src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
author wenzelm
Sat, 07 Apr 2012 16:41:59 +0200
changeset 47389 e8552cba702d
parent 44890 22f665a2e91c
child 49374 b08c6312782b
permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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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   Translation from HOL light: Robert Himmelmann, TU Muenchen *)
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header {* Results connected with topological dimension. *}
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theory Brouwer_Fixpoint
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  imports Convex_Euclidean_Space
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begin
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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::_::euclidean_space)))"
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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
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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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lemma kuhn_labelling_lemma: fixes type::"'a::euclidean_space"
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  assumes "(\<forall>x::'a. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i<DIM('a). Q i \<longrightarrow> 0 \<le> x$$i \<and> x$$i \<le> 1)"
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  shows "\<exists>l. (\<forall>x.\<forall> i<DIM('a). l x i \<le> (1::nat)) \<and>
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             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (x$$i = 0) \<longrightarrow> (l x i = 0)) \<and>
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             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (x$$i = 1) \<longrightarrow> (l x i = 1)) \<and>
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             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$$i \<le> f(x)$$i) \<and>
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             (\<forall>x.\<forall> i<DIM('a). P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$$i \<le> x$$i)" proof-
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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
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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" "xa<DIM('a)" 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 "xa<DIM('a) \<Longrightarrow> ?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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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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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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  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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    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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    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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  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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    75
    apply(rule_tac[!] setsum_multicount) using assms by auto
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    76
  have lem3:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices =
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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    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. 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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    79
    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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    80
  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}"
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    82
    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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  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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    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
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    85
           {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 0)} +
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
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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)
hoelzl
parents:
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    87
           {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 2)}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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    88
    apply(rule setsum_Un_disjoint') using assms(2,6) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
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    89
  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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parents:
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    90
    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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    91
    int (card {s \<in> simplices. \<not> compo s \<and> card {f \<in> faces. face f s \<and> compo' f} = 2} * 2)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
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    92
    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
    93
  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
    94
  have odd_minus_even:"\<And>x y. odd x \<Longrightarrow> even (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
35729
3cd1e4b65111 replaced card_def by card_eq_setsum
haftmann
parents: 34964
diff changeset
    95
  show ?thesis unfolding even_nat_def unfolding card_eq_setsum and lem4[THEN sym] and *[unfolded card_eq_setsum]
44821
a92f65e174cf avoid using legacy theorem names
huffman
parents: 44457
diff changeset
    96
    unfolding card_eq_setsum[THEN sym] apply (rule odd_minus_even) unfolding of_nat_add apply(rule odd_plus_even)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
    97
    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
    98
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
    99
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
   100
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   101
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
   102
  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
   103
  fix x assume as:"x \<in> s" "\<forall>y. y \<in> s \<longrightarrow> y = x"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   104
  have *:"s = insert x {}" apply- apply(rule set_eqI,rule) unfolding singleton_iff
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   105
    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
   106
  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
   107
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   108
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
   109
  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
   110
  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
   111
  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
   112
  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
   113
  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
   114
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   115
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
   116
  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
   117
  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
   118
  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
   119
    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
   120
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   121
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
   122
  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
   123
  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
   124
  have inj:"inj_on f s" apply(rule eq_card_imp_inj_on) using assms(1-4) by auto
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   125
  have *:"{a \<in> s. f ` (s - {a}) = t - {b}} = {a}" apply(rule set_eqI) unfolding singleton_iff
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   126
    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
   127
  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
   128
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   129
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
   130
  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
   131
         (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}} = {}")
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36340
diff changeset
   132
  case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   133
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
   134
  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
   135
  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
   136
  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
   137
  then obtain c where c:"c \<in> s" "a \<noteq> c" "f a = f c" by auto
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   138
  hence *:"f ` (s - {c}) = f ` (s - {a})" apply-apply(rule set_eqI,rule) proof-
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   139
    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
   140
    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
   141
  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
   142
  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
   143
    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
   144
    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
   145
    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
   146
    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
   147
      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]]
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   148
        using `a\<in>s` `c\<in>s` `f a = f c` `a\<noteq>c` by auto qed qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   149
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   150
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
   151
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   152
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
   153
  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
   154
  "\<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
   155
  "\<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
   156
  "\<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
   157
  "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
   158
  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
   159
  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
   160
  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
   161
  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
   162
  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
   163
    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
   164
  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
   165
    (\<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
   166
  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
   167
    have "{0..n + 1} - {n + 1} = {0..n}" by auto
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   168
    hence S:"?S = {s'. \<exists>a\<in>s. s' = s - {a} \<and> rl ` s' = {0..n + 1} - {n + 1}}" apply- apply(rule set_eqI)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   169
      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
   170
    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
   171
      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
   172
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   173
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
   174
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   175
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
   176
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   177
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
   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_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
   180
  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
   181
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   182
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
   183
  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
   184
  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
   185
  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
   186
  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
   187
    "\<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
   188
        \<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
   189
    "\<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
   190
  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
   191
    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
   192
    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
   193
      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
   194
    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
   195
      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
   196
      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
