro: differentiable_within_subset) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
155 |
|
68284 | 156 |
lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on S" |
61204 | 157 |
by (simp add: differentiable_imp_piecewise_differentiable) |
158 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
159 |
lemma piecewise_differentiable_compose: |
68284 | 160 |
"\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on (f ` S); |
161 |
\<And>x. finite (S \<inter> f-`{x})\<rbrakk> |
|
68339 | 162 |
\<Longrightarrow> (g \<circ> f) piecewise_differentiable_on S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
163 |
apply (simp add: piecewise_differentiable_on_def, safe) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
164 |
apply (blast intro: continuous_on_compose2) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
165 |
apply (rename_tac A B) |
68284 | 166 |
apply (rule_tac x="A \<union> (\<Union>x\<in>B. S \<inter> f-`{x})" in exI) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
167 |
apply (blast intro!: differentiable_chain_within) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
168 |
done |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
169 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
170 |
lemma piecewise_differentiable_affine: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
171 |
fixes m::real |
68284 | 172 |
assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` S)" |
68339 | 173 |
shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
174 |
proof (cases "m = 0") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
175 |
case True |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
176 |
then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
177 |
unfolding o_def |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
178 |
by (force intro: differentiable_imp_piecewise_differentiable differentiable_const) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
179 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
180 |
case False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
181 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
182 |
apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
183 |
apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
184 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
185 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
186 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
187 |
lemma piecewise_differentiable_cases: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
188 |
fixes c::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
189 |
assumes "f piecewise_differentiable_on {a..c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
190 |
"g piecewise_differentiable_on {c..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
191 |
"a \<le> c" "c \<le> b" "f c = g c" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
192 |
shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
193 |
proof - |
68284 | 194 |
obtain S T where st: "finite S" "finite T" |
195 |
and fd: "\<And>x. x \<in> {a..c} - S \<Longrightarrow> f differentiable at x within {a..c}" |
|
196 |
and gd: "\<And>x. x \<in> {c..b} - T \<Longrightarrow> g differentiable at x within {c..b}" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
197 |
using assms |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
198 |
by (auto simp: piecewise_differentiable_on_def) |
68284 | 199 |
have finabc: "finite ({a,b,c} \<union> (S \<union> T))" |
200 |
by (metis \<open>finite S\<close> \<open>finite T\<close> finite_Un finite_insert finite.emptyI) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
201 |
have "continuous_on {a..c} f" "continuous_on {c..b} g" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
202 |
using assms piecewise_differentiable_on_def by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
203 |
then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
204 |
using continuous_on_cases [OF closed_real_atLeastAtMost [of a c], |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
205 |
OF closed_real_atLeastAtMost [of c b], |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
206 |
of f g "\<lambda>x. x\<le>c"] assms |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
207 |
by (force simp: ivl_disj_un_two_touch) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
208 |
moreover |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
209 |
{ fix x |
68284 | 210 |
assume x: "x \<in> {a..b} - ({a,b,c} \<union> (S \<union> T))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
211 |
have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg") |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
212 |
proof (cases x c rule: le_cases) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
213 |
case le show ?diff_fg |
68284 | 214 |
proof (rule differentiable_transform_within [where d = "dist x c"]) |
215 |
have "f differentiable at x" |
|
216 |
using x le fd [of x] at_within_interior [of x "{a..c}"] by simp |
|
217 |
then show "f differentiable at x within {a..b}" |
|
218 |
by (simp add: differentiable_at_withinI) |
|
63955 | 219 |
qed (use x le st dist_real_def in auto) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
220 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
221 |
case ge show ?diff_fg |
68284 | 222 |
proof (rule differentiable_transform_within [where d = "dist x c"]) |
223 |
have "g differentiable at x" |
|
224 |
using x ge gd [of x] at_within_interior [of x "{c..b}"] by simp |
|
225 |
then show "g differentiable at x within {a..b}" |
|
226 |
by (simp add: differentiable_at_withinI) |
|
63955 | 227 |
qed (use x ge st dist_real_def in auto) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
228 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
229 |
} |
68284 | 230 |
then have "\<exists>S. finite S \<and> |
231 |
(\<forall>x\<in>{a..b} - S. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})" |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
232 |
by (meson finabc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
233 |
ultimately show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
234 |
by (simp add: piecewise_differentiable_on_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
235 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
236 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
237 |
lemma piecewise_differentiable_neg: |
68284 | 238 |
"f piecewise_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
239 |
by (auto simp: piecewise_differentiable_on_def continuous_on_minus) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
240 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
241 |
lemma piecewise_differentiable_add: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
242 |
assumes "f piecewise_differentiable_on i" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
243 |
"g piecewise_differentiable_on i" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
244 |
shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
245 |
proof - |
68284 | 246 |
obtain S T where st: "finite S" "finite T" |
247 |
"\<forall>x\<in>i - S. f differentiable at x within i" |
|
248 |
"\<forall>x\<in>i - T. g differentiable at x within i" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
249 |
using assms by (auto simp: piecewise_differentiable_on_def) |
68284 | 250 |
then have "finite (S \<union> T) \<and> (\<forall>x\<in>i - (S \<union> T). (\<lambda>x. f x + g x) differentiable at x within i)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
251 |
by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
252 |
moreover have "continuous_on i f" "continuous_on i g" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
253 |
using assms piecewise_differentiable_on_def by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
254 |
ultimately show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
255 |
by (auto simp: piecewise_differentiable_on_def continuous_on_add) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
256 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
257 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
258 |
lemma piecewise_differentiable_diff: |
68284 | 259 |
"\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on S\<rbrakk> |
260 |
\<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on S" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
261 |
unfolding diff_conv_add_uminus |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
262 |
by (metis piecewise_differentiable_add piecewise_differentiable_neg) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
263 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
264 |
lemma continuous_on_joinpaths_D1: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
265 |
"continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1" |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
266 |
apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> ((*)(inverse 2))"]) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
267 |
apply (rule continuous_intros | simp)+ |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
268 |
apply (auto elim!: continuous_on_subset simp: joinpaths_def) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
269 |
done |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
270 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
271 |
lemma continuous_on_joinpaths_D2: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
272 |
"\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2" |
68339 | 273 |
apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> (\<lambda>x. inverse 2*x + 1/2)"]) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
274 |
apply (rule continuous_intros | simp)+ |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
275 |
apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
276 |
done |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
277 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
278 |
lemma piecewise_differentiable_D1: |
68284 | 279 |
assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}" |
280 |
shows "g1 piecewise_differentiable_on {0..1}" |
|
281 |
proof - |
|
282 |
obtain S where cont: "continuous_on {0..1} g1" and "finite S" |
|
283 |
and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}" |
|
284 |
using assms unfolding piecewise_differentiable_on_def |
|
285 |
by (blast dest!: continuous_on_joinpaths_D1) |
|
286 |
show ?thesis |
|
287 |
unfolding piecewise_differentiable_on_def |
|
288 |
proof (intro exI conjI ballI cont) |
|
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
289 |
show "finite (insert 1 (((*)2) ` S))" |
68284 | 290 |
by (simp add: \<open>finite S\<close>) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
291 |
show "g1 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x |
68284 | 292 |
proof (rule_tac d="dist (x/2) (1/2)" in differentiable_transform_within) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
293 |
have "g1 +++ g2 differentiable at (x / 2) within {0..1/2}" |
68284 | 294 |
by (rule differentiable_subset [OF S [of "x/2"]] | use that in force)+ |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
295 |
then show "g1 +++ g2 \<circ> (*) (inverse 2) differentiable at x within {0..1}" |
69661 | 296 |
using image_affinity_atLeastAtMost_div [of 2 0 "0::real" 1] |
68284 | 297 |
by (auto intro: differentiable_chain_within) |
298 |
qed (use that in \<open>auto simp: joinpaths_def\<close>) |
|
299 |
qed |
|
300 |
qed |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
301 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
302 |
lemma piecewise_differentiable_D2: |
68284 | 303 |
assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}" and eq: "pathfinish g1 = pathstart g2" |
304 |
shows "g2 piecewise_differentiable_on {0..1}" |
|
305 |
proof - |
|
306 |
have [simp]: "g1 1 = g2 0" |
|
307 |
using eq by (simp add: pathfinish_def pathstart_def) |
|
308 |
obtain S where cont: "continuous_on {0..1} g2" and "finite S" |
|
309 |
and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}" |
|
310 |
using assms unfolding piecewise_differentiable_on_def |
|
311 |
by (blast dest!: continuous_on_joinpaths_D2) |
|
312 |
show ?thesis |
|
313 |
unfolding piecewise_differentiable_on_def |
|
314 |
proof (intro exI conjI ballI cont) |
|
315 |
show "finite (insert 0 ((\<lambda>x. 2*x-1)`S))" |
|
316 |
by (simp add: \<open>finite S\<close>) |
|
317 |
show "g2 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1)`S)" for x |
|
318 |
proof (rule_tac d="dist ((x+1)/2) (1/2)" in differentiable_transform_within) |
|
319 |
have x2: "(x + 1) / 2 \<notin> S" |
|
320 |
using that |
|
321 |
apply (clarsimp simp: image_iff) |
|
68527
2f4e2aab190a
Generalising and renaming some basic results
paulson <lp15@cam.ac.uk>
parents:
68493
diff
changeset
|
322 |
by (metis add.commute add_diff_cancel_left' mult_2 field_sum_of_halves) |
68284 | 323 |
have "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}" |
324 |
by (rule differentiable_chain_within differentiable_subset [OF S [of "(x+1)/2"]] | use x2 that in force)+ |
|
325 |
then show "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}" |
|
326 |
by (auto intro: differentiable_chain_within) |
|
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
327 |
show "(g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'" if "x' \<in> {0..1}" "dist x' x < dist ((x + 1) / 2) (1/2)" for x' |
68284 | 328 |
proof - |
329 |
have [simp]: "(2*x'+2)/2 = x'+1" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
330 |
by (simp add: field_split_simps) |
68284 | 331 |
show ?thesis |
332 |
using that by (auto simp: joinpaths_def) |
|
333 |
qed |
|
334 |
qed (use that in \<open>auto simp: joinpaths_def\<close>) |
|
335 |
qed |
|
336 |
qed |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
337 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
338 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
339 |
subsection\<open>The concept of continuously differentiable\<close> |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
340 |
|
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
341 |
text \<open> |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
342 |
John Harrison writes as follows: |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
343 |
|
62456 | 344 |
``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
345 |
continuously differentiable, which ensures that the path integral exists at least for any continuous |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
346 |
f, since all piecewise continuous functions are integrable. However, our notion of validity is |
68341 | 347 |
weaker, just piecewise differentiability\ldots{} [namely] continuity plus differentiability except on a |
348 |
finite set\ldots{} [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to |
|
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
349 |
the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
350 |
can integrate all derivatives.'' |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
351 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
352 |
"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
353 |
Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165. |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
354 |
|
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
355 |
And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
356 |
difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
357 |
asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close> |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
358 |
|
70136 | 359 |
definition\<^marker>\<open>tag important\<close> C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
360 |
(infix "C1'_differentiable'_on" 50) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
361 |
where |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
362 |
"f C1_differentiable_on S \<longleftrightarrow> |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
363 |
(\<exists>D. (\<forall>x \<in> S. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on S D)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
364 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
365 |
lemma C1_differentiable_on_eq: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
366 |
"f C1_differentiable_on S \<longleftrightarrow> |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
367 |
(\<forall>x \<in> S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
368 |
(is "?lhs = ?rhs") |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
369 |
proof |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
370 |
assume ?lhs |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
371 |
then show ?rhs |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
372 |
unfolding C1_differentiable_on_def |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
373 |
by (metis (no_types, lifting) continuous_on_eq differentiableI_vector vector_derivative_at) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
374 |
next |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
375 |
assume ?rhs |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
376 |
then show ?lhs |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
377 |
using C1_differentiable_on_def vector_derivative_works by fastforce |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
378 |
qed |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
379 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
380 |
lemma C1_differentiable_on_subset: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
381 |
"f C1_differentiable_on T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> f C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
382 |
unfolding C1_differentiable_on_def continuous_on_eq_continuous_within |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
383 |
by (blast intro: continuous_within_subset) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
384 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
385 |
lemma C1_differentiable_compose: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
386 |
assumes fg: "f C1_differentiable_on S" "g C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})" |
68339 | 387 |
shows "(g \<circ> f) C1_differentiable_on S" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
388 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
389 |
have "\<And>x. x \<in> S \<Longrightarrow> g \<circ> f differentiable at x" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
390 |
by (meson C1_differentiable_on_eq assms differentiable_chain_at imageI) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
391 |
moreover have "continuous_on S (\<lambda>x. vector_derivative (g \<circ> f) (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
392 |
proof (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"]) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
393 |
show "continuous_on S (\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
394 |
using fg |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
395 |
apply (clarsimp simp add: C1_differentiable_on_eq) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
396 |
apply (rule Limits.continuous_on_scaleR, assumption) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
397 |
by (metis (mono_tags, lifting) continuous_at_imp_continuous_on continuous_on_compose continuous_on_cong differentiable_imp_continuous_within o_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
398 |
show "\<And>x. x \<in> S \<Longrightarrow> vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)) = vector_derivative (g \<circ> f) (at x)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
399 |
by (metis (mono_tags, hide_lams) C1_differentiable_on_eq fg imageI vector_derivative_chain_at) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
400 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
401 |
ultimately show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
402 |
by (simp add: C1_differentiable_on_eq) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
403 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
404 |
|
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
405 |
lemma C1_diff_imp_diff: "f C1_differentiable_on S \<Longrightarrow> f differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
406 |
by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
407 |
|
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
408 |
lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
409 |
by (auto simp: C1_differentiable_on_eq continuous_on_const) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
410 |
|
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
411 |
lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
412 |
by (auto simp: C1_differentiable_on_eq continuous_on_const) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
413 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
414 |
lemma C1_differentiable_on_add [simp, derivative_intros]: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
415 |
"f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
416 |
unfolding C1_differentiable_on_eq by (auto intro: continuous_intros) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
417 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
418 |
lemma C1_differentiable_on_minus [simp, derivative_intros]: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
419 |
"f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
420 |
unfolding C1_differentiable_on_eq by (auto intro: continuous_intros) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
421 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
422 |
lemma C1_differentiable_on_diff [simp, derivative_intros]: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
423 |
"f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
424 |
unfolding C1_differentiable_on_eq by (auto intro: continuous_intros) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
425 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
426 |
lemma C1_differentiable_on_mult [simp, derivative_intros]: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
427 |
fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
428 |
shows "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
429 |
unfolding C1_differentiable_on_eq |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
430 |
by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
431 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
432 |
lemma C1_differentiable_on_scaleR [simp, derivative_intros]: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
433 |
"f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
434 |
unfolding C1_differentiable_on_eq |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
435 |
by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+ |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
436 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
437 |
|
70136 | 438 |
definition\<^marker>\<open>tag important\<close> piecewise_C1_differentiable_on |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
439 |
(infixr "piecewise'_C1'_differentiable'_on" 50) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
440 |
where "f piecewise_C1_differentiable_on i \<equiv> |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
441 |
continuous_on i f \<and> |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
442 |
(\<exists>S. finite S \<and> (f C1_differentiable_on (i - S)))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
443 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
444 |
lemma C1_differentiable_imp_piecewise: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
445 |
"f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
446 |
by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
447 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
448 |
lemma piecewise_C1_imp_differentiable: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
449 |
"f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
450 |
by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
451 |
C1_differentiable_on_def differentiable_def has_vector_derivative_def |
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
452 |
intro: has_derivative_at_withinI) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
453 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
454 |
lemma piecewise_C1_differentiable_compose: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
455 |
assumes fg: "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})" |
68339 | 456 |
shows "(g \<circ> f) piecewise_C1_differentiable_on S" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
457 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
458 |
have "continuous_on S (\<lambda>x. g (f x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
459 |
by (metis continuous_on_compose2 fg order_refl piecewise_C1_differentiable_on_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
460 |
moreover have "\<exists>T. finite T \<and> g \<circ> f C1_differentiable_on S - T" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
461 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
462 |
obtain F where "finite F" and F: "f C1_differentiable_on S - F" and f: "f piecewise_C1_differentiable_on S" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
463 |
using fg by (auto simp: piecewise_C1_differentiable_on_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
464 |
obtain G where "finite G" and G: "g C1_differentiable_on f ` S - G" and g: "g piecewise_C1_differentiable_on f ` S" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
465 |
using fg by (auto simp: piecewise_C1_differentiable_on_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
466 |
show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
467 |
proof (intro exI conjI) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
468 |
show "finite (F \<union> (\<Union>x\<in>G. S \<inter> f-`{x}))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
469 |
using fin by (auto simp only: Int_Union \<open>finite F\<close> \<open>finite G\<close> finite_UN finite_imageI) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
470 |
show "g \<circ> f C1_differentiable_on S - (F \<union> (\<Union>x\<in>G. S \<inter> f -` {x}))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
471 |
apply (rule C1_differentiable_compose) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
472 |
apply (blast intro: C1_differentiable_on_subset [OF F]) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
473 |
apply (blast intro: C1_differentiable_on_subset [OF G]) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
474 |
by (simp add: C1_differentiable_on_subset G Diff_Int_distrib2 fin) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
475 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
476 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
477 |
ultimately show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
478 |
by (simp add: piecewise_C1_differentiable_on_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
479 |
qed |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
480 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
481 |
lemma piecewise_C1_differentiable_on_subset: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
482 |
"f piecewise_C1_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_C1_differentiable_on T" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
483 |
by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
484 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
485 |
lemma C1_differentiable_imp_continuous_on: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
486 |
"f C1_differentiable_on S \<Longrightarrow> continuous_on S f" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
487 |
unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
488 |
using differentiable_at_withinI differentiable_imp_continuous_within by blast |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
489 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
490 |
lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
491 |
unfolding C1_differentiable_on_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
492 |
by auto |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
493 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
494 |
lemma piecewise_C1_differentiable_affine: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
495 |
fixes m::real |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
496 |
assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` S)" |
68339 | 497 |
shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
498 |
proof (cases "m = 0") |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
499 |
case True |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
500 |
then show ?thesis |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
501 |
unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
502 |
next |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
503 |
case False |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
504 |
have *: "\<And>x. finite (S \<inter> {y. m * y + c = x})" |
68493 | 505 |
using False not_finite_existsD by fastforce |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
506 |
show ?thesis |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
507 |
apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise]) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
508 |
apply (rule * assms derivative_intros | simp add: False vimage_def)+ |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
509 |
done |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
510 |
qed |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
511 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
512 |
lemma piecewise_C1_differentiable_cases: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
513 |
fixes c::real |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
514 |
assumes "f piecewise_C1_differentiable_on {a..c}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
515 |
"g piecewise_C1_differentiable_on {c..b}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
516 |
"a \<le> c" "c \<le> b" "f c = g c" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
517 |
shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
518 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
519 |
obtain S T where st: "f C1_differentiable_on ({a..c} - S)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
520 |
"g C1_differentiable_on ({c..b} - T)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
521 |
"finite S" "finite T" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
522 |
using assms |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
523 |
by (force simp: piecewise_C1_differentiable_on_def) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
524 |
then have f_diff: "f differentiable_on {a..<c} - S" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
525 |
and g_diff: "g differentiable_on {c<..b} - T" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
526 |
by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
527 |
have "continuous_on {a..c} f" "continuous_on {c..b} g" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
528 |
using assms piecewise_C1_differentiable_on_def by auto |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
529 |
then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
530 |
using continuous_on_cases [OF closed_real_atLeastAtMost [of a c], |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
531 |
OF closed_real_atLeastAtMost [of c b], |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
532 |
of f g "\<lambda>x. x\<le>c"] assms |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
533 |
by (force simp: ivl_disj_un_two_touch) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
534 |
{ fix x |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
535 |
assume x: "x \<in> {a..b} - insert c (S \<union> T)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
536 |
have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg") |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
537 |
proof (cases x c rule: le_cases) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
538 |
case le show ?diff_fg |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
539 |
apply (rule differentiable_transform_within [where f=f and d = "dist x c"]) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
540 |
using x dist_real_def le st by (auto simp: C1_differentiable_on_eq) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
541 |
next |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
542 |
case ge show ?diff_fg |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
543 |
apply (rule differentiable_transform_within [where f=g and d = "dist x c"]) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
544 |
using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
545 |
qed |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
546 |
} |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
547 |
then have "(\<forall>x \<in> {a..b} - insert c (S \<union> T). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
548 |
by auto |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
549 |
moreover |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
550 |
{ assume fcon: "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative f (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
551 |
and gcon: "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative g (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
552 |
have "open ({a<..<c} - S)" "open ({c<..<b} - T)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
553 |
using st by (simp_all add: open_Diff finite_imp_closed) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
554 |
moreover have "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
555 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
556 |
have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative f (at x)) (at x)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
557 |
if "a < x" "x < c" "x \<notin> S" for x |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
558 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
559 |
have f: "f differentiable at x" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
560 |
by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_eq_real_def st(1) that) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
561 |
show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
562 |
using that |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
563 |
apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within) |
68339 | 564 |
apply (auto simp: dist_norm vector_derivative_works [symmetric] f) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
565 |
done |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
566 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
567 |
then show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
568 |
by (metis (no_types, lifting) continuous_on_eq [OF fcon] DiffE greaterThanLessThan_iff vector_derivative_at) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
569 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
570 |
moreover have "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
571 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
572 |
have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative g (at x)) (at x)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
573 |
if "c < x" "x < b" "x \<notin> T" for x |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
574 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
575 |
have g: "g differentiable at x" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
576 |
by (metis C1_differentiable_on_eq DiffD1 DiffI atLeastAtMost_diff_ends greaterThanLessThan_iff st(2) that) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
577 |
show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
578 |
using that |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
579 |
apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within) |
68339 | 580 |
apply (auto simp: dist_norm vector_derivative_works [symmetric] g) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
581 |
done |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
582 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
583 |
then show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
584 |
by (metis (no_types, lifting) continuous_on_eq [OF gcon] DiffE greaterThanLessThan_iff vector_derivative_at) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
585 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
586 |
ultimately have "continuous_on ({a<..<b} - insert c (S \<union> T)) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
587 |
(\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
588 |
by (rule continuous_on_subset [OF continuous_on_open_Un], auto) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
589 |
} note * = this |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
590 |
have "continuous_on ({a<..<b} - insert c (S \<union> T)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
591 |
using st |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
592 |
by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
593 |
ultimately have "\<exists>S. finite S \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - S)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
594 |
apply (rule_tac x="{a,b,c} \<union> S \<union> T" in exI) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
595 |
using st by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
596 |
with cab show ?thesis |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
597 |
by (simp add: piecewise_C1_differentiable_on_def) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
598 |
qed |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
599 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
600 |
lemma piecewise_C1_differentiable_neg: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
601 |
"f piecewise_C1_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
602 |
unfolding piecewise_C1_differentiable_on_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
603 |
by (auto intro!: continuous_on_minus C1_differentiable_on_minus) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
604 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
605 |
lemma piecewise_C1_differentiable_add: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
606 |
assumes "f piecewise_C1_differentiable_on i" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
607 |
"g piecewise_C1_differentiable_on i" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
608 |
shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
609 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
610 |
obtain S t where st: "finite S" "finite t" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
611 |
"f C1_differentiable_on (i-S)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
612 |
"g C1_differentiable_on (i-t)" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
613 |
using assms by (auto simp: piecewise_C1_differentiable_on_def) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
614 |
then have "finite (S \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (S \<union> t)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
615 |
by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
616 |
moreover have "continuous_on i f" "continuous_on i g" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
617 |
using assms piecewise_C1_differentiable_on_def by auto |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
618 |
ultimately show ?thesis |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
619 |
by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
620 |
qed |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
621 |
|
61204 | 622 |
lemma piecewise_C1_differentiable_diff: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
623 |
"\<lbrakk>f piecewise_C1_differentiable_on S; g piecewise_C1_differentiable_on S\<rbrakk> |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
624 |
\<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
625 |
unfolding diff_conv_add_uminus |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
626 |
by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
627 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
628 |
lemma piecewise_C1_differentiable_D1: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
629 |
fixes g1 :: "real \<Rightarrow> 'a::real_normed_field" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
630 |
assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
631 |
shows "g1 piecewise_C1_differentiable_on {0..1}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
632 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
633 |
obtain S where "finite S" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
634 |
and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
635 |
and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
636 |
using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
637 |
have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
638 |
proof (rule differentiable_transform_within) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
639 |
show "g1 +++ g2 \<circ> (*) (inverse 2) differentiable at x" |
68493 | 640 |
using that g12D |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
641 |
apply (simp only: joinpaths_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
642 |
by (rule differentiable_chain_at derivative_intros | force)+ |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
643 |
show "\<And>x'. \<lbrakk>dist x' x < dist (x/2) (1/2)\<rbrakk> |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
644 |
\<Longrightarrow> (g1 +++ g2 \<circ> (*) (inverse 2)) x' = g1 x'" |
68339 | 645 |
using that by (auto simp: dist_real_def joinpaths_def) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
646 |
qed (use that in \<open>auto simp: dist_real_def\<close>) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
647 |
have [simp]: "vector_derivative (g1 \<circ> (*) 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)" |
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
648 |
if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
649 |
apply (subst vector_derivative_chain_at) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
650 |
using that |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
651 |
apply (rule derivative_eq_intros g1D | simp)+ |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
652 |
done |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
653 |
have "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
654 |
using co12 by (rule continuous_on_subset) force |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
655 |
then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 \<circ> (*)2) (at x))" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
656 |
proof (rule continuous_on_eq [OF _ vector_derivative_at]) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
657 |
show "(g1 +++ g2 has_vector_derivative vector_derivative (g1 \<circ> (*) 2) (at x)) (at x)" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
658 |
if "x \<in> {0..1/2} - insert (1/2) S" for x |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
659 |
proof (rule has_vector_derivative_transform_within) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
660 |
show "(g1 \<circ> (*) 2 has_vector_derivative vector_derivative (g1 \<circ> (*) 2) (at x)) (at x)" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
661 |
using that |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
662 |
by (force intro: g1D differentiable_chain_at simp: vector_derivative_works [symmetric]) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
663 |
show "\<And>x'. \<lbrakk>dist x' x < dist x (1/2)\<rbrakk> \<Longrightarrow> (g1 \<circ> (*) 2) x' = (g1 +++ g2) x'" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
664 |
using that by (auto simp: dist_norm joinpaths_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
665 |
qed (use that in \<open>auto simp: dist_norm\<close>) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
666 |
qed |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
667 |
have "continuous_on ({0..1} - insert 1 ((*) 2 ` S)) |
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
668 |
((\<lambda>x. 1/2 * vector_derivative (g1 \<circ> (*)2) (at x)) \<circ> (*)(1/2))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
669 |
apply (rule continuous_intros)+ |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
670 |
using coDhalf |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
671 |
apply (simp add: scaleR_conv_of_real image_set_diff image_image) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
672 |
done |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
673 |
then have con_g1: "continuous_on ({0..1} - insert 1 ((*) 2 ` S)) (\<lambda>x. vector_derivative g1 (at x))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
674 |
by (rule continuous_on_eq) (simp add: scaleR_conv_of_real) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
675 |
have "continuous_on {0..1} g1" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
676 |
using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
677 |
with \<open>finite S\<close> show ?thesis |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
678 |
apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
679 |
apply (rule_tac x="insert 1 (((*)2)`S)" in exI) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
680 |
apply (simp add: g1D con_g1) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
681 |
done |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
682 |
qed |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
683 |
|
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
684 |
lemma piecewise_C1_differentiable_D2: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
685 |
fixes g2 :: "real \<Rightarrow> 'a::real_normed_field" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
686 |
assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
687 |
shows "g2 piecewise_C1_differentiable_on {0..1}" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
688 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
689 |
obtain S where "finite S" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
690 |
and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
691 |
and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
692 |
using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
693 |
have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
694 |
proof (rule differentiable_transform_within) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
695 |
show "g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2) differentiable at x" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
696 |
using g12D that |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
697 |
apply (simp only: joinpaths_def) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
698 |
apply (drule_tac x= "(x+1) / 2" in bspec, force simp: field_split_simps) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
699 |
apply (rule differentiable_chain_at derivative_intros | force)+ |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
700 |
done |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
701 |
show "\<And>x'. dist x' x < dist ((x + 1) / 2) (1/2) \<Longrightarrow> (g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
702 |
using that by (auto simp: dist_real_def joinpaths_def field_simps) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
703 |
qed (use that in \<open>auto simp: dist_norm\<close>) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
704 |
have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
705 |
if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
706 |
using that by (auto simp: vector_derivative_chain_at field_split_simps g2D) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
707 |
have "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
708 |
using co12 by (rule continuous_on_subset) force |
68339 | 709 |
then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x))" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
710 |
proof (rule continuous_on_eq [OF _ vector_derivative_at]) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
711 |
show "(g1 +++ g2 has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x)) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
712 |
(at x)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
713 |
if "x \<in> {1 / 2..1} - insert (1 / 2) S" for x |
68339 | 714 |
proof (rule_tac f="g2 \<circ> (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
715 |
show "(g2 \<circ> (\<lambda>x. 2 * x - 1) has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x)) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
716 |
(at x)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
717 |
using that by (force intro: g2D differentiable_chain_at simp: vector_derivative_works [symmetric]) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
718 |
show "\<And>x'. \<lbrakk>dist x' x < dist (3 / 4) ((x + 1) / 2)\<rbrakk> \<Longrightarrow> (g2 \<circ> (\<lambda>x. 2 * x - 1)) x' = (g1 +++ g2) x'" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
719 |
using that by (auto simp: dist_norm joinpaths_def add_divide_distrib) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
720 |
qed (use that in \<open>auto simp: dist_norm\<close>) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
721 |
qed |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
722 |
have [simp]: "((\<lambda>x. (x+1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` S))) = ({1/2..1} - insert (1/2) S)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
723 |
apply (simp add: image_set_diff inj_on_def image_image) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
724 |
apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
725 |
done |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
726 |
have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)) |
68339 | 727 |
((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) \<circ> (\<lambda>x. (x+1)/2))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
728 |
by (rule continuous_intros | simp add: coDhalf)+ |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
729 |
then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)) (\<lambda>x. vector_derivative g2 (at x))" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
730 |
by (rule continuous_on_eq) (simp add: scaleR_conv_of_real) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
731 |
have "continuous_on {0..1} g2" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
732 |
using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
733 |
with \<open>finite S\<close> show ?thesis |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
734 |
apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
735 |
apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` S)" in exI) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
736 |
apply (simp add: g2D con_g2) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
737 |
done |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
738 |
qed |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
739 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
740 |
subsection \<open>Valid paths, and their start and finish\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
741 |
|
70136 | 742 |
definition\<^marker>\<open>tag important\<close> valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
743 |
where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
744 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
745 |
definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
746 |
where "closed_path g \<equiv> g 0 = g 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
747 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
748 |
text\<open>In particular, all results for paths apply\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
749 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
750 |
lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g" |
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
751 |
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
752 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
753 |
lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
754 |
by (metis connected_path_image valid_path_imp_path) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
755 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
756 |
lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
757 |
by (metis compact_path_image valid_path_imp_path) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
758 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
759 |
lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
760 |
by (metis bounded_path_image valid_path_imp_path) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
761 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
762 |
lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
763 |
by (metis closed_path_image valid_path_imp_path) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
764 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
765 |
lemma valid_path_compose: |
62623
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
766 |
assumes "valid_path g" |
64394 | 767 |
and der: "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable (at x)" |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
768 |
and con: "continuous_on (path_image g) (deriv f)" |
68339 | 769 |
shows "valid_path (f \<circ> g)" |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
770 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
771 |
obtain S where "finite S" and g_diff: "g C1_differentiable_on {0..1} - S" |
62837 | 772 |
using \<open>valid_path g\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
773 |
have "f \<circ> g differentiable at t" when "t\<in>{0..1} - S" for t |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
774 |
proof (rule differentiable_chain_at) |
62837 | 775 |
show "g differentiable at t" using \<open>valid_path g\<close> |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
776 |
by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - S\<close> that) |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
777 |
next |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
778 |
have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis |
68493 | 779 |
then show "f differentiable at (g t)" |
64394 | 780 |
using der[THEN field_differentiable_imp_differentiable] by auto |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
781 |
qed |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
782 |
moreover have "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (f \<circ> g) (at x))" |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
783 |
proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"], |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
784 |
rule continuous_intros) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
785 |
show "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative g (at x))" |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
786 |
using g_diff C1_differentiable_on_eq by auto |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
787 |
next |
62623
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
788 |
have "continuous_on {0..1} (\<lambda>x. deriv f (g x))" |
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
789 |
using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def] |
62837 | 790 |
\<open>valid_path g\<close> piecewise_C1_differentiable_on_def valid_path_def |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
791 |
by blast |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
792 |
then show "continuous_on ({0..1} - S) (\<lambda>x. deriv f (g x))" |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
793 |
using continuous_on_subset by blast |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
794 |
next |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
795 |
show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
796 |
when "t \<in> {0..1} - S" for t |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
797 |
proof (rule vector_derivative_chain_at_general[symmetric]) |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
798 |
show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that) |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
799 |
next |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
800 |
have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis |
64394 | 801 |
then show "f field_differentiable at (g t)" using der by auto |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
802 |
qed |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
803 |
qed |
68339 | 804 |
ultimately have "f \<circ> g C1_differentiable_on {0..1} - S" |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
805 |
using C1_differentiable_on_eq by blast |
68493 | 806 |
moreover have "path (f \<circ> g)" |
64394 | 807 |
apply (rule path_continuous_image[OF valid_path_imp_path[OF \<open>valid_path g\<close>]]) |
808 |
using der |
|
809 |
by (simp add: continuous_at_imp_continuous_on field_differentiable_imp_continuous_at) |
|
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
810 |
ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
811 |
using \<open>finite S\<close> by auto |
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
812 |
qed |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
813 |
|
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
814 |
lemma valid_path_uminus_comp[simp]: |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
815 |
fixes g::"real \<Rightarrow> 'a ::real_normed_field" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
816 |
shows "valid_path (uminus \<circ> g) \<longleftrightarrow> valid_path g" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
817 |
proof |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
818 |
show "valid_path g \<Longrightarrow> valid_path (uminus \<circ> g)" for g::"real \<Rightarrow> 'a" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
819 |
by (auto intro!: valid_path_compose derivative_intros simp add: deriv_linear[of "-1",simplified]) |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
820 |
then show "valid_path g" when "valid_path (uminus \<circ> g)" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
821 |
by (metis fun.map_comp group_add_class.minus_comp_minus id_comp that) |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
822 |
qed |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
823 |
|
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
824 |
lemma valid_path_offset[simp]: |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
825 |
shows "valid_path (\<lambda>t. g t - z) \<longleftrightarrow> valid_path g" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
826 |
proof |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
827 |
show *: "valid_path (g::real\<Rightarrow>'a) \<Longrightarrow> valid_path (\<lambda>t. g t - z)" for g z |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
828 |
unfolding valid_path_def |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
829 |
by (fastforce intro:derivative_intros C1_differentiable_imp_piecewise piecewise_C1_differentiable_diff) |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
830 |
show "valid_path (\<lambda>t. g t - z) \<Longrightarrow> valid_path g" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
831 |
using *[of "\<lambda>t. g t - z" "-z",simplified] . |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
832 |
qed |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
833 |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
834 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
835 |
subsection\<open>Contour Integrals along a path\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
836 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
837 |
text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
838 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
839 |
text\<open>piecewise differentiable function on [0,1]\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
840 |
|
70136 | 841 |
definition\<^marker>\<open>tag important\<close> has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
842 |
(infixr "has'_contour'_integral" 50) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
843 |
where "(f has_contour_integral i) g \<equiv> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
844 |
((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1})) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
845 |
has_integral i) {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
846 |
|
70136 | 847 |
definition\<^marker>\<open>tag important\<close> contour_integrable_on |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
848 |
(infixr "contour'_integrable'_on" 50) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
849 |
where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
850 |
|
70136 | 851 |
definition\<^marker>\<open>tag important\<close> contour_integral |
67613 | 852 |
where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0" |
853 |
||
854 |
lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0" |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
855 |
unfolding contour_integrable_on_def contour_integral_def by blast |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
856 |
|
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
857 |
lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i" |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
858 |
apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def) |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
859 |
using has_integral_unique by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
860 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
861 |
lemma has_contour_integral_eqpath: |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
862 |
"\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>; |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
863 |
contour_integral p f = contour_integral \<gamma> f\<rbrakk> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
864 |
\<Longrightarrow> (f has_contour_integral y) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
865 |
using contour_integrable_on_def contour_integral_unique by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
866 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
867 |
lemma has_contour_integral_integral: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
868 |
"f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
869 |
by (metis contour_integral_unique contour_integrable_on_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
870 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
871 |
lemma has_contour_integral_unique: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
872 |
"(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
873 |
using has_integral_unique |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
874 |
by (auto simp: has_contour_integral_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
875 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
876 |
lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
877 |
using contour_integrable_on_def by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
878 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
879 |
text\<open>Show that we can forget about the localized derivative.\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
880 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
881 |
lemma has_integral_localized_vector_derivative: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
882 |
"((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
883 |
((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
884 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
885 |
have *: "{a..b} - {a,b} = interior {a..b}" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
886 |
by (simp add: atLeastAtMost_diff_ends) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
887 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
888 |
apply (rule has_integral_spike_eq [of "{a,b}"]) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
889 |
apply (auto simp: at_within_interior [of _ "{a..b}"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
890 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
891 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
892 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
893 |
lemma integrable_on_localized_vector_derivative: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
894 |
"(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
895 |
(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
896 |
by (simp add: integrable_on_def has_integral_localized_vector_derivative) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
897 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
898 |
lemma has_contour_integral: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
899 |
"(f has_contour_integral i) g \<longleftrightarrow> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
900 |
((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
901 |
by (simp add: has_integral_localized_vector_derivative has_contour_integral_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
902 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
903 |
lemma contour_integrable_on: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
904 |
"f contour_integrable_on g \<longleftrightarrow> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
905 |
(\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
906 |
by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
907 |
|
70136 | 908 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
909 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
910 |
lemma valid_path_imp_reverse: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
911 |
assumes "valid_path g" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
912 |
shows "valid_path(reversepath g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
913 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
914 |
obtain S where "finite S" and S: "g C1_differentiable_on ({0..1} - S)" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
915 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
916 |
then have "finite ((-) 1 ` S)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
917 |
by auto |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
918 |
moreover have "(reversepath g C1_differentiable_on ({0..1} - (-) 1 ` S))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
919 |
unfolding reversepath_def |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
920 |
apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def]) |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
921 |
using S |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
922 |
by (force simp: finite_vimageI inj_on_def C1_differentiable_on_eq continuous_on_const elim!: continuous_on_subset)+ |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
923 |
ultimately show ?thesis using assms |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
924 |
by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
925 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
926 |
|
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
927 |
lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \<longleftrightarrow> valid_path g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
928 |
using valid_path_imp_reverse by force |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
929 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
930 |
lemma has_contour_integral_reversepath: |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
931 |
assumes "valid_path g" and f: "(f has_contour_integral i) g" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
932 |
shows "(f has_contour_integral (-i)) (reversepath g)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
933 |
proof - |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
934 |
{ fix S x |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
935 |
assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
936 |
have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) = |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
937 |
- vector_derivative g (at (1 - x) within {0..1})" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
938 |
proof - |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
939 |
obtain f' where f': "(g has_vector_derivative f') (at (1 - x))" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
940 |
using xs |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
941 |
by (force simp: has_vector_derivative_def C1_differentiable_on_def) |
68339 | 942 |
have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
943 |
by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+ |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
944 |
then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
945 |
by (simp add: o_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
946 |
show ?thesis |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
947 |
using xs |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
948 |
by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f']) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
949 |
qed |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
950 |
} note * = this |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
951 |
obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
952 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
953 |
have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
954 |
{0..1}" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
955 |
using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]] |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
956 |
by (simp add: has_integral_neg) |
68493 | 957 |
then show ?thesis |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
958 |
using S |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
959 |
apply (clarsimp simp: reversepath_def has_contour_integral_def) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
960 |
apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
961 |
apply (auto simp: *) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
962 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
963 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
964 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
965 |
lemma contour_integrable_reversepath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
966 |
"valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
967 |
using has_contour_integral_reversepath contour_integrable_on_def by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
968 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
969 |
lemma contour_integrable_reversepath_eq: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
970 |
"valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
971 |
using contour_integrable_reversepath valid_path_reversepath by fastforce |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
972 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
973 |
lemma contour_integral_reversepath: |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
974 |
assumes "valid_path g" |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
975 |
shows "contour_integral (reversepath g) f = - (contour_integral g f)" |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
976 |
proof (cases "f contour_integrable_on g") |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
977 |
case True then show ?thesis |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
978 |
by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath) |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
979 |
next |
69508 | 980 |
case False then have "\<not> f contour_integrable_on (reversepath g)" |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
981 |
by (simp add: assms contour_integrable_reversepath_eq) |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
982 |
with False show ?thesis by (simp add: not_integrable_contour_integral) |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
983 |
qed |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
984 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
985 |
|
70136 | 986 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Joining two paths together\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
987 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
988 |
lemma valid_path_join: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
989 |
assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
990 |
shows "valid_path(g1 +++ g2)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
991 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
992 |
have "g1 1 = g2 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
993 |
using assms by (auto simp: pathfinish_def pathstart_def) |
68339 | 994 |
moreover have "(g1 \<circ> (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
995 |
apply (rule piecewise_C1_differentiable_compose) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
996 |
using assms |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
997 |
apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
998 |
apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
999 |
done |
68339 | 1000 |
moreover have "(g2 \<circ> (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1001 |
apply (rule piecewise_C1_differentiable_compose) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1002 |
using assms unfolding valid_path_def piecewise_C1_differentiable_on_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1003 |
by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1004 |
simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1005 |
ultimately show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1006 |
apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1007 |
apply (rule piecewise_C1_differentiable_cases) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1008 |
apply (auto simp: o_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1009 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1010 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1011 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1012 |
lemma valid_path_join_D1: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1013 |
fixes g1 :: "real \<Rightarrow> 'a::real_normed_field" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1014 |
shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1015 |
unfolding valid_path_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1016 |
by (rule piecewise_C1_differentiable_D1) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1017 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1018 |
lemma valid_path_join_D2: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1019 |
fixes g2 :: "real \<Rightarrow> 'a::real_normed_field" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1020 |
shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1021 |
unfolding valid_path_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1022 |
by (rule piecewise_C1_differentiable_D2) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1023 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1024 |
lemma valid_path_join_eq [simp]: |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1025 |
fixes g2 :: "real \<Rightarrow> 'a::real_normed_field" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1026 |
shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1027 |
using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1028 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1029 |
lemma has_contour_integral_join: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1030 |
assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1031 |
"valid_path g1" "valid_path g2" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1032 |
shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1033 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1034 |
obtain s1 s2 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1035 |
where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1036 |
and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1037 |
using assms |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1038 |
by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1039 |
have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1040 |
and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1041 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1042 |
by (auto simp: has_contour_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1043 |
have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1044 |
and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1045 |
using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1046 |
has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1047 |
by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1048 |
have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1049 |
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1050 |
2 *\<^sub>R vector_derivative g1 (at (z*2))" for z |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1051 |
apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]]) |
62390 | 1052 |
apply (simp_all add: dist_real_def abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1053 |
apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1054 |
apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1055 |
using s1 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1056 |
apply (auto simp: algebra_simps vector_derivative_works) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1057 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1058 |
have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1059 |
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1060 |
2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1061 |
apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]]) |
62390 | 1062 |
apply (simp_all add: dist_real_def abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1063 |
apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1064 |
apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1065 |
using s2 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1066 |
apply (auto simp: algebra_simps vector_derivative_works) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1067 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1068 |
have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}" |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
1069 |
apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1070 |
using s1 |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
1071 |
apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1072 |
apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1073 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1074 |
moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1075 |
apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1076 |
using s2 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1077 |
apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1078 |
apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1079 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1080 |
ultimately |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1081 |
show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1082 |
apply (simp add: has_contour_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1083 |
apply (rule has_integral_combine [where c = "1/2"], auto) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1084 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1085 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1086 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1087 |
lemma contour_integrable_joinI: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1088 |
assumes "f contour_integrable_on g1" "f contour_integrable_on g2" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1089 |
"valid_path g1" "valid_path g2" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1090 |
shows "f contour_integrable_on (g1 +++ g2)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1091 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1092 |
by (meson has_contour_integral_join contour_integrable_on_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1093 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1094 |
lemma contour_integrable_joinD1: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1095 |
assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1096 |
shows "f contour_integrable_on g1" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1097 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1098 |
obtain s1 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1099 |
where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1100 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1101 |
have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1102 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1103 |
apply (auto simp: contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1104 |
apply (drule integrable_on_subcbox [where a=0 and b="1/2"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1105 |
apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1106 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1107 |
then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1108 |
by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1109 |
have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1110 |
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1111 |
2 *\<^sub>R vector_derivative g1 (at z)" for z |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1112 |
apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]]) |
62390 | 1113 |
apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1114 |
apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1115 |
using s1 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1116 |
apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1117 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1118 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1119 |
using s1 |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1120 |
apply (auto simp: contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1121 |
apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1122 |
apply (auto simp: joinpaths_def scaleR_conv_of_real g1) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1123 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1124 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1125 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1126 |
lemma contour_integrable_joinD2: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1127 |
assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1128 |
shows "f contour_integrable_on g2" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1129 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1130 |
obtain s2 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1131 |
where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1132 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1133 |
have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1134 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1135 |
apply (auto simp: contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1136 |
apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1137 |
apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1138 |
apply (simp add: image_affinity_atLeastAtMost_diff) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1139 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1140 |
then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1141 |
integrable_on {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1142 |
by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1143 |
have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1144 |
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1145 |
2 *\<^sub>R vector_derivative g2 (at z)" for z |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1146 |
apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]]) |
62390 | 1147 |
apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1148 |
apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1149 |
using s2 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1150 |
apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1151 |
vector_derivative_works add_divide_distrib) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1152 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1153 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1154 |
using s2 |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1155 |
apply (auto simp: contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1156 |
apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1157 |
apply (auto simp: joinpaths_def scaleR_conv_of_real g2) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1158 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1159 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1160 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1161 |
lemma contour_integrable_join [simp]: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1162 |
shows |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1163 |
"\<lbrakk>valid_path g1; valid_path g2\<rbrakk> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1164 |
\<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1165 |
using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1166 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1167 |
lemma contour_integral_join [simp]: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1168 |
shows |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1169 |
"\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1170 |
\<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1171 |
by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1172 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1173 |
|
70136 | 1174 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Shifting the starting point of a (closed) path\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1175 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1176 |
lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1177 |
by (auto simp: shiftpath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1178 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1179 |
lemma valid_path_shiftpath [intro]: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1180 |
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1181 |
shows "valid_path(shiftpath a g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1182 |
using assms |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1183 |
apply (auto simp: valid_path_def shiftpath_alt_def) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1184 |
apply (rule piecewise_C1_differentiable_cases) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1185 |
apply (auto simp: algebra_simps) |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1186 |
apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one]) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1187 |
apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1188 |
apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps]) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1189 |
apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1190 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1191 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1192 |
lemma has_contour_integral_shiftpath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1193 |
assumes f: "(f has_contour_integral i) g" "valid_path g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1194 |
and a: "a \<in> {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1195 |
shows "(f has_contour_integral i) (shiftpath a g)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1196 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1197 |
obtain s |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1198 |
where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1199 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1200 |
have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1201 |
using assms by (auto simp: has_contour_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1202 |
then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1203 |
integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1204 |
apply (rule has_integral_unique) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1205 |
apply (subst add.commute) |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63589
diff
changeset
|
1206 |
apply (subst integral_combine) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1207 |
using assms * integral_unique by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1208 |
{ fix x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1209 |
have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1210 |
vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1211 |
unfolding shiftpath_def |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1212 |
apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]]) |
62390 | 1213 |
apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1214 |
apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1215 |
apply (intro derivative_eq_intros | simp)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1216 |
using g |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1217 |
apply (drule_tac x="x+a" in bspec) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1218 |
using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1219 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1220 |
} note vd1 = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1221 |
{ fix x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1222 |
have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1223 |
vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1224 |
unfolding shiftpath_def |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1225 |
apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]]) |
62390 | 1226 |
apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1227 |
apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1228 |
apply (intro derivative_eq_intros | simp)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1229 |
using g |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1230 |
apply (drule_tac x="x+a-1" in bspec) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1231 |
using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1232 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1233 |
} note vd2 = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1234 |
have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1235 |
using * a by (fastforce intro: integrable_subinterval_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1236 |
have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1237 |
apply (rule integrable_subinterval_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1238 |
using * a by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1239 |
have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1240 |
has_integral integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x))) {0..1 - a}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1241 |
apply (rule has_integral_spike_finite |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
1242 |
[where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1243 |
using s apply blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1244 |
using a apply (auto simp: algebra_simps vd1) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1245 |
apply (force simp: shiftpath_def add.commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1246 |
using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1247 |
apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1248 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1249 |
moreover |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1250 |
have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1251 |
has_integral integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))) {1 - a..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1252 |
apply (rule has_integral_spike_finite |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
1253 |
[where S = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1254 |
using s apply blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1255 |
using a apply (auto simp: algebra_simps vd2) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1256 |
apply (force simp: shiftpath_def add.commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1257 |
using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1258 |
apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1259 |
apply (simp add: algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1260 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1261 |
ultimately show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1262 |
using a |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1263 |
by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1264 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1265 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1266 |
lemma has_contour_integral_shiftpath_D: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1267 |
assumes "(f has_contour_integral i) (shiftpath a g)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1268 |
"valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1269 |
shows "(f has_contour_integral i) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1270 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1271 |
obtain s |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1272 |
where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1273 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1274 |
{ fix x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1275 |
assume x: "0 < x" "x < 1" "x \<notin> s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1276 |
then have gx: "g differentiable at x" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1277 |
using g by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1278 |
have "vector_derivative g (at x within {0..1}) = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1279 |
vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1280 |
apply (rule vector_derivative_at_within_ivl |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1281 |
[OF has_vector_derivative_transform_within_open |
68239 | 1282 |
[where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1283 |
using s g assms x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1284 |
apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1285 |
at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric]) |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
1286 |
apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"]) |
62390 | 1287 |
apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1288 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1289 |
} note vd = this |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1290 |
have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1291 |
using assms by (auto intro!: has_contour_integral_shiftpath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1292 |
show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1293 |
apply (simp add: has_contour_integral_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1294 |
apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _ fi [unfolded has_contour_integral_def]]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1295 |
using s assms vd |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1296 |
apply (auto simp: Path_Connected.shiftpath_shiftpath) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1297 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1298 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1299 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1300 |
lemma has_contour_integral_shiftpath_eq: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1301 |
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1302 |
shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1303 |
using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1304 |
|
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1305 |
lemma contour_integrable_on_shiftpath_eq: |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1306 |
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1307 |
shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g" |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1308 |
using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1309 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1310 |
lemma contour_integral_shiftpath: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1311 |
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1312 |
shows "contour_integral (shiftpath a g) f = contour_integral g f" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1313 |
using assms |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1314 |
by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1315 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1316 |
|
70136 | 1317 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>More about straight-line paths\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1318 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1319 |
lemma has_vector_derivative_linepath_within: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1320 |
"(linepath a b has_vector_derivative (b - a)) (at x within s)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1321 |
apply (simp add: linepath_def has_vector_derivative_def algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1322 |
apply (rule derivative_eq_intros | simp)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1323 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1324 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1325 |
lemma vector_derivative_linepath_within: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1326 |
"x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a" |
67685
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67613
diff
changeset
|
1327 |
apply (rule vector_derivative_within_cbox [of 0 "1::real", simplified]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1328 |
apply (auto simp: has_vector_derivative_linepath_within) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1329 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1330 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1331 |
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1332 |
by (simp add: has_vector_derivative_linepath_within vector_derivative_at) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1333 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1334 |
lemma valid_path_linepath [iff]: "valid_path (linepath a b)" |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1335 |
apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1336 |
apply (rule_tac x="{}" in exI) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1337 |
apply (simp add: differentiable_on_def differentiable_def) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1338 |
using has_vector_derivative_def has_vector_derivative_linepath_within |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1339 |
apply (fastforce simp add: continuous_on_eq_continuous_within) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1340 |
done |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1341 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1342 |
lemma has_contour_integral_linepath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1343 |
shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1344 |
((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1345 |
by (simp add: has_contour_integral vector_derivative_linepath_at) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1346 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1347 |
lemma linepath_in_path: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1348 |
shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1349 |
by (auto simp: segment linepath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1350 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1351 |
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1352 |
by (auto simp: segment linepath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1353 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1354 |
lemma linepath_in_convex_hull: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1355 |
fixes x::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1356 |
assumes a: "a \<in> convex hull s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1357 |
and b: "b \<in> convex hull s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1358 |
and x: "0\<le>x" "x\<le>1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1359 |
shows "linepath a b x \<in> convex hull s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1360 |
apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1361 |
using x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1362 |
apply (auto simp: linepath_image_01 [symmetric]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1363 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1364 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1365 |
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1366 |
by (simp add: linepath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1367 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1368 |
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1369 |
by (simp add: linepath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1370 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1371 |
lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1372 |
by (simp add: has_contour_integral_linepath) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1373 |
|
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1374 |
lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0" |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1375 |
using has_contour_integral_unique by blast |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1376 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1377 |
lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1378 |
using has_contour_integral_trivial contour_integral_unique by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1379 |
|
68721 | 1380 |
lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A" |
1381 |
by (auto simp: linepath_def) |
|
1382 |
||
1383 |
lemma bounded_linear_linepath: |
|
1384 |
assumes "bounded_linear f" |
|
1385 |
shows "f (linepath a b x) = linepath (f a) (f b) x" |
|
1386 |
proof - |
|
1387 |
interpret f: bounded_linear f by fact |
|
1388 |
show ?thesis by (simp add: linepath_def f.add f.scale) |
|
1389 |
qed |
|
1390 |
||
1391 |
lemma bounded_linear_linepath': |
|
1392 |
assumes "bounded_linear f" |
|
1393 |
shows "f \<circ> linepath a b = linepath (f a) (f b)" |
|
1394 |
using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff) |
|
1395 |
||
1396 |
lemma cnj_linepath: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x" |
|
1397 |
by (simp add: linepath_def) |
|
1398 |
||
1399 |
lemma cnj_linepath': "cnj \<circ> linepath a b = linepath (cnj a) (cnj b)" |
|
1400 |
by (simp add: linepath_def fun_eq_iff) |
|
1401 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1402 |
subsection\<open>Relation to subpath construction\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1403 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1404 |
lemma valid_path_subpath: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1405 |
fixes g :: "real \<Rightarrow> 'a :: real_normed_vector" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1406 |
assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1407 |
shows "valid_path(subpath u v g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1408 |
proof (cases "v=u") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1409 |
case True |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1410 |
then show ?thesis |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1411 |
unfolding valid_path_def subpath_def |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1412 |
by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1413 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1414 |
case False |
68339 | 1415 |
have "(g \<circ> (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1416 |
apply (rule piecewise_C1_differentiable_compose) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1417 |
apply (simp add: C1_differentiable_imp_piecewise) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1418 |
apply (simp add: image_affinity_atLeastAtMost) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1419 |
using assms False |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1420 |
apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1421 |
apply (subst Int_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1422 |
apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1423 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1424 |
then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1425 |
by (auto simp: o_def valid_path_def subpath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1426 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1427 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1428 |
lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1429 |
by (simp add: has_contour_integral subpath_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1430 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1431 |
lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1432 |
using has_contour_integral_subpath_refl contour_integrable_on_def by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1433 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1434 |
lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1435 |
by (simp add: has_contour_integral_subpath_refl contour_integral_unique) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1436 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1437 |
lemma has_contour_integral_subpath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1438 |
assumes f: "f contour_integrable_on g" and g: "valid_path g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1439 |
and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1440 |
shows "(f has_contour_integral integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1441 |
(subpath u v g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1442 |
proof (cases "v=u") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1443 |
case True |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1444 |
then show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1445 |
using f by (simp add: contour_integrable_on_def subpath_def has_contour_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1446 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1447 |
case False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1448 |
obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1449 |
using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1450 |
have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u))) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1451 |
has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t))) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1452 |
{0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1453 |
using f uv |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1454 |
apply (simp add: contour_integrable_on subpath_def has_contour_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1455 |
apply (drule integrable_on_subcbox [where a=u and b=v, simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1456 |
apply (simp_all add: has_integral_integral) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1457 |
apply (drule has_integral_affinity [where m="v-u" and c=u, simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1458 |
apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1459 |
apply (simp add: divide_simps False) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1460 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1461 |
{ fix x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1462 |
have "x \<in> {0..1} \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1463 |
x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1464 |
vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1465 |
apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1466 |
apply (intro derivative_eq_intros | simp)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1467 |
apply (cut_tac s [of "(v - u) * x + u"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1468 |
using uv mult_left_le [of x "v-u"] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1469 |
apply (auto simp: vector_derivative_works) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1470 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1471 |
} note vd = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1472 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1473 |
apply (cut_tac has_integral_cmul [OF *, where c = "v-u"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1474 |
using fs assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1475 |
apply (simp add: False subpath_def has_contour_integral) |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
1476 |
apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1477 |
apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1478 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1479 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1480 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1481 |
lemma contour_integrable_subpath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1482 |
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1483 |
shows "f contour_integrable_on (subpath u v g)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1484 |
apply (cases u v rule: linorder_class.le_cases) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1485 |
apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1486 |
apply (subst reversepath_subpath [symmetric]) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1487 |
apply (rule contour_integrable_reversepath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1488 |
using assms apply (blast intro: valid_path_subpath) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1489 |
apply (simp add: contour_integrable_on_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1490 |
using assms apply (blast intro: has_contour_integral_subpath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1491 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1492 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1493 |
lemma has_integral_contour_integral_subpath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1494 |
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1495 |
shows "(((\<lambda>x. f(g x) * vector_derivative g (at x))) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1496 |
has_integral contour_integral (subpath u v g) f) {u..v}" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1497 |
using assms |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1498 |
apply (auto simp: has_integral_integrable_integral) |
66507
678774070c9b
renamed s to S to work with previous change
paulson <lp15@cam.ac.uk>
parents:
66294
diff
changeset
|
1499 |
apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified]) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1500 |
apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1501 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1502 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1503 |
lemma contour_integral_subcontour_integral: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1504 |
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1505 |
shows "contour_integral (subpath u v g) f = |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1506 |
integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1507 |
using assms has_contour_integral_subpath contour_integral_unique by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1508 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1509 |
lemma contour_integral_subpath_combine_less: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1510 |
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1511 |
"u<v" "v<w" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1512 |
shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1513 |
contour_integral (subpath u w g) f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1514 |
using assms apply (auto simp: contour_integral_subcontour_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1515 |
apply (rule integral_combine, auto) |
66507
678774070c9b
renamed s to S to work with previous change
paulson <lp15@cam.ac.uk>
parents:
66294
diff
changeset
|
1516 |
apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified]) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1517 |
apply (auto simp: contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1518 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1519 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1520 |
lemma contour_integral_subpath_combine: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1521 |
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1522 |
shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1523 |
contour_integral (subpath u w g) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1524 |
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1525 |
case True |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1526 |
have *: "subpath v u g = reversepath(subpath u v g) \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1527 |
subpath w u g = reversepath(subpath u w g) \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1528 |
subpath w v g = reversepath(subpath v w g)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1529 |
by (auto simp: reversepath_subpath) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1530 |
have "u < v \<and> v < w \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1531 |
u < w \<and> w < v \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1532 |
v < u \<and> u < w \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1533 |
v < w \<and> w < u \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1534 |
w < u \<and> u < v \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1535 |
w < v \<and> v < u" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1536 |
using True assms by linarith |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1537 |
with assms show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1538 |
using contour_integral_subpath_combine_less [of f g u v w] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1539 |
contour_integral_subpath_combine_less [of f g u w v] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1540 |
contour_integral_subpath_combine_less [of f g v u w] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1541 |
contour_integral_subpath_combine_less [of f g v w u] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1542 |
contour_integral_subpath_combine_less [of f g w u v] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1543 |
contour_integral_subpath_combine_less [of f g w v u] |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1544 |
apply simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1545 |
apply (elim disjE) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1546 |
apply (auto simp: * contour_integral_reversepath contour_integrable_subpath |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1547 |
valid_path_reversepath valid_path_subpath algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1548 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1549 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1550 |
case False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1551 |
then show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1552 |
apply (auto simp: contour_integral_subpath_refl) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1553 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1554 |
by (metis eq_neg_iff_add_eq_0 contour_integrable_subpath contour_integral_reversepath reversepath_subpath valid_path_subpath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1555 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1556 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1557 |
lemma contour_integral_integral: |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1558 |
"contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))" |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1559 |
by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1560 |
|
68721 | 1561 |
lemma contour_integral_cong: |
1562 |
assumes "g = g'" "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x" |
|
1563 |
shows "contour_integral g f = contour_integral g' f'" |
|
1564 |
unfolding contour_integral_integral using assms |
|
1565 |
by (intro integral_cong) (auto simp: path_image_def) |
|
1566 |
||
1567 |
||
1568 |
text \<open>Contour integral along a segment on the real axis\<close> |
|
1569 |
||
1570 |
lemma has_contour_integral_linepath_Reals_iff: |
|
1571 |
fixes a b :: complex and f :: "complex \<Rightarrow> complex" |
|
1572 |
assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b" |
|
1573 |
shows "(f has_contour_integral I) (linepath a b) \<longleftrightarrow> |
|
1574 |
((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}" |
|
1575 |
proof - |
|
1576 |
from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b" |
|
1577 |
by (simp_all add: complex_eq_iff) |
|
1578 |
from assms have "a \<noteq> b" by auto |
|
1579 |
have "((\<lambda>x. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) \<longleftrightarrow> |
|
1580 |
((\<lambda>x. f (a + b * of_real x - a * of_real x)) has_integral I /\<^sub>R (Re b - Re a)) {0..1}" |
|
1581 |
by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric]) |
|
1582 |
(insert assms, simp_all add: field_simps scaleR_conv_of_real) |
|
1583 |
also have "(\<lambda>x. f (a + b * of_real x - a * of_real x)) = |
|
1584 |
(\<lambda>x. (f (a + b * of_real x - a * of_real x) * (b - a)) /\<^sub>R (Re b - Re a))" |
|
1585 |
using \<open>a \<noteq> b\<close> by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real) |
|
1586 |
also have "(\<dots> has_integral I /\<^sub>R (Re b - Re a)) {0..1} \<longleftrightarrow> |
|
1587 |
((\<lambda>x. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms |
|
1588 |
by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps) |
|
1589 |
also have "\<dots> \<longleftrightarrow> (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def |
|
1590 |
by (intro has_integral_cong) (simp add: vector_derivative_linepath_within) |
|
1591 |
finally show ?thesis by simp |
|
1592 |
qed |
|
1593 |
||
1594 |
lemma contour_integrable_linepath_Reals_iff: |
|
1595 |
fixes a b :: complex and f :: "complex \<Rightarrow> complex" |
|
1596 |
assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b" |
|
1597 |
shows "(f contour_integrable_on linepath a b) \<longleftrightarrow> |
|
1598 |
(\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}" |
|
1599 |
using has_contour_integral_linepath_Reals_iff[OF assms, of f] |
|
1600 |
by (auto simp: contour_integrable_on_def integrable_on_def) |
|
1601 |
||
1602 |
lemma contour_integral_linepath_Reals_eq: |
|
1603 |
fixes a b :: complex and f :: "complex \<Rightarrow> complex" |
|
1604 |
assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b" |
|
1605 |
shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))" |
|
1606 |
proof (cases "f contour_integrable_on linepath a b") |
|
1607 |
case True |
|
1608 |
thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f] |
|
1609 |
using has_contour_integral_integral has_contour_integral_unique by blast |
|
1610 |
next |
|
1611 |
case False |
|
1612 |
thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f] |
|
1613 |
by (simp add: not_integrable_contour_integral not_integrable_integral) |
|
1614 |
qed |
|
1615 |
||
1616 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1617 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1618 |
text\<open>Cauchy's theorem where there's a primitive\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1619 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1620 |
lemma contour_integral_primitive_lemma: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1621 |
fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1622 |
assumes "a \<le> b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1623 |
and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1624 |
and "g piecewise_differentiable_on {a..b}" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1625 |
shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b})) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1626 |
has_integral (f(g b) - f(g a))) {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1627 |
proof - |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
1628 |
obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1629 |
using assms by (auto simp: piecewise_differentiable_on_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1630 |
have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1631 |
apply (rule continuous_on_compose [OF cg, unfolded o_def]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1632 |
using assms |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1633 |
apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1634 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1635 |
{ fix x::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1636 |
assume a: "a < x" and b: "x < b" and xk: "x \<notin> k" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1637 |
then have "g differentiable at x within {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1638 |
using k by (simp add: differentiable_at_withinI) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1639 |
then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1640 |
by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1641 |
then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1642 |
by (simp add: has_vector_derivative_def scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1643 |
have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1644 |
using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
1645 |
then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1646 |
by (simp add: has_field_derivative_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1647 |
have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1648 |
using diff_chain_within [OF gdiff fdiff] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1649 |
by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1650 |
} note * = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1651 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1652 |
apply (rule fundamental_theorem_of_calculus_interior_strong) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1653 |
using k assms cfg * |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66708
diff
changeset
|
1654 |
apply (auto simp: at_within_Icc_at) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1655 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1656 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1657 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1658 |
lemma contour_integral_primitive: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1659 |
assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1660 |
and "valid_path g" "path_image g \<subseteq> s" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1661 |
shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1662 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1663 |
apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1664 |
apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1665 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1666 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1667 |
corollary Cauchy_theorem_primitive: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1668 |
assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1669 |
and "valid_path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1670 |
shows "(f' has_contour_integral 0) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1671 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1672 |
by (metis diff_self contour_integral_primitive) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1673 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1674 |
text\<open>Existence of path integral for continuous function\<close> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1675 |
lemma contour_integrable_continuous_linepath: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1676 |
assumes "continuous_on (closed_segment a b) f" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1677 |
shows "f contour_integrable_on (linepath a b)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1678 |
proof - |
68339 | 1679 |
have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1680 |
apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1681 |
apply (rule continuous_intros | simp add: assms)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1682 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1683 |
then show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1684 |
apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1685 |
apply (rule integrable_continuous [of 0 "1::real", simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1686 |
apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1687 |
apply (auto simp: vector_derivative_linepath_within) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1688 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1689 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1690 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1691 |
lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1692 |
by (rule has_derivative_imp_has_field_derivative) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1693 |
(rule derivative_intros | simp)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1694 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1695 |
lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1696 |
apply (rule contour_integral_unique) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1697 |
using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"] |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1698 |
apply (auto simp: field_simps has_field_der_id) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1699 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1700 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1701 |
lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1702 |
by (simp add: continuous_on_const contour_integrable_continuous_linepath) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1703 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1704 |
lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1705 |
by (simp add: continuous_on_id contour_integrable_continuous_linepath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1706 |
|
70136 | 1707 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetical combining theorems\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1708 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1709 |
lemma has_contour_integral_neg: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1710 |
"(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1711 |
by (simp add: has_integral_neg has_contour_integral_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1712 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1713 |
lemma has_contour_integral_add: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1714 |
"\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1715 |
\<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1716 |
by (simp add: has_integral_add has_contour_integral_def algebra_simps) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1717 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1718 |
lemma has_contour_integral_diff: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1719 |
"\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1720 |
\<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g" |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
65587
diff
changeset
|
1721 |
by (simp add: has_integral_diff has_contour_integral_def algebra_simps) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1722 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1723 |
lemma has_contour_integral_lmul: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1724 |
"(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1725 |
apply (simp add: has_contour_integral_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1726 |
apply (drule has_integral_mult_right) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1727 |
apply (simp add: algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1728 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1729 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1730 |
lemma has_contour_integral_rmul: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1731 |
"(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1732 |
apply (drule has_contour_integral_lmul) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1733 |
apply (simp add: mult.commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1734 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1735 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1736 |
lemma has_contour_integral_div: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1737 |
"(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1738 |
by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1739 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1740 |
lemma has_contour_integral_eq: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1741 |
"\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1742 |
apply (simp add: path_image_def has_contour_integral_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1743 |
by (metis (no_types, lifting) image_eqI has_integral_eq) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1744 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1745 |
lemma has_contour_integral_bound_linepath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1746 |
assumes "(f has_contour_integral i) (linepath a b)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1747 |
"0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1748 |
shows "norm i \<le> B * norm(b - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1749 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1750 |
{ fix x::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1751 |
assume x: "0 \<le> x" "x \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1752 |
have "norm (f (linepath a b x)) * |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1753 |
norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1754 |
by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1755 |
} note * = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1756 |
have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1757 |
apply (rule has_integral_bound |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1758 |
[of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"]) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1759 |
using assms * unfolding has_contour_integral_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1760 |
apply (auto simp: norm_mult) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1761 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1762 |
then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1763 |
by (auto simp: content_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1764 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1765 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1766 |
(*UNUSED |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1767 |
lemma has_contour_integral_bound_linepath_strong: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1768 |
fixes a :: real and f :: "complex \<Rightarrow> real" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1769 |
assumes "(f has_contour_integral i) (linepath a b)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1770 |
"finite k" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1771 |
"0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1772 |
shows "norm i \<le> B*norm(b - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1773 |
*) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1774 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1775 |
lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1776 |
unfolding has_contour_integral_linepath |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1777 |
by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1778 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1779 |
lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1780 |
by (simp add: has_contour_integral_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1781 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1782 |
lemma has_contour_integral_is_0: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1783 |
"(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1784 |
by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1785 |
|
64267 | 1786 |
lemma has_contour_integral_sum: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1787 |
"\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk> |
64267 | 1788 |
\<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1789 |
by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1790 |
|
70136 | 1791 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Operations on path integrals\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1792 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1793 |
lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1794 |
by (rule contour_integral_unique [OF has_contour_integral_const_linepath]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1795 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1796 |
lemma contour_integral_neg: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1797 |
"f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1798 |
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1799 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1800 |
lemma contour_integral_add: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1801 |
"f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1802 |
contour_integral g f1 + contour_integral g f2" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1803 |
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1804 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1805 |
lemma contour_integral_diff: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1806 |
"f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1807 |
contour_integral g f1 - contour_integral g f2" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1808 |
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1809 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1810 |
lemma contour_integral_lmul: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1811 |
shows "f contour_integrable_on g |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1812 |
\<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1813 |
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1814 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1815 |
lemma contour_integral_rmul: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1816 |
shows "f contour_integrable_on g |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1817 |
\<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1818 |
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1819 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1820 |
lemma contour_integral_div: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1821 |
shows "f contour_integrable_on g |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1822 |
\<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1823 |
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1824 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1825 |
lemma contour_integral_eq: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1826 |
"(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g" |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1827 |
apply (simp add: contour_integral_def) |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1828 |
using has_contour_integral_eq |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
1829 |
by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1830 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1831 |
lemma contour_integral_eq_0: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1832 |
"(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1833 |
by (simp add: has_contour_integral_is_0 contour_integral_unique) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1834 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1835 |
lemma contour_integral_bound_linepath: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1836 |
shows |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1837 |
"\<lbrakk>f contour_integrable_on (linepath a b); |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1838 |
0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1839 |
\<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1840 |
apply (rule has_contour_integral_bound_linepath [of f]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1841 |
apply (auto simp: has_contour_integral_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1842 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1843 |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
1844 |
lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1845 |
by (simp add: contour_integral_unique has_contour_integral_0) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1846 |
|
64267 | 1847 |
lemma contour_integral_sum: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1848 |
"\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk> |
64267 | 1849 |
\<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s" |
1850 |
by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral) |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1851 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1852 |
lemma contour_integrable_eq: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1853 |
"\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1854 |
unfolding contour_integrable_on_def |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1855 |
by (metis has_contour_integral_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1856 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1857 |
|
70136 | 1858 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic theorems for path integrability\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1859 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1860 |
lemma contour_integrable_neg: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1861 |
"f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1862 |
using has_contour_integral_neg contour_integrable_on_def by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1863 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1864 |
lemma contour_integrable_add: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1865 |
"\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1866 |
using has_contour_integral_add contour_integrable_on_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1867 |
by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1868 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1869 |
lemma contour_integrable_diff: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1870 |
"\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1871 |
using has_contour_integral_diff contour_integrable_on_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1872 |
by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1873 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1874 |
lemma contour_integrable_lmul: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1875 |
"f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1876 |
using has_contour_integral_lmul contour_integrable_on_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1877 |
by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1878 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1879 |
lemma contour_integrable_rmul: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1880 |
"f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1881 |
using has_contour_integral_rmul contour_integrable_on_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1882 |
by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1883 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1884 |
lemma contour_integrable_div: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1885 |
"f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1886 |
using has_contour_integral_div contour_integrable_on_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1887 |
by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1888 |
|
64267 | 1889 |
lemma contour_integrable_sum: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1890 |
"\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk> |
64267 | 1891 |
\<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1892 |
unfolding contour_integrable_on_def |
64267 | 1893 |
by (metis has_contour_integral_sum) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1894 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1895 |
|
70136 | 1896 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path integral\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1897 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1898 |
lemma has_contour_integral_reverse_linepath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1899 |
"(f has_contour_integral i) (linepath a b) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1900 |
\<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1901 |
using has_contour_integral_reversepath valid_path_linepath by fastforce |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1902 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1903 |
lemma contour_integral_reverse_linepath: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1904 |
"continuous_on (closed_segment a b) f |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1905 |
\<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1906 |
apply (rule contour_integral_unique) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1907 |
apply (rule has_contour_integral_reverse_linepath) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1908 |
by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1909 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1910 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1911 |
(* Splitting a path integral in a flat way.*) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1912 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1913 |
lemma has_contour_integral_split: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1914 |
assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1915 |
and k: "0 \<le> k" "k \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1916 |
and c: "c - a = k *\<^sub>R (b - a)" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1917 |
shows "(f has_contour_integral (i + j)) (linepath a b)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1918 |
proof (cases "k = 0 \<or> k = 1") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1919 |
case True |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1920 |
then show ?thesis |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1921 |
using assms by auto |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1922 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1923 |
case False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1924 |
then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1" |
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
1925 |
using assms by auto |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1926 |
have c': "c = k *\<^sub>R (b - a) + a" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1927 |
by (metis diff_add_cancel c) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1928 |
have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1929 |
by (simp add: algebra_simps c') |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1930 |
{ assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1931 |
have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b" |
68302 | 1932 |
using False apply (simp add: c' algebra_simps) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
1933 |
apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1934 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1935 |
have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1936 |
using k has_integral_affinity01 [OF *, of "inverse k" "0"] |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1937 |
apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1938 |
apply (auto dest: has_integral_cmul [where c = "inverse k"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1939 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1940 |
} note fi = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1941 |
{ assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1942 |
have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1943 |
using k |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1944 |
apply (simp add: c' field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1945 |
apply (simp add: scaleR_conv_of_real divide_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1946 |
apply (simp add: field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1947 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1948 |
have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}" |
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1949 |
using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"] |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1950 |
apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc) |
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
1951 |
apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1952 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1953 |
} note fj = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1954 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1955 |
using f k |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1956 |
apply (simp add: has_contour_integral_linepath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1957 |
apply (simp add: linepath_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1958 |
apply (rule has_integral_combine [OF _ _ fi fj], simp_all) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1959 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1960 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1961 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1962 |
lemma continuous_on_closed_segment_transform: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1963 |
assumes f: "continuous_on (closed_segment a b) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1964 |
and k: "0 \<le> k" "k \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1965 |
