author | paulson <lp15@cam.ac.uk> |
Thu, 15 Jun 2017 17:22:23 +0100 | |
changeset 66089 | def95e0bc529 |
parent 65587 | 16a8991ab398 |
child 66252 | b73f94b366b7 |
permissions | -rw-r--r-- |
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(* Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno |
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Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014) |
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*) |
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||
60420 | 5 |
section \<open>Complex Analysis Basics\<close> |
56215 | 6 |
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theory Complex_Analysis_Basics |
|
63941
f353674c2528
move absolutely_integrable_on to Equivalence_Lebesgue_Henstock_Integration, now based on the Lebesgue integral
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parents:
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diff
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8 |
imports Equivalence_Lebesgue_Henstock_Integration "~~/src/HOL/Library/Nonpos_Ints" |
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begin |
10 |
||
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The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
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changeset
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11 |
|
62131
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nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
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diff
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subsection\<open>General lemmas\<close> |
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b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
13 |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
14 |
lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
15 |
by (simp add: complex_nonneg_Reals_iff cmod_eq_Re) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
56369
diff
changeset
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16 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
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|
17 |
lemma has_derivative_mult_right: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
18 |
fixes c:: "'a :: real_normed_algebra" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
19 |
shows "((op * c) has_derivative (op * c)) F" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
20 |
by (rule has_derivative_mult_right [OF has_derivative_id]) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
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21 |
|
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:
61531
diff
changeset
|
22 |
lemma has_derivative_of_real[derivative_intros, simp]: |
56370
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reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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23 |
"(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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changeset
|
24 |
using bounded_linear.has_derivative[OF bounded_linear_of_real] . |
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reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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|
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reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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|
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lemma has_vector_derivative_real_complex: |
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Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
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diff
changeset
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"DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
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diff
changeset
|
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using has_derivative_compose[of of_real of_real a _ f "op * f'"] |
56370
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reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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changeset
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by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def) |
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|
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a few new lemmas and generalisations of old ones
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parents:
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lemma fact_cancel: |
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a few new lemmas and generalisations of old ones
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parents:
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changeset
|
32 |
fixes c :: "'a::real_field" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
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shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)" |
56369
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moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
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parents:
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diff
changeset
|
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by (simp add: of_nat_mult del: of_nat_Suc times_nat.simps) |
56889
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avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
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parents:
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|
56215 | 36 |
lemma bilinear_times: |
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moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
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parents:
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|
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fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)" |
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
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parents:
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changeset
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by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI) |
56215 | 39 |
|
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lemma linear_cnj: "linear cnj" |
|
56369
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moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
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diff
changeset
|
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using bounded_linear.linear[OF bounded_linear_cnj] . |
56215 | 42 |
|
43 |
lemma tendsto_Re_upper: |
|
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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:
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diff
changeset
|
44 |
assumes "~ (trivial_limit F)" |
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"(f \<longlongrightarrow> l) F" |
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"eventually (\<lambda>x. Re(f x) \<le> b) F" |
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shows "Re(l) \<le> b" |
|
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by (metis assms tendsto_le [OF _ tendsto_const] tendsto_Re) |
|
49 |
||
50 |
lemma tendsto_Re_lower: |
|
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:
61531
diff
changeset
|
51 |
assumes "~ (trivial_limit F)" |
61973 | 52 |
"(f \<longlongrightarrow> l) F" |
56215 | 53 |
"eventually (\<lambda>x. b \<le> Re(f x)) F" |
54 |
shows "b \<le> Re(l)" |
|
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by (metis assms tendsto_le [OF _ _ tendsto_const] tendsto_Re) |
|
56 |
||
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lemma tendsto_Im_upper: |
|
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:
61531
diff
changeset
|
58 |
assumes "~ (trivial_limit F)" |
61973 | 59 |
"(f \<longlongrightarrow> l) F" |
56215 | 60 |
"eventually (\<lambda>x. Im(f x) \<le> b) F" |
61 |
shows "Im(l) \<le> b" |
|
62 |
by (metis assms tendsto_le [OF _ tendsto_const] tendsto_Im) |
|
63 |
||
64 |
lemma tendsto_Im_lower: |
|
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:
61531
diff
changeset
|
65 |
assumes "~ (trivial_limit F)" |
61973 | 66 |
"(f \<longlongrightarrow> l) F" |
56215 | 67 |
"eventually (\<lambda>x. b \<le> Im(f x)) F" |
68 |
shows "b \<le> Im(l)" |
|
69 |
by (metis assms tendsto_le [OF _ _ tendsto_const] tendsto_Im) |
|
70 |
||
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reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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diff
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|
71 |
lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
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diff
changeset
|
72 |
by auto |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
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73 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
56369
diff
changeset
|
74 |
lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
75 |
by auto |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
76 |
|
56215 | 77 |
lemma continuous_mult_left: |
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:
61531
diff
changeset
|
78 |
fixes c::"'a::real_normed_algebra" |
56215 | 79 |
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)" |
80 |
by (rule continuous_mult [OF continuous_const]) |
|
81 |
||
82 |
lemma continuous_mult_right: |
|
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:
61531
diff
changeset
|
83 |
fixes c::"'a::real_normed_algebra" |
56215 | 84 |
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)" |
85 |
by (rule continuous_mult [OF _ continuous_const]) |
|
86 |
||
87 |
lemma continuous_on_mult_left: |
|
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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:
61531
diff
changeset
|
88 |
fixes c::"'a::real_normed_algebra" |
56215 | 89 |
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)" |
90 |
by (rule continuous_on_mult [OF continuous_on_const]) |
|
91 |
||
92 |
lemma continuous_on_mult_right: |
|
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:
61531
diff
changeset
|
93 |
fixes c::"'a::real_normed_algebra" |
56215 | 94 |
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)" |
95 |
by (rule continuous_on_mult [OF _ continuous_on_const]) |
|
96 |
||
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fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
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parents:
56370
diff
changeset
|
97 |
lemma uniformly_continuous_on_cmul_right [continuous_intros]: |
56215 | 98 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra" |
56332 | 99 |
shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)" |
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:
61531
diff
changeset
|
100 |
using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] . |
56215 | 101 |
|
56371
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extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56370
diff
changeset
|
102 |