   197
      assume "\<forall>j. x j \<le> a j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using a apply -
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   198
        apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed moreover
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   199
    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
   200
      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
   201
      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
   202
      assume "\<forall>j. b j \<le> x j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using b apply -
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   203
        apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   204
    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
   205
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   206
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
   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 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
   209
  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
   210
  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
   211
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   212
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
   213
  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
   214
  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
   215
    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
   216
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   217
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
   218
  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
   219
  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
   220
  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
   221
  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
   222
    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
   223
    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
   224
    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
   225
    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
   226
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   227
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
   228
  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
   229
  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
   230
    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
   231
  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
   232
    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
   233
      assume "kle n x a" hence "x = a" apply- unfolding pointwise_antisym[THEN sym]
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   234
        apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   235
      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
   236
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   237
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
   238
  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
   239
  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
   240
    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
   241
  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
   242
    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
   243
      assume "kle n a x" hence "x = a" apply- unfolding pointwise_antisym[THEN sym]
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   244
        apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   245
      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
   246
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   247
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
   248
  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
   249
  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
   250
  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
   251
  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
   252
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   253
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
   254
  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
   255
  "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
   256
  "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
   257
  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
   258
  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
   259
  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
   260
  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
   261
  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
   262
  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
   263
    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
   264
    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
   265
    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
   266
    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
   267
    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
   268
    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
   269
    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
   270
    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
   271
    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
   272
    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
   273
  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
   274
  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
   275
  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
   276
  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
   277
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   278
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
   279
  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
   280
  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
   281
  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
   282
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   283
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
   284
  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
   285
  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
   286
  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
   287
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   288
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
   289
  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
   290
  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
   291
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
   292
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
   293
  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
   294
  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
   295
  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
   296
    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
   297
    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
   298
      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
   299
    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
   300
      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
   301
    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
   302
      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
   303
    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
   304
    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
   305
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   306
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
   307
  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
   308
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   309
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
   310
  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
   311
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   312
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
   313
  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
   314
  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
   315
  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
   316
    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
   317
      case True hence "c i \<ge> a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   318
        unfolding kk1[THEN conjunct2,rule_format,of i] kk2[THEN conjunct2,rule_format,of i] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   319
      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
   320
      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
   321
      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
   322
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   323
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
   324
  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
   325
  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
   326
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   327
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
   328
  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
   329
  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
   330
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   331
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
   332
  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
   333
  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
   334
  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
   335
    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
   336
      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
   337
    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
   338
  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
   339
    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
   340
      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
   341
    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
   342
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   343
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
   344
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   345
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
   346
        (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
   347
        (\<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
   348
        (\<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
   349
        (\<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
   350
36318
3567d0571932 eliminated spurious schematic statements;
wenzelm
parents: 35729
diff changeset
   351
lemma ksimplexI:"card s = n + 1 \<Longrightarrow>  \<forall>x\<in>s. \<forall>j. x j \<le> p \<Longrightarrow> \<forall>x\<in>s. \<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x \<Longrightarrow> ksimplex p n s"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   352
  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
   353
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   354
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
   355
        (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
   356
        (\<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
   357
        (\<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
   358
        (\<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
   359
  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
   360
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   361
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
   362
  "\<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
   363
  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
   364
    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
   365
  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
   366
  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
   367
  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
   368
  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
   369
  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
   370
  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
   371
    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
   372
  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
   373
  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
   374
  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
   375
  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
   376
  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
   377
    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
   378
    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
   379
      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
   380
      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
   381
      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
   382
      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
   383
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   384
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
   385
  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
   386
  "\<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
   387
  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
   388
    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
   389
  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
   390
  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
   391
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   392
lemma ksimplexD:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   393
  assumes "ksimplex p n s"
36318