and c: "c - a = k *\<^sub>R (b - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1966 |
shows "continuous_on (closed_segment a c) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1967 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1968 |
have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1969 |
using c by (simp add: algebra_simps) |
68302 | 1970 |
have "closed_segment a c \<subseteq> closed_segment a b" |
1971 |
by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment) |
|
1972 |
then show "continuous_on (closed_segment a c) f" |
|
1973 |
by (rule continuous_on_subset [OF f]) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1974 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1975 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1976 |
lemma contour_integral_split: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1977 |
assumes f: "continuous_on (closed_segment a b) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1978 |
and k: "0 \<le> k" "k \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1979 |
and c: "c - a = k *\<^sub>R (b - a)" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1980 |
shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1981 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1982 |
have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1983 |
using c by (simp add: algebra_simps) |
68302 | 1984 |
have "closed_segment a c \<subseteq> closed_segment a b" |
1985 |
by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment) |
|
1986 |
moreover have "closed_segment c b \<subseteq> closed_segment a b" |
|
1987 |
by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment) |
|
1988 |
ultimately |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1989 |
have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f" |
68302 | 1990 |
by (auto intro: continuous_on_subset [OF f]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1991 |
show ?thesis |
68302 | 1992 |
by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1993 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1994 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1995 |
lemma contour_integral_split_linepath: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1996 |
assumes f: "continuous_on (closed_segment a b) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
1997 |
and c: "c \<in> closed_segment a b" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1998 |
shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f" |
68302 | 1999 |
using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2000 |
|
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
2001 |
text\<open>The special case of midpoints used in the main quadrisection\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2002 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2003 |
lemma has_contour_integral_midpoint: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2004 |
assumes "(f has_contour_integral i) (linepath a (midpoint a b))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2005 |
"(f has_contour_integral j) (linepath (midpoint a b) b)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2006 |
shows "(f has_contour_integral (i + j)) (linepath a b)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2007 |
apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2008 |
using assms |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2009 |
apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2010 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2011 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2012 |
lemma contour_integral_midpoint: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2013 |
"continuous_on (closed_segment a b) f |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2014 |
\<Longrightarrow> contour_integral (linepath a b) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2015 |
contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2016 |
apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2017 |
apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2018 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2019 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2020 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2021 |
text\<open>A couple of special case lemmas that are useful below\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2022 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2023 |
lemma triangle_linear_has_chain_integral: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2024 |
"((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2025 |
apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2026 |
apply (auto intro!: derivative_eq_intros) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2027 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2028 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2029 |
lemma has_chain_integral_chain_integral3: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2030 |
"(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2031 |
\<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2032 |
apply (subst contour_integral_unique [symmetric], assumption) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2033 |
apply (drule has_contour_integral_integrable) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2034 |
apply (simp add: valid_path_join) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2035 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2036 |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2037 |
lemma has_chain_integral_chain_integral4: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2038 |
"(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2039 |
\<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f + contour_integral (linepath d e) f = i" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2040 |
apply (subst contour_integral_unique [symmetric], assumption) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2041 |
apply (drule has_contour_integral_integrable) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2042 |
apply (simp add: valid_path_join) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2043 |
done |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
2044 |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2045 |
subsection\<open>Reversing the order in a double path integral\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2046 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2047 |
text\<open>The condition is stronger than needed but it's often true in typical situations\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2048 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2049 |
lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2050 |
by (auto simp: cbox_Pair_eq) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2051 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2052 |
lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2053 |
by (auto simp: cbox_Pair_eq) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2054 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
2055 |
proposition contour_integral_swap: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2056 |
assumes fcon: "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2057 |
and vp: "valid_path g" "valid_path h" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2058 |
and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2059 |
and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2060 |
shows "contour_integral g (\<lambda>w. contour_integral h (f w)) = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2061 |
contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2062 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2063 |
have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
2064 |
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def) |
68339 | 2065 |
have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2066 |
by (rule ext) simp |
68339 | 2067 |
have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2068 |
by (rule ext) simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2069 |
have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2070 |
by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2071 |
have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2072 |
by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def) |
68302 | 2073 |
have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))" |
2074 |
by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+ |
|
2075 |
then have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}" |
|
2076 |
using continuous_on_mult gvcon integrable_continuous_real by blast |
|
68339 | 2077 |
have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2078 |
by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2079 |
then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2080 |
apply (rule ssubst) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2081 |
apply (rule continuous_intros | simp add: gvcon)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2082 |
done |
68339 | 2083 |
have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2084 |
by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2085 |
then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2086 |
apply (rule ssubst) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2087 |
apply (rule continuous_intros | simp add: hvcon)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2088 |
done |
68339 | 2089 |
have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2090 |
by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2091 |
then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2092 |
apply (rule ssubst) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2093 |
apply (rule gcon hcon continuous_intros | simp)+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2094 |
apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2095 |
done |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2096 |
have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2097 |
integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))" |
68302 | 2098 |
proof (rule integral_cong [OF contour_integral_rmul [symmetric]]) |
2099 |
show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h" |
|
2100 |
unfolding contour_integrable_on |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2101 |
apply (rule integrable_continuous_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2102 |
apply (rule continuous_on_mult [OF _ hvcon]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2103 |
apply (subst fgh1) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2104 |
apply (rule fcon_im1 hcon continuous_intros | simp)+ |
68302 | 2105 |
done |
2106 |
qed |
|
68339 | 2107 |
also have "\<dots> = integral {0..1} |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2108 |
(\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))" |
68302 | 2109 |
unfolding contour_integral_integral |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2110 |
apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified]) |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
2111 |
apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+ |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
2112 |
unfolding integral_mult_left [symmetric] |
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
2113 |
apply (simp only: mult_ac) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2114 |
done |
68339 | 2115 |
also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))" |
68302 | 2116 |
unfolding contour_integral_integral |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2117 |
apply (rule integral_cong) |
62463
547c5c6e66d4
the integral is 0 when otherwise it would be undefined (also for contour integrals)
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
2118 |
unfolding integral_mult_left [symmetric] |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2119 |
apply (simp add: algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2120 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2121 |
finally show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2122 |
by (simp add: contour_integral_integral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2123 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2124 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2125 |
|
70136 | 2126 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>The key quadrisection step\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2127 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2128 |
lemma norm_sum_half: |
68302 | 2129 |
assumes "norm(a + b) \<ge> e" |
2130 |
shows "norm a \<ge> e/2 \<or> norm b \<ge> e/2" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2131 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2132 |
have "e \<le> norm (- a - b)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2133 |
by (simp add: add.commute assms norm_minus_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2134 |
thus ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2135 |
using norm_triangle_ineq4 order_trans by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2136 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2137 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2138 |
lemma norm_sum_lemma: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2139 |
assumes "e \<le> norm (a + b + c + d)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2140 |
shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2141 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2142 |
have "e \<le> norm ((a + b) + (c + d))" using assms |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2143 |
by (simp add: algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2144 |
then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2145 |
by (auto dest!: norm_sum_half) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2146 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2147 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2148 |
lemma Cauchy_theorem_quadrisection: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2149 |
assumes f: "continuous_on (convex hull {a,b,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2150 |
and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2151 |
and e: "e * K^2 \<le> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2152 |
norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2153 |
shows "\<exists>a' b' c'. |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2154 |
a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2155 |
dist a' b' \<le> K/2 \<and> dist b' c' \<le> K/2 \<and> dist c' a' \<le> K/2 \<and> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2156 |
e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)" |
68302 | 2157 |
(is "\<exists>x y z. ?\<Phi> x y z") |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2158 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2159 |
note divide_le_eq_numeral1 [simp del] |
63040 | 2160 |
define a' where "a' = midpoint b c" |
2161 |
define b' where "b' = midpoint c a" |
|
2162 |
define c' where "c' = midpoint a b" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2163 |
have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2164 |
using f continuous_on_subset segments_subset_convex_hull by metis+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2165 |
have fcont': "continuous_on (closed_segment c' b') f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2166 |
"continuous_on (closed_segment a' c') f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2167 |
"continuous_on (closed_segment b' a') f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2168 |
unfolding a'_def b'_def c'_def |
68302 | 2169 |
by (rule continuous_on_subset [OF f], |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2170 |
metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+ |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2171 |
let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2172 |
have *: "?pathint a b + ?pathint b c + ?pathint c a = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2173 |
(?pathint a c' + ?pathint c' b' + ?pathint b' a) + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2174 |
(?pathint a' c' + ?pathint c' b + ?pathint b a') + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2175 |
(?pathint a' c + ?pathint c b' + ?pathint b' a') + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2176 |
(?pathint a' b' + ?pathint b' c' + ?pathint c' a')" |
68302 | 2177 |
by (simp add: fcont' contour_integral_reverse_linepath) (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2178 |
have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2179 |
by (metis left_diff_distrib mult.commute norm_mult_numeral1) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2180 |
have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2181 |
by (simp add: norm_minus_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2182 |
consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" | |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2183 |
"e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" | |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2184 |
"e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" | |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2185 |
"e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')" |
68302 | 2186 |
using assms unfolding * by (blast intro: that dest!: norm_sum_lemma) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2187 |
then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2188 |
proof cases |
68302 | 2189 |
case 1 then have "?\<Phi> a c' b'" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2190 |
using assms |
68302 | 2191 |
apply (clarsimp simp: c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
2192 |
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps) |
68302 | 2193 |
done |
2194 |
then show ?thesis by blast |
|
2195 |
next |
|
2196 |
case 2 then have "?\<Phi> a' c' b" |
|
2197 |
using assms |
|
2198 |
apply (clarsimp simp: a'_def c'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
2199 |
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2200 |
done |
68302 | 2201 |
then show ?thesis by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2202 |
next |
68302 | 2203 |
case 3 then have "?\<Phi> a' c b'" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2204 |
using assms |
68302 | 2205 |
apply (clarsimp simp: a'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
2206 |
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2207 |
done |
68302 | 2208 |
then show ?thesis by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2209 |
next |
68302 | 2210 |
case 4 then have "?\<Phi> a' b' c'" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2211 |
using assms |
68302 | 2212 |
apply (clarsimp simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
2213 |
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2214 |
done |
68302 | 2215 |
then show ?thesis by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2216 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2217 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2218 |
|
70136 | 2219 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Cauchy's theorem for triangles\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2220 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2221 |
lemma triangle_points_closer: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2222 |
fixes a::complex |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2223 |
shows "\<lbrakk>x \<in> convex hull {a,b,c}; y \<in> convex hull {a,b,c}\<rbrakk> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2224 |
\<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2225 |
norm(x - y) \<le> norm(b - c) \<or> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2226 |
norm(x - y) \<le> norm(c - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2227 |
using simplex_extremal_le [of "{a,b,c}"] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2228 |
by (auto simp: norm_minus_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2229 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2230 |
lemma holomorphic_point_small_triangle: |
68302 | 2231 |
assumes x: "x \<in> S" |
2232 |
and f: "continuous_on S f" |
|
2233 |
and cd: "f field_differentiable (at x within S)" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2234 |
and e: "0 < e" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2235 |
shows "\<exists>k>0. \<forall>a b c. dist a b \<le> k \<and> dist b c \<le> k \<and> dist c a \<le> k \<and> |
68302 | 2236 |
x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> S |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2237 |
\<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2238 |
contour_integral(linepath c a) f) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2239 |
\<le> e*(dist a b + dist b c + dist c a)^2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2240 |
(is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2241 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2242 |
have le_of_3: "\<And>a x y z. \<lbrakk>0 \<le> x*y; 0 \<le> x*z; 0 \<le> y*z; a \<le> (e*(x + y + z))*x + (e*(x + y + z))*y + (e*(x + y + z))*z\<rbrakk> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2243 |
\<Longrightarrow> a \<le> e*(x + y + z)^2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2244 |
by (simp add: algebra_simps power2_eq_square) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2245 |
have disj_le: "\<lbrakk>x \<le> a \<or> x \<le> b \<or> x \<le> c; 0 \<le> a; 0 \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> x \<le> a + b + c" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2246 |
for x::real and a b c |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2247 |
by linarith |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2248 |
have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a" |
68302 | 2249 |
if "convex hull {a, b, c} \<subseteq> S" for a b c |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2250 |
using segments_subset_convex_hull that |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2251 |
by (metis continuous_on_subset f contour_integrable_continuous_linepath)+ |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2252 |
note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral] |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2253 |
{ fix f' a b c d |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2254 |
assume d: "0 < d" |
68302 | 2255 |
and f': "\<And>y. \<lbrakk>cmod (y - x) \<le> d; y \<in> S\<rbrakk> \<Longrightarrow> cmod (f y - f x - f' * (y - x)) \<le> e * cmod (y - x)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2256 |
and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2257 |
and xc: "x \<in> convex hull {a, b, c}" |
68302 | 2258 |
and S: "convex hull {a, b, c} \<subseteq> S" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2259 |
have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2260 |
contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2261 |
contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2262 |
contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))" |
68302 | 2263 |
apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF S]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2264 |
apply (simp add: field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2265 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2266 |
{ fix y |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2267 |
assume yc: "y \<in> convex hull {a,b,c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2268 |
have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)" |
68302 | 2269 |
proof (rule f') |
2270 |
show "cmod (y - x) \<le> d" |
|
2271 |
by (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans) |
|
2272 |
qed (use S yc in blast) |
|
68339 | 2273 |
also have "\<dots> \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2274 |
by (simp add: yc e xc disj_le [OF triangle_points_closer]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2275 |
finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" . |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2276 |
} note cm_le = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2277 |
have "?normle a b c" |
68302 | 2278 |
unfolding dist_norm pa |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2279 |
apply (rule le_of_3) |
68302 | 2280 |
using f' xc S e |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2281 |
apply simp_all |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2282 |
apply (intro norm_triangle_le add_mono path_bound) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2283 |
apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2284 |
apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2285 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2286 |
} note * = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2287 |
show ?thesis |
68493 | 2288 |
using cd e |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
2289 |
apply (simp add: field_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2290 |
apply (clarify dest!: spec mp) |
68302 | 2291 |
using * unfolding dist_norm |
68339 | 2292 |
apply blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2293 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2294 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2295 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2296 |
|
68310 | 2297 |
text\<open>Hence the most basic theorem for a triangle.\<close> |
2298 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2299 |
locale Chain = |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2300 |
fixes x0 At Follows |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2301 |
assumes At0: "At x0 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2302 |
and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2303 |
begin |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2304 |
primrec f where |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2305 |
"f 0 = x0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2306 |
| "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2307 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2308 |
lemma At: "At (f n) n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2309 |
proof (induct n) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2310 |
case 0 show ?case |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2311 |
by (simp add: At0) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2312 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2313 |
case (Suc n) show ?case |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2314 |
by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2315 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2316 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2317 |
lemma Follows: "Follows (f(Suc n)) (f n)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2318 |
by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2319 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2320 |
declare f.simps(2) [simp del] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2321 |
end |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2322 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2323 |
lemma Chain3: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2324 |
assumes At0: "At x0 y0 z0 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2325 |
and AtSuc: "\<And>x y z n. At x y z n \<Longrightarrow> \<exists>x' y' z'. At x' y' z' (Suc n) \<and> Follows x' y' z' x y z" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2326 |
obtains f g h where |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2327 |
"f 0 = x0" "g 0 = y0" "h 0 = z0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2328 |
"\<And>n. At (f n) (g n) (h n) n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2329 |
"\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2330 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2331 |
interpret three: Chain "(x0,y0,z0)" "\<lambda>(x,y,z). At x y z" "\<lambda>(x',y',z'). \<lambda>(x,y,z). Follows x' y' z' x y z" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2332 |
apply unfold_locales |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2333 |
using At0 AtSuc by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2334 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2335 |
apply (rule that [of "\<lambda>n. fst (three.f n)" "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"]) |
68302 | 2336 |
using three.At three.Follows |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2337 |
apply simp_all |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2338 |
apply (simp_all add: split_beta') |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2339 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2340 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2341 |
|
70136 | 2342 |
proposition\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_triangle: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2343 |
assumes "f holomorphic_on (convex hull {a,b,c})" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2344 |
shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2345 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2346 |
have contf: "continuous_on (convex hull {a,b,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2347 |
by (metis assms holomorphic_on_imp_continuous_on) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2348 |
let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2349 |
{ fix y::complex |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2350 |
assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2351 |
and ynz: "y \<noteq> 0" |
63040 | 2352 |
define K where "K = 1 + max (dist a b) (max (dist b c) (dist c a))" |
2353 |
define e where "e = norm y / K^2" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2354 |
have K1: "K \<ge> 1" by (simp add: K_def max.coboundedI1) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2355 |
then have K: "K > 0" by linarith |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2356 |
have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2357 |
by (simp_all add: K_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2358 |
have e: "e > 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2359 |
unfolding e_def using ynz K1 by simp |
63040 | 2360 |
define At where "At x y z n \<longleftrightarrow> |
2361 |
convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and> |
|
2362 |
dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and> |
|
2363 |
norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2" |
|
2364 |
for x y z n |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2365 |
have At0: "At a b c 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2366 |
using fy |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2367 |
by (simp add: At_def e_def has_chain_integral_chain_integral3) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2368 |
{ fix x y z n |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2369 |
assume At: "At x y z n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2370 |
then have contf': "continuous_on (convex hull {x,y,z}) f" |
63938 | 2371 |
using contf At_def continuous_on_subset by metis |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2372 |
have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}" |
68302 | 2373 |
using At Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e] |
2374 |
apply (simp add: At_def algebra_simps) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2375 |
apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2376 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2377 |
} note AtSuc = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2378 |
obtain fa fb fc |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2379 |
where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2380 |
and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2381 |
and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2382 |
"\<And>n. dist (fb n) (fc n) \<le> K/2^n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2383 |
"\<And>n. dist (fc n) (fa n) \<le> K/2^n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2384 |
and no: "\<And>n. norm(?pathint (fa n) (fb n) + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2385 |
?pathint (fb n) (fc n) + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2386 |
?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2387 |
and conv_le: "\<And>n. convex hull {fa(Suc n), fb(Suc n), fc(Suc n)} \<subseteq> convex hull {fa n, fb n, fc n}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2388 |
apply (rule Chain3 [of At, OF At0 AtSuc]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2389 |
apply (auto simp: At_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2390 |
done |
68302 | 2391 |
obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" |
2392 |
proof (rule bounded_closed_nest) |
|
2393 |
show "\<And>n. closed (convex hull {fa n, fb n, fc n})" |
|
2394 |
by (simp add: compact_imp_closed finite_imp_compact_convex_hull) |
|
2395 |
show "\<And>m n. m \<le> n \<Longrightarrow> convex hull {fa n, fb n, fc n} \<subseteq> convex hull {fa m, fb m, fc m}" |
|
2396 |
by (erule transitive_stepwise_le) (auto simp: conv_le) |
|
2397 |
qed (fastforce intro: finite_imp_bounded_convex_hull)+ |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2398 |
then have xin: "x \<in> convex hull {a,b,c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2399 |
using assms f0 by blast |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
2400 |
then have fx: "f field_differentiable at x within (convex hull {a,b,c})" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2401 |
using assms holomorphic_on_def by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2402 |
{ fix k n |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2403 |
assume k: "0 < k" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2404 |
and le: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2405 |
"\<And>x' y' z'. |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2406 |
\<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k; |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2407 |
x \<in> convex hull {x',y',z'}; |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2408 |
convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2409 |
\<Longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2410 |
cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2411 |
\<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2412 |
and Kk: "K / k < 2 ^ n" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2413 |
have "K / 2 ^ n < k" using Kk k |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2414 |
by (auto simp: field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2415 |
then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2416 |
using dist [of n] k |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2417 |
by linarith+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2418 |
have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2419 |
\<le> (3 * K / 2 ^ n)\<^sup>2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2420 |
using dist [of n] e K |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2421 |
by (simp add: abs_le_square_iff [symmetric]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2422 |
have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2423 |
by linarith |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2424 |
have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \<le> e * (3 * K / 2 ^ n)\<^sup>2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2425 |
using ynz dle e mult_le_cancel_left_pos by blast |
68339 | 2426 |
also have "\<dots> < |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2427 |
cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2428 |
using no [of n] e K |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2429 |
apply (simp add: e_def field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2430 |
apply (simp only: zero_less_norm_iff [symmetric]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2431 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2432 |
finally have False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2433 |
using le [OF DD x cosb] by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2434 |
} then |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2435 |
have ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2436 |
using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2437 |
apply clarsimp |
68339 | 2438 |
apply (rule_tac y1="K/k" in exE [OF real_arch_pow[of 2]], force+) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2439 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2440 |
} |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2441 |
moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2442 |
by simp (meson contf continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(1) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2443 |
segments_subset_convex_hull(3) segments_subset_convex_hull(5)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2444 |
ultimately show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2445 |
using has_contour_integral_integral by fastforce |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2446 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2447 |
|
70136 | 2448 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Version needing function holomorphic in interior only\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2449 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2450 |
lemma Cauchy_theorem_flat_lemma: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2451 |
assumes f: "continuous_on (convex hull {a,b,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2452 |
and c: "c - a = k *\<^sub>R (b - a)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2453 |
and k: "0 \<le> k" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2454 |
shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2455 |
contour_integral (linepath c a) f = 0" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2456 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2457 |
have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2458 |
using f continuous_on_subset segments_subset_convex_hull by metis+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2459 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2460 |
proof (cases "k \<le> 1") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2461 |
case True show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2462 |
by (simp add: contour_integral_split [OF fabc(1) k True c] contour_integral_reverse_linepath fabc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2463 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2464 |
case False then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2465 |
using fabc c |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2466 |
apply (subst contour_integral_split [of a c f "1/k" b, symmetric]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2467 |
apply (metis closed_segment_commute fabc(3)) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2468 |
apply (auto simp: k contour_integral_reverse_linepath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2469 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2470 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2471 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2472 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2473 |
lemma Cauchy_theorem_flat: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2474 |
assumes f: "continuous_on (convex hull {a,b,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2475 |
and c: "c - a = k *\<^sub>R (b - a)" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2476 |
shows "contour_integral (linepath a b) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2477 |
contour_integral (linepath b c) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2478 |
contour_integral (linepath c a) f = 0" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2479 |
proof (cases "0 \<le> k") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2480 |
case True with assms show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2481 |
by (blast intro: Cauchy_theorem_flat_lemma) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2482 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2483 |
case False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2484 |
have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2485 |
using f continuous_on_subset segments_subset_convex_hull by metis+ |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2486 |
moreover have "contour_integral (linepath b a) f + contour_integral (linepath a c) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2487 |
contour_integral (linepath c b) f = 0" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2488 |
apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2489 |
using False c |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2490 |
apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2491 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2492 |
ultimately show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2493 |
apply (auto simp: contour_integral_reverse_linepath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2494 |
using add_eq_0_iff by force |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2495 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2496 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
2497 |
lemma Cauchy_theorem_triangle_interior: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2498 |
assumes contf: "continuous_on (convex hull {a,b,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2499 |
and holf: "f holomorphic_on interior (convex hull {a,b,c})" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2500 |
shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2501 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2502 |
have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2503 |
using contf continuous_on_subset segments_subset_convex_hull by metis+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2504 |
have "bounded (f ` (convex hull {a,b,c}))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2505 |
by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2506 |
then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2507 |
by (auto simp: dest!: bounded_pos [THEN iffD1]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2508 |
have "bounded (convex hull {a,b,c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2509 |
by (simp add: bounded_convex_hull) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2510 |
then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2511 |
using bounded_pos_less by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2512 |
then have diff_2C: "norm(x - y) \<le> 2*C" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2513 |
if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2514 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2515 |
have "cmod x \<le> C" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2516 |
using x by (meson Cno not_le not_less_iff_gr_or_eq) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2517 |
hence "cmod (x - y) \<le> C + C" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2518 |
using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2519 |
thus "cmod (x - y) \<le> 2 * C" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2520 |
by (metis mult_2) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2521 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2522 |
have contf': "continuous_on (convex hull {b,a,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2523 |
using contf by (simp add: insert_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2524 |
{ fix y::complex |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2525 |
assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2526 |
and ynz: "y \<noteq> 0" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2527 |
have pi_eq_y: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = y" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2528 |
by (rule has_chain_integral_chain_integral3 [OF fy]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2529 |
have ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2530 |
proof (cases "c=a \<or> a=b \<or> b=c") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2531 |
case True then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2532 |
using Cauchy_theorem_flat [OF contf, of 0] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2533 |
using has_chain_integral_chain_integral3 [OF fy] ynz |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2534 |
by (force simp: fabc contour_integral_reverse_linepath) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2535 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2536 |
case False |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2537 |
then have car3: "card {a, b, c} = Suc (DIM(complex))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2538 |
by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2539 |
{ assume "interior(convex hull {a,b,c}) = {}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2540 |
then have "collinear{a,b,c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2541 |
using interior_convex_hull_eq_empty [OF car3] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2542 |
by (simp add: collinear_3_eq_affine_dependent) |
68302 | 2543 |
with False obtain d where "c \<noteq> a" "a \<noteq> b" "b \<noteq> c" "c - b = d *\<^sub>R (a - b)" |
68339 | 2544 |
by (auto simp: collinear_3 collinear_lemma) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2545 |
then have "False" |
68302 | 2546 |
using False Cauchy_theorem_flat [OF contf'] pi_eq_y ynz |
2547 |
by (simp add: fabc add_eq_0_iff contour_integral_reverse_linepath) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2548 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2549 |
then obtain d where d: "d \<in> interior (convex hull {a, b, c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2550 |
by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2551 |
{ fix d1 |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2552 |
assume d1_pos: "0 < d1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2553 |
and d1: "\<And>x x'. \<lbrakk>x\<in>convex hull {a, b, c}; x'\<in>convex hull {a, b, c}; cmod (x' - x) < d1\<rbrakk> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2554 |
\<Longrightarrow> cmod (f x' - f x) < cmod y / (24 * C)" |
63040 | 2555 |
define e where "e = min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))" |
2556 |
define shrink where "shrink x = x - e *\<^sub>R (x - d)" for x |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2557 |
let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2558 |
have e: "0 < e" "e \<le> 1" "e \<le> d1 / (4 * C)" "e \<le> cmod y / 24 / C / B" |
61222 | 2559 |
using d1_pos \<open>C>0\<close> \<open>B>0\<close> ynz by (simp_all add: e_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2560 |
then have eCB: "24 * e * C * B \<le> cmod y" |
61222 | 2561 |
using \<open>C>0\<close> \<open>B>0\<close> by (simp add: field_simps) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2562 |
have e_le_d1: "e * (4 * C) \<le> d1" |
61222 | 2563 |
using e \<open>C>0\<close> by (simp add: field_simps) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2564 |
have "shrink a \<in> interior(convex hull {a,b,c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2565 |
"shrink b \<in> interior(convex hull {a,b,c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2566 |
"shrink c \<in> interior(convex hull {a,b,c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2567 |
using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2568 |
then have fhp0: "(f has_contour_integral 0) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2569 |
(linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))" |
68310 | 2570 |
by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2571 |
then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2572 |
by (simp add: has_chain_integral_chain_integral3) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2573 |
have fpi_abc: "f contour_integrable_on linepath (shrink a) (shrink b)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2574 |
"f contour_integrable_on linepath (shrink b) (shrink c)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2575 |
"f contour_integrable_on linepath (shrink c) (shrink a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2576 |
using fhp0 by (auto simp: valid_path_join dest: has_contour_integral_integrable) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2577 |
have cmod_shr: "\<And>x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2578 |
using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2579 |
have sh_eq: "\<And>a b d::complex. (b - e *\<^sub>R (b - d)) - (a - e *\<^sub>R (a - d)) - (b - a) = e *\<^sub>R (a - b)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2580 |
by (simp add: algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2581 |
have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12" |
61222 | 2582 |
using False \<open>C>0\<close> diff_2C [of b a] ynz |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
2583 |
by (auto simp: field_split_simps hull_inc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2584 |
have less_C: "\<lbrakk>u \<in> convex hull {a, b, c}; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> x * cmod u < C" for x u |
61222 | 2585 |
apply (cases "x=0", simp add: \<open>0<C\<close>) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2586 |
using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2587 |
{ fix u v |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2588 |
assume uv: "u \<in> convex hull {a, b, c}" "v \<in> convex hull {a, b, c}" "u\<noteq>v" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2589 |
and fpi_uv: "f contour_integrable_on linepath (shrink u) (shrink v)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2590 |
have shr_uv: "shrink u \<in> interior(convex hull {a,b,c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2591 |
"shrink v \<in> interior(convex hull {a,b,c})" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2592 |
using d e uv |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2593 |
by (auto simp: hull_inc mem_interior_convex_shrink shrink_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2594 |
have cmod_fuv: "\<And>x. 0\<le>x \<Longrightarrow> x\<le>1 \<Longrightarrow> cmod (f (linepath (shrink u) (shrink v) x)) \<le> B" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2595 |
using shr_uv by (blast intro: Bnf linepath_in_convex_hull interior_subset [THEN subsetD]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2596 |
have By_uv: "B * (12 * (e * cmod (u - v))) \<le> cmod y" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2597 |
apply (rule order_trans [OF _ eCB]) |
61222 | 2598 |
using e \<open>B>0\<close> diff_2C [of u v] uv |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2599 |
by (auto simp: field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2600 |
{ fix x::real assume x: "0\<le>x" "x\<le>1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2601 |
have cmod_less_4C: "cmod ((1 - x) *\<^sub>R u - (1 - x) *\<^sub>R d) + cmod (x *\<^sub>R v - x *\<^sub>R d) < (C+C) + (C+C)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2602 |
apply (rule add_strict_mono; rule norm_triangle_half_l [of _ 0]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2603 |
using uv x d interior_subset |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2604 |
apply (auto simp: hull_inc intro!: less_C) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2605 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2606 |
have ll: "linepath (shrink u) (shrink v) x - linepath u v x = -e * ((1 - x) *\<^sub>R (u - d) + x *\<^sub>R (v - d))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2607 |
by (simp add: linepath_def shrink_def algebra_simps scaleR_conv_of_real) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2608 |
have cmod_less_dt: "cmod (linepath (shrink u) (shrink v) x - linepath u v x) < d1" |
68310 | 2609 |
apply (simp only: ll norm_mult scaleR_diff_right) |
2610 |
using \<open>e>0\<close> cmod_less_4C apply (force intro: norm_triangle_lt less_le_trans [OF _ e_le_d1]) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2611 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2612 |
have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) < cmod y / (24 * C)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2613 |
using x uv shr_uv cmod_less_dt |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2614 |
by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull) |
68339 | 2615 |
also have "\<dots> \<le> cmod y / cmod (v - u) / 12" |
61222 | 2616 |
using False uv \<open>C>0\<close> diff_2C [of v u] ynz |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
2617 |
by (auto simp: field_split_simps hull_inc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2618 |
finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2619 |
by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2620 |
then have cmod_12_le: "cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) * 12 \<le> cmod y" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2621 |
using uv False by (auto simp: field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2622 |
have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) + |
68302 | 2623 |
cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) |
2624 |
\<le> B * (cmod y / 24 / C / B * 2 * C) + 2 * C * (cmod y / 24 / C)" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2625 |
apply (rule add_mono [OF mult_mono]) |
68302 | 2626 |
using By_uv e \<open>0 < B\<close> \<open>0 < C\<close> x apply (simp_all add: cmod_fuv cmod_shr cmod_12_le) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2627 |
apply (simp add: field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2628 |
done |
68339 | 2629 |
also have "\<dots> \<le> cmod y / 6" |
2630 |
by simp |
|
68302 | 2631 |
finally have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) + |
2632 |
cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) |
|
2633 |
\<le> cmod y / 6" . |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2634 |
} note cmod_diff_le = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2635 |
have f_uv: "continuous_on (closed_segment u v) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2636 |
by (blast intro: uv continuous_on_subset [OF contf closed_segment_subset_convex_hull]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2637 |
have **: "\<And>f' x' f x::complex. f'*x' - f*x = f'*(x' - x) + x*(f' - f)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2638 |
by (simp add: algebra_simps) |
68493 | 2639 |
have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) |
68310 | 2640 |
\<le> (B*(norm y /24/C/B)*2*C + (2*C)*(norm y/24/C)) * content (cbox 0 (1::real))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2641 |
apply (rule has_integral_bound |
68310 | 2642 |
[of _ "\<lambda>x. f(linepath (shrink u) (shrink v) x) * (shrink v - shrink u) - f(linepath u v x)*(v - u)" |
2643 |
_ 0 1]) |
|
61222 | 2644 |
using ynz \<open>0 < B\<close> \<open>0 < C\<close> |
68310 | 2645 |
apply (simp_all del: le_divide_eq_numeral1) |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
65587
diff
changeset
|
2646 |
apply (simp add: has_integral_diff has_contour_integral_linepath [symmetric] has_contour_integral_integral |
68310 | 2647 |
fpi_uv f_uv contour_integrable_continuous_linepath) |
68339 | 2648 |
apply (auto simp: ** norm_triangle_le norm_mult cmod_diff_le simp del: le_divide_eq_numeral1) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2649 |
done |
68339 | 2650 |
also have "\<dots> \<le> norm y / 6" |
68310 | 2651 |
by simp |
2652 |
finally have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) \<le> norm y / 6" . |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2653 |
} note * = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2654 |
have "norm (?pathint (shrink a) (shrink b) - ?pathint a b) \<le> norm y / 6" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2655 |
using False fpi_abc by (rule_tac *) (auto simp: hull_inc) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2656 |
moreover |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2657 |
have "norm (?pathint (shrink b) (shrink c) - ?pathint b c) \<le> norm y / 6" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2658 |
using False fpi_abc by (rule_tac *) (auto simp: hull_inc) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2659 |
moreover |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2660 |
have "norm (?pathint (shrink c) (shrink a) - ?pathint c a) \<le> norm y / 6" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2661 |
using False fpi_abc by (rule_tac *) (auto simp: hull_inc) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2662 |
ultimately |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2663 |
have "norm((?pathint (shrink a) (shrink b) - ?pathint a b) + |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2664 |
(?pathint (shrink b) (shrink c) - ?pathint b c) + (?pathint (shrink c) (shrink a) - ?pathint c a)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2665 |
\<le> norm y / 6 + norm y / 6 + norm y / 6" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2666 |
by (metis norm_triangle_le add_mono) |
68339 | 2667 |
also have "\<dots> = norm y / 2" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2668 |
by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2669 |
finally have "norm((?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a)) - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2670 |
(?pathint a b + ?pathint b c + ?pathint c a)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2671 |
\<le> norm y / 2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2672 |
by (simp add: algebra_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2673 |
then |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2674 |
have "norm(?pathint a b + ?pathint b c + ?pathint c a) \<le> norm y / 2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2675 |
by (simp add: f_0_shrink) (metis (mono_tags) add.commute minus_add_distrib norm_minus_cancel uminus_add_conv_diff) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2676 |
then have "False" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2677 |
using pi_eq_y ynz by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2678 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2679 |
moreover have "uniformly_continuous_on (convex hull {a,b,c}) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2680 |
by (simp add: contf compact_convex_hull compact_uniformly_continuous) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2681 |
ultimately have "False" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2682 |
unfolding uniformly_continuous_on_def |
61222 | 2683 |
by (force simp: ynz \<open>0 < C\<close> dist_norm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2684 |
then show ?thesis .. |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2685 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2686 |
} |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2687 |
moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2688 |
using fabc contour_integrable_continuous_linepath by auto |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2689 |
ultimately show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2690 |
using has_contour_integral_integral by fastforce |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2691 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2692 |
|
70136 | 2693 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Version allowing finite number of exceptional points\<close> |
2694 |
||
2695 |
proposition\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_triangle_cofinite: |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2696 |
assumes "continuous_on (convex hull {a,b,c}) f" |
68310 | 2697 |
and "finite S" |
2698 |
and "(\<And>x. x \<in> interior(convex hull {a,b,c}) - S \<Longrightarrow> f field_differentiable (at x))" |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2699 |
shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2700 |
using assms |
68310 | 2701 |
proof (induction "card S" arbitrary: a b c S rule: less_induct) |
2702 |
case (less S a b c) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2703 |
show ?case |
68310 | 2704 |
proof (cases "S={}") |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2705 |
case True with less show ?thesis |
68310 | 2706 |
by (fastforce simp: holomorphic_on_def field_differentiable_at_within Cauchy_theorem_triangle_interior) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2707 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2708 |
case False |
68310 | 2709 |
then obtain d S' where d: "S = insert d S'" "d \<notin> S'" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2710 |
by (meson Set.set_insert all_not_in_conv) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2711 |
then show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2712 |
proof (cases "d \<in> convex hull {a,b,c}") |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2713 |
case False |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2714 |
show "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
68310 | 2715 |
proof (rule less.hyps) |
2716 |
show "\<And>x. x \<in> interior (convex hull {a, b, c}) - S' \<Longrightarrow> f field_differentiable at x" |
|
2717 |
using False d interior_subset by (auto intro!: less.prems) |
|
2718 |
qed (use d less.prems in auto) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2719 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2720 |
case True |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2721 |
have *: "convex hull {a, b, d} \<subseteq> convex hull {a, b, c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2722 |
by (meson True hull_subset insert_subset convex_hull_subset) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2723 |
have abd: "(f has_contour_integral 0) (linepath a b +++ linepath b d +++ linepath d a)" |
68310 | 2724 |
proof (rule less.hyps) |
2725 |
show "\<And>x. x \<in> interior (convex hull {a, b, d}) - S' \<Longrightarrow> f field_differentiable at x" |
|
2726 |
using d not_in_interior_convex_hull_3 |
|
2727 |
by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono) |
|
2728 |
qed (use d continuous_on_subset [OF _ *] less.prems in auto) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2729 |
have *: "convex hull {b, c, d} \<subseteq> convex hull {a, b, c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2730 |
by (meson True hull_subset insert_subset convex_hull_subset) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2731 |
have bcd: "(f has_contour_integral 0) (linepath b c +++ linepath c d +++ linepath d b)" |
68310 | 2732 |
proof (rule less.hyps) |
2733 |
show "\<And>x. x \<in> interior (convex hull {b, c, d}) - S' \<Longrightarrow> f field_differentiable at x" |
|
2734 |
using d not_in_interior_convex_hull_3 |
|
2735 |
by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono) |
|
2736 |
qed (use d continuous_on_subset [OF _ *] less.prems in auto) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2737 |
have *: "convex hull {c, a, d} \<subseteq> convex hull {a, b, c}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2738 |
by (meson True hull_subset insert_subset convex_hull_subset) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2739 |
have cad: "(f has_contour_integral 0) (linepath c a +++ linepath a d +++ linepath d c)" |
68310 | 2740 |
proof (rule less.hyps) |
2741 |
show "\<And>x. x \<in> interior (convex hull {c, a, d}) - S' \<Longrightarrow> f field_differentiable at x" |
|
2742 |
using d not_in_interior_convex_hull_3 |
|
2743 |
by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono) |
|
2744 |
qed (use d continuous_on_subset [OF _ *] less.prems in auto) |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2745 |
have "f contour_integrable_on linepath a b" |
68310 | 2746 |
using less.prems abd contour_integrable_joinD1 contour_integrable_on_def by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2747 |
moreover have "f contour_integrable_on linepath b c" |
68310 | 2748 |
using less.prems bcd contour_integrable_joinD1 contour_integrable_on_def by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2749 |
moreover have "f contour_integrable_on linepath c a" |
68310 | 2750 |
using less.prems cad contour_integrable_joinD1 contour_integrable_on_def by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2751 |
ultimately have fpi: "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2752 |
by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2753 |
{ fix y::complex |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2754 |
assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2755 |
and ynz: "y \<noteq> 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2756 |
have cont_ad: "continuous_on (closed_segment a d) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2757 |
by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(3)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2758 |
have cont_bd: "continuous_on (closed_segment b d) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2759 |
by (meson True closed_segment_subset_convex_hull continuous_on_subset hull_subset insert_subset less.prems(1)) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2760 |
have cont_cd: "continuous_on (closed_segment c d) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2761 |
by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(2)) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2762 |
have "contour_integral (linepath a b) f = - (contour_integral (linepath b d) f + (contour_integral (linepath d a) f))" |
68310 | 2763 |
"contour_integral (linepath b c) f = - (contour_integral (linepath c d) f + (contour_integral (linepath d b) f))" |
2764 |
"contour_integral (linepath c a) f = - (contour_integral (linepath a d) f + contour_integral (linepath d c) f)" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2765 |
using has_chain_integral_chain_integral3 [OF abd] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2766 |
has_chain_integral_chain_integral3 [OF bcd] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2767 |
has_chain_integral_chain_integral3 [OF cad] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2768 |
by (simp_all add: algebra_simps add_eq_0_iff) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2769 |
then have ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2770 |
using cont_ad cont_bd cont_cd fy has_chain_integral_chain_integral3 contour_integral_reverse_linepath by fastforce |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2771 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2772 |
then show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2773 |
using fpi contour_integrable_on_def by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2774 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2775 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2776 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2777 |
|
70136 | 2778 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Cauchy's theorem for an open starlike set\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2779 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2780 |
lemma starlike_convex_subset: |
68310 | 2781 |
assumes S: "a \<in> S" "closed_segment b c \<subseteq> S" and subs: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S" |
2782 |
shows "convex hull {a,b,c} \<subseteq> S" |
|
2783 |
using S |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2784 |
apply (clarsimp simp add: convex_hull_insert [of "{b,c}" a] segment_convex_hull) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2785 |
apply (meson subs convexD convex_closed_segment ends_in_segment(1) ends_in_segment(2) subsetCE) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2786 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2787 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2788 |
lemma triangle_contour_integrals_starlike_primitive: |
68310 | 2789 |
assumes contf: "continuous_on S f" |
2790 |
and S: "a \<in> S" "open S" |
|
2791 |
and x: "x \<in> S" |
|
2792 |
and subs: "\<And>y. y \<in> S \<Longrightarrow> closed_segment a y \<subseteq> S" |
|
2793 |
and zer: "\<And>b c. closed_segment b c \<subseteq> S |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2794 |
\<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2795 |
contour_integral (linepath c a) f = 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2796 |
shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2797 |
proof - |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2798 |
let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2799 |
{ fix e y |
68310 | 2800 |
assume e: "0 < e" and bxe: "ball x e \<subseteq> S" and close: "cmod (y - x) < e" |
2801 |
have y: "y \<in> S" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2802 |
using bxe close by (force simp: dist_norm norm_minus_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2803 |
have cont_ayf: "continuous_on (closed_segment a y) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2804 |
using contf continuous_on_subset subs y by blast |
68310 | 2805 |
have xys: "closed_segment x y \<subseteq> S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2806 |
apply (rule order_trans [OF _ bxe]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2807 |
using close |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2808 |
by (auto simp: dist_norm ball_def norm_minus_commute dest: segment_bound) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2809 |
have "?pathint a y - ?pathint a x = ?pathint x y" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2810 |
using zer [OF xys] contour_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2811 |
} note [simp] = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2812 |
{ fix e::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2813 |
assume e: "0 < e" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2814 |
have cont_atx: "continuous (at x) f" |
68310 | 2815 |
using x S contf continuous_on_eq_continuous_at by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2816 |
then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. cmod (y - x) < d1 \<Longrightarrow> cmod (f y - f x) < e/2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2817 |
unfolding continuous_at Lim_at dist_norm using e |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2818 |
by (drule_tac x="e/2" in spec) force |
68310 | 2819 |
obtain d2 where d2: "d2>0" "ball x d2 \<subseteq> S" using \<open>open S\<close> x |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2820 |
by (auto simp: open_contains_ball) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2821 |
have dpos: "min d1 d2 > 0" using d1 d2 by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2822 |
{ fix y |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2823 |
assume yx: "y \<noteq> x" and close: "cmod (y - x) < min d1 d2" |
68310 | 2824 |
have y: "y \<in> S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2825 |
using d2 close by (force simp: dist_norm norm_minus_commute) |
68310 | 2826 |
have "closed_segment x y \<subseteq> S" |
2827 |
using close d2 by (auto simp: dist_norm norm_minus_commute dest!: segment_bound(1)) |
|
2828 |
then have fxy: "f contour_integrable_on linepath x y" |
|
2829 |
by (metis contour_integrable_continuous_linepath continuous_on_subset [OF contf]) |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2830 |
then obtain i where i: "(f has_contour_integral i) (linepath x y)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2831 |
by (auto simp: contour_integrable_on_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2832 |
then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2833 |
by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2834 |
then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)" |
68310 | 2835 |
proof (rule has_contour_integral_bound_linepath) |
2836 |
show "\<And>u. u \<in> closed_segment x y \<Longrightarrow> cmod (f u - f x) \<le> e / 2" |
|
2837 |
by (meson close d1_less le_less_trans less_imp_le min.strict_boundedE segment_bound1) |
|
2838 |
qed (use e in simp) |
|
68339 | 2839 |
also have "\<dots> < e * cmod (y - x)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2840 |
by (simp add: e yx) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2841 |
finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2842 |
using i yx by (simp add: contour_integral_unique divide_less_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2843 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2844 |
then have "\<exists>d>0. \<forall>y. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2845 |
using dpos by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2846 |
} |
61976 | 2847 |
then have *: "(\<lambda>y. (?pathint x y - f x * (y - x)) /\<^sub>R cmod (y - x)) \<midarrow>x\<rightarrow> 0" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2848 |
by (simp add: Lim_at dist_norm inverse_eq_divide) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2849 |
show ?thesis |
67685
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67613
diff
changeset
|
2850 |
apply (simp add: has_field_derivative_def has_derivative_at2 bounded_linear_mult_right) |
