src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
author huffman
Wed Sep 07 09:02:58 2011 -0700 (2011-09-07)
changeset 44821 a92f65e174cf
parent 44457 d366fa5551ef
child 44890 22f665a2e91c
permissions -rw-r--r--
avoid using legacy theorem names
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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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    apply(rule_tac[!] setsum_multicount) using assms by auto
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  have lem3:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices =
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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}"
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    apply(rule setsum_Un_disjoint') using assms(2) by auto
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  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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    apply(rule setsum_cong2) using assms(5) by auto
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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} =
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    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
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           {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 0)} +
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    setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
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           {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 2)}"
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    apply(rule setsum_Un_disjoint') using assms(2,6) by auto
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  have *:"int (\<Sum>s\<in>{s \<in> simplices. compo s}. card {f \<in> faces. face f s \<and> compo' f}) =
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    int (card {f \<in> faces. compo' f \<and> bnd f} + 2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}) - 
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    int (card {s \<in> simplices. \<not> compo s \<and> card {f \<in> faces. face f s \<and> compo' f} = 2} * 2)"
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    using lem1[unfolded lem3 lem2 lem5] by auto
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  have even_minus_odd:"\<And>x y. even x \<Longrightarrow> odd (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
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  have odd_minus_even:"\<And>x y. odd x \<Longrightarrow> even (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
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  show ?thesis unfolding even_nat_def unfolding card_eq_setsum and lem4[THEN sym] and *[unfolded card_eq_setsum]
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    unfolding card_eq_setsum[THEN sym] apply (rule odd_minus_even) unfolding of_nat_add apply(rule odd_plus_even)
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    apply(rule assms(7)[unfolded even_nat_def]) unfolding int_mult by auto qed
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subsection {* The odd/even result for faces of complete vertices, generalized. *}
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lemma card_1_exists: "card s = 1 \<longleftrightarrow> (\<exists>!x. x \<in> s)" unfolding One_nat_def
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  apply rule apply(drule card_eq_SucD) defer apply(erule ex1E) proof-
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  fix x assume as:"x \<in> s" "\<forall>y. y \<in> s \<longrightarrow> y = x"
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  have *:"s = insert x {}" apply- apply(rule set_eqI,rule) unfolding singleton_iff
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    apply(rule as(2)[rule_format]) using as(1) by auto
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  show "card s = Suc 0" unfolding * using card_insert by auto qed auto
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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
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  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
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  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
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  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  ..
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  with `x\<in>s` have *:"s = {x, y}" "x\<noteq>y" by auto
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  from this(2) show "card s = 2" unfolding * by auto qed
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lemma image_lemma_0: assumes "card {a\<in>s. f ` (s - {a}) = t - {b}} = n"
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  shows "card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> (f ` s' = t - {b})} = n" proof-
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  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
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  show ?thesis unfolding * unfolding assms[THEN sym] apply(rule card_image) unfolding inj_on_def 
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    apply(rule,rule,rule) unfolding mem_Collect_eq by auto qed
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lemma image_lemma_1: assumes "finite s" "finite t" "card s = card t" "f ` s = t" "b \<in> t"
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  shows "card {s'. \<exists>a\<in>s. s' = s - {a} \<and>  f ` s' = t - {b}} = 1" proof-
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  obtain a where a:"b = f a" "a\<in>s" using assms(4-5) by auto
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  have inj:"inj_on f s" apply(rule eq_card_imp_inj_on) using assms(1-4) by auto
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  have *:"{a \<in> s. f ` (s - {a}) = t - {b}} = {a}" apply(rule set_eqI) unfolding singleton_iff
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    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
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  show ?thesis apply(rule image_lemma_0) unfolding *  by auto qed
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lemma image_lemma_2: assumes "finite s" "finite t" "card s = card t" "f ` s \<subseteq> t" "f ` s \<noteq> t" "b \<in> t"
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  shows "(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 0) \<or>
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         (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}} = {}")
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  case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by auto
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next let ?M = "{a\<in>s. f ` (s - {a}) = t - {b}}"
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  case False then obtain a where "a\<in>?M" by auto hence a:"a\<in>s" "f ` (s - {a}) = t - {b}" by auto
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  have "f a \<in> t - {b}" using a and assms by auto
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  hence "\<exists>c \<in> s - {a}. f a = f c" unfolding image_iff[symmetric] and a by auto
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  then obtain c where c:"c \<in> s" "a \<noteq> c" "f a = f c" by auto
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  hence *:"f ` (s - {c}) = f ` (s - {a})" apply-apply(rule set_eqI,rule) proof-
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    fix x assume "x \<in> f ` (s - {a})" then obtain y where y:"f y = x" "y\<in>s- {a}" by auto
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    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
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  have "c\<in>?M" unfolding mem_Collect_eq and * using a and c(1) by auto
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  show ?thesis apply(rule disjI2, rule image_lemma_0) unfolding card_2_exists
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    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)
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    fix z assume as:"z \<in> s" "f ` (s - {z}) = t - {b}"
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    have inj:"inj_on f (s - {z})" apply(rule eq_card_imp_inj_on) unfolding as using as(1) and assms by auto
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    show "z = a \<or> z = c" proof(rule ccontr)
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      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]]
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        using `a\<in>s` `c\<in>s` `f a = f c` `a\<noteq>c` by auto qed qed qed
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subsection {* Combine this with the basic counting lemma. *}
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lemma kuhn_complete_lemma:
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  assumes "finite simplices"
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  "\<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}"
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  "\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. bnd f  \<longrightarrow> (card {s\<in>simplices. face f s} = 1)"
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  "\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. \<not>bnd f \<longrightarrow> (card {s\<in>simplices. face f s} = 2)"
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  "odd(card {f\<in>{f. \<exists>s\<in>simplices. face f s}. rl ` f = {0..n} \<and> bnd f})"
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  shows "odd (card {s\<in>simplices. (rl ` s = {0..n+1})})" 
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  apply(rule kuhn_counting_lemma) defer apply(rule assms)+ prefer 3 apply(rule assms) proof(rule_tac[1-2] ballI impI)+ 
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  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
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  have **: "\<forall>s\<in>simplices. card s = n + 2 \<and> finite s" using assms(3) by (auto intro: card_ge_0_finite)
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  show "finite {f. \<exists>s\<in>simplices. face f s}" unfolding assms(2)[rule_format] and *
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    apply(rule finite_UN_I[OF assms(1)]) using ** by auto
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  have *:"\<And> P f s. s\<in>simplices \<Longrightarrow> (f \<in> {f. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s - {a}}) \<and>
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    (\<exists>a\<in>s. (f = s - {a})) \<and> P f \<longleftrightarrow> (\<exists>a\<in>s. (f = s - {a}) \<and> P f)" by auto
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  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}}"
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    have "{0..n + 1} - {n + 1} = {0..n}" by auto
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    hence S:"?S = {s'. \<exists>a\<in>s. s' = s - {a} \<and> rl ` s' = {0..n + 1} - {n + 1}}" apply- apply(rule set_eqI)
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      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
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    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
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      apply(rule_tac[!] image_lemma_1 image_lemma_2) using ** assms(4) and s by auto qed
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subsection {*We use the following notion of ordering rather than pointwise indexing. *}
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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)))"
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   177
lemma kle_refl[intro]: "kle n x x" unfolding kle_def by auto
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   178
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   179
lemma kle_antisym: "kle n x y \<and> kle n y x \<longleftrightarrow> (x = y)"
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   180
  unfolding kle_def apply rule apply(rule ext) by auto
hoelzl@33741
   181
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   182
lemma pointwise_minimal_pointwise_maximal: fixes s::"(nat\<Rightarrow>nat) set"
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   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)"
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   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"
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   185
  using assms unfolding atomize_conj apply- proof(induct s rule:finite_induct)
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   186
  fix x and F::"(nat\<Rightarrow>nat) set" assume as:"finite F" "x \<notin> F" 
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   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>
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   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> {}"
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   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)"
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   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={}")
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   191
    case True thus ?thesis apply-apply(rule,rule_tac[!] x=x in bexI) by auto next
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   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
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   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
hoelzl@33741
   194
    have "\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. a j \<le> x j"
hoelzl@33741
   195
      using as(5)[rule_format,OF a(1) insertI1] apply- proof(erule disjE)
hoelzl@33741
   196
      assume "\<forall>j. a j \<le> x j" thus ?thesis apply(rule_tac x=a in bexI) using a by auto next
hoelzl@33741
   197
      assume "\<forall>j. x j \<le> a j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using a apply -
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   198
        apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed moreover
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   199
    have "\<exists>b\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. x j \<le> b j"
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   200
      using as(5)[rule_format,OF b(1) insertI1] apply- proof(erule disjE)
hoelzl@33741
   201
      assume "\<forall>j. x j \<le> b j" thus ?thesis apply(rule_tac x=b in bexI) using b by auto next
hoelzl@33741
   202
      assume "\<forall>j. b j \<le> x j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using b apply -
wenzelm@41958
   203
        apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed
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   204
    ultimately show  ?thesis by auto qed qed(auto)
hoelzl@33741
   205
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   206
lemma kle_imp_pointwise: "kle n x y \<Longrightarrow> (\<forall>j. x j \<le> y j)" unfolding kle_def by auto
hoelzl@33741
   207
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   208
lemma pointwise_antisym: fixes x::"nat \<Rightarrow> nat"
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   209
  shows "(\<forall>j. x j \<le> y j) \<and> (\<forall>j. y j \<le> x j) \<longleftrightarrow> (x = y)"
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   210
  apply(rule, rule ext,erule conjE) apply(erule_tac x=xa in allE)+ by auto
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   211
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   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"
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   213
  using assms apply- apply(erule disjE) apply assumption proof- case goal1
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   214
  hence "x=z" apply- apply(rule ext) apply(drule kle_imp_pointwise)+ 
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   215
    apply(erule_tac x=xa in allE)+ by auto thus ?case by auto qed
hoelzl@33741
   216
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   217
lemma kle_strict: assumes "kle n x y" "x \<noteq> y"
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   218
  shows "\<forall>j. x j \<le> y j"  "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> x(k) < y(k)"
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   219
  apply(rule kle_imp_pointwise[OF assms(1)]) proof-
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   220
  guess k using assms(1)[unfolded kle_def] .. note k = this
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   221
  show "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> x(k) < y(k)" proof(cases "k={}")
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   222
    case True hence "x=y" apply-apply(rule ext) using k by auto
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   223
    thus ?thesis using assms(2) by auto next
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   224
    case False hence "(SOME k'. k' \<in> k) \<in> k" apply-apply(rule someI_ex) by auto
hoelzl@33741
   225
    thus ?thesis apply(rule_tac x="SOME k'. k' \<in> k" in exI) using k by auto qed qed
hoelzl@33741
   226
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   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"
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   228
  shows "\<exists>a\<in>s. \<forall>x\<in>s. kle n a x" proof-
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   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)])
hoelzl@33741
   230
    apply(rule,rule) apply(drule_tac assms(3)[rule_format],assumption) using kle_imp_pointwise by auto
hoelzl@33741
   231
  then guess a .. note a=this show ?thesis apply(rule_tac x=a in bexI) proof fix x assume "x\<in>s"
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   232
    show "kle n a x" using assms(3)[rule_format,OF a(1) `x\<in>s`] apply- proof(erule disjE)
hoelzl@33741
   233
      assume "kle n x a" hence "x = a" apply- unfolding pointwise_antisym[THEN sym]
wenzelm@41958
   234
        apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto
hoelzl@33741
   235
      thus ?thesis using kle_refl by auto  qed qed(insert a, auto) qed
hoelzl@33741
   236
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   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"
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   238
  shows "\<exists>a\<in>s. \<forall>x\<in>s. kle n x a" proof-
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   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)])
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   240
    apply(rule,rule) apply(drule_tac assms(3)[rule_format],assumption) using kle_imp_pointwise by auto
hoelzl@33741
   241
  then guess a .. note a=this show ?thesis apply(rule_tac x=a in bexI) proof fix x assume "x\<in>s"
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   242
    show "kle n x a" using assms(3)[rule_format,OF a(1) `x\<in>s`] apply- proof(erule disjE)
hoelzl@33741
   243
      assume "kle n a x" hence "x = a" apply- unfolding pointwise_antisym[THEN sym]
wenzelm@41958
   244
        apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x\<in>s`] by auto
hoelzl@33741
   245
      thus ?thesis using kle_refl by auto  qed qed(insert a, auto) qed
hoelzl@33741
   246
hoelzl@33741
   247
lemma kle_strict_set: assumes "kle n x y" "x \<noteq> y"
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   248
  shows "1 \<le> card {k\<in>{1..n}. x k < y k}" proof-
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   249
  guess i using kle_strict(2)[OF assms] ..
