author  haftmann 
Thu, 11 Mar 2010 15:52:34 +0100  
changeset 35729  3cd1e4b65111 
parent 34964  4e8be3c04d37 
child 36318  3567d0571932 
permissions  rwrr 
33741
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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1 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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2 
(* ========================================================================= *) 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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3 
(* Results connected with topological dimension. *) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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4 
(* *) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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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 HOLlight)
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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 HOLlight)
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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 HOLlight)
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8 
(* *) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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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 HOLlight)
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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 HOLlight)
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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 HOLlight)
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12 
(* *) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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13 
(* (c) Copyright, John Harrison 19982008 *) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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14 
(* ========================================================================= *) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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15 

33759
b369324fc244
Added the contributions of Robert Himmelmann to CONTRIBUTIONS and NEWS
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16 
(* Author: John Harrison 
b369324fc244
Added the contributions of Robert Himmelmann to CONTRIBUTIONS and NEWS
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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 HOLlight)
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18 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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19 
header {* Results connected with topological dimension. *} 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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20 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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21 
theory Brouwer_Fixpoint 
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22 
imports Convex_Euclidean_Space 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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23 
begin 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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24 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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25 
declare norm_scaleR[simp] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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26 

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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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27 
lemma brouwer_compactness_lemma: 
34291
4e896680897e
finite annotation on cartesian product is now implicit.
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28 
assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = (0::real^'n)))" 
33741
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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29 
obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)" proof(cases "s={}") case False 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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30 
have "continuous_on s (norm \<circ> f)" by(rule continuous_on_intros continuous_on_norm assms(2))+ 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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31 
then obtain x where x:"x\<in>s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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32 
using continuous_attains_inf[OF assms(1), of "norm \<circ> f"] and False unfolding o_def by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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33 
have "(norm \<circ> f) x > 0" using assms(3) and x(1) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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34 
thus ?thesis apply(rule that) using x(2) unfolding o_def by auto qed(rule that[of 1], auto) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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35 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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36 
lemma kuhn_labelling_lemma: 
34289  37 
assumes "(\<forall>x::real^_. P x \<longrightarrow> P (f x))" "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)" 
33741
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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38 
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 HOLlight)
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39 
(\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and> 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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40 
(\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and> 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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41 
(\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and> 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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42 
(\<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 HOLlight)
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43 
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 HOLlight)
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44 
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 HOLlight)
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45 
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 HOLlight)
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46 
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 HOLlight)
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47 
(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 HOLlight)
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48 
{ 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 HOLlight)
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49 
apply(drule_tac assms(1)[rule_format]) by auto } 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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50 
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 HOLlight)
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51 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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52 
subsection {* The key "counting" observation, somewhat abstracted. *} 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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53 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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54 
lemma setsum_Un_disjoint':assumes "finite A" "finite B" "A \<inter> B = {}" "A \<union> B = C" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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55 
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 HOLlight)
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56 
using setsum_Un_disjoint[OF assms(13)] and assms(4) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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57 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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58 
lemma kuhn_counting_lemma: assumes "finite faces" "finite simplices" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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59 
"\<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 HOLlight)
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60 
"\<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 HOLlight)
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61 
"\<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 HOLlight)
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62 
"\<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 HOLlight)
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63 
(card {f \<in> faces. face f s \<and> compo' f} = 2)" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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64 
"odd(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 HOLlight)
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65 
shows "odd(card {s \<in> simplices. compo s})" proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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66 
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} = {f\<in>faces. compo' f \<and> face f x}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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67 
"\<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} = {}" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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68 
hence lem1:"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 HOLlight)
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69 
setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f s}) simplices + 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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70 
setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> \<not> (bnd f)}. face f s}) simplices" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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71 
unfolding setsum_addf[THEN sym] apply apply(rule setsum_cong2) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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72 
using assms(1) by(auto simp add: card_Un_Int, auto simp add:conj_commute) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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73 
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 HOLlight)
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74 
1 * card {f \<in> faces. compo' f \<and> bnd f}" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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75 
"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 HOLlight)
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76 
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 HOLlight)
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77 
apply(rule_tac[!] setsum_multicount) using assms by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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78 
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 HOLlight)
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79 
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 HOLlight)
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80 
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 HOLlight)
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81 
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 HOLlight)
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82 
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 HOLlight)
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83 
= 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 HOLlight)
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84 
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 HOLlight)
hoelzl
parents:
diff
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85 
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 HOLlight)
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parents:
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86 
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 HOLlight)
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87 
{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 HOLlight)
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88 
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 HOLlight)
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89 
{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 HOLlight)
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90 
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 HOLlight)
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91 
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 HOLlight)
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92 
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 HOLlight)
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93 
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 HOLlight)
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94 
using lem1[unfolded lem3 lem2 lem5] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
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changeset

95 
have even_minus_odd:"\<And>x y. even x \<Longrightarrow> odd (y::int) \<Longrightarrow> odd (x  y)" using assms by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

96 
have odd_minus_even:"\<And>x y. odd x \<Longrightarrow> even (y::int) \<Longrightarrow> odd (x  y)" using assms by auto 
35729  97 
show ?thesis unfolding even_nat_def unfolding card_eq_setsum and lem4[THEN sym] and *[unfolded card_eq_setsum] 
98 
unfolding card_eq_setsum[THEN sym] apply (rule odd_minus_even) unfolding zadd_int[THEN sym] apply(rule odd_plus_even) 

33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

99 
apply(rule assms(7)[unfolded even_nat_def]) unfolding int_mult by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

100 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

101 
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 HOLlight)
hoelzl
parents:
diff
changeset

102 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

103 
lemma card_1_exists: "card s = 1 \<longleftrightarrow> (\<exists>!x. x \<in> s)" unfolding One_nat_def 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

104 
apply rule apply(drule card_eq_SucD) defer apply(erule ex1E) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

105 
fix x assume as:"x \<in> s" "\<forall>y. y \<in> s \<longrightarrow> y = x" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

106 
have *:"s = insert x {}" apply apply(rule set_ext,rule) unfolding singleton_iff 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

107 
apply(rule as(2)[rule_format]) using as(1) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

108 
show "card s = Suc 0" unfolding * using card_insert by auto qed auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

109 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

110 
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)))" proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

111 
assume "card s = 2" then obtain x y where obt:"s = {x, y}" "x\<noteq>y" unfolding numeral_2_eq_2 apply  apply(erule exE conjEdrule card_eq_SucD)+ by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

112 
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 next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

113 
assume "\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. z = x \<or> z = y)" then guess x .. from this(2) guess y .. 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

114 
with `x\<in>s` have *:"s = {x, y}" "x\<noteq>y" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

115 
from this(2) show "card s = 2" unfolding * by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

116 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

117 
lemma image_lemma_0: assumes "card {a\<in>s. f ` (s  {a}) = t  {b}} = n" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

118 
shows "card {s'. \<exists>a\<in>s. (s' = s  {a}) \<and> (f ` s' = t  {b})} = n" proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

119 
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}}" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

120 
show ?thesis unfolding * unfolding assms[THEN sym] apply(rule card_image) unfolding inj_on_def 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

121 
apply(rule,rule,rule) unfolding mem_Collect_eq by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

122 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

123 
lemma image_lemma_1: assumes "finite s" "finite t" "card s = card t" "f ` s = t" "b \<in> t" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

124 
shows "card {s'. \<exists>a\<in>s. s' = s  {a} \<and> f ` s' = t  {b}} = 1" proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

125 
obtain a where a:"b = f a" "a\<in>s" using assms(45) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

126 
have inj:"inj_on f s" apply(rule eq_card_imp_inj_on) using assms(14) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

127 
have *:"{a \<in> s. f ` (s  {a}) = t  {b}} = {a}" apply(rule set_ext) unfolding singleton_iff 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

128 
apply(rule,rule inj[unfolded inj_on_def,rule_format]) unfolding a using a(2) and assms and inj[unfolded inj_on_def] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

129 
show ?thesis apply(rule image_lemma_0) unfolding * by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

130 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

131 
lemma image_lemma_2: assumes "finite s" "finite t" "card s = card t" "f ` s \<subseteq> t" "f ` s \<noteq> t" "b \<in> t" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
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diff
changeset

132 
shows "(card {s'. \<exists>a\<in>s. (s' = s  {a}) \<and> f ` s' = t  {b}} = 0) \<or> 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
changeset

133 
(card {s'. \<exists>a\<in>s. (s' = s  {a}) \<and> f ` s' = t  {b}} = 2)" proof(cases "{a\<in>s. f ` (s  {a}) = t  {b}} = {}") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

134 
case True thus ?thesis applyapply(rule disjI1, rule image_lemma_0) using assms(1) by(auto simp add:card_0_eq) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

135 
next let ?M = "{a\<in>s. f ` (s  {a}) = t  {b}}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

136 
case False then obtain a where "a\<in>?M" by auto hence a:"a\<in>s" "f ` (s  {a}) = t  {b}" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

137 
have "f a \<in> t  {b}" using a and assms by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

138 
hence "\<exists>c \<in> s  {a}. f a = f c" unfolding image_iff[symmetric] and a by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

139 
then obtain c where c:"c \<in> s" "a \<noteq> c" "f a = f c" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

140 
hence *:"f ` (s  {c}) = f ` (s  {a})" applyapply(rule set_ext,rule) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

141 
fix x assume "x \<in> f ` (s  {a})" then obtain y where y:"f y = x" "y\<in>s {a}" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

142 
thus "x \<in> f ` (s  {c})" unfolding image_iff apply(rule_tac x="if y = c then a else y" in bexI) using c a by auto qed auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

143 
have "c\<in>?M" unfolding mem_Collect_eq and * using a and c(1) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

144 
show ?thesis apply(rule disjI2, rule image_lemma_0) unfolding card_2_exists 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

145 
apply(rule bexI[OF _ `a\<in>?M`], rule bexI[OF _ `c\<in>?M`],rule,rule `a\<noteq>c`) proof(rule,unfold mem_Collect_eq,erule conjE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

146 
fix z assume as:"z \<in> s" "f ` (s  {z}) = t  {b}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

147 
have inj:"inj_on f (s  {z})" apply(rule eq_card_imp_inj_on) unfolding as using as(1) and assms by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