70365
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70196
diff
changeset
|
2851 |
apply (rule Lim_transform [OF * tendsto_eventually]) |
68310 | 2852 |
using \<open>open S\<close> x apply (force simp: dist_norm open_contains_ball inverse_eq_divide [symmetric] eventually_at) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2853 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2854 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2855 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2856 |
(** Existence of a primitive.*) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2857 |
lemma holomorphic_starlike_primitive: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62464
diff
changeset
|
2858 |
fixes f :: "complex \<Rightarrow> complex" |
68310 | 2859 |
assumes contf: "continuous_on S f" |
2860 |
and S: "starlike S" and os: "open S" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2861 |
and k: "finite k" |
68310 | 2862 |
and fcd: "\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x" |
2863 |
shows "\<exists>g. \<forall>x \<in> S. (g has_field_derivative f x) (at x)" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2864 |
proof - |
68310 | 2865 |
obtain a where a: "a\<in>S" and a_cs: "\<And>x. x\<in>S \<Longrightarrow> closed_segment a x \<subseteq> S" |
2866 |
using S by (auto simp: starlike_def) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2867 |
{ fix x b c |
68310 | 2868 |
assume "x \<in> S" "closed_segment b c \<subseteq> S" |
2869 |
then have abcs: "convex hull {a, b, c} \<subseteq> S" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2870 |
by (simp add: a a_cs starlike_convex_subset) |
68310 | 2871 |
then have "continuous_on (convex hull {a, b, c}) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2872 |
by (simp add: continuous_on_subset [OF contf]) |
68310 | 2873 |
then have "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
2874 |
using abcs interior_subset by (force intro: fcd Cauchy_theorem_triangle_cofinite [OF _ k]) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2875 |
} note 0 = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2876 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2877 |
apply (intro exI ballI) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2878 |
apply (rule triangle_contour_integrals_starlike_primitive [OF contf a os], assumption) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2879 |
apply (metis a_cs) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2880 |
apply (metis has_chain_integral_chain_integral3 0) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2881 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2882 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2883 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
2884 |
lemma Cauchy_theorem_starlike: |
68310 | 2885 |
"\<lbrakk>open S; starlike S; finite k; continuous_on S f; |
2886 |
\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x; |
|
2887 |
valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk> |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2888 |
\<Longrightarrow> (f has_contour_integral 0) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2889 |
by (metis holomorphic_starlike_primitive Cauchy_theorem_primitive at_within_open) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2890 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
2891 |
lemma Cauchy_theorem_starlike_simple: |
68310 | 2892 |
"\<lbrakk>open S; starlike S; f holomorphic_on S; valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2893 |
\<Longrightarrow> (f has_contour_integral 0) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2894 |
apply (rule Cauchy_theorem_starlike [OF _ _ finite.emptyI]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2895 |
apply (simp_all add: holomorphic_on_imp_continuous_on) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2896 |
apply (metis at_within_open holomorphic_on_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2897 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2898 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2899 |
subsection\<open>Cauchy's theorem for a convex set\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2900 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2901 |
text\<open>For a convex set we can avoid assuming openness and boundary analyticity\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2902 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2903 |
lemma triangle_contour_integrals_convex_primitive: |
68310 | 2904 |
assumes contf: "continuous_on S f" |
2905 |
and S: "a \<in> S" "convex S" |
|
2906 |
and x: "x \<in> S" |
|
2907 |
and zer: "\<And>b c. \<lbrakk>b \<in> S; c \<in> S\<rbrakk> |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2908 |
\<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2909 |
contour_integral (linepath c a) f = 0" |
68310 | 2910 |
shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x within S)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2911 |
proof - |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2912 |
let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2913 |
{ fix y |
68310 | 2914 |
assume y: "y \<in> S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2915 |
have cont_ayf: "continuous_on (closed_segment a y) f" |
68310 | 2916 |
using S y by (meson contf continuous_on_subset convex_contains_segment) |
2917 |
have xys: "closed_segment x y \<subseteq> S" (*?*) |
|
2918 |
using convex_contains_segment S x y by auto |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2919 |
have "?pathint a y - ?pathint a x = ?pathint x y" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2920 |
using zer [OF x y] contour_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2921 |
} note [simp] = this |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2922 |
{ fix e::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2923 |
assume e: "0 < e" |
68310 | 2924 |
have cont_atx: "continuous (at x within S) f" |
2925 |
using x S contf by (simp add: continuous_on_eq_continuous_within) |
|
2926 |
then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. \<lbrakk>y \<in> S; cmod (y - x) < d1\<rbrakk> \<Longrightarrow> cmod (f y - f x) < e/2" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2927 |
unfolding continuous_within Lim_within dist_norm using e |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2928 |
by (drule_tac x="e/2" in spec) force |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2929 |
{ fix y |
68310 | 2930 |
assume yx: "y \<noteq> x" and close: "cmod (y - x) < d1" and y: "y \<in> S" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2931 |
have fxy: "f contour_integrable_on linepath x y" |
68310 | 2932 |
using convex_contains_segment S x y |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2933 |
by (blast intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2934 |
then obtain i where i: "(f has_contour_integral i) (linepath x y)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2935 |
by (auto simp: contour_integrable_on_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2936 |
then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2937 |
by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2938 |
then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)" |
68310 | 2939 |
proof (rule has_contour_integral_bound_linepath) |
2940 |
show "\<And>u. u \<in> closed_segment x y \<Longrightarrow> cmod (f u - f x) \<le> e / 2" |
|
2941 |
by (meson assms(3) close convex_contains_segment d1_less le_less_trans less_imp_le segment_bound1 subset_iff x y) |
|
2942 |
qed (use e in simp) |
|
68339 | 2943 |
also have "\<dots> < e * cmod (y - x)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2944 |
by (simp add: e yx) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2945 |
finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
2946 |
using i yx by (simp add: contour_integral_unique divide_less_eq) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2947 |
} |
68310 | 2948 |
then have "\<exists>d>0. \<forall>y\<in>S. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2949 |
using d1 by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2950 |
} |
68310 | 2951 |
then have *: "((\<lambda>y. (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) \<longlongrightarrow> 0) (at x within S)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2952 |
by (simp add: Lim_within dist_norm inverse_eq_divide) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2953 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2954 |
apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right) |
70365
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70196
diff
changeset
|
2955 |
apply (rule Lim_transform [OF * tendsto_eventually]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2956 |
using linordered_field_no_ub |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2957 |
apply (force simp: inverse_eq_divide [symmetric] eventually_at) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2958 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2959 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2960 |
|
61848 | 2961 |
lemma contour_integral_convex_primitive: |
68493 | 2962 |
assumes "convex S" "continuous_on S f" |
68310 | 2963 |
"\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" |
2964 |
obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)" |
|
2965 |
proof (cases "S={}") |
|
2966 |
case False |
|
2967 |
with assms that show ?thesis |
|
2968 |
by (blast intro: triangle_contour_integrals_convex_primitive has_chain_integral_chain_integral3) |
|
2969 |
qed auto |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2970 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2971 |
lemma holomorphic_convex_primitive: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62464
diff
changeset
|
2972 |
fixes f :: "complex \<Rightarrow> complex" |
68493 | 2973 |
assumes "convex S" "finite K" and contf: "continuous_on S f" |
68310 | 2974 |
and fd: "\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x" |
2975 |
obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)" |
|
2976 |
proof (rule contour_integral_convex_primitive [OF \<open>convex S\<close> contf Cauchy_theorem_triangle_cofinite]) |
|
2977 |
have *: "convex hull {a, b, c} \<subseteq> S" if "a \<in> S" "b \<in> S" "c \<in> S" for a b c |
|
2978 |
by (simp add: \<open>convex S\<close> hull_minimal that) |
|
2979 |
show "continuous_on (convex hull {a, b, c}) f" if "a \<in> S" "b \<in> S" "c \<in> S" for a b c |
|
2980 |
by (meson "*" contf continuous_on_subset that) |
|
2981 |
show "f field_differentiable at x" if "a \<in> S" "b \<in> S" "c \<in> S" "x \<in> interior (convex hull {a, b, c}) - K" for a b c x |
|
2982 |
by (metis "*" DiffD1 DiffD2 DiffI fd interior_mono subsetCE that) |
|
2983 |
qed (use assms in \<open>force+\<close>) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
2984 |
|
67107
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
2985 |
lemma holomorphic_convex_primitive': |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
2986 |
fixes f :: "complex \<Rightarrow> complex" |
68310 | 2987 |
assumes "convex S" and "open S" and "f holomorphic_on S" |
2988 |
obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)" |
|
67107
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
2989 |
proof (rule holomorphic_convex_primitive) |
68310 | 2990 |
fix x assume "x \<in> interior S - {}" |
67107
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
2991 |
with assms show "f field_differentiable at x" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
2992 |
by (auto intro!: holomorphic_on_imp_differentiable_at simp: interior_open) |
68310 | 2993 |
qed (use assms in \<open>auto intro: holomorphic_on_imp_continuous_on\<close>) |
2994 |
||
70136 | 2995 |
corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_convex: |
68310 | 2996 |
"\<lbrakk>continuous_on S f; convex S; finite K; |
2997 |
\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x; |
|
68493 | 2998 |
valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk> |
68310 | 2999 |
\<Longrightarrow> (f has_contour_integral 0) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3000 |
by (metis holomorphic_convex_primitive Cauchy_theorem_primitive) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3001 |
|
68310 | 3002 |
corollary Cauchy_theorem_convex_simple: |
3003 |
"\<lbrakk>f holomorphic_on S; convex S; |
|
3004 |
valid_path g; path_image g \<subseteq> S; |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3005 |
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g" |
68310 | 3006 |
apply (rule Cauchy_theorem_convex [where K = "{}"]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3007 |
apply (simp_all add: holomorphic_on_imp_continuous_on) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3008 |
using at_within_interior holomorphic_on_def interior_subset by fastforce |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3009 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3010 |
text\<open>In particular for a disc\<close> |
70136 | 3011 |
corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_disc: |
68310 | 3012 |
"\<lbrakk>finite K; continuous_on (cball a e) f; |
3013 |
\<And>x. x \<in> ball a e - K \<Longrightarrow> f field_differentiable at x; |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3014 |
valid_path g; path_image g \<subseteq> cball a e; |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3015 |
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g" |
68310 | 3016 |
by (auto intro: Cauchy_theorem_convex) |
3017 |
||
70136 | 3018 |
corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_disc_simple: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3019 |
"\<lbrakk>f holomorphic_on (ball a e); valid_path g; path_image g \<subseteq> ball a e; |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3020 |
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3021 |
by (simp add: Cauchy_theorem_convex_simple) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3022 |
|
70136 | 3023 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Generalize integrability to local primitives\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3024 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3025 |
lemma contour_integral_local_primitive_lemma: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3026 |
fixes f :: "complex\<Rightarrow>complex" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3027 |
shows |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3028 |
"\<lbrakk>g piecewise_differentiable_on {a..b}; |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3029 |
\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s); |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3030 |
\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s\<rbrakk> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3031 |
\<Longrightarrow> (\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b})) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3032 |
integrable_on {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3033 |
apply (cases "cbox a b = {}", force) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3034 |
apply (simp add: integrable_on_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3035 |
apply (rule exI) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3036 |
apply (rule contour_integral_primitive_lemma, assumption+) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3037 |
using atLeastAtMost_iff by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3038 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3039 |
lemma contour_integral_local_primitive_any: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3040 |
fixes f :: "complex \<Rightarrow> complex" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3041 |
assumes gpd: "g piecewise_differentiable_on {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3042 |
and dh: "\<And>x. x \<in> s |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3043 |
\<Longrightarrow> \<exists>d h. 0 < d \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3044 |
(\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3045 |
and gs: "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3046 |
shows "(\<lambda>x. f(g x) * vector_derivative g (at x)) integrable_on {a..b}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3047 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3048 |
{ fix x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3049 |
assume x: "a \<le> x" "x \<le> b" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3050 |
obtain d h where d: "0 < d" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3051 |
and h: "(\<And>y. norm(y - g x) < d \<Longrightarrow> (h has_field_derivative f y) (at y within s))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3052 |
using x gs dh by (metis atLeastAtMost_iff) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3053 |
have "continuous_on {a..b} g" using gpd piecewise_differentiable_on_def by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3054 |
then obtain e where e: "e>0" and lessd: "\<And>x'. x' \<in> {a..b} \<Longrightarrow> \<bar>x' - x\<bar> < e \<Longrightarrow> cmod (g x' - g x) < d" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3055 |
using x d |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3056 |
apply (auto simp: dist_norm continuous_on_iff) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3057 |
apply (drule_tac x=x in bspec) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3058 |
using x apply simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3059 |
apply (drule_tac x=d in spec, auto) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3060 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3061 |
have "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> {u..v} \<subseteq> ball x d \<and> (u \<le> v \<longrightarrow> a \<le> u \<and> v \<le> b) \<longrightarrow> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3062 |
(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {u..v}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3063 |
apply (rule_tac x=e in exI) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3064 |
using e |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3065 |
apply (simp add: integrable_on_localized_vector_derivative [symmetric], clarify) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3066 |
apply (rule_tac f = h and s = "g ` {u..v}" in contour_integral_local_primitive_lemma) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3067 |
apply (meson atLeastatMost_subset_iff gpd piecewise_differentiable_on_subset) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3068 |
apply (force simp: ball_def dist_norm intro: lessd gs DERIV_subset [OF h], force) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3069 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3070 |
} then |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3071 |
show ?thesis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3072 |
by (force simp: intro!: integrable_on_little_subintervals [of a b, simplified]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3073 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3074 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3075 |
lemma contour_integral_local_primitive: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3076 |
fixes f :: "complex \<Rightarrow> complex" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3077 |
assumes g: "valid_path g" "path_image g \<subseteq> s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3078 |
and dh: "\<And>x. x \<in> s |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3079 |
\<Longrightarrow> \<exists>d h. 0 < d \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3080 |
(\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3081 |
shows "f contour_integrable_on g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3082 |
using g |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3083 |
apply (simp add: valid_path_def path_image_def contour_integrable_on_def has_contour_integral_def |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3084 |
has_integral_localized_vector_derivative integrable_on_def [symmetric]) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3085 |
using contour_integral_local_primitive_any [OF _ dh] |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3086 |
by (meson image_subset_iff piecewise_C1_imp_differentiable) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3087 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3088 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3089 |
text\<open>In particular if a function is holomorphic\<close> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3090 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3091 |
lemma contour_integrable_holomorphic: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3092 |
assumes contf: "continuous_on s f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3093 |
and os: "open s" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3094 |
and k: "finite k" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3095 |
and g: "valid_path g" "path_image g \<subseteq> s" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
3096 |
and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f field_differentiable at x" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3097 |
shows "f contour_integrable_on g" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3098 |
proof - |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3099 |
{ fix z |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3100 |
assume z: "z \<in> s" |
68310 | 3101 |
obtain d where "d>0" and d: "ball z d \<subseteq> s" using \<open>open s\<close> z |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3102 |
by (auto simp: open_contains_ball) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3103 |
then have contfb: "continuous_on (ball z d) f" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3104 |
using contf continuous_on_subset by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3105 |
obtain h where "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within ball z d)" |
69712 | 3106 |
by (metis holomorphic_convex_primitive [OF convex_ball k contfb fcd] d interior_subset Diff_iff subsetD) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3107 |
then have "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within s)" |
68310 | 3108 |
by (metis open_ball at_within_open d os subsetCE) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3109 |
then have "\<exists>h. (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3110 |
by (force simp: dist_norm norm_minus_commute) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3111 |
then have "\<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))" |
68310 | 3112 |
using \<open>0 < d\<close> by blast |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3113 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3114 |
then show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3115 |
by (rule contour_integral_local_primitive [OF g]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3116 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3117 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3118 |
lemma contour_integrable_holomorphic_simple: |
68310 | 3119 |
assumes fh: "f holomorphic_on S" |
3120 |
and os: "open S" |
|
3121 |
and g: "valid_path g" "path_image g \<subseteq> S" |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3122 |
shows "f contour_integrable_on g" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
3123 |
apply (rule contour_integrable_holomorphic [OF _ os Finite_Set.finite.emptyI g]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
3124 |
apply (simp add: fh holomorphic_on_imp_continuous_on) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
3125 |
using fh by (simp add: field_differentiable_def holomorphic_on_open os) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3126 |
|
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3127 |
lemma continuous_on_inversediff: |
68310 | 3128 |
fixes z:: "'a::real_normed_field" shows "z \<notin> S \<Longrightarrow> continuous_on S (\<lambda>w. 1 / (w - z))" |
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3129 |
by (rule continuous_intros | force)+ |
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3130 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
3131 |
lemma contour_integrable_inversediff: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3132 |
"\<lbrakk>valid_path g; z \<notin> path_image g\<rbrakk> \<Longrightarrow> (\<lambda>w. 1 / (w-z)) contour_integrable_on g" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
3133 |
apply (rule contour_integrable_holomorphic_simple [of _ "UNIV-{z}"]) |
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3134 |
apply (auto simp: holomorphic_on_open open_delete intro!: derivative_eq_intros) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3135 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3136 |
|
61222 | 3137 |
text\<open>Key fact that path integral is the same for a "nearby" path. This is the |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3138 |
main lemma for the homotopy form of Cauchy's theorem and is also useful |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3139 |
if we want "without loss of generality" to assume some nice properties of a |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3140 |
path (e.g. smoothness). It can also be used to define the integrals of |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3141 |
analytic functions over arbitrary continuous paths. This is just done for |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3142 |
winding numbers now. |
61222 | 3143 |
\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3144 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3145 |
text\<open>A technical definition to avoid duplication of similar proofs, |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3146 |
for paths joined at the ends versus looping paths\<close> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3147 |
definition linked_paths :: "bool \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3148 |
where "linked_paths atends g h == |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3149 |
(if atends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3150 |
else pathfinish g = pathstart g \<and> pathfinish h = pathstart h)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3151 |
|
69597 | 3152 |
text\<open>This formulation covers two cases: \<^term>\<open>g\<close> and \<^term>\<open>h\<close> share their |
3153 |
start and end points; \<^term>\<open>g\<close> and \<^term>\<open>h\<close> both loop upon themselves.\<close> |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3154 |
lemma contour_integral_nearby: |
68310 | 3155 |
assumes os: "open S" and p: "path p" "path_image p \<subseteq> S" |
3156 |
shows "\<exists>d. 0 < d \<and> |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3157 |
(\<forall>g h. valid_path g \<and> valid_path h \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3158 |
(\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and> |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3159 |
linked_paths atends g h |
68310 | 3160 |
\<longrightarrow> path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> |
3161 |
(\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f))" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3162 |
proof - |
68310 | 3163 |
have "\<forall>z. \<exists>e. z \<in> path_image p \<longrightarrow> 0 < e \<and> ball z e \<subseteq> S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3164 |
using open_contains_ball os p(2) by blast |
68310 | 3165 |
then obtain ee where ee: "\<And>z. z \<in> path_image p \<Longrightarrow> 0 < ee z \<and> ball z (ee z) \<subseteq> S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3166 |
by metis |
63040 | 3167 |
define cover where "cover = (\<lambda>z. ball z (ee z/3)) ` (path_image p)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3168 |
have "compact (path_image p)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3169 |
by (metis p(1) compact_path_image) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3170 |
moreover have "path_image p \<subseteq> (\<Union>c\<in>path_image p. ball c (ee c / 3))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3171 |
using ee by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3172 |
ultimately have "\<exists>D \<subseteq> cover. finite D \<and> path_image p \<subseteq> \<Union>D" |
69529 | 3173 |
by (simp add: compact_eq_Heine_Borel cover_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3174 |
then obtain D where D: "D \<subseteq> cover" "finite D" "path_image p \<subseteq> \<Union>D" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3175 |
by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3176 |
then obtain k where k: "k \<subseteq> {0..1}" "finite k" and D_eq: "D = ((\<lambda>z. ball z (ee z / 3)) \<circ> p) ` k" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3177 |
apply (simp add: cover_def path_image_def image_comp) |
61222 | 3178 |
apply (blast dest!: finite_subset_image [OF \<open>finite D\<close>]) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3179 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3180 |
then have kne: "k \<noteq> {}" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3181 |
using D by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3182 |
have pi: "\<And>i. i \<in> k \<Longrightarrow> p i \<in> path_image p" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3183 |
using k by (auto simp: path_image_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3184 |
then have eepi: "\<And>i. i \<in> k \<Longrightarrow> 0 < ee((p i))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3185 |
by (metis ee) |
68339 | 3186 |
define e where "e = Min((ee \<circ> p) ` k)" |
3187 |
have fin_eep: "finite ((ee \<circ> p) ` k)" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3188 |
using k by blast |
68310 | 3189 |
have "0 < e" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3190 |
using ee k by (simp add: kne e_def Min_gr_iff [OF fin_eep] eepi) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3191 |
have "uniformly_continuous_on {0..1} p" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3192 |
using p by (simp add: path_def compact_uniformly_continuous) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3193 |
then obtain d::real where d: "d>0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3194 |
and de: "\<And>x x'. \<bar>x' - x\<bar> < d \<Longrightarrow> x\<in>{0..1} \<Longrightarrow> x'\<in>{0..1} \<Longrightarrow> cmod (p x' - p x) < e/3" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3195 |
unfolding uniformly_continuous_on_def dist_norm real_norm_def |
68310 | 3196 |
by (metis divide_pos_pos \<open>0 < e\<close> zero_less_numeral) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3197 |
then obtain N::nat where N: "N>0" "inverse N < d" |
62623
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
3198 |
using real_arch_inverse [of d] by auto |
68310 | 3199 |
show ?thesis |
3200 |
proof (intro exI conjI allI; clarify?) |
|
3201 |
show "e/3 > 0" |
|
3202 |
using \<open>0 < e\<close> by simp |
|
3203 |
fix g h |
|
3204 |
assume g: "valid_path g" and ghp: "\<forall>t\<in>{0..1}. cmod (g t - p t) < e / 3 \<and> cmod (h t - p t) < e / 3" |
|
3205 |
and h: "valid_path h" |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3206 |
and joins: "linked_paths atends g h" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3207 |
{ fix t::real |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3208 |
assume t: "0 \<le> t" "t \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3209 |
then obtain u where u: "u \<in> k" and ptu: "p t \<in> ball(p u) (ee(p u) / 3)" |
61222 | 3210 |
using \<open>path_image p \<subseteq> \<Union>D\<close> D_eq by (force simp: path_image_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3211 |
then have ele: "e \<le> ee (p u)" using fin_eep |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3212 |
by (simp add: e_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3213 |
have "cmod (g t - p t) < e / 3" "cmod (h t - p t) < e / 3" |
68310 | 3214 |
using ghp t by auto |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3215 |
with ele have "cmod (g t - p t) < ee (p u) / 3" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3216 |
"cmod (h t - p t) < ee (p u) / 3" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3217 |
by linarith+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3218 |
then have "g t \<in> ball(p u) (ee(p u))" "h t \<in> ball(p u) (ee(p u))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3219 |
using norm_diff_triangle_ineq [of "g t" "p t" "p t" "p u"] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3220 |
norm_diff_triangle_ineq [of "h t" "p t" "p t" "p u"] ptu eepi u |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3221 |
by (force simp: dist_norm ball_def norm_minus_commute)+ |
68310 | 3222 |
then have "g t \<in> S" "h t \<in> S" using ee u k |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3223 |
by (auto simp: path_image_def ball_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3224 |
} |
68310 | 3225 |
then have ghs: "path_image g \<subseteq> S" "path_image h \<subseteq> S" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3226 |
by (auto simp: path_image_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3227 |
moreover |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3228 |
{ fix f |
68310 | 3229 |
assume fhols: "f holomorphic_on S" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3230 |
then have fpa: "f contour_integrable_on g" "f contour_integrable_on h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3231 |
using g ghs h holomorphic_on_imp_continuous_on os contour_integrable_holomorphic_simple |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3232 |
by blast+ |
68310 | 3233 |
have contf: "continuous_on S f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3234 |
by (simp add: fhols holomorphic_on_imp_continuous_on) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3235 |
{ fix z |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3236 |
assume z: "z \<in> path_image p" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3237 |
have "f holomorphic_on ball z (ee z)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3238 |
using fhols ee z holomorphic_on_subset by blast |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3239 |
then have "\<exists>ff. (\<forall>w \<in> ball z (ee z). (ff has_field_derivative f w) (at w))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3240 |
using holomorphic_convex_primitive [of "ball z (ee z)" "{}" f, simplified] |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3241 |
by (metis open_ball at_within_open holomorphic_on_def holomorphic_on_imp_continuous_on mem_ball) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3242 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3243 |
then obtain ff where ff: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3244 |
"\<And>z w. \<lbrakk>z \<in> path_image p; w \<in> ball z (ee z)\<rbrakk> \<Longrightarrow> (ff z has_field_derivative f w) (at w)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3245 |
by metis |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3246 |
{ fix n |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3247 |
assume n: "n \<le> N" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3248 |
then have "contour_integral(subpath 0 (n/N) h) f - contour_integral(subpath 0 (n/N) g) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3249 |
contour_integral(linepath (g(n/N)) (h(n/N))) f - contour_integral(linepath (g 0) (h 0)) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3250 |
proof (induct n) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3251 |
case 0 show ?case by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3252 |
next |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3253 |
case (Suc n) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3254 |
obtain t where t: "t \<in> k" and "p (n/N) \<in> ball(p t) (ee(p t) / 3)" |
61222 | 3255 |
using \<open>path_image p \<subseteq> \<Union>D\<close> [THEN subsetD, where c="p (n/N)"] D_eq N Suc.prems |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3256 |
by (force simp: path_image_def) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3257 |
then have ptu: "cmod (p t - p (n/N)) < ee (p t) / 3" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3258 |
by (simp add: dist_norm) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3259 |
have e3le: "e/3 \<le> ee (p t) / 3" using fin_eep t |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3260 |
by (simp add: e_def) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3261 |
{ fix x |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3262 |
assume x: "n/N \<le> x" "x \<le> (1 + n)/N" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3263 |
then have nN01: "0 \<le> n/N" "(1 + n)/N \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3264 |
using Suc.prems by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3265 |
then have x01: "0 \<le> x" "x \<le> 1" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3266 |
using x by linarith+ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3267 |
have "cmod (p t - p x) < ee (p t) / 3 + e/3" |
68310 | 3268 |
proof (rule norm_diff_triangle_less [OF ptu de]) |
3269 |
show "\<bar>real n / real N - x\<bar> < d" |
|
3270 |
using x N by (auto simp: field_simps) |
|
3271 |
qed (use x01 Suc.prems in auto) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3272 |
then have ptx: "cmod (p t - p x) < 2*ee (p t)/3" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3273 |
using e3le eepi [OF t] by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3274 |
have "cmod (p t - g x) < 2*ee (p t)/3 + e/3 " |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3275 |
apply (rule norm_diff_triangle_less [OF ptx]) |
68310 | 3276 |
using ghp x01 by (simp add: norm_minus_commute) |
68339 | 3277 |
also have "\<dots> \<le> ee (p t)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3278 |
using e3le eepi [OF t] by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3279 |
finally have gg: "cmod (p t - g x) < ee (p t)" . |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3280 |
have "cmod (p t - h x) < 2*ee (p t)/3 + e/3 " |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3281 |
apply (rule norm_diff_triangle_less [OF ptx]) |
68310 | 3282 |
using ghp x01 by (simp add: norm_minus_commute) |
68339 | 3283 |
also have "\<dots> \<le> ee (p t)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3284 |
using e3le eepi [OF t] by simp |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3285 |
finally have "cmod (p t - g x) < ee (p t)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3286 |
"cmod (p t - h x) < ee (p t)" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3287 |
using gg by auto |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3288 |
} note ptgh_ee = this |
68310 | 3289 |
have "closed_segment (g (real n / real N)) (h (real n / real N)) = path_image (linepath (h (n/N)) (g (n/N)))" |
3290 |
by (simp add: closed_segment_commute) |
|
3291 |
also have pi_hgn: "\<dots> \<subseteq> ball (p t) (ee (p t))" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3292 |
using ptgh_ee [of "n/N"] Suc.prems |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
3293 |
by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"]) |
68310 | 3294 |
finally have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> S" |
3295 |
using ee pi t by blast |
|
3296 |
have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N))) \<subseteq> ball (p t) (ee (p t))" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3297 |
using ptgh_ee [of "(1+n)/N"] Suc.prems |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
3298 |
by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"]) |
68310 | 3299 |
then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> S" |
61222 | 3300 |
using \<open>N>0\<close> Suc.prems ee pi t |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3301 |
by (auto simp: Path_Connected.path_image_join field_simps) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3302 |
have pi_subset_ball: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3303 |
"path_image (subpath (n/N) ((1+n) / N) g +++ linepath (g ((1+n) / N)) (h ((1+n) / N)) +++ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3304 |
subpath ((1+n) / N) (n/N) h +++ linepath (h (n/N)) (g (n/N))) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3305 |
\<subseteq> ball (p t) (ee (p t))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3306 |
apply (intro subset_path_image_join pi_hgn pi_ghn') |
61222 | 3307 |
using \<open>N>0\<close> Suc.prems |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3308 |
apply (auto simp: path_image_subpath dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3309 |
done |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3310 |
have pi0: "(f has_contour_integral 0) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3311 |
(subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++ |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3312 |
subpath ((Suc n) / N) (n/N) h +++ linepath(h (n/N)) (g (n/N)))" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3313 |
apply (rule Cauchy_theorem_primitive [of "ball(p t) (ee(p t))" "ff (p t)" "f"]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3314 |
apply (metis ff open_ball at_within_open pi t) |
68310 | 3315 |
using Suc.prems pi_subset_ball apply (simp_all add: valid_path_join valid_path_subpath g h) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3316 |
done |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3317 |
have fpa1: "f contour_integrable_on subpath (real n / real N) (real (Suc n) / real N) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3318 |
using Suc.prems by (simp add: contour_integrable_subpath g fpa) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3319 |
have fpa2: "f contour_integrable_on linepath (g (real (Suc n) / real N)) (h (real (Suc n) / real N))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3320 |
using gh_n's |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3321 |
by (auto intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3322 |
have fpa3: "f contour_integrable_on linepath (h (real n / real N)) (g (real n / real N))" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3323 |
using gh_ns |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3324 |
by (auto simp: closed_segment_commute intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3325 |
have eq0: "contour_integral (subpath (n/N) ((Suc n) / real N) g) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3326 |
contour_integral (linepath (g ((Suc n) / N)) (h ((Suc n) / N))) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3327 |
contour_integral (subpath ((Suc n) / N) (n/N) h) f + |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3328 |
contour_integral (linepath (h (n/N)) (g (n/N))) f = 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3329 |
using contour_integral_unique [OF pi0] Suc.prems |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3330 |
by (simp add: g h fpa valid_path_subpath contour_integrable_subpath |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
3331 |
fpa1 fpa2 fpa3 algebra_simps del: of_nat_Suc) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3332 |
have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'. |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3333 |
\<lbrakk>hn - gn = ghn - gh0; |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3334 |
gd + ghn' + he + hgn = (0::complex); |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3335 |
hn - he = hn'; gn + gd = gn'; hgn = -ghn\<rbrakk> \<Longrightarrow> hn' - gn' = ghn' - gh0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3336 |
by (auto simp: algebra_simps) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3337 |
have "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3338 |
contour_integral (subpath 0 (n/N) h) f + contour_integral (subpath (n/N) ((Suc n) / N) h) f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3339 |
unfolding reversepath_subpath [symmetric, of "((Suc n) / N)"] |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3340 |
using Suc.prems by (simp add: h fpa contour_integral_reversepath valid_path_subpath contour_integrable_subpath) |
68339 | 3341 |
also have "\<dots> = contour_integral (subpath 0 ((Suc n) / N) h) f" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3342 |
using Suc.prems by (simp add: contour_integral_subpath_combine h fpa) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3343 |
finally have pi0_eq: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3344 |
"contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3345 |
contour_integral (subpath 0 ((Suc n) / N) h) f" . |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3346 |
show ?case |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3347 |
apply (rule * [OF Suc.hyps eq0 pi0_eq]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3348 |
using Suc.prems |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3349 |
apply (simp_all add: g h fpa contour_integral_subpath_combine |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3350 |
contour_integral_reversepath [symmetric] contour_integrable_continuous_linepath |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3351 |
continuous_on_subset [OF contf gh_ns]) |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3352 |
done |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3353 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3354 |
} note ind = this |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3355 |
have "contour_integral h f = contour_integral g f" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3356 |
using ind [OF order_refl] N joins |
62390 | 3357 |
by (simp add: linked_paths_def pathstart_def pathfinish_def split: if_split_asm) |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3358 |
} |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3359 |
ultimately |
68310 | 3360 |
show "path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f)" |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3361 |
by metis |
68310 | 3362 |
qed |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3363 |
qed |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3364 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3365 |
|
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3366 |
lemma |
68310 | 3367 |
assumes "open S" "path p" "path_image p \<subseteq> S" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3368 |
shows contour_integral_nearby_ends: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3369 |
"\<exists>d. 0 < d \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3370 |
(\<forall>g h. valid_path g \<and> valid_path h \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3371 |
(\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3372 |
pathstart h = pathstart g \<and> pathfinish h = pathfinish g |
68310 | 3373 |
\<longrightarrow> path_image g \<subseteq> S \<and> |
3374 |
path_image h \<subseteq> S \<and> |
|
3375 |
(\<forall>f. f holomorphic_on S |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3376 |
\<longrightarrow> contour_integral h f = contour_integral g f))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3377 |
and contour_integral_nearby_loops: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3378 |
"\<exists>d. 0 < d \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3379 |
(\<forall>g h. valid_path g \<and> valid_path h \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3380 |
(\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and> |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3381 |
pathfinish g = pathstart g \<and> pathfinish h = pathstart h |
68310 | 3382 |
\<longrightarrow> path_image g \<subseteq> S \<and> |
3383 |
path_image h \<subseteq> S \<and> |
|
3384 |
(\<forall>f. f holomorphic_on S |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3385 |
\<longrightarrow> contour_integral h f = contour_integral g f))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3386 |
using contour_integral_nearby [OF assms, where atends=True] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3387 |
using contour_integral_nearby [OF assms, where atends=False] |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3388 |
unfolding linked_paths_def by simp_all |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
3389 |
|
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3390 |
lemma C1_differentiable_polynomial_function: |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3391 |
fixes p :: "real \<Rightarrow> 'a::euclidean_space" |
68310 | 3392 |
shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on S" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3393 |
by (metis continuous_on_polymonial_function C1_differentiable_on_def has_vector_derivative_polynomial_function) |
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3394 |
|
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3395 |
lemma valid_path_polynomial_function: |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3396 |
fixes p :: "real \<Rightarrow> 'a::euclidean_space" |
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3397 |
shows "polynomial_function p \<Longrightarrow> valid_path p" |
61190
2bd401e364f9
Massive revisions, as a valid path must now be continously differentiable (C!)
paulson <lp15@cam.ac.uk>
parents:
61104
diff
changeset
|
3398 |
by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function) |
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3399 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3400 |
lemma valid_path_subpath_trivial [simp]: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3401 |
fixes g :: "real \<Rightarrow> 'a::euclidean_space" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3402 |
shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3403 |
by (simp add: subpath_def valid_path_polynomial_function) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3404 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3405 |
lemma contour_integral_bound_exists: |
68310 | 3406 |
assumes S: "open S" |
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3407 |
and g: "valid_path g" |
68310 | 3408 |
and pag: "path_image g \<subseteq> S" |
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3409 |
shows "\<exists>L. 0 < L \<and> |
68310 | 3410 |
(\<forall>f B. f holomorphic_on S \<and> (\<forall>z \<in> S. norm(f z) \<le> B) |
3411 |
\<longrightarrow> norm(contour_integral g f) \<le> L*B)" |
|
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3412 |
proof - |
68310 | 3413 |
have "path g" using g |
3414 |
by (simp add: valid_path_imp_path) |
|
3415 |
then obtain d::real and p |
|
3416 |
where d: "0 < d" |
|
3417 |
and p: "polynomial_function p" "path_image p \<subseteq> S" |
|
3418 |
and pi: "\<And>f. f holomorphic_on S \<Longrightarrow> contour_integral g f = contour_integral p f" |
|
3419 |
using contour_integral_nearby_ends [OF S \<open>path g\<close> pag] |
|
3420 |
apply clarify |
|
3421 |
apply (drule_tac x=g in spec) |
|
3422 |
apply (simp only: assms) |
|
3423 |
apply (force simp: valid_path_polynomial_function dest: path_approx_polynomial_function) |
|
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3424 |
done |
68310 | 3425 |
then obtain p' where p': "polynomial_function p'" |
3426 |
"\<And>x. (p has_vector_derivative (p' x)) (at x)" |
|
68339 | 3427 |
by (blast intro: has_vector_derivative_polynomial_function that) |
68310 | 3428 |
then have "bounded(p' ` {0..1})" |
3429 |
using continuous_on_polymonial_function |
|
3430 |
by (force simp: intro!: compact_imp_bounded compact_continuous_image) |
|
3431 |
then obtain L where L: "L>0" and nop': "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> norm (p' x) \<le> L" |
|
3432 |
by (force simp: bounded_pos) |
|
3433 |
{ fix f B |
|
3434 |
assume f: "f holomorphic_on S" and B: "\<And>z. z\<in>S \<Longrightarrow> cmod (f z) \<le> B" |
|
3435 |
then have "f contour_integrable_on p \<and> valid_path p" |
|
3436 |
using p S |
|
3437 |
by (blast intro: valid_path_polynomial_function contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on) |
|
3438 |
moreover have "cmod (vector_derivative p (at x)) * cmod (f (p x)) \<le> L * B" if "0 \<le> x" "x \<le> 1" for x |
|
3439 |
proof (rule mult_mono) |
|
3440 |
show "cmod (vector_derivative p (at x)) \<le> L" |
|
3441 |
by (metis nop' p'(2) that vector_derivative_at) |
|
3442 |
show "cmod (f (p x)) \<le> B" |
|
3443 |
by (metis B atLeastAtMost_iff imageI p(2) path_defs(4) subset_eq that) |
|
3444 |
qed (use \<open>L>0\<close> in auto) |
|
3445 |
ultimately have "cmod (contour_integral g f) \<le> L * B" |
|
3446 |
apply (simp only: pi [OF f]) |
|
3447 |
apply (simp only: contour_integral_integral) |
|
3448 |
apply (rule order_trans [OF integral_norm_bound_integral]) |
|
3449 |
apply (auto simp: mult.commute integral_norm_bound_integral contour_integrable_on [symmetric] norm_mult) |
|
3450 |
done |
|
3451 |
} then |
|
3452 |
show ?thesis |
|
3453 |
by (force simp: L contour_integral_integral) |
|
61104
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3454 |
qed |
3c2d4636cebc
new lemmas about vector_derivative, complex numbers, paths, etc.
paulson
parents:
60809
diff
changeset
|
3455 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3456 |
text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3457 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
3458 |
subsection \<open>Winding Numbers\<close> |
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
3459 |
|
70136 | 3460 |
definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where |
68326 | 3461 |
"winding_number_prop \<gamma> z e p n \<equiv> |
3462 |
valid_path p \<and> z \<notin> path_image p \<and> |
|
3463 |
pathstart p = pathstart \<gamma> \<and> |
|
3464 |
pathfinish p = pathfinish \<gamma> \<and> |
|
3465 |
(\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and> |
|
3466 |
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n" |
|
3467 |
||
70136 | 3468 |
definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where |
68326 | 3469 |
"winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n" |
3470 |
||
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3471 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3472 |
lemma winding_number: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3473 |
assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e" |
68326 | 3474 |
shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3475 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3476 |
have "path_image \<gamma> \<subseteq> UNIV - {z}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3477 |
using assms by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3478 |
then obtain d |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3479 |
where d: "d>0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3480 |
and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3481 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3482 |
pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3483 |
path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3484 |
(\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3485 |
using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3486 |
then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3487 |
(\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)" |
61808 | 3488 |
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto |
63589 | 3489 |
define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))" |
68326 | 3490 |
have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3491 |
proof (rule_tac x=nn in exI, clarify) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3492 |
fix e::real |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3493 |
assume e: "e>0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3494 |
obtain p where p: "polynomial_function p \<and> |
68359 | 3495 |
pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))" |
61808 | 3496 |
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3497 |
have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3498 |
by (auto simp: intro!: holomorphic_intros) |
68326 | 3499 |
then show "\<exists>p. winding_number_prop \<gamma> z e p nn" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3500 |
apply (rule_tac x=p in exI) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3501 |
using pi_eq [of h p] h p d |
68326 | 3502 |
apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3503 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3504 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3505 |
then show ?thesis |
68326 | 3506 |
unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3507 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3508 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3509 |
lemma winding_number_unique: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3510 |
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>" |
68326 | 3511 |
and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3512 |
shows "winding_number \<gamma> z = n" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3513 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3514 |
have "path_image \<gamma> \<subseteq> UNIV - {z}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3515 |
using assms by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3516 |
then obtain e |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3517 |
where e: "e>0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3518 |
and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2; |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3519 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e); |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3520 |
pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3521 |
contour_integral h2 f = contour_integral h1 f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3522 |
using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete) |
68326 | 3523 |
obtain p where p: "winding_number_prop \<gamma> z e p n" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3524 |
using pi [OF e] by blast |
68326 | 3525 |
obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3526 |
using winding_number [OF \<gamma> e] by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3527 |
have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))" |
68326 | 3528 |
using p by (auto simp: winding_number_prop_def) |
68339 | 3529 |
also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))" |
68310 | 3530 |
proof (rule pi_eq) |
3531 |
show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}" |
|
3532 |
by (auto intro!: holomorphic_intros) |
|
68326 | 3533 |
qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>) |
68339 | 3534 |
also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" |
68326 | 3535 |
using q by (auto simp: winding_number_prop_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3536 |
finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" . |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3537 |
then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3538 |
by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3539 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3540 |
|
68326 | 3541 |
(*NB not winding_number_prop here due to the loop in p*) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3542 |
lemma winding_number_unique_loop: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3543 |
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3544 |
and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3545 |
and pi: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3546 |
"\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3547 |
pathfinish p = pathstart p \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3548 |
(\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and> |
63589 | 3549 |
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3550 |
shows "winding_number \<gamma> z = n" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3551 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3552 |
have "path_image \<gamma> \<subseteq> UNIV - {z}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3553 |
using assms by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3554 |
then obtain e |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3555 |
where e: "e>0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3556 |
and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2; |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3557 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e); |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3558 |
pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3559 |
contour_integral h2 f = contour_integral h1 f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3560 |
using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3561 |
obtain p where p: |
68326 | 3562 |
"valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3563 |
(\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and> |
63589 | 3564 |
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3565 |
using pi [OF e] by blast |
68326 | 3566 |
obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3567 |
using winding_number [OF \<gamma> e] by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3568 |
have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3569 |
using p by auto |
68339 | 3570 |
also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))" |
68310 | 3571 |
proof (rule pi_eq) |
3572 |
show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}" |
|
3573 |
by (auto intro!: holomorphic_intros) |
|
68326 | 3574 |
qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>) |
68339 | 3575 |
also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" |
68326 | 3576 |
using q by (auto simp: winding_number_prop_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3577 |
finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" . |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3578 |
then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3579 |
by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3580 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3581 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
3582 |
proposition winding_number_valid_path: |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3583 |
assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
68326 | 3584 |
shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))" |
3585 |
by (rule winding_number_unique) |
|
3586 |
(use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>) |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3587 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
3588 |
proposition has_contour_integral_winding_number: |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3589 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
63589 | 3590 |
shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3591 |
by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3592 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3593 |
lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3594 |
by (simp add: winding_number_valid_path) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3595 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3596 |
lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0" |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3597 |
by (simp add: path_image_subpath winding_number_valid_path) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3598 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3599 |
lemma winding_number_join: |
68326 | 3600 |
assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1" |
3601 |
and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2" |
|
3602 |
and "pathfinish \<gamma>1 = pathstart \<gamma>2" |
|
3603 |
shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z" |
|
3604 |
proof (rule winding_number_unique) |
|
3605 |
show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p |
|
3606 |
(winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e |
|
3607 |
proof - |
|
3608 |
obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)" |
|
3609 |
using \<open>0 < e\<close> \<gamma>1 winding_number by blast |
|
3610 |
moreover |
|
3611 |
obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)" |
|
3612 |
using \<open>0 < e\<close> \<gamma>2 winding_number by blast |
|
3613 |
ultimately |
|
3614 |
have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" |
|
3615 |
using assms |
|
3616 |
apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps) |
|
3617 |
apply (auto simp: joinpaths_def) |
|
3618 |
done |
|
3619 |
then show ?thesis |
|
3620 |
by blast |
|
3621 |
qed |
|
3622 |
qed (use assms in \<open>auto simp: not_in_path_image_join\<close>) |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3623 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3624 |
lemma winding_number_reversepath: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3625 |
assumes "path \<gamma>" "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3626 |
shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)" |
68326 | 3627 |
proof (rule winding_number_unique) |
3628 |
show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e |
|
3629 |
proof - |
|
3630 |
obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)" |
|
3631 |
using \<open>0 < e\<close> assms winding_number by blast |
|
3632 |
then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)" |
|
3633 |
using assms |
|
3634 |
apply (simp add: winding_number_prop_def contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse) |
|
3635 |
apply (auto simp: reversepath_def) |
|
3636 |
done |
|
3637 |
then show ?thesis |
|
3638 |
by blast |
|
3639 |
qed |
|
3640 |
qed (use assms in auto) |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3641 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3642 |
lemma winding_number_shiftpath: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3643 |
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3644 |
and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3645 |
shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z" |
68326 | 3646 |
proof (rule winding_number_unique_loop) |
3647 |
show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> |
|
3648 |
(\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and> |
|
3649 |
contour_integral p (\<lambda>w. 1 / (w - z)) = |
|
3650 |
complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
|
3651 |
if "e > 0" for e |
|
3652 |
proof - |
|
3653 |
obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)" |
|
3654 |
using \<open>0 < e\<close> assms winding_number by blast |
|
3655 |
then show ?thesis |
|
3656 |
apply (rule_tac x="shiftpath a p" in exI) |
|
3657 |
using assms that |
|
3658 |
apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath) |
|
3659 |
apply (simp add: shiftpath_def) |
|
3660 |
done |
|
3661 |
qed |
|
3662 |
qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>) |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3663 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3664 |
lemma winding_number_split_linepath: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3665 |
assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3666 |
shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3667 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3668 |
have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b" |
68310 | 3669 |
using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+ |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3670 |
then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3671 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3672 |
by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3673 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3674 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3675 |
lemma winding_number_cong: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3676 |
"(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z" |
68326 | 3677 |
by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3678 |
|
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3679 |
lemma winding_number_constI: |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3680 |
assumes "c\<noteq>z" "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3681 |
shows "winding_number g z = 0" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3682 |
proof - |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3683 |
have "winding_number g z = winding_number (linepath c c) z" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3684 |
apply (rule winding_number_cong) |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3685 |
using assms unfolding linepath_def by auto |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3686 |
moreover have "winding_number (linepath c c) z =0" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3687 |
apply (rule winding_number_trivial) |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3688 |
using assms by auto |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3689 |
ultimately show ?thesis by auto |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3690 |
qed |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
3691 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3692 |
lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0" |
68339 | 3693 |
unfolding winding_number_def |
3694 |
proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe) |
|
3695 |
fix n e g |
|
3696 |
assume "0 < e" and g: "winding_number_prop p z e g n" |
|
3697 |
then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n" |
|
3698 |
by (rule_tac x="\<lambda>t. g t - z" in exI) |
|
68493 | 3699 |
(force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs |
68339 | 3700 |
vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise) |
3701 |
next |
|
3702 |
fix n e g |
|
3703 |
assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n" |
|
3704 |
then show "\<exists>r. winding_number_prop p z e r n" |
|
3705 |
apply (rule_tac x="\<lambda>t. g t + z" in exI) |
|
68493 | 3706 |
apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs |
68339 | 3707 |
piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise) |
3708 |
apply (force simp: algebra_simps) |
|
3709 |
done |
|
3710 |
qed |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3711 |
|
70136 | 3712 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some lemmas about negating a path\<close> |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3713 |
|
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3714 |
lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3715 |
unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3716 |
|
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3717 |
lemma has_contour_integral_negatepath: |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3718 |
assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3719 |
shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3720 |
proof - |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3721 |
obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3722 |
using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3723 |
have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3724 |
using cint by (auto simp: has_contour_integral_def dest: has_integral_neg) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3725 |
then |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3726 |
have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3727 |
proof (rule rev_iffD1 [OF _ has_integral_spike_eq]) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3728 |
show "negligible S" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3729 |
by (simp add: \<open>finite S\<close> negligible_finite) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3730 |
show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) = |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3731 |
- (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3732 |
if "x \<in> {0..1} - S" for x |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3733 |
proof - |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3734 |
have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)" |
68310 | 3735 |
proof (rule vector_derivative_within_cbox) |
3736 |
show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)" |
|
3737 |
using that unfolding o_def |
|
3738 |
by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works) |
|
3739 |
qed (use that in auto) |
|
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3740 |
then show ?thesis |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3741 |
by simp |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3742 |
qed |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3743 |
qed |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3744 |
then show ?thesis by (simp add: has_contour_integral_def) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3745 |
qed |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3746 |
|
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3747 |
lemma winding_number_negatepath: |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3748 |
assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3749 |
shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3750 |
