author | paulson <lp15@cam.ac.uk> |
Fri, 27 Apr 2018 12:38:30 +0100 | |
changeset 68050 | 7eacc812ad1c |
parent 68004 | a8a20be7053a |
child 68054 | ebd179b82e20 |
permissions | -rw-r--r-- |
53572 | 1 |
(* Author: John Harrison |
2 |
Author: Robert Himmelmann, TU Muenchen (translation from HOL light) |
|
3 |
*) |
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36432 | 4 |
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section \<open>Fashoda meet theorem\<close> |
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|
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HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
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theory Fashoda_Theorem |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
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8 |
imports Brouwer_Fixpoint Path_Connected Cartesian_Euclidean_Space |
36432 | 9 |
begin |
10 |
||
67968 | 11 |
subsection \<open>Bijections between intervals\<close> |
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def3bbe6f2a5
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definition interval_bij :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a::euclidean_space" |
def3bbe6f2a5
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14 |
where "interval_bij = |
def3bbe6f2a5
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parents:
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(\<lambda>(a, b) (u, v) x. (\<Sum>i\<in>Basis. (u\<bullet>i + (x\<bullet>i - a\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (v\<bullet>i - u\<bullet>i)) *\<^sub>R i))" |
def3bbe6f2a5
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parents:
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16 |
|
def3bbe6f2a5
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parents:
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17 |
lemma interval_bij_affine: |
def3bbe6f2a5
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18 |
"interval_bij (a,b) (u,v) = (\<lambda>x. (\<Sum>i\<in>Basis. ((v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (x\<bullet>i)) *\<^sub>R i) + |
def3bbe6f2a5
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(\<Sum>i\<in>Basis. (u\<bullet>i - (v\<bullet>i - u\<bullet>i) / (b\<bullet>i - a\<bullet>i) * (a\<bullet>i)) *\<^sub>R i))" |
64267 | 20 |
by (auto simp: sum.distrib[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff |
21 |
field_simps inner_simps add_divide_distrib[symmetric] intro!: sum.cong) |
|
56273
def3bbe6f2a5
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22 |
|
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
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parents:
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23 |
lemma continuous_interval_bij: |
def3bbe6f2a5
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24 |
fixes a b :: "'a::euclidean_space" |
def3bbe6f2a5
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25 |
shows "continuous (at x) (interval_bij (a, b) (u, v))" |
64267 | 26 |
by (auto simp add: divide_inverse interval_bij_def intro!: continuous_sum continuous_intros) |
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parents:
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27 |
|
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
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28 |
lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a, b) (u, v))" |
def3bbe6f2a5
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29 |
apply(rule continuous_at_imp_continuous_on) |
def3bbe6f2a5
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parents:
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30 |
apply (rule, rule continuous_interval_bij) |
def3bbe6f2a5
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31 |
done |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
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32 |
|
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
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lemma in_interval_interval_bij: |
def3bbe6f2a5
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hoelzl
parents:
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34 |
fixes a b u v x :: "'a::euclidean_space" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
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parents:
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changeset
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35 |
assumes "x \<in> cbox a b" |
def3bbe6f2a5
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parents:
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36 |
and "cbox u v \<noteq> {}" |
def3bbe6f2a5
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hoelzl
parents:
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changeset
|
37 |
shows "interval_bij (a, b) (u, v) x \<in> cbox u v" |
64267 | 38 |
apply (simp only: interval_bij_def split_conv mem_box inner_sum_left_Basis cong: ball_cong) |
56273
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
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|
39 |
apply safe |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
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|
40 |
proof - |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
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changeset
|
41 |
fix i :: 'a |
def3bbe6f2a5
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parents:
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42 |
assume i: "i \<in> Basis" |
def3bbe6f2a5
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hoelzl
parents:
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changeset
|
43 |
have "cbox a b \<noteq> {}" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
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changeset
|
44 |
using assms by auto |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
45 |
with i have *: "a\<bullet>i \<le> b\<bullet>i" "u\<bullet>i \<le> v\<bullet>i" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
