(* Author: John Harrison
Author: Robert Himmelmann, TU Muenchen (Translation from HOL light)
*)
(* ========================================================================= *)
(* Results connected with topological dimension. *)
(* *)
(* At the moment this is just Brouwer's fixpoint theorem. The proof is from *)
(* Kuhn: "some combinatorial lemmas in topology", IBM J. v4. (1960) p. 518 *)
(* See "http://www.research.ibm.com/journal/rd/045/ibmrd0405K.pdf". *)
(* *)
(* The script below is quite messy, but at least we avoid formalizing any *)
(* topological machinery; we don't even use barycentric subdivision; this is *)
(* the big advantage of Kuhn's proof over the usual Sperner's lemma one. *)
(* *)
(* (c) Copyright, John Harrison 1998-2008 *)
(* ========================================================================= *)
header {* Results connected with topological dimension. *}
theory Brouwer_Fixpoint
imports Convex_Euclidean_Space
begin
lemma bij_betw_singleton_eq:
assumes f: "bij_betw f A B" and g: "bij_betw g A B" and a: "a \<in> A"
assumes eq: "(\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x = g x)"
shows "f a = g a"
proof -
have "f ` (A - {a}) = g ` (A - {a})"
by (intro image_cong) (simp_all add: eq)
then have "B - {f a} = B - {g a}"
using f g a by (auto simp: bij_betw_def inj_on_image_set_diff set_eq_iff)
moreover have "f a \<in> B" "g a \<in> B"
using f g a by (auto simp: bij_betw_def)
ultimately show ?thesis
by auto
qed
lemma swap_image:
"Fun.swap i j f ` A = (if i \<in> A then (if j \<in> A then f ` A else f ` ((A - {i}) \<union> {j}))
else (if j \<in> A then f ` ((A - {j}) \<union> {i}) else f ` A))"
apply (auto simp: Fun.swap_def image_iff)
apply metis
apply (metis member_remove remove_def)
apply (metis member_remove remove_def)
done
lemma swap_apply1: "Fun.swap x y f x = f y"
by (simp add: swap_def)
lemma swap_apply2: "Fun.swap x y f y = f x"
by (simp add: swap_def)
lemma (in -) lessThan_empty_iff: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
by auto
lemma Zero_notin_Suc: "0 \<notin> Suc ` A"
by auto
lemma atMost_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
apply auto
apply (case_tac x)
apply auto
done
lemma divide_nonneg_nonneg:
fixes a b :: "'a :: {linordered_field, field_inverse_zero}"
shows "a \<ge> 0 \<Longrightarrow> b \<ge> 0 \<Longrightarrow> 0 \<le> a / b"
by (cases "b = 0") (auto intro!: divide_nonneg_pos)
lemma setsum_Un_disjoint':
assumes "finite A"
and "finite B"
and "A \<inter> B = {}"
and "A \<union> B = C"
shows "setsum g C = setsum g A + setsum g B"
using setsum_Un_disjoint[OF assms(1-3)] and assms(4) by auto
lemma pointwise_minimal_pointwise_maximal:
fixes s :: "(nat \<Rightarrow> nat) set"
assumes "finite s"
and "s \<noteq> {}"
and "\<forall>x\<in>s. \<forall>y\<in>s. x \<le> y \<or> y \<le> x"
shows "\<exists>a\<in>s. \<forall>x\<in>s. a \<le> x"
and "\<exists>a\<in>s. \<forall>x\<in>s. x \<le> a"
using assms
proof (induct s rule: finite_ne_induct)
case (insert b s)
assume *: "\<forall>x\<in>insert b s. \<forall>y\<in>insert b s. x \<le> y \<or> y \<le> x"
moreover then obtain u l where "l \<in> s" "\<forall>b\<in>s. l \<le> b" "u \<in> s" "\<forall>b\<in>s. b \<le> u"
using insert by auto
ultimately show "\<exists>a\<in>insert b s. \<forall>x\<in>insert b s. a \<le> x" "\<exists>a\<in>insert b s. \<forall>x\<in>insert b s. x \<le> a"
using *[rule_format, of b u] *[rule_format, of b l] by (metis insert_iff order.trans)+