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   250
  hence "card {i} \<le> card {k\<in>{1..n}. x k < y k}" apply- apply(rule card_mono) by auto
hoelzl@33741
   251
  thus ?thesis by auto qed
hoelzl@33741
   252
hoelzl@33741
   253
lemma kle_range_combine:
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   254
  assumes "kle n x y" "kle n y z" "kle n x z \<or> kle n z x"
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   255
  "m1 \<le> card {k\<in>{1..n}. x k < y k}"
hoelzl@33741
   256
  "m2 \<le> card {k\<in>{1..n}. y k < z k}"
hoelzl@33741
   257
  shows "kle n x z \<and> m1 + m2 \<le> card {k\<in>{1..n}. x k < z k}"
hoelzl@33741
   258
  apply(rule,rule kle_trans[OF assms(1-3)]) proof-
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   262
  have **:"{k \<in> {1..n}. x k < y k} \<inter> {k \<in> {1..n}. y k < z k} = {}" unfolding disjoint_iff_not_equal
hoelzl@33741
   263
    apply(rule,rule,unfold mem_Collect_eq,rule ccontr) apply(erule conjE)+ proof-
hoelzl@33741
   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"
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   265
    guess kx using assms(1)[unfolded kle_def] .. note kx=this
hoelzl@33741
   266
    have "x i < y i" using as by auto hence "i \<in> kx" using as(1) kx apply(rule_tac ccontr) by auto 
hoelzl@33741
   267
    hence "x i + 1 = y i" using kx by auto moreover
hoelzl@33741
   268
    guess ky using assms(2)[unfolded kle_def] .. note ky=this
hoelzl@33741
   269
    have "y i < z i" using as by auto hence "i \<in> ky" using as(1) ky apply(rule_tac ccontr) by auto 
hoelzl@33741
   270
    hence "y i + 1 = z i" using ky by auto ultimately
hoelzl@33741
   271
    have "z i = x i + 2" by auto
hoelzl@33741
   272
    thus False using assms(3) unfolding kle_def by(auto simp add: split_if_eq1) qed
hoelzl@33741
   273
  have fin:"\<And>P. finite {x\<in>{1..n::nat}. P x}" by auto
hoelzl@33741
   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
hoelzl@33741
   275
  also have "\<dots> \<le>  card {k\<in>{1..n}. x k < z k}" unfolding card_Un_Int[OF fin fin] unfolding * ** by auto
hoelzl@33741
   276
  finally show " m1 + m2 \<le> card {k \<in> {1..n}. x k < z k}" by auto qed
hoelzl@33741
   277
hoelzl@33741
   278
lemma kle_range_combine_l:
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   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)}"
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   280
  shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}"
hoelzl@33741
   281
  using kle_range_combine[OF assms(1-3) _ assms(4), of 0] by auto
hoelzl@33741
   282
hoelzl@33741
   283
lemma kle_range_combine_r:
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   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}"
hoelzl@33741
   285
  shows "kle n x z \<and> m \<le> card {k\<in>{1..n}. x(k) < z(k)}"
hoelzl@33741
   286
  using kle_range_combine[OF assms(1-3) assms(4), of 0] by auto
hoelzl@33741
   287
hoelzl@33741
   288
lemma kle_range_induct:
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   289
  assumes "card s = Suc m" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x"
hoelzl@33741
   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-
hoelzl@33741
   291
have "finite s" "s\<noteq>{}" using assms(1) by (auto intro: card_ge_0_finite)
hoelzl@33741
   292
thus ?thesis using assms apply- proof(induct m arbitrary: s)
hoelzl@33741
   293
  case 0 thus ?case using kle_refl by auto next
hoelzl@33741
   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
hoelzl@33741
   295
  show ?case proof(cases "s \<subseteq> {a}") case False
hoelzl@33741
   296
    hence "card (s - {a}) = Suc m" "s - {a} \<noteq> {}" using card_Diff_singleton[OF _ a(1)] Suc(4) `finite s` by auto
hoelzl@33741
   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}" 
hoelzl@33741
   298
      using Suc(1)[of "s - {a}"] using Suc(5) `finite s` by auto
hoelzl@33741
   299
    have "1 \<le> card {k \<in> {1..n}. a k < x k}" "m \<le> card {k \<in> {1..n}. x k < b k}"
hoelzl@33741
   300
      apply(rule kle_strict_set) apply(rule a(2)[rule_format]) using a and xb by auto
hoelzl@33741
   301
    thus ?thesis apply(rule_tac x=a in bexI, rule_tac x=b in bexI) 
hoelzl@33741
   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
hoelzl@33741
   303
    case True hence "s = {a}" using Suc(3) by auto hence "card s = 1" by auto
hoelzl@33741
   304
    hence False using Suc(4) `finite s` by auto thus ?thesis by auto qed qed qed
hoelzl@33741
   305
hoelzl@33741
   306
lemma kle_Suc: "kle n x y \<Longrightarrow> kle (n + 1) x y"
hoelzl@33741
   307
  unfolding kle_def apply(erule exE) apply(rule_tac x=k in exI) by auto
hoelzl@33741
   308
hoelzl@33741
   309
lemma kle_trans_1: assumes "kle n x y" shows "x j \<le> y j" "y j \<le> x j + 1"
hoelzl@33741
   310
  using assms[unfolded kle_def] by auto 
hoelzl@33741
   311
hoelzl@33741
   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-
hoelzl@33741
   313
  guess kk1 using assms(1)[unfolded kle_def] .. note kk1=this
hoelzl@33741
   314
  guess kk2 using assms(2)[unfolded kle_def] .. note kk2=this
hoelzl@33741
   315
  show ?thesis unfolding kle_def apply(rule_tac x="kk1 \<union> kk2" in exI) apply(rule) defer proof
hoelzl@33741
   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")
hoelzl@33741
   317
      case True hence "c i \<ge> a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)"
wenzelm@41958
   318
        unfolding kk1[THEN conjunct2,rule_format,of i] kk2[THEN conjunct2,rule_format,of i] by auto
hoelzl@33741
   319
      moreover have "c i \<le> a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)" using True assms(3) by auto  
hoelzl@33741
   320
      ultimately show ?thesis by auto next
hoelzl@33741
   321
      case False thus ?thesis using kk1 kk2 by auto qed qed(insert kk1 kk2, auto) qed
hoelzl@33741
   322
hoelzl@33741
   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"
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   326
hoelzl@33741
   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"
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   330
hoelzl@33741
   331
lemma kle_adjacent:
hoelzl@33741
   332
  assumes "\<forall>j. b j = (if j = k then a(j) + 1 else a j)" "kle n a x" "kle n x b"
hoelzl@33741
   333
  shows "(x = a) \<or> (x = b)" proof(cases "x k = a k")
hoelzl@33741
   334
  case True show ?thesis apply(rule disjI1,rule ext) proof- fix j
hoelzl@33741
   335
    have "x j \<le> a j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]] 
hoelzl@33741
   336
      unfolding assms(1)[rule_format] apply-apply(cases "j=k") using True by auto
hoelzl@33741
   337
    thus "x j = a j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]] by auto qed next
hoelzl@33741
   338
  case False show ?thesis apply(rule disjI2,rule ext) proof- fix j
hoelzl@33741
   339
    have "x j \<ge> b j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]]
hoelzl@33741
   340
      unfolding assms(1)[rule_format] apply-apply(cases "j=k") using False by auto
hoelzl@33741
   341
    thus "x j = b j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]] by auto qed qed
hoelzl@33741
   342
hoelzl@33741
   343
subsection {* kuhn's notion of a simplex (a reformulation to avoid so much indexing). *}
hoelzl@33741
   344
hoelzl@33741
   345
definition "ksimplex p n (s::(nat \<Rightarrow> nat) set) \<longleftrightarrow>
hoelzl@33741
   346
        (card s = n + 1 \<and>
hoelzl@33741
   347
        (\<forall>x\<in>s. \<forall>j. x(j) \<le> p) \<and>
hoelzl@33741
   348
        (\<forall>x\<in>s. \<forall>j. j\<notin>{1..n} \<longrightarrow> (x j = p)) \<and>
hoelzl@33741
   349
        (\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x))"
hoelzl@33741
   350
wenzelm@36318
   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"
hoelzl@33741
   352
  unfolding ksimplex_def by auto
hoelzl@33741
   353
hoelzl@33741
   354
lemma ksimplex_eq: "ksimplex p n (s::(nat \<Rightarrow> nat) set) \<longleftrightarrow>
hoelzl@33741
   355
        (card s = n + 1 \<and> finite s \<and>
hoelzl@33741
   356
        (\<forall>x\<in>s. \<forall>j. x(j) \<le> p) \<and>
hoelzl@33741
   357
        (\<forall>x\<in>s. \<forall>j. j\<notin>{1..n} \<longrightarrow> (x j = p)) \<and>
hoelzl@33741
   358
        (\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x))"
hoelzl@33741
   359
  unfolding ksimplex_def by (auto intro: card_ge_0_finite)
hoelzl@33741
   360
hoelzl@33741
   361
lemma ksimplex_extrema: assumes "ksimplex p n s" obtains a b where "a \<in> s" "b \<in> s"
hoelzl@33741
   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")
hoelzl@33741
   363
  case True obtain x where *:"s = {x}" using assms[unfolded ksimplex_eq True,THEN conjunct1]
hoelzl@33741
   364
    unfolding add_0_left card_1_exists by auto
hoelzl@33741
   365
  show ?thesis apply(rule that[of x x]) unfolding * True by auto next
hoelzl@33741
   366
  note assm = assms[unfolded ksimplex_eq]
hoelzl@33741
   367
  case False have "s\<noteq>{}" using assm by auto
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   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}"
hoelzl@33741
   371
    using kle_range_induct[of s n n] using assm by auto
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   374
  moreover have "card {1..n} \<ge> card {k\<in>{1..n}. a(k) < b(k)}" apply(rule card_mono) by auto
hoelzl@33741
   375
  ultimately have *:"{k\<in>{1 .. n}. a k < b k} = {1..n}" apply- apply(rule card_subset_eq) by auto
hoelzl@33741
   376
  show ?thesis apply(rule that[OF a(1) b(1)]) defer apply(subst *[THEN sym]) unfolding mem_Collect_eq proof
hoelzl@33741
   377
    guess k using a(2)[rule_format,OF b(1),unfolded kle_def] .. note k=this
hoelzl@33741
   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}")
hoelzl@33741
   379
      case True thus ?thesis unfolding k[THEN conjunct2,rule_format] by auto next
hoelzl@33741
   380
      case False have "a i = p" using assm and False `a\<in>s` by auto
hoelzl@33741
   381
      moreover   have "b i = p" using assm and False `b\<in>s` by auto
hoelzl@33741
   382
      ultimately show ?thesis by auto qed qed(insert a(2) b(2) assm,auto) qed
hoelzl@33741
   383
hoelzl@33741
   384
lemma ksimplex_extrema_strong:
hoelzl@33741
   385
  assumes "ksimplex p n s" "n \<noteq> 0" obtains a b where "a \<in> s" "b \<in> s" "a \<noteq> b"
hoelzl@33741
   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-
hoelzl@33741
   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))" 
hoelzl@33741
   388
    apply(rule ksimplex_extrema[OF assms(1)]) by auto 
hoelzl@33741
   389
  have "a \<noteq> b" apply(rule ccontr) unfolding not_not apply(drule cong[of _ _ 1 1]) using ab(4) assms(2) by auto
hoelzl@33741
   390
  thus ?thesis apply(rule_tac that[of a b]) using ab by auto qed
hoelzl@33741
   391
hoelzl@33741
   392
lemma ksimplexD:
hoelzl@33741
   393
  assumes "ksimplex p n s"
wenzelm@36318
   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"
hoelzl@33741
   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
hoelzl@33741
   396
hoelzl@33741
   397
lemma ksimplex_successor:
hoelzl@33741
   398
  assumes "ksimplex p n s" "a \<in> s"
hoelzl@33741
   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)))"
hoelzl@33741
   400
proof(cases "\<forall>x\<in>s. kle n x a") case True thus ?thesis by auto next note assm = ksimplexD[OF assms(1)]
hoelzl@33741
   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"
hoelzl@33741
   402
    using kle_minimal[of "{x\<in>s. \<not> kle n x a}" n] and assm by auto
hoelzl@33741
   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)
hoelzl@33741
   404
hoelzl@33741
   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
hoelzl@33741
   406
  hence sizekle1: "card ?kle1 = Suc (card ?kle1 - 1)" using assm(2) by auto
hoelzl@33741
   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}"
hoelzl@33741
   408
    using kle_range_induct[OF sizekle1, of n] using assm by auto
hoelzl@33741
   409
hoelzl@33741
   410
  let ?kle2 = "{x \<in> s. kle n x a}"
hoelzl@33741
   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
hoelzl@33741
   412
  hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)" using assm(2) by auto
hoelzl@33741
   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}"
hoelzl@33741
   414
    using kle_range_induct[OF sizekle2, of n] using assm by auto
hoelzl@33741
   415
hoelzl@33741
   416
  have "card {k\<in>{1..n}. a k < b k} = 1" proof(rule ccontr) case goal1
hoelzl@33741
   417
    hence as:"card {k\<in>{1..n}. a k < b k} \<ge> 2" using ** by auto
hoelzl@33741
   418
    have *:"finite ?kle2" "finite ?kle1" "?kle2 \<union> ?kle1 = s" "?kle2 \<inter> ?kle1 = {}" using assm(2) by auto
hoelzl@33741
   419
    have "(card ?kle2 - 1) + 2 + (card ?kle1 - 1) = card ?kle2 + card ?kle1" using sizekle1 sizekle2 by auto
hoelzl@33741
   420
    also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto
hoelzl@33741
   421
    finally have n:"(card ?kle2 - 1) + (2 + (card ?kle1 - 1)) = n + 1" by auto
hoelzl@33741
   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}"
hoelzl@33741
   423
      apply(rule kle_range_combine_r[where y=f]) using e_f using `a\<in>s` assm(6) by auto
hoelzl@33741
   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}"
hoelzl@33741
   425
      apply(rule kle_range_combine_l[where y=c]) using c_d using assm(6) and b by auto
hoelzl@33741
   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-
hoelzl@33741
   427
      apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a\<in>s` `d\<in>s` apply- by blast+
hoelzl@33741
   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-
hoelzl@33741
   429
      apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply - by blast+ 
hoelzl@33741
   430
    moreover have "card {k \<in> {1..n}. e k < d k} \<le> card {1..n}" apply(rule card_mono) by auto
hoelzl@33741
   431
    ultimately show False unfolding n by auto qed
hoelzl@33741
   432
  then guess k unfolding card_1_exists .. note k=this[unfolded mem_Collect_eq]
hoelzl@33741
   433
hoelzl@33741
   434
  show ?thesis apply(rule disjI2) apply(rule_tac x=b in bexI,rule_tac x=k in bexI) proof
hoelzl@33741
   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
hoelzl@33741
   436
    then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format]
hoelzl@33741
   437
    have kkk:"k\<in>kk" apply(rule ccontr) using k(1) unfolding kk by auto 
hoelzl@33741
   438
    show "b j = (if j = k then a j + 1 else a j)" proof(cases "j\<in>kk")
hoelzl@33741
   439
      case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto
hoelzl@33741
   440
      thus ?thesis unfolding kk using kkk by auto next
hoelzl@33741
   441
      case False hence "j\<noteq>k" using k(2)[rule_format, of j k] using kk_raw kkk by auto
hoelzl@33741
   442
      thus ?thesis unfolding kk using kkk using False by auto qed qed(insert k(1) `b\<in>s`, auto) qed
hoelzl@33741
   443
hoelzl@33741
   444
lemma ksimplex_predecessor:
hoelzl@33741
   445
  assumes "ksimplex p n s" "a \<in> s"
hoelzl@33741
   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)))"
hoelzl@33741
   447
proof(cases "\<forall>x\<in>s. kle n a x") case True thus ?thesis by auto next note assm = ksimplexD[OF assms(1)]
hoelzl@33741
   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" 
hoelzl@33741
   449
    using kle_maximal[of "{x\<in>s. \<not> kle n a x}" n] and assm by auto
hoelzl@33741
   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)
hoelzl@33741
   451
hoelzl@33741
   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
hoelzl@33741
   453
  hence sizekle1:"card ?kle1 = Suc (card ?kle1 - 1)" using assm(2) by auto
hoelzl@33741
   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}"
hoelzl@33741
   455
    using kle_range_induct[OF sizekle1, of n] using assm by auto
hoelzl@33741
   456
hoelzl@33741
   457
  let ?kle2 = "{x \<in> s. kle n a x}"
hoelzl@33741
   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
hoelzl@33741
   459
  hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)" using assm(2) by auto
hoelzl@33741
   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}"
hoelzl@33741
   461
    using kle_range_induct[OF sizekle2, of n] using assm by auto
hoelzl@33741
   462
hoelzl@33741
   463
  have "card {k\<in>{1..n}. a k > b k} = 1" proof(rule ccontr) case goal1
hoelzl@33741
   464
    hence as:"card {k\<in>{1..n}. a k > b k} \<ge> 2" using ** by auto
hoelzl@33741
   465
    have *:"finite ?kle2" "finite ?kle1" "?kle2 \<union> ?kle1 = s" "?kle2 \<inter> ?kle1 = {}" using assm(2) by auto
hoelzl@33741
   466
