author | hoelzl |
Tue, 05 Mar 2013 15:43:15 +0100 | |
changeset 51344 | b3d246c92dfb |
parent 50884 | 2b21b4e2d7cb |
child 51478 | 270b21f3ae0a |
permissions | -rw-r--r-- |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
2 |
(* ========================================================================= *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
3 |
(* Results connected with topological dimension. *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
4 |
(* *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
5 |
(* At the moment this is just Brouwer's fixpoint theorem. The proof is from *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
6 |
(* Kuhn: "some combinatorial lemmas in topology", IBM J. v4. (1960) p. 518 *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
7 |
(* See "http://www.research.ibm.com/journal/rd/045/ibmrd0405K.pdf". *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
8 |
(* *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
9 |
(* The script below is quite messy, but at least we avoid formalizing any *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
10 |
(* topological machinery; we don't even use barycentric subdivision; this is *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
11 |
(* the big advantage of Kuhn's proof over the usual Sperner's lemma one. *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
12 |
(* *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
13 |
(* (c) Copyright, John Harrison 1998-2008 *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
14 |
(* ========================================================================= *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
15 |
|
33759
b369324fc244
Added the contributions of Robert Himmelmann to CONTRIBUTIONS and NEWS
hoelzl
parents:
33758
diff
changeset
|
16 |
(* Author: John Harrison |
b369324fc244
Added the contributions of Robert Himmelmann to CONTRIBUTIONS and NEWS
hoelzl
parents:
33758
diff
changeset
|
17 |
Translation from HOL light: Robert Himmelmann, TU Muenchen *) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
18 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
19 |
header {* Results connected with topological dimension. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
20 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
21 |
theory Brouwer_Fixpoint |
36432
1ad1cfeaec2d
move proof of Fashoda meet theorem into separate file
huffman
parents:
36431
diff
changeset
|
22 |
imports Convex_Euclidean_Space |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
23 |
begin |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
24 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
25 |
(** move this **) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
26 |
lemma divide_nonneg_nonneg:assumes "a \<ge> 0" "b \<ge> 0" shows "0 \<le> a / (b::real)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
27 |
apply(cases "b=0") defer apply(rule divide_nonneg_pos) using assms by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
28 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
29 |
lemma continuous_setsum: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
30 |
fixes f :: "'i \<Rightarrow> 'a::t2_space \<Rightarrow> 'b::real_normed_vector" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
31 |
assumes f: "\<And>i. i \<in> I \<Longrightarrow> continuous F (f i)" shows "continuous F (\<lambda>x. \<Sum>i\<in>I. f i x)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
32 |
proof cases |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
33 |
assume "finite I" from this f show ?thesis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
34 |
by (induct I) (auto intro!: continuous_intros) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
35 |
qed (auto intro!: continuous_intros) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
36 |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
37 |
lemma brouwer_compactness_lemma: |
50884
2b21b4e2d7cb
differentiate (cover) compactness and sequential compactness
hoelzl
parents:
50526
diff
changeset
|
38 |
fixes f :: "'a::metric_space \<Rightarrow> 'b::euclidean_space" |
2b21b4e2d7cb
differentiate (cover) compactness and sequential compactness
hoelzl
parents:
50526
diff
changeset
|
39 |
assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = 0))" |
49374 | 40 |
obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)" |
41 |
proof (cases "s={}") |
|
42 |
case False |
|
43 |
have "continuous_on s (norm \<circ> f)" |
|
49555 | 44 |
by (rule continuous_on_intros continuous_on_norm assms(2))+ |
49374 | 45 |
with False obtain x where x: "x\<in>s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y" |
46 |
using continuous_attains_inf[OF assms(1), of "norm \<circ> f"] unfolding o_def by auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
47 |
have "(norm \<circ> f) x > 0" using assms(3) and x(1) by auto |
49374 | 48 |
then show ?thesis by (rule that) (insert x(2), auto simp: o_def) |
49555 | 49 |
next |
50 |
case True |
|
51 |
show thesis by (rule that [of 1]) (auto simp: True) |
|
52 |
qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
53 |
|
49555 | 54 |
lemma kuhn_labelling_lemma: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
55 |
fixes P Q :: "'a::euclidean_space \<Rightarrow> bool" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
56 |
assumes "(\<forall>x. P x \<longrightarrow> P (f x))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
57 |
and "\<forall>x. P x \<longrightarrow> (\<forall>i\<in>Basis. Q i \<longrightarrow> 0 \<le> x\<bullet>i \<and> x\<bullet>i \<le> 1)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
58 |
shows "\<exists>l. (\<forall>x.\<forall>i\<in>Basis. l x i \<le> (1::nat)) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
59 |
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 0) \<longrightarrow> (l x i = 0)) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
60 |
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 1) \<longrightarrow> (l x i = 1)) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
61 |
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x\<bullet>i \<le> f(x)\<bullet>i) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
62 |
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)\<bullet>i \<le> x\<bullet>i)" |
49374 | 63 |
proof - |
64 |
have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" |
|
65 |
by auto |
|
49555 | 66 |
have *: "\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" |
49374 | 67 |
by auto |
68 |
show ?thesis |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
69 |
unfolding and_forall_thm Ball_def |
49374 | 70 |
apply(subst choice_iff[THEN sym])+ |
71 |
apply rule |
|
72 |
apply rule |
|
73 |
proof - |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
74 |
case (goal1 x) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
75 |
let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x \<bullet> xa = 0 \<longrightarrow> y = (0::nat)) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
76 |
(P x \<and> Q xa \<and> x \<bullet> xa = 1 \<longrightarrow> y = 1) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
77 |
(P x \<and> Q xa \<and> y = 0 \<longrightarrow> x \<bullet> xa \<le> f x \<bullet> xa) \<and> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
78 |
(P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x \<bullet> xa \<le> x \<bullet> xa)" |
49374 | 79 |
{ |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
80 |
assume "P x" "Q xa" "xa\<in>Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
81 |
then have "0 \<le> f x \<bullet> xa \<and> f x \<bullet> xa \<le> 1" |
49374 | 82 |
using assms(2)[rule_format,of "f x" xa] |
83 |
apply (drule_tac assms(1)[rule_format]) |
|
84 |
apply auto |
|
85 |
done |
|
86 |
} |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
87 |
then have "xa\<in>Basis \<Longrightarrow> ?R 0 \<or> ?R 1" by auto |
49374 | 88 |
then show ?case by auto |
89 |
qed |
|
90 |
qed |
|
91 |
||
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
92 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
93 |
subsection {* The key "counting" observation, somewhat abstracted. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
94 |
|
49374 | 95 |
lemma setsum_Un_disjoint': |
96 |
assumes "finite A" "finite B" "A \<inter> B = {}" "A \<union> B = C" |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
97 |
shows "setsum g C = setsum g A + setsum g B" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
98 |
using setsum_Un_disjoint[OF assms(1-3)] and assms(4) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
99 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
100 |
lemma kuhn_counting_lemma: assumes "finite faces" "finite simplices" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
101 |
"\<forall>f\<in>faces. bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 1)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
102 |
"\<forall>f\<in>faces. \<not> bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 2)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
103 |
"\<forall>s\<in>simplices. compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 1)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
104 |
"\<forall>s\<in>simplices. \<not> compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 0) \<or> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
105 |
(card {f \<in> faces. face f s \<and> compo' f} = 2)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
106 |
"odd(card {f \<in> faces. compo' f \<and> bnd f})" |
49374 | 107 |
shows "odd(card {s \<in> simplices. compo s})" |
108 |
proof - |
|
109 |
have "\<And>x. {f\<in>faces. compo' f \<and> bnd f \<and> face f x} \<union> {f\<in>faces. compo' f \<and> \<not>bnd f \<and> face f x} = |
|
110 |
{f\<in>faces. compo' f \<and> face f x}" |
|
111 |
"\<And>x. {f \<in> faces. compo' f \<and> bnd f \<and> face f x} \<inter> {f \<in> faces. compo' f \<and> \<not> bnd f \<and> face f x} = {}" |
|
112 |
by auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
113 |
hence lem1:"setsum (\<lambda>s. (card {f \<in> faces. face f s \<and> compo' f})) simplices = |
49374 | 114 |
setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f s}) simplices + |
115 |
setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> \<not> (bnd f)}. face f s}) simplices" |
|
116 |
unfolding setsum_addf[THEN sym] |
|
117 |
apply - |
|
118 |
apply(rule setsum_cong2) |
|
119 |
using assms(1) |
|
120 |
apply (auto simp add: card_Un_Int, auto simp add:conj_commute) |
|
121 |
done |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
122 |
have lem2:"setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f j}) simplices = |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
123 |
1 * card {f \<in> faces. compo' f \<and> bnd f}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
124 |
"setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> \<not> bnd f}. face f j}) simplices = |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
125 |
2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}" |
49374 | 126 |
apply(rule_tac[!] setsum_multicount) |
127 |
using assms |
|
128 |
apply auto |
|
129 |
done |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
130 |
have lem3:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices = |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
131 |
setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. compo s}+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
132 |
setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
133 |
apply(rule setsum_Un_disjoint') using assms(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
134 |
have lem4:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. compo s} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
135 |
= setsum (\<lambda>s. 1) {s \<in> simplices. compo s}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
136 |
apply(rule setsum_cong2) using assms(5) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
137 |
have lem5: "setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s} = |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
138 |
setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
139 |
{s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 0)} + |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
140 |
setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
141 |
{s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 2)}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
142 |
apply(rule setsum_Un_disjoint') using assms(2,6) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
143 |
have *:"int (\<Sum>s\<in>{s \<in> simplices. compo s}. card {f \<in> faces. face f s \<and> compo' f}) = |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
144 |
int (card {f \<in> faces. compo' f \<and> bnd f} + 2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}) - |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
145 |
int (card {s \<in> simplices. \<not> compo s \<and> card {f \<in> faces. face f s \<and> compo' f} = 2} * 2)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
146 |
using lem1[unfolded lem3 lem2 lem5] by auto |
49374 | 147 |
have even_minus_odd:"\<And>x y. even x \<Longrightarrow> odd (y::int) \<Longrightarrow> odd (x - y)" |
148 |
using assms by auto |
|
149 |
have odd_minus_even:"\<And>x y. odd x \<Longrightarrow> even (y::int) \<Longrightarrow> odd (x - y)" |
|
150 |
using assms by auto |
|
151 |
show ?thesis |
|
152 |
unfolding even_nat_def card_eq_setsum and lem4[THEN sym] and *[unfolded card_eq_setsum] |
|
153 |
unfolding card_eq_setsum[THEN sym] |
|
154 |
apply (rule odd_minus_even) |
|
155 |
unfolding of_nat_add |
|
156 |
apply(rule odd_plus_even) |
|
157 |
apply(rule assms(7)[unfolded even_nat_def]) |
|
158 |
unfolding int_mult |
|
159 |
apply auto |
|
160 |
done |
|
161 |
qed |
|
162 |
||
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
163 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
164 |
subsection {* The odd/even result for faces of complete vertices, generalized. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
165 |
|
49374 | 166 |
lemma card_1_exists: "card s = 1 \<longleftrightarrow> (\<exists>!x. x \<in> s)" |
167 |
unfolding One_nat_def |
|
168 |
apply rule |
|
169 |
apply (drule card_eq_SucD) |
|
170 |
defer |
|
49555 | 171 |
apply (erule ex1E) |
49374 | 172 |
proof - |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
173 |
fix x assume as:"x \<in> s" "\<forall>y. y \<in> s \<longrightarrow> y = x" |
49374 | 174 |
have *: "s = insert x {}" |
49555 | 175 |
apply (rule set_eqI, rule) unfolding singleton_iff |
49374 | 176 |
apply (rule as(2)[rule_format]) using as(1) |
177 |
apply auto |
|
178 |
done |
|
179 |
show "card s = Suc 0" unfolding * using card_insert by auto |
|
180 |
qed auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
181 |
|
49374 | 182 |
lemma card_2_exists: "card s = 2 \<longleftrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. (z = x) \<or> (z = y)))" |
183 |
proof |
|
184 |
assume "card s = 2" |
|
185 |
then obtain x y where obt:"s = {x, y}" "x\<noteq>y" unfolding numeral_2_eq_2 |
|
49555 | 186 |
apply - apply (erule exE conjE | drule card_eq_SucD)+ apply auto done |
187 |
show "\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. z = x \<or> z = y)" using obt by auto |
|
188 |
next |
|
49374 | 189 |
assume "\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. z = x \<or> z = y)" |
49555 | 190 |
then obtain x y where "x\<in>s" "y\<in>s" "x \<noteq> y" "\<forall>z\<in>s. z = x \<or> z = y" by auto |
191 |
then have "s = {x, y}" by auto |
|
192 |
with `x \<noteq> y` show "card s = 2" by auto |
|
193 |
qed |
|
194 |
||
195 |
lemma image_lemma_0: |
|
196 |
assumes "card {a\<in>s. f ` (s - {a}) = t - {b}} = n" |
|
197 |
shows "card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> (f ` s' = t - {b})} = n" |
|
198 |
proof - |
|
199 |
have *:"{s'. \<exists>a\<in>s. (s' = s - {a}) \<and> (f ` s' = t - {b})} = (\<lambda>a. s - {a}) ` {a\<in>s. f ` (s - {a}) = t - {b}}" |
|
200 |
by auto |
|
201 |
show ?thesis unfolding * unfolding assms[THEN sym] apply(rule card_image) unfolding inj_on_def |
|
202 |
apply (rule, rule, rule) unfolding mem_Collect_eq apply auto done |
|
203 |
qed |
|
204 |
||
205 |
lemma image_lemma_1: |
|
206 |
assumes "finite s" "finite t" "card s = card t" "f ` s = t" "b \<in> t" |
|
207 |
shows "card {s'. \<exists>a\<in>s. s' = s - {a} \<and> f ` s' = t - {b}} = 1" |
|
208 |
proof - |
|
209 |
obtain a where a: "b = f a" "a\<in>s" using assms(4-5) by auto |
|
210 |
have inj: "inj_on f s" apply (rule eq_card_imp_inj_on) using assms(1-4) apply auto done |
|
211 |
have *: "{a \<in> s. f ` (s - {a}) = t - {b}} = {a}" apply (rule set_eqI) unfolding singleton_iff |
|
212 |
apply (rule, rule inj[unfolded inj_on_def, rule_format]) |
|
213 |
unfolding a using a(2) and assms and inj[unfolded inj_on_def] apply auto |
|
214 |
done |
|
215 |
show ?thesis apply (rule image_lemma_0) unfolding * apply auto done |
|
49374 | 216 |
qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
217 |
|
49555 | 218 |
lemma image_lemma_2: |
219 |
assumes "finite s" "finite t" "card s = card t" "f ` s \<subseteq> t" "f ` s \<noteq> t" "b \<in> t" |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
220 |
shows "(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 0) \<or> |
49555 | 221 |
(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 2)" |
222 |
proof (cases "{a\<in>s. f ` (s - {a}) = t - {b}} = {}") |
|
223 |
case True |
|
224 |
then show ?thesis |
|
225 |
apply - |
|
226 |
apply (rule disjI1, rule image_lemma_0) using assms(1) apply auto done |
|
227 |
next |
|
228 |
let ?M = "{a\<in>s. f ` (s - {a}) = t - {b}}" |
|
229 |
case False |
|
230 |
then obtain a where "a\<in>?M" by auto |
|
231 |
then have a: "a\<in>s" "f ` (s - {a}) = t - {b}" by auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
232 |
have "f a \<in> t - {b}" using a and assms by auto |
49555 | 233 |
then have "\<exists>c \<in> s - {a}. f a = f c" unfolding image_iff[symmetric] and a by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
234 |
then obtain c where c:"c \<in> s" "a \<noteq> c" "f a = f c" by auto |
49555 | 235 |
then have *: "f ` (s - {c}) = f ` (s - {a})" |
236 |
apply - |
|
237 |
apply (rule set_eqI, rule) |
|
238 |
proof - |
|
239 |
fix x |
|
240 |
assume "x \<in> f ` (s - {a})" |
|
241 |
then obtain y where y: "f y = x" "y\<in>s- {a}" by auto |
|
242 |
then show "x \<in> f ` (s - {c})" |
|
243 |
unfolding image_iff |
|
244 |
apply (rule_tac x = "if y = c then a else y" in bexI) |
|
245 |
using c a apply auto done |
|
246 |
qed auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
247 |
have "c\<in>?M" unfolding mem_Collect_eq and * using a and c(1) by auto |
49555 | 248 |
show ?thesis |
249 |
apply (rule disjI2, rule image_lemma_0) unfolding card_2_exists |
|
250 |
apply (rule bexI[OF _ `a\<in>?M`], rule bexI[OF _ `c\<in>?M`],rule,rule `a\<noteq>c`) |
|
251 |
proof (rule, unfold mem_Collect_eq, erule conjE) |
|
252 |
fix z |
|
253 |
assume as: "z \<in> s" "f ` (s - {z}) = t - {b}" |
|
254 |
have inj: "inj_on f (s - {z})" |
|
255 |
apply (rule eq_card_imp_inj_on) |
|
256 |
unfolding as using as(1) and assms apply auto |
|
257 |
done |
|
258 |
show "z = a \<or> z = c" |
|
259 |
proof (rule ccontr) |
|
260 |
assume "\<not> ?thesis" |
|
261 |
then show False |
|
262 |
using inj[unfolded inj_on_def, THEN bspec[where x=a], THEN bspec[where x=c]] |
|
263 |
using `a\<in>s` `c\<in>s` `f a = f c` `a\<noteq>c` apply auto |
|
264 |
done |
|
265 |
qed |
|
266 |
qed |
|
267 |
qed |
|
268 |
||
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
269 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
270 |
subsection {* Combine this with the basic counting lemma. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
271 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
272 |
lemma kuhn_complete_lemma: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
273 |
assumes "finite simplices" |
49555 | 274 |
"\<forall>f s. face f s \<longleftrightarrow> (\<exists>a\<in>s. f = s - {a})" |
275 |
"\<forall>s\<in>simplices. card s = n + 2" "\<forall>s\<in>simplices. (rl ` s) \<subseteq> {0..n+1}" |
|
276 |
"\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. bnd f \<longrightarrow> (card {s\<in>simplices. face f s} = 1)" |
|
277 |
"\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. \<not>bnd f \<longrightarrow> (card {s\<in>simplices. face f s} = 2)" |
|
278 |
"odd(card {f\<in>{f. \<exists>s\<in>simplices. face f s}. rl ` f = {0..n} \<and> bnd f})" |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
279 |
shows "odd (card {s\<in>simplices. (rl ` s = {0..n+1})})" |
49555 | 280 |
apply (rule kuhn_counting_lemma) |
281 |
defer |
|
282 |
apply (rule assms)+ |
|
283 |
prefer 3 |
|
284 |
apply (rule assms) |
|
285 |
proof (rule_tac[1-2] ballI impI)+ |
|
286 |
have *: "{f. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s - {a}} = (\<Union>s\<in>simplices. {f. \<exists>a\<in>s. (f = s - {a})})" |
|
287 |
by auto |
|
288 |
have **: "\<forall>s\<in>simplices. card s = n + 2 \<and> finite s" |
|
289 |
using assms(3) by (auto intro: card_ge_0_finite) |
|
290 |
show "finite {f. \<exists>s\<in>simplices. face f s}" |
|
291 |
unfolding assms(2)[rule_format] and * |
|
292 |
apply (rule finite_UN_I[OF assms(1)]) using ** apply auto |
|
293 |
done |
|
294 |
have *: "\<And>P f s. s\<in>simplices \<Longrightarrow> (f \<in> {f. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s - {a}}) \<and> |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
295 |
(\<exists>a\<in>s. (f = s - {a})) \<and> P f \<longleftrightarrow> (\<exists>a\<in>s. (f = s - {a}) \<and> P f)" by auto |
49555 | 296 |
fix s assume s: "s\<in>simplices" |
297 |
let ?S = "{f \<in> {f. \<exists>s\<in>simplices. face f s}. face f s \<and> rl ` f = {0..n}}" |
|
298 |
have "{0..n + 1} - {n + 1} = {0..n}" by auto |
|
299 |