lemma uniformly_continuous_on_cmul_left[continuous_intros]: |
56215 | 103 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra" |
104 |
assumes "uniformly_continuous_on s f" |
|
105 |
shows "uniformly_continuous_on s (\<lambda>x. c * f x)" |
|
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by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right) |
|
107 |
||
108 |
lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm" |
|
109 |
by (rule continuous_norm [OF continuous_ident]) |
|
110 |
||
111 |
lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm" |
|
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moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
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parents:
56332
diff
changeset
|
112 |
by (intro continuous_on_id continuous_on_norm) |
56215 | 113 |
|
60420 | 114 |
subsection\<open>DERIV stuff\<close> |
56215 | 115 |
|
116 |
lemma DERIV_zero_connected_constant: |
|
117 |
fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a" |
|
118 |
assumes "connected s" |
|
119 |
and "open s" |
|
120 |
and "finite k" |
|
121 |
and "continuous_on s f" |
|
122 |
and "\<forall>x\<in>(s - k). DERIV f x :> 0" |
|
123 |
obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c" |
|
124 |
using has_derivative_zero_connected_constant [OF assms(1-4)] assms |
|
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56332
diff
changeset
|
125 |
by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def) |
56215 | 126 |
|
127 |
lemma DERIV_zero_constant: |
|
56370
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reorder Complex_Analysis_Basics; rename DD to deriv
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parents:
56369
diff
changeset
|
128 |
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a" |
56215 | 129 |
shows "\<lbrakk>convex s; |
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:
61531
diff
changeset
|
130 |
\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk> |
56215 | 131 |
\<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
132 |
by (auto simp: has_field_derivative_def lambda_zero intro: has_derivative_zero_constant) |
56215 | 133 |
|
134 |
lemma DERIV_zero_unique: |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
135 |
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a" |
56215 | 136 |
assumes "convex s" |
137 |
and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)" |
|
138 |
and "a \<in> s" |
|
139 |
and "x \<in> s" |
|
140 |
shows "f x = f a" |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
141 |
by (rule has_derivative_zero_unique [OF assms(1) _ assms(4,3)]) |
56332 | 142 |
(metis d0 has_field_derivative_imp_has_derivative lambda_zero) |
56215 | 143 |
|
144 |
lemma DERIV_zero_connected_unique: |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
145 |
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a" |
56215 | 146 |
assumes "connected s" |
147 |
and "open s" |
|
148 |
and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0" |
|
149 |
and "a \<in> s" |
|
150 |
and "x \<in> s" |
|
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:
61531
diff
changeset
|
151 |
shows "f x = f a" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
152 |
by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)]) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
153 |
(metis has_field_derivative_def lambda_zero d0) |
56215 | 154 |
|
155 |
lemma DERIV_transform_within: |
|
156 |
assumes "(f has_field_derivative f') (at a within s)" |
|
157 |
and "0 < d" "a \<in> s" |
|
158 |
and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x" |
|
159 |
shows "(g has_field_derivative f') (at a within s)" |
|
160 |
using assms unfolding has_field_derivative_def |
|
56332 | 161 |
by (blast intro: has_derivative_transform_within) |
56215 | 162 |
|
163 |
lemma DERIV_transform_within_open: |
|
164 |
assumes "DERIV f a :> f'" |
|
165 |
and "open s" "a \<in> s" |
|
166 |
and "\<And>x. x\<in>s \<Longrightarrow> f x = g x" |
|
167 |
shows "DERIV g a :> f'" |
|
168 |
using assms unfolding has_field_derivative_def |
|
169 |
by (metis has_derivative_transform_within_open) |
|
170 |
||
171 |
lemma DERIV_transform_at: |
|
172 |
assumes "DERIV f a :> f'" |
|
173 |
and "0 < d" |
|
174 |
and "\<And>x. dist x a < d \<Longrightarrow> f x = g x" |
|
175 |
shows "DERIV g a :> f'" |
|
176 |
by (blast intro: assms DERIV_transform_within) |
|
177 |
||
59615
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59554
diff
changeset
|
178 |
(*generalising DERIV_isconst_all, which requires type real (using the ordering)*) |
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59554
diff
changeset
|
179 |
lemma DERIV_zero_UNIV_unique: |
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59554
diff
changeset
|
180 |
fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a" |
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:
61531
diff
changeset
|
181 |
shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a" |
63092 | 182 |
by (metis DERIV_zero_unique UNIV_I convex_UNIV) |
59615
fdfdf89a83a6
A few new lemmas and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
59554
diff
changeset
|
183 |
|
60420 | 184 |
subsection \<open>Some limit theorems about real part of real series etc.\<close> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
185 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
186 |
(*MOVE? But not to Finite_Cartesian_Product*) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
187 |
lemma sums_vec_nth : |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
188 |
assumes "f sums a" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
189 |
shows "(\<lambda>x. f x $ i) sums a $ i" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
190 |
using assms unfolding sums_def |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
191 |
by (auto dest: tendsto_vec_nth [where i=i]) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
192 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
193 |
lemma summable_vec_nth : |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
194 |
assumes "summable f" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
195 |
shows "summable (\<lambda>x. f x $ i)" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
196 |
using assms unfolding summable_def |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
197 |
by (blast intro: sums_vec_nth) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
198 |
|
60420 | 199 |
subsection \<open>Complex number lemmas\<close> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
200 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
201 |
lemma |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
202 |
shows open_halfspace_Re_lt: "open {z. Re(z) < b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
203 |
and open_halfspace_Re_gt: "open {z. Re(z) > b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
204 |
and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
205 |
and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
206 |
and closed_halfspace_Re_eq: "closed {z. Re(z) = b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
207 |
and open_halfspace_Im_lt: "open {z. Im(z) < b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
208 |
and open_halfspace_Im_gt: "open {z. Im(z) > b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
209 |
and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
210 |
and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
211 |
and closed_halfspace_Im_eq: "closed {z. Im(z) = b}" |
63332 | 212 |
by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re |
213 |
continuous_on_Im continuous_on_id continuous_on_const)+ |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
214 |
|
61070 | 215 |
lemma closed_complex_Reals: "closed (\<real> :: complex set)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
216 |
proof - |
61070 | 217 |
have "(\<real> :: complex set) = {z. Im z = 0}" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
218 |
by (auto simp: complex_is_Real_iff) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
219 |
then show ?thesis |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
220 |
by (metis closed_halfspace_Im_eq) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
221 |
qed |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
222 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
223 |
lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
224 |
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
225 |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
226 |
corollary closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
227 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
228 |
have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
229 |
using complex_nonpos_Reals_iff complex_is_Real_iff by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
230 |
then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
231 |
by (metis closed_Real_halfspace_Re_le) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
232 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
233 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
234 |
lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
235 |
using closed_halfspace_Re_ge |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
236 |
by (simp add: closed_Int closed_complex_Reals) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
237 |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
238 |
corollary closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
239 |
proof - |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
240 |
have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}" |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
241 |
using complex_nonneg_Reals_iff complex_is_Real_iff by auto |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
242 |
then show ?thesis |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
243 |
by (metis closed_Real_halfspace_Re_ge) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
244 |
qed |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
245 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
246 |
lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
247 |
proof - |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
248 |
have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
249 |
by auto |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
250 |
then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
251 |
by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
252 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
253 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
254 |
lemma real_lim: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
255 |
fixes l::complex |
61973 | 256 |
assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
257 |
shows "l \<in> \<real>" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
258 |
proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)]) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
259 |
show "eventually (\<lambda>x. f x \<in> \<real>) F" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
260 |
using assms(3, 4) by (auto intro: eventually_mono) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