148 
show "z = a \<or> z = c" proof(rule ccontr) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

149 
assume "\<not> (z = a \<or> z = c)" thus False using inj[unfolded inj_on_def,THEN bspec[where x=a],THEN bspec[where x=c]] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

150 
using `a\<in>s` `c\<in>s` `f a = f c` `a\<noteq>c` by auto qed qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

151 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

152 
subsection {* Combine this with the basic counting lemma. *} 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

153 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

154 
lemma kuhn_complete_lemma: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

155 
assumes "finite simplices" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

156 
"\<forall>f s. face f s \<longleftrightarrow> (\<exists>a\<in>s. f = s  {a})" "\<forall>s\<in>simplices. card s = n + 2" "\<forall>s\<in>simplices. (rl ` s) \<subseteq> {0..n+1}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

157 
"\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. 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 HOLlight)
hoelzl
parents:
diff
changeset

158 
"\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. \<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 HOLlight)
hoelzl
parents:
diff
changeset

159 
"odd(card {f\<in>{f. \<exists>s\<in>simplices. face f s}. rl ` f = {0..n} \<and> bnd f})" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

160 
shows "odd (card {s\<in>simplices. (rl ` s = {0..n+1})})" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

161 
apply(rule kuhn_counting_lemma) defer apply(rule assms)+ prefer 3 apply(rule assms) proof(rule_tac[12] ballI impI)+ 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

162 
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})})" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

163 
have **: "\<forall>s\<in>simplices. card s = n + 2 \<and> finite s" using assms(3) by (auto intro: card_ge_0_finite) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

164 
show "finite {f. \<exists>s\<in>simplices. face f s}" unfolding assms(2)[rule_format] and * 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

165 
apply(rule finite_UN_I[OF assms(1)]) using ** by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

166 
have *:"\<And> P f s. s\<in>simplices \<Longrightarrow> (f \<in> {f. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s  {a}}) \<and> 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

167 
(\<exists>a\<in>s. (f = s  {a})) \<and> P f \<longleftrightarrow> (\<exists>a\<in>s. (f = s  {a}) \<and> P f)" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

168 
fix s assume s:"s\<in>simplices" let ?S = "{f \<in> {f. \<exists>s\<in>simplices. face f s}. face f s \<and> rl ` f = {0..n}}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

169 
have "{0..n + 1}  {n + 1} = {0..n}" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

170 
hence S:"?S = {s'. \<exists>a\<in>s. s' = s  {a} \<and> rl ` s' = {0..n + 1}  {n + 1}}" apply apply(rule set_ext) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

171 
unfolding assms(2)[rule_format] mem_Collect_eq and *[OF s, unfolded mem_Collect_eq, where P="\<lambda>x. rl ` x = {0..n}"] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

172 
show "rl ` s = {0..n+1} \<Longrightarrow> card ?S = 1" "rl ` s \<noteq> {0..n+1} \<Longrightarrow> card ?S = 0 \<or> card ?S = 2" unfolding S 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

173 
apply(rule_tac[!] image_lemma_1 image_lemma_2) using ** assms(4) and s by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

174 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

175 
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 HOLlight)
hoelzl
parents:
diff
changeset

176 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

177 
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 HOLlight)
hoelzl
parents:
diff
changeset

178 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

179 
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 HOLlight)
hoelzl
parents:
diff
changeset

180 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

181 
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 HOLlight)
hoelzl
parents:
diff
changeset

182 
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 HOLlight)
hoelzl
parents:
diff
changeset

183 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

184 
lemma pointwise_minimal_pointwise_maximal: fixes s::"(nat\<Rightarrow>nat) set" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

185 
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)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

186 
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" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

187 
using assms unfolding atomize_conj apply proof(induct s rule:finite_induct) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

188 
fix x and F::"(nat\<Rightarrow>nat) set" assume as:"finite F" "x \<notin> F" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

189 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

190 
\<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 HOLlight)
hoelzl
parents:
diff
changeset

191 
"\<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)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

192 
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)" proof(cases "F={}") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

193 
case True thus ?thesis applyapply(rule,rule_tac[!] x=x in bexI) by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

194 
case False obtain a b where a:"a\<in>insert x F" "\<forall>x\<in>F. \<forall>j. a j \<le> x j" and 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

195 
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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

196 
have "\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. a j \<le> x j" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

197 
using as(5)[rule_format,OF a(1) insertI1] apply proof(erule disjE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

198 
assume "\<forall>j. a j \<le> x j" thus ?thesis apply(rule_tac x=a in bexI) using a by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

199 
assume "\<forall>j. x j \<le> a j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using a apply  
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

200 
apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

201 
have "\<exists>b\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. x j \<le> b j" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

202 
using as(5)[rule_format,OF b(1) insertI1] apply proof(erule disjE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

203 
assume "\<forall>j. x j \<le> b j" thus ?thesis apply(rule_tac x=b in bexI) using b by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

204 
assume "\<forall>j. b j \<le> x j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using b apply  
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

205 
apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

206 
ultimately show ?thesis by auto qed qed(auto) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

207 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

208 
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 HOLlight)
hoelzl
parents:
diff
changeset

209 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

210 
lemma pointwise_antisym: fixes x::"nat \<Rightarrow> nat" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

211 
shows "(\<forall>j. x j \<le> y j) \<and> (\<forall>j. y j \<le> x j) \<longleftrightarrow> (x = y)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

212 
apply(rule, rule ext,erule conjE) apply(erule_tac x=xa in allE)+ by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

213 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

214 
lemma kle_trans: assumes "kle n x y" "kle n y z" "kle n x z \<or> kle n z x" shows "kle n x z" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

215 
using assms apply apply(erule disjE) apply assumption proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

216 
hence "x=z" apply apply(rule ext) apply(drule kle_imp_pointwise)+ 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

217 
apply(erule_tac x=xa in allE)+ by auto thus ?case by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

218 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

219 
lemma kle_strict: assumes "kle n x y" "x \<noteq> y" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

220 
shows "\<forall>j. x j \<le> y j" "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> x(k) < y(k)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

221 
apply(rule kle_imp_pointwise[OF assms(1)]) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

222 
guess k using assms(1)[unfolded kle_def] .. note k = this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

223 
show "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> x(k) < y(k)" proof(cases "k={}") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

224 
case True hence "x=y" applyapply(rule ext) using k by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

225 
thus ?thesis using assms(2) by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

226 
case False hence "(SOME k'. k' \<in> k) \<in> k" applyapply(rule someI_ex) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

227 
thus ?thesis apply(rule_tac x="SOME k'. k' \<in> k" in exI) using k by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

228 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

229 
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 HOLlight)
hoelzl
parents:
diff
changeset

230 
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 HOLlight)
hoelzl
parents:
diff
changeset

231 
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(12)]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

232 
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 HOLlight)
hoelzl
parents:
diff
changeset

233 
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 HOLlight)
hoelzl
parents:
diff
changeset

234 
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 HOLlight)
hoelzl
parents:
diff
changeset

235 
assume "kle n x a" hence "x = a" apply unfolding pointwise_antisym[THEN sym] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

236 
apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

237 
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 HOLlight)
hoelzl
parents:
diff
changeset

238 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

239 
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 HOLlight)
hoelzl
parents:
diff
changeset

240 
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 HOLlight)
hoelzl
parents:
diff
changeset

241 
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(12)]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

242 
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 HOLlight)
hoelzl
parents:
diff
changeset

243 
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 HOLlight)
hoelzl
parents:
diff
changeset

244 
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 HOLlight)
hoelzl
parents:
diff
changeset

245 
assume "kle n a x" hence "x = a" apply unfolding pointwise_antisym[THEN sym] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

246 
apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

247 
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 HOLlight)
hoelzl
parents:
diff
changeset

248 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

249 
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 HOLlight)
hoelzl
parents:
diff
changeset

250 
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 HOLlight)
hoelzl
parents:
diff
changeset

251 
guess i using kle_strict(2)[OF assms] .. 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

252 
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 HOLlight)
hoelzl
parents:
diff
changeset

253 
thus ?thesis by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

254 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

255 
lemma kle_range_combine: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

256 
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 HOLlight)
hoelzl
parents:
diff
changeset

257 
"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 HOLlight)
hoelzl
parents:
diff
changeset

258 
"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 HOLlight)
hoelzl
parents:
diff
changeset

259 
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 HOLlight)
hoelzl
parents:
diff
changeset

260 
apply(rule,rule kle_trans[OF assms(13)]) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

261 
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 HOLlight)
hoelzl
parents:
diff
changeset

262 
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 HOLlight)
hoelzl
parents:
diff
changeset

263 
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 HOLlight)
hoelzl
parents:
diff
changeset

264 
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 HOLlight)
hoelzl
parents:
diff
changeset

265 
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 HOLlight)
hoelzl
parents:
diff
changeset

266 
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 HOLlight)
hoelzl
parents:
diff
changeset

267 
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 HOLlight)
hoelzl
parents:
diff
changeset

268 
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 HOLlight)
hoelzl
parents:
diff
changeset

269 
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 HOLlight)
hoelzl
parents:
diff
changeset

270 
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 HOLlight)
hoelzl
parents:
diff
changeset

271 
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 HOLlight)
hoelzl
parents:
diff
changeset

272 
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 HOLlight)
hoelzl
parents:
diff
changeset

273 
have "z i = x i + 2" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

274 
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 HOLlight)
hoelzl
parents:
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275 
have fin:"\<And>P. finite {x\<in>{1..n::nat}. P x}" by auto 
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276 
have "m1 + m2 \<le> card {k\<in>{1..n}. x k < y k} + card {k\<in>{1..n}. y k < z k}" using assms(45) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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277 
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 HOLlight)
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278 
finally show " m1 + m2 \<le> card {k \<in> {1..n}. x k < z k}" by auto qed 
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279 

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280 
lemma kle_range_combine_l: 
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281 
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 HOLlight)
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282 
shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}" 
4c414d0835ab
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283 
using kle_range_combine[OF assms(13) _ assms(4), of 0] by auto 
4c414d0835ab
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parents:
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284 

4c414d0835ab
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285 
lemma kle_range_combine_r: 
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286 
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 HOLlight)
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parents:
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287 
shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}" 
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288 
using kle_range_combine[OF assms(13) assms(4), of 0] by auto 
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parents:
diff
changeset