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changeset
|
46 |
using assms(2) by (auto simp add: box_eq_empty) |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
47 |
have x: "a\<bullet>i\<le>x\<bullet>i" "x\<bullet>i\<le>b\<bullet>i" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
48 |
using assms(1)[unfolded mem_box] using i by auto |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
49 |
have "0 \<le> (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)" |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56371
diff
changeset
|
50 |
using * x by auto |
56273
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
51 |
then show "u \<bullet> i \<le> u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
52 |
using * by auto |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
53 |
have "((x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (v \<bullet> i - u \<bullet> i) \<le> 1 * (v \<bullet> i - u \<bullet> i)" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
54 |
apply (rule mult_right_mono) |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
55 |
unfolding divide_le_eq_1 |
def3bbe6f2a5
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hoelzl
parents:
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|
56 |
using * x |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
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|
57 |
apply auto |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
58 |
done |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
59 |
then show "u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i) \<le> v \<bullet> i" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
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|
60 |
using * by auto |
def3bbe6f2a5
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hoelzl
parents:
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|
61 |
qed |
def3bbe6f2a5
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parents:
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62 |
|
def3bbe6f2a5
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hoelzl
parents:
56189
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changeset
|
63 |
lemma interval_bij_bij: |
def3bbe6f2a5
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hoelzl
parents:
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changeset
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64 |
"\<forall>(i::'a::euclidean_space)\<in>Basis. a\<bullet>i < b\<bullet>i \<and> u\<bullet>i < v\<bullet>i \<Longrightarrow> |
def3bbe6f2a5
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parents:
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65 |
interval_bij (a, b) (u, v) (interval_bij (u, v) (a, b) x) = x" |
def3bbe6f2a5
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hoelzl
parents:
56189
diff
changeset
|
66 |
by (auto simp: interval_bij_def euclidean_eq_iff[where 'a='a]) |
def3bbe6f2a5
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hoelzl
parents:
56189
diff
changeset
|
67 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
68 |
lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" |
56273
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
69 |
shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x" |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
70 |
using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis) |
def3bbe6f2a5
cleanup auxiliary proofs for Brouwer fixpoint theorem (removes ~2400 lines)
hoelzl
parents:
56189
diff
changeset
|
71 |
|
53572 | 72 |
|
60420 | 73 |
subsection \<open>Fashoda meet theorem\<close> |
36432 | 74 |
|
53572 | 75 |
lemma infnorm_2: |
76 |
fixes x :: "real^2" |
|
61945 | 77 |
shows "infnorm x = max \<bar>x$1\<bar> \<bar>x$2\<bar>" |
53572 | 78 |
unfolding infnorm_cart UNIV_2 by (rule cSup_eq) auto |
36432 | 79 |
|
53572 | 80 |
lemma infnorm_eq_1_2: |
81 |
fixes x :: "real^2" |
|
82 |
shows "infnorm x = 1 \<longleftrightarrow> |
|
61945 | 83 |
\<bar>x$1\<bar> \<le> 1 \<and> \<bar>x$2\<bar> \<le> 1 \<and> (x$1 = -1 \<or> x$1 = 1 \<or> x$2 = -1 \<or> x$2 = 1)" |
36432 | 84 |
unfolding infnorm_2 by auto |
85 |
||
53572 | 86 |
lemma infnorm_eq_1_imp: |
87 |
fixes x :: "real^2" |
|
88 |
assumes "infnorm x = 1" |
|
61945 | 89 |
shows "\<bar>x$1\<bar> \<le> 1" and "\<bar>x$2\<bar> \<le> 1" |
36432 | 90 |
using assms unfolding infnorm_eq_1_2 by auto |
91 |
||
53572 | 92 |
lemma fashoda_unit: |
93 |
fixes f g :: "real \<Rightarrow> real^2" |
|
56188 | 94 |
assumes "f ` {-1 .. 1} \<subseteq> cbox (-1) 1" |
95 |
and "g ` {-1 .. 1} \<subseteq> cbox (-1) 1" |
|
96 |
and "continuous_on {-1 .. 1} f" |
|
97 |
and "continuous_on {-1 .. 1} g" |
|
53572 | 98 |
and "f (- 1)$1 = - 1" |
99 |
and "f 1$1 = 1" "g (- 1) $2 = -1" |
|
100 |
and "g 1 $2 = 1" |
|
56188 | 101 |
shows "\<exists>s\<in>{-1 .. 1}. \<exists>t\<in>{-1 .. 1}. f s = g t" |
53572 | 102 |
proof (rule ccontr) |
103 |
assume "\<not> ?thesis" |
|
104 |
note as = this[unfolded bex_simps,rule_format] |
|
63040 | 105 |
define sqprojection |
106 |
where [abs_def]: "sqprojection z = (inverse (infnorm z)) *\<^sub>R z" for z :: "real^2" |
|
107 |
define negatex :: "real^2 \<Rightarrow> real^2" |
|
108 |
where "negatex x = (vector [-(x$1), x$2])" for x |
|
53572 | 109 |
have lem1: "\<forall>z::real^2. infnorm (negatex z) = infnorm z" |
36432 | 110 |
unfolding negatex_def infnorm_2 vector_2 by auto |
53572 | 111 |
have lem2: "\<forall>z. z \<noteq> 0 \<longrightarrow> infnorm (sqprojection z) = 1" |
112 |
unfolding sqprojection_def |
|
113 |
unfolding infnorm_mul[unfolded scalar_mult_eq_scaleR] |
|
114 |
unfolding abs_inverse real_abs_infnorm |
|
53628 | 115 |
apply (subst infnorm_eq_0[symmetric]) |
53572 | 116 |
apply auto |
117 |
done |
|
118 |
let ?F = "\<lambda>w::real^2. (f \<circ> (\<lambda>x. x$1)) w - (g \<circ> (\<lambda>x. x$2)) w" |
|
56188 | 119 |
have *: "\<And>i. (\<lambda>x::real^2. x $ i) ` cbox (- 1) 1 = {-1 .. 1}" |
53572 | 120 |