261 |
qed |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
262 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
263 |
lemma real_lim_sequentially: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
264 |
fixes l::complex |
61973 | 265 |
shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
266 |
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
267 |
|
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:
61531
diff
changeset
|
268 |
lemma real_series: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
269 |
fixes l::complex |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
270 |
shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
271 |
unfolding sums_def |
64267 | 272 |
by (metis real_lim_sequentially sum_in_Reals) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
273 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
274 |
lemma Lim_null_comparison_Re: |
61973 | 275 |
assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
276 |
by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp |
56215 | 277 |
|
60420 | 278 |
subsection\<open>Holomorphic functions\<close> |
56215 | 279 |
|
64394 | 280 |
subsection\<open>Holomorphic functions\<close> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
281 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
282 |
definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
283 |
(infixl "(holomorphic'_on)" 50) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
284 |
where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)" |
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:
61531
diff
changeset
|
285 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
286 |
named_theorems holomorphic_intros "structural introduction rules for holomorphic_on" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
287 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
288 |
lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
289 |
by (simp add: holomorphic_on_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
290 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
291 |
lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
292 |
by (simp add: holomorphic_on_def) |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
293 |
|
64394 | 294 |
lemma holomorphic_on_imp_differentiable_on: |
295 |
"f holomorphic_on s \<Longrightarrow> f differentiable_on s" |
|
296 |
unfolding holomorphic_on_def differentiable_on_def |
|
297 |
by (simp add: field_differentiable_imp_differentiable) |
|
298 |
||
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
299 |
lemma holomorphic_on_imp_differentiable_at: |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
300 |
"\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)" |
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
301 |
using at_within_open holomorphic_on_def by fastforce |
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62087
diff
changeset
|
302 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
303 |
lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
304 |
by (simp add: holomorphic_on_def) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
305 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
306 |
lemma holomorphic_on_open: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
307 |
"open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> 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
|
308 |
by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s]) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
309 |
|
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:
61531
diff
changeset
|
310 |
lemma holomorphic_on_imp_continuous_on: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
311 |
"f holomorphic_on s \<Longrightarrow> 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
|
312 |
by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
313 |
|
62540
f2fc5485e3b0
Wenda Li's new material: residue theorem, argument_principle, Rouche_theorem
paulson <lp15@cam.ac.uk>
parents:
62534
diff
changeset
|
314 |
lemma holomorphic_on_subset [elim]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
315 |
"f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
316 |
unfolding 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
|
317 |
by (metis field_differentiable_within_subset subsetD) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
318 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
319 |
lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g 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
|
320 |
by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
321 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
322 |
lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
323 |
by (metis holomorphic_transform) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
324 |
|
62217 | 325 |
lemma holomorphic_on_linear [simp, holomorphic_intros]: "(op * c) 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
|
326 |
unfolding holomorphic_on_def by (metis field_differentiable_linear) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
327 |
|
62217 | 328 |
lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) 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
|
329 |
unfolding holomorphic_on_def by (metis field_differentiable_const) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
330 |
|
62217 | 331 |
lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) 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
|
332 |
unfolding holomorphic_on_def by (metis field_differentiable_ident) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
333 |
|
62217 | 334 |
lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
335 |
unfolding id_def by (rule holomorphic_on_ident) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
336 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
337 |
lemma holomorphic_on_compose: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
338 |
"f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o 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
|
339 |
using field_differentiable_compose_within[of f _ s g] |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
340 |
by (auto simp: holomorphic_on_def) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
341 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
342 |
lemma holomorphic_on_compose_gen: |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
343 |
"f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
344 |
by (metis holomorphic_on_compose holomorphic_on_subset) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
345 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
346 |
lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) 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
|
347 |
by (metis field_differentiable_minus holomorphic_on_def) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
348 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
349 |
lemma holomorphic_on_add [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
350 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) 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
|
351 |
unfolding holomorphic_on_def by (metis field_differentiable_add) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
352 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
353 |
lemma holomorphic_on_diff [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
354 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) 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
|
355 |
unfolding holomorphic_on_def by (metis field_differentiable_diff) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
356 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
357 |
lemma holomorphic_on_mult [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
358 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) 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
|
359 |
unfolding holomorphic_on_def by (metis field_differentiable_mult) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
360 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
361 |
lemma holomorphic_on_inverse [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
362 |
"\<lbrakk>f holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) 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
|
363 |
unfolding holomorphic_on_def by (metis field_differentiable_inverse) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
364 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
365 |
lemma holomorphic_on_divide [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
366 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) 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
|
367 |
unfolding holomorphic_on_def by (metis field_differentiable_divide) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
368 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
369 |
lemma holomorphic_on_power [holomorphic_intros]: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
370 |
"f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) 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
|
371 |
unfolding holomorphic_on_def by (metis field_differentiable_power) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
372 |
|
64267 | 373 |
lemma holomorphic_on_sum [holomorphic_intros]: |
374 |
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s" |
|
375 |
unfolding holomorphic_on_def by (metis field_differentiable_sum) |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
376 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
377 |
lemma DERIV_deriv_iff_field_differentiable: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
378 |
"DERIV f x :> deriv f x \<longleftrightarrow> f 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
|
379 |
unfolding field_differentiable_def by (metis DERIV_imp_deriv) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
380 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
381 |
lemma holomorphic_derivI: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
382 |
"\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
383 |
\<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
384 |
by (metis DERIV_deriv_iff_field_differentiable at_within_open holomorphic_on_def has_field_derivative_at_within) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
385 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
386 |
lemma complex_derivative_chain: |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
387 |
"f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
388 |
\<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f 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
|
389 |
by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
390 |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62217
diff
changeset
|
391 |
lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
392 |
by (metis DERIV_imp_deriv DERIV_cmult_Id) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
393 |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62217
diff
changeset
|
394 |
lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
395 |
by (metis DERIV_imp_deriv DERIV_ident) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