289 

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290 
lemma kle_range_induct: 
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291 
assumes "card s = Suc m" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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292 
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 HOLlight)
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293 
have "finite s" "s\<noteq>{}" using assms(1) by (auto intro: card_ge_0_finite) 
4c414d0835ab
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294 
thus ?thesis using assms apply proof(induct m arbitrary: s) 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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295 
case 0 thus ?case using kle_refl by auto next 
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296 
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 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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297 
show ?case proof(cases "s \<subseteq> {a}") case False 
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298 
hence "card (s  {a}) = Suc m" "s  {a} \<noteq> {}" using card_Diff_singleton[OF _ a(1)] Suc(4) `finite s` by auto 
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299 
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}" 
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300 
using Suc(1)[of "s  {a}"] using Suc(5) `finite s` by auto 
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301 
have "1 \<le> card {k \<in> {1..n}. a k < x k}" "m \<le> card {k \<in> {1..n}. x k < b k}" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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302 
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 HOLlight)
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303 
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 HOLlight)
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304 
using kle_range_combine[OF a(2)[rule_format] xb(3) Suc(5)[rule_format], of 1 "m"] using a(1) xb(12) by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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305 
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 HOLlight)
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306 
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 HOLlight)
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parents:
diff
changeset

307 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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308 
lemma kle_Suc: "kle n x y \<Longrightarrow> kle (n + 1) x y" 
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309 
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 HOLlight)
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parents:
diff
changeset

310 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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311 
lemma kle_trans_1: assumes "kle n x y" shows "x j \<le> y j" "y j \<le> x j + 1" 
4c414d0835ab
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312 
using assms[unfolded kle_def] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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changeset

313 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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314 
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 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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315 
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 HOLlight)
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diff
changeset

316 
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 HOLlight)
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317 
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 HOLlight)
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diff
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318 
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 HOLlight)
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diff
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319 
case True hence "c i \<ge> a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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320 
unfolding kk1[THEN conjunct2,rule_format,of i] kk2[THEN conjunct2,rule_format,of i] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
changeset

321 
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 HOLlight)
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diff
changeset

322 
ultimately show ?thesis by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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323 
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 HOLlight)
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parents:
diff
changeset

324 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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325 
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" 
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Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
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326 
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 HOLlight)
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parents:
diff
changeset

327 
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 HOLlight)
hoelzl
parents:
diff
changeset

328 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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329 
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 HOLlight)
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parents:
diff
changeset

330 
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 HOLlight)
hoelzl
parents:
diff
changeset

331 
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 HOLlight)
hoelzl
parents:
diff
changeset

332 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
changeset

333 
lemma kle_adjacent: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
changeset

334 
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 HOLlight)
hoelzl
parents:
diff
changeset

335 
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 HOLlight)
hoelzl
parents:
diff
changeset

336 
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 HOLlight)
hoelzl
parents:
diff
changeset

337 
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 HOLlight)
hoelzl
parents:
diff
changeset

338 
unfolding assms(1)[rule_format] applyapply(cases "j=k") using True by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
changeset

339 
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 HOLlight)
hoelzl
parents:
diff
changeset

340 
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 HOLlight)
hoelzl
parents:
diff
changeset

341 
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 HOLlight)
hoelzl
parents:
diff
changeset

342 
unfolding assms(1)[rule_format] applyapply(cases "j=k") using False by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

343 
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 HOLlight)
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parents:
diff
changeset

344 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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345 
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 HOLlight)
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diff
changeset

346 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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347 
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 HOLlight)
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diff
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348 
(card s = n + 1 \<and> 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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349 
(\<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 HOLlight)
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350 
(\<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 HOLlight)
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diff
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351 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

352 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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353 
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" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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354 
unfolding ksimplex_def by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
changeset

355 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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diff
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356 
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 HOLlight)
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parents:
diff
changeset

357 
(card s = n + 1 \<and> finite s \<and> 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
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parents:
diff
changeset

358 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

359 
(\<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 HOLlight)
hoelzl
parents:
diff
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360 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

361 
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 HOLlight)
hoelzl
parents:
diff
changeset

362 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

363 
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 HOLlight)
hoelzl
parents:
diff
changeset

364 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

365 
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 HOLlight)
hoelzl
parents:
diff
changeset

366 
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 HOLlight)
hoelzl
parents:
diff
changeset

367 
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 HOLlight)
hoelzl
parents:
diff
changeset

368 
note assm = assms[unfolded ksimplex_eq] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

369 
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 HOLlight)
hoelzl
parents:
diff
changeset

370 
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 HOLlight)
hoelzl
parents:
diff
changeset

371 
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 HOLlight)
hoelzl
parents:
diff
changeset

372 
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 HOLlight)
hoelzl
parents:
diff
changeset

373 
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 HOLlight)
hoelzl
parents:
diff
changeset

374 
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 HOLlight)
hoelzl
parents:
diff
changeset

375 
hence "kle n a b \<and> n \<le> card {k\<in>{1..n}. a(k) < b(k)}" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

376 
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 HOLlight)
hoelzl
parents:
diff
changeset

377 
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 HOLlight)
hoelzl
parents:
diff
changeset

378 
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 HOLlight)
hoelzl
parents:
diff
changeset

379 
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 HOLlight)
hoelzl
parents:
diff
changeset

380 
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 HOLlight)
hoelzl
parents:
diff
changeset

381 
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 HOLlight)
hoelzl
parents:
diff
changeset

382 
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 HOLlight)
hoelzl
parents:
diff
changeset

383 
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 HOLlight)
hoelzl
parents:
diff
changeset

384 
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 HOLlight)
hoelzl
parents:
diff
changeset

385 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

386 
lemma ksimplex_extrema_strong: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

387 
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 HOLlight)
hoelzl
parents:
diff
changeset

388 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

389 
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 HOLlight)
hoelzl
parents:
diff
changeset

390 
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 HOLlight)
hoelzl
parents:
diff
changeset

391 
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 HOLlight)
hoelzl
parents:
diff
changeset

392 
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 HOLlight)
hoelzl
parents:
diff
changeset

393 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

394 
lemma ksimplexD: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

395 
assumes "ksimplex p n s" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

396 
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" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

397 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

398 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

399 
lemma ksimplex_successor: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

400 
assumes "ksimplex p n s" "a \<in> s" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

401 
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 HOLlight)
hoelzl
parents:
diff
changeset

402 
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 HOLlight)
hoelzl
parents:
diff
changeset

403 
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 HOLlight)
hoelzl
parents:
diff
changeset

404 
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 HOLlight)
hoelzl
parents:
diff
changeset

405 
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 HOLlight)
hoelzl
parents:
diff
changeset

406 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

407 
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 HOLlight)
hoelzl
parents:
diff
changeset

408 
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 HOLlight)
hoelzl
parents:
diff
changeset

409 
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 HOLlight)
hoelzl
parents:
diff
changeset

410 
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 HOLlight)
hoelzl
parents:
diff
changeset

411 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

412 
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 HOLlight)
hoelzl
parents:
diff
changeset

413 
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 HOLlight)
hoelzl
parents:
diff
changeset

414 
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 HOLlight)
hoelzl
parents:
diff
changeset

415 
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 HOLlight)
hoelzl
parents:
diff
changeset

416 
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 HOLlight)
hoelzl
parents:
diff
changeset

417 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

418 
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 HOLlight)
hoelzl
parents:
diff
changeset

419 
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 HOLlight)
hoelzl
parents:
diff
changeset

420 
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 HOLlight)
hoelzl
parents:
diff
changeset

421 
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 HOLlight)
hoelzl
parents:
diff
changeset

422 
also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(12)] *(3) using assm(3) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

423 
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 HOLlight)
hoelzl
parents:
diff
changeset

424 
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 HOLlight)
hoelzl
parents:
diff
changeset

425 
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 HOLlight)
hoelzl
parents:
diff
changeset

426 
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 HOLlight)
hoelzl
parents:
diff
changeset

427 
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 HOLlight)
hoelzl
parents:
diff
changeset

428 
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 HOLlight)
hoelzl
parents:
diff
changeset

429 
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 HOLlight)
hoelzl
parents:
diff
changeset

430 
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 HOLlight)
hoelzl
parents:
diff
changeset

431 
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 HOLlight)
hoelzl
parents:
diff
changeset

432 
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 HOLlight)
hoelzl
parents:
diff
changeset

433 
ultimately show False unfolding n by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

434 
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 HOLlight)
hoelzl
parents:
diff
changeset

435 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

436 
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 HOLlight)
hoelzl
parents:
diff
changeset

437 
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 HOLlight)
hoelzl
parents:
diff
changeset

438 
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 HOLlight)
hoelzl
parents:
diff
changeset

439 
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 HOLlight)
hoelzl
parents:
diff
changeset

440 
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 HOLlight)
hoelzl
parents:
diff
changeset

441 
case True hence "j=k" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

442 
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 HOLlight)
hoelzl
parents:
diff
changeset

443 
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 HOLlight)
hoelzl
parents:
diff
changeset

444 
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 HOLlight)
hoelzl
parents:
diff
changeset

445 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

446 
lemma ksimplex_predecessor: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

447 
assumes "ksimplex p n s" "a \<in> s" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

448 
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 HOLlight)
hoelzl
parents:
diff
changeset

449 
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 HOLlight)
hoelzl
parents:
diff
changeset

450 
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 HOLlight)
hoelzl
parents:
diff
changeset

451 
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 HOLlight)
hoelzl
parents:
diff
changeset

452 
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 HOLlight)
hoelzl
parents:
diff
changeset

453 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

454 
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 HOLlight)
hoelzl
parents:
diff
changeset

455 
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 HOLlight)
hoelzl
parents:
diff
changeset

456 
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 HOLlight)
hoelzl
parents:
diff
changeset

457 
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 HOLlight)
hoelzl
parents:
diff
changeset

458 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

459 
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 HOLlight)
hoelzl
parents:
diff
changeset

460 
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 HOLlight)
hoelzl
parents:
diff
changeset

461 
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 HOLlight)
hoelzl
parents:
diff
changeset

462 
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 HOLlight)
hoelzl
parents:
diff
changeset