proof - |
67399 | 3751 |
have "(/) 1 contour_integrable_on \<gamma>" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3752 |
using "0" \<gamma> contour_integrable_inversediff by fastforce |
67399 | 3753 |
then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3754 |
by (rule has_contour_integral_integral) |
67399 | 3755 |
then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3756 |
using has_contour_integral_neg by auto |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3757 |
then show ?thesis |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3758 |
using assms |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3759 |
apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3760 |
apply (simp add: contour_integral_unique has_contour_integral_negatepath) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3761 |
done |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3762 |
qed |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3763 |
|
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3764 |
lemma contour_integrable_negatepath: |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3765 |
assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3766 |
shows "f contour_integrable_on (uminus \<circ> \<gamma>)" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3767 |
by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi) |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
65578
diff
changeset
|
3768 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3769 |
(* A combined theorem deducing several things piecewise.*) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3770 |
lemma winding_number_join_pos_combined: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3771 |
"\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z); |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3772 |
valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3773 |
\<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3774 |
by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3775 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3776 |
|
70136 | 3777 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3778 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3779 |
lemma Re_winding_number: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3780 |
"\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3781 |
\<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3782 |
by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3783 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3784 |
lemma winding_number_pos_le: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3785 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3786 |
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3787 |
shows "0 \<le> Re(winding_number \<gamma> z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3788 |
proof - |
66539
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3789 |
have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3790 |
using ge by (simp add: Complex.Im_divide algebra_simps x) |
68310 | 3791 |
let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)" |
3792 |
let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))" |
|
3793 |
have hi: "(?vd has_integral ?int z) (cbox 0 1)" |
|
66539
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3794 |
unfolding box_real |
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3795 |
apply (subst has_contour_integral [symmetric]) |
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3796 |
using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral) |
68310 | 3797 |
have "0 \<le> Im (?int z)" |
3798 |
proof (rule has_integral_component_nonneg [of \<i>, simplified]) |
|
3799 |
show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)" |
|
3800 |
by (force simp: ge0) |
|
3801 |
show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)" |
|
3802 |
by (rule has_integral_spike_interior [OF hi]) simp |
|
68493 | 3803 |
qed |
66539
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3804 |
then show ?thesis |
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3805 |
by (simp add: Re_winding_number [OF \<gamma>] field_simps) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3806 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3807 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3808 |
lemma winding_number_pos_lt_lemma: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3809 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3810 |
and e: "0 < e" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3811 |
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3812 |
shows "0 < Re(winding_number \<gamma> z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3813 |
proof - |
68310 | 3814 |
let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)" |
3815 |
let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))" |
|
3816 |
have hi: "(?vd has_integral ?int z) (cbox 0 1)" |
|
68493 | 3817 |
unfolding box_real |
66539
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3818 |
apply (subst has_contour_integral [symmetric]) |
0ad3fc48c9ec
final cleanup of negligible_standard_hyperplane and other things
paulson <lp15@cam.ac.uk>
parents:
66507
diff
changeset
|
3819 |
using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3820 |
have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))" |
68310 | 3821 |
proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified]) |
3822 |
show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e) has_integral ?int z) {0..1}" |
|
3823 |
by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp) |
|
3824 |
show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> |
|
3825 |
e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e)" |
|
3826 |
by (simp add: ge) |
|
3827 |
qed (use has_integral_const_real [of _ 0 1] in auto) |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3828 |
with e show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3829 |
by (simp add: Re_winding_number [OF \<gamma>] field_simps) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3830 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3831 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3832 |
lemma winding_number_pos_lt: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3833 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3834 |
and e: "0 < e" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3835 |
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3836 |
shows "0 < Re (winding_number \<gamma> z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3837 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3838 |
have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3839 |
using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3840 |
then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3841 |
using bounded_pos [THEN iffD1, OF bm] by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3842 |
{ fix x::real assume x: "0 < x" "x < 1" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3843 |
then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3844 |
by (simp add: path_image_def power2_eq_square mult_mono') |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3845 |
with x have "\<gamma> x \<noteq> z" using \<gamma> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3846 |
using path_image_def by fastforce |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3847 |
then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3848 |
using B ge [OF x] B2 e |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3849 |
apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3850 |
apply (auto simp: divide_left_mono divide_right_mono) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3851 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3852 |
then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
68339 | 3853 |
by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3854 |
} note * = this |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3855 |
show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3856 |
using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3857 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3858 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3859 |
subsection\<open>The winding number is an integer\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3860 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3861 |
text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1, |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3862 |
Also on page 134 of Serge Lang's book with the name title, etc.\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3863 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3864 |
lemma exp_fg: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3865 |
fixes z::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3866 |
assumes g: "(g has_vector_derivative g') (at x within s)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3867 |
and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3868 |
and z: "g x \<noteq> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3869 |
shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3870 |
proof - |
68339 | 3871 |
have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3872 |
using assms unfolding has_vector_derivative_def scaleR_conv_of_real |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3873 |
by (auto intro!: derivative_eq_intros) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3874 |
show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3875 |
apply (rule has_vector_derivative_eq_rhs) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3876 |
using z |
68339 | 3877 |
apply (auto intro!: derivative_eq_intros * [unfolded o_def] g) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3878 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3879 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3880 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3881 |
lemma winding_number_exp_integral: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3882 |
fixes z::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3883 |
assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3884 |
and ab: "a \<le> b" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3885 |
and z: "z \<notin> \<gamma> ` {a..b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3886 |
shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3887 |
(is "?thesis1") |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3888 |
"exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3889 |
(is "?thesis2") |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3890 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3891 |
let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3892 |
have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3893 |
using z by force |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3894 |
have cong: "continuous_on {a..b} \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3895 |
using \<gamma> by (simp add: piecewise_C1_differentiable_on_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3896 |
obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3897 |
using \<gamma> by (force simp: piecewise_C1_differentiable_on_def) |
68339 | 3898 |
have \<circ>: "open ({a<..<b} - k)" |
61808 | 3899 |
using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3900 |
moreover have "{a<..<b} - k \<subseteq> {a..b} - k" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3901 |
by force |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3902 |
ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x" |
63955 | 3903 |
by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3904 |
{ fix w |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3905 |
assume "w \<noteq> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3906 |
have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3907 |
by (auto simp: dist_norm intro!: continuous_intros) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3908 |
moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3909 |
by (auto simp: intro!: derivative_eq_intros) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3910 |
ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3911 |
using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"] |
68339 | 3912 |
by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3913 |
} |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3914 |
then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3915 |
by meson |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3916 |
have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3917 |
unfolding integrable_on_def [symmetric] |
66708 | 3918 |
proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]]) |
3919 |
show "\<exists>d h. 0 < d \<and> |
|
68493 | 3920 |
(\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))" |
66708 | 3921 |
if "w \<in> - {z}" for w |
3922 |
apply (rule_tac x="norm(w - z)" in exI) |
|
3923 |
using that inverse_eq_divide has_field_derivative_at_within h |
|
3924 |
by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff) |
|
3925 |
qed simp |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3926 |
have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3927 |
unfolding box_real [symmetric] divide_inverse_commute |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3928 |
by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3929 |
with ab show ?thesis1 |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3930 |
by (simp add: divide_inverse_commute integral_def integrable_on_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3931 |
{ fix t |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3932 |
assume t: "t \<in> {a..b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3933 |
have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3934 |
using z by (auto intro!: continuous_intros simp: dist_norm) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
3935 |
have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x" |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
3936 |
unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3937 |
obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3938 |
(h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3939 |
using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3940 |
by simp (auto simp: ball_def dist_norm that) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3941 |
{ fix x D |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3942 |
assume x: "x \<notin> k" "a < x" "x < b" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3943 |
then have "x \<in> interior ({a..b} - k)" |
68339 | 3944 |
using open_subset_interior [OF \<circ>] by fastforce |
66708 | 3945 |
then have con: "isCont ?D\<gamma> x" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3946 |
using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3947 |
then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3948 |
by (rule continuous_at_imp_continuous_within) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3949 |
have gdx: "\<gamma> differentiable at x" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3950 |
using x by (simp add: g_diff_at) |
66708 | 3951 |
have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b; |
3952 |
(\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk> |
|
3953 |
\<Longrightarrow> ((\<lambda>x. integral {a..x} |
|
3954 |
(\<lambda>x. ?D\<gamma> x / |
|
3955 |
(\<gamma> x - z))) has_vector_derivative |
|
3956 |
d / (\<gamma> x - z)) |
|
3957 |
(at x within {a..b})" |
|
3958 |
apply (rule has_vector_derivative_eq_rhs) |
|
3959 |
apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified]) |
|
3960 |
apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+ |
|
3961 |
done |
|
3962 |
then have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0)) |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3963 |
(at x within {a..b})" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3964 |
using x gdx t |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3965 |
apply (clarsimp simp add: differentiable_iff_scaleR) |
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
3966 |
apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3967 |
apply (simp_all add: has_vector_derivative_def [symmetric]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3968 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3969 |
} note * = this |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3970 |
have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3971 |
apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3972 |
using t |
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
3973 |
apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int] simp add: ab)+ |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3974 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3975 |
} |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3976 |
with ab show ?thesis2 |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3977 |
by (simp add: divide_inverse_commute integral_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3978 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3979 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
3980 |
lemma winding_number_exp_2pi: |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3981 |
"\<lbrakk>path p; z \<notin> path_image p\<rbrakk> |
63589 | 3982 |
\<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)" |
68326 | 3983 |
using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3984 |
by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3985 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3986 |
lemma integer_winding_number_eq: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3987 |
assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3988 |
shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3989 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3990 |
obtain p where p: "valid_path p" "z \<notin> path_image p" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3991 |
"pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>" |
68339 | 3992 |
and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
68326 | 3993 |
using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto |
68339 | 3994 |
then have wneq: "winding_number \<gamma> z = winding_number p z" |
3995 |
using eq winding_number_valid_path by force |
|
3996 |
have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)" |
|
3997 |
using eq by (simp add: exp_eq_1 complex_is_Int_iff) |
|
3998 |
have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)" |
|
3999 |
using p winding_number_exp_integral(2) [of p 0 1 z] |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
4000 |
apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps) |
68339 | 4001 |
by (metis path_image_def pathstart_def pathstart_in_path_image) |
4002 |
then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p" |
|
4003 |
using p wneq iff by (auto simp: path_defs) |
|
4004 |
then show ?thesis using p eq |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4005 |
by (auto simp: winding_number_valid_path) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4006 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4007 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4008 |
theorem integer_winding_number: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4009 |
"\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4010 |
by (metis integer_winding_number_eq) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4011 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4012 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4013 |
text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4014 |
We can thus bound the winding number of a path that doesn't intersect a given ray. \<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4015 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4016 |
lemma winding_number_pos_meets: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4017 |
fixes z::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4018 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4019 |
and w: "w \<noteq> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4020 |
shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4021 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4022 |
have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4023 |
using z by (auto simp: path_image_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4024 |
have [simp]: "z \<notin> \<gamma> ` {0..1}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4025 |
using path_image_def z by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4026 |
have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4027 |
using \<gamma> valid_path_def by blast |
63040 | 4028 |
define r where "r = (w - z) / (\<gamma> 0 - z)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4029 |
have [simp]: "r \<noteq> 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4030 |
using w z by (auto simp: r_def) |
68339 | 4031 |
have cont: "continuous_on {0..1} |
4032 |
(\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))" |
|
4033 |
by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp) |
|
68493 | 4034 |
have "Arg2pi r \<le> 2*pi" |
4035 |
by (simp add: Arg2pi less_eq_real_def) |
|
68339 | 4036 |
also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4037 |
using 1 |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63589
diff
changeset
|
4038 |
apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4039 |
apply (simp add: Complex.Re_divide field_simps power2_eq_square) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4040 |
done |
68493 | 4041 |
finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" . |
4042 |
then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r" |
|
4043 |
by (simp add: Arg2pi_ge_0 cont IVT') |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4044 |
then obtain t where t: "t \<in> {0..1}" |
68493 | 4045 |
and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4046 |
by blast |
63040 | 4047 |
define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
68493 | 4048 |
have iArg: "Arg2pi r = Im i" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4049 |
using eqArg by (simp add: i_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4050 |
have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4051 |
by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t) |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
4052 |
have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4053 |
unfolding i_def |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4054 |
apply (rule winding_number_exp_integral [OF gpdt]) |
68339 | 4055 |
using t z unfolding path_image_def by force+ |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4056 |
then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4057 |
by (simp add: exp_minus field_simps) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4058 |
then have "(w - z) = r * (\<gamma> 0 - z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4059 |
by (simp add: r_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4060 |
then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t" |
68339 | 4061 |
apply simp |
68493 | 4062 |
apply (subst Complex_Transcendental.Arg2pi_eq [of r]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4063 |
apply (simp add: iArg) |
68339 | 4064 |
using * apply (simp add: exp_eq_polar field_simps) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4065 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4066 |
with t show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4067 |
by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4068 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4069 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4070 |
lemma winding_number_big_meets: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4071 |
fixes z::complex |
61945 | 4072 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4073 |
and w: "w \<noteq> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4074 |
shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4075 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4076 |
{ assume "Re (winding_number \<gamma> z) \<le> - 1" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4077 |
then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4078 |
by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4079 |
moreover have "valid_path (reversepath \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4080 |
using \<gamma> valid_path_imp_reverse by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4081 |
moreover have "z \<notin> path_image (reversepath \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4082 |
by (simp add: z) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4083 |
ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4084 |
using winding_number_pos_meets w by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4085 |
then have ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4086 |
by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4087 |
} |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4088 |
then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4089 |
using assms |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63589
diff
changeset
|
4090 |
by (simp add: abs_if winding_number_pos_meets split: if_split_asm) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4091 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4092 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4093 |
lemma winding_number_less_1: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4094 |
fixes z::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4095 |
shows |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4096 |
"\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z; |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4097 |
\<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk> |
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
4098 |
\<Longrightarrow> Re(winding_number \<gamma> z) < 1" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4099 |
by (auto simp: not_less dest: winding_number_big_meets) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4100 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4101 |
text\<open>One way of proving that WN=1 for a loop.\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4102 |
lemma winding_number_eq_1: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4103 |
fixes z::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4104 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4105 |
and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4106 |
shows "winding_number \<gamma> z = 1" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4107 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4108 |
have "winding_number \<gamma> z \<in> Ints" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4109 |
by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4110 |
then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4111 |
using 0 2 by (auto simp: Ints_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4112 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4113 |
|
67968 | 4114 |
subsection\<open>Continuity of winding number and invariance on connected sets\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4115 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4116 |
lemma continuous_at_winding_number: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4117 |
fixes z::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4118 |
assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4119 |
shows "continuous (at z) (winding_number \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4120 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4121 |
obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4122 |
using open_contains_cball [of "- path_image \<gamma>"] z |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4123 |
by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4124 |
then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4125 |
by (force simp: cball_def dist_norm) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4126 |
have oc: "open (- cball z (e / 2))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4127 |
by (simp add: closed_def [symmetric]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4128 |
obtain d where "d>0" and pi_eq: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4129 |
"\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2; |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4130 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d); |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4131 |
pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4132 |
\<Longrightarrow> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4133 |
path_image h1 \<subseteq> - cball z (e / 2) \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4134 |
path_image h2 \<subseteq> - cball z (e / 2) \<and> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4135 |
(\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4136 |
using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4137 |
obtain p where p: "valid_path p" "z \<notin> path_image p" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4138 |
"pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4139 |
and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4140 |
and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
68326 | 4141 |
using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4142 |
{ fix w |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4143 |
assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4144 |
then have wnotp: "w \<notin> path_image p" |
61808 | 4145 |
using cbg \<open>d>0\<close> \<open>e>0\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4146 |
apply (simp add: path_image_def cball_def dist_norm, clarify) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4147 |
apply (frule pg) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4148 |
apply (drule_tac c="\<gamma> x" in subsetD) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4149 |
apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4150 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4151 |
have wnotg: "w \<notin> path_image \<gamma>" |
61808 | 4152 |
using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4153 |
{ fix k::real |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4154 |
assume k: "k>0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4155 |
then obtain q where q: "valid_path q" "w \<notin> path_image q" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4156 |
"pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4157 |
and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4158 |
and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w" |
61808 | 4159 |
using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k |
68326 | 4160 |
by (force simp: min_divide_distrib_right winding_number_prop_def) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4161 |
have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))" |
61808 | 4162 |
apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4163 |
apply (frule pg) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4164 |
apply (frule qg) |
61808 | 4165 |
using p q \<open>d>0\<close> e2 |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4166 |
apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4167 |
done |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4168 |
then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4169 |
by (simp add: pi qi) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4170 |
} note pip = this |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4171 |
have "path p" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4172 |
using p by (simp add: valid_path_imp_path) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4173 |
then have "winding_number p w = winding_number \<gamma> w" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4174 |
apply (rule winding_number_unique [OF _ wnotp]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4175 |
apply (rule_tac x=p in exI) |
68326 | 4176 |
apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4177 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4178 |
} note wnwn = this |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4179 |
obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4180 |
using p open_contains_cball [of "- path_image p"] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4181 |
by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4182 |
obtain L |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4183 |
where "L>0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4184 |
and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe); |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4185 |
\<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4186 |
cmod (contour_integral p f) \<le> L * B" |
61808 | 4187 |
using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by force |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4188 |
{ fix e::real and w::complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4189 |
assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4190 |
then have [simp]: "w \<notin> path_image p" |
61808 | 4191 |
using cbp p(2) \<open>0 < pe\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4192 |
by (force simp: dist_norm norm_minus_commute path_image_def cball_def) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4193 |
have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) = |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4194 |
contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4195 |
by (simp add: p contour_integrable_inversediff contour_integral_diff) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4196 |
{ fix x |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4197 |
assume pe: "3/4 * pe < cmod (z - x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4198 |
have "cmod (w - x) < pe/4 + cmod (z - x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4199 |
by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1)) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4200 |
then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4201 |
have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4202 |
using norm_diff_triangle_le by blast |
68339 | 4203 |
also have "\<dots> < pe/4 + cmod (w - x)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4204 |
using w by (simp add: norm_minus_commute) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4205 |
finally have "pe/2 < cmod (w - x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4206 |
using pe by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4207 |
then have "(pe/2)^2 < cmod (w - x) ^ 2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4208 |
apply (rule power_strict_mono) |
61808 | 4209 |
using \<open>pe>0\<close> by auto |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4210 |
then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
4211 |
by (simp add: power_divide) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4212 |
have "8 * L * cmod (w - z) < e * pe\<^sup>2" |
61808 | 4213 |
using w \<open>L>0\<close> by (simp add: field_simps) |
68339 | 4214 |
also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)" |
61808 | 4215 |
using pe2 \<open>e>0\<close> by (simp add: power2_eq_square) |
68339 | 4216 |
also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4217 |
using wx |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4218 |
apply (rule mult_strict_left_mono) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4219 |
using pe2 e not_less_iff_gr_or_eq by fastforce |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4220 |
finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4221 |
by simp |
68339 | 4222 |
also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4223 |
using e by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4224 |
finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" . |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4225 |
have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4226 |
apply (cases "x=z \<or> x=w") |
61808 | 4227 |
using pe \<open>pe>0\<close> w \<open>L>0\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4228 |
apply (force simp: norm_minus_commute) |
61808 | 4229 |
using wx w(2) \<open>L>0\<close> pe pe2 Lwz |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4230 |
apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4231 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4232 |
} note L_cmod_le = this |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4233 |
have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4234 |
apply (rule L) |
61808 | 4235 |
using \<open>pe>0\<close> w |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4236 |
apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros) |
61808 | 4237 |
using \<open>pe>0\<close> w \<open>L>0\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4238 |
apply (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4239 |
done |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4240 |
have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e" |
68339 | 4241 |
apply simp |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4242 |
apply (rule le_less_trans [OF *]) |
61808 | 4243 |
using \<open>L>0\<close> e |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4244 |
apply (force simp: field_simps) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4245 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4246 |
then have "cmod (winding_number p w - winding_number p z) < e" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4247 |
using pi_ge_two e |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4248 |
by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4249 |
} note cmod_wn_diff = this |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4250 |
then have "isCont (winding_number p) z" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4251 |
apply (simp add: continuous_at_eps_delta, clarify) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4252 |
apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI) |
61808 | 4253 |
using \<open>pe>0\<close> \<open>L>0\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4254 |
apply (simp add: dist_norm cmod_wn_diff) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4255 |
done |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4256 |
then show ?thesis |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4257 |
apply (rule continuous_transform_within [where d = "min d e / 2"]) |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4258 |
apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn) |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4259 |
done |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4260 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4261 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4262 |
corollary continuous_on_winding_number: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4263 |
"path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4264 |
by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4265 |
|
70136 | 4266 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4267 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4268 |
lemma winding_number_constant: |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4269 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}" |
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4270 |
shows "winding_number \<gamma> constant_on S" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4271 |
proof - |
65037
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4272 |
have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)" |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4273 |
if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z |
65037
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4274 |
proof - |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4275 |
have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>" |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4276 |
using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto |
65037
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4277 |
with ne show ?thesis |
68403 | 4278 |
by (auto simp: Ints_def simp flip: of_int_diff) |
65037
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4279 |
qed |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4280 |
have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4281 |
using continuous_on_winding_number [OF \<gamma>] sg |
65037
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4282 |
by (meson continuous_on_subset disjoint_eq_subset_Compl) |
2cf841ff23be
some new material, also recasting some theorems using “obtains”
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
4283 |
show ?thesis |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4284 |
using "*" zero_less_one |
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4285 |
by (blast intro: continuous_discrete_range_constant [OF cs cont]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4286 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4287 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4288 |
lemma winding_number_eq: |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4289 |
"\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4290 |
\<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z" |
68493 | 4291 |
using winding_number_constant by (metis constant_on_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4292 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4293 |
lemma open_winding_number_levelsets: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4294 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4295 |
shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4296 |
proof - |
67237 | 4297 |
have opn: "open (- path_image \<gamma>)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4298 |
by (simp add: closed_path_image \<gamma> open_Compl) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4299 |
{ fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4300 |
obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>" |
67237 | 4301 |
using open_contains_ball [of "- path_image \<gamma>"] opn z |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4302 |
by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4303 |
have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4304 |
apply (rule_tac x=e in exI) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4305 |
using e apply (simp add: dist_norm ball_def norm_minus_commute) |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4306 |
apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4307 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4308 |
} then |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4309 |
show ?thesis |
62101 | 4310 |
by (auto simp: open_dist) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4311 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4312 |
|
71031 | 4313 |
subsection\<open>Winding number is zero "outside" a curve\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4314 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66708
diff
changeset
|
4315 |
proposition winding_number_zero_in_outside: |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4316 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4317 |
shows "winding_number \<gamma> z = 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4318 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4319 |
obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4320 |
using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4321 |
obtain w::complex where w: "w \<notin> ball 0 (B + 1)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4322 |
by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4323 |
have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4324 |
apply (rule outside_subset_convex) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4325 |
using B subset_ball by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4326 |
then have wout: "w \<in> outside (path_image \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4327 |
using w by blast |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4328 |
moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4329 |
using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4330 |
by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4331 |
ultimately have "winding_number \<gamma> z = winding_number \<gamma> w" |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
4332 |
by (metis (no_types, hide_lams) constant_on_def z) |
68339 | 4333 |
also have "\<dots> = 0" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4334 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4335 |
have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4336 |
{ fix e::real assume "0<e" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4337 |
obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4338 |
and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4339 |
and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)" |
61808 | 4340 |
using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4341 |
have pip: "path_image p \<subseteq> ball 0 (B + 1)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4342 |
using B |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4343 |
apply (clarsimp simp add: path_image_def dist_norm ball_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4344 |
apply (frule (1) pg1) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4345 |
apply (fastforce dest: norm_add_less) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4346 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4347 |
then have "w \<notin> path_image p" using w by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4348 |
then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4349 |
pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4350 |
(\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4351 |
apply (rule_tac x=p in exI) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4352 |
apply (simp add: p valid_path_polynomial_function) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4353 |
apply (intro conjI) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4354 |
using pge apply (simp add: norm_minus_commute) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4355 |
apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4356 |
apply (rule holomorphic_intros | simp add: dist_norm)+ |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4357 |
using mem_ball_0 w apply blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4358 |
using p apply (simp_all add: valid_path_polynomial_function loop pip) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4359 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4360 |
} |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4361 |
then show ?thesis |
68326 | 4362 |
by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4363 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4364 |
finally show ?thesis . |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4365 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4366 |
|
70136 | 4367 |
corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0" |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66708
diff
changeset
|
4368 |
by (rule winding_number_zero_in_outside) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66708
diff
changeset
|
4369 |
(auto simp: pathfinish_def pathstart_def path_polynomial_function) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66708
diff
changeset
|
4370 |
|
70136 | 4371 |
corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside: |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4372 |
"\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4373 |
by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4374 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4375 |
lemma winding_number_zero_at_infinity: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4376 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4377 |
shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4378 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4379 |
obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4380 |
using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4381 |
then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4382 |
apply (rule_tac x="B+1" in exI, clarify) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4383 |
apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4384 |
apply (meson less_add_one mem_cball_0 not_le order_trans) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4385 |
using ball_subset_cball by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4386 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4387 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4388 |
lemma winding_number_zero_point: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4389 |
"\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4390 |
\<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4391 |
using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4392 |
by (fastforce simp add: compact_path_image) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4393 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4394 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4395 |
text\<open>If a path winds round a set, it winds rounds its inside.\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4396 |
lemma winding_number_around_inside: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4397 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4398 |
and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4399 |
and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4400 |
shows "winding_number \<gamma> w = winding_number \<gamma> z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4401 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4402 |
have ssb: "s \<subseteq> inside(path_image \<gamma>)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4403 |
proof |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4404 |
fix x :: complex |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4405 |
assume "x \<in> s" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4406 |
hence "x \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4407 |
by (meson disjoint_iff_not_equal s_disj) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4408 |
thus "x \<in> inside (path_image \<gamma>)" |
61808 | 4409 |
using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4410 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4411 |
show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4412 |
apply (rule winding_number_eq [OF \<gamma> loop w]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4413 |
using z apply blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4414 |
apply (simp add: cls connected_with_inside cos) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4415 |
apply (simp add: Int_Un_distrib2 s_disj, safe) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4416 |
by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4417 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4418 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4419 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4420 |
text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4421 |
lemma winding_number_subpath_continuous: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4422 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4423 |
shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4424 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4425 |
have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) = |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4426 |
winding_number (subpath 0 x \<gamma>) z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4427 |
if x: "0 \<le> x" "x \<le> 1" for x |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4428 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4429 |
have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) = |
63589 | 4430 |
1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4431 |
using assms x |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4432 |
apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4433 |
done |
68339 | 4434 |
also have "\<dots> = winding_number (subpath 0 x \<gamma>) z" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4435 |
apply (subst winding_number_valid_path) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4436 |
using assms x |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
4437 |
apply (simp_all add: path_image_subpath valid_path_subpath) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
4438 |
by (force simp: path_image_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4439 |
finally show ?thesis . |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4440 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4441 |
show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4442 |
apply (rule continuous_on_eq |
63589 | 4443 |
[where f = "\<lambda>x. 1 / (2*pi*\<i>) * |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4444 |
integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4445 |
apply (rule continuous_intros)+ |
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
4446 |
apply (rule indefinite_integral_continuous_1) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4447 |
apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4448 |
using assms |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4449 |
apply (simp add: *) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4450 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4451 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4452 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4453 |
lemma winding_number_ivt_pos: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4454 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4455 |
shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w" |
68339 | 4456 |
apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4457 |
apply (rule winding_number_subpath_continuous [OF \<gamma> z]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4458 |
using assms |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4459 |
apply (auto simp: path_image_def image_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4460 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4461 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4462 |
lemma winding_number_ivt_neg: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4463 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4464 |
shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w" |
68339 | 4465 |
apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4466 |
apply (rule winding_number_subpath_continuous [OF \<gamma> z]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4467 |
using assms |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4468 |
apply (auto simp: path_image_def image_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4469 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4470 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4471 |
lemma winding_number_ivt_abs: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4472 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4473 |
shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4474 |
using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4475 |
by force |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4476 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4477 |
lemma winding_number_lt_half_lemma: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4478 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4479 |
shows "Re(winding_number \<gamma> z) < 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4480 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4481 |
{ assume "Re(winding_number \<gamma> z) \<ge> 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4482 |
then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4483 |
using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4484 |
have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4485 |
using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4486 |
apply (simp add: t \<gamma> valid_path_imp_path) |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
4487 |
using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4488 |
have "b < a \<bullet> \<gamma> 0" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4489 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4490 |
have "\<gamma> 0 \<in> {c. b < a \<bullet> c}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4491 |
by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4492 |
thus ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4493 |
by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4494 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4495 |
moreover have "b < a \<bullet> \<gamma> t" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4496 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4497 |
have "\<gamma> t \<in> {c. b < a \<bullet> c}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4498 |
by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4499 |
thus ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4500 |
by blast |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4501 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4502 |
ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4503 |
by (simp add: inner_diff_right)+ |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4504 |
then have False |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4505 |
by (simp add: gt inner_mult_right mult_less_0_iff) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4506 |
} |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4507 |
then show ?thesis by force |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4508 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4509 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4510 |
lemma winding_number_lt_half: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4511 |
assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4512 |
shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4513 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4514 |
have "z \<notin> path_image \<gamma>" using assms by auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4515 |
with assms show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4516 |
apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4517 |
apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4518 |
winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4519 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4520 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4521 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4522 |
lemma winding_number_le_half: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4523 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4524 |
and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4525 |
shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4526 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4527 |
{ assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4528 |
have "isCont (winding_number \<gamma>) z" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4529 |
by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z) |
61945 | 4530 |
then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2" |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
4531 |
using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast |
63040 | 4532 |
define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4533 |
have *: "a \<bullet> z' \<le> b - d / 3 * cmod a" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4534 |
unfolding z'_def inner_mult_right' divide_inverse |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
4535 |
apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz) |
61808 | 4536 |
apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4537 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4538 |
have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2" |
61808 | 4539 |
using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4540 |
then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4541 |
by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4542 |
then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4543 |
using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4544 |
then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4545 |
by linarith |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4546 |
moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4547 |
apply (rule winding_number_lt_half [OF \<gamma> *]) |
61808 | 4548 |
using azb \<open>d>0\<close> pag |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
4549 |
apply (auto simp: add_strict_increasing anz field_split_simps algebra_simps dest!: subsetD) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4550 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4551 |
ultimately have False |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4552 |
by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4553 |
} |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4554 |
then show ?thesis by force |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4555 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4556 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4557 |
lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4558 |
using separating_hyperplane_closed_point [of "closed_segment a b" z] |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4559 |
apply auto |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4560 |
apply (simp add: closed_segment_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4561 |
apply (drule less_imp_le) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4562 |
apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4563 |
apply (auto simp: segment) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4564 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4565 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4566 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4567 |
text\<open> Positivity of WN for a linepath.\<close> |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4568 |
lemma winding_number_linepath_pos_lt: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4569 |
assumes "0 < Im ((b - a) * cnj (b - z))" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4570 |
shows "0 < Re(winding_number(linepath a b) z)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4571 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4572 |
have z: "z \<notin> path_image (linepath a b)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4573 |
using assms |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4574 |
by (simp add: closed_segment_def) (force simp: algebra_simps) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4575 |
show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4576 |
apply (rule winding_number_pos_lt [OF valid_path_linepath z assms]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4577 |
apply (simp add: linepath_def algebra_simps) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4578 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4579 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4580 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4581 |
|
67968 | 4582 |
subsection\<open>Cauchy's integral formula, again for a convex enclosing set\<close> |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4583 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
4584 |
lemma Cauchy_integral_formula_weak: |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
4585 |
assumes s: "convex s" and "finite k" and conf: "continuous_on s f" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
4586 |
and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
4587 |
and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4588 |
and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
63589 | 4589 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4590 |
proof - |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4591 |
obtain f' where f': "(f has_field_derivative f') (at z)" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
4592 |
using fcd [OF z] by (auto simp: field_differentiable_def) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4593 |
have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+ |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4594 |
have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4595 |
proof (cases "x = z") |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4596 |
case True then show ?thesis |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4597 |
apply (simp add: continuous_within) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4598 |
apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"]) |
68239 | 4599 |
using has_field_derivative_at_within has_field_derivative_iff f' |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4600 |
apply (fastforce simp add:)+ |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4601 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4602 |
next |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4603 |
case False |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4604 |
then have dxz: "dist x z > 0" by auto |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4605 |
have cf: "continuous (at x within s) f" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4606 |
using conf continuous_on_eq_continuous_within that by blast |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4607 |
have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))" |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4608 |
by (rule cf continuous_intros | simp add: False)+ |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4609 |
then show ?thesis |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4610 |
apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"]) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4611 |
apply (force simp: dist_commute) |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
4612 |
done |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4613 |
qed |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4614 |
have fink': "finite (insert z k)" using \<open>finite k\<close> by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4615 |
have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4616 |
apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4617 |
using c apply (force simp: continuous_on_eq_continuous_within) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4618 |
apply (rename_tac w) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
4619 |
apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4620 |
apply (simp_all add: dist_pos_lt dist_commute) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4621 |
apply (metis less_irrefl) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4622 |
apply (rule derivative_intros fcd | simp)+ |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4623 |
done |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4624 |
show ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4625 |
apply (rule has_contour_integral_eq) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4626 |
using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *] |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
4627 |
apply (auto simp: ac_simps divide_simps) |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4628 |
done |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
4629 |
qed |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
4630 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
4631 |
theorem Cauchy_integral_formula_convex_simple: |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4632 |
"\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z}; |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4633 |
pathfinish \<gamma> = pathstart \<gamma>\<rbrakk> |
63589 | 4634 |
\<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4635 |
apply (rule Cauchy_integral_formula_weak [where k = "{}"]) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4636 |
using holomorphic_on_imp_continuous_on |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4637 |
by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
4638 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4639 |
subsection\<open>Homotopy forms of Cauchy's theorem\<close> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4640 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
4641 |
lemma Cauchy_theorem_homotopic: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4642 |
assumes hom: "if atends then homotopic_paths s g h else homotopic_loops s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4643 |
and "open s" and f: "f holomorphic_on s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4644 |
and vpg: "valid_path g" and vph: "valid_path h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4645 |
shows "contour_integral g f = contour_integral h f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4646 |
proof - |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4647 |
have pathsf: "linked_paths atends g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4648 |
using hom by (auto simp: linked_paths_def homotopic_paths_imp_pathstart homotopic_paths_imp_pathfinish homotopic_loops_imp_loop) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4649 |
obtain k :: "real \<times> real \<Rightarrow> complex" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4650 |
where contk: "continuous_on ({0..1} \<times> {0..1}) k" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4651 |
and ks: "k ` ({0..1} \<times> {0..1}) \<subseteq> s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4652 |
and k [simp]: "\<forall>x. k (0, x) = g x" "\<forall>x. k (1, x) = h x" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4653 |
and ksf: "\<forall>t\<in>{0..1}. linked_paths atends g (\<lambda>x. k (t, x))" |
62390 | 4654 |
using hom pathsf by (auto simp: linked_paths_def homotopic_paths_def homotopic_loops_def homotopic_with_def split: if_split_asm) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4655 |
have ucontk: "uniformly_continuous_on ({0..1} \<times> {0..1}) k" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4656 |
by (blast intro: compact_Times compact_uniformly_continuous [OF contk]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4657 |
{ fix t::real assume t: "t \<in> {0..1}" |
68339 | 4658 |
have pak: "path (k \<circ> (\<lambda>u. (t, u)))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4659 |
unfolding path_def |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4660 |
apply (rule continuous_intros continuous_on_subset [OF contk])+ |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4661 |
using t by force |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4662 |
have pik: "path_image (k \<circ> Pair t) \<subseteq> s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4663 |
using ks t by (auto simp: path_image_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4664 |
obtain e where "e>0" and e: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4665 |
"\<And>g h. \<lbrakk>valid_path g; valid_path h; |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4666 |
\<forall>u\<in>{0..1}. cmod (g u - (k \<circ> Pair t) u) < e \<and> cmod (h u - (k \<circ> Pair t) u) < e; |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4667 |
linked_paths atends g h\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4668 |
\<Longrightarrow> contour_integral h f = contour_integral g f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4669 |
using contour_integral_nearby [OF \<open>open s\<close> pak pik, of atends] f by metis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4670 |
obtain d where "d>0" and d: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4671 |
"\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> {0..1}; x' \<in> {0..1} \<times> {0..1}; norm (x'-x) < d\<rbrakk> \<Longrightarrow> norm (k x' - k x) < e/4" |
61808 | 4672 |
by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm \<open>e>0\<close>) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4673 |
{ fix t1 t2 |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4674 |
assume t1: "0 \<le> t1" "t1 \<le> 1" and t2: "0 \<le> t2" "t2 \<le> 1" and ltd: "\<bar>t1 - t\<bar> < d" "\<bar>t2 - t\<bar> < d" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4675 |
have no2: "\<And>g1 k1 kt. \<lbrakk>norm(g1 - k1) < e/4; norm(k1 - kt) < e/4\<rbrakk> \<Longrightarrow> norm(g1 - kt) < e" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4676 |
using \<open>e > 0\<close> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4677 |
apply (rule_tac y = k1 in norm_triangle_half_l) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4678 |
apply (auto simp: norm_minus_commute intro: order_less_trans) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4679 |
done |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4680 |
have "\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4681 |
(\<forall>u\<in>{0..1}. cmod (g1 u - k (t1, u)) < d \<and> cmod (g2 u - k (t2, u)) < d) \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4682 |
linked_paths atends g1 g2 \<longrightarrow> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4683 |