apply (rule set_eqI) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
121 |
unfolding image_iff Bex_def mem_box_cart interval_cbox_cart |
53572 | 122 |
apply rule |
123 |
defer |
|
124 |
apply (rule_tac x="vec x" in exI) |
|
125 |
apply auto |
|
126 |
done |
|
127 |
{ |
|
128 |
fix x |
|
56188 | 129 |
assume "x \<in> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w) ` (cbox (- 1) (1::real^2))" |
55675 | 130 |
then obtain w :: "real^2" where w: |
56188 | 131 |
"w \<in> cbox (- 1) 1" |
55675 | 132 |
"x = (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w" |
133 |
unfolding image_iff .. |
|
53572 | 134 |
then have "x \<noteq> 0" |
135 |
using as[of "w$1" "w$2"] |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
136 |
unfolding mem_box_cart atLeastAtMost_iff |
53572 | 137 |
by auto |
138 |
} note x0 = this |
|
139 |
have 21: "\<And>i::2. i \<noteq> 1 \<Longrightarrow> i = 2" |
|
140 |
using UNIV_2 by auto |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
53628
diff
changeset
|
141 |
have 1: "box (- 1) (1::real^2) \<noteq> {}" |
53572 | 142 |
unfolding interval_eq_empty_cart by auto |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
143 |
have 2: "continuous_on (cbox (- 1) 1) (negatex \<circ> sqprojection \<circ> ?F)" |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56273
diff
changeset
|
144 |
apply (intro continuous_intros continuous_on_component) |
53572 | 145 |
unfolding * |
146 |
apply (rule assms)+ |
|
147 |
apply (subst sqprojection_def) |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56273
diff
changeset
|
148 |
apply (intro continuous_intros) |
53572 | 149 |
apply (simp add: infnorm_eq_0 x0) |
150 |
apply (rule linear_continuous_on) |
|
151 |
proof - |
|
152 |
show "bounded_linear negatex" |
|
153 |
apply (rule bounded_linearI') |
|
154 |
unfolding vec_eq_iff |
|
155 |
proof (rule_tac[!] allI) |
|
156 |
fix i :: 2 |
|
157 |
fix x y :: "real^2" |
|
158 |
fix c :: real |
|
159 |
show "negatex (x + y) $ i = |
|
160 |
(negatex x + negatex y) $ i" "negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i" |
|
161 |
apply - |
|
162 |
apply (case_tac[!] "i\<noteq>1") |
|
163 |
prefer 3 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
164 |
apply (drule_tac[1-2] 21) |
53572 | 165 |
unfolding negatex_def |
166 |
apply (auto simp add:vector_2) |
|
167 |
done |
|
168 |
qed |
|
44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44531
diff
changeset
|
169 |
qed |
56188 | 170 |
have 3: "(negatex \<circ> sqprojection \<circ> ?F) ` cbox (-1) 1 \<subseteq> cbox (-1) 1" |
53572 | 171 |
unfolding subset_eq |
61166
5976fe402824
renamed method "goals" to "goal_cases" to emphasize its meaning;
wenzelm
parents:
61165
diff
changeset
|
172 |
proof (rule, goal_cases) |
61165 | 173 |
case (1 x) |
55675 | 174 |
then obtain y :: "real^2" where y: |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
175 |
"y \<in> cbox (- 1) 1" |
55675 | 176 |
"x = (negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) y" |
177 |
unfolding image_iff .. |
|
53572 | 178 |
have "?F y \<noteq> 0" |
179 |
apply (rule x0) |
|
180 |
using y(1) |
|
181 |
apply auto |
|
182 |
done |
|
183 |
then have *: "infnorm (sqprojection (?F y)) = 1" |
|
53628 | 184 |
unfolding y o_def |
185 |
by - (rule lem2[rule_format]) |
|
53572 | 186 |
have "infnorm x = 1" |
53628 | 187 |
unfolding *[symmetric] y o_def |
188 |
by (rule lem1[rule_format]) |
|
56188 | 189 |
then show "x \<in> cbox (-1) 1" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
190 |
unfolding mem_box_cart interval_cbox_cart infnorm_2 |
53572 | 191 |
apply - |
192 |
apply rule |
|
193 |
proof - |
|
61165 | 194 |
fix i |
195 |
assume "max \<bar>x $ 1\<bar> \<bar>x $ 2\<bar> = 1" |
|
196 |
then show "(- 1) $ i \<le> x $ i \<and> x $ i \<le> 1 $ i" |
|
53572 | 197 |
apply (cases "i = 1") |
198 |
defer |
|
199 |
apply (drule 21) |
|
200 |
apply auto |
|
201 |
done |
|
202 |
qed |
|
203 |
qed |
|
55675 | 204 |
obtain x :: "real^2" where x: |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
205 |
"x \<in> cbox (- 1) 1" |
55675 | 206 |
"(negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) x = x" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
207 |
apply (rule brouwer_weak[of "cbox (- 1) (1::real^2)" "negatex \<circ> sqprojection \<circ> ?F"]) |
56188 | 208 |
apply (rule compact_cbox convex_box)+ |
209 |
unfolding interior_cbox |
|
53572 | 210 |
apply (rule 1 2 3)+ |
55675 | 211 |
apply blast |
53572 | 212 |
done |
213 |
have "?F x \<noteq> 0" |
|
214 |
apply (rule x0) |
|
215 |
using x(1) |
|
216 |
apply auto |
|
217 |
done |
|
218 |
then have *: "infnorm (sqprojection (?F x)) = 1" |
|
53628 | 219 |
unfolding o_def |
220 |
by (rule lem2[rule_format]) |
|
53572 | 221 |
have nx: "infnorm x = 1" |
53628 | 222 |
apply (subst x(2)[symmetric]) |
223 |
unfolding *[symmetric] o_def |
|
53572 | 224 |
apply (rule lem1[rule_format]) |
225 |
done |
|
226 |
have "\<forall>x i. x \<noteq> 0 \<longrightarrow> (0 < (sqprojection x)$i \<longleftrightarrow> 0 < x$i)" |
|
227 |
and "\<forall>x i. x \<noteq> 0 \<longrightarrow> ((sqprojection x)$i < 0 \<longleftrightarrow> x$i < 0)" |
|
228 |
apply - |
|
229 |
apply (rule_tac[!] allI impI)+ |
|
230 |
proof - |
|
231 |
fix x :: "real^2" |
|
232 |
fix i :: 2 |
|
233 |
assume x: "x \<noteq> 0" |
|
234 |
have "inverse (infnorm x) > 0" |
|
53628 | 235 |
using x[unfolded infnorm_pos_lt[symmetric]] by auto |
53572 | 236 |
then show "(0 < sqprojection x $ i) = (0 < x $ i)" |
237 |
and "(sqprojection x $ i < 0) = (x $ i < 0)" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44136
diff
changeset
|
238 |
unfolding sqprojection_def vector_component_simps vector_scaleR_component real_scaleR_def |
53572 | 239 |
unfolding zero_less_mult_iff mult_less_0_iff |
240 |
by (auto simp add: field_simps) |
|
241 |
qed |
|
36432 | 242 |
note lem3 = this[rule_format] |
53572 | 243 |
have x1: "x $ 1 \<in> {- 1..1::real}" "x $ 2 \<in> {- 1..1::real}" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
244 |