then have S: "?S = {s'. \<exists>a\<in>s. s' = s - {a} \<and> rl ` s' = {0..n + 1} - {n + 1}}" |
|
300 |
apply - |
|
301 |
apply (rule set_eqI) |
|
302 |
unfolding assms(2)[rule_format] mem_Collect_eq and *[OF s, unfolded mem_Collect_eq, where P="\<lambda>x. rl ` x = {0..n}"] |
|
303 |
apply auto |
|
304 |
done |
|
305 |
show "rl ` s = {0..n+1} \<Longrightarrow> card ?S = 1" "rl ` s \<noteq> {0..n+1} \<Longrightarrow> card ?S = 0 \<or> card ?S = 2" |
|
306 |
unfolding S |
|
307 |
apply(rule_tac[!] image_lemma_1 image_lemma_2) |
|
308 |
using ** assms(4) and s apply auto |
|
309 |
done |
|
310 |
qed |
|
311 |
||
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
312 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
313 |
subsection {*We use the following notion of ordering rather than pointwise indexing. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
314 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
315 |
definition "kle n x y \<longleftrightarrow> (\<exists>k\<subseteq>{1..n::nat}. (\<forall>j. y(j) = x(j) + (if j \<in> k then (1::nat) else 0)))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
316 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
317 |
lemma kle_refl[intro]: "kle n x x" unfolding kle_def by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
318 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
319 |
lemma kle_antisym: "kle n x y \<and> kle n y x \<longleftrightarrow> (x = y)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
320 |
unfolding kle_def apply rule apply(rule ext) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
321 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
322 |
lemma pointwise_minimal_pointwise_maximal: fixes s::"(nat\<Rightarrow>nat) set" |
49555 | 323 |
assumes "finite s" "s \<noteq> {}" "\<forall>x\<in>s. \<forall>y\<in>s. (\<forall>j. x j \<le> y j) \<or> (\<forall>j. y j \<le> x j)" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
324 |
shows "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. a j \<le> x j" "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. x j \<le> a j" |
49555 | 325 |
using assms unfolding atomize_conj |
326 |
proof (induct s rule:finite_induct) |
|
327 |
fix x and F::"(nat\<Rightarrow>nat) set" |
|
328 |
assume as:"finite F" "x \<notin> F" |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
329 |
"\<lbrakk>F \<noteq> {}; \<forall>x\<in>F. \<forall>y\<in>F. (\<forall>j. x j \<le> y j) \<or> (\<forall>j. y j \<le> x j)\<rbrakk> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
330 |
\<Longrightarrow> (\<exists>a\<in>F. \<forall>x\<in>F. \<forall>j. a j \<le> x j) \<and> (\<exists>a\<in>F. \<forall>x\<in>F. \<forall>j. x j \<le> a j)" "insert x F \<noteq> {}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
331 |
"\<forall>xa\<in>insert x F. \<forall>y\<in>insert x F. (\<forall>j. xa j \<le> y j) \<or> (\<forall>j. y j \<le> xa j)" |
49555 | 332 |
show "(\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. a j \<le> x j) \<and> (\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. x j \<le> a j)" |
333 |
proof (cases "F = {}") |
|
334 |
case True |
|
335 |
then show ?thesis |
|
336 |
apply - |
|
337 |
apply (rule, rule_tac[!] x=x in bexI) |
|
338 |
apply auto |
|
339 |
done |
|
340 |
next |
|
341 |
case False |
|
342 |
obtain a b where a: "a\<in>insert x F" "\<forall>x\<in>F. \<forall>j. a j \<le> x j" and |
|
343 |
b: "b\<in>insert x F" "\<forall>x\<in>F. \<forall>j. x j \<le> b j" using as(3)[OF False] using as(5) by auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
344 |
have "\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. a j \<le> x j" |
49555 | 345 |
using as(5)[rule_format,OF a(1) insertI1] |
346 |
apply - |
|
347 |
proof (erule disjE) |
|
348 |
assume "\<forall>j. a j \<le> x j" |
|
349 |
then show ?thesis |
|
350 |
apply (rule_tac x=a in bexI) using a apply auto done |
|
351 |
next |
|
352 |
assume "\<forall>j. x j \<le> a j" |
|
353 |
then show ?thesis |
|
354 |
apply (rule_tac x=x in bexI) |
|
355 |
apply (rule, rule) using a apply - |
|
356 |
apply (erule_tac x=xa in ballE) |
|
357 |
apply (erule_tac x=j in allE)+ |
|
358 |
apply auto |
|
359 |
done |
|
360 |
qed |
|
361 |
moreover |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
362 |
have "\<exists>b\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. x j \<le> b j" |
49555 | 363 |
using as(5)[rule_format,OF b(1) insertI1] apply- |
364 |
proof (erule disjE) |
|
365 |
assume "\<forall>j. x j \<le> b j" |
|
366 |
then show ?thesis |
|
367 |
apply(rule_tac x=b in bexI) using b |
|
368 |
apply auto |
|
369 |
done |
|
370 |
next |
|
371 |
assume "\<forall>j. b j \<le> x j" |
|
372 |
then show ?thesis |
|
373 |
apply (rule_tac x=x in bexI) |
|
374 |
apply (rule, rule) using b apply - |
|
375 |
apply (erule_tac x=xa in ballE) |
|
376 |
apply (erule_tac x=j in allE)+ |
|
377 |
apply auto |
|
378 |
done |
|
379 |
qed |
|
380 |
ultimately show ?thesis by auto |
|
381 |
qed |
|
382 |
qed auto |
|
383 |
||
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
384 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
385 |
lemma kle_imp_pointwise: "kle n x y \<Longrightarrow> (\<forall>j. x j \<le> y j)" unfolding kle_def by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
386 |
|
49555 | 387 |
lemma pointwise_antisym: |
388 |
fixes x :: "nat \<Rightarrow> nat" |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
389 |
shows "(\<forall>j. x j \<le> y j) \<and> (\<forall>j. y j \<le> x j) \<longleftrightarrow> (x = y)" |
49555 | 390 |
apply (rule, rule ext, erule conjE) |
391 |
apply (erule_tac x=xa in allE)+ |
|
392 |
apply auto |
|
393 |
done |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
394 |
|
49555 | 395 |
lemma kle_trans: |
396 |
assumes "kle n x y" "kle n y z" "kle n x z \<or> kle n z x" |
|
397 |
shows "kle n x z" |
|
398 |
using assms |
|
399 |
apply - |
|
400 |
apply (erule disjE) |
|
401 |
apply assumption |
|
402 |
proof - |
|
403 |
case goal1 |
|
404 |
then have "x = z" |
|
405 |
apply - |
|
406 |
apply (rule ext) |
|
407 |
apply (drule kle_imp_pointwise)+ |
|
408 |
apply (erule_tac x=xa in allE)+ |
|
409 |
apply auto |
|
410 |
done |
|
411 |
then show ?case by auto |
|
412 |
qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
413 |
|
49555 | 414 |
lemma kle_strict: |
415 |
assumes "kle n x y" "x \<noteq> y" |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
416 |
shows "\<forall>j. x j \<le> y j" "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> x(k) < y(k)" |
49555 | 417 |
apply (rule kle_imp_pointwise[OF assms(1)]) |
418 |
proof - |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
419 |
guess k using assms(1)[unfolded kle_def] .. note k = this |
49555 | 420 |
show "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> x(k) < y(k)" |
421 |
proof (cases "k={}") |
|
422 |
case True |
|
423 |
then have "x = y" |
|
424 |
apply - |
|
425 |
apply (rule ext) |
|
426 |
using k apply auto |
|
427 |
done |
|
428 |
then show ?thesis using assms(2) by auto |
|
429 |
next |
|
430 |
case False |
|
431 |
then have "(SOME k'. k' \<in> k) \<in> k" |
|
432 |
apply - |
|
433 |
apply (rule someI_ex) |
|
434 |
apply auto |
|
435 |
done |
|
436 |
then show ?thesis |
|
437 |
apply (rule_tac x = "SOME k'. k' \<in> k" in exI) |
|
438 |
using k apply auto |
|
439 |
done |
|
440 |
qed |
|
441 |
qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
442 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
443 |
lemma kle_minimal: assumes "finite s" "s \<noteq> {}" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
444 |
shows "\<exists>a\<in>s. \<forall>x\<in>s. kle n a x" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
445 |
have "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. a j \<le> x j" apply(rule pointwise_minimal_pointwise_maximal(1)[OF assms(1-2)]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
446 |
apply(rule,rule) apply(drule_tac assms(3)[rule_format],assumption) using kle_imp_pointwise by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
447 |
then guess a .. note a=this show ?thesis apply(rule_tac x=a in bexI) proof fix x assume "x\<in>s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
448 |
show "kle n a x" using assms(3)[rule_format,OF a(1) `x\<in>s`] apply- proof(erule disjE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
449 |
assume "kle n x a" hence "x = a" apply- unfolding pointwise_antisym[THEN sym] |
41958 | 450 |
apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
451 |
thus ?thesis using kle_refl by auto qed qed(insert a, auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
452 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
453 |
lemma kle_maximal: assumes "finite s" "s \<noteq> {}" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
454 |
shows "\<exists>a\<in>s. \<forall>x\<in>s. kle n x a" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
455 |
have "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. a j \<ge> x j" apply(rule pointwise_minimal_pointwise_maximal(2)[OF assms(1-2)]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
456 |
apply(rule,rule) apply(drule_tac assms(3)[rule_format],assumption) using kle_imp_pointwise by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
457 |
then guess a .. note a=this show ?thesis apply(rule_tac x=a in bexI) proof fix x assume "x\<in>s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
458 |
show "kle n x a" using assms(3)[rule_format,OF a(1) `x\<in>s`] apply- proof(erule disjE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
459 |
assume "kle n a x" hence "x = a" apply- unfolding pointwise_antisym[THEN sym] |
41958 | 460 |
apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
461 |
thus ?thesis using kle_refl by auto qed qed(insert a, auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
462 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
463 |
lemma kle_strict_set: assumes "kle n x y" "x \<noteq> y" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
464 |
shows "1 \<le> card {k\<in>{1..n}. x k < y k}" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
465 |
guess i using kle_strict(2)[OF assms] .. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
466 |
hence "card {i} \<le> card {k\<in>{1..n}. x k < y k}" apply- apply(rule card_mono) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
467 |
thus ?thesis by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
468 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
469 |
lemma kle_range_combine: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
470 |
assumes "kle n x y" "kle n y z" "kle n x z \<or> kle n z x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
471 |
"m1 \<le> card {k\<in>{1..n}. x k < y k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
472 |
"m2 \<le> card {k\<in>{1..n}. y k < z k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
473 |
shows "kle n x z \<and> m1 + m2 \<le> card {k\<in>{1..n}. x k < z k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
474 |
apply(rule,rule kle_trans[OF assms(1-3)]) proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
475 |
have "\<And>j. x j < y j \<Longrightarrow> x j < z j" apply(rule less_le_trans) using kle_imp_pointwise[OF assms(2)] by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
476 |
have "\<And>j. y j < z j \<Longrightarrow> x j < z j" apply(rule le_less_trans) using kle_imp_pointwise[OF assms(1)] by auto ultimately |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
477 |
have *:"{k\<in>{1..n}. x k < y k} \<union> {k\<in>{1..n}. y k < z k} = {k\<in>{1..n}. x k < z k}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
478 |
have **:"{k \<in> {1..n}. x k < y k} \<inter> {k \<in> {1..n}. y k < z k} = {}" unfolding disjoint_iff_not_equal |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
479 |
apply(rule,rule,unfold mem_Collect_eq,rule ccontr) apply(erule conjE)+ proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
480 |
fix i j assume as:"i \<in> {1..n}" "x i < y i" "j \<in> {1..n}" "y j < z j" "\<not> i \<noteq> j" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
481 |
guess kx using assms(1)[unfolded kle_def] .. note kx=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
482 |
have "x i < y i" using as by auto hence "i \<in> kx" using as(1) kx apply(rule_tac ccontr) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
483 |
hence "x i + 1 = y i" using kx by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
484 |
guess ky using assms(2)[unfolded kle_def] .. note ky=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
485 |
have "y i < z i" using as by auto hence "i \<in> ky" using as(1) ky apply(rule_tac ccontr) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
486 |
hence "y i + 1 = z i" using ky by auto ultimately |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
487 |
have "z i = x i + 2" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
488 |
thus False using assms(3) unfolding kle_def by(auto simp add: split_if_eq1) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
489 |
have fin:"\<And>P. finite {x\<in>{1..n::nat}. P x}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
490 |
have "m1 + m2 \<le> card {k\<in>{1..n}. x k < y k} + card {k\<in>{1..n}. y k < z k}" using assms(4-5) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
491 |
also have "\<dots> \<le> card {k\<in>{1..n}. x k < z k}" unfolding card_Un_Int[OF fin fin] unfolding * ** by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
492 |
finally show " m1 + m2 \<le> card {k \<in> {1..n}. x k < z k}" by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
493 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
494 |
lemma kle_range_combine_l: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
495 |
assumes "kle n x y" "kle n y z" "kle n x z \<or> kle n z x" "m \<le> card {k\<in>{1..n}. y(k) < z(k)}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
496 |
shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
497 |
using kle_range_combine[OF assms(1-3) _ assms(4), of 0] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
498 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
499 |
lemma kle_range_combine_r: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
500 |
assumes "kle n x y" "kle n y z" "kle n x z \<or> kle n z x" "m \<le> card {k\<in>{1..n}. x k < y k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
501 |
shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
502 |
using kle_range_combine[OF assms(1-3) assms(4), of 0] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
503 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
504 |
lemma kle_range_induct: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
505 |
assumes "card s = Suc m" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
506 |
shows "\<exists>x\<in>s. \<exists>y\<in>s. kle n x y \<and> m \<le> card {k\<in>{1..n}. x k < y k}" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
507 |
have "finite s" "s\<noteq>{}" using assms(1) by (auto intro: card_ge_0_finite) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
508 |
thus ?thesis using assms apply- proof(induct m arbitrary: s) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
509 |
case 0 thus ?case using kle_refl by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
510 |
case (Suc m) then obtain a where a:"a\<in>s" "\<forall>x\<in>s. kle n a x" using kle_minimal[of s n] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
511 |
show ?case proof(cases "s \<subseteq> {a}") case False |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
512 |
hence "card (s - {a}) = Suc m" "s - {a} \<noteq> {}" using card_Diff_singleton[OF _ a(1)] Suc(4) `finite s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
513 |
then obtain x b where xb:"x\<in>s - {a}" "b\<in>s - {a}" "kle n x b" "m \<le> card {k \<in> {1..n}. x k < b k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
514 |
using Suc(1)[of "s - {a}"] using Suc(5) `finite s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
515 |
have "1 \<le> card {k \<in> {1..n}. a k < x k}" "m \<le> card {k \<in> {1..n}. x k < b k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
516 |
apply(rule kle_strict_set) apply(rule a(2)[rule_format]) using a and xb by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
517 |
thus ?thesis apply(rule_tac x=a in bexI, rule_tac x=b in bexI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
518 |
using kle_range_combine[OF a(2)[rule_format] xb(3) Suc(5)[rule_format], of 1 "m"] using a(1) xb(1-2) by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
519 |
case True hence "s = {a}" using Suc(3) by auto hence "card s = 1" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
520 |
hence False using Suc(4) `finite s` by auto thus ?thesis by auto qed qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
521 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
522 |
lemma kle_Suc: "kle n x y \<Longrightarrow> kle (n + 1) x y" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
523 |
unfolding kle_def apply(erule exE) apply(rule_tac x=k in exI) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
524 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
525 |
lemma kle_trans_1: assumes "kle n x y" shows "x j \<le> y j" "y j \<le> x j + 1" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
526 |
using assms[unfolded kle_def] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
527 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
528 |
lemma kle_trans_2: assumes "kle n a b" "kle n b c" "\<forall>j. c j \<le> a j + 1" shows "kle n a c" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
529 |
guess kk1 using assms(1)[unfolded kle_def] .. note kk1=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
530 |
guess kk2 using assms(2)[unfolded kle_def] .. note kk2=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
531 |
show ?thesis unfolding kle_def apply(rule_tac x="kk1 \<union> kk2" in exI) apply(rule) defer proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
532 |
fix i show "c i = a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)" proof(cases "i\<in>kk1 \<union> kk2") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
533 |
case True hence "c i \<ge> a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)" |
41958 | 534 |
unfolding kk1[THEN conjunct2,rule_format,of i] kk2[THEN conjunct2,rule_format,of i] by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
535 |
moreover have "c i \<le> a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)" using True assms(3) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
536 |
ultimately show ?thesis by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
537 |
case False thus ?thesis using kk1 kk2 by auto qed qed(insert kk1 kk2, auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
538 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
539 |
lemma kle_between_r: assumes "kle n a b" "kle n b c" "kle n a x" "kle n c x" shows "kle n b x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
540 |
apply(rule kle_trans_2[OF assms(2,4)]) proof have *:"\<And>c b x::nat. x \<le> c + 1 \<Longrightarrow> c \<le> b \<Longrightarrow> x \<le> b + 1" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
541 |
fix j show "x j \<le> b j + 1" apply(rule *)using kle_trans_1[OF assms(1),of j] kle_trans_1[OF assms(3), of j] by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
542 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
543 |
lemma kle_between_l: assumes "kle n a b" "kle n b c" "kle n x a" "kle n x c" shows "kle n x b" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
544 |
apply(rule kle_trans_2[OF assms(3,1)]) proof have *:"\<And>c b x::nat. c \<le> x + 1 \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> x + 1" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
545 |
fix j show "b j \<le> x j + 1" apply(rule *) using kle_trans_1[OF assms(2),of j] kle_trans_1[OF assms(4), of j] by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
546 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
547 |
lemma kle_adjacent: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
548 |
assumes "\<forall>j. b j = (if j = k then a(j) + 1 else a j)" "kle n a x" "kle n x b" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
549 |
shows "(x = a) \<or> (x = b)" proof(cases "x k = a k") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
550 |
case True show ?thesis apply(rule disjI1,rule ext) proof- fix j |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
551 |
have "x j \<le> a j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
552 |
unfolding assms(1)[rule_format] apply-apply(cases "j=k") using True by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
553 |
thus "x j = a j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]] by auto qed next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
554 |
case False show ?thesis apply(rule disjI2,rule ext) proof- fix j |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
555 |
have "x j \<ge> b j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
556 |
unfolding assms(1)[rule_format] apply-apply(cases "j=k") using False by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
557 |
thus "x j = b j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]] by auto qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
558 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
559 |
subsection {* kuhn's notion of a simplex (a reformulation to avoid so much indexing). *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
560 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
561 |
definition "ksimplex p n (s::(nat \<Rightarrow> nat) set) \<longleftrightarrow> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
562 |
(card s = n + 1 \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
563 |
(\<forall>x\<in>s. \<forall>j. x(j) \<le> p) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
564 |
(\<forall>x\<in>s. \<forall>j. j\<notin>{1..n} \<longrightarrow> (x j = p)) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