qed auto
lemma brouwer_compactness_lemma:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "compact s"
and "continuous_on s f"
and "\<not> (\<exists>x\<in>s. f x = 0)"
obtains d where "0 < d" and "\<forall>x\<in>s. d \<le> norm (f x)"
proof (cases "s = {}")
case True
show thesis
by (rule that [of 1]) (auto simp: True)
next
case False
have "continuous_on s (norm \<circ> f)"
by (rule continuous_intros continuous_on_norm assms(2))+
with False obtain x where x: "x \<in> s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y"
using continuous_attains_inf[OF assms(1), of "norm \<circ> f"]
unfolding o_def
by auto
have "(norm \<circ> f) x > 0"
using assms(3) and x(1)
by auto
then show ?thesis
by (rule that) (insert x(2), auto simp: o_def)
qed
lemma kuhn_labelling_lemma:
fixes P Q :: "'a::euclidean_space \<Rightarrow> bool"
assumes "\<forall>x. P x \<longrightarrow> P (f x)"
and "\<forall>x. P x \<longrightarrow> (\<forall>i\<in>Basis. Q i \<longrightarrow> 0 \<le> x\<bullet>i \<and> x\<bullet>i \<le> 1)"
shows "\<exists>l. (\<forall>x.\<forall>i\<in>Basis. l x i \<le> (1::nat)) \<and>
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 0) \<longrightarrow> (l x i = 0)) \<and>
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 1) \<longrightarrow> (l x i = 1)) \<and>
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x\<bullet>i \<le> f x\<bullet>i) \<and>
(\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f x\<bullet>i \<le> x\<bullet>i)"
proof -
{ fix x i
let ?R = "\<lambda>y. (P x \<and> Q i \<and> x \<bullet> i = 0 \<longrightarrow> y = (0::nat)) \<and>
(P x \<and> Q i \<and> x \<bullet> i = 1 \<longrightarrow> y = 1) \<and>
(P x \<and> Q i \<and> y = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i) \<and>
(P x \<and> Q i \<and> y = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i)"
{ assume "P x" "Q i" "i \<in> Basis" with assms have "0 \<le> f x \<bullet> i \<and> f x \<bullet> i \<le> 1" by auto }
then have "i \<in> Basis \<Longrightarrow> ?R 0 \<or> ?R 1" by auto }
then show ?thesis
unfolding all_conj_distrib[symmetric] Ball_def (* FIXME: shouldn't this work by metis? *)
by (subst choice_iff[symmetric])+ blast
qed
subsection {* The key "counting" observation, somewhat abstracted. *}
lemma kuhn_counting_lemma:
fixes bnd compo compo' face S F
defines "nF s == card {f\<in>F. face f s \<and> compo' f}"
assumes [simp, intro]: "finite F" -- "faces" and [simp, intro]: "finite S" -- "simplices"
and "\<And>f. f \<in> F \<Longrightarrow> bnd f \<Longrightarrow> card {s\<in>S. face f s} = 1"
and "\<And>f. f \<in> F \<Longrightarrow> \<not> bnd f \<Longrightarrow> card {s\<in>S. face f s} = 2"
and "\<And>s. s \<in> S \<Longrightarrow> compo s \<Longrightarrow> nF s = 1"
and "\<And>s. s \<in> S \<Longrightarrow> \<not> compo s \<Longrightarrow> nF s = 0 \<or> nF s = 2"
and "odd (card {f\<in>F. compo' f \<and> bnd f})"
shows "odd (card {s\<in>S. compo s})"
proof -
have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + (\<Sum>s | s \<in> S \<and> compo s. nF s) = (\<Sum>s\<in>S. nF s)"
by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
also have "\<dots> = (\<Sum>s\<in>S. card {f \<in> {f\<in>F. compo' f \<and> bnd f}. face f s}) +
(\<Sum>s\<in>S. card {f \<in> {f\<in>F. compo' f \<and> \<not> bnd f}. face f s})"
unfolding setsum_addf[symmetric]
by (subst card_Un_disjoint[symmetric])