3567d0571932 eliminated spurious schematic statements;
wenzelm
parents: 35729
diff changeset
   394
  shows "card s = n + 1" "finite s" "card s = n + 1" "\<forall>x\<in>s. \<forall>j. x j \<le> p" "\<forall>x\<in>s. \<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   395
  "\<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
   396
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   397
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
   398
  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
   399
  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
   400
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
   401
  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
   402
    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
   403
  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
   404
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   405
  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
   406
  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
   407
  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
   408
    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
   409
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   410
  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
   411
  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
   412
  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
   413
  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
   414
    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
   415
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   416
  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
   417
    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
   418
    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
   419
    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
   420
    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
   421
    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
   422
    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
   423
      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
   424
    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
   425
      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
   426
    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
   427
      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
   428
    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
   429
      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
   430
    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
   431
    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
   432
  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
   433
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   434
  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
   435
    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
   436
    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
   437
    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
   438
    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
   439
      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
   440
      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
   441
      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
   442
      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
   443
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   444
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
   445
  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
   446
  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
   447
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
   448
  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
   449
    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
   450
  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
   451
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   452
  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
   453
  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
   454
  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
   455
    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
   456
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   457
  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
   458
  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
   459
  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
   460
  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
   461
    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
   462
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   463
  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
   464
    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
   465
    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
   466
    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
   467
    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
   468
    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
   469
    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
   470
      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
   471
    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
   472
      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
   473
    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
   474
      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
   475
    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
   476
      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
   477
    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
   478
    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
   479
  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
   480
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   481
  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
   482
    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
   483
    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
   484
    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
   485
    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
   486
      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
   487
      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
   488
      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
   489
      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
   490
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   491
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
   492
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   493
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
   494
  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
   495
  using assms apply - proof(induct m arbitrary: s)
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   496
  have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" apply(rule set_eqI,rule)unfolding mem_Collect_eq apply(rule,rule ext) by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   497
  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
   498
  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
   499
    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
   500
  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
   501
  let ?l = "(\<lambda>(b,g) x. if x = a then b else g x)" have *:"?M (insert a s0) = ?l ` {(b,g). b\<in>t \<and> g\<in>?M s0}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   502
    apply(rule set_eqI,rule) unfolding mem_Collect_eq image_iff apply(erule conjE)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   503
    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
   504
    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
   505
    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
   506
      "\<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
   507
  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
   508
    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
   509
    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
   510
    moreover have "ya = yb" proof(rule ext) fix x show "ya x = yb x" proof(cases "x = a") 
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   511
        case False thus ?thesis using as(1)[THEN cong[of _ _ x x]] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   512
        case True thus ?thesis using as(5,7) using as0(2) by auto qed qed 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   513
    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
   514
  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
   515
  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
   516
    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
   517
    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
   518
    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
   519
qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   520
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   521
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
   522
  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
   523
  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
   524
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   525
lemma 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
   526
  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
   527
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
   528
  case True
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   529
  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
   530
    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
   531
  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
   532
next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   533
  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
   534
  show ?thesis
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   535
  proof (cases "s = {}")
44457
d366fa5551ef declare euclidean_simps [simp] at the point they are proved;
huffman
parents: 44282
diff changeset
   536
    have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   537
    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
   538
  next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   539
    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
   540
  qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   541
qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   542
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   543
lemma 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
   544
  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
   545
  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
   546
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   547
lemma 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
   548
  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
   549
  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
   550
  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
   551
    fix x y assume as:"x \<in>s - {a}" "y \<in>s - {a}" have xyp:"x (n + 1) = y (n + 1)"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   552
        using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   553
        using as(2)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   554
    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
   555
      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
   556
      have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   557
        fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   558
        thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   559
      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
   560
      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
   561
      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
   562
      hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply-apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   563
        fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   564
        thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   565
      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
   566
    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
   567
    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
   568
      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
   569
  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
   570
  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
   571
  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
   572
  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
   573
    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
   574
  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
   575
    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
   576
      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
   577
    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
   578
    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
   579
    { fix z assume z:"z \<in> insert c f" hence "kle (n + 1) c z" apply(cases "z = c") (*defer apply(rule kle_Suc)*) proof-
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   580
        case False hence "z\<in>f" using z by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   581
        then guess k apply(drule_tac ab(3)[THEN bspec[where x=z], THEN conjunct1]) unfolding kle_def apply(erule exE) .