    have "(card ?kle2 - 1) + 2 + (card ?kle1 - 1) = card ?kle2 + card ?kle1" using sizekle1 sizekle2 by auto
hoelzl@33741
   467
    also have "\<dots> = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto
hoelzl@33741
   468
    finally have n:"(card ?kle1 - 1) + 2 + (card ?kle2 - 1) = n + 1" by auto
hoelzl@33741
   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}"
hoelzl@33741
   470
      apply(rule kle_range_combine_l[where y=f]) using e_f using `a\<in>s` assm(6) by auto
hoelzl@33741
   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}"
hoelzl@33741
   472
      apply(rule kle_range_combine_r[where y=c]) using c_d using assm(6) and b by auto
hoelzl@33741
   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-
hoelzl@33741
   474
      apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a\<in>s` `d\<in>s` by blast+
hoelzl@33741
   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-
hoelzl@33741
   476
      apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply - by blast+
hoelzl@33741
   477
    moreover have "card {k \<in> {1..n}. e k > d k} \<le> card {1..n}" apply(rule card_mono) by auto
hoelzl@33741
   478
    ultimately show False unfolding n by auto qed
hoelzl@33741
   479
  then guess k unfolding card_1_exists .. note k=this[unfolded mem_Collect_eq]
hoelzl@33741
   480
hoelzl@33741
   481
  show ?thesis apply(rule disjI2) apply(rule_tac x=b in bexI,rule_tac x=k in bexI) proof
hoelzl@33741
   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
hoelzl@33741
   483
    then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format]
hoelzl@33741
   484
    have kkk:"k\<in>kk" apply(rule ccontr) using k(1) unfolding kk by auto 
hoelzl@33741
   485
    show "a j = (if j = k then b j + 1 else b j)" proof(cases "j\<in>kk")
hoelzl@33741
   486
      case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto
hoelzl@33741
   487
      thus ?thesis unfolding kk using kkk by auto next
hoelzl@33741
   488
      case False hence "j\<noteq>k" using k(2)[rule_format, of j k] using kk_raw kkk by auto
hoelzl@33741
   489
      thus ?thesis unfolding kk using kkk using False by auto qed qed(insert k(1) `b\<in>s`, auto) qed
hoelzl@33741
   490
hoelzl@33741
   491
subsection {* The lemmas about simplices that we need. *}
hoelzl@33741
   492
hoelzl@33741
   493
lemma card_funspace': assumes "finite s" "finite t" "card s = m" "card t = n"
hoelzl@33741
   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) = _")
hoelzl@33741
   495
  using assms apply - proof(induct m arbitrary: s)
nipkow@39302
   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
hoelzl@33741
   497
  case 0 thus ?case by(auto simp add: *) next
hoelzl@33741
   498
  case (Suc m) guess a using card_eq_SucD[OF Suc(4)] .. then guess s0
hoelzl@33741
   499
    apply(erule_tac exE) apply(erule conjE)+ . note as0 = this
hoelzl@33741
   500
  have **:"card s0 = m" using as0 using Suc(2) Suc(4) by auto
hoelzl@33741
   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}"
nipkow@39302
   502
    apply(rule set_eqI,rule) unfolding mem_Collect_eq image_iff apply(erule conjE)
hoelzl@33741
   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
hoelzl@33741
   504
    apply(erule bexE,rule) unfolding mem_Collect_eq apply(erule splitE)+ apply(erule conjE)+ proof-
hoelzl@33741
   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"
hoelzl@33741
   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
hoelzl@33741
   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-
hoelzl@33741
   508
    case goal1 note as = this(1,4-)[unfolded goal1 split_conv]
hoelzl@33741
   509
    have "xa = xb" using as(1)[THEN cong[of _ _ a]] by auto
hoelzl@33741
   510
    moreover have "ya = yb" proof(rule ext) fix x show "ya x = yb x" proof(cases "x = a") 
wenzelm@41958
   511
        case False thus ?thesis using as(1)[THEN cong[of _ _ x x]] by auto next
wenzelm@41958
   512
        case True thus ?thesis using as(5,7) using as0(2) by auto qed qed 
hoelzl@33741
   513
    ultimately show ?case unfolding goal1 by auto qed
hoelzl@33741
   514
  have "finite s0" using `finite s` unfolding as0 by simp
hoelzl@33741
   515
  show ?case unfolding as0 * card_image[OF inj] using assms
hoelzl@33741
   516
    unfolding SetCompr_Sigma_eq apply-
hoelzl@33741
   517
    unfolding card_cartesian_product
hoelzl@33741
   518
    using Suc(1)[OF `finite s0` `finite t` ** `card t = n`] by auto
hoelzl@33741
   519
qed
hoelzl@33741
   520
hoelzl@33741
   521
lemma card_funspace: assumes  "finite s" "finite t"
hoelzl@33741
   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)"
hoelzl@33741
   523
  using assms by (auto intro: card_funspace')
hoelzl@33741
   524
hoelzl@33741
   525
lemma finite_funspace: assumes "finite s" "finite t"
hoelzl@33741
   526
  shows "finite {f. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>UNIV - s. f x = d)}" (is "finite ?S")
hoelzl@33741
   527
proof (cases "card t > 0")
hoelzl@33741
   528
  case True
hoelzl@33741
   529
  have "card ?S = (card t) ^ (card s)"
hoelzl@33741
   530
    using assms by (auto intro!: card_funspace)
hoelzl@33741
   531
  thus ?thesis using True by (auto intro: card_ge_0_finite)
hoelzl@33741
   532
next
hoelzl@33741
   533
  case False hence "t = {}" using `finite t` by auto
hoelzl@33741
   534
  show ?thesis
hoelzl@33741
   535
  proof (cases "s = {}")
huffman@44457
   536
    have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" by auto
hoelzl@33741
   537
    case True thus ?thesis using `t = {}` by (auto simp: *)
hoelzl@33741
   538
  next
hoelzl@33741
   539
    case False thus ?thesis using `t = {}` by simp
hoelzl@33741
   540
  qed
hoelzl@33741
   541
qed
hoelzl@33741
   542
hoelzl@33741
   543
lemma finite_simplices: "finite {s. ksimplex p n s}"
hoelzl@33741
   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)}}"])
hoelzl@33741
   545
  unfolding ksimplex_def defer apply(rule finite_Collect_subsets) apply(rule finite_funspace) by auto
hoelzl@33741
   546
hoelzl@33741
   547
lemma simplex_top_face: assumes "0<p" "\<forall>x\<in>f. x (n + 1) = p"
hoelzl@33741
   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
hoelzl@33741
   549
  assume ?ls then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
hoelzl@33741
   550
  show ?rs unfolding ksimplex_def sa(3) apply(rule) defer apply rule defer apply(rule,rule,rule,rule) defer apply(rule,rule) proof-
hoelzl@33741
   551
    fix x y assume as:"x \<in>s - {a}" "y \<in>s - {a}" have xyp:"x (n + 1) = y (n + 1)"
wenzelm@41958
   552
        using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]]
wenzelm@41958
   553
        using as(2)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] by auto
hoelzl@33741
   554
    show "kle n x y \<or> kle n y x" proof(cases "kle (n + 1) x y")
hoelzl@33741
   555
      case True then guess k unfolding kle_def .. note k=this hence *:"n+1 \<notin> k" using xyp by auto
hoelzl@33741
   556
      have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
wenzelm@41958
   557
        fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto
wenzelm@41958
   558
        thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed
hoelzl@33741
   559
      thus ?thesis apply-apply(rule disjI1) unfolding kle_def using k apply(rule_tac x=k in exI) by auto next
hoelzl@33741
   560
      case False hence "kle (n + 1) y x" using ksimplexD(6)[OF sa(1),rule_format, of x y] using as by auto
hoelzl@33741
   561
      then guess k unfolding kle_def .. note k=this hence *:"n+1 \<notin> k" using xyp by auto
hoelzl@33741
   562
      hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply-apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
wenzelm@41958
   563
        fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> n+1" using as and * by auto
wenzelm@41958
   564
        thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed
hoelzl@33741
   565
      thus ?thesis apply-apply(rule disjI2) unfolding kle_def using k apply(rule_tac x=k in exI) by auto qed next
hoelzl@33741
   566
    fix x j assume as:"x\<in>s - {a}" "j\<notin>{1..n}"
hoelzl@33741
   567
    thus "x j = p" using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]]
hoelzl@33741
   568
      apply(cases "j = n+1") using sa(1)[unfolded ksimplex_def] by auto qed(insert sa ksimplexD[OF sa(1)], auto) next
hoelzl@33741
   569
  assume ?rs note rs=ksimplexD[OF this] guess a b apply(rule ksimplex_extrema[OF `?rs`]) . note ab = this
hoelzl@33741
   570
  def c \<equiv> "\<lambda>i. if i = (n + 1) then p - 1 else a i"
hoelzl@33741
   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
hoelzl@33741
   572
  thus ?ls apply(rule_tac x="insert c f" in exI,rule_tac x=c in exI) unfolding ksimplex_def conj_assoc
hoelzl@33741
   573
    apply(rule conjI) defer apply(rule conjI) defer apply(rule conjI) defer apply(rule conjI) defer  
hoelzl@33741
   574
  proof(rule_tac[3-5] ballI allI)+
hoelzl@33741
   575
    fix x j assume x:"x \<in> insert c f" thus "x j \<le> p" proof (cases "x=c")
hoelzl@33741
   576
      case True show ?thesis unfolding True c_def apply(cases "j=n+1") using ab(1) and rs(4) by auto 
hoelzl@33741
   577
    qed(insert x rs(4), auto simp add:c_def)
hoelzl@33741
   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
hoelzl@33741
   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-
wenzelm@41958
   580
        case False hence "z\<in>f" using z by auto
wenzelm@41958
   581
        then guess k apply(drule_tac ab(3)[THEN bspec[where x=z], THEN conjunct1]) unfolding kle_def apply(erule exE) .
wenzelm@41958
   582
        thus "kle (n + 1) c z" unfolding kle_def apply(rule_tac x="insert (n + 1) k" in exI) unfolding c_def
wenzelm@41958
   583
          using ab using rs(5)[rule_format,OF ab(1),of "n + 1"] assms(1) by auto qed auto } note * = this
hoelzl@33741
   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")
hoelzl@33741
   585
      case False hence **:"x\<in>f" "y\<in>f" using x y by auto
hoelzl@33741
   586
      show ?thesis using rs(6)[rule_format,OF **] by(auto dest: kle_Suc) qed(insert * x y, auto)
hoelzl@33741
   587
  qed(insert rs, auto) qed
hoelzl@33741
   588
hoelzl@33741
   589
lemma ksimplex_fix_plane:
hoelzl@33741
   590
  assumes "a \<in> s" "j\<in>{1..n::nat}" "\<forall>x\<in>s - {a}. x j = q" "a0 \<in> s" "a1 \<in> s"
hoelzl@33741
   591
  "\<forall>i. a1 i = ((if i\<in>{1..n} then a0 i + 1 else a0 i)::nat)"
hoelzl@33741
   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
hoelzl@33741
   593
  show ?thesis apply(rule ccontr) using *[OF assms(3), of a0 a1] unfolding assms(6)[THEN spec[where x=j]]
hoelzl@33741
   594
    using assms(1-2,4-5) by auto qed
hoelzl@33741
   595
hoelzl@33741
   596
lemma ksimplex_fix_plane_0:
hoelzl@33741
   597
  assumes "a \<in> s" "j\<in>{1..n::nat}" "\<forall>x\<in>s - {a}. x j = 0" "a0 \<in> s" "a1 \<in> s"
hoelzl@33741
   598
  "\<forall>i. a1 i = ((if i\<in>{1..n} then a0 i + 1 else a0 i)::nat)"
hoelzl@33741
   599
  shows "a = a1" apply(rule ccontr) using ksimplex_fix_plane[OF assms]
hoelzl@33741
   600
  using assms(3)[THEN bspec[where x=a1]] using assms(2,5)  
hoelzl@33741
   601
  unfolding assms(6)[THEN spec[where x=j]] by simp
hoelzl@33741
   602
hoelzl@33741
   603
lemma ksimplex_fix_plane_p:
hoelzl@33741
   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"
hoelzl@33741
   605
  "\<forall>i. a1 i = (if i\<in>{1..n} then a0 i + 1 else a0 i)"
hoelzl@33741
   606
  shows "a = a0" proof(rule ccontr) note s = ksimplexD[OF assms(1),rule_format]
hoelzl@33741
   607
  assume as:"a \<noteq> a0" hence *:"a0 \<in> s - {a}" using assms(5) by auto
hoelzl@33741
   608
  hence "a1 = a" using ksimplex_fix_plane[OF assms(2-)] by auto
hoelzl@33741
   609
  thus False using as using assms(3,5) and assms(7)[rule_format,of j]
hoelzl@33741
   610
    unfolding assms(4)[rule_format,OF *] using s(4)[OF assms(6), of j] by auto qed
hoelzl@33741
   611
hoelzl@33741
   612
lemma ksimplex_replace_0:
hoelzl@33741
   613
  assumes "ksimplex p n s" "a \<in> s" "n \<noteq> 0" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = 0"
hoelzl@33741
   614
  shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1" proof-
hoelzl@33741
   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
hoelzl@33741
   616
  have **:"\<And>s' a'. ksimplex p n s' \<Longrightarrow> a' \<in> s' \<Longrightarrow> s' - {a'} = s - {a} \<Longrightarrow> s' = s" proof- case goal1
hoelzl@33741
   617
    guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note exta = this[rule_format]
hoelzl@33741
   618
    have a:"a = a1" apply(rule ksimplex_fix_plane_0[OF assms(2,4-5)]) using exta(1-2,5) by auto moreover
hoelzl@33741
   619
    guess b0 b1 apply(rule ksimplex_extrema_strong[OF goal1(1) assms(3)]) . note extb = this[rule_format]
hoelzl@33741
   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
hoelzl@33741
   621
    have "b0 = a0" unfolding kle_antisym[THEN sym, of b0 a0 n] using exta extb using goal1(3) unfolding a a' by blast
hoelzl@33741
   622
    hence "b1 = a1" apply-apply(rule ext) unfolding exta(5) extb(5) by auto ultimately
hoelzl@33741
   623
    show "s' = s" apply-apply(rule *[of _ a1 b1]) using exta(1-2) extb(1-2) goal1 by auto qed
hoelzl@33741
   624
  show ?thesis unfolding card_1_exists apply-apply(rule ex1I[of _ s])
hoelzl@33741
   625
    unfolding mem_Collect_eq defer apply(erule conjE bexE)+ apply(rule_tac a'=b in **) using assms(1,2) by auto qed
hoelzl@33741
   626
hoelzl@33741
   627
lemma ksimplex_replace_1:
hoelzl@33741
   628
  assumes "ksimplex p n s" "a \<in> s" "n \<noteq> 0" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = p"
hoelzl@33741
   629
  shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1" proof-
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   632
    guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note exta = this[rule_format]
hoelzl@33741
   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
hoelzl@33741
   634
    guess b0 b1 apply(rule ksimplex_extrema_strong[OF goal1(1) assms(3)]) . note extb = this[rule_format]
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   637
    have "a0 = b0" apply(rule ext) proof- case goal1 show "a0 x = b0 x"
wenzelm@41958
   638
        using *[THEN cong, of x x] unfolding exta extb apply-apply(cases "x\<in>{1..n}") by auto qed
hoelzl@33741
   639
    ultimately show "s' = s" apply-apply(rule lem[OF goal1(3) _ goal1(2) assms(2)]) by auto qed 
hoelzl@33741
   640
  show ?thesis unfolding card_1_exists apply(rule ex1I[of _ s]) unfolding mem_Collect_eq apply(rule,rule assms(1))
hoelzl@33741
   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
hoelzl@33741
   642
hoelzl@33741
   643
lemma ksimplex_replace_2:
hoelzl@33741
   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)"
hoelzl@33741
   645
  shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 2" (is "card ?A = 2")  proof-
hoelzl@33741
   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
hoelzl@33741
   647
  have lem2:"\<And>a b. a\<in>s \<Longrightarrow> b\<noteq>a \<Longrightarrow> s \<noteq> insert b (s - {a})" proof case goal1
hoelzl@33741
   648
    hence "a\<in>insert b (s - {a})" by auto hence "a\<in> s - {a}" unfolding insert_iff using goal1 by auto
hoelzl@33741
   649
    thus False by auto qed
hoelzl@33741
   650
  guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note a0a1=this
hoelzl@33741
   651
  { assume "a=a0"
hoelzl@33741
   652
    have *:"\<And>P Q. (P \<or> Q) \<Longrightarrow> \<not> P \<Longrightarrow> Q" by auto
hoelzl@33741
   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"
hoelzl@33741
   654
      show False using kle_imp_pointwise[OF as,THEN spec[where x=1]] unfolding a0a1(5)[THEN spec[where x=1]]
hoelzl@33741
   655
        using assms(3) by auto qed(insert a0a1,auto)
hoelzl@33741
   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)"
hoelzl@33741
   657
      apply(rule_tac *[OF ksimplex_successor[OF assms(1-2),unfolded `a=a0`]]) by auto
hoelzl@33741
   658
    then guess a2 .. from this(2) guess k .. note k=this note a2=`a2\<in>s`
hoelzl@33741
   659
    def a3 \<equiv> "\<lambda>j. if j = k then a1 j + 1 else a1 j"
hoelzl@33741
   660
    have "a3 \<notin> s" proof assume "a3\<in>s" hence "kle n a3 a1" using a0a1(4) by auto
hoelzl@33741
   661
      thus False apply(drule_tac kle_imp_pointwise) unfolding a3_def
hoelzl@33741
   662
        apply(erule_tac x=k in allE) by auto qed
hoelzl@33741
   663
    hence "a3 \<noteq> a0" "a3 \<noteq> a1" using a0a1 by auto
hoelzl@33741
   664
    have "a2 \<noteq> a0" using k(2)[THEN spec[where x=k]] by auto
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   667
