src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
author hoelzl
Tue, 17 Nov 2009 18:52:30 +0100
changeset 33741 4c414d0835ab
child 33758 53078b0d21f5
permissions -rw-r--r--
Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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(* ========================================================================= *)
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(* Results connected with topological dimension.                             *)
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(*                                                                           *)
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(* At the moment this is just Brouwer's fixpoint theorem. The proof is from  *)
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(* Kuhn: "some combinatorial lemmas in topology", IBM J. v4. (1960) p. 518   *)
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(* See "http://www.research.ibm.com/journal/rd/045/ibmrd0405K.pdf".          *)
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(*                                                                           *)
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(* The script below is quite messy, but at least we avoid formalizing any    *)
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(* topological machinery; we don't even use barycentric subdivision; this is *)
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(* the big advantage of Kuhn's proof over the usual Sperner's lemma one.     *)
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(*                                                                           *)
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(*              (c) Copyright, John Harrison 1998-2008                       *)
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(* ========================================================================= *)
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(* Author:                       John Harrison
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   Translated to from HOL light: Robert Himmelmann, TU Muenchen *)
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4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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header {* Results connected with topological dimension. *}
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4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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theory Brouwer_Fixpoint
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  imports Convex_Euclidean_Space
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begin
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declare norm_scaleR[simp]
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lemma brouwer_compactness_lemma:
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  assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = (0::real^'n::finite)))"
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  obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)" proof(cases "s={}") case False
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  have "continuous_on s (norm \<circ> f)" by(rule continuous_on_intros continuous_on_norm assms(2))+
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  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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    using continuous_attains_inf[OF assms(1), of "norm \<circ> f"] and False unfolding o_def by auto
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  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 HOL-light)
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  thus ?thesis apply(rule that) using x(2) unfolding o_def by auto qed(rule that[of 1], auto)
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4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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lemma kuhn_labelling_lemma:
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  assumes "(\<forall>x::real^'n. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i::'n. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"
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  shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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             (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
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             (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
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             (\<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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             (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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  have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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  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
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  show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
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    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
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        (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)"
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    { 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]
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        apply(drule_tac assms(1)[rule_format]) by auto }
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    hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed 
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subsection {* The key "counting" observation, somewhat abstracted. *}
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4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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lemma setsum_Un_disjoint':assumes "finite A" "finite B" "A \<inter> B = {}" "A \<union> B = C"
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  shows "setsum g C = setsum g A + setsum g B"
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  using setsum_Un_disjoint[OF assms(1-3)] and assms(4) by auto
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4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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lemma kuhn_counting_lemma: assumes "finite faces" "finite simplices"
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  "\<forall>f\<in>faces. bnd f  \<longrightarrow> (card {s \<in> simplices. face f s} = 1)"
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  "\<forall>f\<in>faces. \<not> bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 2)"
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  "\<forall>s\<in>simplices. compo s  \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 1)"
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  "\<forall>s\<in>simplices. \<not> compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 0) \<or>
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                             (card {f \<in> faces. face f s \<and> compo' f} = 2)"
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    "odd(card {f \<in> faces. compo' f \<and> bnd f})"
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  shows "odd(card {s \<in> simplices. compo s})" proof-
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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}"
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    "\<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
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  hence lem1:"setsum (\<lambda>s. (card {f \<in> faces. face f s \<and> compo' f})) simplices =
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    69
    setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f s}) simplices +
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    setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> \<not> (bnd f)}. face f s}) simplices"
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    71