contour_integral g2 f = contour_integral g1 f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4684 |
apply (rule_tac x="e/4" in exI) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4685 |
using t t1 t2 ltd \<open>e > 0\<close> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4686 |
apply (auto intro!: e simp: d no2 simp del: less_divide_eq_numeral1) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4687 |
done |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4688 |
} |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4689 |
then have "\<exists>e. 0 < e \<and> |
61945 | 4690 |
(\<forall>t1 t2. t1 \<in> {0..1} \<and> t2 \<in> {0..1} \<and> \<bar>t1 - t\<bar> < e \<and> \<bar>t2 - t\<bar> < e |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4691 |
\<longrightarrow> (\<exists>d. 0 < d \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4692 |
(\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4693 |
(\<forall>u \<in> {0..1}. |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4694 |
norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4695 |
linked_paths atends g1 g2 |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4696 |
\<longrightarrow> contour_integral g2 f = contour_integral g1 f)))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4697 |
by (rule_tac x=d in exI) (simp add: \<open>d > 0\<close>) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4698 |
} |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4699 |
then obtain ee where ee: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4700 |
"\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0 \<and> |
61945 | 4701 |
(\<forall>t1 t2. t1 \<in> {0..1} \<longrightarrow> t2 \<in> {0..1} \<longrightarrow> \<bar>t1 - t\<bar> < ee t \<longrightarrow> \<bar>t2 - t\<bar> < ee t |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4702 |
\<longrightarrow> (\<exists>d. 0 < d \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4703 |
(\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4704 |
(\<forall>u \<in> {0..1}. |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4705 |
norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4706 |
linked_paths atends g1 g2 |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4707 |
\<longrightarrow> contour_integral g2 f = contour_integral g1 f)))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4708 |
by metis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4709 |
note ee_rule = ee [THEN conjunct2, rule_format] |
63040 | 4710 |
define C where "C = (\<lambda>t. ball t (ee t / 3)) ` {0..1}" |
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
4711 |
obtain C' where C': "C' \<subseteq> C" "finite C'" and C'01: "{0..1} \<subseteq> \<Union>C'" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
4712 |
proof (rule compactE [OF compact_interval]) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
4713 |
show "{0..1} \<subseteq> \<Union>C" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
4714 |
using ee [THEN conjunct1] by (auto simp: C_def dist_norm) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
4715 |
qed (use C_def in auto) |
63040 | 4716 |
define kk where "kk = {t \<in> {0..1}. ball t (ee t / 3) \<in> C'}" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4717 |
have kk01: "kk \<subseteq> {0..1}" by (auto simp: kk_def) |
63040 | 4718 |
define e where "e = Min (ee ` kk)" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4719 |
have C'_eq: "C' = (\<lambda>t. ball t (ee t / 3)) ` kk" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4720 |
using C' by (auto simp: kk_def C_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4721 |
have ee_pos[simp]: "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4722 |
by (simp add: kk_def ee) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4723 |
moreover have "finite kk" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4724 |
using \<open>finite C'\<close> kk01 by (force simp: C'_eq inj_on_def ball_eq_ball_iff dest: ee_pos finite_imageD) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4725 |
moreover have "kk \<noteq> {}" using \<open>{0..1} \<subseteq> \<Union>C'\<close> C'_eq by force |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4726 |
ultimately have "e > 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4727 |
using finite_less_Inf_iff [of "ee ` kk" 0] kk01 by (force simp: e_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4728 |
then obtain N::nat where "N > 0" and N: "1/N < e/3" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4729 |
by (meson divide_pos_pos nat_approx_posE zero_less_Suc zero_less_numeral) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4730 |
have e_le_ee: "\<And>i. i \<in> kk \<Longrightarrow> e \<le> ee i" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4731 |
using \<open>finite kk\<close> by (simp add: e_def Min_le_iff [of "ee ` kk"]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4732 |
have plus: "\<exists>t \<in> kk. x \<in> ball t (ee t / 3)" if "x \<in> {0..1}" for x |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4733 |
using C' subsetD [OF C'01 that] unfolding C'_eq by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4734 |
have [OF order_refl]: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4735 |
"\<exists>d. 0 < d \<and> (\<forall>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (n/N, u)) < d) \<and> linked_paths atends g j |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4736 |
\<longrightarrow> contour_integral j f = contour_integral g f)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4737 |
if "n \<le> N" for n |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4738 |
using that |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4739 |
proof (induct n) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4740 |
case 0 show ?case using ee_rule [of 0 0 0] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4741 |
apply clarsimp |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4742 |
apply (rule_tac x=d in exI, safe) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4743 |
by (metis diff_self vpg norm_zero) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4744 |
next |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4745 |
case (Suc n) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4746 |
then have N01: "n/N \<in> {0..1}" "(Suc n)/N \<in> {0..1}" by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4747 |
then obtain t where t: "t \<in> kk" "n/N \<in> ball t (ee t / 3)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4748 |
using plus [of "n/N"] by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4749 |
then have nN_less: "\<bar>n/N - t\<bar> < ee t" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4750 |
by (simp add: dist_norm del: less_divide_eq_numeral1) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4751 |
have n'N_less: "\<bar>real (Suc n) / real N - t\<bar> < ee t" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4752 |
using t N \<open>N > 0\<close> e_le_ee [of t] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4753 |
by (simp add: dist_norm add_divide_distrib abs_diff_less_iff del: less_divide_eq_numeral1) (simp add: field_simps) |
61808 | 4754 |
have t01: "t \<in> {0..1}" using \<open>kk \<subseteq> {0..1}\<close> \<open>t \<in> kk\<close> by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4755 |
obtain d1 where "d1 > 0" and d1: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4756 |
"\<And>g1 g2. \<lbrakk>valid_path g1; valid_path g2; |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4757 |
\<forall>u\<in>{0..1}. cmod (g1 u - k (n/N, u)) < d1 \<and> cmod (g2 u - k ((Suc n) / N, u)) < d1; |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4758 |
linked_paths atends g1 g2\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4759 |
\<Longrightarrow> contour_integral g2 f = contour_integral g1 f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4760 |
using ee [THEN conjunct2, rule_format, OF t01 N01 nN_less n'N_less] by fastforce |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4761 |
have "n \<le> N" using Suc.prems by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4762 |
with Suc.hyps |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4763 |
obtain d2 where "d2 > 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4764 |
and d2: "\<And>j. \<lbrakk>valid_path j; \<forall>u\<in>{0..1}. cmod (j u - k (n/N, u)) < d2; linked_paths atends g j\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4765 |
\<Longrightarrow> contour_integral j f = contour_integral g f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4766 |
by auto |
68339 | 4767 |
have "continuous_on {0..1} (k \<circ> (\<lambda>u. (n/N, u)))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4768 |
apply (rule continuous_intros continuous_on_subset [OF contk])+ |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4769 |
using N01 by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4770 |
then have pkn: "path (\<lambda>u. k (n/N, u))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4771 |
by (simp add: path_def) |
61808 | 4772 |
have min12: "min d1 d2 > 0" by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4773 |
obtain p where "polynomial_function p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4774 |
and psf: "pathstart p = pathstart (\<lambda>u. k (n/N, u))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4775 |
"pathfinish p = pathfinish (\<lambda>u. k (n/N, u))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4776 |
and pk_le: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (p t - k (n/N, t)) < min d1 d2" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4777 |
using path_approx_polynomial_function [OF pkn min12] by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4778 |
then have vpp: "valid_path p" using valid_path_polynomial_function by blast |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4779 |
have lpa: "linked_paths atends g p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4780 |
by (metis (mono_tags, lifting) N01(1) ksf linked_paths_def pathfinish_def pathstart_def psf) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4781 |
show ?case |
68359 | 4782 |
proof (intro exI; safe) |
4783 |
fix j |
|
4784 |
assume "valid_path j" "linked_paths atends g j" |
|
4785 |
and "\<forall>u\<in>{0..1}. cmod (j u - k (real (Suc n) / real N, u)) < min d1 d2" |
|
4786 |
then have "contour_integral j f = contour_integral p f" |
|
4787 |
using pk_le N01(1) ksf by (force intro!: vpp d1 simp add: linked_paths_def psf) |
|
4788 |
also have "... = contour_integral g f" |
|
4789 |
using pk_le by (force intro!: vpp d2 lpa) |
|
4790 |
finally show "contour_integral j f = contour_integral g f" . |
|
4791 |
qed (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>) |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4792 |
qed |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4793 |
then obtain d where "0 < d" |
68359 | 4794 |
"\<And>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (1,u)) < d) \<and> linked_paths atends g j |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4795 |
\<Longrightarrow> contour_integral j f = contour_integral g f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4796 |
using \<open>N>0\<close> by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4797 |
then have "linked_paths atends g h \<Longrightarrow> contour_integral h f = contour_integral g f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4798 |
using \<open>N>0\<close> vph by fastforce |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4799 |
then show ?thesis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4800 |
by (simp add: pathsf) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4801 |
qed |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4802 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4803 |
proposition Cauchy_theorem_homotopic_paths: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4804 |
assumes hom: "homotopic_paths s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4805 |
and "open s" and f: "f holomorphic_on s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4806 |
and vpg: "valid_path g" and vph: "valid_path h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4807 |
shows "contour_integral g f = contour_integral h f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4808 |
using Cauchy_theorem_homotopic [of True s g h] assms by simp |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4809 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4810 |
proposition Cauchy_theorem_homotopic_loops: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4811 |
assumes hom: "homotopic_loops s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4812 |
and "open s" and f: "f holomorphic_on s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4813 |
and vpg: "valid_path g" and vph: "valid_path h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4814 |
shows "contour_integral g f = contour_integral h f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4815 |
using Cauchy_theorem_homotopic [of False s g h] assms by simp |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4816 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4817 |
lemma has_contour_integral_newpath: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4818 |
"\<lbrakk>(f has_contour_integral y) h; f contour_integrable_on g; contour_integral g f = contour_integral h f\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4819 |
\<Longrightarrow> (f has_contour_integral y) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4820 |
using has_contour_integral_integral contour_integral_unique by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4821 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4822 |
lemma Cauchy_theorem_null_homotopic: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4823 |
"\<lbrakk>f holomorphic_on s; open s; valid_path g; homotopic_loops s g (linepath a a)\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4824 |
apply (rule has_contour_integral_newpath [where h = "linepath a a"], simp) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4825 |
using contour_integrable_holomorphic_simple |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4826 |
apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4827 |
by (simp add: Cauchy_theorem_homotopic_loops) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4828 |
|
70136 | 4829 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close> |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4830 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4831 |
text\<open>including the fact that it's +-1 inside a simple closed curve.\<close> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4832 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4833 |
lemma winding_number_homotopic_paths: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4834 |
assumes "homotopic_paths (-{z}) g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4835 |
shows "winding_number g z = winding_number h z" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4836 |
proof - |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4837 |
have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4838 |
moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4839 |
using homotopic_paths_imp_subset [OF assms] by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4840 |
ultimately obtain d e where "d > 0" "e > 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4841 |
and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4842 |
\<Longrightarrow> homotopic_paths (-{z}) g p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4843 |
and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4844 |
\<Longrightarrow> homotopic_paths (-{z}) h q" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4845 |
using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4846 |
obtain p where p: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4847 |
"valid_path p" "z \<notin> path_image p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4848 |
"pathstart p = pathstart g" "pathfinish p = pathfinish g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4849 |
and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4850 |
and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z" |
68326 | 4851 |
using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4852 |
obtain q where q: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4853 |
"valid_path q" "z \<notin> path_image q" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4854 |
"pathstart q = pathstart h" "pathfinish q = pathfinish h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4855 |
and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4856 |
and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z" |
68326 | 4857 |
using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast |
68359 | 4858 |
have "homotopic_paths (- {z}) g p" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4859 |
by (simp add: d p valid_path_imp_path norm_minus_commute gp_less) |
68359 | 4860 |
moreover have "homotopic_paths (- {z}) h q" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4861 |
by (simp add: e q valid_path_imp_path norm_minus_commute hq_less) |
68359 | 4862 |
ultimately have "homotopic_paths (- {z}) p q" |
4863 |
by (blast intro: homotopic_paths_trans homotopic_paths_sym assms) |
|
4864 |
then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))" |
|
4865 |
by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q) |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4866 |
then show ?thesis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4867 |
by (simp add: pap paq) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4868 |
qed |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4869 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4870 |
lemma winding_number_homotopic_loops: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4871 |
assumes "homotopic_loops (-{z}) g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4872 |
shows "winding_number g z = winding_number h z" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4873 |
proof - |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4874 |
have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4875 |
moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4876 |
using homotopic_loops_imp_subset [OF assms] by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4877 |
moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4878 |
using homotopic_loops_imp_loop [OF assms] by auto |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4879 |
ultimately obtain d e where "d > 0" "e > 0" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4880 |
and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4881 |
\<Longrightarrow> homotopic_loops (-{z}) g p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4882 |
and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4883 |
\<Longrightarrow> homotopic_loops (-{z}) h q" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4884 |
using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4885 |
obtain p where p: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4886 |
"valid_path p" "z \<notin> path_image p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4887 |
"pathstart p = pathstart g" "pathfinish p = pathfinish g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4888 |
and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4889 |
and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z" |
68326 | 4890 |
using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4891 |
obtain q where q: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4892 |
"valid_path q" "z \<notin> path_image q" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4893 |
"pathstart q = pathstart h" "pathfinish q = pathfinish h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4894 |
and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4895 |
and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z" |
68326 | 4896 |
using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4897 |
have gp: "homotopic_loops (- {z}) g p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4898 |
by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4899 |
have hq: "homotopic_loops (- {z}) h q" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4900 |
by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4901 |
have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))" |
68310 | 4902 |
proof (rule Cauchy_theorem_homotopic_loops) |
4903 |
show "homotopic_loops (- {z}) p q" |
|
4904 |
by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms) |
|
4905 |
qed (auto intro!: holomorphic_intros simp: p q) |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4906 |
then show ?thesis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4907 |
by (simp add: pap paq) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4908 |
qed |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4909 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4910 |
lemma winding_number_paths_linear_eq: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4911 |
"\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g; |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4912 |
\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4913 |
\<Longrightarrow> winding_number h z = winding_number g z" |
68339 | 4914 |
by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4915 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4916 |
lemma winding_number_loops_linear_eq: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4917 |
"\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h; |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4918 |
\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4919 |
\<Longrightarrow> winding_number h z = winding_number g z" |
68339 | 4920 |
by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4921 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4922 |
lemma winding_number_nearby_paths_eq: |
68359 | 4923 |
"\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g; |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4924 |
\<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4925 |
\<Longrightarrow> winding_number h z = winding_number g z" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4926 |
by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4927 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4928 |
lemma winding_number_nearby_loops_eq: |
68359 | 4929 |
"\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h; |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4930 |
\<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4931 |
\<Longrightarrow> winding_number h z = winding_number g z" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4932 |
by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
4933 |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4934 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
4935 |
lemma winding_number_subpath_combine: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4936 |
"\<lbrakk>path g; z \<notin> path_image g; |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4937 |
u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4938 |
\<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z = |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4939 |
winding_number (subpath u w g) z" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4940 |
apply (rule trans [OF winding_number_join [THEN sym] |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4941 |
winding_number_homotopic_paths [OF homotopic_join_subpaths]]) |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
4942 |
using path_image_subpath_subset by auto |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4943 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4944 |
subsection\<open>Partial circle path\<close> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4945 |
|
70136 | 4946 |
definition\<^marker>\<open>tag important\<close> part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex" |
63589 | 4947 |
where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4948 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4949 |
lemma pathstart_part_circlepath [simp]: |
63589 | 4950 |
"pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4951 |
by (metis part_circlepath_def pathstart_def pathstart_linepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4952 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4953 |
lemma pathfinish_part_circlepath [simp]: |
63589 | 4954 |
"pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4955 |
by (metis part_circlepath_def pathfinish_def pathfinish_linepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4956 |
|
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
4957 |
lemma reversepath_part_circlepath[simp]: |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
4958 |
"reversepath (part_circlepath z r s t) = part_circlepath z r t s" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
4959 |
unfolding part_circlepath_def reversepath_def linepath_def |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
4960 |
by (auto simp:algebra_simps) |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68527
diff
changeset
|
4961 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
4962 |
lemma has_vector_derivative_part_circlepath [derivative_intros]: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4963 |
"((part_circlepath z r s t) has_vector_derivative |
63589 | 4964 |
(\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x))) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4965 |
(at x within X)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4966 |
apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real) |
70707
125705f5965f
A little-known material, and some tidying up
paulson <lp15@cam.ac.uk>
parents:
70642
diff
changeset
|
4967 |
apply (rule has_vector_derivative_real_field) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4968 |
apply (rule derivative_eq_intros | simp)+ |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4969 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4970 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
4971 |
lemma differentiable_part_circlepath: |
68721 | 4972 |
"part_circlepath c r a b differentiable at x within A" |
4973 |
using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast |
|
4974 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
4975 |
lemma vector_derivative_part_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4976 |
"vector_derivative (part_circlepath z r s t) (at x) = |
63589 | 4977 |
\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4978 |
using has_vector_derivative_part_circlepath vector_derivative_at by blast |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4979 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
4980 |
lemma vector_derivative_part_circlepath01: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4981 |
"\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4982 |
\<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) = |
63589 | 4983 |
\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4984 |
using has_vector_derivative_part_circlepath |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4985 |
by (auto simp: vector_derivative_at_within_ivl) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4986 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4987 |
lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4988 |
apply (simp add: valid_path_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4989 |
apply (rule C1_differentiable_imp_piecewise) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4990 |
apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4991 |
intro!: continuous_intros) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4992 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4993 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4994 |
lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4995 |
by (simp add: valid_path_imp_path) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4996 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4997 |
proposition path_image_part_circlepath: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
4998 |
assumes "s \<le> t" |
63589 | 4999 |
shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5000 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5001 |
{ fix z::real |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5002 |
assume "0 \<le> z" "z \<le> 1" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5003 |
with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5004 |
apply (rule_tac x="(1 - z) * s + z * t" in exI) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5005 |
apply (simp add: linepath_def scaleR_conv_of_real algebra_simps) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5006 |
apply (rule conjI) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5007 |
using mult_right_mono apply blast |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5008 |
using affine_ineq by (metis "mult.commute") |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5009 |
} |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5010 |
moreover |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5011 |
{ fix z |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5012 |
assume "s \<le> z" "z \<le> t" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5013 |
then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5014 |
apply (rule_tac x="(z - s)/(t - s)" in image_eqI) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5015 |
apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5016 |
apply (auto simp: field_split_simps) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5017 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5018 |
} |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5019 |
ultimately show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5020 |
by (fastforce simp add: path_image_def part_circlepath_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5021 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5022 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5023 |
lemma path_image_part_circlepath': |
68721 | 5024 |
"path_image (part_circlepath z r s t) = (\<lambda>x. z + r * cis x) ` closed_segment s t" |
5025 |
proof - |
|
5026 |
have "path_image (part_circlepath z r s t) = |
|
5027 |
(\<lambda>x. z + r * exp(\<i> * of_real x)) ` linepath s t ` {0..1}" |
|
5028 |
by (simp add: image_image path_image_def part_circlepath_def) |
|
5029 |
also have "linepath s t ` {0..1} = closed_segment s t" |
|
5030 |
by (rule linepath_image_01) |
|
5031 |
finally show ?thesis by (simp add: cis_conv_exp) |
|
5032 |
qed |
|
5033 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5034 |
lemma path_image_part_circlepath_subset: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5035 |
"\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5036 |
by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5037 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5038 |
lemma in_path_image_part_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5039 |
assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5040 |
shows "norm(w - z) = r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5041 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5042 |
have "w \<in> {c. dist z c = r}" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5043 |
by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5044 |
thus ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5045 |
by (simp add: dist_norm norm_minus_commute) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5046 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5047 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5048 |
lemma path_image_part_circlepath_subset': |
68721 | 5049 |
assumes "r \<ge> 0" |
5050 |
shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r" |
|
5051 |
proof (cases "s \<le> t") |
|
5052 |
case True |
|
5053 |
thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp |
|
5054 |
next |
|
5055 |
case False |
|
5056 |
thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms |
|
5057 |
by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all |
|
5058 |
qed |
|
5059 |
||
5060 |
lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x" |
|
5061 |
by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps) |
|
5062 |
||
5063 |
lemma contour_integral_bound_part_circlepath: |
|
5064 |
assumes "f contour_integrable_on part_circlepath c r a b" |
|
5065 |
assumes "B \<ge> 0" "r \<ge> 0" "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> norm (f x) \<le> B" |
|
5066 |
shows "norm (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>" |
|
5067 |
proof - |
|
5068 |
let ?I = "integral {0..1} (\<lambda>x. f (part_circlepath c r a b x) * \<i> * of_real (r * (b - a)) * |
|
5069 |
exp (\<i> * linepath a b x))" |
|
5070 |
have "norm ?I \<le> integral {0..1} (\<lambda>x::real. B * 1 * (r * \<bar>b - a\<bar>) * 1)" |
|
5071 |
proof (rule integral_norm_bound_integral, goal_cases) |
|
5072 |
case 1 |
|
5073 |
with assms(1) show ?case |
|
5074 |
by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac) |
|
5075 |
next |
|
5076 |
case (3 x) |
|
5077 |
with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult |
|
5078 |
by (intro mult_mono) (auto simp: path_image_def) |
|
5079 |
qed auto |
|
5080 |
also have "?I = contour_integral (part_circlepath c r a b) f" |
|
5081 |
by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac) |
|
5082 |
finally show ?thesis by simp |
|
5083 |
qed |
|
5084 |
||
5085 |
lemma has_contour_integral_part_circlepath_iff: |
|
5086 |
assumes "a < b" |
|
5087 |
shows "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow> |
|
5088 |
((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" |
|
5089 |
proof - |
|
5090 |
have "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow> |
|
5091 |
((\<lambda>x. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b) |
|
5092 |
(at x within {0..1})) has_integral I) {0..1}" |
|
5093 |
unfolding has_contour_integral_def .. |
|
5094 |
also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (part_circlepath c r a b x) * r * (b - a) * \<i> * |
|
5095 |
cis (linepath a b x)) has_integral I) {0..1}" |
|
5096 |
by (intro has_integral_cong, subst vector_derivative_part_circlepath01) |
|
5097 |
(simp_all add: cis_conv_exp) |
|
5098 |
also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (c + r * exp (\<i> * linepath (of_real a) (of_real b) x)) * |
|
5099 |
r * \<i> * exp (\<i> * linepath (of_real a) (of_real b) x) * |
|
5100 |
vector_derivative (linepath (of_real a) (of_real b)) |
|
5101 |
(at x within {0..1})) has_integral I) {0..1}" |
|
5102 |
by (intro has_integral_cong, subst vector_derivative_linepath_within) |
|
5103 |
(auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric]) |
|
5104 |
also have "\<dots> \<longleftrightarrow> ((\<lambda>z. f (c + r * exp (\<i> * z)) * r * \<i> * exp (\<i> * z)) has_contour_integral I) |
|
5105 |
(linepath (of_real a) (of_real b))" |
|
5106 |
by (simp add: has_contour_integral_def) |
|
5107 |
also have "\<dots> \<longleftrightarrow> ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" using assms |
|
5108 |
by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp) |
|
5109 |
finally show ?thesis . |
|
5110 |
qed |
|
5111 |
||
5112 |
lemma contour_integrable_part_circlepath_iff: |
|
5113 |
assumes "a < b" |
|
5114 |
shows "f contour_integrable_on (part_circlepath c r a b) \<longleftrightarrow> |
|
5115 |
(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}" |
|
5116 |
using assms by (auto simp: contour_integrable_on_def integrable_on_def |
|
5117 |
has_contour_integral_part_circlepath_iff) |
|
5118 |
||
5119 |
lemma contour_integral_part_circlepath_eq: |
|
5120 |
assumes "a < b" |
|
5121 |
shows "contour_integral (part_circlepath c r a b) f = |
|
5122 |
integral {a..b} (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t)" |
|
5123 |
proof (cases "f contour_integrable_on part_circlepath c r a b") |
|
5124 |
case True |
|
5125 |
hence "(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}" |
|
5126 |
using assms by (simp add: contour_integrable_part_circlepath_iff) |
|
5127 |
with True show ?thesis |
|
5128 |
using has_contour_integral_part_circlepath_iff[OF assms] |
|
5129 |
contour_integral_unique has_integral_integrable_integral by blast |
|
5130 |
next |
|
5131 |
case False |
|
5132 |
hence "\<not>(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}" |
|
5133 |
using assms by (simp add: contour_integrable_part_circlepath_iff) |
|
5134 |
with False show ?thesis |
|
5135 |
by (simp add: not_integrable_contour_integral not_integrable_integral) |
|
5136 |
qed |
|
5137 |
||
5138 |
lemma contour_integral_part_circlepath_reverse: |
|
5139 |
"contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f" |
|
5140 |
by (subst reversepath_part_circlepath [symmetric], subst contour_integral_reversepath) simp_all |
|
5141 |
||
5142 |
lemma contour_integral_part_circlepath_reverse': |
|
5143 |
"b < a \<Longrightarrow> contour_integral (part_circlepath c r a b) f = |
|
5144 |
-contour_integral (part_circlepath c r b a) f" |
|
5145 |
by (rule contour_integral_part_circlepath_reverse) |
|
5146 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5147 |
lemma finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5148 |
proof (cases "w = 0") |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5149 |
case True then show ?thesis by auto |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5150 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5151 |
case False |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5152 |
have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5153 |
apply (simp add: norm_mult finite_int_iff_bounded_le) |
61942 | 5154 |
apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5155 |
apply (auto simp: field_split_simps le_floor_iff) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5156 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5157 |
have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5158 |
by blast |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5159 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5160 |
apply (subst exp_Ln [OF False, symmetric]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5161 |
apply (simp add: exp_eq) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5162 |
using norm_add_leD apply (fastforce intro: finite_subset [OF _ *]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5163 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5164 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5165 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5166 |
lemma finite_bounded_log2: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5167 |
fixes a::complex |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5168 |
assumes "a \<noteq> 0" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5169 |
shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5170 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5171 |
have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5172 |
by (rule finite_imageI [OF finite_bounded_log]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5173 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5174 |
by (rule finite_subset [OF _ *]) (force simp: assms norm_mult) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5175 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5176 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5177 |
lemma has_contour_integral_bound_part_circlepath_strong: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5178 |
assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5179 |
and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5180 |
and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5181 |
shows "cmod i \<le> B * r * (t - s)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5182 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5183 |
consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5184 |
then show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5185 |
proof cases |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5186 |
case 1 with fi [unfolded has_contour_integral] |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5187 |
have "i = 0" by (simp add: vector_derivative_part_circlepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5188 |
with assms show ?thesis by simp |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5189 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5190 |
case 2 |
61945 | 5191 |
have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5192 |
have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5193 |
by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5194 |
have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5195 |
proof - |
63589 | 5196 |
define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5197 |
have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5198 |
apply (rule finite_vimageI [OF finite_bounded_log2]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5199 |
using \<open>s < t\<close> apply (auto simp: inj_of_real) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5200 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5201 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5202 |
apply (simp add: part_circlepath_def linepath_def vimage_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5203 |
apply (rule finite_subset [OF _ fin]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5204 |
using le |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5205 |
apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5206 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5207 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5208 |
then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5209 |
by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5210 |
have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0 |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5211 |
else f(part_circlepath z r s t x) * |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5212 |
vector_derivative (part_circlepath z r s t) (at x)) has_integral i) {0..1}" |
68359 | 5213 |
by (rule has_integral_spike [OF negligible_finite [OF fin01]]) (use fi has_contour_integral in auto) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5214 |
have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5215 |
by (auto intro!: B [unfolded path_image_def image_def, simplified]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5216 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5217 |
apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5218 |
using assms apply force |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5219 |
apply (simp add: norm_mult vector_derivative_part_circlepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5220 |
using le * "2" \<open>r > 0\<close> by auto |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5221 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5222 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5223 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5224 |
lemma has_contour_integral_bound_part_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5225 |
"\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t); |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5226 |
0 \<le> B; 0 < r; s \<le> t; |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5227 |
\<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5228 |
\<Longrightarrow> norm i \<le> B*r*(t - s)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5229 |
by (auto intro: has_contour_integral_bound_part_circlepath_strong) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5230 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5231 |
lemma contour_integrable_continuous_part_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5232 |
"continuous_on (path_image (part_circlepath z r s t)) f |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5233 |
\<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5234 |
apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5235 |
apply (rule integrable_continuous_real) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5236 |
apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5237 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5238 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5239 |
proposition winding_number_part_circlepath_pos_less: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5240 |
assumes "s < t" and no: "norm(w - z) < r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5241 |
shows "0 < Re (winding_number(part_circlepath z r s t) w)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5242 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5243 |
have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5244 |
note valid_path_part_circlepath |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5245 |
moreover have " w \<notin> path_image (part_circlepath z r s t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5246 |
using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5247 |
moreover have "0 < r * (t - s) * (r - cmod (w - z))" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5248 |
using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5249 |
ultimately show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5250 |
apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5251 |
apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5252 |
apply (rule mult_left_mono)+ |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5253 |
using Re_Im_le_cmod [of "w-z" "linepath s t x" for x] |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5254 |
apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5255 |
using assms \<open>0 < r\<close> by auto |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5256 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5257 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5258 |
lemma simple_path_part_circlepath: |
61945 | 5259 |
"simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5260 |
proof (cases "r = 0 \<or> s = t") |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5261 |
case True |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5262 |
then show ?thesis |
68359 | 5263 |
unfolding part_circlepath_def simple_path_def |
5264 |
by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+ |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5265 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5266 |
case False then have "r \<noteq> 0" "s \<noteq> t" by auto |
63589 | 5267 |
have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z \<longleftrightarrow> \<i>*(x - y) * (t - s) = z" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5268 |
by (simp add: algebra_simps) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5269 |
have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 |
61945 | 5270 |
\<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5271 |
by auto |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5272 |
have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow> |
61945 | 5273 |
(\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5274 |
by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real] |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5275 |
intro: exI [where x = "-n" for n]) |
68359 | 5276 |
have 1: "\<bar>s - t\<bar> \<le> 2 * pi" |
5277 |
if "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1" |
|
5278 |
proof (rule ccontr) |
|
5279 |
assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi" |
|
5280 |
then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5281 |
using False that [of "2*pi / \<bar>t - s\<bar>"] |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5282 |
by (simp add: abs_minus_commute divide_simps) |
68359 | 5283 |
show False |
5284 |
using * [of 1] * [of "-1"] by auto |
|
5285 |
qed |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5286 |
have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5287 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5288 |
have "t-s = 2 * (real_of_int n * pi)/x" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5289 |
using that by (simp add: field_simps) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5290 |
then show ?thesis by (metis abs_minus_commute) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5291 |
qed |
68359 | 5292 |
have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)" |
5293 |
by force |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5294 |
show ?thesis using False |
68359 | 5295 |
apply (simp add: simple_path_def) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5296 |
apply (simp add: part_circlepath_def linepath_def exp_eq * ** abs01 del: Set.insert_iff) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5297 |
apply (subst abs_away) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5298 |
apply (auto simp: 1) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5299 |
apply (rule ccontr) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5300 |
apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5301 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5302 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5303 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5304 |
lemma arc_part_circlepath: |
61945 | 5305 |
assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5306 |
shows "arc (part_circlepath z r s t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5307 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5308 |
have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>" |
68359 | 5309 |
and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n |
5310 |
proof (rule ccontr) |
|
5311 |
assume "x \<noteq> y" |
|
5312 |
have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi" |
|
5313 |
by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq) |
|
5314 |
then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))" |
|
5315 |
by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re]) |
|
5316 |
with \<open>x \<noteq> y\<close> have st: "s-t = (of_int n * (pi * 2) / (y-x))" |
|
5317 |
by (force simp: field_simps) |
|
5318 |
have "\<bar>real_of_int n\<bar> < \<bar>y - x\<bar>" |
|
5319 |
using assms \<open>x \<noteq> y\<close> by (simp add: st abs_mult field_simps) |
|
5320 |
then show False |
|
5321 |
using assms x y st by (auto dest: of_int_lessD) |
|
5322 |
qed |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5323 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5324 |
using assms |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5325 |
apply (simp add: arc_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5326 |
apply (simp add: part_circlepath_def inj_on_def exp_eq) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5327 |
apply (blast intro: *) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5328 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5329 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5330 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5331 |
subsection\<open>Special case of one complete circle\<close> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5332 |
|
70136 | 5333 |
definition\<^marker>\<open>tag important\<close> circlepath :: "[complex, real, real] \<Rightarrow> complex" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5334 |
where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5335 |
|
63589 | 5336 |
lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5337 |
by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5338 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5339 |
lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5340 |
by (simp add: circlepath_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5341 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5342 |
lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5343 |
by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5344 |
|
61848 | 5345 |
lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)" |
5346 |
proof - |
|
68296
69d680e94961
tidying and reorganisation around Cauchy Integral Theorem
paulson <lp15@cam.ac.uk>
parents:
68284
diff
changeset
|
5347 |
have "z + of_real r * exp (2 * pi * \<i> * (x + 1/2)) = |
61848 | 5348 |
z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)" |
5349 |
by (simp add: divide_simps) (simp add: algebra_simps) |
|
68339 | 5350 |
also have "\<dots> = z - r * exp (2 * pi * \<i> * x)" |
61848 | 5351 |
by (simp add: exp_add) |
5352 |
finally show ?thesis |
|
5353 |
by (simp add: circlepath path_image_def sphere_def dist_norm) |
|
5354 |
qed |
|
5355 |
||
5356 |
lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x" |
|
5357 |
using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x] |
|
5358 |
by (simp add: add.commute) |
|
5359 |
||
5360 |
lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)" |
|
5361 |
using circlepath_add1 [of z r "x-1/2"] |
|
5362 |
by (simp add: add.commute) |
|
5363 |
||
5364 |
lemma path_image_circlepath_minus_subset: |
|
5365 |
"path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)" |
|
5366 |
apply (simp add: path_image_def image_def circlepath_minus, clarify) |
|
5367 |
apply (case_tac "xa \<le> 1/2", force) |
|
68339 | 5368 |
apply (force simp: circlepath_add_half)+ |
61848 | 5369 |
done |
5370 |
||
5371 |
lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)" |
|
5372 |
using path_image_circlepath_minus_subset by fastforce |
|
5373 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5374 |
lemma has_vector_derivative_circlepath [derivative_intros]: |
63589 | 5375 |
"((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x))) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5376 |
(at x within X)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5377 |
apply (simp add: circlepath_def scaleR_conv_of_real) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5378 |
apply (rule derivative_eq_intros) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5379 |
apply (simp add: algebra_simps) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5380 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5381 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5382 |
lemma vector_derivative_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5383 |
"vector_derivative (circlepath z r) (at x) = |
63589 | 5384 |
2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5385 |
using has_vector_derivative_circlepath vector_derivative_at by blast |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5386 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5387 |
lemma vector_derivative_circlepath01: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5388 |
"\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5389 |
\<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) = |
63589 | 5390 |
2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5391 |
using has_vector_derivative_circlepath |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5392 |
by (auto simp: vector_derivative_at_within_ivl) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5393 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5394 |
lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5395 |
by (simp add: circlepath_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5396 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5397 |
lemma path_circlepath [simp]: "path (circlepath z r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5398 |
by (simp add: valid_path_imp_path) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5399 |
|
61848 | 5400 |
lemma path_image_circlepath_nonneg: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5401 |
assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5402 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5403 |
have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5404 |
proof (cases "x = z") |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5405 |
case True then show ?thesis by force |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5406 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5407 |
case False |
63040 | 5408 |
define w where "w = x - z" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5409 |
then have "w \<noteq> 0" by (simp add: False) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5410 |
have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5411 |
using cis_conv_exp complex_eq_iff by auto |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5412 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5413 |
apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5414 |
apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5415 |
apply (rule_tac x="t / (2*pi)" in image_eqI) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5416 |
apply (simp add: field_simps \<open>w \<noteq> 0\<close>) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5417 |
using False ** |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5418 |
apply (auto simp: w_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5419 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5420 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5421 |
show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5422 |
unfolding circlepath path_image_def sphere_def dist_norm |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5423 |
by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5424 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5425 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5426 |
lemma path_image_circlepath [simp]: |
61945 | 5427 |
"path_image (circlepath z r) = sphere z \<bar>r\<bar>" |
61848 | 5428 |
using path_image_circlepath_minus |
68339 | 5429 |
by (force simp: path_image_circlepath_nonneg abs_if) |
61848 | 5430 |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5431 |
lemma has_contour_integral_bound_circlepath_strong: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5432 |
"\<lbrakk>(f has_contour_integral i) (circlepath z r); |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5433 |
finite k; 0 \<le> B; 0 < r; |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5434 |
\<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5435 |
\<Longrightarrow> norm i \<le> B*(2*pi*r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5436 |
unfolding circlepath_def |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5437 |
by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5438 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5439 |
lemma has_contour_integral_bound_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5440 |
"\<lbrakk>(f has_contour_integral i) (circlepath z r); |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5441 |
0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5442 |
\<Longrightarrow> norm i \<le> B*(2*pi*r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5443 |
by (auto intro: has_contour_integral_bound_circlepath_strong) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5444 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5445 |
lemma contour_integrable_continuous_circlepath: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5446 |
"continuous_on (path_image (circlepath z r)) f |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5447 |
\<Longrightarrow> f contour_integrable_on (circlepath z r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5448 |
by (simp add: circlepath_def contour_integrable_continuous_part_circlepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5449 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5450 |
lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5451 |
by (simp add: circlepath_def simple_path_part_circlepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5452 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5453 |
lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)" |
61848 | 5454 |
by (simp add: sphere_def dist_norm norm_minus_commute) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5455 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5456 |
lemma contour_integral_circlepath: |
68359 | 5457 |
assumes "r > 0" |
5458 |
shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>" |
|
5459 |
proof (rule contour_integral_unique) |
|
5460 |
show "((\<lambda>w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \<i>) (circlepath z r)" |
|
5461 |
unfolding has_contour_integral_def using assms |
|
5462 |
apply (subst has_integral_cong) |
|
5463 |
apply (simp add: vector_derivative_circlepath01) |
|
5464 |
using has_integral_const_real [of _ 0 1] apply (force simp: circlepath) |
|
5465 |
done |
|
5466 |
qed |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5467 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5468 |
lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5469 |
apply (rule winding_number_unique_loop) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5470 |
apply (simp_all add: sphere_def valid_path_imp_path) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5471 |
apply (rule_tac x="circlepath z r" in exI) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5472 |
apply (simp add: sphere_def contour_integral_circlepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5473 |
done |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5474 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5475 |
proposition winding_number_circlepath: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5476 |
assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5477 |
proof (cases "w = z") |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5478 |
case True then show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5479 |
using assms winding_number_circlepath_centre by auto |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5480 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5481 |
case False |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5482 |
have [simp]: "r > 0" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5483 |
using assms le_less_trans norm_ge_zero by blast |
63040 | 5484 |
define r' where "r' = norm(w - z)" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5485 |
have "r' < r" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5486 |
by (simp add: assms r'_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5487 |
have disjo: "cball z r' \<inter> sphere z r = {}" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5488 |
using \<open>r' < r\<close> by (force simp: cball_def sphere_def) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5489 |
have "winding_number(circlepath z r) w = winding_number(circlepath z r) z" |
68359 | 5490 |
proof (rule winding_number_around_inside [where s = "cball z r'"]) |
5491 |
show "winding_number (circlepath z r) z \<noteq> 0" |
|
5492 |
by (simp add: winding_number_circlepath_centre) |
|
5493 |
show "cball z r' \<inter> path_image (circlepath z r) = {}" |
|
5494 |
by (simp add: disjo less_eq_real_def) |
|
5495 |
qed (auto simp: r'_def dist_norm norm_minus_commute) |
|
68339 | 5496 |
also have "\<dots> = 1" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5497 |
by (simp add: winding_number_circlepath_centre) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5498 |
finally show ?thesis . |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5499 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5500 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5501 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5502 |
text\<open> Hence the Cauchy formula for points inside a circle.\<close> |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5503 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5504 |
theorem Cauchy_integral_circlepath: |
68359 | 5505 |
assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r" |
63589 | 5506 |
shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w)) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5507 |
(circlepath z r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5508 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5509 |
have "r > 0" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5510 |
using assms le_less_trans norm_ge_zero by blast |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5511 |
have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5512 |
(circlepath z r)" |
68359 | 5513 |
proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"]) |
5514 |
show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow> |
|
5515 |
f field_differentiable at x" |
|
5516 |
using holf holomorphic_on_imp_differentiable_at by auto |
|
5517 |
have "w \<notin> sphere z r" |
|
5518 |
by simp (metis dist_commute dist_norm not_le order_refl wz) |
|
5519 |
then show "path_image (circlepath z r) \<subseteq> cball z r - {w}" |
|
5520 |
using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def) |
|
5521 |
qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>) |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5522 |
then show ?thesis |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5523 |
by (simp add: winding_number_circlepath assms) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5524 |
qed |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5525 |
|
70136 | 5526 |
corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5527 |
assumes "f holomorphic_on cball z r" "norm(w - z) < r" |
63589 | 5528 |
shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w)) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5529 |
(circlepath z r)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5530 |
using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5531 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5532 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5533 |
lemma no_bounded_connected_component_imp_winding_number_zero: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5534 |
assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5535 |
and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5536 |
shows "winding_number g z = 0" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5537 |
apply (rule winding_number_zero_in_outside) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5538 |
apply (simp_all add: assms) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5539 |
by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5540 |
|
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5541 |
lemma no_bounded_path_component_imp_winding_number_zero: |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5542 |
assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5543 |
and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5544 |
shows "winding_number g z = 0" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5545 |
apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5546 |
by (simp add: bounded_subset nb path_component_subset_connected_component) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
5547 |
|
61848 | 5548 |
|
5549 |
subsection\<open> Uniform convergence of path integral\<close> |
|
5550 |
||
5551 |
text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close> |
|
5552 |
||
5553 |
proposition contour_integral_uniform_limit: |
|
5554 |
assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F" |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5555 |
and ul_f: "uniform_limit (path_image \<gamma>) f l F" |
61848 | 5556 |
and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B" |
5557 |
and \<gamma>: "valid_path \<gamma>" |
|
69508 | 5558 |
and [simp]: "\<not> trivial_limit F" |
61973 | 5559 |
shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F" |
61848 | 5560 |
proof - |
5561 |
have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one) |
|
5562 |
{ fix e::real |
|
5563 |
assume "0 < e" |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5564 |
then have "0 < e / (\<bar>B\<bar> + 1)" by simp |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5565 |
then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5566 |
using ul_f [unfolded uniform_limit_iff dist_norm] by auto |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5567 |
with ev_fint |
61848 | 5568 |
obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)" |
5569 |
and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}" |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5570 |
using eventually_happens [OF eventually_conj] |
61848 | 5571 |
by (fastforce simp: contour_integrable_on path_image_def) |
5572 |
have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5573 |
using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps) |
61848 | 5574 |
have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}" |
68359 | 5575 |
proof (intro exI conjI ballI) |
68493 | 5576 |
show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e" |
68359 | 5577 |
if "x \<in> {0..1}" for x |
5578 |
apply (rule order_trans [OF _ Ble]) |
|
5579 |
using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close> |
|
5580 |
apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps) |
|
5581 |
apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le]) |
|
5582 |
done |
|
5583 |
qed (rule inta) |
|
61848 | 5584 |
} |
5585 |
then show lintg: "l contour_integrable_on \<gamma>" |
|
68493 | 5586 |
unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real) |
61848 | 5587 |
{ fix e::real |
63040 | 5588 |
define B' where "B' = B + 1" |
61848 | 5589 |
have B': "B' > 0" "B' > B" using \<open>0 \<le> B\<close> by (auto simp: B'_def) |
5590 |
assume "0 < e" |
|
5591 |
then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'" |
|
68493 | 5592 |
using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B' |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5593 |
by (simp add: field_simps) |
61848 | 5594 |
have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp |
5595 |
have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2" |
|
5596 |
if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t |
|
5597 |
proof - |
|
5598 |
have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')" |
|
5599 |
using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto |
|
68339 | 5600 |
also have "\<dots> < e" |
61848 | 5601 |
by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less) |
5602 |
finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" . |
|
5603 |
then show ?thesis |
|
5604 |
by (simp add: left_diff_distrib [symmetric] norm_mult) |
|
5605 |
qed |
|
68359 | 5606 |
have le_e: "\<And>x. \<lbrakk>\<forall>xa\<in>{0..1}. 2 * cmod (f x (\<gamma> xa) - l (\<gamma> xa)) < e / B'; f x contour_integrable_on \<gamma>\<rbrakk> |
5607 |
\<Longrightarrow> cmod (integral {0..1} |
|
5608 |
(\<lambda>u. f x (\<gamma> u) * vector_derivative \<gamma> (at u) - l (\<gamma> u) * vector_derivative \<gamma> (at u))) < e" |
|
5609 |
apply (rule le_less_trans [OF integral_norm_bound_integral ie]) |
|
5610 |
apply (simp add: lintg integrable_diff contour_integrable_on [symmetric]) |
|
5611 |
apply (blast intro: *)+ |
|
5612 |
done |
|
61848 | 5613 |
have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e" |
5614 |
apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]]) |
|
5615 |
apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral) |
|
68359 | 5616 |
apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e) |
61848 | 5617 |
done |
5618 |
} |
|
61973 | 5619 |
then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F" |
61848 | 5620 |
by (rule tendstoI) |
5621 |
qed |
|
5622 |
||
70136 | 5623 |
corollary\<^marker>\<open>tag unimportant\<close> contour_integral_uniform_limit_circlepath: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5624 |
assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5625 |
and "uniform_limit (sphere z r) f l F" |
69508 | 5626 |
and "\<not> trivial_limit F" "0 < r" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5627 |
shows "l contour_integrable_on (circlepath z r)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5628 |
"((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5629 |
using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit) |
61848 | 5630 |
|
5631 |
||
70136 | 5632 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close> |
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5633 |
|
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5634 |
lemma Cauchy_next_derivative: |
61848 | 5635 |
assumes "continuous_on (path_image \<gamma>) f'" |
5636 |
and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B" |
|
5637 |
and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>" |
|
5638 |
and k: "k \<noteq> 0" |
|
5639 |
and "open s" |
|
5640 |
and \<gamma>: "valid_path \<gamma>" |
|
5641 |
and w: "w \<in> s - path_image \<gamma>" |
|
5642 |
shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>" |
|
5643 |
and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k)))) |
|
5644 |
(at w)" (is "?thes2") |
|
5645 |
proof - |
|
5646 |
have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast |
|
5647 |
then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w |
|
5648 |
using open_contains_ball by blast |
|
5649 |
have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n" |
|
5650 |
by (metis norm_of_nat of_nat_Suc) |
|
68359 | 5651 |
have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk> |
5652 |
\<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>" |
|
5653 |
apply (rule contour_integrable_div [OF contour_integrable_diff]) |
|
5654 |
using int w d |
|
5655 |
by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+ |
|
61848 | 5656 |
have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) |
5657 |
contour_integrable_on \<gamma>" |
|
68359 | 5658 |
unfolding eventually_at |
61848 | 5659 |
apply (rule_tac x=d in exI) |
68359 | 5660 |
apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint) |