      have "kle n a0 x" using a0a1(4) as by auto
hoelzl@33741
   668
      ultimately have "x = a0 \<or> x = a2" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto
hoelzl@33741
   669
      hence "x = a2" using as by auto thus False using goal1(2) using kle_refl by auto qed
hoelzl@33741
   670
    let ?s = "insert a3 (s - {a0})" have "ksimplex p n ?s" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI)
hoelzl@33741
   671
      show "card ?s = n + 1" using ksimplexD(2-3)[OF assms(1)]
hoelzl@33741
   672
        using `a3\<noteq>a0` `a3\<notin>s` `a0\<in>s` by(auto simp add:card_insert_if)
hoelzl@33741
   673
      fix x assume x:"x \<in> insert a3 (s - {a0})"
hoelzl@33741
   674
      show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3")
wenzelm@41958
   675
        fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
wenzelm@41958
   676
        fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") 
wenzelm@41958
   677
          case False thus "a3 j \<le>p" unfolding True a3_def using `a1\<in>s` ksimplexD(4)[OF assms(1)] by auto next
wenzelm@41958
   678
          guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this
wenzelm@41958
   679
          have "a2 k \<le> a4 k" using lem3[OF a4(1)[unfolded `a=a0`],THEN kle_imp_pointwise] by auto
wenzelm@41958
   680
          also have "\<dots> < p" using ksimplexD(4)[OF assms(1),rule_format,of a4 k] using a4 by auto
wenzelm@41958
   681
          finally have *:"a0 k + 1 < p" unfolding k(2)[rule_format] by auto
wenzelm@41958
   682
          case True thus "a3 j \<le>p" unfolding a3_def unfolding a0a1(5)[rule_format]
wenzelm@41958
   683
            using k(1) k(2)assms(5) using * by simp qed qed
hoelzl@33741
   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}"
wenzelm@41958
   685
        { case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
wenzelm@41958
   686
        case True show "x j = p" unfolding True a3_def using j k(1) 
wenzelm@41958
   687
          using ksimplexD(5)[OF assms(1),rule_format,OF `a1\<in>s` j] by auto qed
hoelzl@33741
   688
      fix y assume y:"y\<in>insert a3 (s - {a0})"
hoelzl@33741
   689
      have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a0 \<Longrightarrow> kle n x a3" proof- case goal1
wenzelm@41958
   690
        guess kk using a0a1(4)[rule_format,OF `x\<in>s`,THEN conjunct2,unfolded kle_def] 
hoelzl@33741
   691
          apply-apply(erule exE,erule conjE) . note kk=this
wenzelm@41958
   692
        have "k\<notin>kk" proof assume "k\<in>kk"
wenzelm@41958
   693
          hence "a1 k = x k + 1" using kk by auto
wenzelm@41958
   694
          hence "a0 k = x k" unfolding a0a1(5)[rule_format] using k(1) by auto
wenzelm@41958
   695
          hence "a2 k = x k + 1" unfolding k(2)[rule_format] by auto moreover
wenzelm@41958
   696
          have "a2 k \<le> x k" using lem3[of x,THEN kle_imp_pointwise] goal1 by auto 
wenzelm@41958
   697
          ultimately show False by auto qed
wenzelm@41958
   698
        thus ?case unfolding kle_def apply(rule_tac x="insert k kk" in exI) using kk(1)
wenzelm@41958
   699
          unfolding a3_def kle_def kk(2)[rule_format] using k(1) by auto qed
hoelzl@33741
   700
      show "kle n x y \<or> kle n y x" proof(cases "y=a3")
wenzelm@41958
   701
        case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI1,rule lem4)
wenzelm@41958
   702
          using x by auto next
wenzelm@41958
   703
        case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True
wenzelm@41958
   704
            apply(rule disjI2,rule lem4) using y False by auto next
wenzelm@41958
   705
          case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
wenzelm@41958
   706
            using x y `y\<noteq>a3` by auto qed qed qed
hoelzl@33741
   707
    hence "insert a3 (s - {a0}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
hoelzl@33741
   708
      apply(rule_tac x="a3" in bexI) unfolding `a=a0` using `a3\<notin>s` by auto moreover
hoelzl@33741
   709
    have "s \<in> ?A" using assms(1,2) by auto ultimately have  "?A \<supseteq> {s, insert a3 (s - {a0})}" by auto
hoelzl@33741
   710
    moreover have "?A \<subseteq> {s, insert a3 (s - {a0})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
hoelzl@33741
   711
      fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}"
hoelzl@33741
   712
      from this(2) guess a' .. note a'=this
hoelzl@33741
   713
      guess a_min a_max apply(rule ksimplex_extrema_strong[OF as assms(3)]) . note min_max=this
hoelzl@33741
   714
      have *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}"
wenzelm@41958
   715
        hence "kle n a2 x" apply-apply(rule lem3) using `a=a0` by auto
wenzelm@41958
   716
        hence "a2 k \<le> x k" apply(drule_tac kle_imp_pointwise) by auto moreover
wenzelm@41958
   717
        have "x k \<le> a2 k" unfolding k(2)[rule_format] using a0a1(4)[rule_format,of x,THEN conjunct1] 
wenzelm@41958
   718
          unfolding kle_def using x by auto ultimately show "x k = a2 k" by auto qed
hoelzl@33741
   719
      have **:"a'=a_min \<or> a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto
hoelzl@33741
   720
      show "s' \<in> {s, insert a3 (s - {a0})}" proof(cases "a'=a_min")
wenzelm@41958
   721
        case True have "a_max = a1" unfolding kle_antisym[THEN sym,of a_max a1 n] apply(rule)
wenzelm@41958
   722
          apply(rule a0a1(4)[rule_format,THEN conjunct2]) defer  proof(rule min_max(4)[rule_format,THEN conjunct2])
wenzelm@41958
   723
          show "a1\<in>s'" using a' unfolding `a=a0` using a0a1 by auto
wenzelm@41958
   724
          show "a_max \<in> s" proof(rule ccontr) assume "a_max\<notin>s"
wenzelm@41958
   725
            hence "a_max = a'" using a' min_max by auto
wenzelm@41958
   726
            thus False unfolding True using min_max by auto qed qed
wenzelm@41958
   727
        hence "\<forall>i. a_max i = a1 i" by auto
wenzelm@41958
   728
        hence "a' = a" unfolding True `a=a0` apply-apply(subst fun_eq_iff,rule)
wenzelm@41958
   729
          apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format]
wenzelm@41958
   730
        proof- case goal1 thus ?case apply(cases "x\<in>{1..n}") by auto qed
wenzelm@41958
   731
        hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>s'` by auto
wenzelm@41958
   732
        thus ?thesis by auto next
wenzelm@41958
   733
        case False hence as:"a' = a_max" using ** by auto
wenzelm@41958
   734
        have "a_min = a2" unfolding kle_antisym[THEN sym, of _ _ n] apply rule
wenzelm@41958
   735
          apply(rule min_max(4)[rule_format,THEN conjunct1]) defer proof(rule lem3)
wenzelm@41958
   736
          show "a_min \<in> s - {a0}" unfolding a'(2)[THEN sym,unfolded `a=a0`] 
wenzelm@41958
   737
            unfolding as using min_max(1-3) by auto
wenzelm@41958
   738
          have "a2 \<noteq> a" unfolding `a=a0` using k(2)[rule_format,of k] by auto
wenzelm@41958
   739
          hence "a2 \<in> s - {a}" using a2 by auto thus "a2 \<in> s'" unfolding a'(2)[THEN sym] by auto qed
wenzelm@41958
   740
        hence "\<forall>i. a_min i = a2 i" by auto
wenzelm@41958
   741
        hence "a' = a3" unfolding as `a=a0` apply-apply(subst fun_eq_iff,rule)
wenzelm@41958
   742
          apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format]
wenzelm@41958
   743
          unfolding a3_def k(2)[rule_format] unfolding a0a1(5)[rule_format] proof- case goal1
wenzelm@41958
   744
          show ?case unfolding goal1 apply(cases "x\<in>{1..n}") defer apply(cases "x=k")
wenzelm@41958
   745
            using `k\<in>{1..n}` by auto qed
wenzelm@41958
   746
        hence "s' = insert a3 (s - {a0})" apply-apply(rule lem1) defer apply assumption
wenzelm@41958
   747
          apply(rule a'(1)) unfolding a' `a=a0` using `a3\<notin>s` by auto
wenzelm@41958
   748
        thus ?thesis by auto qed qed
hoelzl@33741
   749
    ultimately have *:"?A = {s, insert a3 (s - {a0})}" by blast
hoelzl@33741
   750
    have "s \<noteq> insert a3 (s - {a0})" using `a3\<notin>s` by auto
hoelzl@33741
   751
    hence ?thesis unfolding * by auto } moreover
hoelzl@33741
   752
  { assume "a=a1"
hoelzl@33741
   753
    have *:"\<And>P Q. (P \<or> Q) \<Longrightarrow> \<not> P \<Longrightarrow> Q" by auto
hoelzl@33741
   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"
hoelzl@33741
   755
      show False using kle_imp_pointwise[OF as,THEN spec[where x=1]] unfolding a0a1(5)[THEN spec[where x=1]]
hoelzl@33741
   756
        using assms(3) by auto qed(insert a0a1,auto)
hoelzl@33741
   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)"
hoelzl@33741
   758
      apply(rule_tac *[OF ksimplex_predecessor[OF assms(1-2),unfolded `a=a1`]]) by auto
hoelzl@33741
   759
    then guess a2 .. from this(2) guess k .. note k=this note a2=`a2\<in>s`
hoelzl@33741
   760
    def a3 \<equiv> "\<lambda>j. if j = k then a0 j - 1 else a0 j"
hoelzl@33741
   761
    have "a2 \<noteq> a1" using k(2)[THEN spec[where x=k]] by auto
hoelzl@33741
   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
hoelzl@33741
   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
hoelzl@33741
   764
      have "kle n x a1" using a0a1(4) as by auto
hoelzl@33741
   765
      ultimately have "x = a2 \<or> x = a1" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto
hoelzl@33741
   766
      hence "x = a2" using as by auto thus False using goal1(2) using kle_refl by auto qed
hoelzl@33741
   767
    have "a0 k \<noteq> 0" proof-
hoelzl@33741
   768
      guess a4 using assms(4)[unfolded bex_simps ball_simps,rule_format,OF `k\<in>{1..n}`] .. note a4=this
hoelzl@33741
   769
      have "a4 k \<le> a2 k" using lem3[OF a4(1)[unfolded `a=a1`],THEN kle_imp_pointwise] by auto
hoelzl@33741
   770
      moreover have "a4 k > 0" using a4 by auto ultimately have "a2 k > 0" by auto
hoelzl@33741
   771
      hence "a1 k > 1" unfolding k(2)[rule_format] by simp
hoelzl@33741
   772
      thus ?thesis unfolding a0a1(5)[rule_format] using k(1) by simp qed
hoelzl@33741
   773
    hence lem4:"\<forall>j. a0 j = (if j=k then a3 j + 1 else a3 j)" unfolding a3_def by simp
hoelzl@33741
   774
    have "\<not> kle n a0 a3" apply(rule ccontr) unfolding not_not apply(drule kle_imp_pointwise)
hoelzl@33741
   775
      unfolding lem4[rule_format] apply(erule_tac x=k in allE) by auto
hoelzl@33741
   776
    hence "a3 \<notin> s" using a0a1(4) by auto
hoelzl@33741
   777
    hence "a3 \<noteq> a1" "a3 \<noteq> a0" using a0a1 by auto
hoelzl@33741
   778
    let ?s = "insert a3 (s - {a1})" have "ksimplex p n ?s" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI)
hoelzl@33741
   779
      show "card ?s = n+1" using ksimplexD(2-3)[OF assms(1)]
hoelzl@33741
   780
        using `a3\<noteq>a0` `a3\<notin>s` `a1\<in>s` by(auto simp add:card_insert_if)
hoelzl@33741
   781
      fix x assume x:"x \<in> insert a3 (s - {a1})"
hoelzl@33741
   782
      show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3")
wenzelm@41958
   783
        fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
wenzelm@41958
   784
        fix j case True show "x j\<le>p" unfolding True proof(cases "j=k") 
wenzelm@41958
   785
          case False thus "a3 j \<le>p" unfolding True a3_def using `a0\<in>s` ksimplexD(4)[OF assms(1)] by auto next
wenzelm@41958
   786
          guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this
hoelzl@33741
   787
          case True have "a3 k \<le> a0 k" unfolding lem4[rule_format] by auto
hoelzl@33741
   788
          also have "\<dots> \<le> p" using ksimplexD(4)[OF assms(1),rule_format,of a0 k] a0a1 by auto
hoelzl@33741
   789
          finally show "a3 j \<le> p" unfolding True by auto qed qed
hoelzl@33741
   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}"
wenzelm@41958
   791
        { case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
wenzelm@41958
   792
        case True show "x j = p" unfolding True a3_def using j k(1) 
wenzelm@41958
   793
          using ksimplexD(5)[OF assms(1),rule_format,OF `a0\<in>s` j] by auto qed
hoelzl@33741
   794
      fix y assume y:"y\<in>insert a3 (s - {a1})"
hoelzl@33741
   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
hoelzl@33741
   796
        have "kle n a3 a2" proof- have "kle n a0 a1" using a0a1 by auto then guess kk unfolding kle_def ..
hoelzl@33741
   797
          thus ?thesis unfolding kle_def apply(rule_tac x=kk in exI) unfolding lem4[rule_format] k(2)[rule_format]
hoelzl@33741
   798
            apply(rule)defer proof(rule) case goal1 thus ?case apply-apply(erule conjE)
hoelzl@33741
   799
              apply(erule_tac[!] x=j in allE) apply(cases "j\<in>kk") apply(case_tac[!] "j=k") by auto qed auto qed moreover
hoelzl@33741
   800
        have "kle n a3 a0" unfolding kle_def lem4[rule_format] apply(rule_tac x="{k}" in exI) using k(1) by auto
hoelzl@33741
   801
        ultimately show ?case apply-apply(rule kle_between_l[of _ a0 _ a2]) using lem3[OF *]
hoelzl@33741
   802
          using a0a1(4)[rule_format,OF goal1(1)] by auto qed
hoelzl@33741
   803
      show "kle n x y \<or> kle n y x" proof(cases "y=a3")
wenzelm@41958
   804
        case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI2,rule lem4)
wenzelm@41958
   805
          using x by auto next
wenzelm@41958
   806
        case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True
wenzelm@41958
   807
            apply(rule disjI1,rule lem4) using y False by auto next
wenzelm@41958
   808
          case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
wenzelm@41958
   809
            using x y `y\<noteq>a3` by auto qed qed qed
hoelzl@33741
   810
    hence "insert a3 (s - {a1}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
hoelzl@33741
   811
      apply(rule_tac x="a3" in bexI) unfolding `a=a1` using `a3\<notin>s` by auto moreover
hoelzl@33741
   812
    have "s \<in> ?A" using assms(1,2) by auto ultimately have  "?A \<supseteq> {s, insert a3 (s - {a1})}" by auto
hoelzl@33741
   813
    moreover have "?A \<subseteq> {s, insert a3 (s - {a1})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
hoelzl@33741
   814
      fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}"
hoelzl@33741
   815
      from this(2) guess a' .. note a'=this
hoelzl@33741
   816
      guess a_min a_max apply(rule ksimplex_extrema_strong[OF as assms(3)]) . note min_max=this
hoelzl@33741
   817
      have *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}"
wenzelm@41958
   818
        hence "kle n x a2" apply-apply(rule lem3) using `a=a1` by auto
wenzelm@41958
   819
        hence "x k \<le> a2 k" apply(drule_tac kle_imp_pointwise) by auto moreover
wenzelm@41958
   820
        { have "a2 k \<le> a0 k" using k(2)[rule_format,of k] unfolding a0a1(5)[rule_format] using k(1) by simp
wenzelm@41958
   821
          also have "\<dots> \<le> x k" using a0a1(4)[rule_format,of x,THEN conjunct1,THEN kle_imp_pointwise] x by auto
wenzelm@41958
   822
          finally have "a2 k \<le> x k" . } ultimately show "x k = a2 k" by auto qed
hoelzl@33741
   823
      have **:"a'=a_min \<or> a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto
hoelzl@33741
   824
      have "a2 \<noteq> a1" proof assume as:"a2 = a1"
wenzelm@41958
   825
        show False using k(2) unfolding as apply(erule_tac x=k in allE) by auto qed
hoelzl@33741
   826
      hence a2':"a2 \<in> s' - {a'}" unfolding a' using a2 unfolding `a=a1` by auto
hoelzl@33741
   827
      show "s' \<in> {s, insert a3 (s - {a1})}" proof(cases "a'=a_min")
wenzelm@41958
   828
        case True have "a_max \<in> s - {a1}" using min_max unfolding a'(2)[unfolded `a=a1`,THEN sym] True by auto
wenzelm@41958
   829
        hence "a_max = a2" unfolding kle_antisym[THEN sym,of a_max a2 n] apply-apply(rule)
wenzelm@41958
   830
          apply(rule lem3,assumption) apply(rule min_max(4)[rule_format,THEN conjunct2]) using a2' by auto
wenzelm@41958
   831
        hence a_max:"\<forall>i. a_max i = a2 i" by auto
wenzelm@41958
   832
        have *:"\<forall>j. a2 j = (if j\<in>{1..n} then a3 j + 1 else a3 j)" 
wenzelm@41958
   833
          using k(2) unfolding lem4[rule_format] a0a1(5)[rule_format] apply-apply(rule,erule_tac x=j in allE)
wenzelm@41958
   834
        proof- case goal1 thus ?case apply(cases "j\<in>{1..n}",case_tac[!] "j=k") by auto qed
wenzelm@41958
   835
        have "\<forall>i. a_min i = a3 i" using a_max apply-apply(rule,erule_tac x=i in allE)
wenzelm@41958
   836
          unfolding min_max(5)[rule_format] *[rule_format] proof- case goal1
wenzelm@41958
   837
          thus ?case apply(cases "i\<in>{1..n}") by auto qed hence "a_min = a3" unfolding fun_eq_iff .