463 
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 HOLlight)
hoelzl
parents:
diff
changeset

464 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

465 
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 HOLlight)
hoelzl
parents:
diff
changeset

466 
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 HOLlight)
hoelzl
parents:
diff
changeset

467 
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 HOLlight)
hoelzl
parents:
diff
changeset

468 
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 HOLlight)
hoelzl
parents:
diff
changeset

469 
also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(12)] *(3) using assm(3) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

470 
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 HOLlight)
hoelzl
parents:
diff
changeset

471 
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 HOLlight)
hoelzl
parents:
diff
changeset

472 
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 HOLlight)
hoelzl
parents:
diff
changeset

473 
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 HOLlight)
hoelzl
parents:
diff
changeset

474 
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 HOLlight)
hoelzl
parents:
diff
changeset

475 
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 HOLlight)
hoelzl
parents:
diff
changeset

476 
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 HOLlight)
hoelzl
parents:
diff
changeset

477 
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 HOLlight)
hoelzl
parents:
diff
changeset

478 
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 HOLlight)
hoelzl
parents:
diff
changeset

479 
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 HOLlight)
hoelzl
parents:
diff
changeset

480 
ultimately show False unfolding n by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

481 
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 HOLlight)
hoelzl
parents:
diff
changeset

482 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

483 
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 HOLlight)
hoelzl
parents:
diff
changeset

484 
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 HOLlight)
hoelzl
parents:
diff
changeset

485 
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 HOLlight)
hoelzl
parents:
diff
changeset

486 
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 HOLlight)
hoelzl
parents:
diff
changeset

487 
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 HOLlight)
hoelzl
parents:
diff
changeset

488 
case True hence "j=k" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

489 
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 HOLlight)
hoelzl
parents:
diff
changeset

490 
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 HOLlight)
hoelzl
parents:
diff
changeset

491 
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 HOLlight)
hoelzl
parents:
diff
changeset

492 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

493 
subsection {* The lemmas about simplices that we need. *} 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

494 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

495 
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 HOLlight)
hoelzl
parents:
diff
changeset

496 
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 HOLlight)
hoelzl
parents:
diff
changeset

497 
using assms apply  proof(induct m arbitrary: s) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

498 
have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" apply(rule set_ext,rule)unfolding mem_Collect_eq apply(rule,rule ext) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

499 
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 HOLlight)
hoelzl
parents:
diff
changeset

500 
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 HOLlight)
hoelzl
parents:
diff
changeset

501 
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 HOLlight)
hoelzl
parents:
diff
changeset

502 
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 HOLlight)
hoelzl
parents:
diff
changeset

503 
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}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

504 
apply(rule set_ext,rule) unfolding mem_Collect_eq image_iff apply(erule conjE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

505 
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 HOLlight)
hoelzl
parents:
diff
changeset

506 
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 HOLlight)
hoelzl
parents:
diff
changeset

507 
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 HOLlight)
hoelzl
parents:
diff
changeset

508 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

509 
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 HOLlight)
hoelzl
parents:
diff
changeset

510 
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 HOLlight)
hoelzl
parents:
diff
changeset

511 
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 HOLlight)
hoelzl
parents:
diff
changeset

512 
moreover have "ya = yb" proof(rule ext) fix x show "ya x = yb x" proof(cases "x = a") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

513 
case False thus ?thesis using as(1)[THEN cong[of _ _ x x]] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

514 
case True thus ?thesis using as(5,7) using as0(2) by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

515 
ultimately show ?case unfolding goal1 by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

516 
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 HOLlight)
hoelzl
parents:
diff
changeset

517 
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 HOLlight)
hoelzl
parents:
diff
changeset

518 
unfolding SetCompr_Sigma_eq apply 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

519 
unfolding card_cartesian_product 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

520 
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 HOLlight)
hoelzl
parents:
diff
changeset

521 
qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

522 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

523 
lemma card_funspace: assumes "finite s" "finite t" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

524 
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 HOLlight)
hoelzl
parents:
diff
changeset

525 
using assms by (auto intro: card_funspace') 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

526 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

527 
lemma finite_funspace: assumes "finite s" "finite t" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

528 
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 HOLlight)
hoelzl
parents:
diff
changeset

529 
proof (cases "card t > 0") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

530 
case True 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

531 
have "card ?S = (card t) ^ (card s)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

532 
using assms by (auto intro!: card_funspace) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

533 
thus ?thesis using True by (auto intro: card_ge_0_finite) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

534 
next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

535 
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 HOLlight)
hoelzl
parents:
diff
changeset

536 
show ?thesis 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

537 
proof (cases "s = {}") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

538 
have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" by (auto intro: ext) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

539 
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 HOLlight)
hoelzl
parents:
diff
changeset

540 
next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

541 
case False thus ?thesis using `t = {}` by simp 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

542 
qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

543 
qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

544 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

545 
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 HOLlight)
hoelzl
parents:
diff
changeset

546 
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 HOLlight)
hoelzl
parents:
diff
changeset

547 
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 HOLlight)
hoelzl
parents:
diff
changeset

548 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

549 
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 HOLlight)
hoelzl
parents:
diff
changeset

550 
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 HOLlight)
hoelzl
parents:
diff
changeset

551 
assume ?ls then guess s .. then guess a applyapply(erule exE,(erule conjE)+) . note sa=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

552 
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 HOLlight)
hoelzl
parents:
diff
changeset

553 
fix x y assume as:"x \<in>s  {a}" "y \<in>s  {a}" have xyp:"x (n + 1) = y (n + 1)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

554 
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 HOLlight)
hoelzl
parents:
diff
changeset

555 
using as(2)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

556 
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 HOLlight)
hoelzl
parents:
diff
changeset

557 
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 HOLlight)
hoelzl
parents:
diff
changeset

558 
have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply(rule ccontr) unfolding not_not apply(erule bexE) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

559 
fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

560 
thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

561 
thus ?thesis applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

562 
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 HOLlight)
hoelzl
parents:
diff
changeset

563 
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 HOLlight)
hoelzl
parents:
diff
changeset

564 
hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" applyapply(rule ccontr) unfolding not_not apply(erule bexE) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

565 
fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

566 
thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

567 
thus ?thesis applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

568 
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 HOLlight)
hoelzl
parents:
diff
changeset

569 
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 HOLlight)
hoelzl
parents:
diff
changeset

570 
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 HOLlight)
hoelzl
parents:
diff
changeset

571 
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 HOLlight)
hoelzl
parents:
diff
changeset

572 
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 HOLlight)
hoelzl
parents:
diff
changeset

573 
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 HOLlight)
hoelzl
parents:
diff
changeset

574 
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 HOLlight)
hoelzl
parents:
diff
changeset

575 
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 HOLlight)
hoelzl
parents:
diff
changeset

576 
proof(rule_tac[35] ballI allI)+ 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

577 
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 HOLlight)
hoelzl
parents:
diff
changeset

578 
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 HOLlight)
hoelzl
parents:
diff
changeset

579 
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 HOLlight)
hoelzl
parents:
diff
changeset

580 
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 HOLlight)
hoelzl
parents:
diff
changeset

581 
{ 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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

582 
case False hence "z\<in>f" using z by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

583 
then guess k apply(drule_tac ab(3)[THEN bspec[where x=z], THEN conjunct1]) unfolding kle_def apply(erule exE) . 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

584 
thus "kle (n + 1) c z" unfolding kle_def apply(rule_tac x="insert (n + 1) k" in exI) unfolding c_def 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

585 
using ab using rs(5)[rule_format,OF ab(1),of "n + 1"] assms(1) by auto qed auto } note * = this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

586 
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 HOLlight)
hoelzl
parents:
diff
changeset

587 
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 HOLlight)
hoelzl
parents:
diff
changeset

588 
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 HOLlight)
hoelzl
parents:
diff
changeset

589 
qed(insert rs, auto) qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

590 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

591 
lemma ksimplex_fix_plane: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

592 
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 HOLlight)
hoelzl
parents:
diff
changeset

593 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

594 
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 HOLlight)
hoelzl
parents:
diff
changeset

595 
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 HOLlight)
hoelzl
parents:
diff
changeset

596 
using assms(12,45) by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

597 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

598 
lemma ksimplex_fix_plane_0: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

599 
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 HOLlight)
hoelzl
parents:
diff
changeset

600 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

601 
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 HOLlight)
hoelzl
parents:
diff
changeset

602 
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 HOLlight)
hoelzl
parents:
diff
changeset

603 
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 HOLlight)
hoelzl
parents:
diff
changeset

604 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

605 
lemma ksimplex_fix_plane_p: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

606 
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 HOLlight)
hoelzl
parents:
diff
changeset

607 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

608 
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 HOLlight)
hoelzl
parents:
diff
changeset

609 
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 HOLlight)
hoelzl
parents:
diff
changeset

610 
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 HOLlight)
hoelzl
parents:
diff
changeset

611 
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 HOLlight)
hoelzl
parents:
diff
changeset

612 
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 HOLlight)
hoelzl
parents:
diff
changeset

613 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

614 
lemma ksimplex_replace_0: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

615 
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 HOLlight)
hoelzl
parents:
diff
changeset

616 
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 HOLlight)
hoelzl
parents:
diff
changeset

617 
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 HOLlight)
hoelzl
parents:
diff
changeset

618 
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 HOLlight)
hoelzl
parents:
diff
changeset

619 
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 HOLlight)
hoelzl
parents:
diff
changeset

620 
have a:"a = a1" apply(rule ksimplex_fix_plane_0[OF assms(2,45)]) using exta(12,5) by auto moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

621 
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 HOLlight)
hoelzl
parents:
diff
changeset

622 
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 HOLlight)
hoelzl
parents:
diff
changeset

623 
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 HOLlight)
hoelzl
parents:
diff
changeset

624 
hence "b1 = a1" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

625 
show "s' = s" applyapply(rule *[of _ a1 b1]) using exta(12) extb(12) goal1 by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

626 
show ?thesis unfolding card_1_exists applyapply(rule ex1I[of _ s]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

627 
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 HOLlight)
hoelzl
parents:
diff
changeset

628 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

629 
lemma ksimplex_replace_1: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

630 
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 HOLlight)
hoelzl
parents:
diff
changeset