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   582
        thus "kle (n + 1) c z" unfolding kle_def apply(rule_tac x="insert (n + 1) k" in exI) unfolding c_def
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   583
          using ab using rs(5)[rule_format,OF ab(1),of "n + 1"] assms(1) by auto qed auto } note * = this
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   584
    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
   585
      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
   586
      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
   587
  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
   588
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   589
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
   590
  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
   591
  "\<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
   592
  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
   593
  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
   594
    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
   595
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   596
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
   597
  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
   598
  "\<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
   599
  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
   600
  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
   601
  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
   602
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   603
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
   604
  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
   605
  "\<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
   606
  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
   607
  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
   608
  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
   609
  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
   610
    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
   611
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   612
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
   613
  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
   614
  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
   615
  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
   616
  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
   617
    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
   618
    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
   619
    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
   620
    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
   621
    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
   622
    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
   623
    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
   624
  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
   625
    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
   626
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   627
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
   628
  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
   629
  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
   630
  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
   631
  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
   632
    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
   633
    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
   634
    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
   635
    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
   636
    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
   637
    have "a0 = b0" apply(rule ext) proof- case goal1 show "a0 x = b0 x"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   638
        using *[THEN cong, of x x] unfolding exta extb apply-apply(cases "x\<in>{1..n}") by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   639
    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
   640
  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
   641
    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
   642
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   643
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
   644
  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
   645
  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
   646
  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
   647
  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
   648
    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
   649
    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
   650
  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
   651
  { 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
   652
    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
   653
    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
   654
      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
   655
        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
   656
    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
   657
      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
   658
    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
   659
    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
   660
    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
   661
      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
   662
        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
   663
    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
   664
    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
   665
    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
   666
      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
   667
      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
   668
      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
   669
      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
   670
    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
   671
      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
   672
        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
   673
      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
   674
      show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   675
        fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   676
        fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   677
          case False thus "a3 j \<le>p" unfolding True a3_def using `a1\<in>s` ksimplexD(4)[OF assms(1)] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   678
          guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   679
          have "a2 k \<le> a4 k" using lem3[OF a4(1)[unfolded `a=a0`],THEN kle_imp_pointwise] by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   680
          also have "\<dots> < p" using ksimplexD(4)[OF assms(1),rule_format,of a4 k] using a4 by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   681
          finally have *:"a0 k + 1 < p" unfolding k(2)[rule_format] by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   682
          case True thus "a3 j \<le>p" unfolding a3_def unfolding a0a1(5)[rule_format]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   683
            using k(1) k(2)assms(5) using * by simp qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   684
      show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j\<notin>{1..n}"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   685
        { case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   686
        case True show "x j = p" unfolding True a3_def using j k(1) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   687
          using ksimplexD(5)[OF assms(1),rule_format,OF `a1\<in>s` j] by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   688
      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
   689
      have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a0 \<Longrightarrow> kle n x a3" proof- case goal1
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   690
        guess kk using a0a1(4)[rule_format,OF `x\<in>s`,THEN conjunct2,unfolded kle_def] 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   691
          apply-apply(erule exE,erule conjE) . note kk=this
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   692
        have "k\<notin>kk" proof assume "k\<in>kk"
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   693
          hence "a1 k = x k + 1" using kk by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   694
          hence "a0 k = x k" unfolding a0a1(5)[rule_format] using k(1) by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   695
          hence "a2 k = x k + 1" unfolding k(2)[rule_format] by auto moreover
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   696
          have "a2 k \<le> x k" using lem3[of x,THEN kle_imp_pointwise] goal1 by auto 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   697
          ultimately show False by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   698
        thus ?case unfolding kle_def apply(rule_tac x="insert k kk" in exI) using kk(1)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   699
          unfolding a3_def kle_def kk(2)[rule_format] using k(1) by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   700
      show "kle n x y \<or> kle n y x" proof(cases "y=a3")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   701
        case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI1,rule lem4)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   702
          using x by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   703
        case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   704
            apply(rule disjI2,rule lem4) using y False by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   705
          case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   706
            using x y `y\<noteq>a3` by auto qed qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   707
    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
   708
      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
   709
    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
   710
    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
   711
      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
   712
      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
   713
      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
   714
      have *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   715
        hence "kle n a2 x" apply-apply(rule lem3) using `a=a0` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   716
        hence "a2 k \<le> x k" apply(drule_tac kle_imp_pointwise) by auto moreover
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   717
        have "x k \<le> a2 k" unfolding k(2)[rule_format] using a0a1(4)[rule_format,of x,THEN conjunct1] 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   718
          unfolding kle_def using x by auto ultimately show "x k = a2 k" by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   719
      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
   720