(auto simp: nF_def intro!: setsum_cong arg_cong[where f=card])
also have "\<dots> = 1 * card {f\<in>F. compo' f \<and> bnd f} + 2 * card {f\<in>F. compo' f \<and> \<not> bnd f}"
using assms(4,5) by (fastforce intro!: arg_cong2[where f="op +"] setsum_multicount)
finally have "odd ((\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + card {s\<in>S. compo s})"
using assms(6,8) by simp
moreover have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) =
(\<Sum>s | s \<in> S \<and> \<not> compo s \<and> nF s = 0. nF s) + (\<Sum>s | s \<in> S \<and> \<not> compo s \<and> nF s = 2. nF s)"
using assms(7) by (subst setsum_Un_disjoint[symmetric]) (fastforce intro!: setsum_cong)+
ultimately show ?thesis
by auto
qed
subsection {* The odd/even result for faces of complete vertices, generalized. *}
lemma kuhn_complete_lemma:
assumes [simp]: "finite simplices"
and face: "\<And>f s. face f s \<longleftrightarrow> (\<exists>a\<in>s. f = s - {a})"
and card_s[simp]: "\<And>s. s \<in> simplices \<Longrightarrow> card s = n + 2"
and rl_bd: "\<And>s. s \<in> simplices \<Longrightarrow> rl ` s \<subseteq> {..Suc n}"
and bnd: "\<And>f s. s \<in> simplices \<Longrightarrow> face f s \<Longrightarrow> bnd f \<Longrightarrow> card {s\<in>simplices. face f s} = 1"
and nbnd: "\<And>f s. s \<in> simplices \<Longrightarrow> face f s \<Longrightarrow> \<not> bnd f \<Longrightarrow> card {s\<in>simplices. face f s} = 2"
and odd_card: "odd (card {f. (\<exists>s\<in>simplices. face f s) \<and> rl ` f = {..n} \<and> bnd f})"
shows "odd (card {s\<in>simplices. (rl ` s = {..Suc n})})"
proof (rule kuhn_counting_lemma)
have finite_s[simp]: "\<And>s. s \<in> simplices \<Longrightarrow> finite s"
by (metis add_is_0 zero_neq_numeral card_infinite assms(3))
let ?F = "{f. \<exists>s\<in>simplices. face f s}"
have F_eq: "?F = (\<Union>s\<in>simplices. \<Union>a\<in>s. {s - {a}})"
by (auto simp: face)
show "finite ?F"
using `finite simplices` unfolding F_eq by auto
{ fix f assume "f \<in> ?F" "bnd f" then show "card {s \<in> simplices. face f s} = 1"
using bnd by auto }
{ fix f assume "f \<in> ?F" "\<not> bnd f" then show "card {s \<in> simplices. face f s} = 2"
using nbnd by auto }
show "odd (card {f \<in> {f. \<exists>s\<in>simplices. face f s}. rl ` f = {..n} \<and> bnd f})"
using odd_card by simp
fix s assume s[simp]: "s \<in> simplices"
let ?S = "{f \<in> {f. \<exists>s\<in>simplices. face f s}. face f s \<and> rl ` f = {..n}}"
have "?S = (\<lambda>a. s - {a}) ` {a\<in>s. rl ` (s - {a}) = {..n}}"
using s by (fastforce simp: face)
then have card_S: "card ?S = card {a\<in>s. rl ` (s - {a}) = {..n}}"
by (auto intro!: card_image inj_onI)
{ assume rl: "rl ` s = {..Suc n}"
then have inj_rl: "inj_on rl s"
by (intro eq_card_imp_inj_on) auto
moreover obtain a where "rl a = Suc n" "a \<in> s"
by (metis atMost_iff image_iff le_Suc_eq rl)
ultimately have n: "{..n} = rl ` (s - {a})"
by (auto simp add: inj_on_image_set_diff rl)
have "{a\<in>s. rl ` (s - {a}) = {..n}} = {a}"
using inj_rl `a \<in> s` by (auto simp add: n inj_on_image_eq_iff[OF inj_rl] Diff_subset)
then show "card ?S = 1"
unfolding card_S by simp }
{ assume rl: "rl ` s \<noteq> {..Suc n}"
show "card ?S = 0 \<or> card ?S = 2"
proof cases
assume *: "{..n} \<subseteq> rl ` s"
with rl rl_bd[OF s] have rl_s: "rl ` s = {..n}"
by (auto simp add: atMost_Suc subset_insert_iff split: split_if_asm)
then have "\<not> inj_on rl s"