631 
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 HOLlight)
hoelzl
parents:
diff
changeset

632 
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 HOLlight)
hoelzl
parents:
diff
changeset

633 
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 HOLlight)
hoelzl
parents:
diff
changeset

634 
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 HOLlight)
hoelzl
parents:
diff
changeset

635 
have a:"a = a0" apply(rule ksimplex_fix_plane_p[OF assms(12,45) exta(1,2)]) unfolding exta by auto moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

636 
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 HOLlight)
hoelzl
parents:
diff
changeset

637 
have a':"a' = b0" apply(rule ksimplex_fix_plane_p[OF goal1(12) 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 HOLlight)
hoelzl
parents:
diff
changeset

638 
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 HOLlight)
hoelzl
parents:
diff
changeset

639 
have "a0 = b0" apply(rule ext) proof case goal1 show "a0 x = b0 x" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

640 
using *[THEN cong, of x x] unfolding exta extb applyapply(cases "x\<in>{1..n}") by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

641 
ultimately show "s' = s" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

642 
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 HOLlight)
hoelzl
parents:
diff
changeset

643 
apply(rule_tac x=a in bexI) prefer 3 apply(erule conjE bexE)+ apply(rule_tac a'=b in lem) using assms(12) by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

644 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

645 
lemma ksimplex_replace_2: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

646 
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 HOLlight)
hoelzl
parents:
diff
changeset

647 
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 HOLlight)
hoelzl
parents:
diff
changeset

648 
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 HOLlight)
hoelzl
parents:
diff
changeset

649 
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 HOLlight)
hoelzl
parents:
diff
changeset

650 
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 HOLlight)
hoelzl
parents:
diff
changeset

651 
thus False by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

652 
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 HOLlight)
hoelzl
parents:
diff
changeset

653 
{ assume "a=a0" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

654 
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 HOLlight)
hoelzl
parents:
diff
changeset

655 
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 HOLlight)
hoelzl
parents:
diff
changeset

656 
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 HOLlight)
hoelzl
parents:
diff
changeset

657 
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 HOLlight)
hoelzl
parents:
diff
changeset

658 
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 HOLlight)
hoelzl
parents:
diff
changeset

659 
apply(rule_tac *[OF ksimplex_successor[OF assms(12),unfolded `a=a0`]]) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

660 
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 HOLlight)
hoelzl
parents:
diff
changeset

661 
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 HOLlight)
hoelzl
parents:
diff
changeset

662 
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 HOLlight)
hoelzl
parents:
diff
changeset

663 
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 HOLlight)
hoelzl
parents:
diff
changeset

664 
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 HOLlight)
hoelzl
parents:
diff
changeset

665 
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 HOLlight)
hoelzl
parents:
diff
changeset

666 
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 HOLlight)
hoelzl
parents:
diff
changeset

667 
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 HOLlight)
hoelzl
parents:
diff
changeset

668 
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 HOLlight)
hoelzl
parents:
diff
changeset

669 
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 HOLlight)
hoelzl
parents:
diff
changeset

670 
ultimately have "x = a0 \<or> x = a2" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

671 
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 HOLlight)
hoelzl
parents:
diff
changeset

672 
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 HOLlight)
hoelzl
parents:
diff
changeset

673 
show "card ?s = n + 1" using ksimplexD(23)[OF assms(1)] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

674 
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 HOLlight)
hoelzl
parents:
diff
changeset

675 
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 HOLlight)
hoelzl
parents:
diff
changeset

676 
show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

677 
fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

678 
fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

679 
case False thus "a3 j \<le>p" unfolding True a3_def using `a1\<in>s` ksimplexD(4)[OF assms(1)] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

680 
guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

681 
have "a2 k \<le> a4 k" using lem3[OF a4(1)[unfolded `a=a0`],THEN kle_imp_pointwise] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

682 
also have "\<dots> < p" using ksimplexD(4)[OF assms(1),rule_format,of a4 k] using a4 by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

683 
finally have *:"a0 k + 1 < p" unfolding k(2)[rule_format] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

684 
case True thus "a3 j \<le>p" unfolding a3_def unfolding a0a1(5)[rule_format] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

685 
using k(1) k(2)assms(5) using * by simp qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

686 
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}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

687 
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

688 
case True show "x j = p" unfolding True a3_def using j k(1) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

689 
using ksimplexD(5)[OF assms(1),rule_format,OF `a1\<in>s` j] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

690 
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 HOLlight)
hoelzl
parents:
diff
changeset

691 
have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a0 \<Longrightarrow> kle n x a3" proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

692 
guess kk using a0a1(4)[rule_format,OF `x\<in>s`,THEN conjunct2,unfolded kle_def] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

693 
applyapply(erule exE,erule conjE) . note kk=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

694 
have "k\<notin>kk" proof assume "k\<in>kk" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

695 
hence "a1 k = x k + 1" using kk by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

696 
hence "a0 k = x k" unfolding a0a1(5)[rule_format] using k(1) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

697 
hence "a2 k = x k + 1" unfolding k(2)[rule_format] by auto moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

698 
have "a2 k \<le> x k" using lem3[of x,THEN kle_imp_pointwise] goal1 by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

699 
ultimately show False by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

700 
thus ?case unfolding kle_def apply(rule_tac x="insert k kk" in exI) using kk(1) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

701 
unfolding a3_def kle_def kk(2)[rule_format] using k(1) by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

702 
show "kle n x y \<or> kle n y x" proof(cases "y=a3") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

703 
case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI1,rule lem4) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

704 
using x by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

705 
case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

706 
apply(rule disjI2,rule lem4) using y False by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

707 
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

708 
using x y `y\<noteq>a3` by auto qed qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

709 
hence "insert a3 (s  {a0}) \<in> ?A" unfolding mem_Collect_eq applyapply(rule,assumption) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

710 
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 HOLlight)
hoelzl
parents:
diff
changeset

711 
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 HOLlight)
hoelzl
parents:
diff
changeset

712 
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 HOLlight)
hoelzl
parents:
diff
changeset

713 
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 HOLlight)
hoelzl
parents:
diff
changeset

714 
from this(2) guess a' .. note a'=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

715 
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 HOLlight)
hoelzl
parents:
diff
changeset

716 
have *:"\<forall>x\<in>s'  {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s{a}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

717 
hence "kle n a2 x" applyapply(rule lem3) using `a=a0` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

718 
hence "a2 k \<le> x k" apply(drule_tac kle_imp_pointwise) by auto moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

719 
have "x k \<le> a2 k" unfolding k(2)[rule_format] using a0a1(4)[rule_format,of x,THEN conjunct1] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

720 
unfolding kle_def using x by auto ultimately show "x k = a2 k" by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

721 
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 HOLlight)
hoelzl
parents:
diff
changeset

722 
show "s' \<in> {s, insert a3 (s  {a0})}" proof(cases "a'=a_min") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

723 
case True have "a_max = a1" unfolding kle_antisym[THEN sym,of a_max a1 n] apply(rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

724 
apply(rule a0a1(4)[rule_format,THEN conjunct2]) defer proof(rule min_max(4)[rule_format,THEN conjunct2]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

725 
show "a1\<in>s'" using a' unfolding `a=a0` using a0a1 by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

726 
show "a_max \<in> s" proof(rule ccontr) assume "a_max\<notin>s" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

727 
hence "a_max = a'" using a' min_max by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

728 
thus False unfolding True using min_max by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

729 
hence "\<forall>i. a_max i = a1 i" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

730 
hence "a' = a" unfolding True `a=a0` applyapply(subst expand_fun_eq,rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

731 
apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

732 
proof case goal1 thus ?case apply(cases "x\<in>{1..n}") by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

733 
hence "s' = s" applyapply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

734 
thus ?thesis by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

735 
case False hence as:"a' = a_max" using ** by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

736 
have "a_min = a2" unfolding kle_antisym[THEN sym, of _ _ n] apply rule 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

737 
apply(rule min_max(4)[rule_format,THEN conjunct1]) defer proof(rule lem3) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

738 
show "a_min \<in> s  {a0}" unfolding a'(2)[THEN sym,unfolded `a=a0`] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

739 
unfolding as using min_max(13) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

740 
have "a2 \<noteq> a" unfolding `a=a0` using k(2)[rule_format,of k] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

741 
hence "a2 \<in> s  {a}" using a2 by auto thus "a2 \<in> s'" unfolding a'(2)[THEN sym] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

742 
hence "\<forall>i. a_min i = a2 i" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

743 
hence "a' = a3" unfolding as `a=a0` applyapply(subst expand_fun_eq,rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

744 
apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

745 
unfolding a3_def k(2)[rule_format] unfolding a0a1(5)[rule_format] proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

746 
show ?case unfolding goal1 apply(cases "x\<in>{1..n}") defer apply(cases "x=k") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

747 
using `k\<in>{1..n}` by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

748 
hence "s' = insert a3 (s  {a0})" applyapply(rule lem1) defer apply assumption 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

749 
apply(rule a'(1)) unfolding a' `a=a0` using `a3\<notin>s` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

750 
thus ?thesis by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

751 
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 HOLlight)
hoelzl
parents:
diff
changeset

752 
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 HOLlight)
hoelzl
parents:
diff
changeset

753 
hence ?thesis unfolding * by auto } moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

754 
{ assume "a=a1" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

755 
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 HOLlight)
hoelzl
parents:
diff
changeset

756 
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 HOLlight)
hoelzl
parents:
diff
changeset

757 
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 HOLlight)
hoelzl
parents:
diff
changeset

758 
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 HOLlight)
hoelzl
parents:
diff
changeset

759 
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 HOLlight)
hoelzl
parents:
diff
changeset

760 
apply(rule_tac *[OF ksimplex_predecessor[OF assms(12),unfolded `a=a1`]]) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

761 
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 HOLlight)
hoelzl
parents:
diff
changeset

762 
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 HOLlight)
hoelzl
parents:
diff
changeset

763 
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 HOLlight)
hoelzl
parents:
diff
changeset

764 
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 HOLlight)
hoelzl
parents:
diff
changeset

765 
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 HOLlight)
hoelzl
parents:
diff
changeset

766 
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 HOLlight)
hoelzl
parents:
diff
changeset

767 
ultimately have "x = a2 \<or> x = a1" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