      show "s' \<in> {s, insert a3 (s - {a0})}" proof(cases "a'=a_min")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   721
        case True have "a_max = a1" unfolding kle_antisym[THEN sym,of a_max a1 n] apply(rule)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   722
          apply(rule a0a1(4)[rule_format,THEN conjunct2]) defer  proof(rule min_max(4)[rule_format,THEN conjunct2])
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   723
          show "a1\<in>s'" using a' unfolding `a=a0` using a0a1 by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   724
          show "a_max \<in> s" proof(rule ccontr) assume "a_max\<notin>s"
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   725
            hence "a_max = a'" using a' min_max by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   726
            thus False unfolding True using min_max by auto qed qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   727
        hence "\<forall>i. a_max i = a1 i" by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   728
        hence "a' = a" unfolding True `a=a0` apply-apply(subst fun_eq_iff,rule)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   729
          apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   730
        proof- case goal1 thus ?case apply(cases "x\<in>{1..n}") by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   731
        hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   732
        thus ?thesis by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   733
        case False hence as:"a' = a_max" using ** by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   734
        have "a_min = a2" unfolding kle_antisym[THEN sym, of _ _ n] apply rule
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   735
          apply(rule min_max(4)[rule_format,THEN conjunct1]) defer proof(rule lem3)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   736
          show "a_min \<in> s - {a0}" unfolding a'(2)[THEN sym,unfolded `a=a0`] 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   737
            unfolding as using min_max(1-3) by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   738
          have "a2 \<noteq> a" unfolding `a=a0` using k(2)[rule_format,of k] by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   739
          hence "a2 \<in> s - {a}" using a2 by auto thus "a2 \<in> s'" unfolding a'(2)[THEN sym] by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   740
        hence "\<forall>i. a_min i = a2 i" by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   741
        hence "a' = a3" unfolding as `a=a0` apply-apply(subst fun_eq_iff,rule)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   742
          apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   743
          unfolding a3_def k(2)[rule_format] unfolding a0a1(5)[rule_format] proof- case goal1
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   744
          show ?case unfolding goal1 apply(cases "x\<in>{1..n}") defer apply(cases "x=k")
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   745
            using `k\<in>{1..n}` by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   746
        hence "s' = insert a3 (s - {a0})" apply-apply(rule lem1) defer apply assumption
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   747
          apply(rule a'(1)) unfolding a' `a=a0` using `a3\<notin>s` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   748
        thus ?thesis by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   749
    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
   750
    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
   751
    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
   752
  { 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
   753
    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
   754
    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
   755
      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
   756
        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
   757
    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
   758
      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
   759
    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
   760
    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
   761
    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
   762
    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
   763
      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
   764
      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
   765
      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
   766
      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
   767
    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
   768
      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
   769
      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
   770
      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
   771
      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
   772
      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
   773
    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
   774
    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
   775
      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
   776
    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
   777
    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
   778
    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
   779
      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
   780
        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
   781
      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
   782
      show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   783
        fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   784
        fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   785
          case False thus "a3 j \<le>p" unfolding True a3_def using `a0\<in>s` ksimplexD(4)[OF assms(1)] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   786
          guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   787
          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
   788
          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
   789
          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
   790
      show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j\<notin>{1..n}"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   791
        { case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   792
        case True show "x j = p" unfolding True a3_def using j k(1) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   793
          using ksimplexD(5)[OF assms(1),rule_format,OF `a0\<in>s` j] by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   794
      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
   795
      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
   796
        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
   797
          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
   798
            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
   799
              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
   800
        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
   801
        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
   802
          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
   803
      show "kle n x y \<or> kle n y x" proof(cases "y=a3")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   804
        case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI2,rule lem4)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   805
          using x by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   806
        case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   807
            apply(rule disjI1,rule lem4) using y False by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   808
          case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   809
            using x y `y\<noteq>a3` by auto qed qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   810
    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
   811
      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
   812
    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
   813
    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
   814
      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
   815
      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
   816
      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
   817
      have *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   818
        hence "kle n x a2" apply-apply(rule lem3) using `a=a1` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   819
        hence "x k \<le> a2 k" apply(drule_tac kle_imp_pointwise) by auto moreover
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   820
        { have "a2 k \<le> a0 k" using k(2)[rule_format,of k] unfolding a0a1(5)[rule_format] using k(1) by simp
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   821
          also have "\<dots> \<le> x k" using a0a1(4)[rule_format,of x,THEN conjunct1,THEN kle_imp_pointwise] x by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   822
          finally have "a2 k \<le> x k" . } ultimately show "x k = a2 k" by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   823
      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
   824
      have "a2 \<noteq> a1" proof assume as:"a2 = a1"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   825
        show False using k(2) unfolding as apply(erule_tac x=k in allE) by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   826
      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
   827
      show "s' \<in> {s, insert a3 (s - {a1})}" proof(cases "a'=a_min")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   828
        case True have "a_max \<in> s - {a1}" using min_max unfolding a'(2)[unfolded `a=a1`,THEN sym] True by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   829
        hence "a_max = a2" unfolding kle_antisym[THEN sym,of a_max a2 n] apply-apply(rule)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   830
          apply(rule lem3,assumption) apply(rule min_max(4)[rule_format,THEN conjunct2]) using a2' by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   831
        hence a_max:"\<forall>i. a_max i = a2 i" by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   832
        have *:"\<forall>j. a2 j = (if j\<in>{1..n} then a3 j + 1 else a3 j)" 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   833
          using k(2) unfolding lem4[rule_format] a0a1(5)[rule_format] apply-apply(rule,erule_tac x=j in allE)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   834
        proof- case goal1 thus ?case apply(cases "j\<in>{1..n}",case_tac[!] "j=k") by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   835
        have "\<forall>i. a_min i = a3 i" using a_max apply-apply(rule,erule_tac x=i in allE)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   836
          unfolding min_max(5)[rule_format] *[rule_format] proof- case goal1
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   837
          thus ?case apply(cases "i\<in>{1..n}") by auto qed hence "a_min = a3" unfolding fun_eq_iff .