wenzelm@41958
   838
        hence "s' = insert a3 (s - {a1})" using a' unfolding `a=a1` True by auto thus ?thesis by auto next
wenzelm@41958
   839
        case False hence as:"a'=a_max" using ** by auto
wenzelm@41958
   840
        have "a_min = a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
wenzelm@41958
   841
          apply(rule min_max(4)[rule_format,THEN conjunct1]) defer apply(rule a0a1(4)[rule_format,THEN conjunct1]) proof-
wenzelm@41958
   842
          have "a_min \<in> s - {a1}" using min_max(1,3) unfolding a'(2)[THEN sym,unfolded `a=a1`] as by auto
wenzelm@41958
   843
          thus "a_min \<in> s" by auto have "a0 \<in> s - {a1}" using a0a1(1-3) by auto thus "a0 \<in> s'"
wenzelm@41958
   844
            unfolding a'(2)[THEN sym,unfolded `a=a1`] by auto qed
wenzelm@41958
   845
        hence "\<forall>i. a_max i = a1 i" unfolding a0a1(5)[rule_format] min_max(5)[rule_format] by auto
wenzelm@41958
   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
wenzelm@41958
   847
        thus ?thesis by auto qed qed 
hoelzl@33741
   848
    ultimately have *:"?A = {s, insert a3 (s - {a1})}" by blast
hoelzl@33741
   849
    have "s \<noteq> insert a3 (s - {a1})" using `a3\<notin>s` by auto
hoelzl@33741
   850
    hence ?thesis unfolding * by auto } moreover
hoelzl@33741
   851
  { assume as:"a\<noteq>a0" "a\<noteq>a1" have "\<not> (\<forall>x\<in>s. kle n a x)" proof case goal1
hoelzl@33741
   852
      have "a=a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
wenzelm@41958
   853
        using goal1 a0a1 assms(2) by auto thus False using as by auto qed
hoelzl@33741
   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
hoelzl@33741
   855
    then guess u .. from this(2) guess k .. note k = this[rule_format] note u = `u\<in>s`
hoelzl@33741
   856
    have "\<not> (\<forall>x\<in>s. kle n x a)" proof case goal1
hoelzl@33741
   857
      have "a=a1" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
wenzelm@41958
   858
        using goal1 a0a1 assms(2) by auto thus False using as by auto qed
hoelzl@33741
   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
hoelzl@33741
   860
    then guess v .. from this(2) guess l .. note l = this[rule_format] note v = `v\<in>s`
hoelzl@33741
   861
    def a' \<equiv> "\<lambda>j. if j = l then u j + 1 else u j"
hoelzl@33741
   862
    have kl:"k \<noteq> l" proof assume "k=l" have *:"\<And>P. (if P then (1::nat) else 0) \<noteq> 2" by auto
hoelzl@33741
   863
      thus False using ksimplexD(6)[OF assms(1),rule_format,OF u v] unfolding kle_def
wenzelm@41958
   864
        unfolding l(2) k(2) `k=l` apply-apply(erule disjE)apply(erule_tac[!] exE conjE)+
wenzelm@41958
   865
        apply(erule_tac[!] x=l in allE)+ by(auto simp add: *) qed
nipkow@39302
   866
    hence aa':"a'\<noteq>a" apply-apply rule unfolding fun_eq_iff unfolding a'_def k(2)
hoelzl@33741
   867
      apply(erule_tac x=l in allE) by auto
hoelzl@33741
   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'")
hoelzl@33741
   869
      case goal2 hence "kle n a' a" by auto thus False apply(drule_tac kle_imp_pointwise)
wenzelm@41958
   870
        apply(erule_tac x=l in allE) unfolding a'_def k(2) using kl by auto next
hoelzl@33741
   871
      case True thus False apply(drule_tac kle_imp_pointwise)
wenzelm@41958
   872
        apply(erule_tac x=k in allE) unfolding a'_def k(2) using kl by auto qed
hoelzl@33741
   873
    have kle_uv:"kle n u a" "kle n u a'" "kle n a v" "kle n a' v" unfolding kle_def apply-
hoelzl@33741
   874
      apply(rule_tac[1] x="{k}" in exI,rule_tac[2] x="{l}" in exI)
hoelzl@33741
   875
      apply(rule_tac[3] x="{l}" in exI,rule_tac[4] x="{k}" in exI)
hoelzl@33741
   876
      unfolding l(2) k(2) a'_def using l(1) k(1) by auto
hoelzl@33741
   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)"
hoelzl@33741
   878
    proof- case goal1 thus ?case proof(cases "x k = u k", case_tac[!] "x l = u l")
hoelzl@33741
   879
      assume as:"x l = u l" "x k = u k"
nipkow@39302
   880
      have "x = u" unfolding fun_eq_iff
wenzelm@41958
   881
        using goal1(2)[THEN kle_imp_pointwise,unfolded l(2)] unfolding k(2) apply-
wenzelm@41958
   882
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
wenzelm@41958
   883
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
hoelzl@33741
   884
      assume as:"x l \<noteq> u l" "x k = u k"
nipkow@39302
   885
      have "x = a'" unfolding fun_eq_iff unfolding a'_def
wenzelm@41958
   886
        using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
wenzelm@41958
   887
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
wenzelm@41958
   888
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
hoelzl@33741
   889
      assume as:"x l = u l" "x k \<noteq> u k"
nipkow@39302
   890
      have "x = a" unfolding fun_eq_iff
wenzelm@41958
   891
        using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
wenzelm@41958
   892
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
wenzelm@41958
   893
        thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
hoelzl@33741
   894
      assume as:"x l \<noteq> u l" "x k \<noteq> u k"
nipkow@39302
   895
      have "x = v" unfolding fun_eq_iff
wenzelm@41958
   896
        using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
wenzelm@41958
   897
        using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
wenzelm@41958
   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
hoelzl@33741
   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
hoelzl@33741
   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])
hoelzl@33741
   901
      prefer 3 apply(rule kle_trans[OF uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format])
hoelzl@33741
   902
      using kle_uv `u\<in>s` by auto
hoelzl@33741
   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])
hoelzl@33741
   904
      prefer 4 apply(rule kle_trans[OF _ uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format])
hoelzl@33741
   905
      using kle_uv `v\<in>s` by auto
hoelzl@33741
   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
hoelzl@33741
   907
      proof(cases "kle n v x \<or> kle n x u") case True thus ?thesis using goal1 by(auto intro:lem3 lem4) next
wenzelm@41958
   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
wenzelm@41958
   909
        show ?thesis using uxv[OF *] using kle_uv using goal1 by auto qed qed
hoelzl@33741
   910
    have "ksimplex p n (insert a' (s - {a}))" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI)
hoelzl@33741
   911
      show "card (insert a' (s - {a})) = n + 1" using ksimplexD(2-3)[OF assms(1)]
hoelzl@33741
   912
        using `a'\<noteq>a` `a'\<notin>s` `a\<in>s` by(auto simp add:card_insert_if)
hoelzl@33741
   913
      fix x assume x:"x \<in> insert a' (s - {a})"
hoelzl@33741
   914
      show "\<forall>j. x j \<le> p" proof(rule,cases "x = a'")
wenzelm@41958
   915
        fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
wenzelm@41958
   916
        fix j case True show "x j\<le>p" unfolding True proof(cases "j=l") 
wenzelm@41958
   917
          case False thus "a' j \<le>p" unfolding True a'_def using `u\<in>s` ksimplexD(4)[OF assms(1)] by auto next
wenzelm@41958
   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
wenzelm@41958
   919
          have "u l + 1 \<le> p" unfolding *[THEN sym] using ksimplexD(4)[OF assms(1)] using `v\<in>s` by auto
wenzelm@41958
   920
          thus "a' j \<le>p" unfolding a'_def True by auto qed qed
hoelzl@33741
   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}"
wenzelm@41958
   922
        { case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
wenzelm@41958
   923
        case True show "x j = p" unfolding True a'_def using j l(1) 
wenzelm@41958
   924
          using ksimplexD(5)[OF assms(1),rule_format,OF `u\<in>s` j] by auto qed
hoelzl@33741
   925
      fix y assume y:"y\<in>insert a' (s - {a})"
hoelzl@33741
   926
      show "kle n x y \<or> kle n y x" proof(cases "y=a'")
wenzelm@41958
   927
        case True show ?thesis unfolding True apply(cases "x=a'") defer apply(rule lem5) using x by auto next
wenzelm@41958
   928
        case False show ?thesis proof(cases "x=a'") case True show ?thesis unfolding True
wenzelm@41958
   929
            using lem5[of y] using y by auto next
wenzelm@41958
   930
          case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format]) 
wenzelm@41958
   931
            using x y `y\<noteq>a'` by auto qed qed qed
hoelzl@33741
   932
    hence "insert a' (s - {a}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
hoelzl@33741
   933
      apply(rule_tac x="a'" in bexI) using aa' `a'\<notin>s` by auto moreover
hoelzl@33741
   934
    have "s \<in> ?A" using assms(1,2) by auto ultimately have  "?A \<supseteq> {s, insert a' (s - {a})}" by auto
hoelzl@33741
   935
    moreover have "?A \<subseteq> {s, insert a' (s - {a})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
hoelzl@33741
   936
      fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}"
hoelzl@33741
   937
      from this(2) guess a'' .. note a''=this
nipkow@39302
   938
      have "u\<noteq>v" unfolding fun_eq_iff unfolding l(2) k(2) by auto
hoelzl@33741
   939
      hence uv':"\<not> kle n v u" using uv using kle_antisym by auto
nipkow@39302
   940
      have "u\<noteq>a" "v\<noteq>a" unfolding fun_eq_iff k(2) l(2) by auto 
hoelzl@33741
   941
      hence uvs':"u\<in>s'" "v\<in>s'" using `u\<in>s` `v\<in>s` using a'' by auto
hoelzl@33741
   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")
wenzelm@41958
   943
        case False then guess w unfolding ball_simps .. note w=this
wenzelm@41958
   944
        hence "kle n u w" "kle n w v" using ksimplexD(6)[OF as] uvs' by auto
wenzelm@41958
   945
        hence "w = a' \<or> w = a" using uxv[of w] uvs' w by auto thus ?thesis using w by auto next
wenzelm@41958
   946
        case True have "\<not> (\<forall>x\<in>s'. kle n x u)" unfolding ball_simps apply(rule_tac x=v in bexI)
wenzelm@41958
   947
          using uv `u\<noteq>v` unfolding kle_antisym[of n u v,THEN sym] using `v\<in>s'` by auto
wenzelm@41958
   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
wenzelm@41958
   949
        then guess w .. note w=this from this(2) guess kk .. note kk=this[rule_format]
wenzelm@41958
   950
        have "\<not> kle n w u" apply-apply(rule,drule kle_imp_pointwise) 
wenzelm@41958
   951
          apply(erule_tac x=kk in allE) unfolding kk by auto 
wenzelm@41958
   952
        hence *:"kle n v w" using True[rule_format,OF w(1)] by auto
wenzelm@41958
   953
        hence False proof(cases "kk\<noteq>l") case True thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)]
wenzelm@41958
   954
            apply(erule_tac x=l in allE) using `k\<noteq>l` by auto  next
wenzelm@41958
   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)]
wenzelm@41958
   956
            apply(erule_tac x=k in allE) using `k\<noteq>l` by auto qed
wenzelm@41958
   957
        thus ?thesis by auto qed
hoelzl@33741
   958
      thus "s' \<in> {s, insert a' (s - {a})}" proof(cases "a\<in>s'")
wenzelm@41958
   959
        case True hence "s' = s" apply-apply(rule lem1[OF a''(2)]) using a'' `a\<in>s` by auto
wenzelm@41958
   960
        thus ?thesis by auto next case False hence "a'\<in>s'" using lem6 by auto
wenzelm@41958
   961
        hence "s' = insert a' (s - {a})" apply-apply(rule lem1[of _ a'' _ a'])
wenzelm@41958
   962
          unfolding a''(2)[THEN sym] using a'' using `a'\<notin>s` by auto
wenzelm@41958
   963
        thus ?thesis by auto qed qed 
hoelzl@33741
   964
    ultimately have *:"?A = {s, insert a' (s - {a})}" by blast
hoelzl@33741
   965
    have "s \<noteq> insert a' (s - {a})" using `a'\<notin>s` by auto
hoelzl@33741
   966
    hence ?thesis unfolding * by auto } 
hoelzl@33741
   967
  ultimately show ?thesis by auto qed
hoelzl@33741
   968
hoelzl@33741
   969
subsection {* Hence another step towards concreteness. *}
hoelzl@33741
   970
hoelzl@33741
   971
lemma kuhn_simplex_lemma:
hoelzl@33741
   972
  assumes "\<forall>s. ksimplex p (n + 1) s \<longrightarrow> (rl ` s \<subseteq>{0..n+1})"
hoelzl@33741
   973
  "odd (card{f. \<exists>s a. ksimplex p (n + 1) s \<and> a \<in> s \<and> (f = s - {a}) \<and>
hoelzl@33741
   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))})"
hoelzl@33741
   975
  shows "odd(card {s\<in>{s. ksimplex p (n + 1) s}. rl ` s = {0..n+1} })" proof-
hoelzl@33741
   976
  have *:"\<And>x y. x = y \<Longrightarrow> odd (card x) \<Longrightarrow> odd (card y)" by auto
hoelzl@33741
   977
  have *:"odd(card {f\<in>{f. \<exists>s\<in>{s. ksimplex p (n + 1) s}. (\<exists>a\<in>s. f = s - {a})}. 
hoelzl@33741
   978
                (rl ` f = {0..n}) \<and>
hoelzl@33741
   979
               ((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or>
hoelzl@33741
   980
                (\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p))})" apply(rule *[OF _ assms(2)]) by auto
hoelzl@33741
   981
  show ?thesis apply(rule kuhn_complete_lemma[OF finite_simplices]) prefer 6 apply(rule *) apply(rule,rule,rule)
hoelzl@33741
   982
    apply(subst ksimplex_def) defer apply(rule,rule assms(1)[rule_format]) unfolding mem_Collect_eq apply assumption
hoelzl@33741
   983
    apply default+ unfolding mem_Collect_eq apply(erule disjE bexE)+ defer apply(erule disjE bexE)+ defer 
hoelzl@33741
   984
    apply default+ unfolding mem_Collect_eq apply(erule disjE bexE)+ unfolding mem_Collect_eq proof-
hoelzl@33741
   985
    fix f s a assume as:"ksimplex p (n + 1) s" "a\<in>s" "f = s - {a}"
hoelzl@33741
   986
    let ?S = "{s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})}"
hoelzl@33741
   987
    have S:"?S = {s'. ksimplex p (n + 1) s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})}" unfolding as by blast
hoelzl@33741
   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
wenzelm@41958
   989
        apply-apply(rule ksimplex_replace_0) apply(rule as)+ unfolding as by auto }
hoelzl@33741
   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
wenzelm@41958
   991
        apply-apply(rule ksimplex_replace_1) apply(rule as)+ unfolding as by auto }
hoelzl@33741
   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"
hoelzl@33741
   993
      unfolding S apply(rule ksimplex_replace_2) apply(rule as)+ unfolding as by auto qed auto qed
hoelzl@33741
   994
hoelzl@33741
   995
subsection {* Reduced labelling. *}
hoelzl@33741
   996
hoelzl@33741
   997
definition "reduced label (n::nat) (x::nat\<Rightarrow>nat) =
hoelzl@33741
   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)))"
hoelzl@33741
   999
hoelzl@33741
  1000
lemma reduced_labelling: shows "reduced label n x \<le> n" (is ?t1) and
hoelzl@33741
  1001
  "\<forall>i. 1\<le>i \<and> i < reduced label n x + 1 \<longrightarrow> (label x i = 0)" (is ?t2)
hoelzl@33741
  1002
  "(reduced label n x = n) \<or> (label x (reduced label n x + 1) \<noteq> 0)"  (is ?t3) proof-
hoelzl@33741
  1003
  have num_WOP:"\<And>P k. P (k::nat) \<Longrightarrow> \<exists>n. P n \<and> (\<forall>m<n. \<not> P m)"
hoelzl@33741
  1004
    apply(drule ex_has_least_nat[where m="\<lambda>x. x"]) apply(erule exE,rule_tac x=x in exI) by auto
hoelzl@33741
  1005
  have *:"n \<le> n \<and> (label x (n + 1) \<noteq> 0 \<or> n = n)" by auto
hoelzl@33741
  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
hoelzl@33741
  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)
hoelzl@33741
  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)
hoelzl@33741
  1009
  show ?t1 ?t2 ?t3 unfolding reduced_def apply(rule_tac[!] someI2_ex) using N' by(auto intro!: exI[where x=N]) qed
hoelzl@33741
  1010
hoelzl@33741
  1011
lemma reduced_labelling_unique: fixes x::"nat \<Rightarrow> nat"
hoelzl@33741
  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)"
hoelzl@33741
  1013
  shows "reduced label n x = r" apply(rule le_antisym) apply(rule_tac[!] ccontr) unfolding not_le
hoelzl@33741
  1014
  using reduced_labelling[of label n x] using assms by auto
hoelzl@33741
  1015
hoelzl@33741
  1016
lemma reduced_labelling_0: assumes "j\<in>{1..n}" "label x j = 0" shows "reduced label n x \<noteq> j - 1"
hoelzl@33741
  1017
  using reduced_labelling[of label n x] using assms by fastsimp 
hoelzl@33741
  1018
hoelzl@33741
  1019
lemma reduced_labelling_1: assumes "j\<in>{1..n}" "label x j \<noteq> 0" shows "reduced label n x < j"
hoelzl@33741
  1020
  using assms and reduced_labelling[of label n x] apply(erule_tac x=j in allE) by auto
hoelzl@33741
  1021
hoelzl@33741
  1022
lemma reduced_labelling_Suc:
hoelzl@33741
  1023
  assumes "reduced lab (n + 1) x \<noteq> n + 1" shows "reduced lab (n + 1) x = reduced lab n x"
hoelzl@33741
  1024
  apply(subst eq_commute) apply(rule reduced_labelling_unique)
hoelzl@33741
  1025
  using reduced_labelling[of lab "n+1" x] and assms by auto 
hoelzl@33741
  1026
hoelzl@33741
  1027
lemma complete_face_top:
hoelzl@33741
  1028
  assumes "\<forall>x\<in>f. \<forall>j\<in>{1..n+1}. x j = 0 \<longrightarrow> lab x j = 0"
hoelzl@33741
  1029
          "\<forall>x\<in>f. \<forall>j\<in>{1..n+1}. x j = p \<longrightarrow> lab x j = 1"
hoelzl@33741
  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>
hoelzl@33741
  1031
  ((reduced lab (n + 1)) ` f = {0..n}) \<and> (\<forall>x\<in>f. x (n + 1) = p)" (is "?l = ?r") proof
hoelzl@33741
  1032
  assume ?l (is "?as \<and> (?a \<or> ?b)") thus ?r apply-apply(rule,erule conjE,assumption) proof(cases ?a)
hoelzl@33741
  1033
    case True then guess j .. note j=this {fix x assume x:"x\<in>f"
hoelzl@33741
  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 }
hoelzl@33741
  1035
    moreover have "j - 1 \<in> {0..n}" using j by auto
hoelzl@33741
  1036
    then guess y unfolding `?l`[THEN conjunct1,THEN sym] and image_iff .. note y = this
hoelzl@33741
  1037
    ultimately have False by auto thus "\<forall>x\<in>f. x (n + 1) = p" by auto next
hoelzl@33741
  1038
hoelzl@33741
  1039
    case False hence ?b using `?l` by blast then guess j .. note j=this {fix x assume x:"x\<in>f"
hoelzl@33741
  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
hoelzl@33741
  1041
    have "j = n + 1" proof(rule ccontr) case goal1 hence "j < n + 1" using j by auto moreover
hoelzl@33741
  1042
      have "n \<in> {0..n}" by auto then guess y unfolding `?l`[THEN conjunct1,THEN sym] image_iff ..