768 
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 HOLlight)
hoelzl
parents:
diff
changeset

769 
have "a0 k \<noteq> 0" proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

770 
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 HOLlight)
hoelzl
parents:
diff
changeset

771 
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 HOLlight)
hoelzl
parents:
diff
changeset

772 
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 HOLlight)
hoelzl
parents:
diff
changeset

773 
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 HOLlight)
hoelzl
parents:
diff
changeset

774 
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 HOLlight)
hoelzl
parents:
diff
changeset

775 
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 HOLlight)
hoelzl
parents:
diff
changeset

776 
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 HOLlight)
hoelzl
parents:
diff
changeset

777 
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 HOLlight)
hoelzl
parents:
diff
changeset

778 
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 HOLlight)
hoelzl
parents:
diff
changeset

779 
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 HOLlight)
hoelzl
parents:
diff
changeset

780 
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 HOLlight)
hoelzl
parents:
diff
changeset

781 
show "card ?s = n+1" using ksimplexD(23)[OF assms(1)] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

782 
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 HOLlight)
hoelzl
parents:
diff
changeset

783 
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 HOLlight)
hoelzl
parents:
diff
changeset

784 
show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

785 
fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

786 
fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

787 
case False thus "a3 j \<le>p" unfolding True a3_def using `a0\<in>s` ksimplexD(4)[OF assms(1)] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

788 
guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

789 
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 HOLlight)
hoelzl
parents:
diff
changeset

790 
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 HOLlight)
hoelzl
parents:
diff
changeset

791 
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 HOLlight)
hoelzl
parents:
diff
changeset

792 
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}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

793 
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

794 
case True show "x j = p" unfolding True a3_def using j k(1) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

795 
using ksimplexD(5)[OF assms(1),rule_format,OF `a0\<in>s` j] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

796 
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 HOLlight)
hoelzl
parents:
diff
changeset

797 
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 HOLlight)
hoelzl
parents:
diff
changeset

798 
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 HOLlight)
hoelzl
parents:
diff
changeset

799 
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 HOLlight)
hoelzl
parents:
diff
changeset

800 
apply(rule)defer proof(rule) case goal1 thus ?case applyapply(erule conjE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

801 
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 HOLlight)
hoelzl
parents:
diff
changeset

802 
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 HOLlight)
hoelzl
parents:
diff
changeset

803 
ultimately show ?case applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

804 
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 HOLlight)
hoelzl
parents:
diff
changeset

805 
show "kle n x y \<or> kle n y x" proof(cases "y=a3") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

806 
case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI2,rule lem4) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

807 
using x by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

808 
case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

809 
apply(rule disjI1,rule lem4) using y False by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

810 
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

811 
using x y `y\<noteq>a3` by auto qed qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

812 
hence "insert a3 (s  {a1}) \<in> ?A" unfolding mem_Collect_eq applyapply(rule,assumption) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

813 
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 HOLlight)
hoelzl
parents:
diff
changeset

814 
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 HOLlight)
hoelzl
parents:
diff
changeset

815 
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 HOLlight)
hoelzl
parents:
diff
changeset

816 
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 HOLlight)
hoelzl
parents:
diff
changeset

817 
from this(2) guess a' .. note a'=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

818 
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 HOLlight)
hoelzl
parents:
diff
changeset

819 
have *:"\<forall>x\<in>s'  {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s{a}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

820 
hence "kle n x a2" applyapply(rule lem3) using `a=a1` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

821 
hence "x k \<le> a2 k" apply(drule_tac kle_imp_pointwise) by auto moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

822 
{ have "a2 k \<le> a0 k" using k(2)[rule_format,of k] unfolding a0a1(5)[rule_format] using k(1) by simp 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

823 
also have "\<dots> \<le> x k" using a0a1(4)[rule_format,of x,THEN conjunct1,THEN kle_imp_pointwise] x by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

824 
finally have "a2 k \<le> x k" . } ultimately show "x k = a2 k" by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

825 
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 HOLlight)
hoelzl
parents:
diff
changeset

826 
have "a2 \<noteq> a1" proof assume as:"a2 = a1" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

827 
show False using k(2) unfolding as 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 HOLlight)
hoelzl
parents:
diff
changeset

828 
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 HOLlight)
hoelzl
parents:
diff
changeset

829 
show "s' \<in> {s, insert a3 (s  {a1})}" proof(cases "a'=a_min") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

830 
case True have "a_max \<in> s  {a1}" using min_max unfolding a'(2)[unfolded `a=a1`,THEN sym] True by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

831 
hence "a_max = a2" unfolding kle_antisym[THEN sym,of a_max a2 n] applyapply(rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

832 
apply(rule lem3,assumption) apply(rule min_max(4)[rule_format,THEN conjunct2]) using a2' by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

833 
hence a_max:"\<forall>i. a_max i = a2 i" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

834 
have *:"\<forall>j. a2 j = (if j\<in>{1..n} then a3 j + 1 else a3 j)" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

835 
using k(2) unfolding lem4[rule_format] a0a1(5)[rule_format] applyapply(rule,erule_tac x=j in allE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

836 
proof case goal1 thus ?case apply(cases "j\<in>{1..n}",case_tac[!] "j=k") by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

837 
have "\<forall>i. a_min i = a3 i" using a_max applyapply(rule,erule_tac x=i in allE) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

838 
unfolding min_max(5)[rule_format] *[rule_format] proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

839 
thus ?case apply(cases "i\<in>{1..n}") by auto qed hence "a_min = a3" unfolding expand_fun_eq . 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

840 
hence "s' = insert a3 (s  {a1})" using a' unfolding `a=a1` True by auto thus ?thesis by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

841 
case False hence as:"a'=a_max" using ** by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

842 
have "a_min = a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

843 
apply(rule min_max(4)[rule_format,THEN conjunct1]) defer apply(rule a0a1(4)[rule_format,THEN conjunct1]) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

844 
have "a_min \<in> s  {a1}" using min_max(1,3) unfolding a'(2)[THEN sym,unfolded `a=a1`] as by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

845 
thus "a_min \<in> s" by auto have "a0 \<in> s  {a1}" using a0a1(13) by auto thus "a0 \<in> s'" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

846 
unfolding a'(2)[THEN sym,unfolded `a=a1`] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

847 
hence "\<forall>i. a_max i = a1 i" unfolding a0a1(5)[rule_format] min_max(5)[rule_format] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

848 
hence "s' = s" applyapply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` unfolding as `a=a1` unfolding expand_fun_eq by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

849 
thus ?thesis by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

850 
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 HOLlight)
hoelzl
parents:
diff
changeset

851 
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 HOLlight)
hoelzl
parents:
diff
changeset

852 
hence ?thesis unfolding * by auto } moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

853 
{ 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 HOLlight)
hoelzl
parents:
diff
changeset

854 
have "a=a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

855 
using goal1 a0a1 assms(2) by auto thus False using as by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

856 
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(12)] by blast 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

857 
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 HOLlight)
hoelzl
parents:
diff
changeset

858 
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 HOLlight)
hoelzl
parents:
diff
changeset

859 
have "a=a1" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

860 
using goal1 a0a1 assms(2) by auto thus False using as by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

861 
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(12)] by blast 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

862 
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 HOLlight)
hoelzl
parents:
diff
changeset

863 
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 HOLlight)
hoelzl
parents:
diff
changeset

864 
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 HOLlight)
hoelzl
parents:
diff
changeset

865 
thus False using ksimplexD(6)[OF assms(1),rule_format,OF u v] unfolding kle_def 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

866 
unfolding l(2) k(2) `k=l` applyapply(erule disjE)apply(erule_tac[!] exE conjE)+ 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

867 
apply(erule_tac[!] x=l in allE)+ by(auto simp add: *) qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

868 
hence aa':"a'\<noteq>a" applyapply rule unfolding expand_fun_eq unfolding a'_def k(2) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

869 
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 HOLlight)
hoelzl
parents:
diff
changeset

870 
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 HOLlight)
hoelzl
parents:
diff
changeset

871 
case goal2 hence "kle n a' a" by auto thus False apply(drule_tac kle_imp_pointwise) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

872 
apply(erule_tac x=l in allE) unfolding a'_def k(2) using kl by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

873 
case True thus False apply(drule_tac kle_imp_pointwise) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

874 
apply(erule_tac x=k in allE) unfolding a'_def k(2) using kl by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

875 
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 HOLlight)
hoelzl
parents:
diff
changeset

876 
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 HOLlight)
hoelzl
parents:
diff
changeset

877 
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 HOLlight)
hoelzl
parents:
diff
changeset

878 
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 HOLlight)
hoelzl
parents:
diff
changeset

879 
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 HOLlight)
hoelzl
parents:
diff
changeset

880 
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 HOLlight)
hoelzl
parents:
diff
changeset

881 
assume as:"x l = u l" "x k = u k" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

882 
have "x = u" unfolding expand_fun_eq 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

883 
using goal1(2)[THEN kle_imp_pointwise,unfolded l(2)] unfolding k(2) apply 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

884 
using goal1(1)[THEN kle_imp_pointwise] applyapply rule apply(erule_tac x=xa in allE)+ proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

885 
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

886 
assume as:"x l \<noteq> u l" "x k = u k" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

887 
have "x = a'" unfolding expand_fun_eq unfolding a'_def 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

888 
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

889 
using goal1(1)[THEN kle_imp_pointwise] applyapply rule apply(erule_tac x=xa in allE)+ proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

890 
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

891 
assume as:"x l = u l" "x k \<noteq> u k" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

892 
have "x = a" unfolding expand_fun_eq 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

893 
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

894 
using goal1(1)[THEN kle_imp_pointwise] applyapply rule apply(erule_tac x=xa in allE)+ proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

895 
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

896 
assume as:"x l \<noteq> u l" "x k \<noteq> u k" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

897 
have "x = v" unfolding expand_fun_eq 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

898 
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

899 
using goal1(1)[THEN kle_imp_pointwise] applyapply rule apply(erule_tac x=xa in allE)+ proof case goal1 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

900 
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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

901 
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 HOLlight)
hoelzl
parents:
diff
changeset

902 
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 HOLlight)
hoelzl
parents:
diff
changeset