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   838
        hence "s' = insert a3 (s - {a1})" using a' unfolding `a=a1` True by auto thus ?thesis by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   839
        case False hence as:"a'=a_max" using ** by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   840
        have "a_min = a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   841
          apply(rule min_max(4)[rule_format,THEN conjunct1]) defer apply(rule a0a1(4)[rule_format,THEN conjunct1]) proof-
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   842
          have "a_min \<in> s - {a1}" using min_max(1,3) unfolding a'(2)[THEN sym,unfolded `a=a1`] as by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   843
          thus "a_min \<in> s" by auto have "a0 \<in> s - {a1}" using a0a1(1-3) by auto thus "a0 \<in> s'"
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   844
            unfolding a'(2)[THEN sym,unfolded `a=a1`] by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   845
        hence "\<forall>i. a_max i = a1 i" unfolding a0a1(5)[rule_format] min_max(5)[rule_format] by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   846
        hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` unfolding as `a=a1` unfolding fun_eq_iff by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   847
        thus ?thesis by auto qed qed 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   848
    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
   849
    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
   850
    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
   851
  { 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
   852
      have "a=a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   853
        using goal1 a0a1 assms(2) by auto thus False using as by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   854
    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
   855
    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
   856
    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
   857
      have "a=a1" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   858
        using goal1 a0a1 assms(2) by auto thus False using as by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   859
    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
   860
    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
   861
    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
   862
    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
   863
      thus False using ksimplexD(6)[OF assms(1),rule_format,OF u v] unfolding kle_def
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   864
        unfolding l(2) k(2) `k=l` apply-apply(erule disjE)apply(erule_tac[!] exE conjE)+
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   865
        apply(erule_tac[!] x=l in allE)+ by(auto simp add: *) qed
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   866
    hence aa':"a'\<noteq>a" apply-apply rule unfolding fun_eq_iff unfolding a'_def k(2)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   867
      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
   868
    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
   869
      case goal2 hence "kle n a' a" by auto thus False apply(drule_tac kle_imp_pointwise)
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   870
        apply(erule_tac x=l in allE) unfolding a'_def k(2) using kl by auto next
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   871
      case True thus False apply(drule_tac kle_imp_pointwise)
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   872
        apply(erule_tac x=k in allE) unfolding a'_def k(2) using kl by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   873
    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
   874
      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
   875
      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
   876
      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
   877
    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
   878
    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
   879
      assume as:"x l = u l" "x k = u k"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   880
      have "x = u" unfolding fun_eq_iff
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   881
        using goal1(2)[THEN kle_imp_pointwise,unfolded l(2)] unfolding k(2) apply-
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   882
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   883
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   884
      assume as:"x l \<noteq> u l" "x k = u k"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   885
      have "x = a'" unfolding fun_eq_iff unfolding a'_def
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   886
        using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   887
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   888
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   889
      assume as:"x l = u l" "x k \<noteq> u k"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   890
      have "x = a" unfolding fun_eq_iff
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   891
        using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   892
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   893
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   894
      assume as:"x l \<noteq> u l" "x k \<noteq> u k"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   895
      have "x = v" unfolding fun_eq_iff
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   896
        using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   897
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   898
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as `k\<noteq>l` by auto qed thus ?case by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   899
    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
   900
    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
   901
      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
   902
      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
   903
    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
   904
      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
   905
      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
   906
    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
   907
      proof(cases "kle n v x \<or> kle n x u") case True thus ?thesis using goal1 by(auto intro:lem3 lem4) next
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   908
        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
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   909
        show ?thesis using uxv[OF *] using kle_uv using goal1 by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   910
    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
   911
      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
   912
        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
   913
      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
   914
      show "\<forall>j. x j \<le> p" proof(rule,cases "x = a'")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   915
        fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   916
        fix j case True show "x j\<le>p" unfolding True proof(cases "j=l") 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   917
          case False thus "a' j \<le>p" unfolding True a'_def using `u\<in>s` ksimplexD(4)[OF assms(1)] by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   918
          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
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   919
          have "u l + 1 \<le> p" unfolding *[THEN sym] using ksimplexD(4)[OF assms(1)] using `v\<in>s` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   920
          thus "a' j \<le>p" unfolding a'_def True by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   921
      show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a'") fix j::nat assume j:"j\<notin>{1..n}"
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   922
        { case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   923
        case True show "x j = p" unfolding True a'_def using j l(1) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   924
          using ksimplexD(5)[OF assms(1),rule_format,OF `u\<in>s` j] by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   925
      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
   926
      show "kle n x y \<or> kle n y x" proof(cases "y=a'")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   927
        case True show ?thesis unfolding True apply(cases "x=a'") defer apply(rule lem5) using x by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   928
        case False show ?thesis proof(cases "x=a'") case True show ?thesis unfolding True
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   929
            using lem5[of y] using y by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   930
          case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   931
            using x y `y\<noteq>a'` by auto qed qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   932