    unfolding setsum_addf[THEN sym] apply- apply(rule setsum_cong2)
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    using assms(1) by(auto simp add: card_Un_Int, auto simp add:conj_commute)
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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 = 
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              1 * card {f \<in> faces. compo' f \<and> bnd f}"
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       "setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> \<not> bnd f}. face f j}) simplices = 
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              2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}"
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    77
    apply(rule_tac[!] setsum_multicount) using assms by auto
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    78
  have lem3:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices =
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    79
    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices.   compo s}+
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    80
    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s}"
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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 HOL-light)
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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}
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    = setsum (\<lambda>s. 1) {s \<in> simplices. compo s}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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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 HOL-light)
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parents:
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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 HOL-light)
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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 HOL-light)
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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 HOL-light)
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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 HOL-light)
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    89
           {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 2)}"
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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 HOL-light)
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parents:
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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 HOL-light)
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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 HOL-light)
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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 HOL-light)
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parents:
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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 HOL-light)
hoelzl
parents:
diff 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 HOL-light)
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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
    97
  show ?thesis unfolding even_nat_def unfolding card_def and lem4[THEN sym] and *[unfolded card_def]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
    98
    unfolding card_def[THEN sym] apply(rule odd_minus_even) unfolding zadd_int[THEN sym] apply(rule odd_plus_even)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   100
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   102
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   109
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 conjE|drule card_eq_SucD)+ by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   116
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   122
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   125
  obtain a where a:"b = f a" "a\<in>s" using assms(4-5) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   126
  have inj:"inj_on f s" apply(rule eq_card_imp_inj_on) using assms(1-4) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   130
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
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 HOL-light)
hoelzl
parents:
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 HOL-light)
hoelzl
parents:
diff changeset
   134
  case True thus ?thesis apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   140
  hence *:"f ` (s - {c}) = f ` (s - {a})" apply-apply(rule set_ext,rule) proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   151
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   153
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   155
  assumes "finite simplices"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   161
  apply(rule kuhn_counting_lemma) defer apply(rule assms)+ prefer 3 apply(rule assms) proof(rule_tac[1-2] ballI impI)+ 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   174
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   176
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   178
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   180
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   183
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   193
    case True thus ?thesis apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   207
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   209
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   213
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   218
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   224
    case True hence "x=y" apply-apply(rule ext) using k by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   226
    case False hence "(SOME k'. k' \<in> k) \<in> k" apply-apply(rule someI_ex) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   228
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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(1-2)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   238
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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(1-2)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   248
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   254
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   260
  apply(rule,rule kle_trans[OF assms(1-3)]) proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   275
  have fin:"\<And>P. finite {x\<in>{1..n::nat}. P x}" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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(4-5) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   278
  finally show " m1 + m2 \<le> card {k \<in> {1..n}. x k < z k}" by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   279
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   280
lemma kle_range_combine_l:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   282
  shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   283
  using kle_range_combine[OF assms(1-3) _ assms(4), of 0] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   284
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   285
lemma kle_range_combine_r:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   287
  shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   288
  using kle_range_combine[OF assms(1-3) assms(4), of 0] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   289
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   290
lemma kle_range_induct:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   291
  assumes "card s = Suc m" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   293
have "finite s" "s\<noteq>{}" using assms(1) by (auto intro: card_ge_0_finite)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   294
thus ?thesis using assms apply- proof(induct m arbitrary: s)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   295
  case 0 thus ?case using kle_refl by auto next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   297
  show ?case proof(cases "s \<subseteq> {a}") case False
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   298