903 
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 HOLlight)
hoelzl
parents:
diff
changeset

904 
using kle_uv `u\<in>s` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

905 
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 HOLlight)
hoelzl
parents:
diff
changeset

906 
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 HOLlight)
hoelzl
parents:
diff
changeset

907 
using kle_uv `v\<in>s` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

908 
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 HOLlight)
hoelzl
parents:
diff
changeset

909 
proof(cases "kle n v x \<or> kle n x u") case True thus ?thesis using goal1 by(auto intro:lem3 lem4) next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

910 
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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

911 
show ?thesis using uxv[OF *] using kle_uv using goal1 by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

912 
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 HOLlight)
hoelzl
parents:
diff
changeset

913 
show "card (insert a' (s  {a})) = n + 1" using ksimplexD(23)[OF assms(1)] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

914 
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 HOLlight)
hoelzl
parents:
diff
changeset

915 
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 HOLlight)
hoelzl
parents:
diff
changeset

916 
show "\<forall>j. x j \<le> p" proof(rule,cases "x = a'") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

917 
fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

918 
fix j case True show "x j\<le>p" unfolding True proof(cases "j=l") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

919 
case False thus "a' j \<le>p" unfolding True a'_def using `u\<in>s` ksimplexD(4)[OF assms(1)] by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

920 
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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

921 
have "u l + 1 \<le> p" unfolding *[THEN sym] using ksimplexD(4)[OF assms(1)] using `v\<in>s` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

922 
thus "a' j \<le>p" unfolding a'_def True by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

923 
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}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

924 
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

925 
case True show "x j = p" unfolding True a'_def using j l(1) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

926 
using ksimplexD(5)[OF assms(1),rule_format,OF `u\<in>s` j] by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

927 
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 HOLlight)
hoelzl
parents:
diff
changeset

928 
show "kle n x y \<or> kle n y x" proof(cases "y=a'") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

929 
case True show ?thesis unfolding True apply(cases "x=a'") defer apply(rule lem5) using x by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

930 
case False show ?thesis proof(cases "x=a'") case True show ?thesis unfolding True 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

931 
using lem5[of y] using y by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

932 
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

933 
using x y `y\<noteq>a'` by auto qed qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

934 
hence "insert a' (s  {a}) \<in> ?A" unfolding mem_Collect_eq applyapply(rule,assumption) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

935 
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 HOLlight)
hoelzl
parents:
diff
changeset

936 
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 HOLlight)
hoelzl
parents:
diff
changeset

937 
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 HOLlight)
hoelzl
parents:
diff
changeset

938 
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 HOLlight)
hoelzl
parents:
diff
changeset

939 
from this(2) guess a'' .. note a''=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

940 
have "u\<noteq>v" unfolding expand_fun_eq unfolding l(2) k(2) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

941 
hence uv':"\<not> kle n v u" using uv using kle_antisym by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

942 
have "u\<noteq>a" "v\<noteq>a" unfolding expand_fun_eq k(2) l(2) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

943 
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 HOLlight)
hoelzl
parents:
diff
changeset

944 
have lem6:"a \<in> s' \<or> a' \<in> s'" proof(cases "\<forall>x\<in>s'. kle n x u \<or> kle n v x") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

945 
case False then guess w unfolding ball_simps .. note w=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

946 
hence "kle n u w" "kle n w v" using ksimplexD(6)[OF as] uvs' by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

947 
hence "w = a' \<or> w = a" using uxv[of w] uvs' w by auto thus ?thesis using w by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

948 
case True have "\<not> (\<forall>x\<in>s'. kle n x u)" unfolding ball_simps apply(rule_tac x=v in bexI) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

949 
using uv `u\<noteq>v` unfolding kle_antisym[of n u v,THEN sym] using `v\<in>s'` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

950 
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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

951 
then guess w .. note w=this from this(2) guess kk .. note kk=this[rule_format] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

952 
have "\<not> kle n w u" applyapply(rule,drule kle_imp_pointwise) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

953 
apply(erule_tac x=kk in allE) unfolding kk by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

954 
hence *:"kle n v w" using True[rule_format,OF w(1)] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

955 
hence False proof(cases "kk\<noteq>l") case True thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

956 
apply(erule_tac x=l in allE) using `k\<noteq>l` by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

957 
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)] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

958 
apply(erule_tac x=k in allE) using `k\<noteq>l` by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

959 
thus ?thesis by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

960 
thus "s' \<in> {s, insert a' (s  {a})}" proof(cases "a\<in>s'") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

961 
case True hence "s' = s" applyapply(rule lem1[OF a''(2)]) using a'' `a\<in>s` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

962 
thus ?thesis by auto next case False hence "a'\<in>s'" using lem6 by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

963 
hence "s' = insert a' (s  {a})" applyapply(rule lem1[of _ a'' _ a']) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

964 
unfolding a''(2)[THEN sym] using a'' using `a'\<notin>s` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

965 
thus ?thesis by auto qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

966 
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 HOLlight)
hoelzl
parents:
diff
changeset

967 
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 HOLlight)
hoelzl
parents:
diff
changeset

968 
hence ?thesis unfolding * by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

969 
ultimately show ?thesis by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

970 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

971 
subsection {* Hence another step towards concreteness. *} 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

972 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

973 
lemma kuhn_simplex_lemma: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

974 
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 HOLlight)
hoelzl
parents:
diff
changeset

975 
"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 HOLlight)
hoelzl
parents:
diff
changeset

976 
(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 HOLlight)
hoelzl
parents:
diff
changeset

977 
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 HOLlight)
hoelzl
parents:
diff
changeset

978 
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 HOLlight)
hoelzl
parents:
diff
changeset

979 
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 HOLlight)
hoelzl
parents:
diff
changeset

980 
(rl ` f = {0..n}) \<and> 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

981 
((\<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 HOLlight)
hoelzl
parents:
diff
changeset

982 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

983 
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 HOLlight)
hoelzl
parents:
diff
changeset

984 
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 HOLlight)
hoelzl
parents:
diff
changeset

985 
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 HOLlight)
hoelzl
parents:
diff
changeset

986 
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 HOLlight)
hoelzl
parents:
diff
changeset

987 
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 HOLlight)
hoelzl
parents:
diff
changeset

988 
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 HOLlight)
hoelzl
parents:
diff
changeset

989 
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 HOLlight)
hoelzl
parents:
diff
changeset

990 
{ 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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

991 
applyapply(rule ksimplex_replace_0) apply(rule as)+ unfolding as by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

992 
{ 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 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

993 
applyapply(rule ksimplex_replace_1) apply(rule as)+ unfolding as by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

994 
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 HOLlight)
hoelzl
parents:
diff
changeset

995 
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 HOLlight)
hoelzl
parents:
diff
changeset

996 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

997 
subsection {* Reduced labelling. *} 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

998 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

999 
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 HOLlight)
hoelzl
parents:
diff
changeset

1000 
(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 HOLlight)
hoelzl
parents:
diff
changeset

1001 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1002 
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 HOLlight)
hoelzl
parents:
diff
changeset

1003 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1004 
"(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 HOLlight)
hoelzl
parents:
diff
changeset

1005 
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 HOLlight)
hoelzl
parents:
diff
changeset

1006 
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 HOLlight)
hoelzl
parents:
diff
changeset

1007 
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 HOLlight)
hoelzl
parents:
diff
changeset

1008 
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 HOLlight)
hoelzl
parents:
diff
changeset

1009 
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 HOLlight)
hoelzl
parents:
diff
changeset

1010 
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 HOLlight)
hoelzl
parents:
diff
changeset

1011 
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 HOLlight)
hoelzl
parents:
diff
changeset

1012 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1013 
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 HOLlight)
hoelzl
parents:
diff
changeset

1014 
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 HOLlight)
hoelzl
parents:
diff
changeset

1015 
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 HOLlight)
hoelzl
parents:
diff
changeset

1016 
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 HOLlight)
hoelzl
parents:
diff
changeset

1017 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1018 
lemma reduced_labelling_0: assumes "j\<in>{1..n}" "label x j = 0" shows "reduced label n x \<noteq> j  1" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1019 
using reduced_labelling[of label n x] using assms by fastsimp 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1020 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1021 
lemma reduced_labelling_1: assumes "j\<in>{1..n}" "label x j \<noteq> 0" shows "reduced label n x < j" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1022 
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 HOLlight)
hoelzl
parents:
diff
changeset

1023 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1024 
lemma reduced_labelling_Suc: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1025 
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 HOLlight)
hoelzl
parents:
diff
changeset

1026 
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 HOLlight)
hoelzl
parents:
diff
changeset

1027 
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 HOLlight)
hoelzl
parents:
diff
changeset

1028 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1029 
lemma complete_face_top: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1030 
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 HOLlight)
hoelzl
parents:
diff
changeset

1031 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1032 
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 HOLlight)
hoelzl
parents:
diff
changeset

1033 
((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 HOLlight)
hoelzl
parents:
diff
changeset

1034 
assume ?l (is "?as \<and> (?a \<or> ?b)") thus ?r applyapply(rule,erule conjE,assumption) proof(cases ?a) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1035 
case True then guess j .. note j=this {fix x assume x:"x\<in>f" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1036 
have "reduced lab (n+1) x \<noteq> j  1" using j applyapply(rule reduced_labelling_0) defer apply(rule assms(1)[rule_format]) using x by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1037 
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 HOLlight)
hoelzl
parents:
diff
changeset

1038 
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 HOLlight)
hoelzl
parents:
diff
changeset

1039 
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 HOLlight)
hoelzl
parents:
diff
changeset

1040 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1041 
case False hence ?b using `?l` by blast then guess j .. note j=this {fix x assume x:"x\<in>f" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1042 
have "reduced lab (n+1) x < j" using j applyapply(rule reduced_labelling_1) using assms(2)[rule_format,of x j] and x by auto } note * = this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1043 
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 HOLlight)
hoelzl
parents:
diff
changeset

1044 
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 HOLlight)
hoelzl
parents:
diff
changeset

1045 
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 HOLlight)
hoelzl
parents:
diff
changeset

1046 
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 HOLlight)
hoelzl
parents:
diff
changeset

1047 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1048 
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 HOLlight)
hoelzl
parents:
diff
changeset