565 |
(\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
566 |
|
36318 | 567 |
lemma ksimplexI:"card s = n + 1 \<Longrightarrow> \<forall>x\<in>s. \<forall>j. x j \<le> p \<Longrightarrow> \<forall>x\<in>s. \<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x \<Longrightarrow> ksimplex p n s" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
568 |
unfolding ksimplex_def by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
569 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
570 |
lemma ksimplex_eq: "ksimplex p n (s::(nat \<Rightarrow> nat) set) \<longleftrightarrow> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
571 |
(card s = n + 1 \<and> finite s \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
572 |
(\<forall>x\<in>s. \<forall>j. x(j) \<le> p) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
573 |
(\<forall>x\<in>s. \<forall>j. j\<notin>{1..n} \<longrightarrow> (x j = p)) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
574 |
(\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
575 |
unfolding ksimplex_def by (auto intro: card_ge_0_finite) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
576 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
577 |
lemma ksimplex_extrema: assumes "ksimplex p n s" obtains a b where "a \<in> s" "b \<in> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
578 |
"\<forall>x\<in>s. kle n a x \<and> kle n x b" "\<forall>i. b(i) = (if i \<in> {1..n} then a(i) + 1 else a(i))" proof(cases "n=0") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
579 |
case True obtain x where *:"s = {x}" using assms[unfolded ksimplex_eq True,THEN conjunct1] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
580 |
unfolding add_0_left card_1_exists by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
581 |
show ?thesis apply(rule that[of x x]) unfolding * True by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
582 |
note assm = assms[unfolded ksimplex_eq] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
583 |
case False have "s\<noteq>{}" using assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
584 |
obtain a where a:"a\<in>s" "\<forall>x\<in>s. kle n a x" using `s\<noteq>{}` assm using kle_minimal[of s n] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
585 |
obtain b where b:"b\<in>s" "\<forall>x\<in>s. kle n x b" using `s\<noteq>{}` assm using kle_maximal[of s n] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
586 |
obtain c d where c_d:"c\<in>s" "d\<in>s" "kle n c d" "n \<le> card {k \<in> {1..n}. c k < d k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
587 |
using kle_range_induct[of s n n] using assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
588 |
have "kle n c b \<and> n \<le> card {k \<in> {1..n}. c k < b k}" apply(rule kle_range_combine_r[where y=d]) using c_d a b by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
589 |
hence "kle n a b \<and> n \<le> card {k\<in>{1..n}. a(k) < b(k)}" apply-apply(rule kle_range_combine_l[where y=c]) using a `c\<in>s` `b\<in>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
590 |
moreover have "card {1..n} \<ge> card {k\<in>{1..n}. a(k) < b(k)}" apply(rule card_mono) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
591 |
ultimately have *:"{k\<in>{1 .. n}. a k < b k} = {1..n}" apply- apply(rule card_subset_eq) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
592 |
show ?thesis apply(rule that[OF a(1) b(1)]) defer apply(subst *[THEN sym]) unfolding mem_Collect_eq proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
593 |
guess k using a(2)[rule_format,OF b(1),unfolded kle_def] .. note k=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
594 |
fix i show "b i = (if i \<in> {1..n} \<and> a i < b i then a i + 1 else a i)" proof(cases "i \<in> {1..n}") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
595 |
case True thus ?thesis unfolding k[THEN conjunct2,rule_format] by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
596 |
case False have "a i = p" using assm and False `a\<in>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
597 |
moreover have "b i = p" using assm and False `b\<in>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
598 |
ultimately show ?thesis by auto qed qed(insert a(2) b(2) assm,auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
599 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
600 |
lemma ksimplex_extrema_strong: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
601 |
assumes "ksimplex p n s" "n \<noteq> 0" obtains a b where "a \<in> s" "b \<in> s" "a \<noteq> b" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
602 |
"\<forall>x\<in>s. kle n a x \<and> kle n x b" "\<forall>i. b(i) = (if i \<in> {1..n} then a(i) + 1 else a(i))" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
603 |
obtain a b where ab:"a \<in> s" "b \<in> s" "\<forall>x\<in>s. kle n a x \<and> kle n x b" "\<forall>i. b(i) = (if i \<in> {1..n} then a(i) + 1 else a(i))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
604 |
apply(rule ksimplex_extrema[OF assms(1)]) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
605 |
have "a \<noteq> b" apply(rule ccontr) unfolding not_not apply(drule cong[of _ _ 1 1]) using ab(4) assms(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
606 |
thus ?thesis apply(rule_tac that[of a b]) using ab by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
607 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
608 |
lemma ksimplexD: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
609 |
assumes "ksimplex p n s" |
36318 | 610 |
shows "card s = n + 1" "finite s" "card s = n + 1" "\<forall>x\<in>s. \<forall>j. x j \<le> p" "\<forall>x\<in>s. \<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
611 |
"\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x" using assms unfolding ksimplex_eq by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
612 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
613 |
lemma ksimplex_successor: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
614 |
assumes "ksimplex p n s" "a \<in> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
615 |
shows "(\<forall>x\<in>s. kle n x a) \<or> (\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. y(j) = (if j = k then a(j) + 1 else a(j)))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
616 |
proof(cases "\<forall>x\<in>s. kle n x a") case True thus ?thesis by auto next note assm = ksimplexD[OF assms(1)] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
617 |
case False then obtain b where b:"b\<in>s" "\<not> kle n b a" "\<forall>x\<in>{x \<in> s. \<not> kle n x a}. kle n b x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
618 |
using kle_minimal[of "{x\<in>s. \<not> kle n x a}" n] and assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
619 |
hence **:"1 \<le> card {k\<in>{1..n}. a k < b k}" apply- apply(rule kle_strict_set) using assm(6) and `a\<in>s` by(auto simp add:kle_refl) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
620 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
621 |
let ?kle1 = "{x \<in> s. \<not> kle n x a}" have "card ?kle1 > 0" apply(rule ccontr) using assm(2) and False by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
622 |
hence sizekle1: "card ?kle1 = Suc (card ?kle1 - 1)" using assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
623 |
obtain c d where c_d: "c \<in> s" "\<not> kle n c a" "d \<in> s" "\<not> kle n d a" "kle n c d" "card ?kle1 - 1 \<le> card {k \<in> {1..n}. c k < d k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
624 |
using kle_range_induct[OF sizekle1, of n] using assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
625 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
626 |
let ?kle2 = "{x \<in> s. kle n x a}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
627 |
have "card ?kle2 > 0" apply(rule ccontr) using assm(6)[rule_format,of a a] and `a\<in>s` and assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
628 |
hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)" using assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
629 |
obtain e f where e_f: "e \<in> s" "kle n e a" "f \<in> s" "kle n f a" "kle n e f" "card ?kle2 - 1 \<le> card {k \<in> {1..n}. e k < f k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
630 |
using kle_range_induct[OF sizekle2, of n] using assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
631 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
632 |
have "card {k\<in>{1..n}. a k < b k} = 1" proof(rule ccontr) case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
633 |
hence as:"card {k\<in>{1..n}. a k < b k} \<ge> 2" using ** by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
634 |
have *:"finite ?kle2" "finite ?kle1" "?kle2 \<union> ?kle1 = s" "?kle2 \<inter> ?kle1 = {}" using assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
635 |
have "(card ?kle2 - 1) + 2 + (card ?kle1 - 1) = card ?kle2 + card ?kle1" using sizekle1 sizekle2 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
636 |
also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
637 |
finally have n:"(card ?kle2 - 1) + (2 + (card ?kle1 - 1)) = n + 1" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
638 |
have "kle n e a \<and> card {x \<in> s. kle n x a} - 1 \<le> card {k \<in> {1..n}. e k < a k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
639 |
apply(rule kle_range_combine_r[where y=f]) using e_f using `a\<in>s` assm(6) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
640 |
moreover have "kle n b d \<and> card {x \<in> s. \<not> kle n x a} - 1 \<le> card {k \<in> {1..n}. b k < d k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
641 |
apply(rule kle_range_combine_l[where y=c]) using c_d using assm(6) and b by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
642 |
hence "kle n a d \<and> 2 + (card {x \<in> s. \<not> kle n x a} - 1) \<le> card {k \<in> {1..n}. a k < d k}" apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
643 |
apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a\<in>s` `d\<in>s` apply- by blast+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
644 |
ultimately have "kle n e d \<and> (card ?kle2 - 1) + (2 + (card ?kle1 - 1)) \<le> card {k\<in>{1..n}. e k < d k}" apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
645 |
apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply - by blast+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
646 |
moreover have "card {k \<in> {1..n}. e k < d k} \<le> card {1..n}" apply(rule card_mono) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
647 |
ultimately show False unfolding n by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
648 |
then guess k unfolding card_1_exists .. note k=this[unfolded mem_Collect_eq] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
649 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
650 |
show ?thesis apply(rule disjI2) apply(rule_tac x=b in bexI,rule_tac x=k in bexI) proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
651 |
fix j::nat have "kle n a b" using b and assm(6)[rule_format, OF `a\<in>s` `b\<in>s`] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
652 |
then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
653 |
have kkk:"k\<in>kk" apply(rule ccontr) using k(1) unfolding kk by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
654 |
show "b j = (if j = k then a j + 1 else a j)" proof(cases "j\<in>kk") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
655 |
case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
656 |
thus ?thesis unfolding kk using kkk by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
657 |
case False hence "j\<noteq>k" using k(2)[rule_format, of j k] using kk_raw kkk by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
658 |
thus ?thesis unfolding kk using kkk using False by auto qed qed(insert k(1) `b\<in>s`, auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
659 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
660 |
lemma ksimplex_predecessor: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
661 |
assumes "ksimplex p n s" "a \<in> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
662 |
shows "(\<forall>x\<in>s. kle n a x) \<or> (\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. a(j) = (if j = k then y(j) + 1 else y(j)))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
663 |
proof(cases "\<forall>x\<in>s. kle n a x") case True thus ?thesis by auto next note assm = ksimplexD[OF assms(1)] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
664 |
case False then obtain b where b:"b\<in>s" "\<not> kle n a b" "\<forall>x\<in>{x \<in> s. \<not> kle n a x}. kle n x b" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
665 |
using kle_maximal[of "{x\<in>s. \<not> kle n a x}" n] and assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
666 |
hence **:"1 \<le> card {k\<in>{1..n}. a k > b k}" apply- apply(rule kle_strict_set) using assm(6) and `a\<in>s` by(auto simp add:kle_refl) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
667 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
668 |
let ?kle1 = "{x \<in> s. \<not> kle n a x}" have "card ?kle1 > 0" apply(rule ccontr) using assm(2) and False by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
669 |
hence sizekle1:"card ?kle1 = Suc (card ?kle1 - 1)" using assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
670 |
obtain c d where c_d: "c \<in> s" "\<not> kle n a c" "d \<in> s" "\<not> kle n a d" "kle n d c" "card ?kle1 - 1 \<le> card {k \<in> {1..n}. c k > d k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
671 |
using kle_range_induct[OF sizekle1, of n] using assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
672 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
673 |
let ?kle2 = "{x \<in> s. kle n a x}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
674 |
have "card ?kle2 > 0" apply(rule ccontr) using assm(6)[rule_format,of a a] and `a\<in>s` and assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
675 |
hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)" using assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
676 |
obtain e f where e_f: "e \<in> s" "kle n a e" "f \<in> s" "kle n a f" "kle n f e" "card ?kle2 - 1 \<le> card {k \<in> {1..n}. e k > f k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
677 |
using kle_range_induct[OF sizekle2, of n] using assm by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
678 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
679 |
have "card {k\<in>{1..n}. a k > b k} = 1" proof(rule ccontr) case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
680 |
hence as:"card {k\<in>{1..n}. a k > b k} \<ge> 2" using ** by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
681 |
have *:"finite ?kle2" "finite ?kle1" "?kle2 \<union> ?kle1 = s" "?kle2 \<inter> ?kle1 = {}" using assm(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
682 |
have "(card ?kle2 - 1) + 2 + (card ?kle1 - 1) = card ?kle2 + card ?kle1" using sizekle1 sizekle2 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
683 |
also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
684 |
finally have n:"(card ?kle1 - 1) + 2 + (card ?kle2 - 1) = n + 1" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
685 |
have "kle n a e \<and> card {x \<in> s. kle n a x} - 1 \<le> card {k \<in> {1..n}. e k > a k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
686 |
apply(rule kle_range_combine_l[where y=f]) using e_f using `a\<in>s` assm(6) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
687 |
moreover have "kle n d b \<and> card {x \<in> s. \<not> kle n a x} - 1 \<le> card {k \<in> {1..n}. b k > d k}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
688 |
apply(rule kle_range_combine_r[where y=c]) using c_d using assm(6) and b by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
689 |
hence "kle n d a \<and> (card {x \<in> s. \<not> kle n a x} - 1) + 2 \<le> card {k \<in> {1..n}. a k > d k}" apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
690 |
apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a\<in>s` `d\<in>s` by blast+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
691 |
ultimately have "kle n d e \<and> (card ?kle1 - 1 + 2) + (card ?kle2 - 1) \<le> card {k\<in>{1..n}. e k > d k}" apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
692 |
apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply - by blast+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
693 |
moreover have "card {k \<in> {1..n}. e k > d k} \<le> card {1..n}" apply(rule card_mono) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
694 |
ultimately show False unfolding n by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
695 |
then guess k unfolding card_1_exists .. note k=this[unfolded mem_Collect_eq] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
696 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
697 |
show ?thesis apply(rule disjI2) apply(rule_tac x=b in bexI,rule_tac x=k in bexI) proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
698 |
fix j::nat have "kle n b a" using b and assm(6)[rule_format, OF `a\<in>s` `b\<in>s`] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
699 |
then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
700 |
have kkk:"k\<in>kk" apply(rule ccontr) using k(1) unfolding kk by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
701 |
show "a j = (if j = k then b j + 1 else b j)" proof(cases "j\<in>kk") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
702 |
case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
703 |
thus ?thesis unfolding kk using kkk by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
704 |
case False hence "j\<noteq>k" using k(2)[rule_format, of j k] using kk_raw kkk by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
705 |
thus ?thesis unfolding kk using kkk using False by auto qed qed(insert k(1) `b\<in>s`, auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
706 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
707 |
subsection {* The lemmas about simplices that we need. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
708 |
|
50027
7747a9f4c358
adjusting proofs as the set_comprehension_pointfree simproc breaks some existing proofs
bulwahn
parents:
49555
diff
changeset
|
709 |
(* FIXME: These are clones of lemmas in Library/FuncSet *) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
710 |
lemma card_funspace': assumes "finite s" "finite t" "card s = m" "card t = n" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
711 |
shows "card {f. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>UNIV - s. f x = d)} = n ^ m" (is "card (?M s) = _") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
712 |
using assms apply - proof(induct m arbitrary: s) |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
713 |
have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" apply(rule set_eqI,rule)unfolding mem_Collect_eq apply(rule,rule ext) by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
714 |
case 0 thus ?case by(auto simp add: *) next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
715 |
case (Suc m) guess a using card_eq_SucD[OF Suc(4)] .. then guess s0 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
716 |
apply(erule_tac exE) apply(erule conjE)+ . note as0 = this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
717 |
have **:"card s0 = m" using as0 using Suc(2) Suc(4) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
718 |
let ?l = "(\<lambda>(b,g) x. if x = a then b else g x)" have *:"?M (insert a s0) = ?l ` {(b,g). b\<in>t \<and> g\<in>?M s0}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
719 |
apply(rule set_eqI,rule) unfolding mem_Collect_eq image_iff apply(erule conjE) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
720 |
apply(rule_tac x="(x a, \<lambda>y. if y\<in>s0 then x y else d)" in bexI) apply(rule ext) prefer 3 apply rule defer |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
721 |
apply(erule bexE,rule) unfolding mem_Collect_eq apply(erule splitE)+ apply(erule conjE)+ proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
722 |
fix x xa xb xc y assume as:"x = (\<lambda>(b, g) x. if x = a then b else g x) xa" "xb \<in> UNIV - insert a s0" "xa = (xc, y)" "xc \<in> t" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
723 |
"\<forall>x\<in>s0. y x \<in> t" "\<forall>x\<in>UNIV - s0. y x = d" thus "x xb = d" unfolding as by auto qed auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
724 |
have inj:"inj_on ?l {(b,g). b\<in>t \<and> g\<in>?M s0}" unfolding inj_on_def apply(rule,rule,rule) unfolding mem_Collect_eq apply(erule splitE conjE)+ proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
725 |
case goal1 note as = this(1,4-)[unfolded goal1 split_conv] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
726 |
have "xa = xb" using as(1)[THEN cong[of _ _ a]] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
727 |
moreover have "ya = yb" proof(rule ext) fix x show "ya x = yb x" proof(cases "x = a") |
41958 | 728 |
case False thus ?thesis using as(1)[THEN cong[of _ _ x x]] by auto next |
729 |
case True thus ?thesis using as(5,7) using as0(2) by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
730 |
ultimately show ?case unfolding goal1 by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
731 |
have "finite s0" using `finite s` unfolding as0 by simp |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
732 |
show ?case unfolding as0 * card_image[OF inj] using assms |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
733 |
unfolding SetCompr_Sigma_eq apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
734 |
unfolding card_cartesian_product |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
735 |
using Suc(1)[OF `finite s0` `finite t` ** `card t = n`] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
736 |
qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
737 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