hoelzl@33741
  1043
      ultimately show False using *[of y] by auto qed
hoelzl@33741
  1044
    thus "\<forall>x\<in>f. x (n + 1) = p" using j by auto qed qed(auto)
hoelzl@33741
  1045
hoelzl@33741
  1046
subsection {* Hence we get just about the nice induction. *}
hoelzl@33741
  1047
hoelzl@33741
  1048
lemma kuhn_induction:
hoelzl@33741
  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)"
hoelzl@33741
  1050
                  "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
hoelzl@33741
  1051
        "odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})"
hoelzl@33741
  1052
  shows "odd (card {s. ksimplex p (n+1) s \<and>((reduced lab (n+1)) `  s = {0..n+1})})" proof-
hoelzl@33741
  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
hoelzl@33741
  1054
  show ?thesis apply(rule kuhn_simplex_lemma[unfolded mem_Collect_eq]) apply(rule,rule,rule *,rule reduced_labelling)
nipkow@39302
  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)*)
hoelzl@33741
  1056
    fix f assume as:"ksimplex p n f" "reduced lab n ` f = {0..n}"
hoelzl@33741
  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"
hoelzl@33741
  1058
      using assms(2-3) using as(1)[unfolded ksimplex_def] by auto
hoelzl@33741
  1059
    have allp:"\<forall>x\<in>f. x (n + 1) = p" using assms(2) using as(1)[unfolded ksimplex_def] by auto
hoelzl@33741
  1060
    { fix x assume "x\<in>f" hence "reduced lab (n + 1) x < n + 1" apply-apply(rule reduced_labelling_1)
wenzelm@41958
  1061
        defer using assms(3) using as(1)[unfolded ksimplex_def] by auto
hoelzl@33741
  1062
      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc) using reduced_labelling(1) by auto }
nipkow@39302
  1063
    hence "reduced lab (n + 1) ` f = {0..n}" unfolding as(2)[THEN sym] apply- apply(rule set_eqI) unfolding image_iff by auto
hoelzl@33741
  1064
    moreover guess s using as(1)[unfolded simplex_top_face[OF assms(1) allp,THEN sym]] .. then guess a ..
hoelzl@33741
  1065
    ultimately show "\<exists>s a. ksimplex p (n + 1) s \<and>
hoelzl@33741
  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)
hoelzl@33741
  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
hoelzl@33741
  1068
  next fix f assume as:"\<exists>s a. ksimplex p (n + 1) s \<and>
hoelzl@33741
  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)
hoelzl@33741
  1070
    then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
hoelzl@33741
  1071
    { fix x assume "x\<in>f" hence "reduced lab (n + 1) x \<in> reduced lab (n + 1) ` f" by auto
hoelzl@33741
  1072
      hence "reduced lab (n + 1) x < n + 1" using sa(4) by auto 
hoelzl@33741
  1073
      hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc)
wenzelm@41958
  1074
        using reduced_labelling(1) by auto }
nipkow@39302
  1075
    thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[THEN sym] apply-apply(rule set_eqI) unfolding image_iff by auto
hoelzl@33741
  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")
hoelzl@33741
  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
wenzelm@41958
  1078
        apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover
hoelzl@33741
  1079
      have "j - 1 \<in> {0..n}" using `j\<in>{1..n+1}` by auto
hoelzl@33741
  1080
      ultimately have False unfolding sa(4)[THEN sym] unfolding image_iff by fastsimp thus ?thesis by auto next
hoelzl@33741
  1081
      case False hence "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p" using sa(5) by fastsimp then guess j .. note j=this
hoelzl@33741
  1082
      thus ?thesis proof(cases "j = n+1")
wenzelm@41958
  1083
        case False hence *:"j\<in>{1..n}" using j by auto
wenzelm@41958
  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"
wenzelm@41958
  1085
          hence "lab x j = 1" apply-apply(rule assms(3)[rule_format,OF j(1)]) 
wenzelm@41958
  1086
            using sa(1)[unfolded ksimplex_def] using j unfolding sa by auto thus "lab x j \<noteq> 0" by auto qed
wenzelm@41958
  1087
        moreover have "j\<in>{0..n}" using * by auto
wenzelm@41958
  1088
        ultimately have False unfolding part1[THEN sym] using * unfolding image_iff by auto thus ?thesis by auto qed auto qed 
hoelzl@33741
  1089
    thus "ksimplex p n f" using as unfolding simplex_top_face[OF assms(1) *,THEN sym] by auto qed qed
hoelzl@33741
  1090
hoelzl@33741
  1091
lemma kuhn_induction_Suc:
hoelzl@33741
  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)"
hoelzl@33741
  1093
                  "\<forall>x. \<forall>j\<in>{1..Suc n}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
hoelzl@33741
  1094
        "odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})"
hoelzl@33741
  1095
  shows "odd (card {s. ksimplex p (Suc n) s \<and>((reduced lab (Suc n)) `  s = {0..Suc n})})"
hoelzl@33741
  1096
  using assms unfolding Suc_eq_plus1 by(rule kuhn_induction)
hoelzl@33741
  1097
hoelzl@33741
  1098
subsection {* And so we get the final combinatorial result. *}
hoelzl@33741
  1099
hoelzl@33741
  1100
lemma ksimplex_0: "ksimplex p 0 s \<longleftrightarrow> s = {(\<lambda>x. p)}" (is "?l = ?r") proof
hoelzl@33741
  1101
  assume l:?l guess a using ksimplexD(3)[OF l, unfolded add_0] unfolding card_1_exists .. note a=this
hoelzl@33741
  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
hoelzl@33741
  1103
  assume r:?r show ?l unfolding r ksimplex_eq by auto qed
hoelzl@33741
  1104
hoelzl@33741
  1105
lemma reduce_labelling_0[simp]: "reduced lab 0 x = 0" apply(rule reduced_labelling_unique) by auto
hoelzl@33741
  1106
hoelzl@33741
  1107
lemma kuhn_combinatorial:
hoelzl@33741
  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)"
hoelzl@33741
  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)"
hoelzl@33741
  1110
  shows " odd (card {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})})" using assms proof(induct n)
hoelzl@33741
  1111
  let ?M = "\<lambda>n. {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})}"
hoelzl@33741
  1112
  { case 0 have *:"?M 0 = {{(\<lambda>x. p)}}" unfolding ksimplex_0 by auto show ?case unfolding * by auto }
hoelzl@33741
  1113
  case (Suc n) have "odd (card (?M n))" apply(rule Suc(1)[OF Suc(2)]) using Suc(3-) by auto
hoelzl@33741
  1114
  thus ?case apply-apply(rule kuhn_induction_Suc) using Suc(2-) by auto qed
hoelzl@33741
  1115
hoelzl@33741
  1116
lemma kuhn_lemma: assumes "0 < (p::nat)" "0 < (n::nat)"
hoelzl@33741
  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))"
hoelzl@33741
  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))"
hoelzl@33741
  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))"
hoelzl@33741
  1120
  obtains q where "\<forall>i\<in>{1..n}. q i < p"
hoelzl@33741
  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>
hoelzl@33741
  1122
                               (\<forall>j\<in>{1..n}. q(j) \<le> s(j) \<and> s(j) \<le> q(j) + 1) \<and>
hoelzl@33741
  1123
                               ~(label r i = label s i)" proof-
hoelzl@33741
  1124
  let ?A = "{s. ksimplex p n s \<and> reduced label n ` s = {0..n}}" have "n\<noteq>0" using assms by auto
hoelzl@33741
  1125
  have conjD:"\<And>P Q. P \<and> Q \<Longrightarrow> P" "\<And>P Q. P \<and> Q \<Longrightarrow> Q" by auto
hoelzl@33741
  1126
  have "odd (card ?A)" apply(rule kuhn_combinatorial[of p n label]) using assms by auto
hoelzl@33741
  1127
  hence "card ?A \<noteq> 0" apply-apply(rule ccontr) by auto hence "?A \<noteq> {}" unfolding card_eq_0_iff by auto
hoelzl@33741
  1128
  then obtain s where "s\<in>?A" by auto note s=conjD[OF this[unfolded mem_Collect_eq]]
hoelzl@33741
  1129
  guess a b apply(rule ksimplex_extrema_strong[OF s(1) `n\<noteq>0`]) . note ab=this
hoelzl@33741
  1130
  show ?thesis apply(rule that[of a]) proof(rule_tac[!] ballI) fix i assume "i\<in>{1..n}"
hoelzl@33741
  1131
    hence "a i + 1 \<le> p" apply-apply(rule order_trans[of _ "b i"]) apply(subst ab(5)[THEN spec[where x=i]])
hoelzl@33741
  1132
      using s(1)[unfolded ksimplex_def] defer apply- apply(erule conjE)+ apply(drule_tac bspec[OF _ ab(2)])+ by auto
hoelzl@33741
  1133
    thus "a i < p" by auto
hoelzl@33741
  1134
    case goal2 hence "i \<in> reduced label n ` s" using s by auto then guess u unfolding image_iff .. note u=this
hoelzl@33741
  1135
    from goal2 have "i - 1 \<in> reduced label n ` s" using s by auto then guess v unfolding image_iff .. note v=this
hoelzl@33741
  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)
hoelzl@33741
  1137
      show "label u i \<noteq> label v i" using reduced_labelling[of label n u] reduced_labelling[of label n v]
hoelzl@33741
  1138
        unfolding u(2)[THEN sym] v(2)[THEN sym] using goal2 by auto
hoelzl@33741
  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"
hoelzl@33741
  1140
        using conjD[OF ab(4)[rule_format, OF u(1)]] and conjD[OF ab(4)[rule_format, OF v(1)]] apply- 
hoelzl@33741
  1141
        apply(drule_tac[!] kle_imp_pointwise)+ apply(erule_tac[!] x=j in allE)+ unfolding ab(5)[rule_format] using j
hoelzl@33741
  1142
        by auto qed qed qed
hoelzl@33741
  1143
hoelzl@33741
  1144
subsection {* The main result for the unit cube. *}
hoelzl@33741
  1145
hoelzl@33741
  1146
lemma kuhn_labelling_lemma':
hoelzl@33741
  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)"
hoelzl@33741
  1148
  shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
hoelzl@33741
  1149
             (\<forall>x i. P x \<and> Q i \<and> (x i = 0) \<longrightarrow> (l x i = 0)) \<and>
hoelzl@33741
  1150
             (\<forall>x i. P x \<and> Q i \<and> (x i = 1) \<longrightarrow> (l x i = 1)) \<and>
hoelzl@33741
  1151
             (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x i \<le> f(x) i) \<and>
hoelzl@33741
  1152
             (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x) i \<le> x i)" proof-
hoelzl@33741
  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
hoelzl@33741
  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
hoelzl@33741
  1155
  show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
hoelzl@33741
  1156
    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x xa = 0 \<longrightarrow> y = (0::nat)) \<and>
hoelzl@33741
  1157
        (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)"
hoelzl@33741
  1158
    { assume "P x" "Q xa" hence "0 \<le> (f x) xa \<and> (f x) xa \<le> 1" using assms(2)[rule_format,of "f x" xa]
hoelzl@33741
  1159
        apply(drule_tac assms(1)[rule_format]) by auto }
hoelzl@33741
  1160
    hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed 
hoelzl@33741
  1161
hoelzl@37489
  1162
lemma brouwer_cube: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
hoelzl@37489
  1163
  assumes "continuous_on {0..\<chi>\<chi> i. 1} f" "f ` {0..\<chi>\<chi> i. 1} \<subseteq> {0..\<chi>\<chi> i. 1}"
hoelzl@37489
  1164
  shows "\<exists>x\<in>{0..\<chi>\<chi> i. 1}. f x = x" apply(rule ccontr) proof-
hoelzl@37489
  1165
  def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by(auto simp add:Suc_le_eq)
hoelzl@37489
  1166
  assume "\<not> (\<exists>x\<in>{0..\<chi>\<chi> i. 1}. f x = x)" hence *:"\<not> (\<exists>x\<in>{0..\<chi>\<chi> i. 1}. f x - x = 0)" by auto
hoelzl@33741
  1167
  guess d apply(rule brouwer_compactness_lemma[OF compact_interval _ *]) 
hoelzl@33741
  1168
    apply(rule continuous_on_intros assms)+ . note d=this[rule_format]
hoelzl@37489
  1169
  have *:"\<forall>x. x \<in> {0..\<chi>\<chi> i. 1} \<longrightarrow> f x \<in> {0..\<chi>\<chi> i. 1}"  "\<forall>x. x \<in> {0..(\<chi>\<chi> i. 1)::'a} \<longrightarrow>
hoelzl@37489
  1170
    (\<forall>i<DIM('a). True \<longrightarrow> 0 \<le> x $$ i \<and> x $$ i \<le> 1)"
hoelzl@33741
  1171
    using assms(2)[unfolded image_subset_iff Ball_def] unfolding mem_interval by auto
hoelzl@33741
  1172
  guess label using kuhn_labelling_lemma[OF *] apply-apply(erule exE,(erule conjE)+) . note label = this[rule_format]
hoelzl@37489
  1173
  have lem1:"\<forall>x\<in>{0..\<chi>\<chi> i. 1}.\<forall>y\<in>{0..\<chi>\<chi> i. 1}.\<forall>i<DIM('a). label x i \<noteq> label y i
hoelzl@37489
  1174
            \<longrightarrow> abs(f x $$ i - x $$ i) \<le> norm(f y - f x) + norm(y - x)" proof safe
hoelzl@37489
  1175
    fix x y::'a assume xy:"x\<in>{0..\<chi>\<chi> i. 1}" "y\<in>{0..\<chi>\<chi> i. 1}" fix i assume i:"label x i \<noteq> label y i" "i<DIM('a)"
hoelzl@33741
  1176
    have *:"\<And>x y fx fy::real. (x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy)
hoelzl@33741
  1177
             \<Longrightarrow> abs(fx - x) \<le> abs(fy - fx) + abs(y - x)" by auto
hoelzl@37489
  1178
    have "\<bar>(f x - x) $$ i\<bar> \<le> abs((f y - f x)$$i) + abs((y - x)$$i)" unfolding euclidean_simps
hoelzl@33741
  1179
      apply(rule *) apply(cases "label x i = 0") apply(rule disjI1,rule) prefer 3 proof(rule disjI2,rule)
hoelzl@37489
  1180
      assume lx:"label x i = 0" hence ly:"label y i = 1" using i label(1)[of i y] by auto
hoelzl@37489
  1181
      show "x $$ i \<le> f x $$ i" apply(rule label(4)[rule_format]) using xy lx i(2) by auto
hoelzl@37489
  1182
      show "f y $$ i \<le> y $$ i" apply(rule label(5)[rule_format]) using xy ly i(2) by auto next
hoelzl@33741
  1183
      assume "label x i \<noteq> 0" hence l:"label x i = 1" "label y i = 0"
hoelzl@37489
  1184
        using i label(1)[of i x] label(1)[of i y] by auto
hoelzl@37489
  1185
      show "f x $$ i \<le> x $$ i" apply(rule label(5)[rule_format]) using xy l i(2) by auto
hoelzl@37489
  1186
      show "y $$ i \<le> f y $$ i" apply(rule label(4)[rule_format]) using xy l i(2) by auto qed 
hoelzl@33741
  1187
    also have "\<dots> \<le> norm (f y - f x) + norm (y - x)" apply(rule add_mono) by(rule component_le_norm)+
hoelzl@37489
  1188
    finally show "\<bar>f x $$ i - x $$ i\<bar> \<le> norm (f y - f x) + norm (y - x)" unfolding euclidean_simps . qed
hoelzl@37489
  1189
  have "\<exists>e>0. \<forall>x\<in>{0..\<chi>\<chi> i. 1}. \<forall>y\<in>{0..\<chi>\<chi> i. 1}. \<forall>z\<in>{0..\<chi>\<chi> i. 1}. \<forall>i<DIM('a).