    hence "card (s - {a}) = Suc m" "s - {a} \<noteq> {}" using card_Diff_singleton[OF _ a(1)] Suc(4) `finite s` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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}" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   300
      using Suc(1)[of "s - {a}"] using Suc(5) `finite s` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   301
    have "1 \<le> card {k \<in> {1..n}. a k < x k}" "m \<le> card {k \<in> {1..n}. x k < b k}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   304
      using kle_range_combine[OF a(2)[rule_format] xb(3) Suc(5)[rule_format], of 1 "m"] using a(1) xb(1-2) by auto next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   307
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   308
lemma kle_Suc: "kle n x y \<Longrightarrow> kle (n + 1) x y"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   310
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   311
lemma kle_trans_1: assumes "kle n x y" shows "x j \<le> y j" "y j \<le> x j + 1"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   312
  using assms[unfolded kle_def] by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   313
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
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 HOL-light)
hoelzl
parents:
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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   324
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
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 HOL-light)
hoelzl
parents:
diff changeset
   328
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   332
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   333
lemma kle_adjacent:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   338
      unfolding assms(1)[rule_format] apply-apply(cases "j=k") using True by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   342
      unfolding assms(1)[rule_format] apply-apply(cases "j=k") using False by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   344
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   346
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   348
        (card s = n + 1 \<and>
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   352
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   354
  unfolding ksimplex_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   355
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
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 HOL-light)
hoelzl
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   362
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   375
  hence "kle n a b \<and> n \<le> card {k\<in>{1..n}. a(k) < b(k)}" apply-apply(rule kle_range_combine_l[where y=c]) using a `c\<in>s` `b\<in>s` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   385
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   393
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   394
lemma ksimplexD:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   398
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   399
lemma ksimplex_successor:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   406
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   411
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   417
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   422
    also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   435
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   441
      case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   445
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   446
lemma ksimplex_predecessor:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   453
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   458
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   464
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   469
    also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   482
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   488
      case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   492
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   494
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   521
qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   522
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   526
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   530
  case True
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   534
next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   536
  show ?thesis
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   537
  proof (cases "s = {}")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   540
  next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   542
  qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   543
qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   544
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   548
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   551
  assume ?ls then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   561
      thus ?thesis apply-apply(rule disjI1) unfolding kle_def using k apply(rule_tac x=k in exI) by auto next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   564
      hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply-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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   567
      thus ?thesis apply-apply(rule disjI2) unfolding kle_def using k apply(rule_tac x=k in exI) by auto qed next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   576
  proof(rule_tac[3-5] ballI allI)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   590
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   596
    using assms(1-2,4-5) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   597
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   604
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   613
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   620
    have a:"a = a1" apply(rule ksimplex_fix_plane_0[OF assms(2,4-5)]) using exta(1-2,5) by auto moreover
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   624
    hence "b1 = a1" apply-apply(rule ext) unfolding exta(5) extb(5) by auto ultimately
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   625
    show "s' = s" apply-apply(rule *[of _ a1 b1]) using exta(1-2) extb(1-2) goal1 by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   626
  show ?thesis unfolding card_1_exists apply-apply(rule ex1I[of _ s])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   628
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   635
    have a:"a = a0" apply(rule ksimplex_fix_plane_p[OF assms(1-2,4-5) exta(1,2)]) unfolding exta by auto moreover
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
hoelzl
parents:
diff changeset
   637
    have a':"a' = b0" apply(rule ksimplex_fix_plane_p[OF goal1(1-2) assms(4), of _ b1]) unfolding goal1 extb using extb(1,2) assms(5) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   640
	using *[THEN cong, of x x] unfolding exta extb apply-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 HOL-light)
hoelzl
parents:
diff changeset
   641
    ultimately show "s' = s" apply-apply(rule lem[OF goal1(3) _ goal1(2) assms(2)]) by auto qed 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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(1-2) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   644
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   653
  { assume "a=a0"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   659
      apply(rule_tac *[OF ksimplex_successor[OF assms(1-2),unfolded `a=a0`]]) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   670
      ultimately have "x = a0 \<or> x = a2" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   673
      show "card ?s = n + 1" using ksimplexD(2-3)[OF assms(1)]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   693
          apply-apply(erule exE,erule conjE) . note kk=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   709
    hence "insert a3 (s - {a0}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   717