738 |
lemma card_funspace: assumes "finite s" "finite t" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
739 |
shows "card {f. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>UNIV - s. f x = d)} = (card t) ^ (card s)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
740 |
using assms by (auto intro: card_funspace') |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
741 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
742 |
lemma finite_funspace: assumes "finite s" "finite t" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
743 |
shows "finite {f. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>UNIV - s. f x = d)}" (is "finite ?S") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
744 |
proof (cases "card t > 0") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
745 |
case True |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
746 |
have "card ?S = (card t) ^ (card s)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
747 |
using assms by (auto intro!: card_funspace) |
50027
7747a9f4c358
adjusting proofs as the set_comprehension_pointfree simproc breaks some existing proofs
bulwahn
parents:
49555
diff
changeset
|
748 |
thus ?thesis using True by (rule_tac card_ge_0_finite) simp |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
749 |
next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
750 |
case False hence "t = {}" using `finite t` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
751 |
show ?thesis |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
752 |
proof (cases "s = {}") |
44457
d366fa5551ef
declare euclidean_simps [simp] at the point they are proved;
huffman
parents:
44282
diff
changeset
|
753 |
have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
754 |
case True thus ?thesis using `t = {}` by (auto simp: *) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
755 |
next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
756 |
case False thus ?thesis using `t = {}` by simp |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
757 |
qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
758 |
qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
759 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
760 |
lemma finite_simplices: "finite {s. ksimplex p n s}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
761 |
apply(rule finite_subset[of _ "{s. s\<subseteq>{f. (\<forall>i\<in>{1..n}. f i \<in> {0..p}) \<and> (\<forall>i\<in>UNIV-{1..n}. f i = p)}}"]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
762 |
unfolding ksimplex_def defer apply(rule finite_Collect_subsets) apply(rule finite_funspace) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
763 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
764 |
lemma simplex_top_face: assumes "0<p" "\<forall>x\<in>f. x (n + 1) = p" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
765 |
shows "(\<exists>s a. ksimplex p (n + 1) s \<and> a \<in> s \<and> (f = s - {a})) \<longleftrightarrow> ksimplex p n f" (is "?ls = ?rs") proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
766 |
assume ?ls then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
767 |
show ?rs unfolding ksimplex_def sa(3) apply(rule) defer apply rule defer apply(rule,rule,rule,rule) defer apply(rule,rule) proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
768 |
fix x y assume as:"x \<in>s - {a}" "y \<in>s - {a}" have xyp:"x (n + 1) = y (n + 1)" |
41958 | 769 |
using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] |
770 |
using as(2)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] by auto |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
771 |
show "kle n x y \<or> kle n y x" proof(cases "kle (n + 1) x y") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
772 |
case True then guess k unfolding kle_def .. note k=this hence *:"n+1 \<notin> k" using xyp by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
773 |
have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply(rule ccontr) unfolding not_not apply(erule bexE) proof- |
41958 | 774 |
fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto |
775 |
thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
776 |
thus ?thesis apply-apply(rule disjI1) unfolding kle_def using k apply(rule_tac x=k in exI) by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
777 |
case False hence "kle (n + 1) y x" using ksimplexD(6)[OF sa(1),rule_format, of x y] using as by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
778 |
then guess k unfolding kle_def .. note k=this hence *:"n+1 \<notin> k" using xyp by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
779 |
hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply-apply(rule ccontr) unfolding not_not apply(erule bexE) proof- |
41958 | 780 |
fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto |
781 |
thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
782 |
thus ?thesis apply-apply(rule disjI2) unfolding kle_def using k apply(rule_tac x=k in exI) by auto qed next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
783 |
fix x j assume as:"x\<in>s - {a}" "j\<notin>{1..n}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
784 |
thus "x j = p" using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
785 |
apply(cases "j = n+1") using sa(1)[unfolded ksimplex_def] by auto qed(insert sa ksimplexD[OF sa(1)], auto) next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
786 |
assume ?rs note rs=ksimplexD[OF this] guess a b apply(rule ksimplex_extrema[OF `?rs`]) . note ab = this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
787 |
def c \<equiv> "\<lambda>i. if i = (n + 1) then p - 1 else a i" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
788 |
have "c\<notin>f" apply(rule ccontr) unfolding not_not apply(drule assms(2)[rule_format]) unfolding c_def using assms(1) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
789 |
thus ?ls apply(rule_tac x="insert c f" in exI,rule_tac x=c in exI) unfolding ksimplex_def conj_assoc |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
790 |
apply(rule conjI) defer apply(rule conjI) defer apply(rule conjI) defer apply(rule conjI) defer |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
791 |
proof(rule_tac[3-5] ballI allI)+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
792 |
fix x j assume x:"x \<in> insert c f" thus "x j \<le> p" proof (cases "x=c") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
793 |
case True show ?thesis unfolding True c_def apply(cases "j=n+1") using ab(1) and rs(4) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
794 |
qed(insert x rs(4), auto simp add:c_def) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
795 |
show "j \<notin> {1..n + 1} \<longrightarrow> x j = p" apply(cases "x=c") using x ab(1) rs(5) unfolding c_def by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
796 |
{ fix z assume z:"z \<in> insert c f" hence "kle (n + 1) c z" apply(cases "z = c") (*defer apply(rule kle_Suc)*) proof- |
41958 | 797 |
case False hence "z\<in>f" using z by auto |
798 |
then guess k apply(drule_tac ab(3)[THEN bspec[where x=z], THEN conjunct1]) unfolding kle_def apply(erule exE) . |
|
799 |
thus "kle (n + 1) c z" unfolding kle_def apply(rule_tac x="insert (n + 1) k" in exI) unfolding c_def |
|
800 |
using ab using rs(5)[rule_format,OF ab(1),of "n + 1"] assms(1) by auto qed auto } note * = this |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
801 |
fix y assume y:"y \<in> insert c f" show "kle (n + 1) x y \<or> kle (n + 1) y x" proof(cases "x = c \<or> y = c") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
802 |
case False hence **:"x\<in>f" "y\<in>f" using x y by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
803 |
show ?thesis using rs(6)[rule_format,OF **] by(auto dest: kle_Suc) qed(insert * x y, auto) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
804 |
qed(insert rs, auto) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
805 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
806 |
lemma ksimplex_fix_plane: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
807 |
assumes "a \<in> s" "j\<in>{1..n::nat}" "\<forall>x\<in>s - {a}. x j = q" "a0 \<in> s" "a1 \<in> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
808 |
"\<forall>i. a1 i = ((if i\<in>{1..n} then a0 i + 1 else a0 i)::nat)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
809 |
shows "(a = a0) \<or> (a = a1)" proof- have *:"\<And>P A x y. \<forall>x\<in>A. P x \<Longrightarrow> x\<in>A \<Longrightarrow> y\<in>A \<Longrightarrow> P x \<and> P y" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
810 |
show ?thesis apply(rule ccontr) using *[OF assms(3), of a0 a1] unfolding assms(6)[THEN spec[where x=j]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
811 |
using assms(1-2,4-5) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
812 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
813 |
lemma ksimplex_fix_plane_0: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
814 |
assumes "a \<in> s" "j\<in>{1..n::nat}" "\<forall>x\<in>s - {a}. x j = 0" "a0 \<in> s" "a1 \<in> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
815 |
"\<forall>i. a1 i = ((if i\<in>{1..n} then a0 i + 1 else a0 i)::nat)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
816 |
shows "a = a1" apply(rule ccontr) using ksimplex_fix_plane[OF assms] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
817 |
using assms(3)[THEN bspec[where x=a1]] using assms(2,5) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
818 |
unfolding assms(6)[THEN spec[where x=j]] by simp |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
819 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
820 |
lemma ksimplex_fix_plane_p: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
821 |
assumes "ksimplex p n s" "a \<in> s" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = p" "a0 \<in> s" "a1 \<in> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
822 |
"\<forall>i. a1 i = (if i\<in>{1..n} then a0 i + 1 else a0 i)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
823 |
shows "a = a0" proof(rule ccontr) note s = ksimplexD[OF assms(1),rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
824 |
assume as:"a \<noteq> a0" hence *:"a0 \<in> s - {a}" using assms(5) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
825 |
hence "a1 = a" using ksimplex_fix_plane[OF assms(2-)] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
826 |
thus False using as using assms(3,5) and assms(7)[rule_format,of j] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
827 |
unfolding assms(4)[rule_format,OF *] using s(4)[OF assms(6), of j] by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
828 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
829 |
lemma ksimplex_replace_0: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
830 |
assumes "ksimplex p n s" "a \<in> s" "n \<noteq> 0" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = 0" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
831 |
shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
832 |
have *:"\<And>s' a a'. s' - {a'} = s - {a} \<Longrightarrow> a' = a \<Longrightarrow> a' \<in> s' \<Longrightarrow> a \<in> s \<Longrightarrow> (s' = s)" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
833 |
have **:"\<And>s' a'. ksimplex p n s' \<Longrightarrow> a' \<in> s' \<Longrightarrow> s' - {a'} = s - {a} \<Longrightarrow> s' = s" proof- case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
834 |
guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note exta = this[rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
835 |
have a:"a = a1" apply(rule ksimplex_fix_plane_0[OF assms(2,4-5)]) using exta(1-2,5) by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
836 |
guess b0 b1 apply(rule ksimplex_extrema_strong[OF goal1(1) assms(3)]) . note extb = this[rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
837 |
have a':"a' = b1" apply(rule ksimplex_fix_plane_0[OF goal1(2) assms(4), of b0]) unfolding goal1(3) using assms extb goal1 by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
838 |
have "b0 = a0" unfolding kle_antisym[THEN sym, of b0 a0 n] using exta extb using goal1(3) unfolding a a' by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
839 |
hence "b1 = a1" apply-apply(rule ext) unfolding exta(5) extb(5) by auto ultimately |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
840 |
show "s' = s" apply-apply(rule *[of _ a1 b1]) using exta(1-2) extb(1-2) goal1 by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
841 |
show ?thesis unfolding card_1_exists apply-apply(rule ex1I[of _ s]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
842 |
unfolding mem_Collect_eq defer apply(erule conjE bexE)+ apply(rule_tac a'=b in **) using assms(1,2) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
843 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
844 |
lemma ksimplex_replace_1: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
845 |
assumes "ksimplex p n s" "a \<in> s" "n \<noteq> 0" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = p" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
846 |
shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
847 |
have lem:"\<And>a a' s'. s' - {a'} = s - {a} \<Longrightarrow> a' = a \<Longrightarrow> a' \<in> s' \<Longrightarrow> a \<in> s \<Longrightarrow> s' = s" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
848 |
have lem:"\<And>s' a'. ksimplex p n s' \<Longrightarrow> a'\<in>s' \<Longrightarrow> s' - {a'} = s - {a} \<Longrightarrow> s' = s" proof- case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
849 |
guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note exta = this[rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
850 |
have a:"a = a0" apply(rule ksimplex_fix_plane_p[OF assms(1-2,4-5) exta(1,2)]) unfolding exta by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
851 |
guess b0 b1 apply(rule ksimplex_extrema_strong[OF goal1(1) assms(3)]) . note extb = this[rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
852 |
have a':"a' = b0" apply(rule ksimplex_fix_plane_p[OF goal1(1-2) assms(4), of _ b1]) unfolding goal1 extb using extb(1,2) assms(5) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
853 |
moreover have *:"b1 = a1" unfolding kle_antisym[THEN sym, of b1 a1 n] using exta extb using goal1(3) unfolding a a' by blast moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
854 |
have "a0 = b0" apply(rule ext) proof- case goal1 show "a0 x = b0 x" |
41958 | 855 |
using *[THEN cong, of x x] unfolding exta extb apply-apply(cases "x\<in>{1..n}") by auto qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
856 |
ultimately show "s' = s" apply-apply(rule lem[OF goal1(3) _ goal1(2) assms(2)]) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
857 |
show ?thesis unfolding card_1_exists apply(rule ex1I[of _ s]) unfolding mem_Collect_eq apply(rule,rule assms(1)) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
858 |
apply(rule_tac x=a in bexI) prefer 3 apply(erule conjE bexE)+ apply(rule_tac a'=b in lem) using assms(1-2) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
859 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
860 |
lemma ksimplex_replace_2: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
861 |
assumes "ksimplex p n s" "a \<in> s" "n \<noteq> 0" "~(\<exists>j\<in>{1..n}. \<forall>x\<in>s - {a}. x j = 0)" "~(\<exists>j\<in>{1..n}. \<forall>x\<in>s - {a}. x j = p)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
862 |
shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 2" (is "card ?A = 2") proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
863 |
have lem1:"\<And>a a' s s'. s' - {a'} = s - {a} \<Longrightarrow> a' = a \<Longrightarrow> a' \<in> s' \<Longrightarrow> a \<in> s \<Longrightarrow> s' = s" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
864 |
have lem2:"\<And>a b. a\<in>s \<Longrightarrow> b\<noteq>a \<Longrightarrow> s \<noteq> insert b (s - {a})" proof case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
865 |
hence "a\<in>insert b (s - {a})" by auto hence "a\<in> s - {a}" unfolding insert_iff using goal1 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
866 |
thus False by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
867 |
guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note a0a1=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
868 |
{ assume "a=a0" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
869 |
have *:"\<And>P Q. (P \<or> Q) \<Longrightarrow> \<not> P \<Longrightarrow> Q" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
870 |
have "\<exists>x\<in>s. \<not> kle n x a0" apply(rule_tac x=a1 in bexI) proof assume as:"kle n a1 a0" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
871 |
show False using kle_imp_pointwise[OF as,THEN spec[where x=1]] unfolding a0a1(5)[THEN spec[where x=1]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
872 |
using assms(3) by auto qed(insert a0a1,auto) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
873 |
hence "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. y j = (if j = k then a0 j + 1 else a0 j)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
874 |
apply(rule_tac *[OF ksimplex_successor[OF assms(1-2),unfolded `a=a0`]]) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
875 |
then guess a2 .. from this(2) guess k .. note k=this note a2=`a2\<in>s` |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
876 |
def a3 \<equiv> "\<lambda>j. if j = k then a1 j + 1 else a1 j" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
877 |
have "a3 \<notin> s" proof assume "a3\<in>s" hence "kle n a3 a1" using a0a1(4) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
878 |
thus False apply(drule_tac kle_imp_pointwise) unfolding a3_def |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
879 |
apply(erule_tac x=k in allE) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
880 |
hence "a3 \<noteq> a0" "a3 \<noteq> a1" using a0a1 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
881 |
have "a2 \<noteq> a0" using k(2)[THEN spec[where x=k]] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
882 |
have lem3:"\<And>x. x\<in>(s - {a0}) \<Longrightarrow> kle n a2 x" proof(rule ccontr) case goal1 hence as:"x\<in>s" "x\<noteq>a0" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
883 |
have "kle n a2 x \<or> kle n x a2" using ksimplexD(6)[OF assms(1)] and as `a2\<in>s` by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
884 |
have "kle n a0 x" using a0a1(4) as by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
885 |
ultimately have "x = a0 \<or> x = a2" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
886 |
hence "x = a2" using as by auto thus False using goal1(2) using kle_refl by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
887 |
let ?s = "insert a3 (s - {a0})" have "ksimplex p n ?s" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
888 |
show "card ?s = n + 1" using ksimplexD(2-3)[OF assms(1)] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
889 |
using `a3\<noteq>a0` `a3\<notin>s` `a0\<in>s` by(auto simp add:card_insert_if) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
890 |
fix x assume x:"x \<in> insert a3 (s - {a0})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
891 |
show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3") |
41958 | 892 |
fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next |
893 |
fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") |
|
894 |
case False thus "a3 j \<le>p" unfolding True a3_def using `a1\<in>s` ksimplexD(4)[OF assms(1)] by auto next |
|
895 |
guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this |
|
896 |
have "a2 k \<le> a4 k" using lem3[OF a4(1)[unfolded `a=a0`],THEN kle_imp_pointwise] by auto |
|
897 |
also have "\<dots> < p" using ksimplexD(4)[OF assms(1),rule_format,of a4 k] using a4 by auto |
|
898 |
finally have *:"a0 k + 1 < p" unfolding k(2)[rule_format] by auto |
|
899 |
case True thus "a3 j \<le>p" unfolding a3_def unfolding a0a1(5)[rule_format] |
|
900 |
using k(1) k(2)assms(5) using * by simp qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
901 |
show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j\<notin>{1..n}" |
41958 | 902 |
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto } |
903 |
case True show "x j = p" unfolding True a3_def using j k(1) |
|
904 |
using ksimplexD(5)[OF assms(1),rule_format,OF `a1\<in>s` j] by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
905 |
fix y assume y:"y\<in>insert a3 (s - {a0})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
906 |
have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a0 \<Longrightarrow> kle n x a3" proof- case goal1 |
41958 | 907 |
guess kk using a0a1(4)[rule_format,OF `x\<in>s`,THEN conjunct2,unfolded kle_def] |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
908 |
apply-apply(erule exE,erule conjE) . note kk=this |
41958 | 909 |
have "k\<notin>kk" proof assume "k\<in>kk" |
910 |
hence "a1 k = x k + 1" using kk by auto |
|
911 |
hence "a0 k = x k" unfolding a0a1(5)[rule_format] using k(1) by auto |
|
912 |
hence "a2 k = x k + 1" unfolding k(2)[rule_format] by auto moreover |
|
913 |
have "a2 k \<le> x k" using lem3[of x,THEN kle_imp_pointwise] goal1 by auto |
|
914 |
ultimately show False by auto qed |
|
915 |
thus ?case unfolding kle_def apply(rule_tac x="insert k kk" in exI) using kk(1) |
|
916 |
unfolding a3_def kle_def kk(2)[rule_format] using k(1) by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
917 |
show "kle n x y \<or> kle n y x" proof(cases "y=a3") |
41958 | 918 |
case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI1,rule lem4) |
919 |
using x by auto next |
|
920 |
case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True |
|
921 |
apply(rule disjI2,rule lem4) using y False by auto next |
|
922 |
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) |
|
923 |
using x y `y\<noteq>a3` by auto qed qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