using x(1) unfolding mem_box_cart by auto |
53572 | 245 |
then have nz: "f (x $ 1) - g (x $ 2) \<noteq> 0" |
246 |
unfolding right_minus_eq |
|
247 |
apply - |
|
248 |
apply (rule as) |
|
249 |
apply auto |
|
250 |
done |
|
251 |
have "x $ 1 = -1 \<or> x $ 1 = 1 \<or> x $ 2 = -1 \<or> x $ 2 = 1" |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
252 |
using nx unfolding infnorm_eq_1_2 by auto |
53572 | 253 |
then show False |
254 |
proof - |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
255 |
fix P Q R S |
53572 | 256 |
presume "P \<or> Q \<or> R \<or> S" |
257 |
and "P \<Longrightarrow> False" |
|
258 |
and "Q \<Longrightarrow> False" |
|
259 |
and "R \<Longrightarrow> False" |
|
260 |
and "S \<Longrightarrow> False" |
|
261 |
then show False by auto |
|
262 |
next |
|
263 |
assume as: "x$1 = 1" |
|
264 |
then have *: "f (x $ 1) $ 1 = 1" |
|
265 |
using assms(6) by auto |
|
36432 | 266 |
have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
41958
diff
changeset
|
267 |
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]] |
53572 | 268 |
unfolding as negatex_def vector_2 |
269 |
by auto |
|
270 |
moreover |
|
56188 | 271 |
from x1 have "g (x $ 2) \<in> cbox (-1) 1" |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
272 |
using assms(2) by blast |
53572 | 273 |
ultimately show False |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
274 |
unfolding lem3[OF nz] vector_component_simps * mem_box_cart |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
275 |
using not_le by auto |
53572 | 276 |
next |
277 |
assume as: "x$1 = -1" |
|
278 |
then have *: "f (x $ 1) $ 1 = - 1" |
|
279 |
using assms(5) by auto |
|
36432 | 280 |
have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
41958
diff
changeset
|
281 |
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]] |
53572 | 282 |
unfolding as negatex_def vector_2 |
283 |
by auto |
|
284 |
moreover |
|
56188 | 285 |
from x1 have "g (x $ 2) \<in> cbox (-1) 1" |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
286 |
using assms(2) by blast |
53572 | 287 |
ultimately show False |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
288 |
unfolding lem3[OF nz] vector_component_simps * mem_box_cart |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
289 |
using not_le by auto |
53572 | 290 |
next |
291 |
assume as: "x$2 = 1" |
|
292 |
then have *: "g (x $ 2) $ 2 = 1" |
|
293 |
using assms(8) by auto |
|
36432 | 294 |
have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
41958
diff
changeset
|
295 |
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]] |
53572 | 296 |
unfolding as negatex_def vector_2 |
297 |
by auto |
|
298 |
moreover |
|
56188 | 299 |
from x1 have "f (x $ 1) \<in> cbox (-1) 1" |
53572 | 300 |
apply - |
301 |
apply (rule assms(1)[unfolded subset_eq,rule_format]) |
|
302 |
apply auto |
|
303 |
done |
|
304 |
ultimately show False |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
305 |
unfolding lem3[OF nz] vector_component_simps * mem_box_cart |
53572 | 306 |
apply (erule_tac x=2 in allE) |
307 |
apply auto |
|
308 |
done |
|
309 |
next |
|
310 |
assume as: "x$2 = -1" |
|
311 |
then have *: "g (x $ 2) $ 2 = - 1" |
|
312 |
using assms(7) by auto |
|
36432 | 313 |
have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
41958
diff
changeset
|
314 |
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]] |
53572 | 315 |
unfolding as negatex_def vector_2 |
316 |
by auto |
|
317 |
moreover |
|
56188 | 318 |
from x1 have "f (x $ 1) \<in> cbox (-1) 1" |
53572 | 319 |
apply - |
320 |
apply (rule assms(1)[unfolded subset_eq,rule_format]) |
|
321 |
apply auto |
|
322 |
done |
|
323 |
ultimately show False |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
324 |
unfolding lem3[OF nz] vector_component_simps * mem_box_cart |
53572 | 325 |
apply (erule_tac x=2 in allE) |
326 |
apply auto |
|
327 |
done |
|
328 |
qed auto |
|
329 |
qed |
|
36432 | 330 |
|
53572 | 331 |
lemma fashoda_unit_path: |
332 |
fixes f g :: "real \<Rightarrow> real^2" |
|
333 |
assumes "path f" |
|
334 |
and "path g" |
|
56188 | 335 |
and "path_image f \<subseteq> cbox (-1) 1" |
336 |
and "path_image g \<subseteq> cbox (-1) 1" |
|
53572 | 337 |
and "(pathstart f)$1 = -1" |
338 |
and "(pathfinish f)$1 = 1" |
|
339 |
and "(pathstart g)$2 = -1" |
|
340 |
and "(pathfinish g)$2 = 1" |
|
341 |
obtains z where "z \<in> path_image f" and "z \<in> path_image g" |
|
342 |
proof - |
|
36432 | 343 |
note assms=assms[unfolded path_def pathstart_def pathfinish_def path_image_def] |
63040 | 344 |
define iscale where [abs_def]: "iscale z = inverse 2 *\<^sub>R (z + 1)" for z :: real |
53572 | 345 |
have isc: "iscale ` {- 1..1} \<subseteq> {0..1}" |
346 |
unfolding iscale_def by auto |
|
347 |
have "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. (f \<circ> iscale) s = (g \<circ> iscale) t" |
|
348 |
proof (rule fashoda_unit) |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
349 |
show "(f \<circ> iscale) ` {- 1..1} \<subseteq> cbox (- 1) 1" "(g \<circ> iscale) ` {- 1..1} \<subseteq> cbox (- 1) 1" |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55675
diff
changeset
|
350 |
using isc and assms(3-4) by (auto simp add: image_comp [symmetric]) |
53572 | 351 |
have *: "continuous_on {- 1..1} iscale" |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56273
diff
changeset
|
352 |
unfolding iscale_def by (rule continuous_intros)+ |
36432 | 353 |
show "continuous_on {- 1..1} (f \<circ> iscale)" "continuous_on {- 1..1} (g \<circ> iscale)" |
53572 | 354 |
apply - |
355 |
apply (rule_tac[!] continuous_on_compose[OF *]) |
|
356 |
apply (rule_tac[!] continuous_on_subset[OF _ isc]) |
|
357 |
apply (rule assms)+ |
|
358 |
done |
|
359 |
have *: "(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" |
|
360 |
unfolding vec_eq_iff by auto |
|
361 |
show "(f \<circ> iscale) (- 1) $ 1 = - 1" |
|
362 |
and "(f \<circ> iscale) 1 $ 1 = 1" |
|
363 |
and "(g \<circ> iscale) (- 1) $ 2 = -1" |
|
364 |
and "(g \<circ> iscale) 1 $ 2 = 1" |
|
365 |