    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
   933
      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
   934
    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
   935
    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
   936
      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
   937
      from this(2) guess a'' .. note a''=this
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   938
      have "u\<noteq>v" unfolding fun_eq_iff unfolding l(2) k(2) by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   939
      hence uv':"\<not> kle n v u" using uv using kle_antisym by auto
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   940
      have "u\<noteq>a" "v\<noteq>a" unfolding fun_eq_iff k(2) l(2) by auto 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   941
      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
   942
      have lem6:"a \<in> s' \<or> a' \<in> s'" proof(cases "\<forall>x\<in>s'. kle n x u \<or> kle n v x")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   943
        case False then guess w unfolding ball_simps .. note w=this
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   944
        hence "kle n u w" "kle n w v" using ksimplexD(6)[OF as] uvs' by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   945
        hence "w = a' \<or> w = a" using uxv[of w] uvs' w by auto thus ?thesis using w by auto next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   946
        case True have "\<not> (\<forall>x\<in>s'. kle n x u)" unfolding ball_simps apply(rule_tac x=v in bexI)
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   947
          using uv `u\<noteq>v` unfolding kle_antisym[of n u v,THEN sym] using `v\<in>s'` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   948
        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
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   949
        then guess w .. note w=this from this(2) guess kk .. note kk=this[rule_format]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   950
        have "\<not> kle n w u" apply-apply(rule,drule kle_imp_pointwise) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   951
          apply(erule_tac x=kk in allE) unfolding kk by auto 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   952
        hence *:"kle n v w" using True[rule_format,OF w(1)] by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   953
        hence False proof(cases "kk\<noteq>l") case True thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   954
            apply(erule_tac x=l in allE) using `k\<noteq>l` by auto  next
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   955
          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)]
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   956
            apply(erule_tac x=k in allE) using `k\<noteq>l` by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   957
        thus ?thesis by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   958
      thus "s' \<in> {s, insert a' (s - {a})}" proof(cases "a\<in>s'")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   959
        case True hence "s' = s" apply-apply(rule lem1[OF a''(2)]) using a'' `a\<in>s` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   960
        thus ?thesis by auto next case False hence "a'\<in>s'" using lem6 by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   961
        hence "s' = insert a' (s - {a})" apply-apply(rule lem1[of _ a'' _ a'])
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   962
          unfolding a''(2)[THEN sym] using a'' using `a'\<notin>s` by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   963
        thus ?thesis by auto qed qed 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   964
    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
   965
    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
   966
    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
   967
  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
   968
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   969
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
   970
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   971
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
   972
  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
   973
  "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
   974
  (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
   975
  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
   976
  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
   977
  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
   978
                (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
   979
               ((\<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
   980
                (\<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
   981
  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
   982
    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
   983
    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
   984
    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
   985
    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
   986
    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
   987
    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
   988
    { fix j assume j:"j \<in> {1..n + 1}" "\<forall>x\<in>f. x j = 0" thus "card {s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})} = 1" unfolding S
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   989
        apply-apply(rule ksimplex_replace_0) apply(rule as)+ unfolding as by auto }
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   990
    { fix j assume j:"j \<in> {1..n + 1}" "\<forall>x\<in>f. x j = p" thus "card {s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})} = 1" unfolding S
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
   991
        apply-apply(rule ksimplex_replace_1) apply(rule as)+ unfolding as by auto }
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   992
    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
   993
      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
   994
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   995
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
   996
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   997
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
   998
  (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
   999
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1000
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
  1001
  "\<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
  1002
  "(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
  1003
  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
  1004
    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
  1005
  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
  1006
  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
  1007
  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
  1008
    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
  1009
  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
  1010
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1011
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
  1012
  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
  1013
  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
  1014
  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
  1015
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1016
lemma reduced_labelling_0: assumes "j\<in>{1..n}" "label x j = 0" shows "reduced label n x \<noteq> j - 1"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44821
diff changeset
  1017
  using reduced_labelling[of label n x] using assms by fastforce 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1018
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1019
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
  1020
  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
  1021
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1022
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
  1023
  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
  1024
  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
  1025
  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
  1026
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1027
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
  1028
  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
  1029
          "\<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
  1030
  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
  1031
  ((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
  1032
  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
  1033
    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
  1034
      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
  1035
    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
  1036
    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
  1037
    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
  1038
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1039
    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
  1040
      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
  1041