924 |
hence "insert a3 (s - {a0}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
925 |
apply(rule_tac x="a3" in bexI) unfolding `a=a0` using `a3\<notin>s` by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
926 |
have "s \<in> ?A" using assms(1,2) by auto ultimately have "?A \<supseteq> {s, insert a3 (s - {a0})}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
927 |
moreover have "?A \<subseteq> {s, insert a3 (s - {a0})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
928 |
fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
929 |
from this(2) guess a' .. note a'=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
930 |
guess a_min a_max apply(rule ksimplex_extrema_strong[OF as assms(3)]) . note min_max=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
931 |
have *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}" |
41958 | 932 |
hence "kle n a2 x" apply-apply(rule lem3) using `a=a0` by auto |
933 |
hence "a2 k \<le> x k" apply(drule_tac kle_imp_pointwise) by auto moreover |
|
934 |
have "x k \<le> a2 k" unfolding k(2)[rule_format] using a0a1(4)[rule_format,of x,THEN conjunct1] |
|
935 |
unfolding kle_def using x by auto ultimately show "x k = a2 k" by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
936 |
have **:"a'=a_min \<or> a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
937 |
show "s' \<in> {s, insert a3 (s - {a0})}" proof(cases "a'=a_min") |
41958 | 938 |
case True have "a_max = a1" unfolding kle_antisym[THEN sym,of a_max a1 n] apply(rule) |
939 |
apply(rule a0a1(4)[rule_format,THEN conjunct2]) defer proof(rule min_max(4)[rule_format,THEN conjunct2]) |
|
940 |
show "a1\<in>s'" using a' unfolding `a=a0` using a0a1 by auto |
|
941 |
show "a_max \<in> s" proof(rule ccontr) assume "a_max\<notin>s" |
|
942 |
hence "a_max = a'" using a' min_max by auto |
|
943 |
thus False unfolding True using min_max by auto qed qed |
|
944 |
hence "\<forall>i. a_max i = a1 i" by auto |
|
945 |
hence "a' = a" unfolding True `a=a0` apply-apply(subst fun_eq_iff,rule) |
|
946 |
apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format] |
|
947 |
proof- case goal1 thus ?case apply(cases "x\<in>{1..n}") by auto qed |
|
948 |
hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` by auto |
|
949 |
thus ?thesis by auto next |
|
950 |
case False hence as:"a' = a_max" using ** by auto |
|
951 |
have "a_min = a2" unfolding kle_antisym[THEN sym, of _ _ n] apply rule |
|
952 |
apply(rule min_max(4)[rule_format,THEN conjunct1]) defer proof(rule lem3) |
|
953 |
show "a_min \<in> s - {a0}" unfolding a'(2)[THEN sym,unfolded `a=a0`] |
|
954 |
unfolding as using min_max(1-3) by auto |
|
955 |
have "a2 \<noteq> a" unfolding `a=a0` using k(2)[rule_format,of k] by auto |
|
956 |
hence "a2 \<in> s - {a}" using a2 by auto thus "a2 \<in> s'" unfolding a'(2)[THEN sym] by auto qed |
|
957 |
hence "\<forall>i. a_min i = a2 i" by auto |
|
958 |
hence "a' = a3" unfolding as `a=a0` apply-apply(subst fun_eq_iff,rule) |
|
959 |
apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format] |
|
960 |
unfolding a3_def k(2)[rule_format] unfolding a0a1(5)[rule_format] proof- case goal1 |
|
961 |
show ?case unfolding goal1 apply(cases "x\<in>{1..n}") defer apply(cases "x=k") |
|
962 |
using `k\<in>{1..n}` by auto qed |
|
963 |
hence "s' = insert a3 (s - {a0})" apply-apply(rule lem1) defer apply assumption |
|
964 |
apply(rule a'(1)) unfolding a' `a=a0` using `a3\<notin>s` by auto |
|
965 |
thus ?thesis by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
966 |
ultimately have *:"?A = {s, insert a3 (s - {a0})}" by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
967 |
have "s \<noteq> insert a3 (s - {a0})" using `a3\<notin>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
968 |
hence ?thesis unfolding * by auto } moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
969 |
{ assume "a=a1" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
970 |
have *:"\<And>P Q. (P \<or> Q) \<Longrightarrow> \<not> P \<Longrightarrow> Q" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
971 |
have "\<exists>x\<in>s. \<not> kle n a1 x" apply(rule_tac x=a0 in bexI) proof assume as:"kle n a1 a0" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
972 |
show False using kle_imp_pointwise[OF as,THEN spec[where x=1]] unfolding a0a1(5)[THEN spec[where x=1]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
973 |
using assms(3) by auto qed(insert a0a1,auto) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
974 |
hence "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. a1 j = (if j = k then y j + 1 else y j)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
975 |
apply(rule_tac *[OF ksimplex_predecessor[OF assms(1-2),unfolded `a=a1`]]) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
976 |
then guess a2 .. from this(2) guess k .. note k=this note a2=`a2\<in>s` |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
977 |
def a3 \<equiv> "\<lambda>j. if j = k then a0 j - 1 else a0 j" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
978 |
have "a2 \<noteq> a1" using k(2)[THEN spec[where x=k]] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
979 |
have lem3:"\<And>x. x\<in>(s - {a1}) \<Longrightarrow> kle n x a2" proof(rule ccontr) case goal1 hence as:"x\<in>s" "x\<noteq>a1" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
980 |
have "kle n a2 x \<or> kle n x a2" using ksimplexD(6)[OF assms(1)] and as `a2\<in>s` by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
981 |
have "kle n x a1" using a0a1(4) as by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
982 |
ultimately have "x = a2 \<or> x = a1" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
983 |
hence "x = a2" using as by auto thus False using goal1(2) using kle_refl by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
984 |
have "a0 k \<noteq> 0" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
985 |
guess a4 using assms(4)[unfolded bex_simps ball_simps,rule_format,OF `k\<in>{1..n}`] .. note a4=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
986 |
have "a4 k \<le> a2 k" using lem3[OF a4(1)[unfolded `a=a1`],THEN kle_imp_pointwise] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
987 |
moreover have "a4 k > 0" using a4 by auto ultimately have "a2 k > 0" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
988 |
hence "a1 k > 1" unfolding k(2)[rule_format] by simp |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
989 |
thus ?thesis unfolding a0a1(5)[rule_format] using k(1) by simp qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
990 |
hence lem4:"\<forall>j. a0 j = (if j=k then a3 j + 1 else a3 j)" unfolding a3_def by simp |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
991 |
have "\<not> kle n a0 a3" apply(rule ccontr) unfolding not_not apply(drule kle_imp_pointwise) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
992 |
unfolding lem4[rule_format] apply(erule_tac x=k in allE) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
993 |
hence "a3 \<notin> s" using a0a1(4) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
994 |
hence "a3 \<noteq> a1" "a3 \<noteq> a0" using a0a1 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
995 |
let ?s = "insert a3 (s - {a1})" have "ksimplex p n ?s" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
996 |
show "card ?s = n+1" using ksimplexD(2-3)[OF assms(1)] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
997 |
using `a3\<noteq>a0` `a3\<notin>s` `a1\<in>s` by(auto simp add:card_insert_if) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
998 |
fix x assume x:"x \<in> insert a3 (s - {a1})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
999 |
show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3") |
41958 | 1000 |
fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next |
1001 |
fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") |
|
1002 |
case False thus "a3 j \<le>p" unfolding True a3_def using `a0\<in>s` ksimplexD(4)[OF assms(1)] by auto next |
|
1003 |
guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1004 |
case True have "a3 k \<le> a0 k" unfolding lem4[rule_format] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1005 |
also have "\<dots> \<le> p" using ksimplexD(4)[OF assms(1),rule_format,of a0 k] a0a1 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1006 |
finally show "a3 j \<le> p" unfolding True by auto qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1007 |
show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j\<notin>{1..n}" |
41958 | 1008 |
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto } |
1009 |
case True show "x j = p" unfolding True a3_def using j k(1) |
|
1010 |
using ksimplexD(5)[OF assms(1),rule_format,OF `a0\<in>s` j] by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1011 |
fix y assume y:"y\<in>insert a3 (s - {a1})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1012 |
have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a1 \<Longrightarrow> kle n a3 x" proof- case goal1 hence *:"x\<in>s - {a1}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1013 |
have "kle n a3 a2" proof- have "kle n a0 a1" using a0a1 by auto then guess kk unfolding kle_def .. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1014 |
thus ?thesis unfolding kle_def apply(rule_tac x=kk in exI) unfolding lem4[rule_format] k(2)[rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1015 |
apply(rule)defer proof(rule) case goal1 thus ?case apply-apply(erule conjE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1016 |
apply(erule_tac[!] x=j in allE) apply(cases "j\<in>kk") apply(case_tac[!] "j=k") by auto qed auto qed moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1017 |
have "kle n a3 a0" unfolding kle_def lem4[rule_format] apply(rule_tac x="{k}" in exI) using k(1) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1018 |
ultimately show ?case apply-apply(rule kle_between_l[of _ a0 _ a2]) using lem3[OF *] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1019 |
using a0a1(4)[rule_format,OF goal1(1)] by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1020 |
show "kle n x y \<or> kle n y x" proof(cases "y=a3") |
41958 | 1021 |
case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI2,rule lem4) |
1022 |
using x by auto next |
|
1023 |
case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True |
|
1024 |
apply(rule disjI1,rule lem4) using y False by auto next |
|
1025 |
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) |
|
1026 |
using x y `y\<noteq>a3` by auto qed qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1027 |
hence "insert a3 (s - {a1}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1028 |
apply(rule_tac x="a3" in bexI) unfolding `a=a1` using `a3\<notin>s` by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1029 |
have "s \<in> ?A" using assms(1,2) by auto ultimately have "?A \<supseteq> {s, insert a3 (s - {a1})}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1030 |
moreover have "?A \<subseteq> {s, insert a3 (s - {a1})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1031 |
fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1032 |
from this(2) guess a' .. note a'=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1033 |
guess a_min a_max apply(rule ksimplex_extrema_strong[OF as assms(3)]) . note min_max=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1034 |
have *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}" |
41958 | 1035 |
hence "kle n x a2" apply-apply(rule lem3) using `a=a1` by auto |
1036 |
hence "x k \<le> a2 k" apply(drule_tac kle_imp_pointwise) by auto moreover |
|
1037 |
{ have "a2 k \<le> a0 k" using k(2)[rule_format,of k] unfolding a0a1(5)[rule_format] using k(1) by simp |
|
1038 |
also have "\<dots> \<le> x k" using a0a1(4)[rule_format,of x,THEN conjunct1,THEN kle_imp_pointwise] x by auto |
|
1039 |
finally have "a2 k \<le> x k" . } ultimately show "x k = a2 k" by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1040 |
have **:"a'=a_min \<or> a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1041 |
have "a2 \<noteq> a1" proof assume as:"a2 = a1" |
41958 | 1042 |
show False using k(2) unfolding as apply(erule_tac x=k in allE) by auto qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1043 |
hence a2':"a2 \<in> s' - {a'}" unfolding a' using a2 unfolding `a=a1` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1044 |
show "s' \<in> {s, insert a3 (s - {a1})}" proof(cases "a'=a_min") |
41958 | 1045 |
case True have "a_max \<in> s - {a1}" using min_max unfolding a'(2)[unfolded `a=a1`,THEN sym] True by auto |
1046 |
hence "a_max = a2" unfolding kle_antisym[THEN sym,of a_max a2 n] apply-apply(rule) |
|
1047 |
apply(rule lem3,assumption) apply(rule min_max(4)[rule_format,THEN conjunct2]) using a2' by auto |
|
1048 |
hence a_max:"\<forall>i. a_max i = a2 i" by auto |
|
1049 |
have *:"\<forall>j. a2 j = (if j\<in>{1..n} then a3 j + 1 else a3 j)" |
|
1050 |
using k(2) unfolding lem4[rule_format] a0a1(5)[rule_format] apply-apply(rule,erule_tac x=j in allE) |
|
1051 |
proof- case goal1 thus ?case apply(cases "j\<in>{1..n}",case_tac[!] "j=k") by auto qed |
|
1052 |
have "\<forall>i. a_min i = a3 i" using a_max apply-apply(rule,erule_tac x=i in allE) |
|
1053 |
unfolding min_max(5)[rule_format] *[rule_format] proof- case goal1 |
|
1054 |
thus ?case apply(cases "i\<in>{1..n}") by auto qed hence "a_min = a3" unfolding fun_eq_iff . |
|
1055 |
hence "s' = insert a3 (s - {a1})" using a' unfolding `a=a1` True by auto thus ?thesis by auto next |
|
1056 |
case False hence as:"a'=a_max" using ** by auto |
|
1057 |
have "a_min = a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule) |
|
1058 |
apply(rule min_max(4)[rule_format,THEN conjunct1]) defer apply(rule a0a1(4)[rule_format,THEN conjunct1]) proof- |
|
1059 |
have "a_min \<in> s - {a1}" using min_max(1,3) unfolding a'(2)[THEN sym,unfolded `a=a1`] as by auto |
|
1060 |
thus "a_min \<in> s" by auto have "a0 \<in> s - {a1}" using a0a1(1-3) by auto thus "a0 \<in> s'" |
|
1061 |
unfolding a'(2)[THEN sym,unfolded `a=a1`] by auto qed |
|
1062 |
hence "\<forall>i. a_max i = a1 i" unfolding a0a1(5)[rule_format] min_max(5)[rule_format] by auto |
|
1063 |
hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` unfolding as `a=a1` unfolding fun_eq_iff by auto |
|
1064 |
thus ?thesis by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1065 |
ultimately have *:"?A = {s, insert a3 (s - {a1})}" by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1066 |
have "s \<noteq> insert a3 (s - {a1})" using `a3\<notin>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1067 |
hence ?thesis unfolding * by auto } moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1068 |
{ assume as:"a\<noteq>a0" "a\<noteq>a1" have "\<not> (\<forall>x\<in>s. kle n a x)" proof case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1069 |
have "a=a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule) |
41958 | 1070 |
using goal1 a0a1 assms(2) by auto thus False using as by auto qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1071 |
hence "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. a j = (if j = k then y j + 1 else y j)" using ksimplex_predecessor[OF assms(1-2)] by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1072 |
then guess u .. from this(2) guess k .. note k = this[rule_format] note u = `u\<in>s` |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1073 |
have "\<not> (\<forall>x\<in>s. kle n x a)" proof case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1074 |
have "a=a1" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule) |
41958 | 1075 |
using goal1 a0a1 assms(2) by auto thus False using as by auto qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1076 |
hence "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. y j = (if j = k then a j + 1 else a j)" using ksimplex_successor[OF assms(1-2)] by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1077 |
then guess v .. from this(2) guess l .. note l = this[rule_format] note v = `v\<in>s` |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1078 |
def a' \<equiv> "\<lambda>j. if j = l then u j + 1 else u j" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1079 |
have kl:"k \<noteq> l" proof assume "k=l" have *:"\<And>P. (if P then (1::nat) else 0) \<noteq> 2" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1080 |
thus False using ksimplexD(6)[OF assms(1),rule_format,OF u v] unfolding kle_def |
41958 | 1081 |
unfolding l(2) k(2) `k=l` apply-apply(erule disjE)apply(erule_tac[!] exE conjE)+ |
1082 |
apply(erule_tac[!] x=l in allE)+ by(auto simp add: *) qed |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1083 |
hence aa':"a'\<noteq>a" apply-apply rule unfolding fun_eq_iff unfolding a'_def k(2) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1084 |
apply(erule_tac x=l in allE) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1085 |
have "a' \<notin> s" apply(rule) apply(drule ksimplexD(6)[OF assms(1),rule_format,OF `a\<in>s`]) proof(cases "kle n a a'") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1086 |
case goal2 hence "kle n a' a" by auto thus False apply(drule_tac kle_imp_pointwise) |
41958 | 1087 |
apply(erule_tac x=l in allE) unfolding a'_def k(2) using kl by auto next |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1088 |
case True thus False apply(drule_tac kle_imp_pointwise) |
41958 | 1089 |
apply(erule_tac x=k in allE) unfolding a'_def k(2) using kl by auto qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1090 |
have kle_uv:"kle n u a" "kle n u a'" "kle n a v" "kle n a' v" unfolding kle_def apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1091 |
apply(rule_tac[1] x="{k}" in exI,rule_tac[2] x="{l}" in exI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1092 |
apply(rule_tac[3] x="{l}" in exI,rule_tac[4] x="{k}" in exI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1093 |
unfolding l(2) k(2) a'_def using l(1) k(1) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1094 |
have uxv:"\<And>x. kle n u x \<Longrightarrow> kle n x v \<Longrightarrow> (x = u) \<or> (x = a) \<or> (x = a') \<or> (x = v)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1095 |
proof- case goal1 thus ?case proof(cases "x k = u k", case_tac[!] "x l = u l") |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1096 |
assume as:"x l = u l" "x k = u k" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1097 |
have "x = u" unfolding fun_eq_iff |
41958 | 1098 |
using goal1(2)[THEN kle_imp_pointwise,unfolded l(2)] unfolding k(2) apply- |
1099 |
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1 |
|
1100 |
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1101 |
assume as:"x l \<noteq> u l" "x k = u k" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1102 |
have "x = a'" unfolding fun_eq_iff unfolding a'_def |
41958 | 1103 |
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply- |
1104 |
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1 |
|
1105 |
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1106 |
assume as:"x l = u l" "x k \<noteq> u k" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1107 |
have "x = a" unfolding fun_eq_iff |
41958 | 1108 |
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply- |
1109 |
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1 |
|
1110 |
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1111 |
assume as:"x l \<noteq> u l" "x k \<noteq> u k" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1112 |
have "x = v" unfolding fun_eq_iff |
41958 | 1113 |
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply- |
1114 |
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1 |
|
1115 |
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as `k\<noteq>l` by auto qed thus ?case by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1116 |
have uv:"kle n u v" apply(rule kle_trans[OF kle_uv(1,3)]) using ksimplexD(6)[OF assms(1)] using u v by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1117 |
have lem3:"\<And>x. x\<in>s \<Longrightarrow> kle n v x \<Longrightarrow> kle n a' x" apply(rule kle_between_r[of _ u _ v]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1118 |
prefer 3 apply(rule kle_trans[OF uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1119 |
using kle_uv `u\<in>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1120 |
have lem4:"\<And>x. x\<in>s \<Longrightarrow> kle n x u \<Longrightarrow> kle n x a'" apply(rule kle_between_l[of _ u _ v]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1121 |
prefer 4 apply(rule kle_trans[OF _ uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1122 |
using kle_uv `v\<in>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1123 |
have lem5:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a \<Longrightarrow> kle n x a' \<or> kle n a' x" proof- case goal1 thus ?case |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1124 |
proof(cases "kle n v x \<or> kle n x u") case True thus ?thesis using goal1 by(auto intro:lem3 lem4) next |
41958 | 1125 |
case False hence *:"kle n u x" "kle n x v" using ksimplexD(6)[OF assms(1)] using goal1 `u\<in>s` `v\<in>s` by auto |
1126 |
show ?thesis using uxv[OF *] using kle_uv using goal1 by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1127 |
have "ksimplex p n (insert a' (s - {a}))" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1128 |
show "card (insert a' (s - {a})) = n + 1" using ksimplexD(2-3)[OF assms(1)] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1129 |
using `a'\<noteq>a` `a'\<notin>s` `a\<in>s` by(auto simp add:card_insert_if) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1130 |
fix x assume x:"x \<in> insert a' (s - {a})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1131 |
show "\<forall>j. x j \<le> p" proof(rule,cases "x = a'") |
41958 | 1132 |
fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next |
1133 |
fix j case True show "x j\<le>p" unfolding True proof(cases "j=l") |
|
1134 |
case False thus "a' j \<le>p" unfolding True a'_def using `u\<in>s` ksimplexD(4)[OF assms(1)] by auto next |