unfolding o_def iscale_def |
|
366 |
using assms |
|
367 |
by (auto simp add: *) |
|
368 |
qed |
|
55675 | 369 |
then obtain s t where st: |
370 |
"s \<in> {- 1..1}" |
|
371 |
"t \<in> {- 1..1}" |
|
372 |
"(f \<circ> iscale) s = (g \<circ> iscale) t" |
|
56188 | 373 |
by auto |
53572 | 374 |
show thesis |
53628 | 375 |
apply (rule_tac z = "f (iscale s)" in that) |
55675 | 376 |
using st |
53572 | 377 |
unfolding o_def path_image_def image_iff |
378 |
apply - |
|
379 |
apply (rule_tac x="iscale s" in bexI) |
|
380 |
prefer 3 |
|
381 |
apply (rule_tac x="iscale t" in bexI) |
|
382 |
using isc[unfolded subset_eq, rule_format] |
|
383 |
apply auto |
|
384 |
done |
|
385 |
qed |
|
36432 | 386 |
|
53627 | 387 |
lemma fashoda: |
388 |
fixes b :: "real^2" |
|
389 |
assumes "path f" |
|
390 |
and "path g" |
|
56188 | 391 |
and "path_image f \<subseteq> cbox a b" |
392 |
and "path_image g \<subseteq> cbox a b" |
|
53627 | 393 |
and "(pathstart f)$1 = a$1" |
394 |
and "(pathfinish f)$1 = b$1" |
|
395 |
and "(pathstart g)$2 = a$2" |
|
396 |
and "(pathfinish g)$2 = b$2" |
|
397 |
obtains z where "z \<in> path_image f" and "z \<in> path_image g" |
|
398 |
proof - |
|
399 |
fix P Q S |
|
400 |
presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" and "Q \<Longrightarrow> thesis" and "S \<Longrightarrow> thesis" |
|
401 |
then show thesis |
|
402 |
by auto |
|
403 |
next |
|
56188 | 404 |
have "cbox a b \<noteq> {}" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
53628
diff
changeset
|
405 |
using assms(3) using path_image_nonempty[of f] by auto |
53627 | 406 |
then have "a \<le> b" |
407 |
unfolding interval_eq_empty_cart less_eq_vec_def by (auto simp add: not_less) |
|
408 |
then show "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" |
|
409 |
unfolding less_eq_vec_def forall_2 by auto |
|
410 |
next |
|
411 |
assume as: "a$1 = b$1" |
|
412 |
have "\<exists>z\<in>path_image g. z$2 = (pathstart f)$2" |
|
413 |
apply (rule connected_ivt_component_cart) |
|
414 |
apply (rule connected_path_image assms)+ |
|
415 |
apply (rule pathstart_in_path_image) |
|
416 |
apply (rule pathfinish_in_path_image) |
|
36432 | 417 |
unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"] |
53627 | 418 |
unfolding pathstart_def |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
419 |
apply (auto simp add: less_eq_vec_def mem_box_cart) |
53627 | 420 |
done |
55675 | 421 |
then obtain z :: "real^2" where z: "z \<in> path_image g" "z $ 2 = pathstart f $ 2" .. |
56188 | 422 |
have "z \<in> cbox a b" |
53627 | 423 |
using z(1) assms(4) |
424 |
unfolding path_image_def |
|
56188 | 425 |
by blast |
53627 | 426 |
then have "z = f 0" |
427 |
unfolding vec_eq_iff forall_2 |
|
428 |
unfolding z(2) pathstart_def |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
429 |
using assms(3)[unfolded path_image_def subset_eq mem_box_cart,rule_format,of "f 0" 1] |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
430 |
unfolding mem_box_cart |
53627 | 431 |
apply (erule_tac x=1 in allE) |
432 |
using as |
|
433 |
apply auto |
|
434 |
done |
|
435 |
then show thesis |
|
436 |
apply - |
|
437 |
apply (rule that[OF _ z(1)]) |
|
438 |
unfolding path_image_def |
|
439 |
apply auto |
|
440 |
done |
|
441 |
next |
|
442 |
assume as: "a$2 = b$2" |
|
443 |
have "\<exists>z\<in>path_image f. z$1 = (pathstart g)$1" |
|
444 |
apply (rule connected_ivt_component_cart) |
|
445 |
apply (rule connected_path_image assms)+ |
|
446 |
apply (rule pathstart_in_path_image) |
|
447 |
apply (rule pathfinish_in_path_image) |
|
448 |
unfolding assms |
|
449 |
using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"] |
|
450 |
unfolding pathstart_def |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
451 |
apply (auto simp add: less_eq_vec_def mem_box_cart) |
53627 | 452 |
done |
55675 | 453 |
then obtain z where z: "z \<in> path_image f" "z $ 1 = pathstart g $ 1" .. |
56188 | 454 |
have "z \<in> cbox a b" |
53627 | 455 |
using z(1) assms(3) |
456 |
unfolding path_image_def |
|
56188 | 457 |
by blast |
53627 | 458 |
then have "z = g 0" |
459 |
unfolding vec_eq_iff forall_2 |
|
460 |
unfolding z(2) pathstart_def |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
461 |
using assms(4)[unfolded path_image_def subset_eq mem_box_cart,rule_format,of "g 0" 2] |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
462 |
unfolding mem_box_cart |
53627 | 463 |
apply (erule_tac x=2 in allE) |
464 |
using as |
|
465 |
apply auto |
|
466 |
done |
|
467 |
then show thesis |
|
468 |
apply - |
|
469 |
apply (rule that[OF z(1)]) |
|
470 |
unfolding path_image_def |
|
471 |
apply auto |
|
472 |
done |
|
473 |
next |
|
474 |
assume as: "a $ 1 < b $ 1 \<and> a $ 2 < b $ 2" |
|
56188 | 475 |
have int_nem: "cbox (-1) (1::real^2) \<noteq> {}" |
53627 | 476 |
unfolding interval_eq_empty_cart by auto |
55675 | 477 |
obtain z :: "real^2" where z: |
478 |
"z \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}" |
|
479 |
"z \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}" |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
480 |
apply (rule fashoda_unit_path[of "interval_bij (a,b) (- 1,1) \<circ> f" "interval_bij (a,b) (- 1,1) \<circ> g"]) |
36432 | 481 |
unfolding path_def path_image_def pathstart_def pathfinish_def |
53627 | 482 |
apply (rule_tac[1-2] continuous_on_compose) |
483 |
apply (rule assms[unfolded path_def] continuous_on_interval_bij)+ |
|
484 |
unfolding subset_eq |
|
485 |
apply(rule_tac[1-2] ballI) |
|
486 |
proof - |
|
487 |
fix x |
|
488 |
assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}" |
|
55675 | 489 |
then obtain y where y: |
490 |
"y \<in> {0..1}" |
|
491 |
"x = (interval_bij (a, b) (- 1, 1) \<circ> f) y" |
|
492 |
unfolding image_iff .. |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
493 |
show "x \<in> cbox (- 1) 1" |
53627 | 494 |
unfolding y o_def |
495 |
apply (rule in_interval_interval_bij) |
|
496 |
using y(1) |
|
497 |
using assms(3)[unfolded path_image_def subset_eq] int_nem |
|
498 |
apply auto |
|