61848 | 5661 |
done |
5662 |
have bim_g: "bounded (image f' (path_image \<gamma>))" |
|
5663 |
by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms) |
|
5664 |
then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C" |
|
5665 |
by (force simp: bounded_pos path_image_def) |
|
5666 |
have twom: "\<forall>\<^sub>F n in at w. |
|
5667 |
\<forall>x\<in>path_image \<gamma>. |
|
5668 |
cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e" |
|
5669 |
if "0 < e" for e |
|
5670 |
proof - |
|
5671 |
have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k) < e" |
|
5672 |
if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2" |
|
68359 | 5673 |
and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)" |
61848 | 5674 |
for u x |
5675 |
proof - |
|
63040 | 5676 |
define ff where [abs_def]: |
5677 |
"ff n w = |
|
5678 |
(if n = 0 then inverse(x - w)^k |
|
5679 |
else if n = 1 then k / (x - w)^(Suc k) |
|
5680 |
else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w |
|
61848 | 5681 |
have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z" |
5682 |
by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc) |
|
68359 | 5683 |
have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))" |
5684 |
if "z \<in> ball w (d/2)" "i \<le> 1" for i z |
|
61848 | 5685 |
proof - |
5686 |
have "z \<notin> path_image \<gamma>" |
|
5687 |
using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast |
|
5688 |
then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast |
|
5689 |
then have neq: "x * x + z * z \<noteq> x * (z * 2)" |
|
5690 |
by (blast intro: dest!: sum_sqs_eq) |
|
5691 |
with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto |
|
5692 |
then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))" |
|
5693 |
by (simp add: algebra_simps) |
|
5694 |
show ?thesis using \<open>i \<le> 1\<close> |
|
5695 |
apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe) |
|
5696 |
apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+ |
|
5697 |
done |
|
5698 |
qed |
|
5699 |
{ fix a::real and b::real assume ab: "a > 0" "b > 0" |
|
5700 |
then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5701 |
by (subst mult_le_cancel_left_pos) |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5702 |
(use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>) |
61848 | 5703 |
with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a" |
5704 |
by (simp add: field_simps) |
|
5705 |
} note canc = this |
|
68359 | 5706 |
have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)" |
5707 |
if "v \<in> ball w (d/2)" for v |
|
61848 | 5708 |
proof - |
68359 | 5709 |
have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d" |
5710 |
by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball) |
|
61848 | 5711 |
have "d/2 \<le> cmod (x - v)" using d x that |
68359 | 5712 |
using lessd d x |
5713 |
by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps) |
|
61848 | 5714 |
then have "d \<le> cmod (x - v) * 2" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5715 |
by (simp add: field_split_simps) |
61848 | 5716 |
then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)" |
5717 |
using \<open>0 < d\<close> order_less_imp_le power_mono by blast |
|
5718 |
have "x \<noteq> v" using that |
|
5719 |
using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce |
|
5720 |
then show ?thesis |
|
68359 | 5721 |
using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
5722 |
using dpow_le apply (simp add: algebra_simps field_split_simps mult_less_0_iff) |
61848 | 5723 |
done |
5724 |
qed |
|
68359 | 5725 |
have ub: "u \<in> ball w (d/2)" |
61848 | 5726 |
using uwd by (simp add: dist_commute dist_norm) |
5727 |
have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k))) |
|
68359 | 5728 |
\<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))" |
69530 | 5729 |
using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified] |
61848 | 5730 |
by (simp add: ff_def \<open>0 < d\<close>) |
5731 |
then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k))) |
|
68359 | 5732 |
\<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)" |
61848 | 5733 |
by (simp add: field_simps) |
5734 |
then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k))) |
|
5735 |
/ (cmod (u - w) * real k) |
|
68359 | 5736 |
\<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)" |
61848 | 5737 |
using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq) |
68339 | 5738 |
also have "\<dots> < e" |
61848 | 5739 |
using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps) |
5740 |
finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k))) |
|
5741 |
/ cmod ((u - w) * real k) < e" |
|
5742 |
by (simp add: norm_mult) |
|
5743 |
have "x \<noteq> u" |
|
5744 |
using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute) |
|
5745 |
show ?thesis |
|
5746 |
apply (rule le_less_trans [OF _ e]) |
|
68359 | 5747 |
using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close> |
61848 | 5748 |
apply (simp add: field_simps norm_divide [symmetric]) |
5749 |
done |
|
5750 |
qed |
|
5751 |
show ?thesis |
|
5752 |
unfolding eventually_at |
|
5753 |
apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI) |
|
5754 |
apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *) |
|
5755 |
done |
|
5756 |
qed |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5757 |
have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5758 |
unfolding uniform_limit_iff dist_norm |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5759 |
proof clarify |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5760 |
fix e::real |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5761 |
assume "0 < e" |
61848 | 5762 |
have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
5763 |
f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e" |
|
5764 |
if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
|
5765 |
inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C" |
|
5766 |
and x: "0 \<le> x" "x \<le> 1" |
|
5767 |
for u x |
|
5768 |
proof (cases "(f' (\<gamma> x)) = 0") |
|
5769 |
case True then show ?thesis by (simp add: \<open>0 < e\<close>) |
|
5770 |
next |
|
5771 |
case False |
|
5772 |
have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
|
5773 |
f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) = |
|
5774 |
cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
|
5775 |
inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))" |
|
5776 |
by (simp add: field_simps) |
|
68339 | 5777 |
also have "\<dots> = cmod (f' (\<gamma> x)) * |
61848 | 5778 |
cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) - |
5779 |
inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)" |
|
5780 |
by (simp add: norm_mult) |
|
68339 | 5781 |
also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)" |
68359 | 5782 |
using False mult_strict_left_mono [OF ec] by force |
68339 | 5783 |
also have "\<dots> \<le> e" using C |
61848 | 5784 |
by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff) |
5785 |
finally show ?thesis . |
|
5786 |
qed |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5787 |
show "\<forall>\<^sub>F n in at w. |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5788 |
\<forall>x\<in>path_image \<gamma>. |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5789 |
cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
5790 |
using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]] unfolding path_image_def |
61848 | 5791 |
by (force intro: * elim: eventually_mono) |
5792 |
qed |
|
5793 |
show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>" |
|
5794 |
by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto |
|
5795 |
have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k)) |
|
61976 | 5796 |
\<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))" |
61848 | 5797 |
by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto |
5798 |
have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) = |
|
5799 |
(f u - f w) / (u - w) / k" |
|
68359 | 5800 |
if "dist u w < d" for u |
5801 |
proof - |
|
5802 |
have u: "u \<in> s - path_image \<gamma>" |
|
5803 |
by (metis subsetD d dist_commute mem_ball that) |
|
5804 |
show ?thesis |
|
5805 |
apply (rule contour_integral_unique) |
|
5806 |
apply (simp add: diff_divide_distrib algebra_simps) |
|
5807 |
apply (intro has_contour_integral_diff has_contour_integral_div) |
|
5808 |
using u w apply (simp_all add: field_simps int) |
|
5809 |
done |
|
5810 |
qed |
|
61848 | 5811 |
show ?thes2 |
68239 | 5812 |
apply (simp add: has_field_derivative_iff del: power_Suc) |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
5813 |
apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ]) |
61848 | 5814 |
apply (simp add: \<open>k \<noteq> 0\<close> **) |
5815 |
done |
|
5816 |
qed |
|
5817 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5818 |
lemma Cauchy_next_derivative_circlepath: |
61848 | 5819 |
assumes contf: "continuous_on (path_image (circlepath z r)) f" |
5820 |
and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)" |
|
5821 |
and k: "k \<noteq> 0" |
|
5822 |
and w: "w \<in> ball z r" |
|
5823 |
shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)" |
|
5824 |
(is "?thes1") |
|
5825 |
and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)" |
|
5826 |
(is "?thes2") |
|
5827 |
proof - |
|
5828 |
have "r > 0" using w |
|
5829 |
using ball_eq_empty by fastforce |
|
5830 |
have wim: "w \<in> ball z r - path_image (circlepath z r)" |
|
5831 |
using w by (auto simp: dist_norm) |
|
5832 |
show ?thes1 ?thes2 |
|
5833 |
by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"]; |
|
5834 |
auto simp: vector_derivative_circlepath norm_mult)+ |
|
5835 |
qed |
|
5836 |
||
5837 |
||
5838 |
text\<open> In particular, the first derivative formula.\<close> |
|
5839 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
5840 |
lemma Cauchy_derivative_integral_circlepath: |
61848 | 5841 |
assumes contf: "continuous_on (cball z r) f" |
5842 |
and holf: "f holomorphic_on ball z r" |
|
5843 |
and w: "w \<in> ball z r" |
|
5844 |
shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)" |
|
5845 |
(is "?thes1") |
|
63589 | 5846 |
and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)" |
61848 | 5847 |
(is "?thes2") |
5848 |
proof - |
|
5849 |
have [simp]: "r \<ge> 0" using w |
|
5850 |
using ball_eq_empty by fastforce |
|
5851 |
have f: "continuous_on (path_image (circlepath z r)) f" |
|
68339 | 5852 |
by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def) |
61848 | 5853 |
have int: "\<And>w. dist z w < r \<Longrightarrow> |
63589 | 5854 |
((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)" |
61848 | 5855 |
by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute) |
5856 |
show ?thes1 |
|
5857 |
apply (simp add: power2_eq_square) |
|
5858 |
apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified]) |
|
5859 |
apply (blast intro: int) |
|
5860 |
done |
|
5861 |
have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)" |
|
5862 |
apply (simp add: power2_eq_square) |
|
63589 | 5863 |
apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified]) |
61848 | 5864 |
apply (blast intro: int) |
5865 |
done |
|
5866 |
then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)" |
|
5867 |
by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified]) |
|
5868 |
show ?thes2 |
|
5869 |
by simp (rule fder) |
|
5870 |
qed |
|
5871 |
||
67968 | 5872 |
subsection\<open>Existence of all higher derivatives\<close> |
61848 | 5873 |
|
5874 |
proposition derivative_is_holomorphic: |
|
68359 | 5875 |
assumes "open S" |
5876 |
and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)" |
|
5877 |
shows "f' holomorphic_on S" |
|
61848 | 5878 |
proof - |
68359 | 5879 |
have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z |
61848 | 5880 |
proof - |
68359 | 5881 |
obtain r where "r > 0" and r: "cball z r \<subseteq> S" |
5882 |
using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast |
|
61848 | 5883 |
then have holf_cball: "f holomorphic_on cball z r" |
5884 |
apply (simp add: holomorphic_on_def) |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
5885 |
using field_differentiable_at_within field_differentiable_def fder by blast |
61848 | 5886 |
then have "continuous_on (path_image (circlepath z r)) f" |
5887 |
using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on]) |
|
63589 | 5888 |
then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)" |
61848 | 5889 |
by (auto intro: continuous_intros)+ |
5890 |
have contf_cball: "continuous_on (cball z r) f" using holf_cball |
|
5891 |
by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset) |
|
5892 |
have holf_ball: "f holomorphic_on ball z r" using holf_cball |
|
5893 |
using ball_subset_cball holomorphic_on_subset by blast |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
5894 |
{ fix w assume w: "w \<in> ball z r" |
61848 | 5895 |
have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r" |
5896 |
by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball]) |
|
5897 |
have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2)) |
|
5898 |
(at w)" |
|
5899 |
by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball]) |
|
5900 |
have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)" |
|
5901 |
using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder]) |
|
5902 |
have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral |
|
5903 |
contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)) |
|
5904 |
(circlepath z r)" |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63589
diff
changeset
|
5905 |
by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]]) |
61848 | 5906 |
then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral |
5907 |
contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)) |
|
5908 |
(circlepath z r)" |
|
5909 |
by (simp add: algebra_simps) |
|
5910 |
then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)" |
|
5911 |
by (simp add: f'_eq) |
|
5912 |
} note * = this |
|
5913 |
show ?thesis |
|
5914 |
apply (rule exI) |
|
5915 |
apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified]) |
|
5916 |
apply (simp_all add: \<open>0 < r\<close> * dist_norm) |
|
5917 |
done |
|
5918 |
qed |
|
5919 |
show ?thesis |
|
68359 | 5920 |
by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *) |
61848 | 5921 |
qed |
5922 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
5923 |
lemma holomorphic_deriv [holomorphic_intros]: |
68359 | 5924 |
"\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
5925 |
by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def) |
61848 | 5926 |
|
68359 | 5927 |
lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S" |
61848 | 5928 |
using analytic_on_holomorphic holomorphic_deriv by auto |
5929 |
||
68359 | 5930 |
lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S" |
61848 | 5931 |
by (induction n) (auto simp: holomorphic_deriv) |
5932 |
||
68359 | 5933 |
lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S" |
61848 | 5934 |
unfolding analytic_on_def using holomorphic_higher_deriv by blast |
5935 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
5936 |
lemma has_field_derivative_higher_deriv: |
68359 | 5937 |
"\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> |
61848 | 5938 |
\<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
5939 |
by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply |
61848 | 5940 |
funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def) |
5941 |
||
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
5942 |
lemma valid_path_compose_holomorphic: |
68359 | 5943 |
assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S" |
68339 | 5944 |
shows "valid_path (f \<circ> g)" |
62837 | 5945 |
proof (rule valid_path_compose[OF \<open>valid_path g\<close>]) |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
5946 |
fix x assume "x \<in> path_image g" |
64394 | 5947 |
then show "f field_differentiable at x" |
5948 |
using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast |
|
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
5949 |
next |
68359 | 5950 |
have "deriv f holomorphic_on S" |
5951 |
using holomorphic_deriv holo \<open>open S\<close> by auto |
|
62623
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
5952 |
then show "continuous_on (path_image g) (deriv f)" |
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
5953 |
using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
5954 |
qed |
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
5955 |
|
61848 | 5956 |
|
67968 | 5957 |
subsection\<open>Morera's theorem\<close> |
61848 | 5958 |
|
5959 |
lemma Morera_local_triangle_ball: |
|
68359 | 5960 |
assumes "\<And>z. z \<in> S |
61848 | 5961 |
\<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and> |
5962 |
(\<forall>b c. closed_segment b c \<subseteq> ball a e |
|
5963 |
\<longrightarrow> contour_integral (linepath a b) f + |
|
5964 |
contour_integral (linepath b c) f + |
|
5965 |
contour_integral (linepath c a) f = 0)" |
|
68359 | 5966 |
shows "f analytic_on S" |
61848 | 5967 |
proof - |
68359 | 5968 |
{ fix z assume "z \<in> S" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
5969 |
with assms obtain e a where |
61848 | 5970 |
"0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f" |
5971 |
and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e |
|
5972 |
\<Longrightarrow> contour_integral (linepath a b) f + |
|
5973 |
contour_integral (linepath b c) f + |
|
5974 |
contour_integral (linepath c a) f = 0" |
|
5975 |
by fastforce |
|
5976 |
have az: "dist a z < e" using mem_ball z by blast |
|
5977 |
have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e" |
|
5978 |
by (simp add: dist_commute ball_subset_ball_iff) |
|
5979 |
have "\<exists>e>0. f holomorphic_on ball z e" |
|
68359 | 5980 |
proof (intro exI conjI) |
5981 |
have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e" |
|
5982 |
by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball) |
|
5983 |
show "f holomorphic_on ball z (e - dist a z)" |
|
5984 |
apply (rule holomorphic_on_subset [OF _ sb_ball]) |
|
5985 |
apply (rule derivative_is_holomorphic[OF open_ball]) |
|
5986 |
apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a]) |
|
5987 |
apply (simp_all add: 0 \<open>0 < e\<close> sub_ball) |
|
5988 |
done |
|
5989 |
qed (simp add: az) |
|
61848 | 5990 |
} |
5991 |
then show ?thesis |
|
5992 |
by (simp add: analytic_on_def) |
|
5993 |
qed |
|
5994 |
||
5995 |
lemma Morera_local_triangle: |
|
68359 | 5996 |
assumes "\<And>z. z \<in> S |
61848 | 5997 |
\<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and> |
5998 |
(\<forall>a b c. convex hull {a,b,c} \<subseteq> t |
|
5999 |
\<longrightarrow> contour_integral (linepath a b) f + |
|
6000 |
contour_integral (linepath b c) f + |
|
6001 |
contour_integral (linepath c a) f = 0)" |
|
68359 | 6002 |
shows "f analytic_on S" |
61848 | 6003 |
proof - |
68359 | 6004 |
{ fix z assume "z \<in> S" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6005 |
with assms obtain t where |
61848 | 6006 |
"open t" and z: "z \<in> t" and contf: "continuous_on t f" |
6007 |
and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t |
|
6008 |
\<Longrightarrow> contour_integral (linepath a b) f + |
|
6009 |
contour_integral (linepath b c) f + |
|
6010 |
contour_integral (linepath c a) f = 0" |
|
6011 |
by force |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6012 |
then obtain e where "e>0" and e: "ball z e \<subseteq> t" |
61848 | 6013 |
using open_contains_ball by blast |
6014 |
have [simp]: "continuous_on (ball z e) f" using contf |
|
6015 |
using continuous_on_subset e by blast |
|
68359 | 6016 |
have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow> |
6017 |
contour_integral (linepath z b) f + |
|
6018 |
contour_integral (linepath b c) f + |
|
6019 |
contour_integral (linepath c z) f = 0" |
|
6020 |
by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset) |
|
6021 |
have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and> |
|
6022 |
(\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow> |
|
6023 |
contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)" |
|
6024 |
using \<open>e > 0\<close> eq0 by force |
|
61848 | 6025 |
} |
6026 |
then show ?thesis |
|
6027 |
by (simp add: Morera_local_triangle_ball) |
|
6028 |
qed |
|
6029 |
||
6030 |
proposition Morera_triangle: |
|
68359 | 6031 |
"\<lbrakk>continuous_on S f; open S; |
6032 |
\<And>a b c. convex hull {a,b,c} \<subseteq> S |
|
61848 | 6033 |
\<longrightarrow> contour_integral (linepath a b) f + |
6034 |
contour_integral (linepath b c) f + |
|
6035 |
contour_integral (linepath c a) f = 0\<rbrakk> |
|
68359 | 6036 |
\<Longrightarrow> f analytic_on S" |
61848 | 6037 |
using Morera_local_triangle by blast |
6038 |
||
67968 | 6039 |
subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close> |
61848 | 6040 |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6041 |
lemma higher_deriv_linear [simp]: |
61848 | 6042 |
"(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)" |
68359 | 6043 |
by (induction n) auto |
61848 | 6044 |
|
6045 |
lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)" |
|
68359 | 6046 |
by (induction n) auto |
61848 | 6047 |
|
6048 |
lemma higher_deriv_ident [simp]: |
|
6049 |
"(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)" |
|
62217 | 6050 |
apply (induction n, simp) |
6051 |
apply (metis higher_deriv_linear lambda_one) |
|
61848 | 6052 |
done |
6053 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6054 |
lemma higher_deriv_id [simp]: |
61848 | 6055 |
"(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)" |
6056 |
by (simp add: id_def) |
|
6057 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6058 |
lemma has_complex_derivative_funpow_1: |
61848 | 6059 |
"\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)" |
68339 | 6060 |
apply (induction n, auto) |
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
6061 |
apply (simp add: id_def) |
61848 | 6062 |
by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral) |
6063 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6064 |
lemma higher_deriv_uminus: |
68359 | 6065 |
assumes "f holomorphic_on S" "open S" and z: "z \<in> S" |
61848 | 6066 |
shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)" |
6067 |
using z |
|
6068 |
proof (induction n arbitrary: z) |
|
6069 |
case 0 then show ?case by simp |
|
6070 |
next |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6071 |
case (Suc n z) |
61848 | 6072 |
have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)" |
6073 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
|
68359 | 6074 |
have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)" |
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
6075 |
apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"]) |
68359 | 6076 |
apply (rule derivative_eq_intros | rule * refl assms)+ |
6077 |
apply (auto simp add: Suc) |
|
61848 | 6078 |
done |
68359 | 6079 |
then show ?case |
6080 |
by (simp add: DERIV_imp_deriv) |
|
61848 | 6081 |
qed |
6082 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6083 |
lemma higher_deriv_add: |
61848 | 6084 |
fixes z::complex |
68359 | 6085 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
61848 | 6086 |
shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z" |
6087 |
using z |
|
6088 |
proof (induction n arbitrary: z) |
|
6089 |
case 0 then show ?case by simp |
|
6090 |
next |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6091 |
case (Suc n z) |
61848 | 6092 |
have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)" |
6093 |
"((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)" |
|
6094 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
|
68359 | 6095 |
have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative |
6096 |
deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)" |
|
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
6097 |
apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"]) |
68359 | 6098 |
apply (rule derivative_eq_intros | rule * refl assms)+ |
6099 |
apply (auto simp add: Suc) |
|
61848 | 6100 |
done |
68359 | 6101 |
then show ?case |
6102 |
by (simp add: DERIV_imp_deriv) |
|
61848 | 6103 |
qed |
6104 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6105 |
lemma higher_deriv_diff: |
61848 | 6106 |
fixes z::complex |
68359 | 6107 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
61848 | 6108 |
shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z" |
6109 |
apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add) |
|
6110 |
apply (subst higher_deriv_add) |
|
6111 |
using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus) |
|
6112 |
done |
|
6113 |
||
6114 |
lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))" |
|
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63262
diff
changeset
|
6115 |
by (cases k) simp_all |
61848 | 6116 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6117 |
lemma higher_deriv_mult: |
61848 | 6118 |
fixes z::complex |
68359 | 6119 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6120 |
shows "(deriv ^^ n) (\<lambda>w. f w * g w) z = |
61848 | 6121 |
(\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)" |
6122 |
using z |
|
6123 |
proof (induction n arbitrary: z) |
|
6124 |
case 0 then show ?case by simp |
|
6125 |
next |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6126 |
case (Suc n z) |
61848 | 6127 |
have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)" |
6128 |
"\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)" |
|
6129 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
|
6130 |
have sumeq: "(\<Sum>i = 0..n. |
|
6131 |
of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) = |
|
6132 |
g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))" |
|
64267 | 6133 |
apply (simp add: bb algebra_simps sum.distrib) |
6134 |
apply (subst (4) sum_Suc_reindex) |
|
6135 |
apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong) |
|
61848 | 6136 |
done |
68359 | 6137 |
have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative |
6138 |
(\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z)) |
|
6139 |
(at z)" |
|
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
6140 |
apply (rule has_field_derivative_transform_within_open |
68359 | 6141 |
[of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"]) |
6142 |
apply (simp add: algebra_simps) |
|
6143 |
apply (rule DERIV_cong [OF DERIV_sum]) |
|
6144 |
apply (rule DERIV_cmult) |
|
6145 |
apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric]) |
|
61848 | 6146 |
done |
68359 | 6147 |
then show ?case |
6148 |
unfolding funpow.simps o_apply |
|
6149 |
by (simp add: DERIV_imp_deriv) |
|
61848 | 6150 |
qed |
6151 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6152 |
lemma higher_deriv_transform_within_open: |
61848 | 6153 |
fixes z::complex |
68359 | 6154 |
assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S" |
6155 |
and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w" |
|
61848 | 6156 |
shows "(deriv ^^ i) f z = (deriv ^^ i) g z" |
6157 |
using z |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6158 |
by (induction i arbitrary: z) |
61848 | 6159 |
(auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms) |
6160 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6161 |
lemma higher_deriv_compose_linear: |
61848 | 6162 |
fixes z::complex |
68359 | 6163 |
assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S" |
6164 |
and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T" |
|
61848 | 6165 |
shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)" |
6166 |
using z |
|
6167 |
proof (induction n arbitrary: z) |
|
6168 |
case 0 then show ?case by simp |
|
6169 |
next |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6170 |
case (Suc n z) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
6171 |
have holo0: "f holomorphic_on (*) u ` S" |
61848 | 6172 |
by (meson fg f holomorphic_on_subset image_subset_iff) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
6173 |
have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S" |
68359 | 6174 |
by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T) |
6175 |
have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S" |
|
6176 |
by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros) |
|
6177 |
have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S" |
|
61848 | 6178 |
apply (rule holomorphic_on_compose [where g=f, unfolded o_def]) |
6179 |
apply (rule holo0 holomorphic_intros)+ |
|
6180 |
done |
|
6181 |
have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z" |
|
68359 | 6182 |
apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems]) |
6183 |
apply (rule holomorphic_higher_deriv [OF holo1 S]) |
|
61848 | 6184 |
apply (simp add: Suc.IH) |
6185 |
done |
|
68339 | 6186 |
also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
6187 |
apply (rule deriv_cmult) |
61848 | 6188 |
apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems]) |
68359 | 6189 |
apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def]) |
68255
009f783d1bac
small clean-up of Complex_Analysis_Basics
paulson <lp15@cam.ac.uk>
parents:
68239
diff
changeset
|
6190 |
apply (simp add: analytic_on_linear) |
68359 | 6191 |
apply (simp add: analytic_on_open f holomorphic_higher_deriv T) |
61848 | 6192 |
apply (blast intro: fg) |
6193 |
done |
|
68339 | 6194 |
also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)" |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68721
diff
changeset
|
6195 |
apply (subst complex_derivative_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def]) |
61848 | 6196 |
apply (rule derivative_intros) |
68359 | 6197 |
using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast |
61848 | 6198 |
apply (simp add: deriv_linear) |
6199 |
done |
|
6200 |
finally show ?case |
|
6201 |
by simp |
|
6202 |
qed |
|
6203 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6204 |
lemma higher_deriv_add_at: |
61848 | 6205 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
6206 |
shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z" |
|
6207 |
proof - |
|
6208 |
have "f analytic_on {z} \<and> g analytic_on {z}" |
|
6209 |
using assms by blast |
|
6210 |
with higher_deriv_add show ?thesis |
|
6211 |
by (auto simp: analytic_at_two) |
|
6212 |
qed |
|
6213 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6214 |
lemma higher_deriv_diff_at: |
61848 | 6215 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
6216 |
shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z" |
|
6217 |
proof - |
|
6218 |
have "f analytic_on {z} \<and> g analytic_on {z}" |
|
6219 |
using assms by blast |
|
6220 |
with higher_deriv_diff show ?thesis |
|
6221 |
by (auto simp: analytic_at_two) |
|
6222 |
qed |
|
6223 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6224 |
lemma higher_deriv_uminus_at: |
61848 | 6225 |
"f analytic_on {z} \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)" |
6226 |
using higher_deriv_uminus |
|
6227 |
by (auto simp: analytic_at) |
|
6228 |
||
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6229 |
lemma higher_deriv_mult_at: |
61848 | 6230 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6231 |
shows "(deriv ^^ n) (\<lambda>w. f w * g w) z = |
61848 | 6232 |
(\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)" |
6233 |
proof - |
|
6234 |
have "f analytic_on {z} \<and> g analytic_on {z}" |
|
6235 |
using assms by blast |
|
6236 |
with higher_deriv_mult show ?thesis |
|
6237 |
by (auto simp: analytic_at_two) |
|
6238 |
qed |
|
6239 |
||
6240 |
||
6241 |
text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close> |
|
6242 |
||
6243 |
proposition no_isolated_singularity: |
|
6244 |
fixes z::complex |
|
68359 | 6245 |
assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K" |
6246 |
shows "f holomorphic_on S" |
|
61848 | 6247 |
proof - |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6248 |
{ fix z |
68371 | 6249 |
assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
6250 |
have "f field_differentiable at z" |
68359 | 6251 |
proof (cases "z \<in> K") |
6252 |
case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>) |
|
61848 | 6253 |
next |
6254 |
case True |
|
68359 | 6255 |
with finite_set_avoid [OF K, of z] |
6256 |
obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x" |
|
61848 | 6257 |
by blast |
68359 | 6258 |
obtain e where "e>0" and e: "ball z e \<subseteq> S" |
6259 |
using S \<open>z \<in> S\<close> by (force simp: open_contains_ball) |
|
61848 | 6260 |
have fde: "continuous_on (ball z (min d e)) f" |
6261 |
by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI) |
|
68371 | 6262 |
have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c |
6263 |
by (simp add: hull_minimal continuous_on_subset [OF fde]) |
|
6264 |
have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk> |
|
6265 |
\<Longrightarrow> f field_differentiable at x" for a b c x |
|
6266 |
by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull) |
|
68310 | 6267 |
obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))" |
68493 | 6268 |
apply (rule contour_integral_convex_primitive |
68371 | 6269 |
[OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]]) |
6270 |
using cont fd by auto |
|
61848 | 6271 |
then have "f holomorphic_on ball z (min d e)" |
6272 |
by (metis open_ball at_within_open derivative_is_holomorphic) |
|
6273 |
then show ?thesis |
|
6274 |
unfolding holomorphic_on_def |
|
6275 |
by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj) |
|
6276 |
qed |
|
6277 |
} |
|
68359 | 6278 |
with holf S K show ?thesis |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
6279 |
by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric]) |
61848 | 6280 |
qed |
6281 |
||
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6282 |
lemma no_isolated_singularity': |
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6283 |
fixes z::complex |
68493 | 6284 |
assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)" |
68359 | 6285 |
and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K" |
6286 |
shows "f holomorphic_on S" |
|
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6287 |
proof (rule no_isolated_singularity[OF _ assms(2-)]) |
68359 | 6288 |
show "continuous_on S f" unfolding continuous_on_def |
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6289 |
proof |
68359 | 6290 |
fix z assume z: "z \<in> S" |
6291 |
show "(f \<longlongrightarrow> f z) (at z within S)" |
|
6292 |
proof (cases "z \<in> K") |
|
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6293 |
case False |
68493 | 6294 |
from holf have "continuous_on (S - K) f" |
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6295 |
by (rule holomorphic_on_imp_continuous_on) |
68493 | 6296 |
with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))" |
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6297 |
by (simp add: continuous_on_def) |
68359 | 6298 |
also from z K S False have "at z within (S - K) = at z within S" |
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6299 |
by (subst (1 2) at_within_open) (auto intro: finite_imp_closed) |
68359 | 6300 |
finally show "(f \<longlongrightarrow> f z) (at z within S)" . |
66286
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6301 |
qed (insert assms z, simp_all) |
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6302 |
qed |
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6303 |
qed |
1c977b13414f
poles and residues of the Gamma function
eberlm <eberlm@in.tum.de>
parents:
66193
diff
changeset
|
6304 |
|
61848 | 6305 |
proposition Cauchy_integral_formula_convex: |
68371 | 6306 |
assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f" |
6307 |
and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)" |
|
6308 |
and z: "z \<in> interior S" and vpg: "valid_path \<gamma>" |
|
6309 |
and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
|
6310 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
|
6311 |
proof - |
|
6312 |
have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x" |
|
6313 |
unfolding holomorphic_on_open [symmetric] field_differentiable_def |
|
6314 |
using no_isolated_singularity [where S = "interior S"] |
|
68493 | 6315 |
by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd |
68371 | 6316 |
field_differentiable_at_within field_differentiable_def holomorphic_onI |
6317 |
holomorphic_on_imp_differentiable_at open_interior) |
|
6318 |
show ?thesis |
|
6319 |
by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto) |
|
6320 |
qed |
|
61848 | 6321 |
|
6322 |
text\<open> Formula for higher derivatives.\<close> |
|
6323 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6324 |
lemma Cauchy_has_contour_integral_higher_derivative_circlepath: |
61848 | 6325 |
assumes contf: "continuous_on (cball z r) f" |
6326 |
and holf: "f holomorphic_on ball z r" |
|
6327 |
and w: "w \<in> ball z r" |
|
63589 | 6328 |
shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w)) |
61848 | 6329 |
(circlepath z r)" |
6330 |
using w |
|
6331 |
proof (induction k arbitrary: w) |
|
6332 |
case 0 then show ?case |
|
6333 |
using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm) |
|
6334 |
next |
|
6335 |
case (Suc k) |
|
6336 |
have [simp]: "r > 0" using w |
|
6337 |
using ball_eq_empty by fastforce |
|
6338 |
have f: "continuous_on (path_image (circlepath z r)) f" |
|
68339 | 6339 |
by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le) |
61848 | 6340 |
obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)" |
6341 |
using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems] |
|
6342 |
by (auto simp: contour_integrable_on_def) |
|
6343 |
then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X" |
|
6344 |
by (rule contour_integral_unique) |
|
6345 |
have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)" |
|
6346 |
using Suc.prems assms has_field_derivative_higher_deriv by auto |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
6347 |
then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)" |
68339 | 6348 |
by (force simp: field_differentiable_def) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6349 |
have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w = |
61848 | 6350 |
of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6351 |
by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems]) |
68339 | 6352 |
also have "\<dots> = of_nat (Suc k) * X" |
61848 | 6353 |
by (simp only: con) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6354 |
finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" . |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6355 |
then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
6356 |
by (metis deriv_cmult dnf_diff) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6357 |
then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))" |
61848 | 6358 |
by (simp add: field_simps) |
6359 |
then show ?case |
|
6360 |
using of_nat_eq_0_iff X by fastforce |
|
6361 |
qed |
|
6362 |
||
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6363 |
lemma Cauchy_higher_derivative_integral_circlepath: |
61848 | 6364 |
assumes contf: "continuous_on (cball z r) f" |
6365 |
and holf: "f holomorphic_on ball z r" |
|
6366 |
and w: "w \<in> ball z r" |
|
6367 |
shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)" |
|
6368 |
(is "?thes1") |
|
63589 | 6369 |
and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))" |
61848 | 6370 |
(is "?thes2") |
6371 |
proof - |
|
6372 |
have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) |
|
6373 |
(circlepath z r)" |
|
6374 |
using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms] |
|
6375 |
by simp |
|
6376 |
show ?thes1 using * |
|
6377 |
using contour_integrable_on_def by blast |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6378 |
show ?thes2 |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6379 |
unfolding contour_integral_unique [OF *] by (simp add: field_split_simps) |
61848 | 6380 |
qed |
6381 |
||
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6382 |
corollary Cauchy_contour_integral_circlepath: |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6383 |
assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r" |
63589 | 6384 |
shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6385 |
by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6386 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6387 |
lemma Cauchy_contour_integral_circlepath_2: |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6388 |
assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r" |
63589 | 6389 |
shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6390 |
using Cauchy_contour_integral_circlepath [OF assms, of 1] |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6391 |
by (simp add: power2_eq_square) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6392 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6393 |
|
67968 | 6394 |
subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close> |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6395 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6396 |
theorem holomorphic_power_series: |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6397 |
assumes holf: "f holomorphic_on ball z r" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6398 |
and w: "w \<in> ball z r" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6399 |
shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6400 |
proof - |
69597 | 6401 |
\<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close> |
68371 | 6402 |
obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r" |
68493 | 6403 |
proof |
68371 | 6404 |
have "cball z ((r + dist w z) / 2) \<subseteq> ball z r" |
68527
2f4e2aab190a
Generalising and renaming some basic results
paulson <lp15@cam.ac.uk>
parents:
68493
diff
changeset
|
6405 |
using w by (simp add: dist_commute field_sum_of_halves subset_eq) |
68371 | 6406 |
then show "f holomorphic_on cball z ((r + dist w z) / 2)" |
6407 |
by (rule holomorphic_on_subset [OF holf]) |
|
6408 |
have "r > 0" |
|
6409 |
using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero) |
|
6410 |
then show "0 < (r + dist w z) / 2" |
|
6411 |
by simp (use zero_le_dist [of w z] in linarith) |
|
6412 |
qed (use w in \<open>auto simp: dist_commute\<close>) |
|
68493 | 6413 |
then have holf: "f holomorphic_on ball z r" |
68371 | 6414 |
using ball_subset_cball holomorphic_on_subset by blast |
6415 |
have contf: "continuous_on (cball z r) f" |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6416 |
by (simp add: holfc holomorphic_on_imp_continuous_on) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6417 |
have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r" |
68371 | 6418 |
by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6419 |
obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6420 |
by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI) |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6421 |
obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6422 |
and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)" |
68493 | 6423 |
proof |
68371 | 6424 |
show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)" |
6425 |
by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq) |
|
6426 |
qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>) |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6427 |
have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially" |
68493 | 6428 |
unfolding uniform_limit_iff dist_norm |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6429 |
proof clarify |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6430 |
fix e::real |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6431 |
assume "0 < e" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6432 |
have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using k by auto |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6433 |
obtain n where n: "((r - k) / r) ^ n < e / B * k" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6434 |
using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6435 |
have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6436 |
if "n \<le> N" and r: "r = dist z u" for N u |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6437 |
proof - |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6438 |
have N: "((r - k) / r) ^ N < e / B * k" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6439 |
apply (rule le_less_trans [OF power_decreasing n]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6440 |
using \<open>n \<le> N\<close> k by auto |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6441 |
have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6442 |
using \<open>0 < r\<close> r w by auto |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6443 |
have wzu_not1: "(w - z) / (u - z) \<noteq> 1" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6444 |
by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6445 |
have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6446 |
= norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)" |
64267 | 6447 |
unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps) |
68339 | 6448 |
also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6449 |
using \<open>0 < B\<close> |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6450 |
apply (auto simp: geometric_sum [OF wzu_not1]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6451 |
apply (simp add: field_simps norm_mult [symmetric]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6452 |
done |
68339 | 6453 |
also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6454 |
using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute) |
68339 | 6455 |
also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6456 |
by (simp add: algebra_simps) |
68339 | 6457 |
also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6458 |
by (simp add: norm_mult norm_power norm_minus_commute) |
68339 | 6459 |
also have "\<dots> \<le> (((r - k)/r)^N) * B" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6460 |
using \<open>0 < r\<close> w k |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6461 |
apply (simp add: divide_simps) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6462 |
apply (rule mult_mono [OF power_mono]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6463 |
apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6464 |
done |
68339 | 6465 |
also have "\<dots> < e * k" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6466 |
using \<open>0 < B\<close> N by (simp add: divide_simps) |
68339 | 6467 |
also have "\<dots> \<le> e * norm (u - w)" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6468 |
using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6469 |
finally show ?thesis |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6470 |
by (simp add: field_split_simps norm_divide del: power_Suc) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6471 |
qed |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6472 |
with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r. |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6473 |
norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6474 |
by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6475 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6476 |
have eq: "\<forall>\<^sub>F x in sequentially. |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6477 |
contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) = |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6478 |
(\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6479 |
apply (rule eventuallyI) |
64267 | 6480 |
apply (subst contour_integral_sum, simp) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6481 |
using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6482 |
apply (simp only: contour_integral_lmul cint algebra_simps) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6483 |
done |
68371 | 6484 |
have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r" |
6485 |
apply (intro contour_integrable_sum contour_integrable_lmul, simp) |
|
6486 |
using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf]) |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6487 |
have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6488 |
sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6489 |
unfolding sums_def |
70532
fcf3b891ccb1
new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents:
70365
diff
changeset
|
6490 |
apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic) |
68371 | 6491 |
using \<open>0 < r\<close> apply auto |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6492 |
done |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6493 |
then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k) |
63589 | 6494 |
sums (2 * of_real pi * \<i> * f w)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6495 |
using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]]) |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6496 |
then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2))) |
63589 | 6497 |
sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6498 |
by (rule sums_divide) |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6499 |
then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2))) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6500 |
sums f w" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6501 |
by (simp add: field_simps) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6502 |
then show ?thesis |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6503 |
by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6504 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6505 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6506 |
|
67968 | 6507 |
subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close> |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6508 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6509 |
text\<open> These weak Liouville versions don't even need the derivative formula.\<close> |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6510 |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6511 |
lemma Liouville_weak_0: |
61973 | 6512 |
assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6513 |
shows "f z = 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6514 |
proof (rule ccontr) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6515 |
assume fz: "f z \<noteq> 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6516 |
with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"] |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6517 |
obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6518 |
by (auto simp: dist_norm) |
63040 | 6519 |
define R where "R = 1 + \<bar>B\<bar> + norm z" |
63262 | 6520 |
have "R > 0" unfolding R_def |
62626
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
6521 |
proof - |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
6522 |
have "0 \<le> cmod z + \<bar>B\<bar>" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
6523 |
by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
6524 |
then show "0 < 1 + \<bar>B\<bar> + cmod z" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
6525 |
by linarith |
63262 | 6526 |
qed |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6527 |
have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6528 |
apply (rule Cauchy_integral_circlepath) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6529 |
using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+ |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6530 |
done |
68371 | 6531 |
have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x |
6532 |
unfolding R_def |
|
6533 |
by (rule B) (use norm_triangle_ineq4 [of x z] in auto) |
|
6534 |
with \<open>R > 0\<close> fz show False |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6535 |
using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"] |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6536 |
by (auto simp: less_imp_le norm_mult norm_divide field_split_simps) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6537 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6538 |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6539 |
proposition Liouville_weak: |
61973 | 6540 |
assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6541 |
shows "f z = l" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6542 |
using Liouville_weak_0 [of "\<lambda>z. f z - l"] |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6543 |
by (simp add: assms holomorphic_on_const holomorphic_on_diff LIM_zero) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6544 |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6545 |
proposition Liouville_weak_inverse: |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6546 |
assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6547 |
obtains z where "f z = 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6548 |
proof - |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6549 |
{ assume f: "\<And>z. f z \<noteq> 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6550 |
have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6551 |
by (simp add: holomorphic_on_divide holomorphic_on_const assms f) |
61973 | 6552 |
have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6553 |
apply (rule tendstoI [OF eventually_mono]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6554 |
apply (rule_tac B="2/e" in unbounded) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6555 |
apply (simp add: dist_norm norm_divide field_split_simps mult_ac) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6556 |
done |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6557 |
have False |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6558 |
using Liouville_weak_0 [OF 1 2] f by simp |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6559 |
} |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6560 |
then show ?thesis |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6561 |
using that by blast |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6562 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6563 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6564 |
text\<open> In particular we get the Fundamental Theorem of Algebra.\<close> |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6565 |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6566 |
theorem fundamental_theorem_of_algebra: |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6567 |
fixes a :: "nat \<Rightarrow> complex" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6568 |
assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6569 |
obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6570 |
using assms |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6571 |
proof (elim disjE bexE) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6572 |
assume "a 0 = 0" then show ?thesis |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6573 |
by (auto simp: that [of 0]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6574 |
next |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6575 |
fix i |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6576 |
assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6577 |
have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6578 |
by (rule holomorphic_intros)+ |
68371 | 6579 |
show thesis |
68420 | 6580 |
proof (rule Liouville_weak_inverse [OF 1]) |
6581 |
show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B |
|
6582 |
using i polyfun_extremal nz by force |
|
6583 |
qed (use that in auto) |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6584 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6585 |
|
67968 | 6586 |
subsection\<open>Weierstrass convergence theorem\<close> |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6587 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6588 |
lemma holomorphic_uniform_limit: |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6589 |
assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6590 |
and ulim: "uniform_limit (cball z r) f g F" |
69508 | 6591 |
and F: "\<not> trivial_limit F" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6592 |
obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6593 |
proof (cases r "0::real" rule: linorder_cases) |
68339 | 6594 |
case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6595 |
next |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6596 |
case equal then show ?thesis |
68420 | 6597 |
by (force simp: holomorphic_on_def intro: that) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6598 |
next |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6599 |
case greater |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6600 |
have contg: "continuous_on (cball z r) g" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6601 |
using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast |
68420 | 6602 |
have "path_image (circlepath z r) \<subseteq> cball z r" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6603 |
using \<open>0 < r\<close> by auto |
68420 | 6604 |
then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)" |
6605 |
by (intro continuous_intros continuous_on_subset [OF contg]) |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6606 |
have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6607 |
if w: "w \<in> ball z r" for w |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6608 |
proof - |
63040 | 6609 |
define d where "d = (r - norm(w - z))" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6610 |
have "0 < d" "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6611 |
have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6612 |
unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6613 |
have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6614 |
apply (rule eventually_mono [OF cont]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6615 |
using w |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6616 |
apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6617 |
done |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6618 |
have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F" |
68493 | 6619 |
using greater \<open>0 < d\<close> |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6620 |
apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6621 |
apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]]) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6622 |
apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+ |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6623 |
done |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6624 |
have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6625 |
by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>]) |
61973 | 6626 |
have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6627 |
by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>]) |
63589 | 6628 |
have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F" |
68420 | 6629 |
proof (rule Lim_transform_eventually) |
68493 | 6630 |
show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w)) |
68420 | 6631 |
= 2 * of_real pi * \<i> * f x w" |
6632 |
apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]]) |
|
6633 |
using w\<open>0 < d\<close> d_def by auto |
|
6634 |
qed (auto simp: cif_tends_cig) |
|
6635 |
have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e" |
|
6636 |
by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto) |
|
6637 |
then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F" |
|
6638 |
by (rule tendsto_mult_left [OF tendstoI]) |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6639 |
then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6640 |
using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w |
70804
4eef7c6ef7bf
More theorems about limits, including cancellation simprules
paulson <lp15@cam.ac.uk>
parents:
70707
diff
changeset
|
6641 |
by fastforce |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6642 |
then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6643 |
using has_contour_integral_div [where c = "2 * of_real pi * \<i>"] |
68339 | 6644 |
by (force simp: field_simps) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6645 |
then show ?thesis |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6646 |
by (simp add: dist_norm) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6647 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6648 |
show ?thesis |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6649 |
using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified] |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6650 |
by (fastforce simp add: holomorphic_on_open contg intro: that) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6651 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6652 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6653 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6654 |
text\<open> Version showing that the limit is the limit of the derivatives.\<close> |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6655 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6656 |
proposition has_complex_derivative_uniform_limit: |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6657 |
fixes z::complex |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6658 |
assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6659 |
(\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6660 |
and ulim: "uniform_limit (cball z r) f g F" |
69508 | 6661 |
and F: "\<not> trivial_limit F" and "0 < r" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6662 |
obtains g' where |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6663 |
"continuous_on (cball z r) g" |
61973 | 6664 |
"\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6665 |
proof - |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6666 |
let ?conint = "contour_integral (circlepath z r)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6667 |
have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6668 |
by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F]; |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
6669 |
auto simp: holomorphic_on_open field_differentiable_def)+ |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6670 |
then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6671 |
using DERIV_deriv_iff_has_field_derivative |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6672 |
by (fastforce simp add: holomorphic_on_open) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6673 |
then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6674 |
by (simp add: DERIV_imp_deriv) |
61973 | 6675 |
have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6676 |
proof - |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6677 |
have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)" |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6678 |
if cont_fn: "continuous_on (cball z r) (f n)" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6679 |
and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6680 |
proof - |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6681 |
have hol_fn: "f n holomorphic_on ball z r" |
68339 | 6682 |
using fnd by (force simp: holomorphic_on_open) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6683 |
have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6684 |
by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w]) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6685 |
then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6686 |
using DERIV_unique [OF fnd] w by blast |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6687 |
show ?thesis |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6688 |
by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6689 |
qed |
63040 | 6690 |
define d where "d = (r - norm(w - z))^2" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6691 |
have "d > 0" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6692 |
using w by (simp add: dist_commute dist_norm d_def) |
68420 | 6693 |
have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y |
6694 |
proof - |
|
6695 |
have "w \<in> ball z (cmod (z - y))" |
|
6696 |
using that w by fastforce |
|
6697 |
then have "cmod (w - z) \<le> cmod (z - y)" |
|
6698 |
by (simp add: dist_complex_def norm_minus_commute) |
|
6699 |
moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)" |
|
6700 |
by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2) |
|
6701 |
ultimately show ?thesis |
|
6702 |
using that by (simp add: d_def norm_power power_mono) |
|
6703 |
qed |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6704 |
have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r" |
68339 | 6705 |
by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont]) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6706 |
have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6707 |
unfolding uniform_limit_iff |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6708 |
proof clarify |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6709 |
fix e::real |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6710 |
assume "0 < e" |
68420 | 6711 |
with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6712 |
apply (simp add: norm_divide field_split_simps sphere_def dist_norm) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6713 |
apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"]) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6714 |
apply (simp add: \<open>0 < d\<close>) |
68339 | 6715 |
apply (force simp: dist_norm dle intro: less_le_trans) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6716 |
done |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6717 |
qed |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61976
diff
changeset
|
6718 |
have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2)) |
61973 | 6719 |
\<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63589
diff
changeset
|
6720 |
by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>]) |
61973 | 6721 |
then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F" |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6722 |
using Lim_null by (force intro!: tendsto_mult_right_zero) |
61973 | 6723 |
have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F" |
70532
fcf3b891ccb1
new material; rotated premises of Lim_transform_eventually
paulson <lp15@cam.ac.uk>
parents:
70365
diff
changeset
|
6724 |
apply (rule Lim_transform_eventually [OF tendsto_0]) |
68339 | 6725 |
apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont]) |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6726 |
done |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6727 |
then show ?thesis using Lim_null by blast |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6728 |
qed |
61973 | 6729 |
obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" |
68339 | 6730 |
by (blast intro: tends_f'n_g' g') |
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6731 |
then show ?thesis using g |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6732 |
using that by blast |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6733 |
qed |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61848
diff
changeset
|
6734 |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6735 |
|
70136 | 6736 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6737 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6738 |
lemma holomorphic_uniform_sequence: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6739 |
assumes S: "open S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6740 |
and hol_fn: "\<And>n. (f n) holomorphic_on S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6741 |
and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6742 |
shows "g holomorphic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6743 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6744 |
have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6745 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6746 |
obtain r where "0 < r" and r: "cball z r \<subseteq> S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6747 |
and ul: "uniform_limit (cball z r) f g sequentially" |
68493 | 6748 |
using ulim_g [OF \<open>z \<in> S\<close>] by blast |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6749 |
have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r" |
68420 | 6750 |
proof (intro eventuallyI conjI) |
6751 |
show "continuous_on (cball z r) (f x)" for x |
|
6752 |
using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast |
|
6753 |
show "f x holomorphic_on ball z r" for x |
|
6754 |
by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r) |
|
6755 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6756 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6757 |
apply (rule holomorphic_uniform_limit [OF *]) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6758 |
using \<open>0 < r\<close> centre_in_ball ul |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6759 |
apply (auto simp: holomorphic_on_open) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6760 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6761 |
qed |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6762 |
with S show ?thesis |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6763 |
by (simp add: holomorphic_on_open) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6764 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6765 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6766 |
lemma has_complex_derivative_uniform_sequence: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6767 |
fixes S :: "complex set" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6768 |
assumes S: "open S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6769 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6770 |
and ulim_g: "\<And>x. x \<in> S |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6771 |
\<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6772 |
shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6773 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6774 |
have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6775 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6776 |
obtain r where "0 < r" and r: "cball z r \<subseteq> S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6777 |
and ul: "uniform_limit (cball z r) f g sequentially" |
68493 | 6778 |
using ulim_g [OF \<open>z \<in> S\<close>] by blast |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6779 |
have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6780 |
(\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))" |
68420 | 6781 |
proof (intro eventuallyI conjI ballI) |
6782 |
show "continuous_on (cball z r) (f x)" for x |
|
6783 |
by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r) |
|
6784 |
show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x |
|
6785 |
using ball_subset_cball hfd r by blast |
|
6786 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6787 |
show ?thesis |
68420 | 6788 |
by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6789 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6790 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6791 |
by (rule bchoice) (blast intro: y) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6792 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6793 |
|
67968 | 6794 |
subsection\<open>On analytic functions defined by a series\<close> |
68493 | 6795 |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6796 |
lemma series_and_derivative_comparison: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6797 |
fixes S :: "complex set" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6798 |
assumes S: "open S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6799 |
and h: "summable h" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6800 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6801 |
and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6802 |
obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6803 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6804 |
obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially" |
69529 | 6805 |
using Weierstrass_m_test_ev [OF to_g h] by force |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6806 |
have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6807 |
if "x \<in> S" for x |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6808 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6809 |
obtain d where "d>0" and d: "cball x d \<subseteq> S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6810 |
using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast |
68420 | 6811 |
show ?thesis |
6812 |
proof (intro conjI exI) |
|
6813 |
show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially" |
|
6814 |
using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially) |
|
6815 |
qed (use \<open>d > 0\<close> d in auto) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6816 |
qed |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6817 |
have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6818 |
by (metis tendsto_uniform_limitI [OF g]) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6819 |
moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6820 |
by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+ |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6821 |
ultimately show ?thesis |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6822 |
by (metis sums_def that) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6823 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6824 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6825 |
text\<open>A version where we only have local uniform/comparative convergence.\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6826 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6827 |
lemma series_and_derivative_comparison_local: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6828 |