|
1135 |
case True have *:"a l = u l" "v l = a l + 1" using k(2)[of l] l(2)[of l] `k\<noteq>l` by auto |
|
1136 |
have "u l + 1 \<le> p" unfolding *[THEN sym] using ksimplexD(4)[OF assms(1)] using `v\<in>s` by auto |
|
1137 |
thus "a' j \<le>p" unfolding a'_def True by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1138 |
show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a'") fix j::nat assume j:"j\<notin>{1..n}" |
41958 | 1139 |
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto } |
1140 |
case True show "x j = p" unfolding True a'_def using j l(1) |
|
1141 |
using ksimplexD(5)[OF assms(1),rule_format,OF `u\<in>s` j] by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1142 |
fix y assume y:"y\<in>insert a' (s - {a})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1143 |
show "kle n x y \<or> kle n y x" proof(cases "y=a'") |
41958 | 1144 |
case True show ?thesis unfolding True apply(cases "x=a'") defer apply(rule lem5) using x by auto next |
1145 |
case False show ?thesis proof(cases "x=a'") case True show ?thesis unfolding True |
|
1146 |
using lem5[of y] using y by auto next |
|
1147 |
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) |
|
1148 |
using x y `y\<noteq>a'` by auto qed qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1149 |
hence "insert a' (s - {a}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1150 |
apply(rule_tac x="a'" in bexI) using aa' `a'\<notin>s` by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1151 |
have "s \<in> ?A" using assms(1,2) by auto ultimately have "?A \<supseteq> {s, insert a' (s - {a})}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1152 |
moreover have "?A \<subseteq> {s, insert a' (s - {a})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1153 |
fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1154 |
from this(2) guess a'' .. note a''=this |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1155 |
have "u\<noteq>v" unfolding fun_eq_iff unfolding l(2) k(2) by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1156 |
hence uv':"\<not> kle n v u" using uv using kle_antisym by auto |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1157 |
have "u\<noteq>a" "v\<noteq>a" unfolding fun_eq_iff k(2) l(2) by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1158 |
hence uvs':"u\<in>s'" "v\<in>s'" using `u\<in>s` `v\<in>s` using a'' by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1159 |
have lem6:"a \<in> s' \<or> a' \<in> s'" proof(cases "\<forall>x\<in>s'. kle n x u \<or> kle n v x") |
41958 | 1160 |
case False then guess w unfolding ball_simps .. note w=this |
1161 |
hence "kle n u w" "kle n w v" using ksimplexD(6)[OF as] uvs' by auto |
|
1162 |
hence "w = a' \<or> w = a" using uxv[of w] uvs' w by auto thus ?thesis using w by auto next |
|
1163 |
case True have "\<not> (\<forall>x\<in>s'. kle n x u)" unfolding ball_simps apply(rule_tac x=v in bexI) |
|
1164 |
using uv `u\<noteq>v` unfolding kle_antisym[of n u v,THEN sym] using `v\<in>s'` by auto |
|
1165 |
hence "\<exists>y\<in>s'. \<exists>k\<in>{1..n}. \<forall>j. y j = (if j = k then u j + 1 else u j)" using ksimplex_successor[OF as `u\<in>s'`] by blast |
|
1166 |
then guess w .. note w=this from this(2) guess kk .. note kk=this[rule_format] |
|
1167 |
have "\<not> kle n w u" apply-apply(rule,drule kle_imp_pointwise) |
|
1168 |
apply(erule_tac x=kk in allE) unfolding kk by auto |
|
1169 |
hence *:"kle n v w" using True[rule_format,OF w(1)] by auto |
|
1170 |
hence False proof(cases "kk\<noteq>l") case True thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)] |
|
1171 |
apply(erule_tac x=l in allE) using `k\<noteq>l` by auto next |
|
1172 |
case False hence "kk\<noteq>k" using `k\<noteq>l` by auto thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)] |
|
1173 |
apply(erule_tac x=k in allE) using `k\<noteq>l` by auto qed |
|
1174 |
thus ?thesis by auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1175 |
thus "s' \<in> {s, insert a' (s - {a})}" proof(cases "a\<in>s'") |
41958 | 1176 |
case True hence "s' = s" apply-apply(rule lem1[OF a''(2)]) using a'' `a\<in>s` by auto |
1177 |
thus ?thesis by auto next case False hence "a'\<in>s'" using lem6 by auto |
|
1178 |
hence "s' = insert a' (s - {a})" apply-apply(rule lem1[of _ a'' _ a']) |
|
1179 |
unfolding a''(2)[THEN sym] using a'' using `a'\<notin>s` by auto |
|
1180 |
thus ?thesis by auto qed qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1181 |
ultimately have *:"?A = {s, insert a' (s - {a})}" by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1182 |
have "s \<noteq> insert a' (s - {a})" using `a'\<notin>s` by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1183 |
hence ?thesis unfolding * by auto } |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1184 |
ultimately show ?thesis by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1185 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1186 |
subsection {* Hence another step towards concreteness. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1187 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1188 |
lemma kuhn_simplex_lemma: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1189 |
assumes "\<forall>s. ksimplex p (n + 1) s \<longrightarrow> (rl ` s \<subseteq>{0..n+1})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1190 |
"odd (card{f. \<exists>s a. ksimplex p (n + 1) s \<and> a \<in> s \<and> (f = s - {a}) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1191 |
(rl ` f = {0 .. n}) \<and> ((\<exists>j\<in>{1..n+1}.\<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n+1}.\<forall>x\<in>f. x j = p))})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1192 |
shows "odd(card {s\<in>{s. ksimplex p (n + 1) s}. rl ` s = {0..n+1} })" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1193 |
have *:"\<And>x y. x = y \<Longrightarrow> odd (card x) \<Longrightarrow> odd (card y)" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1194 |
have *:"odd(card {f\<in>{f. \<exists>s\<in>{s. ksimplex p (n + 1) s}. (\<exists>a\<in>s. f = s - {a})}. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1195 |
(rl ` f = {0..n}) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1196 |
((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1197 |
(\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p))})" apply(rule *[OF _ assms(2)]) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1198 |
show ?thesis apply(rule kuhn_complete_lemma[OF finite_simplices]) prefer 6 apply(rule *) apply(rule,rule,rule) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1199 |
apply(subst ksimplex_def) defer apply(rule,rule assms(1)[rule_format]) unfolding mem_Collect_eq apply assumption |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1200 |
apply default+ unfolding mem_Collect_eq apply(erule disjE bexE)+ defer apply(erule disjE bexE)+ defer |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1201 |
apply default+ unfolding mem_Collect_eq apply(erule disjE bexE)+ unfolding mem_Collect_eq proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1202 |
fix f s a assume as:"ksimplex p (n + 1) s" "a\<in>s" "f = s - {a}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1203 |
let ?S = "{s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1204 |
have S:"?S = {s'. ksimplex p (n + 1) s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})}" unfolding as by blast |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1205 |
{ fix j assume j:"j \<in> {1..n + 1}" "\<forall>x\<in>f. x j = 0" thus "card {s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})} = 1" unfolding S |
41958 | 1206 |
apply-apply(rule ksimplex_replace_0) apply(rule as)+ unfolding as by auto } |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1207 |
{ fix j assume j:"j \<in> {1..n + 1}" "\<forall>x\<in>f. x j = p" thus "card {s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})} = 1" unfolding S |
41958 | 1208 |
apply-apply(rule ksimplex_replace_1) apply(rule as)+ unfolding as by auto } |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1209 |
show "\<not> ((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p)) \<Longrightarrow> card ?S = 2" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1210 |
unfolding S apply(rule ksimplex_replace_2) apply(rule as)+ unfolding as by auto qed auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1211 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1212 |
subsection {* Reduced labelling. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1213 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1214 |
definition "reduced label (n::nat) (x::nat\<Rightarrow>nat) = |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1215 |
(SOME k. k \<le> n \<and> (\<forall>i. 1\<le>i \<and> i<k+1 \<longrightarrow> label x i = 0) \<and> (k = n \<or> label x (k + 1) \<noteq> (0::nat)))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1216 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1217 |
lemma reduced_labelling: shows "reduced label n x \<le> n" (is ?t1) and |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1218 |
"\<forall>i. 1\<le>i \<and> i < reduced label n x + 1 \<longrightarrow> (label x i = 0)" (is ?t2) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1219 |
"(reduced label n x = n) \<or> (label x (reduced label n x + 1) \<noteq> 0)" (is ?t3) proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1220 |
have num_WOP:"\<And>P k. P (k::nat) \<Longrightarrow> \<exists>n. P n \<and> (\<forall>m<n. \<not> P m)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1221 |
apply(drule ex_has_least_nat[where m="\<lambda>x. x"]) apply(erule exE,rule_tac x=x in exI) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1222 |
have *:"n \<le> n \<and> (label x (n + 1) \<noteq> 0 \<or> n = n)" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1223 |
then guess N apply(drule_tac num_WOP[of "\<lambda>j. j\<le>n \<and> (label x (j+1) \<noteq> 0 \<or> n = j)"]) apply(erule exE) . note N=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1224 |
have N':"N \<le> n" "\<forall>i. 1 \<le> i \<and> i < N + 1 \<longrightarrow> label x i = 0" "N = n \<or> label x (N + 1) \<noteq> 0" defer proof(rule,rule) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1225 |
fix i assume i:"1\<le>i \<and> i<N+1" thus "label x i = 0" using N[THEN conjunct2,THEN spec[where x="i - 1"]] using N by auto qed(insert N, auto) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1226 |
show ?t1 ?t2 ?t3 unfolding reduced_def apply(rule_tac[!] someI2_ex) using N' by(auto intro!: exI[where x=N]) qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1227 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1228 |
lemma reduced_labelling_unique: fixes x::"nat \<Rightarrow> nat" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1229 |
assumes "r \<le> n" "\<forall>i. 1 \<le> i \<and> i < r + 1 \<longrightarrow> (label x i = 0)" "(r = n) \<or> (label x (r + 1) \<noteq> 0)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1230 |
shows "reduced label n x = r" apply(rule le_antisym) apply(rule_tac[!] ccontr) unfolding not_le |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1231 |
using reduced_labelling[of label n x] using assms by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1232 |
|
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1233 |
lemma reduced_labelling_zero: assumes "j\<in>{1..n}" "label x j = 0" shows "reduced label n x \<noteq> j - 1" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44821
diff
changeset
|
1234 |
using reduced_labelling[of label n x] using assms by fastforce |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1235 |
|
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1236 |
lemma reduced_labelling_nonzero: assumes "j\<in>{1..n}" "label x j \<noteq> 0" shows "reduced label n x < j" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1237 |
using assms and reduced_labelling[of label n x] apply(erule_tac x=j in allE) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1238 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1239 |
lemma reduced_labelling_Suc: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1240 |
assumes "reduced lab (n + 1) x \<noteq> n + 1" shows "reduced lab (n + 1) x = reduced lab n x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1241 |
apply(subst eq_commute) apply(rule reduced_labelling_unique) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1242 |
using reduced_labelling[of lab "n+1" x] and assms by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1243 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1244 |
lemma complete_face_top: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1245 |
assumes "\<forall>x\<in>f. \<forall>j\<in>{1..n+1}. x j = 0 \<longrightarrow> lab x j = 0" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1246 |
"\<forall>x\<in>f. \<forall>j\<in>{1..n+1}. x j = p \<longrightarrow> lab x j = 1" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1247 |
shows "((reduced lab (n + 1)) ` f = {0..n}) \<and> ((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p)) \<longleftrightarrow> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1248 |
((reduced lab (n + 1)) ` f = {0..n}) \<and> (\<forall>x\<in>f. x (n + 1) = p)" (is "?l = ?r") proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1249 |
assume ?l (is "?as \<and> (?a \<or> ?b)") thus ?r apply-apply(rule,erule conjE,assumption) proof(cases ?a) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1250 |
case True then guess j .. note j=this {fix x assume x:"x\<in>f" |
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1251 |
have "reduced lab (n+1) x \<noteq> j - 1" using j apply-apply(rule reduced_labelling_zero) defer apply(rule assms(1)[rule_format]) using x by auto } |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1252 |
moreover have "j - 1 \<in> {0..n}" using j by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1253 |
then guess y unfolding `?l`[THEN conjunct1,THEN sym] and image_iff .. note y = this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1254 |
ultimately have False by auto thus "\<forall>x\<in>f. x (n + 1) = p" by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1255 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1256 |
case False hence ?b using `?l` by blast then guess j .. note j=this {fix x assume x:"x\<in>f" |
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1257 |
have "reduced lab (n+1) x < j" using j apply-apply(rule reduced_labelling_nonzero) using assms(2)[rule_format,of x j] and x by auto } note * = this |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1258 |
have "j = n + 1" proof(rule ccontr) case goal1 hence "j < n + 1" using j by auto moreover |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1259 |
have "n \<in> {0..n}" by auto then guess y unfolding `?l`[THEN conjunct1,THEN sym] image_iff .. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1260 |
ultimately show False using *[of y] by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1261 |
thus "\<forall>x\<in>f. x (n + 1) = p" using j by auto qed qed(auto) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1262 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1263 |
subsection {* Hence we get just about the nice induction. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1264 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1265 |
lemma kuhn_induction: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1266 |
assumes "0 < p" "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = 0) \<longrightarrow> (lab x j = 0)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1267 |
"\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1268 |
"odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1269 |
shows "odd (card {s. ksimplex p (n+1) s \<and>((reduced lab (n+1)) ` s = {0..n+1})})" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1270 |
have *:"\<And>s t. odd (card s) \<Longrightarrow> s = t \<Longrightarrow> odd (card t)" "\<And>s f. (\<And>x. f x \<le> n +1 ) \<Longrightarrow> f ` s \<subseteq> {0..n+1}" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1271 |
show ?thesis apply(rule kuhn_simplex_lemma[unfolded mem_Collect_eq]) apply(rule,rule,rule *,rule reduced_labelling) |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1272 |
apply(rule *(1)[OF assms(4)]) apply(rule set_eqI) unfolding mem_Collect_eq apply(rule,erule conjE) defer apply(rule) proof-(*(rule,rule)*) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1273 |
fix f assume as:"ksimplex p n f" "reduced lab n ` f = {0..n}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1274 |
have *:"\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = 0 \<longrightarrow> lab x j = 0" "\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = p \<longrightarrow> lab x j = 1" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1275 |
using assms(2-3) using as(1)[unfolded ksimplex_def] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1276 |
have allp:"\<forall>x\<in>f. x (n + 1) = p" using assms(2) using as(1)[unfolded ksimplex_def] by auto |
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1277 |
{ fix x assume "x\<in>f" hence "reduced lab (n + 1) x < n + 1" apply-apply(rule reduced_labelling_nonzero) |
41958 | 1278 |
defer using assms(3) using as(1)[unfolded ksimplex_def] by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1279 |
hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc) using reduced_labelling(1) by auto } |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1280 |
hence "reduced lab (n + 1) ` f = {0..n}" unfolding as(2)[THEN sym] apply- apply(rule set_eqI) unfolding image_iff by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1281 |
moreover guess s using as(1)[unfolded simplex_top_face[OF assms(1) allp,THEN sym]] .. then guess a .. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1282 |
ultimately show "\<exists>s a. ksimplex p (n + 1) s \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1283 |
a \<in> s \<and> f = s - {a} \<and> reduced lab (n + 1) ` f = {0..n} \<and> ((\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p))" (is ?ex) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1284 |
apply(rule_tac x=s in exI,rule_tac x=a in exI) unfolding complete_face_top[OF *] using allp as(1) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1285 |
next fix f assume as:"\<exists>s a. ksimplex p (n + 1) s \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1286 |
a \<in> s \<and> f = s - {a} \<and> reduced lab (n + 1) ` f = {0..n} \<and> ((\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p))" (is ?ex) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1287 |
then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1288 |
{ fix x assume "x\<in>f" hence "reduced lab (n + 1) x \<in> reduced lab (n + 1) ` f" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1289 |
hence "reduced lab (n + 1) x < n + 1" using sa(4) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1290 |
hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc) |
41958 | 1291 |
using reduced_labelling(1) by auto } |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1292 |
thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[THEN sym] apply-apply(rule set_eqI) unfolding image_iff by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1293 |
have *:"\<forall>x\<in>f. x (n + 1) = p" proof(cases "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0") |
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1294 |
case True then guess j .. hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab (n + 1) x \<noteq> j - 1" apply-apply(rule reduced_labelling_zero) apply assumption |
41958 | 1295 |
apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1296 |
have "j - 1 \<in> {0..n}" using `j\<in>{1..n+1}` by auto |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44821
diff
changeset
|
1297 |
ultimately have False unfolding sa(4)[THEN sym] unfolding image_iff by fastforce thus ?thesis by auto next |
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44821
diff
changeset
|
1298 |
case False hence "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p" using sa(5) by fastforce then guess j .. note j=this |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1299 |
thus ?thesis proof(cases "j = n+1") |
41958 | 1300 |
case False hence *:"j\<in>{1..n}" using j by auto |
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1301 |
hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab n x < j" apply(rule reduced_labelling_nonzero) proof- fix x assume "x\<in>f" |
41958 | 1302 |
hence "lab x j = 1" apply-apply(rule assms(3)[rule_format,OF j(1)]) |
1303 |
using sa(1)[unfolded ksimplex_def] using j unfolding sa by auto thus "lab x j \<noteq> 0" by auto qed |
|
1304 |
moreover have "j\<in>{0..n}" using * by auto |
|
1305 |
ultimately have False unfolding part1[THEN sym] using * unfolding image_iff by auto thus ?thesis by auto qed auto qed |
|
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1306 |
thus "ksimplex p n f" using as unfolding simplex_top_face[OF assms(1) *,THEN sym] by auto qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1307 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1308 |
lemma kuhn_induction_Suc: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1309 |
assumes "0 < p" "\<forall>x. \<forall>j\<in>{1..Suc n}. (\<forall>j. x j \<le> p) \<and> (x j = 0) \<longrightarrow> (lab x j = 0)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1310 |
"\<forall>x. \<forall>j\<in>{1..Suc n}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1311 |
"odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1312 |
shows "odd (card {s. ksimplex p (Suc n) s \<and>((reduced lab (Suc n)) ` s = {0..Suc n})})" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1313 |
using assms unfolding Suc_eq_plus1 by(rule kuhn_induction) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1314 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1315 |