499 |
done |
|
500 |
next |
|
501 |
fix x |
|
502 |
assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}" |
|
55675 | 503 |
then obtain y where y: |
504 |
"y \<in> {0..1}" |
|
505 |
"x = (interval_bij (a, b) (- 1, 1) \<circ> g) y" |
|
506 |
unfolding image_iff .. |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57418
diff
changeset
|
507 |
show "x \<in> cbox (- 1) 1" |
53627 | 508 |
unfolding y o_def |
509 |
apply (rule in_interval_interval_bij) |
|
510 |
using y(1) |
|
511 |
using assms(4)[unfolded path_image_def subset_eq] int_nem |
|
512 |
apply auto |
|
513 |
done |
|
514 |
next |
|
515 |
show "(interval_bij (a, b) (- 1, 1) \<circ> f) 0 $ 1 = -1" |
|
516 |
and "(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1" |
|
517 |
and "(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1" |
|
518 |
and "(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1" |
|
56188 | 519 |
using assms as |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67968
diff
changeset
|
520 |
by (simp_all add: cart_eq_inner_axis pathstart_def pathfinish_def interval_bij_def) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
44647
diff
changeset
|
521 |
(simp_all add: inner_axis) |
53627 | 522 |
qed |
55675 | 523 |
from z(1) obtain zf where zf: |
524 |
"zf \<in> {0..1}" |
|
525 |
"z = (interval_bij (a, b) (- 1, 1) \<circ> f) zf" |
|
526 |
unfolding image_iff .. |
|
527 |
from z(2) obtain zg where zg: |
|
528 |
"zg \<in> {0..1}" |
|
529 |
"z = (interval_bij (a, b) (- 1, 1) \<circ> g) zg" |
|
530 |
unfolding image_iff .. |
|
53627 | 531 |
have *: "\<forall>i. (- 1) $ i < (1::real^2) $ i \<and> a $ i < b $ i" |
532 |
unfolding forall_2 |
|
533 |
using as |
|
534 |
by auto |
|
535 |
show thesis |
|
536 |
apply (rule_tac z="interval_bij (- 1,1) (a,b) z" in that) |
|
537 |
apply (subst zf) |
|
538 |
defer |
|
539 |
apply (subst zg) |
|
540 |
unfolding o_def interval_bij_bij_cart[OF *] path_image_def |
|
541 |
using zf(1) zg(1) |
|
542 |
apply auto |
|
543 |
done |
|
544 |
qed |
|
36432 | 545 |
|
53627 | 546 |
|
60420 | 547 |
subsection \<open>Some slightly ad hoc lemmas I use below\<close> |
36432 | 548 |
|
53627 | 549 |
lemma segment_vertical: |
550 |
fixes a :: "real^2" |
|
551 |
assumes "a$1 = b$1" |
|
552 |
shows "x \<in> closed_segment a b \<longleftrightarrow> |
|
553 |
x$1 = a$1 \<and> x$1 = b$1 \<and> (a$2 \<le> x$2 \<and> x$2 \<le> b$2 \<or> b$2 \<le> x$2 \<and> x$2 \<le> a$2)" |
|
554 |
(is "_ = ?R") |
|
555 |
proof - |
|
36432 | 556 |
let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1" |
53627 | 557 |
{ |
558 |
presume "?L \<Longrightarrow> ?R" and "?R \<Longrightarrow> ?L" |
|
559 |
then show ?thesis |
|
560 |
unfolding closed_segment_def mem_Collect_eq |
|
53628 | 561 |
unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[symmetric] vector_component_simps |
53627 | 562 |
by blast |
563 |
} |
|
564 |
{ |
|
565 |
assume ?L |
|
55675 | 566 |
then obtain u where u: |
567 |
"x $ 1 = (1 - u) * a $ 1 + u * b $ 1" |
|
568 |
"x $ 2 = (1 - u) * a $ 2 + u * b $ 2" |
|
569 |
"0 \<le> u" |
|
570 |
"u \<le> 1" |
|
571 |
by blast |
|
53627 | 572 |
{ fix b a |
573 |
assume "b + u * a > a + u * b" |
|
574 |
then have "(1 - u) * b > (1 - u) * a" |
|
575 |
by (auto simp add:field_simps) |
|
576 |
then have "b \<ge> a" |
|
59555 | 577 |
apply (drule_tac mult_left_less_imp_less) |
53627 | 578 |
using u |
579 |
apply auto |
|
580 |
done |
|
581 |
then have "u * a \<le> u * b" |
|
582 |
apply - |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
583 |
apply (rule mult_left_mono[OF _ u(3)]) |
53627 | 584 |
using u(3-4) |
585 |
apply (auto simp add: field_simps) |
|
586 |
done |
|
587 |
} note * = this |
|
588 |
{ |
|
589 |
fix a b |
|
590 |
assume "u * b > u * a" |
|
591 |
then have "(1 - u) * a \<le> (1 - u) * b" |
|
592 |
apply - |
|
593 |
apply (rule mult_left_mono) |
|
59555 | 594 |
apply (drule mult_left_less_imp_less) |
53627 | 595 |
using u |
596 |
apply auto |
|
597 |
done |
|
598 |
then have "a + u * b \<le> b + u * a" |
|
599 |
by (auto simp add: field_simps) |
|
600 |
} note ** = this |
|
601 |
then show ?R |
|
602 |
unfolding u assms |
|
603 |
using u |
|
604 |
by (auto simp add:field_simps not_le intro: * **) |
|
605 |
} |
|
606 |
{ |
|
607 |
assume ?R |
|
608 |
then show ?L |
|
609 |
proof (cases "x$2 = b$2") |
|
610 |
case True |
|
611 |
then show ?L |
|
612 |
apply (rule_tac x="(x$2 - a$2) / (b$2 - a$2)" in exI) |
|
613 |
unfolding assms True |
|
60420 | 614 |
using \<open>?R\<close> |
53627 | 615 |
apply (auto simp add: field_simps) |
616 |
done |
|
617 |
next |
|
618 |
case False |
|
619 |
then show ?L |
|
620 |
apply (rule_tac x="1 - (x$2 - b$2) / (a$2 - b$2)" in exI) |
|
621 |
unfolding assms |
|
60420 | 622 |
using \<open>?R\<close> |
53627 | 623 |
apply (auto simp add: field_simps) |
624 |
done |
|
625 |
qed |
|
626 |
} |
|
627 |
qed |
|
36432 | 628 |
|
53627 | 629 |
lemma segment_horizontal: |
630 |
fixes a :: "real^2" |
|
631 |
assumes "a$2 = b$2" |
|
632 |
shows "x \<in> closed_segment a b \<longleftrightarrow> |
|
633 |
x$2 = a$2 \<and> x$2 = b$2 \<and> (a$1 \<le> x$1 \<and> x$1 \<le> b$1 \<or> b$1 \<le> x$1 \<and> x$1 \<le> a$1)" |
|
634 |
(is "_ = ?R") |
|
635 |
proof - |
|
36432 | 636 |
let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1" |
53627 | 637 |
{ |
638 |
presume "?L \<Longrightarrow> ?R" and "?R \<Longrightarrow> ?L" |
|
639 |
then show ?thesis |
|
640 |
unfolding closed_segment_def mem_Collect_eq |
|
53628 | 641 |
unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[symmetric] vector_component_simps |
53627 | 642 |
by blast |
643 |
} |
|
644 |
{ |
|
645 |
assume ?L |
|
55675 | 646 |
then obtain u where u: |
647 |
"x $ 1 = (1 - u) * a $ 1 + u * b $ 1" |
|
648 |
"x $ 2 = (1 - u) * a $ 2 + u * b $ 2" |
|
649 |
"0 \<le> u" |
|
650 |
"u \<le> 1" |
|
651 |
by blast |
|
53627 | 652 |
{ |
653 |
fix b a |
|
654 |
assume "b + u * a > a + u * b" |
|
655 |
then have "(1 - u) * b > (1 - u) * a" |
|
53628 | 656 |
by (auto simp add: field_simps) |
53627 | 657 |