1049 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1050 
lemma kuhn_induction: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1051 
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 HOLlight)
hoelzl
parents:
diff
changeset

1052 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1053 
"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 HOLlight)
hoelzl
parents:
diff
changeset

1054 
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 HOLlight)
hoelzl
parents:
diff
changeset

1055 
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 HOLlight)
hoelzl
parents:
diff
changeset

1056 
show ?thesis apply(rule kuhn_simplex_lemma[unfolded mem_Collect_eq]) apply(rule,rule,rule *,rule reduced_labelling) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1057 
apply(rule *(1)[OF assms(4)]) apply(rule set_ext) unfolding mem_Collect_eq apply(rule,erule conjE) defer apply(rule) proof(*(rule,rule)*) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1058 
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 HOLlight)
hoelzl
parents:
diff
changeset

1059 
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 HOLlight)
hoelzl
parents:
diff
changeset

1060 
using assms(23) using as(1)[unfolded ksimplex_def] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1061 
have allp:"\<forall>x\<in>f. x (n + 1) = p" using assms(2) using as(1)[unfolded ksimplex_def] by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1062 
{ fix x assume "x\<in>f" hence "reduced lab (n + 1) x < n + 1" applyapply(rule reduced_labelling_1) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1063 
defer using assms(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 HOLlight)
hoelzl
parents:
diff
changeset

1064 
hence "reduced lab (n + 1) x = reduced lab n x" applyapply(rule reduced_labelling_Suc) using reduced_labelling(1) by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1065 
hence "reduced lab (n + 1) ` f = {0..n}" unfolding as(2)[THEN sym] apply apply(rule set_ext) unfolding image_iff by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1066 
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 HOLlight)
hoelzl
parents:
diff
changeset

1067 
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 HOLlight)
hoelzl
parents:
diff
changeset

1068 
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 HOLlight)
hoelzl
parents:
diff
changeset

1069 
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 HOLlight)
hoelzl
parents:
diff
changeset

1070 
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 HOLlight)
hoelzl
parents:
diff
changeset

1071 
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 HOLlight)
hoelzl
parents:
diff
changeset

1072 
then guess s .. then guess a applyapply(erule exE,(erule conjE)+) . note sa=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1073 
{ 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 HOLlight)
hoelzl
parents:
diff
changeset

1074 
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 HOLlight)
hoelzl
parents:
diff
changeset

1075 
hence "reduced lab (n + 1) x = reduced lab n x" applyapply(rule reduced_labelling_Suc) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1076 
using reduced_labelling(1) by auto } 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1077 
thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[THEN sym] applyapply(rule set_ext) unfolding image_iff by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1078 
have *:"\<forall>x\<in>f. x (n + 1) = p" proof(cases "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1079 
case True then guess j .. hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab (n + 1) x \<noteq> j  1" applyapply(rule reduced_labelling_0) apply assumption 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1080 
apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1081 
have "j  1 \<in> {0..n}" using `j\<in>{1..n+1}` by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1082 
ultimately have False unfolding sa(4)[THEN sym] unfolding image_iff by fastsimp thus ?thesis by auto next 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1083 
case False hence "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p" using sa(5) by fastsimp then guess j .. note j=this 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1084 
thus ?thesis proof(cases "j = n+1") 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1085 
case False hence *:"j\<in>{1..n}" using j by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1086 
hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab n x < j" apply(rule reduced_labelling_1) proof fix x assume "x\<in>f" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1087 
hence "lab x j = 1" applyapply(rule assms(3)[rule_format,OF j(1)]) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1088 
using sa(1)[unfolded ksimplex_def] using j unfolding sa by auto thus "lab x j \<noteq> 0" by auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1089 
moreover have "j\<in>{0..n}" using * by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1090 
ultimately have False unfolding part1[THEN sym] using * unfolding image_iff by auto thus ?thesis by auto qed auto qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1091 
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 HOLlight)
hoelzl
parents:
diff
changeset

1092 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1093 
lemma kuhn_induction_Suc: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1094 
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 HOLlight)
hoelzl
parents:
diff
changeset

1095 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1096 
"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 HOLlight)
hoelzl
parents:
diff
changeset

1097 
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 HOLlight)
hoelzl
parents:
diff
changeset

1098 
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 HOLlight)
hoelzl
parents:
diff
changeset

1099 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1100 
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 HOLlight)
hoelzl
parents:
diff
changeset

1101 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1102 
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 HOLlight)
hoelzl
parents:
diff
changeset

1103 
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 HOLlight)
hoelzl
parents:
diff
changeset

1104 
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 HOLlight)
hoelzl
parents:
diff
changeset

1105 
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 HOLlight)
hoelzl
parents:
diff
changeset

1106 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1107 
lemma reduce_labelling_0[simp]: "reduced lab 0 x = 0" apply(rule reduced_labelling_unique) by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1108 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1109 
lemma kuhn_combinatorial: 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1110 
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 HOLlight)
hoelzl
parents:
diff
changeset

1111 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1112 
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 HOLlight)
hoelzl
parents:
diff
changeset

1113 
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 HOLlight)
hoelzl
parents:
diff
changeset

1114 
{ 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 HOLlight)
hoelzl
parents:
diff
changeset

1115 
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 HOLlight)
hoelzl
parents:
diff
changeset

1116 
thus ?case applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

1117 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1118 
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 HOLlight)
hoelzl
parents:
diff
changeset

1119 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1120 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1121 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1122 
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 HOLlight)
hoelzl
parents:
diff
changeset

1123 
"\<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 HOLlight)
hoelzl
parents:
diff
changeset

1124 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

1125 
~(label r i = label s i)" proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1126 
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 HOLlight)
hoelzl
parents:
diff
changeset

1127 
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 HOLlight)
hoelzl
parents:
diff
changeset

1128 
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 HOLlight)
hoelzl
parents:
diff
changeset

1129 
hence "card ?A \<noteq> 0" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

1130 
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 HOLlight)
hoelzl
parents:
diff
changeset

1131 
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 HOLlight)
hoelzl
parents:
diff
changeset

1132 
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 HOLlight)
hoelzl
parents:
diff
changeset

1133 
hence "a i + 1 \<le> p" applyapply(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 HOLlight)
hoelzl
parents:
diff
changeset

1134 
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 HOLlight)
hoelzl
parents:
diff
changeset

1135 
thus "a i < p" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1136 
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 HOLlight)
hoelzl
parents:
diff
changeset

1137 
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 HOLlight)
hoelzl
parents:
diff
changeset

1138 
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[12] ballI) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1139 
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 HOLlight)
hoelzl
parents:
diff
changeset

1140 
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 HOLlight)
hoelzl
parents:
diff
changeset

1141 
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 HOLlight)
hoelzl
parents:
diff
changeset

1142 
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 HOLlight)
hoelzl
parents:
diff
changeset

1143 
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 HOLlight)
hoelzl
parents:
diff
changeset

1144 
by auto qed qed qed 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1145 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1146 
subsection {* The main result for the unit cube. *} 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1147 

4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1148 
lemma kuhn_labelling_lemma': 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1149 
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 HOLlight)
hoelzl
parents:
diff
changeset

1150 
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 HOLlight)
hoelzl
parents:
diff
changeset

1151 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

1152 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

1153 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

1154 
(\<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 HOLlight)
hoelzl
parents:
diff
changeset

1155 
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 HOLlight)
hoelzl
parents:
diff
changeset

1156 
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 HOLlight)
hoelzl
parents:
diff
changeset

1157 
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 HOLlight)
hoelzl
parents:
diff
changeset

1158 
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 HOLlight)
hoelzl
parents:
diff
changeset

1159 
(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 HOLlight)
hoelzl
parents:
diff
changeset

1160 
{ 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 HOLlight)
hoelzl
parents:
diff
changeset

1161 
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 HOLlight)
hoelzl
parents:
diff
changeset

1162 
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 HOLlight)
hoelzl
parents:
diff
changeset

1163 

34291
4e896680897e
finite annotation on cartesian product is now implicit.
hoelzl
parents:
34289
diff
changeset

1164 
lemma brouwer_cube: fixes f::"real^'n \<Rightarrow> real^'n" 
33741
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1165 
assumes "continuous_on {0..1} f" "f ` {0..1} \<subseteq> {0..1}" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1166 
shows "\<exists>x\<in>{0..1}. f x = x" apply(rule ccontr) proof 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1167 
def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1168 
assume "\<not> (\<exists>x\<in>{0..1}. f x = x)" hence *:"\<not> (\<exists>x\<in>{0..1}. f x  x = 0)" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1169 
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 HOLlight)
hoelzl
parents:
diff
changeset

1170 
apply(rule continuous_on_intros assms)+ . note d=this[rule_format] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1171 
have *:"\<forall>x. x \<in> {0..1} \<longrightarrow> f x \<in> {0..1}" "\<forall>x. x \<in> {0..1::real^'n} \<longrightarrow> (\<forall>i. True \<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 HOLlight)
hoelzl
parents:
diff
changeset

1172 
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 HOLlight)
hoelzl
parents:
diff
changeset

1173 
guess label using kuhn_labelling_lemma[OF *] applyapply(erule exE,(erule conjE)+) . note label = this[rule_format] 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1174 
have lem1:"\<forall>x\<in>{0..1}.\<forall>y\<in>{0..1}.\<forall>i. label x i \<noteq> label y i 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1175 
\<longrightarrow> abs(f x $ i  x $ i) \<le> norm(f y  f x) + norm(y  x)" proof(rule,rule,rule,rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1176 
fix x y assume xy:"x\<in>{0..1::real^'n}" "y\<in>{0..1::real^'n}" fix i::'n assume i:"label x i \<noteq> label y i" 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1177 
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 HOLlight)
hoelzl
parents:
diff
changeset

1178 
\<Longrightarrow> abs(fx  x) \<le> abs(fy  fx) + abs(y  x)" by auto 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1179 
have "\<bar>(f x  x) $ i\<bar> \<le> abs((f y  f x)$i) + abs((y  x)$i)" unfolding vector_minus_component 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
changeset

1180 
apply(rule *) apply(cases "label x i = 0") apply(rule disjI1,rule) prefer 3 proof(rule disjI2,rule) 
4c414d0835ab
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOLlight)
hoelzl
parents:
diff