subsection {* And so we get the final combinatorial result. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1316 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1317 |
lemma ksimplex_0: "ksimplex p 0 s \<longleftrightarrow> s = {(\<lambda>x. p)}" (is "?l = ?r") proof |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1318 |
assume l:?l guess a using ksimplexD(3)[OF l, unfolded add_0] unfolding card_1_exists .. note a=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1319 |
have "a = (\<lambda>x. p)" using ksimplexD(5)[OF l, rule_format, OF a(1)] by(rule,auto) thus ?r using a by auto next |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1320 |
assume r:?r show ?l unfolding r ksimplex_eq by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1321 |
|
50514
1d1be8bf4cb2
tuned two lemma names, to avoid name hint clash (which confuses the MaSh evaluation, and which anyway isn't nice or necessary)
blanchet
parents:
50027
diff
changeset
|
1322 |
lemma reduce_labelling_zero[simp]: "reduced lab 0 x = 0" apply(rule reduced_labelling_unique) by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1323 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1324 |
lemma kuhn_combinatorial: |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1325 |
assumes "0 < p" "\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n \<and> (x j = 0) \<longrightarrow> (lab x j = 0)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1326 |
"\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n \<and> (x j = p) \<longrightarrow> (lab x j = 1)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1327 |
shows " odd (card {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})})" using assms proof(induct n) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1328 |
let ?M = "\<lambda>n. {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1329 |
{ case 0 have *:"?M 0 = {{(\<lambda>x. p)}}" unfolding ksimplex_0 by auto show ?case unfolding * by auto } |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1330 |
case (Suc n) have "odd (card (?M n))" apply(rule Suc(1)[OF Suc(2)]) using Suc(3-) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1331 |
thus ?case apply-apply(rule kuhn_induction_Suc) using Suc(2-) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1332 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1333 |
lemma kuhn_lemma: assumes "0 < (p::nat)" "0 < (n::nat)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1334 |
"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (label x i = (0::nat)) \<or> (label x i = 1))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1335 |
"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = 0) \<longrightarrow> (label x i = 0))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1336 |
"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = p) \<longrightarrow> (label x i = 1))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1337 |
obtains q where "\<forall>i\<in>{1..n}. q i < p" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1338 |
"\<forall>i\<in>{1..n}. \<exists>r s. (\<forall>j\<in>{1..n}. q(j) \<le> r(j) \<and> r(j) \<le> q(j) + 1) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1339 |
(\<forall>j\<in>{1..n}. q(j) \<le> s(j) \<and> s(j) \<le> q(j) + 1) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1340 |
~(label r i = label s i)" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1341 |
let ?A = "{s. ksimplex p n s \<and> reduced label n ` s = {0..n}}" have "n\<noteq>0" using assms by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1342 |
have conjD:"\<And>P Q. P \<and> Q \<Longrightarrow> P" "\<And>P Q. P \<and> Q \<Longrightarrow> Q" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1343 |
have "odd (card ?A)" apply(rule kuhn_combinatorial[of p n label]) using assms by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1344 |
hence "card ?A \<noteq> 0" apply-apply(rule ccontr) by auto hence "?A \<noteq> {}" unfolding card_eq_0_iff by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1345 |
then obtain s where "s\<in>?A" by auto note s=conjD[OF this[unfolded mem_Collect_eq]] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1346 |
guess a b apply(rule ksimplex_extrema_strong[OF s(1) `n\<noteq>0`]) . note ab=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1347 |
show ?thesis apply(rule that[of a]) proof(rule_tac[!] ballI) fix i assume "i\<in>{1..n}" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1348 |
hence "a i + 1 \<le> p" apply-apply(rule order_trans[of _ "b i"]) apply(subst ab(5)[THEN spec[where x=i]]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1349 |
using s(1)[unfolded ksimplex_def] defer apply- apply(erule conjE)+ apply(drule_tac bspec[OF _ ab(2)])+ by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1350 |
thus "a i < p" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1351 |
case goal2 hence "i \<in> reduced label n ` s" using s by auto then guess u unfolding image_iff .. note u=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1352 |
from goal2 have "i - 1 \<in> reduced label n ` s" using s by auto then guess v unfolding image_iff .. note v=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1353 |
show ?case apply(rule_tac x=u in exI, rule_tac x=v in exI) apply(rule conjI) defer apply(rule conjI) defer 2 proof(rule_tac[1-2] ballI) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1354 |
show "label u i \<noteq> label v i" using reduced_labelling[of label n u] reduced_labelling[of label n v] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1355 |
unfolding u(2)[THEN sym] v(2)[THEN sym] using goal2 by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1356 |
fix j assume j:"j\<in>{1..n}" show "a j \<le> u j \<and> u j \<le> a j + 1" "a j \<le> v j \<and> v j \<le> a j + 1" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1357 |
using conjD[OF ab(4)[rule_format, OF u(1)]] and conjD[OF ab(4)[rule_format, OF v(1)]] apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1358 |
apply(drule_tac[!] kle_imp_pointwise)+ apply(erule_tac[!] x=j in allE)+ unfolding ab(5)[rule_format] using j |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1359 |
by auto qed qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1360 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1361 |
subsection {* The main result for the unit cube. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1362 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1363 |
lemma kuhn_labelling_lemma': |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1364 |
assumes "(\<forall>x::nat\<Rightarrow>real. P x \<longrightarrow> P (f x))" "\<forall>x. P x \<longrightarrow> (\<forall>i::nat. Q i \<longrightarrow> 0 \<le> x i \<and> x i \<le> 1)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1365 |
shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1366 |
(\<forall>x i. P x \<and> Q i \<and> (x i = 0) \<longrightarrow> (l x i = 0)) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1367 |
(\<forall>x i. P x \<and> Q i \<and> (x i = 1) \<longrightarrow> (l x i = 1)) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1368 |
(\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x i \<le> f(x) i) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1369 |
(\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x) i \<le> x i)" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1370 |
have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1371 |
have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1372 |
show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1373 |
let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x xa = 0 \<longrightarrow> y = (0::nat)) \<and> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1374 |
(P x \<and> Q xa \<and> x xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x xa \<le> (f x) xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> (f x) xa \<le> x xa)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1375 |
{ assume "P x" "Q xa" hence "0 \<le> (f x) xa \<and> (f x) xa \<le> 1" using assms(2)[rule_format,of "f x" xa] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1376 |
apply(drule_tac assms(1)[rule_format]) by auto } |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1377 |
hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1378 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1379 |
lemma brouwer_cube: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1380 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1381 |
assumes "continuous_on {0..(\<Sum>Basis)} f" "f ` {0..(\<Sum>Basis)} \<subseteq> {0..(\<Sum>Basis)}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1382 |
shows "\<exists>x\<in>{0..(\<Sum>Basis)}. f x = x" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1383 |
proof (rule ccontr) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1384 |
def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by(auto simp add: Suc_le_eq DIM_positive) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1385 |
assume "\<not> (\<exists>x\<in>{0..\<Sum>Basis}. f x = x)" hence *:"\<not> (\<exists>x\<in>{0..\<Sum>Basis}. f x - x = 0)" by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1386 |
guess d apply(rule brouwer_compactness_lemma[OF compact_interval _ *]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1387 |
apply(rule continuous_on_intros assms)+ . note d=this[rule_format] |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1388 |
have *:"\<forall>x. x \<in> {0..\<Sum>Basis} \<longrightarrow> f x \<in> {0..\<Sum>Basis}" "\<forall>x. x \<in> {0..(\<Sum>Basis)::'a} \<longrightarrow> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1389 |
(\<forall>i\<in>Basis. True \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1390 |
using assms(2)[unfolded image_subset_iff Ball_def] unfolding mem_interval by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1391 |
guess label using kuhn_labelling_lemma[OF *] apply-apply(erule exE,(erule conjE)+) . note label = this[rule_format] |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1392 |
have lem1:"\<forall>x\<in>{0..\<Sum>Basis}.\<forall>y\<in>{0..\<Sum>Basis}.\<forall>i\<in>Basis. label x i \<noteq> label y i |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1393 |
\<longrightarrow> abs(f x \<bullet> i - x \<bullet> i) \<le> norm(f y - f x) + norm(y - x)" proof safe |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1394 |
fix x y::'a assume xy:"x\<in>{0..\<Sum>Basis}" "y\<in>{0..\<Sum>Basis}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1395 |
fix i assume i:"label x i \<noteq> label y i" "i\<in>Basis" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1396 |
have *:"\<And>x y fx fy::real. (x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1397 |
\<Longrightarrow> abs(fx - x) \<le> abs(fy - fx) + abs(y - x)" by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1398 |
have "\<bar>(f x - x) \<bullet> i\<bar> \<le> abs((f y - f x)\<bullet>i) + abs((y - x)\<bullet>i)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1399 |
unfolding inner_simps |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1400 |
apply(rule *) apply(cases "label x i = 0") apply(rule disjI1,rule) prefer 3 proof(rule disjI2,rule) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1401 |
assume lx:"label x i = 0" hence ly:"label y i = 1" using i label(1)[of i y] by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1402 |
show "x \<bullet> i \<le> f x \<bullet> i" apply(rule label(4)[rule_format]) using xy lx i(2) by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1403 |
show "f y \<bullet> i \<le> y \<bullet> i" apply(rule label(5)[rule_format]) using xy ly i(2) by auto next |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1404 |
assume "label x i \<noteq> 0" hence l:"label x i = 1" "label y i = 0" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1405 |
using i label(1)[of i x] label(1)[of i y] by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1406 |
show "f x \<bullet> i \<le> x \<bullet> i" apply(rule label(5)[rule_format]) using xy l i(2) by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1407 |
show "y \<bullet> i \<le> f y \<bullet> i" apply(rule label(4)[rule_format]) using xy l i(2) by auto qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1408 |
also have "\<dots> \<le> norm (f y - f x) + norm (y - x)" apply(rule add_mono) by(rule Basis_le_norm[OF i(2)])+ |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1409 |
finally show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)" unfolding inner_simps . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1410 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1411 |
have "\<exists>e>0. \<forall>x\<in>{0..\<Sum>Basis}. \<forall>y\<in>{0..\<Sum>Basis}. \<forall>z\<in>{0..\<Sum>Basis}. \<forall>i\<in>Basis. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1412 |
norm(x - z) < e \<and> norm(y - z) < e \<and> label x i \<noteq> label y i \<longrightarrow> abs((f(z) - z)\<bullet>i) < d / (real n)" proof- |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1413 |
have d':"d / real n / 8 > 0" apply(rule divide_pos_pos)+ using d(1) unfolding n_def by (auto simp: DIM_positive) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1414 |
have *:"uniformly_continuous_on {0..\<Sum>Basis} f" by(rule compact_uniformly_continuous[OF assms(1) compact_interval]) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1415 |
guess e using *[unfolded uniformly_continuous_on_def,rule_format,OF d'] apply-apply(erule exE,(erule conjE)+) . |
36587 | 1416 |
note e=this[rule_format,unfolded dist_norm] |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1417 |
show ?thesis apply(rule_tac x="min (e/2) (d/real n/8)" in exI) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1418 |
proof safe |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1419 |
show "0 < min (e / 2) (d / real n / 8)" using d' e by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1420 |
fix x y z i assume as:"x \<in> {0..\<Sum>Basis}" "y \<in> {0..\<Sum>Basis}" "z \<in> {0..\<Sum>Basis}" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1421 |
"norm (x - z) < min (e / 2) (d / real n / 8)" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1422 |
"norm (y - z) < min (e / 2) (d / real n / 8)" "label x i \<noteq> label y i" and i:"i\<in>Basis" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1423 |
have *:"\<And>z fz x fx n1 n2 n3 n4 d4 d::real. abs(fx - x) \<le> n1 + n2 \<Longrightarrow> abs(fx - fz) \<le> n3 \<Longrightarrow> abs(x - z) \<le> n4 \<Longrightarrow> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1424 |
n1 < d4 \<Longrightarrow> n2 < 2 * d4 \<Longrightarrow> n3 < d4 \<Longrightarrow> n4 < d4 \<Longrightarrow> (8 * d4 = d) \<Longrightarrow> abs(fz - z) < d" by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1425 |
show "\<bar>(f z - z) \<bullet> i\<bar> < d / real n" unfolding inner_simps proof(rule *) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1426 |
show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y -f x) + norm (y - x)" apply(rule lem1[rule_format]) using as i by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1427 |
show "\<bar>f x \<bullet> i - f z \<bullet> i\<bar> \<le> norm (f x - f z)" "\<bar>x \<bullet> i - z \<bullet> i\<bar> \<le> norm (x - z)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1428 |
unfolding inner_diff_left[THEN sym] by(rule Basis_le_norm[OF i])+ |
36587 | 1429 |
have tria:"norm (y - x) \<le> norm (y - z) + norm (x - z)" using dist_triangle[of y x z,unfolded dist_norm] |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1430 |
unfolding norm_minus_commute by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1431 |
also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_mono) using as(4,5) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1432 |
finally show "norm (f y - f x) < d / real n / 8" apply- apply(rule e(2)) using as by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1433 |
have "norm (y - z) + norm (x - z) < d / real n / 8 + d / real n / 8" apply(rule add_strict_mono) using as by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1434 |
thus "norm (y - x) < 2 * (d / real n / 8)" using tria by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1435 |
show "norm (f x - f z) < d / real n / 8" apply(rule e(2)) using as e(1) by auto qed(insert as, auto) qed qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1436 |
then guess e apply-apply(erule exE,(erule conjE)+) . note e=this[rule_format] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1437 |
guess p using real_arch_simple[of "1 + real n / e"] .. note p=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1438 |
have "1 + real n / e > 0" apply(rule add_pos_pos) defer apply(rule divide_pos_pos) using e(1) n by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1439 |
hence "p > 0" using p by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1440 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1441 |
obtain b :: "nat \<Rightarrow> 'a" where b: "bij_betw b {1..n} Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1442 |
by atomize_elim (auto simp: n_def intro!: finite_same_card_bij) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1443 |
def b' \<equiv> "inv_into {1..n} b" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1444 |
then have b': "bij_betw b' Basis {1..n}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1445 |
using bij_betw_inv_into[OF b] by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1446 |
then have b'_Basis: "\<And>i. i \<in> Basis \<Longrightarrow> b' i \<in> {Suc 0 .. n}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1447 |
unfolding bij_betw_def by (auto simp: set_eq_iff) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1448 |
have bb'[simp]:"\<And>i. i \<in> Basis \<Longrightarrow> b (b' i) = i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1449 |
unfolding b'_def using b by (auto simp: f_inv_into_f bij_betw_def) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1450 |
have b'b[simp]:"\<And>i. i \<in> {1..n} \<Longrightarrow> b' (b i) = i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1451 |
unfolding b'_def using b by (auto simp: inv_into_f_eq bij_betw_def) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1452 |
have *:"\<And>x::nat. x=0 \<or> x=1 \<longleftrightarrow> x\<le>1" by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1453 |
have b'':"\<And>j. j\<in>{Suc 0..n} \<Longrightarrow> b j \<in>Basis" using b unfolding bij_betw_def by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1454 |
have q1:"0 < p" "0 < n" "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1455 |
(\<forall>i\<in>{1..n}. (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0 \<or> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1456 |
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1457 |
unfolding * using `p>0` `n>0` using label(1)[OF b''] by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1458 |
have q2:"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = 0 \<longrightarrow> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1459 |
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1460 |
"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = p \<longrightarrow> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1461 |
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1462 |
apply(rule,rule,rule,rule) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1463 |
defer |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1464 |
proof(rule,rule,rule,rule) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1465 |
fix x i assume as:"\<forall>i\<in>{1..n}. x i \<le> p" "i \<in> {1..n}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1466 |
{ assume "x i = p \<or> x i = 0" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1467 |
have "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> {0::'a..\<Sum>Basis}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1468 |
unfolding mem_interval using as b'_Basis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1469 |
by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) } |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1470 |
note cube=this |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1471 |
{ assume "x i = p" thus "(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1472 |
unfolding o_def using cube as `p>0` |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1473 |
by (intro label(3)) (auto simp add: b'') } |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1474 |
{ assume "x i = 0" thus "(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1475 |
unfolding o_def using cube as `p>0` |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1476 |
by (intro label(2)) (auto simp add: b'') } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1477 |
qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1478 |
guess q apply(rule kuhn_lemma[OF q1 q2]) . note q=this |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1479 |
def z \<equiv> "(\<Sum>i\<in>Basis. (real (q (b' i)) / real p) *\<^sub>R i)::'a" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1480 |
have "\<exists>i\<in>Basis. d / real n \<le> abs((f z - z)\<bullet>i)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1481 |
proof(rule ccontr) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1482 |
have "\<forall>i\<in>Basis. q (b' i) \<in> {0..p}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1483 |
using q(1) b' by (auto intro: less_imp_le simp: bij_betw_def) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1484 |
hence "z\<in>{0..\<Sum>Basis}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1485 |
unfolding z_def mem_interval using b'_Basis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1486 |
by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1487 |
hence d_fz_z:"d \<le> norm (f z - z)" by (rule d) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1488 |