    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
  1042
      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
  1043
      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
  1044
    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
  1045
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1046
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
  1047
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1048
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
  1049
  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
  1050
                  "\<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
  1051
        "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
  1052
  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
  1053
  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
  1054
  show ?thesis apply(rule kuhn_simplex_lemma[unfolded mem_Collect_eq]) apply(rule,rule,rule *,rule reduced_labelling)
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1055
    apply(rule *(1)[OF assms(4)]) apply(rule set_eqI) unfolding mem_Collect_eq apply(rule,erule conjE) defer apply(rule) proof-(*(rule,rule)*)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1056
    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
  1057
    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
  1058
      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
  1059
    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
  1060
    { fix x assume "x\<in>f" hence "reduced lab (n + 1) x < n + 1" apply-apply(rule reduced_labelling_1)
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1061
        defer using assms(3) using as(1)[unfolded ksimplex_def] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1062
      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc) using reduced_labelling(1) by auto }
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1063
    hence "reduced lab (n + 1) ` f = {0..n}" unfolding as(2)[THEN sym] apply- apply(rule set_eqI) unfolding image_iff by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1064
    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
  1065
    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
  1066
      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
  1067
      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
  1068
  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
  1069
      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
  1070
    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
  1071
    { 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
  1072
      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
  1073
      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc)
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1074
        using reduced_labelling(1) by auto }
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1075
    thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[THEN sym] apply-apply(rule set_eqI) unfolding image_iff by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1076
    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
  1077
      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
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1078
        apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1079
      have "j - 1 \<in> {0..n}" using `j\<in>{1..n+1}` by auto
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44821
diff changeset
  1080
      ultimately have False unfolding sa(4)[THEN sym] unfolding image_iff by fastforce thus ?thesis by auto next
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44821
diff changeset
  1081
      case False hence "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p" using sa(5) by fastforce then guess j .. note j=this
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1082
      thus ?thesis proof(cases "j = n+1")
41958
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1083
        case False hence *:"j\<in>{1..n}" using j by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1084
        hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab n x < j" apply(rule reduced_labelling_1) proof- fix x assume "x\<in>f"
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1085
          hence "lab x j = 1" apply-apply(rule assms(3)[rule_format,OF j(1)]) 
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1086
            using sa(1)[unfolded ksimplex_def] using j unfolding sa by auto thus "lab x j \<noteq> 0" by auto qed
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1087
        moreover have "j\<in>{0..n}" using * by auto
5abc60a017e0 eliminated hard tabs;
wenzelm
parents: 39302
diff changeset
  1088
        ultimately have False unfolding part1[THEN sym] using * unfolding image_iff by auto thus ?thesis by auto qed auto qed 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1089
    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
  1090
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1091
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
  1092
  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
  1093
                  "\<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
  1094
        "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
  1095
  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
  1096
  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
  1097
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1098
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
  1099
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1100
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
  1101
  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
  1102
  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
  1103
  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
  1104
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1105
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
  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 kuhn_combinatorial:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1108
  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
  1109
  "\<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
  1110
  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
  1111
  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
  1112
  { 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
  1113
  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
  1114
  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
  1115
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1116
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
  1117
  "\<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
  1118
  "\<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
  1119
  "\<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
  1120
  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
  1121
  "\<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
  1122
                               (\<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
  1123
                               ~(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
  1124
  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
  1125
  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
  1126
  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
  1127
  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
  1128
  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
  1129
  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
  1130
  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
  1131
    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
  1132
      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
  1133
    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
  1134
    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
  1135
    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
  1136
    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
  1137
      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
  1138
        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
  1139
      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
  1140
        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
  1141
        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
  1142
        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
  1143
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1144
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
  1145
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1146
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
  1147
  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
  1148
  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
  1149
             (\<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
  1150
             (\<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
  1151
             (\<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
  1152
             (\<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
  1153
  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
  1154
  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
  1155
  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
  1156
    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x xa = 0 \<longrightarrow> y = (0::nat)) \<and>