then have "b \<ge> a" |
59555 | 658 |
apply (drule_tac mult_left_less_imp_less) |
53627 | 659 |
using u |
660 |
apply auto |
|
661 |
done |
|
662 |
then have "u * a \<le> u * b" |
|
663 |
apply - |
|
664 |
apply (rule mult_left_mono[OF _ u(3)]) |
|
665 |
using u(3-4) |
|
666 |
apply (auto simp add: field_simps) |
|
667 |
done |
|
668 |
} note * = this |
|
669 |
{ |
|
670 |
fix a b |
|
671 |
assume "u * b > u * a" |
|
672 |
then have "(1 - u) * a \<le> (1 - u) * b" |
|
673 |
apply - |
|
674 |
apply (rule mult_left_mono) |
|
59555 | 675 |
apply (drule mult_left_less_imp_less) |
53627 | 676 |
using u |
677 |
apply auto |
|
678 |
done |
|
679 |
then have "a + u * b \<le> b + u * a" |
|
680 |
by (auto simp add: field_simps) |
|
681 |
} note ** = this |
|
682 |
then show ?R |
|
683 |
unfolding u assms |
|
684 |
using u |
|
685 |
by (auto simp add: field_simps not_le intro: * **) |
|
686 |
} |
|
687 |
{ |
|
688 |
assume ?R |
|
689 |
then show ?L |
|
690 |
proof (cases "x$1 = b$1") |
|
691 |
case True |
|
692 |
then show ?L |
|
693 |
apply (rule_tac x="(x$1 - a$1) / (b$1 - a$1)" in exI) |
|
694 |
unfolding assms True |
|
60420 | 695 |
using \<open>?R\<close> |
53627 | 696 |
apply (auto simp add: field_simps) |
697 |
done |
|
698 |
next |
|
699 |
case False |
|
700 |
then show ?L |
|
701 |
apply (rule_tac x="1 - (x$1 - b$1) / (a$1 - b$1)" in exI) |
|
702 |
unfolding assms |
|
60420 | 703 |
using \<open>?R\<close> |
53627 | 704 |
apply (auto simp add: field_simps) |
705 |
done |
|
706 |
qed |
|
707 |
} |
|
708 |
qed |
|
36432 | 709 |
|
53627 | 710 |
|
60420 | 711 |
subsection \<open>Useful Fashoda corollary pointed out to me by Tom Hales\<close> |
36432 | 712 |
|
53627 | 713 |
lemma fashoda_interlace: |
714 |
fixes a :: "real^2" |
|
715 |
assumes "path f" |
|
716 |
and "path g" |
|
56188 | 717 |
and "path_image f \<subseteq> cbox a b" |
718 |
and "path_image g \<subseteq> cbox a b" |
|
53627 | 719 |
and "(pathstart f)$2 = a$2" |
720 |
and "(pathfinish f)$2 = a$2" |
|
721 |
and "(pathstart g)$2 = a$2" |
|
722 |
and "(pathfinish g)$2 = a$2" |
|
723 |
and "(pathstart f)$1 < (pathstart g)$1" |
|
724 |
and "(pathstart g)$1 < (pathfinish f)$1" |
|
725 |
and "(pathfinish f)$1 < (pathfinish g)$1" |
|
726 |
obtains z where "z \<in> path_image f" and "z \<in> path_image g" |
|
727 |
proof - |
|
56188 | 728 |
have "cbox a b \<noteq> {}" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
53628
diff
changeset
|
729 |
using path_image_nonempty[of f] using assms(3) by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36593
diff
changeset
|
730 |
note ab=this[unfolded interval_eq_empty_cart not_ex forall_2 not_less] |
56188 | 731 |
have "pathstart f \<in> cbox a b" |
732 |
and "pathfinish f \<in> cbox a b" |
|
733 |
and "pathstart g \<in> cbox a b" |
|
734 |
and "pathfinish g \<in> cbox a b" |
|
53628 | 735 |
using pathstart_in_path_image pathfinish_in_path_image |
736 |
using assms(3-4) |
|
737 |
by auto |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
738 |
note startfin = this[unfolded mem_box_cart forall_2] |
36432 | 739 |
let ?P1 = "linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2]) +++ |
740 |
linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f) +++ f +++ |
|
741 |
linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2]) +++ |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
742 |
linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2])" |
36432 | 743 |
let ?P2 = "linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g) +++ g +++ |
744 |
linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1]) +++ |
|
745 |
linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1]) +++ |
|
746 |
linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3])" |
|
747 |
let ?a = "vector[a$1 - 2, a$2 - 3]" |
|
748 |
let ?b = "vector[b$1 + 2, b$2 + 3]" |
|
53627 | 749 |
have P1P2: "path_image ?P1 = path_image (linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2])) \<union> |
36432 | 750 |
path_image (linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f)) \<union> path_image f \<union> |
751 |
path_image (linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2])) \<union> |
|
752 |
path_image (linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2]))" |
|
753 |
"path_image ?P2 = path_image(linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g)) \<union> path_image g \<union> |
|
754 |
path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) \<union> |
|
755 |
path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union> |
|
756 |
path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2) |
|
757 |
by(auto simp add: path_image_join path_linepath) |
|
56188 | 758 |
have abab: "cbox a b \<subseteq> cbox ?a ?b" |
759 |
unfolding interval_cbox_cart[symmetric] |
|
53627 | 760 |
by (auto simp add:less_eq_vec_def forall_2 vector_2) |
55675 | 761 |
obtain z where |
762 |
"z \<in> path_image |
|
763 |
(linepath (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) +++ |
|
764 |
linepath (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f) +++ |
|
765 |
f +++ |
|
766 |
linepath (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) +++ |
|
767 |
linepath (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]))" |
|
768 |
"z \<in> path_image |
|
769 |
(linepath (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g) +++ |
|
770 |
g +++ |
|
771 |
linepath (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1]) +++ |
|
772 |
linepath (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1]) +++ |
|
773 |
linepath (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]))" |
|
53627 | 774 |
apply (rule fashoda[of ?P1 ?P2 ?a ?b]) |
775 |
unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2 |
|
776 |
proof - |
|
53628 | 777 |
show "path ?P1" and "path ?P2" |
53627 | 778 |
using assms by auto |
56188 | 779 |
have "path_image ?P1 \<subseteq> cbox ?a ?b" |
53627 | 780 |
unfolding P1P2 path_image_linepath |
781 |
apply (rule Un_least)+ |
|
782 |
defer 3 |
|
56188 | 783 |