396 |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62217
diff
changeset
|
397 |
lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62217
diff
changeset
|
398 |
by (simp add: id_def) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62217
diff
changeset
|
399 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62217
diff
changeset
|
400 |
lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
401 |
by (metis DERIV_imp_deriv DERIV_const) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
402 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
403 |
lemma deriv_add [simp]: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
404 |
"\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
405 |
\<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g 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
|
406 |
unfolding DERIV_deriv_iff_field_differentiable[symmetric] |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
407 |
by (auto intro!: DERIV_imp_deriv derivative_intros) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
408 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
409 |
lemma deriv_diff [simp]: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
410 |
"\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
411 |
\<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g 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
|
412 |
unfolding DERIV_deriv_iff_field_differentiable[symmetric] |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
413 |
by (auto intro!: DERIV_imp_deriv derivative_intros) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
414 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
415 |
lemma deriv_mult [simp]: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
416 |
"\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
417 |
\<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g 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
|
418 |
unfolding DERIV_deriv_iff_field_differentiable[symmetric] |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
419 |
by (auto intro!: DERIV_imp_deriv derivative_eq_intros) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
420 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
421 |
lemma deriv_cmult [simp]: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
422 |
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z" |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
423 |
unfolding DERIV_deriv_iff_field_differentiable[symmetric] |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
424 |
by (auto intro!: DERIV_imp_deriv derivative_eq_intros) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
425 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
426 |
lemma deriv_cmult_right [simp]: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
427 |
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c" |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
428 |
unfolding DERIV_deriv_iff_field_differentiable[symmetric] |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
429 |
by (auto intro!: DERIV_imp_deriv derivative_eq_intros) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
430 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
431 |
lemma deriv_cdivide_right [simp]: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
432 |
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c" |
62217 | 433 |
unfolding Fields.field_class.field_divide_inverse |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
434 |
by (blast intro: deriv_cmult_right) |
62217 | 435 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
436 |
lemma complex_derivative_transform_within_open: |
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:
61531
diff
changeset
|
437 |
"\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
438 |
\<Longrightarrow> deriv f z = deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
439 |
unfolding holomorphic_on_def |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
440 |
by (rule DERIV_imp_deriv) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
441 |
(metis DERIV_deriv_iff_field_differentiable DERIV_transform_within_open at_within_open) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
442 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
443 |
lemma deriv_compose_linear: |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
444 |
"f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
445 |
apply (rule DERIV_imp_deriv) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
446 |
apply (simp add: DERIV_deriv_iff_field_differentiable [symmetric]) |
59554
4044f53326c9
inlined rules to free user-space from technical names
haftmann
parents:
58877
diff
changeset
|
447 |
apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id]) |
4044f53326c9
inlined rules to free user-space from technical names
haftmann
parents:
58877
diff
changeset
|
448 |
apply (simp add: algebra_simps) |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
449 |
done |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
450 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
451 |
lemma nonzero_deriv_nonconstant: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
452 |
assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
453 |
shows "\<not> f constant_on S" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
454 |
unfolding constant_on_def |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
455 |
by (metis \<open>df \<noteq> 0\<close> DERIV_transform_within_open [OF df S] DERIV_const DERIV_unique) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
456 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
457 |
lemma holomorphic_nonconstant: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
458 |
assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
459 |
shows "\<not> f constant_on S" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
460 |
apply (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
461 |
using assms |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
462 |
apply (auto simp: holomorphic_derivI) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
463 |
done |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62408
diff
changeset
|
464 |
|
64394 | 465 |
subsection\<open>Caratheodory characterization\<close> |
466 |
||
467 |
lemma field_differentiable_caratheodory_at: |
|
468 |
"f field_differentiable (at z) \<longleftrightarrow> |
|
469 |
(\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)" |
|
470 |
using CARAT_DERIV [of f] |
|
471 |
by (simp add: field_differentiable_def has_field_derivative_def) |
|
472 |
||
473 |
lemma field_differentiable_caratheodory_within: |
|
474 |
"f field_differentiable (at z within s) \<longleftrightarrow> |
|
475 |
(\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)" |
|
476 |
using DERIV_caratheodory_within [of f] |
|
477 |
by (simp add: field_differentiable_def has_field_derivative_def) |
|
478 |
||
60420 | 479 |
subsection\<open>Analyticity on a set\<close> |
56215 | 480 |
|
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:
61531
diff
changeset
|
481 |
definition analytic_on (infixl "(analytic'_on)" 50) |
56215 | 482 |
where |
483 |
"f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)" |
|
484 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
485 |
named_theorems analytic_intros "introduction rules for proving analyticity" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
486 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
487 |
lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
488 |
by (simp add: at_within_open [OF _ open_ball] analytic_on_def 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
|
489 |
(metis centre_in_ball field_differentiable_at_within) |
56215 | 490 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
491 |
lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s" |
56215 | 492 |
apply (auto simp: analytic_imp_holomorphic) |
493 |
apply (auto simp: analytic_on_def holomorphic_on_def) |
|
494 |
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball) |
|
495 |
||
496 |
lemma analytic_on_imp_differentiable_at: |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
497 |
"f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f field_differentiable (at x)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
498 |
apply (auto simp: analytic_on_def 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
|
499 |
by (metis Topology_Euclidean_Space.open_ball centre_in_ball field_differentiable_within_open) |
56215 | 500 |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
501 |
lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t" |
56215 | 502 |
by (auto simp: analytic_on_def) |
503 |
||
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
504 |
lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t" |
56215 | 505 |
by (auto simp: analytic_on_def) |
506 |
||
60585 | 507 |
lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
508 |
by (auto simp: analytic_on_def) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
509 |
|
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
510 |
lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))" |
56215 | 511 |
by (auto simp: analytic_on_def) |
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:
61531
diff
changeset
|
512 |
|
56215 | 513 |
lemma analytic_on_holomorphic: |
514 |
"f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)" |
|
515 |
(is "?lhs = ?rhs") |
|
516 |
proof - |
|
517 |
have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)" |
|
518 |
proof safe |
|
519 |
assume "f analytic_on s" |
|
520 |
then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t" |
|
521 |
apply (simp add: analytic_on_def) |
|
522 |
apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto) |
|
523 |
apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball) |
|
524 |
by (metis analytic_on_def) |
|
525 |
next |
|
526 |
fix t |
|
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:
61531
diff
changeset
|
527 |
assume "open t" "s \<subseteq> t" "f analytic_on t" |
56215 | 528 |
then show "f analytic_on s" |
529 |
by (metis analytic_on_subset) |
|
530 |
qed |
|
531 |
also have "... \<longleftrightarrow> ?rhs" |
|
532 |
by (auto simp: analytic_on_open) |
|
533 |
finally show ?thesis . |
|
534 |
qed |
|
535 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
536 |
lemma analytic_on_linear [analytic_intros,simp]: "(op * c) analytic_on s" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
537 |
by (auto simp add: analytic_on_holomorphic) |
56215 | 538 |
|
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
539 |
lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on s" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
540 |
by (metis analytic_on_def holomorphic_on_const zero_less_one) |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
541 |
|
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
542 |
lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on s" |
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
543 |
by (simp add: analytic_on_def gt_ex) |
56215 | 544 |
|
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
545 |