	hence "kle n a2 x" apply-apply(rule lem3) using `a=a0` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   730
	hence "a' = a" unfolding True `a=a0` apply-apply(subst expand_fun_eq,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   733
	hence "s' = s" apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   739
	    unfolding as using min_max(1-3) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   743
	hence "a' = a3" unfolding as `a=a0` apply-apply(subst expand_fun_eq,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   748
	hence "s' = insert a3 (s - {a0})" apply-apply(rule lem1) defer apply assumption
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   754
  { assume "a=a1"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   760
      apply(rule_tac *[OF ksimplex_predecessor[OF assms(1-2),unfolded `a=a1`]]) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   767
      ultimately have "x = a2 \<or> x = a1" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   781
      show "card ?s = n+1" using ksimplexD(2-3)[OF assms(1)]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   800
            apply(rule)defer proof(rule) case goal1 thus ?case apply-apply(erule conjE)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   803
        ultimately show ?case apply-apply(rule kle_between_l[of _ a0 _ a2]) using lem3[OF *]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   812
    hence "insert a3 (s - {a1}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   820
	hence "kle n x a2" apply-apply(rule lem3) using `a=a1` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   831
	hence "a_max = a2" unfolding kle_antisym[THEN sym,of a_max a2 n] apply-apply(rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   835
	  using k(2) unfolding lem4[rule_format] a0a1(5)[rule_format] apply-apply(rule,erule_tac x=j in allE)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   837
	have "\<forall>i. a_min i = a3 i" using a_max apply-apply(rule,erule_tac x=i in allE)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   845
	  thus "a_min \<in> s" by auto have "a0 \<in> s - {a1}" using a0a1(1-3) by auto thus "a0 \<in> s'"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   848
	hence "s' = s" apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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(1-2)] by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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(1-2)] by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   866
	unfolding l(2) k(2) `k=l` apply-apply(erule disjE)apply(erule_tac[!] exE conjE)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   868
    hence aa':"a'\<noteq>a" apply-apply 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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   884
	using goal1(1)[THEN kle_imp_pointwise] apply-apply 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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   889
	using goal1(1)[THEN kle_imp_pointwise] apply-apply 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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   894
	using goal1(1)[THEN kle_imp_pointwise] apply-apply 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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   899
	using goal1(1)[THEN kle_imp_pointwise] apply-apply 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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   913
      show "card (insert a' (s - {a})) = n + 1" using ksimplexD(2-3)[OF assms(1)]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   934
    hence "insert a' (s - {a}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   952
	have "\<not> kle n w u" apply-apply(rule,drule kle_imp_pointwise) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   961
	case True hence "s' = s" apply-apply(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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   963
	hence "s' = insert a' (s - {a})" apply-apply(rule lem1[of _ a'' _ a'])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   970
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   972
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   991
	apply-apply(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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   993
	apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
   996
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   997
subsection {* Reduced labelling. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   998
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1001
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1012
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1017
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1020
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1023
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1028
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1034
  assume ?l (is "?as \<and> (?a \<or> ?b)") thus ?r apply-apply(rule,erule conjE,assumption) proof(cases ?a)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
hoelzl
parents:
diff changeset
  1036
      have "reduced lab (n+1) x \<noteq> j - 1" using j apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1040
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1042
      have "reduced lab (n+1) x < j" using j apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1047
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1049
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1050
lemma kuhn_induction:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1060
      using assms(2-3) using as(1)[unfolded ksimplex_def] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
hoelzl
parents:
diff changeset
  1062
    { fix x assume "x\<in>f" hence "reduced lab (n + 1) x < n + 1" apply-apply(rule reduced_labelling_1)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1064
      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1072
    then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1075
      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1076
	using reduced_labelling(1) by auto }
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1077
    thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[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 HOL-light)
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 HOL-light)
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" apply-apply(rule reduced_labelling_0) apply assumption
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1087
	  hence "lab x j = 1" apply-apply(rule assms(3)[rule_format,OF j(1)]) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1092
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1099
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1101
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1106
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1108
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1109
lemma kuhn_combinatorial:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1116
  thus ?case apply-apply(rule kuhn_induction_Suc) using Suc(2-) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1117
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1129
  hence "card ?A \<noteq> 0" apply-apply(rule ccontr) by auto hence "?A \<noteq> {}" unfolding card_eq_0_iff by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1133
    hence "a i + 1 \<le> p" apply-apply(rule order_trans[of _ "b i"]) apply(subst ab(5)[THEN spec[where x=i]])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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[1-2] ballI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1145
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1147
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1163
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1164
lemma brouwer_cube: fixes f::"real^'n::finite \<Rightarrow> real^'n::finite"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
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 HOL-light)
hoelzl
parents:
diff changeset
  1172
    using assms(2)[unfolded image_subset_iff Ball_def] unfolding mem_interval by auto