apply (rule_tac[1-4] convex_box(1)[unfolded convex_contains_segment,rule_format]) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
784 |
unfolding mem_box_cart forall_2 vector_2 |
53627 | 785 |
using ab startfin abab assms(3) |
786 |
using assms(9-) |
|
787 |
unfolding assms |
|
56189
c4daa97ac57a
removed dependencies on theory Ordered_Euclidean_Space
immler
parents:
56188
diff
changeset
|
788 |
apply (auto simp add: field_simps box_def) |
53627 | 789 |
done |
56188 | 790 |
then show "path_image ?P1 \<subseteq> cbox ?a ?b" . |
791 |
have "path_image ?P2 \<subseteq> cbox ?a ?b" |
|
53627 | 792 |
unfolding P1P2 path_image_linepath |
793 |
apply (rule Un_least)+ |
|
794 |
defer 2 |
|
56188 | 795 |
apply (rule_tac[1-4] convex_box(1)[unfolded convex_contains_segment,rule_format]) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
796 |
unfolding mem_box_cart forall_2 vector_2 |
53627 | 797 |
using ab startfin abab assms(4) |
798 |
using assms(9-) |
|
799 |
unfolding assms |
|
56189
c4daa97ac57a
removed dependencies on theory Ordered_Euclidean_Space
immler
parents:
56188
diff
changeset
|
800 |
apply (auto simp add: field_simps box_def) |
53627 | 801 |
done |
56188 | 802 |
then show "path_image ?P2 \<subseteq> cbox ?a ?b" . |
53627 | 803 |
show "a $ 1 - 2 = a $ 1 - 2" |
804 |
and "b $ 1 + 2 = b $ 1 + 2" |
|
805 |
and "pathstart g $ 2 - 3 = a $ 2 - 3" |
|
806 |
and "b $ 2 + 3 = b $ 2 + 3" |
|
807 |
by (auto simp add: assms) |
|
53628 | 808 |
qed |
809 |
note z=this[unfolded P1P2 path_image_linepath] |
|
53627 | 810 |
show thesis |
811 |
apply (rule that[of z]) |
|
812 |
proof - |
|
36432 | 813 |
have "(z \<in> closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) \<or> |
53627 | 814 |
z \<in> closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f)) \<or> |
815 |
z \<in> closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) \<or> |
|
816 |
z \<in> closed_segment (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]) \<Longrightarrow> |
|
817 |
(((z \<in> closed_segment (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g)) \<or> |
|
818 |
z \<in> closed_segment (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1])) \<or> |
|
819 |
z \<in> closed_segment (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1])) \<or> |
|
820 |
z \<in> closed_segment (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]) \<Longrightarrow> False" |
|
61166
5976fe402824
renamed method "goals" to "goal_cases" to emphasize its meaning;
wenzelm
parents:
61165
diff
changeset
|
821 |
proof (simp only: segment_vertical segment_horizontal vector_2, goal_cases) |
61167 | 822 |
case prems: 1 |
56188 | 823 |
have "pathfinish f \<in> cbox a b" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
824 |
using assms(3) pathfinish_in_path_image[of f] by auto |
53628 | 825 |
then have "1 + b $ 1 \<le> pathfinish f $ 1 \<Longrightarrow> False" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
826 |
unfolding mem_box_cart forall_2 by auto |
53627 | 827 |
then have "z$1 \<noteq> pathfinish f$1" |
61167 | 828 |
using prems(2) |
53628 | 829 |
using assms ab |
830 |
by (auto simp add: field_simps) |
|
56188 | 831 |
moreover have "pathstart f \<in> cbox a b" |
53628 | 832 |
using assms(3) pathstart_in_path_image[of f] |
833 |
by auto |
|
53627 | 834 |
then have "1 + b $ 1 \<le> pathstart f $ 1 \<Longrightarrow> False" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
835 |
unfolding mem_box_cart forall_2 |
53628 | 836 |
by auto |
53627 | 837 |
then have "z$1 \<noteq> pathstart f$1" |
61167 | 838 |
using prems(2) using assms ab |
53628 | 839 |
by (auto simp add: field_simps) |
53627 | 840 |
ultimately have *: "z$2 = a$2 - 2" |
61167 | 841 |
using prems(1) |
53628 | 842 |
by auto |
53627 | 843 |
have "z$1 \<noteq> pathfinish g$1" |
61167 | 844 |
using prems(2) |
53628 | 845 |
using assms ab |
846 |
by (auto simp add: field_simps *) |
|
56188 | 847 |
moreover have "pathstart g \<in> cbox a b" |
53628 | 848 |
using assms(4) pathstart_in_path_image[of g] |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63040
diff
changeset
|
849 |
by auto |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
850 |
note this[unfolded mem_box_cart forall_2] |
53627 | 851 |
then have "z$1 \<noteq> pathstart g$1" |
61167 | 852 |
using prems(1) |
53628 | 853 |
using assms ab |
854 |
by (auto simp add: field_simps *) |
|
36432 | 855 |
ultimately have "a $ 2 - 1 \<le> z $ 2 \<and> z $ 2 \<le> b $ 2 + 3 \<or> b $ 2 + 3 \<le> z $ 2 \<and> z $ 2 \<le> a $ 2 - 1" |
61167 | 856 |
using prems(2) |
53628 | 857 |
unfolding * assms |
858 |
by (auto simp add: field_simps) |
|
53627 | 859 |
then show False |
860 |
unfolding * using ab by auto |
|
861 |
qed |
|
862 |
then have "z \<in> path_image f \<or> z \<in> path_image g" |
|
863 |
using z unfolding Un_iff by blast |
|
56188 | 864 |
then have z': "z \<in> cbox a b" |
53628 | 865 |
using assms(3-4) |
866 |
by auto |
|
53627 | 867 |
have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow> |
868 |
z = pathstart f \<or> z = pathfinish f" |
|
53628 | 869 |
unfolding vec_eq_iff forall_2 assms |
870 |
by auto |
|
53627 | 871 |
with z' show "z \<in> path_image f" |
872 |
using z(1) |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
873 |
unfolding Un_iff mem_box_cart forall_2 |
53627 | 874 |
apply - |
875 |
apply (simp only: segment_vertical segment_horizontal vector_2) |
|
876 |
unfolding assms |
|
877 |
apply auto |
|
878 |
done |
|
879 |
have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow> |
|
880 |
z = pathstart g \<or> z = pathfinish g" |
|
53628 | 881 |
unfolding vec_eq_iff forall_2 assms |
882 |
by auto |
|
53627 | 883 |
with z' show "z \<in> path_image g" |
884 |
using z(2) |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
885 |
unfolding Un_iff mem_box_cart forall_2 |
53627 | 886 |
apply (simp only: segment_vertical segment_horizontal vector_2) |
887 |
unfolding assms |
|
888 |
apply auto |
|
889 |
done |
|
890 |
qed |
|
891 |
qed |
|
36432 | 892 |
|
893 |
end |