lemma analytic_on_id [analytic_intros]: "id analytic_on s" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
546 |
unfolding id_def by (rule analytic_on_ident) |
56215 | 547 |
|
548 |
lemma analytic_on_compose: |
|
549 |
assumes f: "f analytic_on s" |
|
550 |
and g: "g analytic_on (f ` s)" |
|
551 |
shows "(g o f) analytic_on s" |
|
552 |
unfolding analytic_on_def |
|
553 |
proof (intro ballI) |
|
554 |
fix x |
|
555 |
assume x: "x \<in> s" |
|
556 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f |
|
557 |
by (metis analytic_on_def) |
|
558 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g |
|
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:
61531
diff
changeset
|
559 |
by (metis analytic_on_def g image_eqI x) |
56215 | 560 |
have "isCont f 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
|
561 |
by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x) |
56215 | 562 |
with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'" |
563 |
by (auto simp: continuous_at_ball) |
|
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:
61531
diff
changeset
|
564 |
have "g \<circ> f holomorphic_on ball x (min d e)" |
56215 | 565 |
apply (rule holomorphic_on_compose) |
566 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
567 |
by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball) |
|
568 |
then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e" |
|
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:
61531
diff
changeset
|
569 |
by (metis d e min_less_iff_conj) |
56215 | 570 |
qed |
571 |
||
572 |
lemma analytic_on_compose_gen: |
|
573 |
"f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t) |
|
574 |
\<Longrightarrow> g o f analytic_on s" |
|
575 |
by (metis analytic_on_compose analytic_on_subset image_subset_iff) |
|
576 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
577 |
lemma analytic_on_neg [analytic_intros]: |
56215 | 578 |
"f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s" |
579 |
by (metis analytic_on_holomorphic holomorphic_on_minus) |
|
580 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
581 |
lemma analytic_on_add [analytic_intros]: |
56215 | 582 |
assumes f: "f analytic_on s" |
583 |
and g: "g analytic_on s" |
|
584 |
shows "(\<lambda>z. f z + g z) analytic_on s" |
|
585 |
unfolding analytic_on_def |
|
586 |
proof (intro ballI) |
|
587 |
fix z |
|
588 |
assume z: "z \<in> s" |
|
589 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
|
590 |
by (metis analytic_on_def) |
|
591 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g |
|
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:
61531
diff
changeset
|
592 |
by (metis analytic_on_def g z) |
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:
61531
diff
changeset
|
593 |
have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" |
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:
61531
diff
changeset
|
594 |
apply (rule holomorphic_on_add) |
56215 | 595 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
596 |
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
597 |
then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e" |
|
598 |
by (metis e e' min_less_iff_conj) |
|
599 |
qed |
|
600 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
601 |
lemma analytic_on_diff [analytic_intros]: |
56215 | 602 |
assumes f: "f analytic_on s" |
603 |
and g: "g analytic_on s" |
|
604 |
shows "(\<lambda>z. f z - g z) analytic_on s" |
|
605 |
unfolding analytic_on_def |
|
606 |
proof (intro ballI) |
|
607 |
fix z |
|
608 |
assume z: "z \<in> s" |
|
609 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
|
610 |
by (metis analytic_on_def) |
|
611 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g |
|
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:
61531
diff
changeset
|
612 |
by (metis analytic_on_def g z) |
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:
61531
diff
changeset
|
613 |
have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" |
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:
61531
diff
changeset
|
614 |
apply (rule holomorphic_on_diff) |
56215 | 615 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
616 |
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
617 |
then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e" |
|
618 |
by (metis e e' min_less_iff_conj) |
|
619 |
qed |
|
620 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
621 |
lemma analytic_on_mult [analytic_intros]: |
56215 | 622 |
assumes f: "f analytic_on s" |
623 |
and g: "g analytic_on s" |
|
624 |
shows "(\<lambda>z. f z * g z) analytic_on s" |
|
625 |
unfolding analytic_on_def |
|
626 |
proof (intro ballI) |
|
627 |
fix z |
|
628 |
assume z: "z \<in> s" |
|
629 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
|
630 |
by (metis analytic_on_def) |
|
631 |
obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g |
|
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:
61531
diff
changeset
|
632 |
by (metis analytic_on_def g z) |
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:
61531
diff
changeset
|
633 |
have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" |
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:
61531
diff
changeset
|
634 |
apply (rule holomorphic_on_mult) |
56215 | 635 |
apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
636 |
by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball) |
|
637 |
then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e" |
|
638 |
by (metis e e' min_less_iff_conj) |
|
639 |
qed |
|
640 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
641 |
lemma analytic_on_inverse [analytic_intros]: |
56215 | 642 |
assumes f: "f analytic_on s" |
643 |
and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)" |
|
644 |
shows "(\<lambda>z. inverse (f z)) analytic_on s" |
|
645 |
unfolding analytic_on_def |
|
646 |
proof (intro ballI) |
|
647 |
fix z |
|
648 |
assume z: "z \<in> s" |
|
649 |
then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f |
|
650 |
by (metis analytic_on_def) |
|
651 |
have "continuous_on (ball z e) f" |
|
652 |
by (metis fh holomorphic_on_imp_continuous_on) |
|
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:
61531
diff
changeset
|
653 |
then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" |
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:
61531
diff
changeset
|
654 |
by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz) |
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:
61531
diff
changeset
|
655 |
have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" |
56215 | 656 |
apply (rule holomorphic_on_inverse) |
657 |
apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball) |
|
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:
61531
diff
changeset
|
658 |
by (metis nz' mem_ball min_less_iff_conj) |
56215 | 659 |
then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e" |
660 |
by (metis e e' min_less_iff_conj) |
|
661 |
qed |
|
662 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
663 |
lemma analytic_on_divide [analytic_intros]: |
56215 | 664 |
assumes f: "f analytic_on s" |
665 |
and g: "g analytic_on s" |
|
666 |
and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)" |
|
667 |
shows "(\<lambda>z. f z / g z) analytic_on s" |
|
668 |
unfolding divide_inverse |
|
669 |
by (metis analytic_on_inverse analytic_on_mult f g nz) |
|
670 |
||
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
671 |
lemma analytic_on_power [analytic_intros]: |
56215 | 672 |
"f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s" |
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
673 |
by (induct n) (auto simp: analytic_on_mult) |
56215 | 674 |
|
65587
16a8991ab398
New material (and some tidying) purely in the Analysis directory
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
675 |
lemma analytic_on_sum [analytic_intros]: |
64267 | 676 |
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on s" |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56332
diff
changeset
|
677 |
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add) |
56215 | 678 |
|
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
679 |
lemma deriv_left_inverse: |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
680 |
assumes "f holomorphic_on S" and "g holomorphic_on T" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
681 |
and "open S" and "open T" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
682 |
and "f ` S \<subseteq> T" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
683 |
and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
684 |
and "w \<in> S" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
685 |
shows "deriv f w * deriv g (f w) = 1" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
686 |
proof - |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
687 |
have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
688 |
by (simp add: algebra_simps) |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
689 |
also have "... = deriv (g o f) w" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
690 |
using assms |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
691 |
by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff) |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
692 |
also have "... = deriv id w" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
693 |
apply (rule complex_derivative_transform_within_open [where s=S]) |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
694 |
apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+ |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
695 |
apply simp |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
696 |
done |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
697 |
also have "... = 1" |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
698 |
by simp |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
699 |
finally show ?thesis . |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
700 |
qed |
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
701 |
|
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
702 |
subsection\<open>analyticity at a point\<close> |
56215 | 703 |
|
704 |
lemma analytic_at_ball: |
|
705 |
"f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)" |
|
706 |
by (metis analytic_on_def singleton_iff) |
|
707 |
||
708 |
lemma analytic_at: |
|
709 |
"f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)" |
|
710 |
by (metis analytic_on_holomorphic empty_subsetI insert_subset) |
|
711 |
||
712 |
lemma analytic_on_analytic_at: |
|
713 |
"f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})" |
|
714 |
by (metis analytic_at_ball analytic_on_def) |
|
715 |
||
716 |
lemma analytic_at_two: |
|
717 |
"f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow> |
|
718 |
(\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)" |
|
719 |
(is "?lhs = ?rhs") |
|
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:
61531
diff
changeset
|
720 |
proof |
56215 | 721 |
assume ?lhs |
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:
61531
diff
changeset
|
722 |
then obtain s t |
56215 | 723 |
where st: "open s" "z \<in> s" "f holomorphic_on s" |
724 |
"open t" "z \<in> t" "g holomorphic_on t" |
|
725 |
by (auto simp: analytic_at) |
|
726 |
show ?rhs |
|