fixes S :: "complex set" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6829 |
assumes S: "open S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6830 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6831 |
and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6832 |
shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6833 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6834 |
have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6835 |
if "z \<in> S" for z |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6836 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6837 |
obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6838 |
using to_g \<open>z \<in> S\<close> by meson |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6839 |
then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6840 |
by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6841 |
have 1: "open (ball z d \<inter> S)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6842 |
by (simp add: open_Int S) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6843 |
have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6844 |
by (auto simp: hfd) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6845 |
obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6846 |
((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6847 |
by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6848 |
then have "(\<lambda>n. f' n z) sums g' z" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6849 |
by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6850 |
moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6851 |
using summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6852 |
by (metis (full_types) Int_iff gg' summable_def that) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6853 |
moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)" |
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
6854 |
proof (rule has_field_derivative_transform_within) |
68420 | 6855 |
show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)" |
6856 |
by (metis subsetD dist_commute gg' mem_ball r sums_unique) |
|
6857 |
qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6858 |
ultimately show ?thesis by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6859 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6860 |
then show ?thesis |
68420 | 6861 |
by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6862 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6863 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6864 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6865 |
text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6866 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6867 |
lemma series_and_derivative_comparison_complex: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6868 |
fixes S :: "complex set" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6869 |
assumes S: "open S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6870 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6871 |
and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6872 |
shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6873 |
apply (rule series_and_derivative_comparison_local [OF S hfd], assumption) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6874 |
apply (rule ex_forward [OF to_g], assumption) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6875 |
apply (erule exE) |
68339 | 6876 |
apply (rule_tac x="Re \<circ> h" in exI) |
6877 |
apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6878 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6879 |
|
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6880 |
text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close> |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6881 |
lemma series_differentiable_comparison_complex: |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6882 |
fixes S :: "complex set" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6883 |
assumes S: "open S" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6884 |
and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6885 |
and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6886 |
obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6887 |
proof - |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6888 |
have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6889 |
using hfd field_differentiable_derivI by blast |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6890 |
have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)" |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6891 |
by (metis series_and_derivative_comparison_complex [OF S hfd' to_g]) |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6892 |
then show ?thesis |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6893 |
using field_differentiable_def that by blast |
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65037
diff
changeset
|
6894 |
qed |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6895 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6896 |
text\<open>In particular, a power series is analytic inside circle of convergence.\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6897 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6898 |
lemma power_series_and_derivative_0: |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6899 |
fixes a :: "nat \<Rightarrow> complex" and r::real |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6900 |
assumes "summable (\<lambda>n. a n * r^n)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6901 |
shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6902 |
((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6903 |
proof (cases "0 < r") |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6904 |
case True |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6905 |
have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6906 |
by (rule derivative_eq_intros | simp)+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6907 |
have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6908 |
using \<open>r > 0\<close> |
68403 | 6909 |
apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6910 |
using norm_triangle_ineq2 [of y z] |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6911 |
apply (simp only: diff_le_eq norm_minus_commute mult_2) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6912 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6913 |
have "summable (\<lambda>n. a n * complex_of_real r ^ n)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6914 |
using assms \<open>r > 0\<close> by simp |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6915 |
moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6916 |
using \<open>r > 0\<close> |
68403 | 6917 |
by (simp flip: of_real_add) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6918 |
ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6919 |
by (rule power_series_conv_imp_absconv_weak) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6920 |
have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n. (a n) * z ^ n) sums g z \<and> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6921 |
(\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6922 |
apply (rule series_and_derivative_comparison_complex [OF open_ball der]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6923 |
apply (rule_tac x="(r - norm z)/2" in exI) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6924 |
apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6925 |
using \<open>r > 0\<close> |
68420 | 6926 |
apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6927 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6928 |
then show ?thesis |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
6929 |
by (simp add: ball_def) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6930 |
next |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6931 |
case False then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6932 |
apply (simp add: not_less) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6933 |
using less_le_trans norm_not_less_zero by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6934 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6935 |
|
70136 | 6936 |
proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative: |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6937 |
fixes a :: "nat \<Rightarrow> complex" and r::real |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6938 |
assumes "summable (\<lambda>n. a n * r^n)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6939 |
obtains g g' where "\<forall>z \<in> ball w r. |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6940 |
((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6941 |
(g has_field_derivative g' z) (at z)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6942 |
using power_series_and_derivative_0 [OF assms] |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6943 |
apply clarify |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6944 |
apply (rule_tac g="(\<lambda>z. g(z - w))" in that) |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
6945 |
using DERIV_shift [where z="-w"] |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6946 |
apply (auto simp: norm_minus_commute Ball_def dist_norm) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6947 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6948 |
|
70136 | 6949 |
proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic: |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6950 |
assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6951 |
shows "f holomorphic_on ball z r" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6952 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6953 |
have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6954 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6955 |
have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6956 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6957 |
have wz: "cmod (w - z) < r" using w |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
6958 |
by (auto simp: field_split_simps dist_norm norm_minus_commute) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6959 |
then have "0 \<le> r" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6960 |
by (meson less_eq_real_def norm_ge_zero order_trans) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6961 |
show ?thesis |
68403 | 6962 |
using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6963 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6964 |
have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))" |
68339 | 6965 |
using assms [OF inb] by (force simp: summable_def dist_norm) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6966 |
obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6967 |
(\<lambda>n. a n * (u - z) ^ n) sums g u \<and> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6968 |
(\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6969 |
by (rule power_series_and_derivative [OF sum, of z]) fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6970 |
have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6971 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6972 |
have less: "cmod (z - u) * 2 < cmod (z - w) + r" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6973 |
using that dist_triangle2 [of z u w] |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6974 |
by (simp add: dist_norm [symmetric] algebra_simps) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6975 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6976 |
apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6977 |
using gg' [of u] less w |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6978 |
apply (auto simp: assms dist_norm) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6979 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6980 |
qed |
68420 | 6981 |
have "(f has_field_derivative g' w) (at w)" |
71029
934e0044e94b
Moved or deleted some out of place material, also eliminating obsolete naming conventions
paulson <lp15@cam.ac.uk>
parents:
70817
diff
changeset
|
6982 |
by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"]) |
68420 | 6983 |
(use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>) |
6984 |
then show ?thesis .. |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6985 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6986 |
then show ?thesis by (simp add: holomorphic_on_open) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6987 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6988 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6989 |
corollary holomorphic_iff_power_series: |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6990 |
"f holomorphic_on ball z r \<longleftrightarrow> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6991 |
(\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)" |
68420 | 6992 |
apply (intro iffI ballI holomorphic_power_series, assumption+) |
6993 |
apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"]) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6994 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6995 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
6996 |
lemma power_series_analytic: |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6997 |
"(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r" |
68339 | 6998 |
by (force simp: analytic_on_open intro!: power_series_holomorphic) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
6999 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
7000 |
lemma analytic_iff_power_series: |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7001 |
"f analytic_on ball z r \<longleftrightarrow> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7002 |
(\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7003 |
by (simp add: analytic_on_open holomorphic_iff_power_series) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7004 |
|
70136 | 7005 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7006 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7007 |
lemma holomorphic_fun_eq_on_ball: |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7008 |
"\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r; |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7009 |
w \<in> ball z r; |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7010 |
\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7011 |
\<Longrightarrow> f w = g w" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7012 |
apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7013 |
apply (auto simp: holomorphic_iff_power_series) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7014 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7015 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7016 |
lemma holomorphic_fun_eq_0_on_ball: |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7017 |
"\<lbrakk>f holomorphic_on ball z r; w \<in> ball z r; |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7018 |
\<And>n. (deriv ^^ n) f z = 0\<rbrakk> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7019 |
\<Longrightarrow> f w = 0" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7020 |
apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7021 |
apply (auto simp: holomorphic_iff_power_series) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7022 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7023 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7024 |
lemma holomorphic_fun_eq_0_on_connected: |
68420 | 7025 |
assumes holf: "f holomorphic_on S" and "open S" |
7026 |
and cons: "connected S" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7027 |
and der: "\<And>n. (deriv ^^ n) f z = 0" |
68420 | 7028 |
and "z \<in> S" "w \<in> S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7029 |
shows "f w = 0" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7030 |
proof - |
68420 | 7031 |
have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
7032 |
if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e |
|
7033 |
proof - |
|
7034 |
have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0" |
|
7035 |
apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv]) |
|
7036 |
apply (rule holomorphic_on_subset [OF holf]) |
|
7037 |
using that apply simp_all |
|
7038 |
by (metis funpow_add o_apply) |
|
7039 |
with that show ?thesis by auto |
|
7040 |
qed |
|
69922
4a9167f377b0
new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
7041 |
have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7042 |
apply (rule open_subset, force) |
68420 | 7043 |
using \<open>open S\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7044 |
apply (simp add: open_contains_ball Ball_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7045 |
apply (erule all_forward) |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62217
diff
changeset
|
7046 |
using "*" by auto blast+ |
69922
4a9167f377b0
new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
7047 |
have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7048 |
using assms |
62843
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62837
diff
changeset
|
7049 |
by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv) |
68420 | 7050 |
obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] . |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7051 |
then have holfb: "f holomorphic_on ball w e" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7052 |
using holf holomorphic_on_subset by blast |
68420 | 7053 |
have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7054 |
using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7055 |
show ?thesis |
68420 | 7056 |
using cons der \<open>z \<in> S\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7057 |
apply (simp add: connected_clopen) |
68420 | 7058 |
apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7059 |
apply (auto simp: 1 2 3) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7060 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7061 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7062 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7063 |
lemma holomorphic_fun_eq_on_connected: |
68420 | 7064 |
assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7065 |
and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z" |
68420 | 7066 |
and "z \<in> S" "w \<in> S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7067 |
shows "f w = g w" |
68420 | 7068 |
proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified]) |
7069 |
show "(\<lambda>x. f x - g x) holomorphic_on S" |
|
7070 |
by (intro assms holomorphic_intros) |
|
7071 |
show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0" |
|
7072 |
using assms higher_deriv_diff by auto |
|
7073 |
qed (use assms in auto) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7074 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7075 |
lemma holomorphic_fun_eq_const_on_connected: |
68420 | 7076 |
assumes holf: "f holomorphic_on S" and "open S" |
7077 |
and cons: "connected S" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7078 |
and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0" |
68420 | 7079 |
and "z \<in> S" "w \<in> S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7080 |
shows "f w = f z" |
68420 | 7081 |
proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified]) |
7082 |
show "(\<lambda>w. f w - f z) holomorphic_on S" |
|
7083 |
by (intro assms holomorphic_intros) |
|
7084 |
show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0" |
|
7085 |
by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>) |
|
7086 |
qed (use assms in auto) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7087 |
|
70136 | 7088 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7089 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7090 |
lemma pole_lemma: |
68420 | 7091 |
assumes holf: "f holomorphic_on S" and a: "a \<in> interior S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7092 |
shows "(\<lambda>z. if z = a then deriv f a |
68420 | 7093 |
else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S") |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7094 |
proof - |
68420 | 7095 |
have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7096 |
proof - |
68420 | 7097 |
have fcd: "f field_differentiable at u within S" |
7098 |
using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>) |
|
7099 |
have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7100 |
by (rule fcd derivative_intros | simp add: that)+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7101 |
have "0 < dist a u" using that dist_nz by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7102 |
then show ?thesis |
68420 | 7103 |
by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7104 |
qed |
68420 | 7105 |
have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7106 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7107 |
have holfb: "f holomorphic_on ball a e" |
68420 | 7108 |
by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>]) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7109 |
have 2: "?F holomorphic_on ball a e - {a}" |
68420 | 7110 |
apply (simp add: holomorphic_on_def flip: field_differentiable_def) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7111 |
using mem_ball that |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7112 |
apply (auto intro: F1 field_differentiable_within_subset) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7113 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7114 |
have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7115 |
if "dist a x < e" for x |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7116 |
proof (cases "x=a") |
68493 | 7117 |
case True |
68420 | 7118 |
then have "f field_differentiable at a" |
7119 |
using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto |
|
7120 |
with True show ?thesis |
|
68634 | 7121 |
by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7122 |
elim: rev_iffD1 [OF _ LIM_equal]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7123 |
next |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7124 |
case False with 2 that show ?thesis |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7125 |
by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7126 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7127 |
then have 1: "continuous_on (ball a e) ?F" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7128 |
by (clarsimp simp: continuous_on_eq_continuous_at) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7129 |
have "?F holomorphic_on ball a e" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7130 |
by (auto intro: no_isolated_singularity [OF 1 2]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7131 |
with that show ?thesis |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7132 |
by (simp add: holomorphic_on_open field_differentiable_def [symmetric] |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7133 |
field_differentiable_at_within) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7134 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7135 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7136 |
proof |
68420 | 7137 |
fix x assume "x \<in> S" show "?F field_differentiable at x within S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7138 |
proof (cases "x=a") |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7139 |
case True then show ?thesis |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7140 |
using a by (auto simp: mem_interior intro: field_differentiable_at_within F2) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7141 |
next |
68420 | 7142 |
case False with F1 \<open>x \<in> S\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7143 |
show ?thesis by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7144 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7145 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7146 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7147 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
7148 |
lemma pole_theorem: |
68420 | 7149 |
assumes holg: "g holomorphic_on S" and a: "a \<in> interior S" |
7150 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7151 |
shows "(\<lambda>z. if z = a then deriv g a |
68420 | 7152 |
else f z - g a/(z - a)) holomorphic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7153 |
using pole_lemma [OF holg a] |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
7154 |
by (rule holomorphic_transform) (simp add: eq field_split_simps) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7155 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7156 |
lemma pole_lemma_open: |
68420 | 7157 |
assumes "f holomorphic_on S" "open S" |
7158 |
shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S" |
|
7159 |
proof (cases "a \<in> S") |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7160 |
case True with assms interior_eq pole_lemma |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7161 |
show ?thesis by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7162 |
next |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7163 |
case False with assms show ?thesis |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7164 |
apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify) |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7165 |
apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1]) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7166 |
apply (rule derivative_intros | force)+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7167 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7168 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7169 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
7170 |
lemma pole_theorem_open: |
68420 | 7171 |
assumes holg: "g holomorphic_on S" and S: "open S" |
7172 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7173 |
shows "(\<lambda>z. if z = a then deriv g a |
68420 | 7174 |
else f z - g a/(z - a)) holomorphic_on S" |
7175 |
using pole_lemma_open [OF holg S] |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7176 |
by (rule holomorphic_transform) (auto simp: eq divide_simps) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7177 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
7178 |
lemma pole_theorem_0: |
68420 | 7179 |
assumes holg: "g holomorphic_on S" and a: "a \<in> interior S" |
7180 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7181 |
and [simp]: "f a = deriv g a" "g a = 0" |
68420 | 7182 |
shows "f holomorphic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7183 |
using pole_theorem [OF holg a eq] |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
7184 |
by (rule holomorphic_transform) (auto simp: eq field_split_simps) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7185 |
|
69423
3922aa1df44e
Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
Wenda Li <wl302@cam.ac.uk>
parents:
69064
diff
changeset
|
7186 |
lemma pole_theorem_open_0: |
68420 | 7187 |
assumes holg: "g holomorphic_on S" and S: "open S" |
7188 |
and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7189 |
and [simp]: "f a = deriv g a" "g a = 0" |
68420 | 7190 |
shows "f holomorphic_on S" |
7191 |
using pole_theorem_open [OF holg S eq] |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
7192 |
by (rule holomorphic_transform) (auto simp: eq field_split_simps) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7193 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7194 |
lemma pole_theorem_analytic: |
68420 | 7195 |
assumes g: "g analytic_on S" |
7196 |
and eq: "\<And>z. z \<in> S |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7197 |
\<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)" |
68420 | 7198 |
shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S") |
7199 |
unfolding analytic_on_def |
|
68493 | 7200 |
proof |
68420 | 7201 |
fix x |
7202 |
assume "x \<in> S" |
|
68493 | 7203 |
with g obtain e where "0 < e" and e: "g holomorphic_on ball x e" |
68420 | 7204 |
by (auto simp add: analytic_on_def) |
7205 |
obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w" |
|
7206 |
using \<open>x \<in> S\<close> eq by blast |
|
7207 |
have "?F holomorphic_on ball x (min d e)" |
|
7208 |
using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open) |
|
7209 |
then show "\<exists>e>0. ?F holomorphic_on ball x e" |
|
7210 |
using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce |
|
7211 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7212 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7213 |
lemma pole_theorem_analytic_0: |
68420 | 7214 |
assumes g: "g analytic_on S" |
7215 |
and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7216 |
and [simp]: "f a = deriv g a" "g a = 0" |
68420 | 7217 |
shows "f analytic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7218 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7219 |
have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7220 |
by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7221 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7222 |
using pole_theorem_analytic [OF g eq] by simp |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7223 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7224 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7225 |
lemma pole_theorem_analytic_open_superset: |
68420 | 7226 |
assumes g: "g analytic_on S" and "S \<subseteq> T" "open T" |
7227 |
and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7228 |
shows "(\<lambda>z. if z = a then deriv g a |
68420 | 7229 |
else f z - g a/(z - a)) analytic_on S" |
7230 |
proof (rule pole_theorem_analytic [OF g]) |
|
7231 |
fix z |
|
7232 |
assume "z \<in> S" |
|
7233 |
then obtain e where "0 < e" and e: "ball z e \<subseteq> T" |
|
7234 |
using assms openE by blast |
|
7235 |
then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w" |
|
7236 |
using eq by auto |
|
7237 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7238 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7239 |
lemma pole_theorem_analytic_open_superset_0: |
68420 | 7240 |
assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7241 |
and [simp]: "f a = deriv g a" "g a = 0" |
68420 | 7242 |
shows "f analytic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7243 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7244 |
have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7245 |
by auto |
68420 | 7246 |
have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7247 |
by (rule pole_theorem_analytic_open_superset [OF g]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7248 |
then show ?thesis by simp |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7249 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7250 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7251 |
|
67968 | 7252 |
subsection\<open>General, homology form of Cauchy's theorem\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7253 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7254 |
text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7255 |
|
62217 | 7256 |
lemma contour_integral_continuous_on_linepath_2D: |
68420 | 7257 |
assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)" |
7258 |
and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)" |
|
7259 |
and abu: "closed_segment a b \<subseteq> U" |
|
7260 |
shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))" |
|
62217 | 7261 |
proof - |
68420 | 7262 |
have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow> |
62217 | 7263 |
dist (contour_integral (linepath a b) (F x')) |
7264 |
(contour_integral (linepath a b) (F w)) \<le> \<epsilon>" |
|
68420 | 7265 |
if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon> |
62217 | 7266 |
proof - |
68420 | 7267 |
obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force |
7268 |
let ?TZ = "cball w \<delta> \<times> closed_segment a b" |
|
62217 | 7269 |
have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)" |
68420 | 7270 |
proof (rule compact_uniformly_continuous) |
7271 |
show "continuous_on ?TZ (\<lambda>(x,y). F x y)" |
|
7272 |
by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast) |
|
7273 |
show "compact ?TZ" |
|
7274 |
by (simp add: compact_Times) |
|
7275 |
qed |
|
62217 | 7276 |
then obtain \<eta> where "\<eta>>0" |
7277 |
and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow> |
|
7278 |
dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)" |
|
7279 |
apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"]) |
|
7280 |
using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
7281 |
have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b; |
62217 | 7282 |
norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk> |
7283 |
\<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)" |
|
7284 |
for x1 x2 x1' x2' |
|
68339 | 7285 |
using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm) |
62217 | 7286 |
have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>" |
68420 | 7287 |
if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>" for x' |
62217 | 7288 |
proof - |
68420 | 7289 |
have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b" |
7290 |
by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that) |
|
7291 |
then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)" |
|
62217 | 7292 |
apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>]) |
68420 | 7293 |
using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute) |
62217 | 7294 |
done |
68339 | 7295 |
also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp |
62217 | 7296 |
finally show ?thesis . |
7297 |
qed |
|
7298 |
show ?thesis |
|
7299 |
apply (rule_tac x="min \<delta> \<eta>" in exI) |
|
7300 |
using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close> |
|
68420 | 7301 |
apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee) |
62217 | 7302 |
done |
7303 |
qed |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
7304 |
show ?thesis |
68420 | 7305 |
proof (cases "a=b") |
7306 |
case True |
|
7307 |
then show ?thesis by simp |
|
7308 |
next |
|
7309 |
case False |
|
7310 |
show ?thesis |
|
7311 |
by (rule continuous_onI) (use False in \<open>auto intro: *\<close>) |
|
7312 |
qed |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
7313 |
qed |
62217 | 7314 |
|
69597 | 7315 |
text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7316 |
lemma Cauchy_integral_formula_global_weak: |
68420 | 7317 |
assumes "open U" and holf: "f holomorphic_on U" |
7318 |
and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>" |
|
7319 |
and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
|
7320 |
and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0" |
|
63589 | 7321 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7322 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7323 |
obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7324 |
using has_vector_derivative_polynomial_function [OF \<gamma>] by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7325 |
then have "bounded(path_image \<gamma>')" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7326 |
by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7327 |
then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7328 |
using bounded_pos by force |
63040 | 7329 |
define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w |
7330 |
define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7331 |
have "path \<gamma>" "valid_path \<gamma>" using \<gamma> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7332 |
by (auto simp: path_polynomial_function valid_path_polynomial_function) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7333 |
then have ov: "open v" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7334 |
by (simp add: v_def open_winding_number_levelsets loop) |
68420 | 7335 |
have uv_Un: "U \<union> v = UNIV" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7336 |
using pasz zero by (auto simp: v_def) |
68420 | 7337 |
have conf: "continuous_on U f" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7338 |
by (metis holf holomorphic_on_imp_continuous_on) |
68420 | 7339 |
have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7340 |
proof - |
68420 | 7341 |
have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U" |
7342 |
by (simp add: holf pole_lemma_open \<open>open U\<close>) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7343 |
then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y" |
68420 | 7344 |
using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce |
7345 |
then have "continuous_on U (d y)" |
|
7346 |
apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7347 |
using * holomorphic_on_def |
68420 | 7348 |
by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>) |
7349 |
moreover have "d y holomorphic_on U - {y}" |
|
7350 |
proof - |
|
7351 |
have "\<And>w. w \<in> U - {y} \<Longrightarrow> |
|
7352 |
(\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w" |
|
7353 |
apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within) |
|
7354 |
apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros) |
|
7355 |
using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast |
|
7356 |
then show ?thesis |
|
7357 |
unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete) |
|
7358 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7359 |
ultimately show ?thesis |
68420 | 7360 |
by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7361 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7362 |
have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y |
68420 | 7363 |
proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"]) |
7364 |
show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}" |
|
7365 |
by (force intro: holomorphic_intros holomorphic_on_subset [OF holf]) |
|
7366 |
show "path_image \<gamma> \<subseteq> U - {y}" |
|
7367 |
using pasz that by blast |
|
7368 |
qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>) |
|
63040 | 7369 |
define h where |
68420 | 7370 |
"h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z |
7371 |
have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z |
|
7372 |
proof - |
|
7373 |
have "d z holomorphic_on U" |
|
7374 |
by (simp add: hol_d that) |
|
7375 |
with that show ?thesis |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7376 |
apply (simp add: h_def) |
68420 | 7377 |
by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans) |
7378 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7379 |
have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7380 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7381 |
have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7382 |
using v_def z by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7383 |
then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7384 |
using z v_def has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7385 |
then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7386 |
using has_contour_integral_lmul by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7387 |
then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
7388 |
by (simp add: field_split_simps) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7389 |
moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7390 |
using z |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7391 |
apply (auto simp: v_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7392 |
apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7393 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7394 |
ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7395 |
by (rule has_contour_integral_add) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7396 |
have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>" |
68420 | 7397 |
if "z \<in> U" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7398 |
using * by (auto simp: divide_simps has_contour_integral_eq) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7399 |
moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>" |
68420 | 7400 |
if "z \<notin> U" |
7401 |
apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]]) |
|
7402 |
using U pasz \<open>valid_path \<gamma>\<close> that |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7403 |
apply (auto intro: holomorphic_on_imp_continuous_on hol_d) |
68420 | 7404 |
apply (rule continuous_intros conf holomorphic_intros holf assms | force)+ |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7405 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7406 |
ultimately show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7407 |
using z by (simp add: h_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7408 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7409 |
have znot: "z \<notin> path_image \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7410 |
using pasz by blast |
68420 | 7411 |
obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y" |
7412 |
using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7413 |
by (fastforce simp add: \<open>path \<gamma>\<close> compact_path_image) |
68420 | 7414 |
obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7415 |
apply (rule that [of "d0/2"]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7416 |
using \<open>0 < d0\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7417 |
apply (auto simp: dist_norm dest: d0) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7418 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7419 |
have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7420 |
apply (rule_tac x=x in exI) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7421 |
apply (rule_tac x="x'-x" in exI) |
68339 | 7422 |
apply (force simp: dist_norm) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7423 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7424 |
then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7425 |
apply (clarsimp simp add: mem_interior) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7426 |
using \<open>0 < dd\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7427 |
apply (rule_tac x="dd/2" in exI, auto) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7428 |
done |
68420 | 7429 |
obtain T where "compact T" and subt: "path_image \<gamma> \<subseteq> interior T" and T: "T \<subseteq> U" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7430 |
apply (rule that [OF _ 1]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7431 |
apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7432 |
apply (rule order_trans [OF _ dd]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7433 |
using \<open>0 < dd\<close> by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7434 |
obtain L where "L>0" |
68420 | 7435 |
and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow> |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7436 |
cmod (contour_integral \<gamma> f) \<le> L * B" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7437 |
using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt] |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7438 |
by blast |
68420 | 7439 |
have "bounded(f ` T)" |
7440 |
by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T) |
|
7441 |
then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7442 |
by (auto simp: bounded_pos) |
68420 | 7443 |
obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C" |
7444 |
using \<open>compact T\<close> bounded_pos compact_imp_bounded by force |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7445 |
have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7446 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7447 |
have "D * L / e > 0" using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7448 |
with le have ybig: "norm y > C" by force |
68420 | 7449 |
with C have "y \<notin> T" by force |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7450 |
then have ynot: "y \<notin> path_image \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7451 |
using subt interior_subset by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7452 |
have [simp]: "winding_number \<gamma> y = 0" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7453 |
apply (rule winding_number_zero_outside [of _ "cball 0 C"]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7454 |
using ybig interior_subset subt |
68339 | 7455 |
apply (force simp: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+ |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7456 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7457 |
have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7458 |
by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V) |
68420 | 7459 |
have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7460 |
apply (rule holomorphic_on_divide) |
68420 | 7461 |
using holf holomorphic_on_subset interior_subset T apply blast |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7462 |
apply (rule holomorphic_intros)+ |
68420 | 7463 |
using \<open>y \<notin> T\<close> interior_subset by auto |
7464 |
have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7465 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7466 |
have "D * L / e + cmod z \<le> cmod y" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7467 |
using le C [of z] z using interior_subset by force |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7468 |
then have DL2: "D * L / e \<le> cmod (z - y)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7469 |
using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7470 |
have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7471 |
by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse) |
68339 | 7472 |
also have "\<dots> \<le> D * (e / L / D)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7473 |
apply (rule mult_mono) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7474 |
using that D interior_subset apply blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7475 |
using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2 |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
7476 |
apply (auto simp: norm_divide field_split_simps algebra_simps) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7477 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7478 |
finally show ?thesis . |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7479 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7480 |
have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7481 |
by (simp add: dist_norm) |
68339 | 7482 |
also have "\<dots> \<le> L * (D * (e / L / D))" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7483 |
by (rule L [OF holint leD]) |
68339 | 7484 |
also have "\<dots> = e" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7485 |
using \<open>L>0\<close> \<open>0 < D\<close> by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7486 |
finally show ?thesis . |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7487 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7488 |
then have "(h \<longlongrightarrow> 0) at_infinity" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7489 |
by (meson Lim_at_infinityI) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7490 |
moreover have "h holomorphic_on UNIV" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7491 |
proof - |
62217 | 7492 |
have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))" |
68420 | 7493 |
if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7494 |
using that conf |
68420 | 7495 |
apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7496 |
apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7497 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7498 |
have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7499 |
by (rule continuous_intros)+ |
68420 | 7500 |
have open_uu_Id: "open (U \<times> U - Id)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7501 |
apply (rule open_Diff) |
68420 | 7502 |
apply (simp add: open_Times \<open>open U\<close>) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7503 |
using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0] |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7504 |
apply (auto simp: Id_fstsnd_eq algebra_simps) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7505 |
done |
68420 | 7506 |
have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z |
7507 |
apply (rule continuous_on_interior [of U]) |
|
7508 |
apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>) |
|
7509 |
by (simp add: interior_open that \<open>open U\<close>) |
|
62217 | 7510 |
have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x) |
7511 |
else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x) |
|
68420 | 7512 |
(at (x, x) within U \<times> U)" if "x \<in> U" for x |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7513 |
proof (rule Lim_withinI) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7514 |
fix e::real assume "0 < e" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7515 |
obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e" |
68420 | 7516 |
using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]] |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7517 |
by (metis UNIV_I dist_norm) |
68493 | 7518 |
obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U" |
68420 | 7519 |
by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7520 |
have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7521 |
if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7522 |
for x' z' |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7523 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7524 |
have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7525 |
apply (drule segment_furthest_le [where y=x]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7526 |
by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7527 |
have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7528 |
by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans) |
68420 | 7529 |
have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)" |
7530 |
by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>) |
|
7531 |
have "closed_segment x' z' \<subseteq> U" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7532 |
by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7533 |
then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7534 |
using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7535 |
then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7536 |
by (rule has_contour_integral_div) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7537 |
have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7538 |
apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7539 |
using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']] |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7540 |
\<open>e > 0\<close> \<open>z' \<noteq> x'\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7541 |
apply (auto simp: norm_divide divide_simps derf_le) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7542 |
done |
68339 | 7543 |
also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7544 |
finally show ?thesis . |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7545 |
qed |
68420 | 7546 |
show "\<exists>d>0. \<forall>xa\<in>U \<times> U. |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7547 |
0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow> |
62217 | 7548 |
dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7549 |
apply (rule_tac x="min k1 k2" in exI) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7550 |
using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7551 |
apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7552 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7553 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7554 |
have con_pa_f: "continuous_on (path_image \<gamma>) f" |
68420 | 7555 |
by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T) |
7556 |
have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7557 |
apply (rule B) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7558 |
using \<gamma>' using path_image_def vector_derivative_at by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7559 |
have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7560 |
by (simp add: V) |
68420 | 7561 |
have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7562 |
apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f') |
68420 | 7563 |
apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify) |
62217 | 7564 |
apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"]) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7565 |
using con_ff |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7566 |
apply (auto simp: continuous_within) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7567 |
done |
68420 | 7568 |
have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7569 |
proof - |
68420 | 7570 |
have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7571 |
by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+ |
68420 | 7572 |
then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7573 |
by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps) |
68420 | 7574 |
have **: "\<And>x. \<lbrakk>x \<in> U; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7575 |
apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within) |
68420 | 7576 |
apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+ |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7577 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7578 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7579 |
unfolding d_def |
68420 | 7580 |
apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"]) |
7581 |
apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7582 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7583 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7584 |
{ fix a b |
68420 | 7585 |
assume abu: "closed_segment a b \<subseteq> U" |
7586 |
then have "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7587 |
by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on) |
68420 | 7588 |
then have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" |
7589 |
apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu]) |
|
68339 | 7590 |
apply (auto intro: continuous_on_swap_args cond_uu) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7591 |
done |
68339 | 7592 |
have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)" |
68420 | 7593 |
proof (rule continuous_on_compose) |
7594 |
show "continuous_on {0..1} \<gamma>" |
|
7595 |
using \<open>path \<gamma>\<close> path_def by blast |
|
7596 |
show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" |
|
7597 |
using pasz unfolding path_image_def |
|
7598 |
by (auto intro!: continuous_on_subset [OF cont_cint_d]) |
|
7599 |
qed |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7600 |
have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7601 |
apply (simp add: contour_integrable_on) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7602 |
apply (rule integrable_continuous_real) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7603 |
apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7604 |
using pf\<gamma>' |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7605 |
by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>']) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7606 |
have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))" |
68339 | 7607 |
using abu by (force simp: h_def intro: contour_integral_eq) |
7608 |
also have "\<dots> = contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7609 |
apply (rule contour_integral_swap) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7610 |
apply (rule continuous_on_subset [OF cond_uu]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7611 |
using abu pasz \<open>valid_path \<gamma>\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7612 |
apply (auto intro!: continuous_intros) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7613 |
by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7614 |
finally have cint_h_eq: |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7615 |
"contour_integral (linepath a b) h = |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7616 |
contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" . |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7617 |
note cint_cint cint_h_eq |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7618 |
} note cint_h = this |
68420 | 7619 |
have conthu: "continuous_on U h" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7620 |
proof (simp add: continuous_on_sequentially, clarify) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7621 |
fix a x |
68420 | 7622 |
assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7623 |
then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7624 |
by (meson U contour_integrable_on_def eventuallyI) |
68420 | 7625 |
obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7626 |
have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7627 |
unfolding uniform_limit_iff dist_norm |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7628 |
proof clarify |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7629 |
fix ee::real |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7630 |
assume "0 < ee" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7631 |
show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7632 |
proof - |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7633 |
let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7634 |
have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7635 |
apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]]) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7636 |
using dd pasz \<open>valid_path \<gamma>\<close> |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7637 |
apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7638 |
done |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7639 |
then obtain kk where "kk>0" |
68420 | 7640 |
and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow> |
62217 | 7641 |
dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee" |
68420 | 7642 |
by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7643 |
have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7644 |
for w z |
68339 | 7645 |
using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7646 |
show ?thesis |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7647 |
using ax unfolding lim_sequentially eventually_sequentially |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7648 |
apply (drule_tac x="min dd kk" in spec) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7649 |
using \<open>dd > 0\<close> \<open>kk > 0\<close> |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7650 |
apply (fastforce simp: kk dist_norm) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7651 |
done |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7652 |
qed |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7653 |
qed |
68420 | 7654 |
have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)" |
7655 |
by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>) |
|
7656 |
then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x" |
|
7657 |
by (simp add: h_def x) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7658 |
then show "(h \<circ> a) \<longlonglongrightarrow> h x" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7659 |
by (simp add: h_def x au o_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7660 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7661 |
show ?thesis |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7662 |
proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7663 |
fix z0 |
68420 | 7664 |
consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7665 |
then show "h field_differentiable at z0" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7666 |
proof cases |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7667 |
assume "z0 \<in> v" then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7668 |
using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close> |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
7669 |
by (auto simp: field_differentiable_def v_def) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7670 |
next |
68420 | 7671 |
assume "z0 \<in> U" then |
7672 |
obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>]) |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7673 |
have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7674 |
if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e" for a b c |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7675 |
proof - |
68420 | 7676 |
have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2" |
7677 |
using hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close> |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7678 |
by (auto intro!: contour_integrable_holomorphic_simple) |
68420 | 7679 |
have abc: "closed_segment a b \<subseteq> U" "closed_segment b c \<subseteq> U" "closed_segment c a \<subseteq> U" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7680 |
using that e segments_subset_convex_hull by fastforce+ |
68420 | 7681 |
have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7682 |
apply (rule contour_integral_unique [OF Cauchy_theorem_triangle]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7683 |
apply (rule holomorphic_on_subset [OF hol_dw]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7684 |
using e abc_subset by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7685 |
have "contour_integral \<gamma> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7686 |
(\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) + |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7687 |
(contour_integral (linepath b c) (\<lambda>z. d z x) + |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7688 |
contour_integral (linepath c a) (\<lambda>z. d z x))) = 0" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7689 |
apply (rule contour_integral_eq_0) |
68420 | 7690 |
using abc pasz U |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7691 |
apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7692 |
done |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7693 |
then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7694 |
by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7695 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7696 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7697 |
using e \<open>e > 0\<close> |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7698 |
by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7699 |
Morera_triangle continuous_on_subset [OF conthu] *) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7700 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7701 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7702 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7703 |
ultimately have [simp]: "h z = 0" for z |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7704 |
by (meson Liouville_weak) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7705 |
have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7706 |
by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7707 |
then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7708 |
by (metis mult.commute has_contour_integral_lmul) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7709 |
then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70804
diff
changeset
|
7710 |
by (simp add: field_split_simps) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7711 |
moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7712 |
using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7713 |
show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7714 |
using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7715 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7716 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7717 |
theorem Cauchy_integral_formula_global: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7718 |
assumes S: "open S" and holf: "f holomorphic_on S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7719 |
and z: "z \<in> S" and vpg: "valid_path \<gamma>" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7720 |
and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7721 |
and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0" |
63589 | 7722 |
shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7723 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7724 |
have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7725 |
have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7726 |
by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7727 |
then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7728 |
by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7729 |
obtain d where "d>0" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7730 |
and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d; |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7731 |
pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk> |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7732 |
\<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7733 |
using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7734 |
obtain p where polyp: "polynomial_function p" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7735 |
and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7736 |
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7737 |
then have ploop: "pathfinish p = pathstart p" using loop by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7738 |
have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7739 |
have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7740 |
have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7741 |
using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7742 |
have wn_eq: "winding_number p z = winding_number \<gamma> z" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7743 |
using vpp paps |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7744 |
by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols) |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7745 |
have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7746 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7747 |
have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7748 |
using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7749 |
have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7750 |
then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7751 |
using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol]) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7752 |
qed |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7753 |
then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7754 |
by (simp add: zero) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7755 |
show ?thesis |
68493 | 7756 |
using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7757 |
by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7758 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7759 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7760 |
theorem Cauchy_theorem_global: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7761 |
assumes S: "open S" and holf: "f holomorphic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7762 |
and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7763 |
and pas: "path_image \<gamma> \<subseteq> S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7764 |
and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7765 |
shows "(f has_contour_integral 0) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7766 |
proof - |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7767 |
obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7768 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7769 |
have "compact (path_image \<gamma>)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7770 |
using compact_valid_path_image vpg by blast |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7771 |
then have "path_image \<gamma> \<noteq> S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7772 |
by (metis (no_types) compact_open path_image_nonempty S) |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7773 |
with pas show ?thesis by (blast intro: that) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7774 |
qed |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7775 |
then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7776 |
have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7777 |
by (rule holomorphic_intros holf)+ |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7778 |
show ?thesis |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7779 |
using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero] |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7780 |
by (auto simp: znot elim!: has_contour_integral_eq) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7781 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7782 |
|
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7783 |
corollary Cauchy_theorem_global_outside: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7784 |
assumes "open S" "f holomorphic_on S" "valid_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7785 |
"\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7786 |
shows "(f has_contour_integral 0) \<gamma>" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7787 |
by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62101
diff
changeset
|
7788 |
|
63955 | 7789 |
lemma simply_connected_imp_winding_number_zero: |
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7790 |
assumes "simply_connected S" "path g" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7791 |
"path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S" |
63955 | 7792 |
shows "winding_number g z = 0" |
7793 |
proof - |
|
68420 | 7794 |
have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))" |
7795 |
by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path) |
|
7796 |
then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))" |
|
7797 |
by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton) |
|
7798 |
then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z" |
|
7799 |
by (rule winding_number_homotopic_paths) |
|
68339 | 7800 |
also have "\<dots> = 0" |
63955 | 7801 |
using assms by (force intro: winding_number_trivial) |
7802 |
finally show ?thesis . |
|
7803 |
qed |
|
7804 |
||
7805 |
lemma Cauchy_theorem_simply_connected: |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7806 |
assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
7807 |
"path_image g \<subseteq> S" "pathfinish g = pathstart g" |
63955 | 7808 |
shows "(f has_contour_integral 0) g" |
7809 |
using assms |
|
7810 |
apply (simp add: simply_connected_eq_contractible_path) |
|
7811 |
apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"] |
|
7812 |
homotopic_paths_imp_homotopic_loops) |
|
7813 |
using valid_path_imp_path by blast |
|
7814 |
||
70136 | 7815 |
proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists: |
68493 | 7816 |
assumes A: "convex A" "open A" |
67107
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7817 |
and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7818 |
and z0: "z0 \<in> A" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7819 |
obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7820 |
proof - |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7821 |
note f' = holomorphic_derivI [OF f(1) A(2)] |
68310 | 7822 |
obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)" |
7823 |
proof (rule holomorphic_convex_primitive' [OF A]) |
|
7824 |
show "(\<lambda>x. deriv f x / f x) holomorphic_on A" |
|
7825 |
by (intro holomorphic_intros f A) |
|
7826 |
qed (auto simp: A at_within_open[of _ A]) |
|
67107
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7827 |
define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7828 |
from g and A have g_holo: "g holomorphic_on A" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7829 |
by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def) |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7830 |
hence h_holo: "h holomorphic_on A" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7831 |
by (auto simp: h_def intro!: holomorphic_intros) |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7832 |
have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c" |
70707
125705f5965f
A little-known material, and some tidying up
paulson <lp15@cam.ac.uk>
parents:
70642
diff
changeset
|
7833 |
proof (rule has_field_derivative_zero_constant, goal_cases) |
67107
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7834 |
case (2 x) |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7835 |
note [simp] = at_within_open[OF _ \<open>open A\<close>] |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7836 |
from 2 and z0 and f show ?case |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7837 |
by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f') |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7838 |
qed fact+ |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7839 |
then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7840 |
by blast |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7841 |
from c[OF z0] and z0 and f have "c = 0" |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7842 |
by (simp add: h_def) |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7843 |
with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7844 |
from that[OF h_holo this] show ?thesis . |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7845 |
qed |
cef76a19125e
Existence of a holomorphic logarithm
eberlm <eberlm@in.tum.de>
parents:
66884
diff
changeset
|
7846 |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
diff
changeset
|
7847 |
end |