hoelzl@37489
  1190
    norm(x - z) < e \<and> norm(y - z) < e \<and> label x i \<noteq> label y i \<longrightarrow> abs((f(z) - z)$$i) < d / (real n)" proof-
hoelzl@33741
  1191
    have d':"d / real n / 8 > 0" apply(rule divide_pos_pos)+ using d(1) unfolding n_def by auto
hoelzl@37489
  1192
    have *:"uniformly_continuous_on {0..\<chi>\<chi> i. 1} f" by(rule compact_uniformly_continuous[OF assms(1) compact_interval])
hoelzl@33741
  1193
    guess e using *[unfolded uniformly_continuous_on_def,rule_format,OF d'] apply-apply(erule exE,(erule conjE)+) .
huffman@36587
  1194
    note e=this[rule_format,unfolded dist_norm]
hoelzl@37489
  1195
    show ?thesis apply(rule_tac x="min (e/2) (d/real n/8)" in exI)
hoelzl@37489
  1196
    proof safe show "0 < min (e / 2) (d / real n / 8)" using d' e by auto
hoelzl@37489
  1197
      fix x y z i assume as:"x \<in> {0..\<chi>\<chi> i. 1}" "y \<in> {0..\<chi>\<chi> i. 1}" "z \<in> {0..\<chi>\<chi> i. 1}"
hoelzl@37489
  1198
        "norm (x - z) < min (e / 2) (d / real n / 8)"
hoelzl@37489
  1199
        "norm (y - z) < min (e / 2) (d / real n / 8)" "label x i \<noteq> label y i" and i:"i<DIM('a)"
hoelzl@33741
  1200
      have *:"\<And>z fz x fx n1 n2 n3 n4 d4 d::real. abs(fx - x) \<le> n1 + n2 \<Longrightarrow> abs(fx - fz) \<le> n3 \<Longrightarrow> abs(x - z) \<le> n4 \<Longrightarrow>
hoelzl@33741
  1201
        n1 < d4 \<Longrightarrow> n2 < 2 * d4 \<Longrightarrow> n3 < d4 \<Longrightarrow> n4 < d4 \<Longrightarrow> (8 * d4 = d) \<Longrightarrow> abs(fz - z) < d" by auto
hoelzl@37489
  1202
      show "\<bar>(f z - z) $$ i\<bar> < d / real n" unfolding euclidean_simps proof(rule *)
hoelzl@37489
  1203
        show "\<bar>f x $$ i - x $$ i\<bar> \<le> norm (f y -f x) + norm (y - x)" apply(rule lem1[rule_format]) using as i  by auto
hoelzl@37489
  1204
        show "\<bar>f x $$ i - f z $$ i\<bar> \<le> norm (f x - f z)" "\<bar>x $$ i - z $$ i\<bar> \<le> norm (x - z)"
huffman@44282
  1205
          unfolding euclidean_component_diff[THEN sym] by(rule component_le_norm)+
huffman@36587
  1206
        have tria:"norm (y - x) \<le> norm (y - z) + norm (x - z)" using dist_triangle[of y x z,unfolded dist_norm]
hoelzl@33741
  1207
          unfolding norm_minus_commute by auto
hoelzl@33741
  1208
        also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_mono) using as(4,5) by auto
hoelzl@33741
  1209
        finally show "norm (f y - f x) < d / real n / 8" apply- apply(rule e(2)) using as by auto
hoelzl@33741
  1210
        have "norm (y - z) + norm (x - z) < d / real n / 8 + d / real n / 8" apply(rule add_strict_mono) using as by auto
hoelzl@33741
  1211
        thus "norm (y - x) < 2 * (d / real n / 8)" using tria by auto
hoelzl@33741
  1212
        show "norm (f x - f z) < d / real n / 8" apply(rule e(2)) using as e(1) by auto qed(insert as, auto) qed qed
hoelzl@33741
  1213
  then guess e apply-apply(erule exE,(erule conjE)+) . note e=this[rule_format] 
hoelzl@33741
  1214
  guess p using real_arch_simple[of "1 + real n / e"] .. note p=this
hoelzl@33741
  1215
  have "1 + real n / e > 0" apply(rule add_pos_pos) defer apply(rule divide_pos_pos) using e(1) n by auto
hoelzl@33741
  1216
  hence "p > 0" using p by auto
hoelzl@37489
  1217
  def b \<equiv> "\<lambda>i. i - 1::nat" have b:"bij_betw b {1..n} {..<DIM('a)}"
hoelzl@37489
  1218
    unfolding bij_betw_def inj_on_def b_def n_def apply rule defer
hoelzl@37489
  1219
    apply safe defer unfolding image_iff apply(rule_tac x="Suc x" in bexI) by auto
hoelzl@33741
  1220
  def b' \<equiv> "inv_into {1..n} b"
hoelzl@37489
  1221
  have b':"bij_betw b' {..<DIM('a)} {1..n}" using bij_betw_inv_into[OF b] unfolding b'_def n_def by auto
hoelzl@37489
  1222
  have bb'[simp]:"\<And>i. i<DIM('a) \<Longrightarrow> b (b' i) = i" unfolding b'_def apply(rule f_inv_into_f) using b  
hoelzl@33741
  1223
    unfolding bij_betw_def by auto
hoelzl@33741
  1224
  have b'b[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> b' (b i) = i" unfolding b'_def apply(rule inv_into_f_eq)
hoelzl@33741
  1225
    using b unfolding n_def bij_betw_def by auto
hoelzl@33741
  1226
  have *:"\<And>x::nat. x=0 \<or> x=1 \<longleftrightarrow> x\<le>1" by auto
hoelzl@37489
  1227
  have b'':"\<And>j. j\<in>{1..n} \<Longrightarrow> b j <DIM('a)" using b unfolding bij_betw_def by auto
hoelzl@33741
  1228
  have q1:"0 < p" "0 < n"  "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow>
hoelzl@37489
  1229
    (\<forall>i\<in>{1..n}. (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0 \<or> (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1)"
hoelzl@37489
  1230
    unfolding * using `p>0` `n>0` using label(1)[OF b'']  by auto
hoelzl@37489
  1231
  have q2:"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = 0 \<longrightarrow> (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0)"
hoelzl@37489
  1232
    "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = p \<longrightarrow> (label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1)"
hoelzl@33741
  1233
    apply(rule,rule,rule,rule) defer proof(rule,rule,rule,rule) fix x i 
hoelzl@33741
  1234
    assume as:"\<forall>i\<in>{1..n}. x i \<le> p" "i \<in> {1..n}"
hoelzl@33741
  1235
    { assume "x i = p \<or> x i = 0" 
hoelzl@37489
  1236
      have "(\<chi>\<chi> i. real (x (b' i)) / real p) \<in> {0::'a..\<chi>\<chi> i. 1}" unfolding mem_interval 
huffman@44282
  1237
        apply safe unfolding euclidean_lambda_beta euclidean_component_zero
hoelzl@37489
  1238
      proof (simp_all only: if_P) fix j assume j':"j<DIM('a)"
hoelzl@37489
  1239
        hence j:"b' j \<in> {1..n}" using b' unfolding n_def bij_betw_def by auto
hoelzl@37489
  1240
        show "0 \<le> real (x (b' j)) / real p"
hoelzl@33741
  1241
          apply(rule divide_nonneg_pos) using `p>0` using as(1)[rule_format,OF j] by auto
hoelzl@37489
  1242
        show "real (x (b' j)) / real p \<le> 1" unfolding divide_le_eq_1
hoelzl@33741
  1243
          using as(1)[rule_format,OF j] by auto qed } note cube=this
hoelzl@37489
  1244
    { assume "x i = p" thus "(label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1" unfolding o_def
hoelzl@37489
  1245
        apply- apply(rule label(3)) apply(rule b'') using cube using as `p>0`
hoelzl@37489
  1246
      proof safe assume i:"i\<in>{1..n}"
hoelzl@37489
  1247
        show "((\<chi>\<chi> ia. real (x (b' ia)) / real (x i))::'a) $$ b i = 1"
hoelzl@37489
  1248
          unfolding euclidean_lambda_beta apply(subst if_P) apply(rule b''[OF i]) unfolding b'b[OF i] 
hoelzl@37489
  1249
          unfolding  `x i = p` using q1(1) by auto
hoelzl@37489
  1250
      qed auto }
hoelzl@37489
  1251
    { assume "x i = 0" thus "(label (\<chi>\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0" unfolding o_def
hoelzl@37489
  1252
        apply-apply(rule label(2)[OF b'']) using cube using as `p>0`
hoelzl@37489
  1253
      proof safe assume i:"i\<in>{1..n}"
hoelzl@37489
  1254
        show "((\<chi>\<chi> ia. real (x (b' ia)) / real p)::'a) $$ b i = 0"
hoelzl@37489
  1255
          unfolding euclidean_lambda_beta apply (subst if_P) apply(rule b''[OF i]) unfolding b'b[OF i] 
hoelzl@37489
  1256
          unfolding `x i = 0` using q1(1) by auto 
hoelzl@37489
  1257
      qed auto }
hoelzl@37489
  1258
  qed
hoelzl@33741
  1259
  guess q apply(rule kuhn_lemma[OF q1 q2]) . note q=this
hoelzl@37489
  1260
  def z \<equiv> "(\<chi>\<chi> i. real (q (b' i)) / real p)::'a"
hoelzl@37489
  1261
  have "\<exists>i<DIM('a). d / real n \<le> abs((f z - z)$$i)" proof(rule ccontr)
hoelzl@37489
  1262
    have "\<forall>i<DIM('a). q (b' i) \<in> {0..<p}" using q(1) b'[unfolded bij_betw_def] by auto 
hoelzl@37489
  1263
    hence "\<forall>i<DIM('a). q (b' i) \<in> {0..p}" apply-apply(rule,erule_tac x=i in allE) by auto
hoelzl@37489
  1264
    hence "z\<in>{0..\<chi>\<chi> i.1}" unfolding z_def mem_interval apply safe unfolding euclidean_lambda_beta
huffman@44282
  1265
      unfolding euclidean_component_zero apply (simp_all only: if_P)
hoelzl@37489
  1266
      apply(rule divide_nonneg_pos) using `p>0` unfolding divide_le_eq_1 by auto
hoelzl@33741
  1267
    hence d_fz_z:"d \<le> norm (f z - z)" apply(drule_tac d) .
hoelzl@37489
  1268
    case goal1 hence as:"\<forall>i<DIM('a). \<bar>f z $$ i - z $$ i\<bar> < d / real n" using `n>0` by(auto simp add:not_le)
huffman@44282
  1269
    have "norm (f z - z) \<le> (\<Sum>i<DIM('a). \<bar>f z $$ i - z $$ i\<bar>)" unfolding euclidean_component_diff[THEN sym] by(rule norm_le_l1)
hoelzl@37489
  1270
    also have "\<dots> < (\<Sum>i<DIM('a). d / real n)" apply(rule setsum_strict_mono) using as by auto
hoelzl@37489
  1271
    also have "\<dots> = d" unfolding real_eq_of_nat n_def using n using DIM_positive[where 'a='a] by auto
hoelzl@33741
  1272
    finally show False using d_fz_z by auto qed then guess i .. note i=this
hoelzl@37489
  1273
  have *:"b' i \<in> {1..n}" using i using b'[unfolded bij_betw_def] by auto
hoelzl@33741
  1274
  guess r using q(2)[rule_format,OF *] .. then guess s apply-apply(erule exE,(erule conjE)+) . note rs=this[rule_format]
hoelzl@37489
  1275
  have b'_im:"\<And>i. i<DIM('a) \<Longrightarrow>  b' i \<in> {1..n}" using b' unfolding bij_betw_def by auto
hoelzl@37489
  1276
  def r' \<equiv> "(\<chi>\<chi> i. real (r (b' i)) / real p)::'a"
hoelzl@37489
  1277
  have "\<And>i. i<DIM('a) \<Longrightarrow> r (b' i) \<le> p" apply(rule order_trans) apply(rule rs(1)[OF b'_im,THEN conjunct2])
hoelzl@33741
  1278
    using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
huffman@44282
  1279
  hence "r' \<in> {0..\<chi>\<chi> i. 1}"  unfolding r'_def mem_interval apply safe unfolding euclidean_lambda_beta euclidean_component_zero
hoelzl@37489
  1280
    apply (simp only: if_P)
hoelzl@37489
  1281
    apply(rule divide_nonneg_pos) using rs(1)[OF b'_im] q(1)[rule_format,OF b'_im] `p>0` by auto
hoelzl@37489
  1282
  def s' \<equiv> "(\<chi>\<chi> i. real (s (b' i)) / real p)::'a"
hoelzl@37489
  1283
  have "\<And>i. i<DIM('a) \<Longrightarrow> s (b' i) \<le> p" apply(rule order_trans) apply(rule rs(2)[OF b'_im,THEN conjunct2])
hoelzl@33741
  1284
    using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
huffman@44282
  1285
  hence "s' \<in> {0..\<chi>\<chi> i.1}" unfolding s'_def mem_interval apply safe unfolding euclidean_lambda_beta euclidean_component_zero
hoelzl@37489
  1286
    apply (simp_all only: if_P) apply(rule divide_nonneg_pos) using rs(1)[OF b'_im] q(1)[rule_format,OF b'_im] `p>0` by auto
huffman@44282
  1287
  have "z\<in>{0..\<chi>\<chi> i.1}" unfolding z_def mem_interval apply safe unfolding euclidean_lambda_beta euclidean_component_zero
hoelzl@37489
  1288
    apply (simp_all only: if_P) apply(rule divide_nonneg_pos) using q(1)[rule_format,OF b'_im] `p>0` by(auto intro:less_imp_le)
hoelzl@33741
  1289
  have *:"\<And>x. 1 + real x = real (Suc x)" by auto
hoelzl@37489
  1290
  { have "(\<Sum>i<DIM('a). \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>i<DIM('a). 1)" 
hoelzl@33741
  1291
      apply(rule setsum_mono) using rs(1)[OF b'_im] by(auto simp add:* field_simps)
hoelzl@33741
  1292
    also have "\<dots> < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
hoelzl@33741
  1293
      by(auto simp add:field_simps)
hoelzl@37489
  1294
    finally have "(\<Sum>i<DIM('a). \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" . } moreover
hoelzl@37489
  1295
  { have "(\<Sum>i<DIM('a). \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>i<DIM('a). 1)" 
hoelzl@33741
  1296
      apply(rule setsum_mono) using rs(2)[OF b'_im] by(auto simp add:* field_simps)
hoelzl@33741
  1297
    also have "\<dots> < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
hoelzl@33741
  1298
      by(auto simp add:field_simps)
hoelzl@37489
  1299
    finally have "(\<Sum>i<DIM('a). \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" . } ultimately
hoelzl@33741
  1300
  have "norm (r' - z) < e" "norm (s' - z) < e" unfolding r'_def s'_def z_def apply-
hoelzl@33741
  1301
    apply(rule_tac[!] le_less_trans[OF norm_le_l1]) using `p>0`
hoelzl@33741
  1302
    by(auto simp add:field_simps setsum_divide_distrib[THEN sym])
hoelzl@37489
  1303
  hence "\<bar>(f z - z) $$ i\<bar> < d / real n" apply-apply(rule e(2)[OF `r'\<in>{0..\<chi>\<chi> i.1}` `s'\<in>{0..\<chi>\<chi> i.1}` `z\<in>{0..\<chi>\<chi> i.1}`])
hoelzl@37489
  1304
    using rs(3) unfolding r'_def[symmetric] s'_def[symmetric] o_def bb' using i by auto
hoelzl@33741
  1305
  thus False using i by auto qed
hoelzl@33741
  1306
hoelzl@33741
  1307
subsection {* Retractions. *}
hoelzl@33741
  1308
hoelzl@37489
  1309
definition "retraction s t r \<longleftrightarrow>
hoelzl@33741
  1310
  t \<subseteq> s \<and> continuous_on s r \<and> (r ` s \<subseteq> t) \<and> (\<forall>x\<in>t. r x = x)"
hoelzl@33741
  1311
hoelzl@33741
  1312
definition retract_of (infixl "retract'_of" 12) where
hoelzl@33741
  1313
  "(t retract_of s) \<longleftrightarrow> (\<exists>r. retraction s t r)"
hoelzl@33741
  1314
hoelzl@33741
  1315
lemma retraction_idempotent: "retraction s t r \<Longrightarrow> x \<in> s \<Longrightarrow>  r(r x) = r x"
hoelzl@33741
  1316
  unfolding retraction_def by auto
hoelzl@33741
  1317
hoelzl@33741
  1318
subsection {*preservation of fixpoints under (more general notion of) retraction. *}
hoelzl@33741
  1319
hoelzl@37489
  1320
lemma invertible_fixpoint_property: fixes s::"('a::euclidean_space) set" and t::"('b::euclidean_space) set" 
hoelzl@33741
  1321
  assumes "continuous_on t i" "i ` t \<subseteq> s" "continuous_on s r" "r ` s \<subseteq> t" "\<forall>y\<in>t. r (i y) = y"
hoelzl@33741
  1322
  "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)" "continuous_on t g" "g ` t \<subseteq> t"
hoelzl@33741
  1323
  obtains y where "y\<in>t" "g y = y" proof-
hoelzl@33741
  1324
  have "\<exists>x\<in>s. (i \<circ> g \<circ> r) x = x" apply(rule assms(6)[rule_format],rule)
hoelzl@33741
  1325
    apply(rule continuous_on_compose assms)+ apply((rule continuous_on_subset)?,rule assms)+
hoelzl@33741
  1326
    using assms(2,4,8) unfolding image_compose by(auto,blast)
hoelzl@33741
  1327
    then guess x .. note x = this hence *:"g (r x) \<in> t" using assms(4,8) by auto
hoelzl@33741
  1328
    have "r ((i \<circ> g \<circ> r) x) = r x" using x by auto
hoelzl@33741
  1329
    thus ?thesis apply(rule_tac that[of "r x"]) using x unfolding o_def
hoelzl@33741
  1330
      unfolding assms(5)[rule_format,OF *] using assms(4) by auto qed
hoelzl@33741
  1331
hoelzl@33741
  1332
lemma homeomorphic_fixpoint_property:
hoelzl@37489
  1333
  fixes s::"('a::euclidean_space) set" and t::"('b::euclidean_space) set" assumes "s homeomorphic t"
hoelzl@33741
  1334
  shows "(\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)) \<longleftrightarrow>
hoelzl@33741
  1335
         (\<forall>g. continuous_on t g \<and> g ` t \<subseteq> t \<longrightarrow> (\<exists>y\<in>t. g y = y))" proof-
hoelzl@33741
  1336
  guess r using assms[unfolded homeomorphic_def homeomorphism_def] .. then guess i ..