727 |
apply (rule_tac x="s \<inter> t" in exI) |
|
728 |
using st |
|
729 |
apply (auto simp: Diff_subset holomorphic_on_subset) |
|
730 |
done |
|
731 |
next |
|
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:
61531
diff
changeset
|
732 |
assume ?rhs |
56215 | 733 |
then show ?lhs |
734 |
by (force simp add: analytic_at) |
|
735 |
qed |
|
736 |
||
60420 | 737 |
subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close> |
56215 | 738 |
|
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:
61531
diff
changeset
|
739 |
lemma |
56215 | 740 |
assumes "f analytic_on {z}" "g analytic_on {z}" |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
741 |
shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
742 |
and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
743 |
and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z = |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
744 |
f z * deriv g z + deriv f z * g z" |
56215 | 745 |
proof - |
746 |
obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s" |
|
747 |
using assms by (metis analytic_at_two) |
|
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
748 |
show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
749 |
apply (rule DERIV_imp_deriv [OF DERIV_add]) |
56215 | 750 |
using 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
|
751 |
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable) |
56215 | 752 |
done |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
753 |
show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
754 |
apply (rule DERIV_imp_deriv [OF DERIV_diff]) |
56215 | 755 |
using 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
|
756 |
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable) |
56215 | 757 |
done |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
758 |
show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z" |
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
759 |
apply (rule DERIV_imp_deriv [OF DERIV_mult']) |
56215 | 760 |
using 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
|
761 |
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable) |
56215 | 762 |
done |
763 |
qed |
|
764 |
||
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
765 |
lemma deriv_cmult_at: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
766 |
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z" |
61848 | 767 |
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const) |
56215 | 768 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
769 |
lemma deriv_cmult_right_at: |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
770 |
"f analytic_on {z} \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c" |
61848 | 771 |
by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const) |
56215 | 772 |
|
60420 | 773 |
subsection\<open>Complex differentiation of sequences and series\<close> |
56215 | 774 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
775 |
(* TODO: Could probably be simplified using Uniform_Limit *) |
56215 | 776 |
lemma has_complex_derivative_sequence: |
777 |
fixes s :: "complex set" |
|
778 |
assumes cvs: "convex s" |
|
779 |
and df: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)" |
|
780 |
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e" |
|
61973 | 781 |
and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially" |
782 |
shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> |
|
56215 | 783 |
(g has_field_derivative (g' x)) (at x within s)" |
784 |
proof - |
|
61973 | 785 |
from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially" |
56215 | 786 |
by blast |
787 |
{ fix e::real assume e: "e > 0" |
|
788 |
then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e" |
|
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:
61531
diff
changeset
|
789 |
by (metis conv) |
56215 | 790 |
have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h" |
791 |
proof (rule exI [of _ N], clarify) |
|
792 |
fix n y h |
|
793 |
assume "N \<le> n" "y \<in> s" |
|
794 |
then have "cmod (f' n y - g' y) \<le> e" |
|
795 |
by (metis N) |
|
796 |
then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e" |
|
797 |
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2) |
|
798 |
then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h" |
|
799 |
by (simp add: norm_mult [symmetric] field_simps) |
|
800 |
qed |
|
801 |
} note ** = this |
|
802 |
show ?thesis |
|
803 |
unfolding has_field_derivative_def |
|
804 |
proof (rule has_derivative_sequence [OF cvs _ _ x]) |
|
805 |
show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)" |
|
806 |
by (metis has_field_derivative_def df) |
|
61969 | 807 |
next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l" |
56215 | 808 |
by (rule tf) |
809 |
next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h" |
|
810 |
by (blast intro: **) |
|
811 |
qed |
|
812 |
qed |
|
813 |
||
814 |
lemma has_complex_derivative_series: |
|
815 |
fixes s :: "complex set" |
|
816 |
assumes cvs: "convex s" |
|
817 |
and df: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)" |
|
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:
61531
diff
changeset
|
818 |
and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s |
56215 | 819 |
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e" |
820 |
and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)" |
|
821 |
shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within s))" |
|
822 |
proof - |
|
823 |
from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)" |
|
824 |
by blast |
|
825 |
{ fix e::real assume e: "e > 0" |
|
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:
61531
diff
changeset
|
826 |
then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s |
56215 | 827 |
\<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e" |
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:
61531
diff
changeset
|
828 |
by (metis conv) |
56215 | 829 |
have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h" |
830 |
proof (rule exI [of _ N], clarify) |
|
831 |
fix n y h |
|
832 |
assume "N \<le> n" "y \<in> s" |
|
833 |
then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e" |
|
834 |
by (metis N) |
|
835 |
then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e" |
|
836 |
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2) |
|
837 |
then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h" |
|
64267 | 838 |
by (simp add: norm_mult [symmetric] field_simps sum_distrib_left) |
56215 | 839 |
qed |
840 |
} note ** = this |
|
841 |
show ?thesis |
|
842 |
unfolding has_field_derivative_def |
|
843 |
proof (rule has_derivative_series [OF cvs _ _ x]) |
|
844 |
fix n x |
|
845 |
assume "x \<in> s" |
|
846 |
then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)" |
|
847 |
by (metis df has_field_derivative_def mult_commute_abs) |
|
848 |
next show " ((\<lambda>n. f n x) sums l)" |
|
849 |
by (rule sf) |
|
850 |
next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h" |
|
851 |
by (blast intro: **) |
|
852 |
qed |
|
853 |
qed |
|
854 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
855 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
856 |
lemma field_differentiable_series: |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
857 |
fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
858 |
assumes "convex s" "open s" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
859 |
assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
860 |
assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
861 |
assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> 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
|
862 |
shows "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
863 |
proof - |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
864 |
from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
865 |
unfolding uniformly_convergent_on_def by blast |
61808 | 866 |
from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
867 |
have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
868 |
by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
869 |
then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
870 |
"\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
871 |
from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
872 |
from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
873 |
by (simp add: has_field_derivative_def s) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
874 |
have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)" |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61975
diff
changeset
|
875 |
by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x]) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
876 |
(insert g, auto simp: sums_iff) |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
877 |
thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
878 |
by (auto simp: summable_def field_differentiable_def has_field_derivative_def) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
879 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
880 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
881 |
lemma field_differentiable_series': |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
882 |
fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
883 |
assumes "convex s" "open s" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
884 |
assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
885 |
assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
886 |
assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" |
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
887 |
shows "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x0)" |
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
888 |
using field_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+ |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
889 |
|
60420 | 890 |
subsection\<open>Bound theorem\<close> |
56215 | 891 |
|
62534
6855b348e828
complex_differentiable -> field_differentiable, etc. (making these theorems also available for type real)
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
892 |
lemma field_differentiable_bound: |
56215 | 893 |
fixes s :: "complex set" |
894 |
assumes cvs: "convex s" |
|
895 |
and df: "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)" |
|
896 |
and dn: "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B" |
|
897 |
and "x \<in> s" "y \<in> s" |
|
898 |
shows "norm(f x - f y) \<le> B * norm(x - y)" |
|
899 |
apply (rule differentiable_bound [OF cvs]) |
|
56223
7696903b9e61
generalize theory of operator norms to work with class real_normed_vector
huffman
parents:
56217
diff
changeset
|
900 |
apply (rule ballI, erule df [unfolded has_field_derivative_def]) |
7696903b9e61
generalize theory of operator norms to work with class real_normed_vector
huffman
parents:
56217
diff
changeset
|
901 |
apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn) |
7696903b9e61
generalize theory of operator norms to work with class real_normed_vector
huffman
parents:
56217
diff
changeset
|
902 |
apply fact |
7696903b9e61
generalize theory of operator norms to work with class real_normed_vector
huffman
parents:
56217
diff
changeset
|
903 |
apply fact |
56215 | 904 |