case goal1 |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1489 |
hence as:"\<forall>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar> < d / real n" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1490 |
using `n>0` by(auto simp add: not_le inner_simps) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1491 |
have "norm (f z - z) \<le> (\<Sum>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar>)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1492 |
unfolding inner_diff_left[symmetric] by(rule norm_le_l1) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1493 |
also have "\<dots> < (\<Sum>(i::'a)\<in>Basis. d / real n)" apply(rule setsum_strict_mono) using as by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1494 |
also have "\<dots> = d" using DIM_positive[where 'a='a] by (auto simp: real_eq_of_nat n_def) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1495 |
finally show False using d_fz_z by auto qed then guess i .. note i=this |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1496 |
have *:"b' i \<in> {1..n}" using i using b'[unfolded bij_betw_def] by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1497 |
guess r using q(2)[rule_format,OF *] .. then guess s apply-apply(erule exE,(erule conjE)+) . note rs=this[rule_format] |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1498 |
have b'_im:"\<And>i. i\<in>Basis \<Longrightarrow> b' i \<in> {1..n}" using b' unfolding bij_betw_def by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1499 |
def r' \<equiv> "(\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i)::'a" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1500 |
have "\<And>i. i\<in>Basis \<Longrightarrow> r (b' i) \<le> p" apply(rule order_trans) apply(rule rs(1)[OF b'_im,THEN conjunct2]) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1501 |
using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1502 |
hence "r' \<in> {0..\<Sum>Basis}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1503 |
unfolding r'_def mem_interval using b'_Basis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1504 |
by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1505 |
def s' \<equiv> "(\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i)::'a" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1506 |
have "\<And>i. i\<in>Basis \<Longrightarrow> s (b' i) \<le> p" apply(rule order_trans) apply(rule rs(2)[OF b'_im,THEN conjunct2]) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1507 |
using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1508 |
hence "s' \<in> {0..\<Sum>Basis}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1509 |
unfolding s'_def mem_interval using b'_Basis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1510 |
by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1511 |
have "z\<in>{0..\<Sum>Basis}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1512 |
unfolding z_def mem_interval using b'_Basis q(1)[rule_format,OF b'_im] `p>0` |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1513 |
by (auto simp add: inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1 less_imp_le) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1514 |
have *:"\<And>x. 1 + real x = real (Suc x)" by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1515 |
{ have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1516 |
apply(rule setsum_mono) using rs(1)[OF b'_im] by(auto simp add:* field_simps) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1517 |
also have "\<dots> < e * real p" using p `e>0` `p>0` |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1518 |
by(auto simp add: field_simps n_def real_of_nat_def) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1519 |
finally have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" . } moreover |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1520 |
{ have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1521 |
apply(rule setsum_mono) using rs(2)[OF b'_im] by(auto simp add:* field_simps) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1522 |
also have "\<dots> < e * real p" using p `e>0` `p>0` |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1523 |
by(auto simp add: field_simps n_def real_of_nat_def) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1524 |
finally have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" . } ultimately |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1525 |
have "norm (r' - z) < e" "norm (s' - z) < e" unfolding r'_def s'_def z_def using `p>0` |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1526 |
by (rule_tac[!] le_less_trans[OF norm_le_l1]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1527 |
(auto simp add: field_simps setsum_divide_distrib[symmetric] inner_diff_left) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1528 |
hence "\<bar>(f z - z) \<bullet> i\<bar> < d / real n" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1529 |
using rs(3) i unfolding r'_def[symmetric] s'_def[symmetric] o_def bb' |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1530 |
by (intro e(2)[OF `r'\<in>{0..\<Sum>Basis}` `s'\<in>{0..\<Sum>Basis}` `z\<in>{0..\<Sum>Basis}`]) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1531 |
thus False using i by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1532 |
qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1533 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1534 |
subsection {* Retractions. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1535 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1536 |
definition "retraction s t r \<longleftrightarrow> |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1537 |
t \<subseteq> s \<and> continuous_on s r \<and> (r ` s \<subseteq> t) \<and> (\<forall>x\<in>t. r x = x)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1538 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1539 |
definition retract_of (infixl "retract'_of" 12) where |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1540 |
"(t retract_of s) \<longleftrightarrow> (\<exists>r. retraction s t r)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1541 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1542 |
lemma retraction_idempotent: "retraction s t r \<Longrightarrow> x \<in> s \<Longrightarrow> r(r x) = r x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1543 |
unfolding retraction_def by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1544 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1545 |
subsection {*preservation of fixpoints under (more general notion of) retraction. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1546 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1547 |
lemma invertible_fixpoint_property: fixes s::"('a::euclidean_space) set" and t::"('b::euclidean_space) set" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1548 |
assumes "continuous_on t i" "i ` t \<subseteq> s" "continuous_on s r" "r ` s \<subseteq> t" "\<forall>y\<in>t. r (i y) = y" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1549 |
"\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)" "continuous_on t g" "g ` t \<subseteq> t" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1550 |
obtains y where "y\<in>t" "g y = y" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1551 |
have "\<exists>x\<in>s. (i \<circ> g \<circ> r) x = x" apply(rule assms(6)[rule_format],rule) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1552 |
apply(rule continuous_on_compose assms)+ apply((rule continuous_on_subset)?,rule assms)+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1553 |
using assms(2,4,8) unfolding image_compose by(auto,blast) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1554 |
then guess x .. note x = this hence *:"g (r x) \<in> t" using assms(4,8) by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1555 |
have "r ((i \<circ> g \<circ> r) x) = r x" using x by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1556 |
thus ?thesis apply(rule_tac that[of "r x"]) using x unfolding o_def |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1557 |
unfolding assms(5)[rule_format,OF *] using assms(4) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1558 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1559 |
lemma homeomorphic_fixpoint_property: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1560 |
fixes s::"('a::euclidean_space) set" and t::"('b::euclidean_space) set" assumes "s homeomorphic t" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1561 |
shows "(\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)) \<longleftrightarrow> |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1562 |
(\<forall>g. continuous_on t g \<and> g ` t \<subseteq> t \<longrightarrow> (\<exists>y\<in>t. g y = y))" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1563 |
guess r using assms[unfolded homeomorphic_def homeomorphism_def] .. then guess i .. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1564 |
thus ?thesis apply- apply rule apply(rule_tac[!] allI impI)+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1565 |
apply(rule_tac g=g in invertible_fixpoint_property[of t i s r]) prefer 10 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1566 |
apply(rule_tac g=f in invertible_fixpoint_property[of s r t i]) by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1567 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1568 |
lemma retract_fixpoint_property: fixes f::"'a::euclidean_space => 'b::euclidean_space" and s::"'a set" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1569 |
assumes "t retract_of s" "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)" "continuous_on t g" "g ` t \<subseteq> t" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1570 |
obtains y where "y \<in> t" "g y = y" proof- guess h using assms(1) unfolding retract_of_def .. |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1571 |
thus ?thesis unfolding retraction_def apply- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1572 |
apply(rule invertible_fixpoint_property[OF continuous_on_id _ _ _ _ assms(2), of t h g]) prefer 7 |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1573 |
apply(rule_tac y=y in that) using assms by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1574 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1575 |
subsection {*So the Brouwer theorem for any set with nonempty interior. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1576 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1577 |
lemma brouwer_weak: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1578 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1579 |
assumes "compact s" "convex s" "interior s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1580 |
obtains x where "x \<in> s" "f x = x" proof- |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1581 |
have *:"interior {0::'a..\<Sum>Basis} \<noteq> {}" unfolding interior_closed_interval interval_eq_empty by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1582 |
have *:"{0::'a..\<Sum>Basis} homeomorphic s" using homeomorphic_convex_compact[OF convex_interval(1) compact_interval * assms(2,1,3)] . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1583 |
have "\<forall>f. continuous_on {0::'a..\<Sum>Basis} f \<and> f ` {0::'a..\<Sum>Basis} \<subseteq> {0::'a..\<Sum>Basis} \<longrightarrow> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1584 |
(\<exists>x\<in>{0::'a..\<Sum>Basis}. f x = x)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1585 |
using brouwer_cube by auto |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1586 |
thus ?thesis unfolding homeomorphic_fixpoint_property[OF *] apply(erule_tac x=f in allE) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1587 |
apply(erule impE) defer apply(erule bexE) apply(rule_tac x=y in that) using assms by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1588 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1589 |
subsection {* And in particular for a closed ball. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1590 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1591 |
lemma brouwer_ball: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1592 |
assumes "0 < e" "continuous_on (cball a e) f" "f ` (cball a e) \<subseteq> (cball a e)" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1593 |
obtains x where "x \<in> cball a e" "f x = x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1594 |
using brouwer_weak[OF compact_cball convex_cball,of a e f] unfolding interior_cball ball_eq_empty |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1595 |
using assms by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1596 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1597 |
text {*Still more general form; could derive this directly without using the |
36334 | 1598 |
rather involved @{text "HOMEOMORPHIC_CONVEX_COMPACT"} theorem, just using |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1599 |
a scaling and translation to put the set inside the unit cube. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1600 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1601 |
lemma brouwer: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1602 |
assumes "compact s" "convex s" "s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1603 |
obtains x where "x \<in> s" "f x = x" proof- |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1604 |
have "\<exists>e>0. s \<subseteq> cball 0 e" using compact_imp_bounded[OF assms(1)] unfolding bounded_pos |
36587 | 1605 |
apply(erule_tac exE,rule_tac x=b in exI) by(auto simp add: dist_norm) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1606 |
then guess e apply-apply(erule exE,(erule conjE)+) . note e=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1607 |
have "\<exists>x\<in> cball 0 e. (f \<circ> closest_point s) x = x" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1608 |
apply(rule_tac brouwer_ball[OF e(1), of 0 "f \<circ> closest_point s"]) apply(rule continuous_on_compose ) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1609 |
apply(rule continuous_on_closest_point[OF assms(2) compact_imp_closed[OF assms(1)] assms(3)]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1610 |
apply(rule continuous_on_subset[OF assms(4)]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1611 |
using closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)] apply - defer |
36587 | 1612 |
using assms(5)[unfolded subset_eq] using e(2)[unfolded subset_eq mem_cball] by(auto simp add: dist_norm) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1613 |
then guess x .. note x=this |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1614 |
have *:"closest_point s x = x" apply(rule closest_point_self) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1615 |
apply(rule assms(5)[unfolded subset_eq,THEN bspec[where x="x"],unfolded image_iff]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1616 |
apply(rule_tac x="closest_point s x" in bexI) using x unfolding o_def |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1617 |
using closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3), of x] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1618 |
show thesis apply(rule_tac x="closest_point s x" in that) unfolding x(2)[unfolded o_def] |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1619 |
apply(rule closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)]) using * by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1620 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1621 |
text {*So we get the no-retraction theorem. *} |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1622 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1623 |
lemma no_retraction_cball: assumes "0 < e" fixes type::"'a::ordered_euclidean_space" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1624 |
shows "\<not> (frontier(cball a e) retract_of (cball (a::'a) e))" proof case goal1 |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1625 |
have *:"\<And>xa. a - (2 *\<^sub>R a - xa) = -(a - xa)" using scaleR_left_distrib[of 1 1 a] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1626 |
guess x apply(rule retract_fixpoint_property[OF goal1, of "\<lambda>x. scaleR 2 a - x"]) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1627 |
apply(rule,rule,erule conjE) apply(rule brouwer_ball[OF assms]) apply assumption+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1628 |
apply(rule_tac x=x in bexI) apply assumption+ apply(rule continuous_on_intros)+ |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1629 |
unfolding frontier_cball subset_eq Ball_def image_iff apply(rule,rule,erule bexE) |
36587 | 1630 |
unfolding dist_norm apply(simp add: * norm_minus_commute) . note x = this |
36350 | 1631 |
hence "scaleR 2 a = scaleR 1 x + scaleR 1 x" by(auto simp add:algebra_simps) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1632 |
hence "a = x" unfolding scaleR_left_distrib[THEN sym] by auto |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1633 |
thus False using x using assms by auto qed |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1634 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1635 |
subsection {*Bijections between intervals. *} |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1636 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1637 |
definition interval_bij :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a::ordered_euclidean_space" where |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1638 |
"interval_bij \<equiv> \<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)" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1639 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1640 |
lemma interval_bij_affine: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1641 |
"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) + |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1642 |
(\<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))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1643 |
by (auto simp: setsum_addf[symmetric] scaleR_add_left[symmetric] interval_bij_def fun_eq_iff |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1644 |
field_simps inner_simps add_divide_distrib[symmetric] intro!: setsum_cong) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1645 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1646 |
lemma continuous_interval_bij: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1647 |
"continuous (at x) (interval_bij (a,b::'a::ordered_euclidean_space) (u,v::'a))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1648 |
by (auto simp add: divide_inverse interval_bij_def intro!: continuous_setsum continuous_intros) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1649 |
|
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1650 |
lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a,b) (u,v))" |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1651 |
apply(rule continuous_at_imp_continuous_on) by(rule, rule continuous_interval_bij) |
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1652 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1653 |
lemma in_interval_interval_bij: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1654 |
fixes a b u v x :: "'a::ordered_euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1655 |
assumes "x \<in> {a..b}" "{u..v} \<noteq> {}" shows "interval_bij (a,b) (u,v) x \<in> {u..v}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1656 |
apply (simp only: interval_bij_def split_conv mem_interval inner_setsum_left_Basis cong: ball_cong) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1657 |
proof safe |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1658 |
fix i :: 'a assume i:"i\<in>Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1659 |
have "{a..b} \<noteq> {}" using assms by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1660 |
with i have *: "a\<bullet>i \<le> b\<bullet>i" "u\<bullet>i \<le> v\<bullet>i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1661 |
using assms(2) by (auto simp add: interval_eq_empty not_less) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1662 |
have x: "a\<bullet>i\<le>x\<bullet>i" "x\<bullet>i\<le>b\<bullet>i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1663 |
using assms(1)[unfolded mem_interval] using i by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1664 |
have "0 \<le> (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1665 |
using * x by (auto intro!: mult_nonneg_nonneg divide_nonneg_nonneg) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1666 |
thus "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)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1667 |
using * by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1668 |
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)" |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1669 |
apply(rule mult_right_mono) unfolding divide_le_eq_1 using * x by auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1670 |
thus "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" using * by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36587
diff
changeset
|
1671 |
qed |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1672 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1673 |
lemma interval_bij_bij: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1674 |
"\<forall>(i::'a::ordered_euclidean_space)\<in>Basis. a\<bullet>i < b\<bullet>i \<and> u\<bullet>i < v\<bullet>i \<Longrightarrow> |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1675 |
interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50514
diff
changeset
|
1676 |
by (auto simp: interval_bij_def euclidean_eq_iff[where 'a='a]) |
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff
changeset
|
1677 |
|
34291
4e896680897e
finite annotation on cartesian product is now implicit.
hoelzl
parents:
34289
diff
changeset
|
1678 |
end |