hoelzl@33741
  1337
  thus ?thesis apply- apply rule apply(rule_tac[!] allI impI)+ 
hoelzl@33741
  1338
    apply(rule_tac g=g in invertible_fixpoint_property[of t i s r]) prefer 10
hoelzl@33741
  1339
    apply(rule_tac g=f in invertible_fixpoint_property[of s r t i]) by auto qed
hoelzl@33741
  1340
hoelzl@37489
  1341
lemma retract_fixpoint_property: fixes f::"'a::euclidean_space => 'b::euclidean_space" and s::"'a set"
hoelzl@33741
  1342
  assumes "t retract_of s"  "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)"  "continuous_on t g" "g ` t \<subseteq> t"
hoelzl@33741
  1343
  obtains y where "y \<in> t" "g y = y" proof- guess h using assms(1) unfolding retract_of_def .. 
hoelzl@33741
  1344
  thus ?thesis unfolding retraction_def apply-
hoelzl@33741
  1345
    apply(rule invertible_fixpoint_property[OF continuous_on_id _ _ _ _ assms(2), of t h g]) prefer 7
hoelzl@33741
  1346
    apply(rule_tac y=y in that) using assms by auto qed
hoelzl@33741
  1347
hoelzl@33741
  1348
subsection {*So the Brouwer theorem for any set with nonempty interior. *}
hoelzl@33741
  1349
hoelzl@37489
  1350
lemma brouwer_weak: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a::ordered_euclidean_space"
hoelzl@33741
  1351
  assumes "compact s" "convex s" "interior s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
hoelzl@33741
  1352
  obtains x where "x \<in> s" "f x = x" proof-
hoelzl@37489
  1353
  have *:"interior {0::'a..\<chi>\<chi> i.1} \<noteq> {}" unfolding interior_closed_interval interval_eq_empty by auto
hoelzl@37489
  1354
  have *:"{0::'a..\<chi>\<chi> i.1} homeomorphic s" using homeomorphic_convex_compact[OF convex_interval(1) compact_interval * assms(2,1,3)] .
hoelzl@37489
  1355
  have "\<forall>f. continuous_on {0::'a..\<chi>\<chi> i.1} f \<and> f ` {0::'a..\<chi>\<chi> i.1} \<subseteq> {0::'a..\<chi>\<chi> i.1} \<longrightarrow> (\<exists>x\<in>{0::'a..\<chi>\<chi> i.1}. f x = x)"
hoelzl@37489
  1356
    using brouwer_cube by auto
hoelzl@33741
  1357
  thus ?thesis unfolding homeomorphic_fixpoint_property[OF *] apply(erule_tac x=f in allE)
hoelzl@33741
  1358
    apply(erule impE) defer apply(erule bexE) apply(rule_tac x=y in that) using assms by auto qed
hoelzl@33741
  1359
hoelzl@33741
  1360
subsection {* And in particular for a closed ball. *}
hoelzl@33741
  1361
hoelzl@37489
  1362
lemma brouwer_ball: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
hoelzl@33741
  1363
  assumes "0 < e" "continuous_on (cball a e) f" "f ` (cball a e) \<subseteq> (cball a e)"
hoelzl@33741
  1364
  obtains x where "x \<in> cball a e" "f x = x"
hoelzl@33741
  1365
  using brouwer_weak[OF compact_cball convex_cball,of a e f] unfolding interior_cball ball_eq_empty
hoelzl@33741
  1366
  using assms by auto
hoelzl@33741
  1367
hoelzl@33741
  1368
text {*Still more general form; could derive this directly without using the 
huffman@36334
  1369
  rather involved @{text "HOMEOMORPHIC_CONVEX_COMPACT"} theorem, just using
hoelzl@33741
  1370
  a scaling and translation to put the set inside the unit cube. *}
hoelzl@33741
  1371
hoelzl@37489
  1372
lemma brouwer: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
hoelzl@33741
  1373
  assumes "compact s" "convex s" "s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
hoelzl@33741
  1374
  obtains x where "x \<in> s" "f x = x" proof-
hoelzl@33741
  1375
  have "\<exists>e>0. s \<subseteq> cball 0 e" using compact_imp_bounded[OF assms(1)] unfolding bounded_pos
huffman@36587
  1376
    apply(erule_tac exE,rule_tac x=b in exI) by(auto simp add: dist_norm) 
hoelzl@33741
  1377
  then guess e apply-apply(erule exE,(erule conjE)+) . note e=this
hoelzl@33741
  1378
  have "\<exists>x\<in> cball 0 e. (f \<circ> closest_point s) x = x"
hoelzl@33741
  1379
    apply(rule_tac brouwer_ball[OF e(1), of 0 "f \<circ> closest_point s"]) apply(rule continuous_on_compose )
hoelzl@33741
  1380
    apply(rule continuous_on_closest_point[OF assms(2) compact_imp_closed[OF assms(1)] assms(3)])
hoelzl@33741
  1381
    apply(rule continuous_on_subset[OF assms(4)])
hoelzl@33741
  1382
    using closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)] apply - defer
huffman@36587
  1383
    using assms(5)[unfolded subset_eq] using e(2)[unfolded subset_eq mem_cball] by(auto simp add: dist_norm)
hoelzl@33741
  1384
  then guess x .. note x=this
hoelzl@33741
  1385
  have *:"closest_point s x = x" apply(rule closest_point_self) 
hoelzl@33741
  1386
    apply(rule assms(5)[unfolded subset_eq,THEN bspec[where x="x"],unfolded image_iff])
hoelzl@33741
  1387
    apply(rule_tac x="closest_point s x" in bexI) using x unfolding o_def
hoelzl@33741
  1388
    using closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3), of x] by auto
hoelzl@33741
  1389
  show thesis apply(rule_tac x="closest_point s x" in that) unfolding x(2)[unfolded o_def]
hoelzl@33741
  1390
    apply(rule closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)]) using * by auto qed
hoelzl@33741
  1391
hoelzl@37489
  1392
text {*So we get the no-retraction theorem. *}
hoelzl@33741
  1393
hoelzl@37489
  1394
lemma no_retraction_cball: assumes "0 < e" fixes type::"'a::ordered_euclidean_space"
hoelzl@37489
  1395
  shows "\<not> (frontier(cball a e) retract_of (cball (a::'a) e))" proof case goal1
hoelzl@33741
  1396
  have *:"\<And>xa. a - (2 *\<^sub>R a - xa) = -(a - xa)" using scaleR_left_distrib[of 1 1 a] by auto
hoelzl@33741
  1397
  guess x apply(rule retract_fixpoint_property[OF goal1, of "\<lambda>x. scaleR 2 a - x"])
hoelzl@33741
  1398
    apply(rule,rule,erule conjE) apply(rule brouwer_ball[OF assms]) apply assumption+
hoelzl@33741
  1399
    apply(rule_tac x=x in bexI) apply assumption+ apply(rule continuous_on_intros)+
hoelzl@33741
  1400
    unfolding frontier_cball subset_eq Ball_def image_iff apply(rule,rule,erule bexE)
huffman@36587
  1401
    unfolding dist_norm apply(simp add: * norm_minus_commute) . note x = this
haftmann@36350
  1402
  hence "scaleR 2 a = scaleR 1 x + scaleR 1 x" by(auto simp add:algebra_simps)
hoelzl@33741
  1403
  hence "a = x" unfolding scaleR_left_distrib[THEN sym] by auto 
hoelzl@33741
  1404
  thus False using x using assms by auto qed
hoelzl@33741
  1405
hoelzl@33741
  1406
subsection {*Bijections between intervals. *}
hoelzl@33741
  1407
hoelzl@37489
  1408
definition "interval_bij = (\<lambda> (a::'a,b::'a) (u::'a,v::'a) (x::'a::ordered_euclidean_space).
hoelzl@37489
  1409
    (\<chi>\<chi> i. u$$i + (x$$i - a$$i) / (b$$i - a$$i) * (v$$i - u$$i))::'a)"
hoelzl@33741
  1410
hoelzl@33741
  1411
lemma interval_bij_affine:
hoelzl@37489
  1412
 "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi>\<chi> i. (v$$i - u$$i) / (b$$i - a$$i) * x$$i) +
hoelzl@37489
  1413
            (\<chi>\<chi> i. u$$i - (v$$i - u$$i) / (b$$i - a$$i) * a$$i))"
hoelzl@37489
  1414
  apply rule apply(subst euclidean_eq,safe) unfolding euclidean_simps interval_bij_def euclidean_lambda_beta
hoelzl@37489
  1415
  by(auto simp add: field_simps add_divide_distrib[THEN sym])
hoelzl@33741
  1416
hoelzl@33741
  1417
lemma continuous_interval_bij:
hoelzl@37489
  1418
  "continuous (at x) (interval_bij (a,b::'a::ordered_euclidean_space) (u,v::'a))" 
hoelzl@33741
  1419
  unfolding interval_bij_affine apply(rule continuous_intros)
hoelzl@33741
  1420
    apply(rule linear_continuous_at) unfolding linear_conv_bounded_linear[THEN sym]
hoelzl@37489
  1421
    unfolding linear_def euclidean_eq[where 'a='a] apply safe unfolding euclidean_lambda_beta prefer 3
hoelzl@33741
  1422
    apply(rule continuous_intros) by(auto simp add:field_simps add_divide_distrib[THEN sym])
hoelzl@33741
  1423
hoelzl@33741
  1424
lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a,b) (u,v))"
hoelzl@33741
  1425
  apply(rule continuous_at_imp_continuous_on) by(rule, rule continuous_interval_bij)
hoelzl@33741
  1426
hoelzl@33741
  1427
(** move this **)
hoelzl@33741
  1428
lemma divide_nonneg_nonneg:assumes "a \<ge> 0" "b \<ge> 0" shows "0 \<le> a / (b::real)"
hoelzl@33741
  1429
  apply(cases "b=0") defer apply(rule divide_nonneg_pos) using assms by auto
hoelzl@33741
  1430
hoelzl@33741
  1431
lemma in_interval_interval_bij: assumes "x \<in> {a..b}" "{u..v} \<noteq> {}"
hoelzl@37489
  1432
  shows "interval_bij (a,b) (u,v) x \<in> {u..v::'a::ordered_euclidean_space}" 
hoelzl@37489
  1433
  unfolding interval_bij_def split_conv mem_interval apply safe unfolding euclidean_lambda_beta
hoelzl@37489
  1434
proof (simp_all only: if_P)
hoelzl@37489
  1435
  fix i assume i:"i<DIM('a)" have "{a..b} \<noteq> {}" using assms by auto
hoelzl@37489
  1436
  hence *:"a$$i \<le> b$$i" "u$$i \<le> v$$i" using assms(2) unfolding interval_eq_empty not_ex apply-
hoelzl@33741
  1437
    apply(erule_tac[!] x=i in allE)+ by auto
hoelzl@37489
  1438
  have x:"a$$i\<le>x$$i" "x$$i\<le>b$$i" using assms(1)[unfolded mem_interval] using i by auto
hoelzl@37489
  1439
  have "0 \<le> (x $$ i - a $$ i) / (b $$ i - a $$ i) * (v $$ i - u $$ i)"
hoelzl@33741
  1440
    apply(rule mult_nonneg_nonneg) apply(rule divide_nonneg_nonneg)
hoelzl@33741
  1441
    using * x by(auto simp add:field_simps)
hoelzl@37489
  1442
  thus "u $$ i \<le> u $$ i + (x $$ i - a $$ i) / (b $$ i - a $$ i) * (v $$ i - u $$ i)" using * by auto
hoelzl@37489
  1443
  have "((x $$ i - a $$ i) / (b $$ i - a $$ i)) * (v $$ i - u $$ i) \<le> 1 * (v $$ i - u $$ i)"
hoelzl@33741
  1444
    apply(rule mult_right_mono) unfolding divide_le_eq_1 using * x by auto
hoelzl@37489
  1445
  thus "u $$ i + (x $$ i - a $$ i) / (b $$ i - a $$ i) * (v $$ i - u $$ i) \<le> v $$ i" using * by auto
hoelzl@37489
  1446
qed
hoelzl@33741
  1447
hoelzl@37489
  1448
lemma interval_bij_bij: fixes x::"'a::ordered_euclidean_space" assumes "\<forall>i. a$$i < b$$i \<and> u$$i < v$$i" 
hoelzl@33741
  1449
  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
hoelzl@37489
  1450
  unfolding interval_bij_def split_conv euclidean_eq[where 'a='a]
hoelzl@33741
  1451
  apply(rule,insert assms,erule_tac x=i in allE) by auto
hoelzl@33741
  1452
hoelzl@34291
  1453
end