done |
905 |
||
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
906 |
subsection\<open>Inverse function theorem for complex derivatives\<close> |
56215 | 907 |
|
908 |
lemma has_complex_derivative_inverse_basic: |
|
909 |
fixes f :: "complex \<Rightarrow> complex" |
|
910 |
shows "DERIV f (g y) :> f' \<Longrightarrow> |
|
911 |
f' \<noteq> 0 \<Longrightarrow> |
|
912 |
continuous (at y) g \<Longrightarrow> |
|
913 |
open t \<Longrightarrow> |
|
914 |
y \<in> t \<Longrightarrow> |
|
915 |
(\<And>z. z \<in> t \<Longrightarrow> f (g z) = z) |
|
916 |
\<Longrightarrow> DERIV g y :> inverse (f')" |
|
917 |
unfolding has_field_derivative_def |
|
918 |
apply (rule has_derivative_inverse_basic) |
|
919 |
apply (auto simp: bounded_linear_mult_right) |
|
920 |
done |
|
921 |
||
922 |
lemma has_complex_derivative_inverse_strong: |
|
923 |
fixes f :: "complex \<Rightarrow> complex" |
|
924 |
shows "DERIV f x :> f' \<Longrightarrow> |
|
925 |
f' \<noteq> 0 \<Longrightarrow> |
|
926 |
open s \<Longrightarrow> |
|
927 |
x \<in> s \<Longrightarrow> |
|
928 |
continuous_on s f \<Longrightarrow> |
|
929 |
(\<And>z. z \<in> s \<Longrightarrow> g (f z) = z) |
|
930 |
\<Longrightarrow> DERIV g (f x) :> inverse (f')" |
|
931 |
unfolding has_field_derivative_def |
|
932 |
apply (rule has_derivative_inverse_strong [of s x f g ]) |
|
933 |
by auto |
|
934 |
||
935 |
lemma has_complex_derivative_inverse_strong_x: |
|
936 |
fixes f :: "complex \<Rightarrow> complex" |
|
937 |
shows "DERIV f (g y) :> f' \<Longrightarrow> |
|
938 |
f' \<noteq> 0 \<Longrightarrow> |
|
939 |
open s \<Longrightarrow> |
|
940 |
continuous_on s f \<Longrightarrow> |
|
941 |
g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow> |
|
942 |
(\<And>z. z \<in> s \<Longrightarrow> g (f z) = z) |
|
943 |
\<Longrightarrow> DERIV g y :> inverse (f')" |
|
944 |
unfolding has_field_derivative_def |
|
945 |
apply (rule has_derivative_inverse_strong_x [of s g y f]) |
|
946 |
by auto |
|
947 |
||
60420 | 948 |
subsection \<open>Taylor on Complex Numbers\<close> |
56215 | 949 |
|
64267 | 950 |
lemma sum_Suc_reindex: |
56215 | 951 |
fixes f :: "nat \<Rightarrow> 'a::ab_group_add" |
64267 | 952 |
shows "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}" |
56215 | 953 |
by (induct n) auto |
954 |
||
955 |
lemma complex_taylor: |
|
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:
61531
diff
changeset
|
956 |
assumes s: "convex s" |
56215 | 957 |
and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)" |
958 |
and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B" |
|
959 |
and w: "w \<in> s" |
|
960 |
and z: "z \<in> s" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
961 |
shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i))) |
56215 | 962 |
\<le> B * cmod(z - w)^(Suc n) / fact n" |
963 |
proof - |
|
964 |
have wzs: "closed_segment w z \<subseteq> s" using assms |
|
965 |
by (metis convex_contains_segment) |
|
966 |
{ fix u |
|
967 |
assume "u \<in> closed_segment w z" |
|
968 |
then have "u \<in> s" |
|
969 |
by (metis wzs subsetD) |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
970 |
have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) + |
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:
61531
diff
changeset
|
971 |
f (Suc i) u * (z-u)^i / (fact i)) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
972 |
f (Suc n) u * (z-u) ^ n / (fact n)" |
56215 | 973 |
proof (induction n) |
974 |
case 0 show ?case by simp |
|
975 |
next |
|
976 |
case (Suc n) |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
977 |
have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) + |
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:
61531
diff
changeset
|
978 |
f (Suc i) u * (z-u) ^ i / (fact i)) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
979 |
f (Suc n) u * (z-u) ^ n / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
980 |
f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
981 |
f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
982 |
using Suc by simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
983 |
also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))" |
56215 | 984 |
proof - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
985 |
have "(fact(Suc n)) * |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
986 |
(f(Suc n) u *(z-u) ^ n / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
987 |
f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
988 |
f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
989 |
((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
990 |
((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
991 |
((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
992 |
by (simp add: algebra_simps del: fact_Suc) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
993 |
also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
994 |
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
995 |
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
996 |
by (simp del: fact_Suc) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
997 |
also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
998 |
(f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
999 |
(f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
1000 |
by (simp only: fact_Suc of_nat_mult ac_simps) simp |
56215 | 1001 |
also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)" |
1002 |
by (simp add: algebra_simps) |
|
1003 |
finally show ?thesis |
|
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63332
diff
changeset
|
1004 |
by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc) |
56215 | 1005 |
qed |
1006 |
finally show ?case . |
|
1007 |
qed |
|
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:
61531
diff
changeset
|
1008 |
then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i))) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1009 |
has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n)) |
56215 | 1010 |
(at u within s)" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1011 |
apply (intro derivative_eq_intros) |
60420 | 1012 |
apply (blast intro: assms \<open>u \<in> s\<close>) |
56215 | 1013 |
apply (rule refl)+ |
1014 |
apply (auto simp: field_simps) |
|
1015 |
done |
|
1016 |
} note sum_deriv = this |
|
1017 |
{ fix u |
|
1018 |
assume u: "u \<in> closed_segment w z" |
|
1019 |
then have us: "u \<in> s" |
|
1020 |
by (metis wzs subsetD) |
|
1021 |
have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n" |
|
1022 |
by (metis norm_minus_commute order_refl) |
|
1023 |
also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n" |
|
1024 |
by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u]) |
|
1025 |
also have "... \<le> B * cmod (z - w) ^ n" |
|
1026 |
by (metis norm_ge_zero zero_le_power mult_right_mono B [OF us]) |
|
1027 |
finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" . |
|
1028 |
} note cmod_bound = this |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1029 |
have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)" |
56215 | 1030 |
by simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1031 |
also have "\<dots> = f 0 z / (fact 0)" |
64267 | 1032 |
by (subst sum_zero_power) simp |
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:
61531
diff
changeset
|
1033 |
finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i))) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1034 |
\<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1035 |
(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))" |
56215 | 1036 |
by (simp add: norm_minus_commute) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1037 |
also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - 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
|
1038 |
apply (rule field_differentiable_bound |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1039 |
[where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)" |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
1040 |
and s = "closed_segment w z", OF convex_closed_segment]) |
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:
61531
diff
changeset
|
1041 |
apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs] |
56215 | 1042 |
norm_divide norm_mult norm_power divide_le_cancel cmod_bound) |
1043 |
done |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1044 |
also have "... \<le> B * cmod (z - w) ^ Suc n / (fact n)" |
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:
61531
diff
changeset
|
1045 |
by (simp add: algebra_simps norm_minus_commute) |
56215 | 1046 |
finally show ?thesis . |
1047 |
qed |
|
1048 |
||
62408
86f27b264d3d
Conformal_mappings: a big development in complex analysis (+ some lemmas)
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1049 |
text\<open>Something more like the traditional MVT for real components\<close> |
56370
7c717ba55a0b
reorder Complex_Analysis_Basics; rename DD to deriv
hoelzl
parents:
56369
diff
changeset
|
1050 |
|
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1051 |
lemma complex_mvt_line: |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56332
diff
changeset
|
1052 |
assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)" |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
1053 |
shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))" |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1054 |
proof - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1055 |
have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1056 |
by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib) |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1057 |
note assms[unfolded has_field_derivative_def, derivative_intros] |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1058 |
show ?thesis |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1059 |
apply (cut_tac mvt_simple |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1060 |
[of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w + t *\<^sub>R z)" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1061 |
"\<lambda>u. Re o (\<lambda>h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\<lambda>t. t *\<^sub>R (z - w))"]) |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1062 |
apply auto |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1063 |
apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
1064 |
apply (auto simp: closed_segment_def twz) [] |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
1065 |
apply (intro derivative_eq_intros has_derivative_at_within, simp_all) |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56332
diff
changeset
|
1066 |
apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61235
diff
changeset
|
1067 |
apply (force simp: twz closed_segment_def) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1068 |
done |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1069 |
qed |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1070 |
|
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1071 |
lemma complex_taylor_mvt: |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1072 |
assumes "\<And>i x. \<lbrakk>x \<in> closed_segment w z; i \<le> n\<rbrakk> \<Longrightarrow> ((f i) has_field_derivative f (Suc i) x) (at x)" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1073 |
shows "\<exists>u. u \<in> closed_segment w z \<and> |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1074 |
Re (f 0 z) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1075 |
Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1076 |
(f (Suc n) u * (z-u)^n / (fact n)) * (z - w))" |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1077 |
proof - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1078 |
{ fix u |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1079 |
assume u: "u \<in> closed_segment w z" |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1080 |
have "(\<Sum>i = 0..n. |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1081 |
(f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1082 |
(fact i)) = |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1083 |
f (Suc 0) u - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1084 |
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1085 |
(fact (Suc n)) + |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1086 |
(\<Sum>i = 0..n. |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1087 |
(f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1088 |
(fact (Suc i)))" |
64267 | 1089 |
by (subst sum_Suc_reindex) simp |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1090 |
also have "... = f (Suc 0) u - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1091 |
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1092 |
(fact (Suc n)) + |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1093 |
(\<Sum>i = 0..n. |
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:
61531
diff
changeset
|
1094 |
f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i)) - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1095 |
f (Suc i) u * (z-u) ^ i / (fact i))" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
56889
diff
changeset
|
1096 |
by (simp only: diff_divide_distrib fact_cancel ac_simps) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1097 |
also have "... = f (Suc 0) u - |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1098 |
(f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) / |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1099 |
(fact (Suc n)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1100 |
f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u" |
64267 | 1101 |
by (subst sum_Suc_diff) auto |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1102 |
also have "... = f (Suc n) u * (z-u) ^ n / (fact n)" |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1103 |
by (simp only: algebra_simps diff_divide_distrib fact_cancel) |
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:
61531
diff
changeset
|
1104 |
finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1105 |
- of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1106 |
f (Suc n) u * (z - u) ^ n / (fact n)" . |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1107 |
then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1108 |
f (Suc n) u * (z - u) ^ n / (fact n)) (at u)" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1109 |
apply (intro derivative_eq_intros)+ |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1110 |
apply (force intro: u assms) |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1111 |
apply (rule refl)+ |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
56889
diff
changeset
|
1112 |
apply (auto simp: ac_simps) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1113 |
done |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1114 |
} |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1115 |
then show ?thesis |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1116 |
apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59615
diff
changeset
|
1117 |
"\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"]) |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1118 |
apply (auto simp add: intro: open_closed_segment) |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1119 |
done |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1120 |
qed |
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56223
diff
changeset
|
1121 |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1122 |
|
60420 | 1123 |
subsection \<open>Polynomal function extremal theorem, from HOL Light\<close> |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1124 |
|
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1125 |
lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1126 |
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1127 |
assumes "0 < e" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1128 |
shows "\<exists>M. \<forall>z. M \<le> norm(z) \<longrightarrow> norm (\<Sum>i\<le>n. c(i) * z^i) \<le> e * norm(z) ^ (Suc n)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1129 |
proof (induct n) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1130 |
case 0 with assms |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1131 |
show ?case |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1132 |
apply (rule_tac x="norm (c 0) / e" in exI) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1133 |
apply (auto simp: field_simps) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1134 |
done |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1135 |
next |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1136 |
case (Suc n) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1137 |
obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1138 |
using Suc assms by blast |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1139 |
show ?case |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1140 |
proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1141 |
fix z::'a |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1142 |
assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1143 |
then have z2: "e + norm (c (Suc n)) \<le> e * norm z" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1144 |
using assms by (simp add: field_simps) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1145 |
have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1146 |
using M [OF z1] by simp |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1147 |
then have "norm (\<Sum>i\<le>n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1148 |
by simp |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1149 |
then have "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1150 |
by (blast intro: norm_triangle_le elim: ) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1151 |
also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1152 |
by (simp add: norm_power norm_mult algebra_simps) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1153 |
also have "... \<le> (e * norm z) * norm z ^ Suc n" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1154 |
by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1155 |
finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)" |
60162 | 1156 |
by simp |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1157 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1158 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1159 |
|
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1160 |
lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1161 |
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1162 |
assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1163 |
shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1164 |
using kn |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1165 |
proof (induction n) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1166 |
case 0 |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1167 |
then show ?case |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1168 |
using k by simp |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1169 |
next |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1170 |
case (Suc m) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1171 |
let ?even = ?case |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1172 |
show ?even |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1173 |
proof (cases "c (Suc m) = 0") |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1174 |
case True |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1175 |
then show ?even using Suc k |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1176 |
by auto (metis antisym_conv less_eq_Suc_le not_le) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1177 |
next |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1178 |
case False |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1179 |
then obtain M where M: |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1180 |
"\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>m. c i * z^i) \<le> norm (c (Suc m)) / 2 * norm z ^ Suc m" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1181 |
using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1182 |
by auto |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1183 |
have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1184 |
proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1185 |
fix z::'a |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1186 |
assume z1: "M \<le> norm z" "1 \<le> norm z" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1187 |
and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1188 |
then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1189 |
using False by (simp add: field_simps) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1190 |
have nz: "norm z \<le> norm z ^ Suc m" |
60420 | 1191 |
by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc) |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1192 |
have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1193 |
by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1194 |
have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1195 |
\<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1196 |
using M [of z] Suc z1 by auto |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1197 |
also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1198 |
using nz by (simp add: mult_mono del: power_Suc) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1199 |
finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)" |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1200 |
using Suc.IH |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1201 |
apply (auto simp: eventually_at_infinity) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1202 |
apply (rule *) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1203 |
apply (simp add: field_simps norm_mult norm_power) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1204 |
done |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1205 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1206 |
then show ?even |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1207 |
by (simp add: eventually_at_infinity) |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1208 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1209 |
qed |
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59730
diff
changeset
|
1210 |
|
56215 | 1211 |
end |