by (intro pigeonhole) simp
then obtain a b where ab: "a \<in> s" "b \<in> s" "rl a = rl b" "a \<noteq> b"
by (auto simp: inj_on_def)
then have eq: "rl ` (s - {a}) = rl ` s"
by auto
with ab have inj: "inj_on rl (s - {a})"
by (intro eq_card_imp_inj_on) (auto simp add: rl_s card_Diff_singleton_if)
{ fix x assume "x \<in> s" "x \<notin> {a, b}"
then have "rl ` s - {rl x} = rl ` ((s - {a}) - {x})"
by (auto simp: eq inj_on_image_set_diff[OF inj])
also have "\<dots> = rl ` (s - {x})"
using ab `x \<notin> {a, b}` by auto
also assume "\<dots> = rl ` s"
finally have False
using `x\<in>s` by auto }
moreover
{ fix x assume "x \<in> {a, b}" with ab have "x \<in> s \<and> rl ` (s - {x}) = rl ` s"
by (simp add: set_eq_iff image_iff Bex_def) metis }
ultimately have "{a\<in>s. rl ` (s - {a}) = {..n}} = {a, b}"
unfolding rl_s[symmetric] by fastforce
with `a \<noteq> b` show "card ?S = 0 \<or> card ?S = 2"
unfolding card_S by simp
next
assume "\<not> {..n} \<subseteq> rl ` s"
then have "\<And>x. rl ` (s - {x}) \<noteq> {..n}"
by auto
then show "card ?S = 0 \<or> card ?S = 2"
unfolding card_S by simp
qed }
qed fact
locale kuhn_simplex =
fixes p n and base upd and s :: "(nat \<Rightarrow> nat) set"
assumes base: "base \<in> {..< n} \<rightarrow> {..< p}"
assumes base_out: "\<And>i. n \<le> i \<Longrightarrow> base i = p"
assumes upd: "bij_betw upd {..< n} {..< n}"
assumes s_pre: "s = (\<lambda>i j. if j \<in> upd`{..< i} then Suc (base j) else base j) ` {.. n}"
begin
definition "enum i j = (if j \<in> upd`{..< i} then Suc (base j) else base j)"
lemma s_eq: "s = enum ` {.. n}"
unfolding s_pre enum_def[abs_def] ..
lemma upd_space: "i < n \<Longrightarrow> upd i < n"
using upd by (auto dest!: bij_betwE)
lemma s_space: "s \<subseteq> {..< n} \<rightarrow> {.. p}"
proof -
{ fix i assume "i \<le> n" then have "enum i \<in> {..< n} \<rightarrow> {.. p}"
proof (induct i)
case 0 then show ?case
using base by (auto simp: Pi_iff less_imp_le enum_def)
next
case (Suc i) with base show ?case
by (auto simp: Pi_iff Suc_le_eq less_imp_le enum_def intro: upd_space)
qed }
then show ?thesis
by (auto simp: s_eq)
qed
lemma inj_upd: "inj_on upd {..< n}"
using upd by (simp add: bij_betw_def)
lemma inj_enum: "inj_on enum {.. n}"
proof -
{ fix x y :: nat assume "x \<noteq> y" "x \<le> n" "y \<le> n"
with upd have "upd ` {..< x} \<noteq> upd ` {..< y}"
by (subst inj_on_image_eq_iff[where C="{..< n}"]) (auto simp: bij_betw_def)
then have "enum x \<noteq> enum y"
by (auto simp add: enum_def fun_eq_iff) }
then show ?thesis
by (auto simp: inj_on_def)
qed
lemma enum_0: "enum 0 = base"
by (simp add: enum_def[abs_def])
lemma base_in_s: "base \<in> s"
unfolding s_eq by (subst enum_0[symmetric]) auto
lemma enum_in: "i \<le> n \<Longrightarrow> enum i \<in> s"
unfolding s_eq by auto
lemma one_step:
assumes a: "a \<in> s" "j < n"
assumes *: "\<And>a'. a' \<in> s \<Longrightarrow> a' \<noteq> a \<Longrightarrow> a' j = p'"
shows "a j \<noteq> p'"
proof
assume "a j = p'"
with * a have "\<And>a'. a' \<in> s \<Longrightarrow> a' j = p'"
by auto
then have "\<And>i. i \<le> n \<Longrightarrow> enum i j = p'"
unfolding s_eq by auto
from this[of 0] this[of n] have "j \<notin> upd ` {..< n}"
by (auto simp: enum_def fun_eq_iff split: split_if_asm)
with upd `j < n` show False
by (auto simp: bij_betw_def)
qed
lemma upd_inj: "i < n \<Longrightarrow> j < n \<Longrightarrow> upd i = upd j \<longleftrightarrow> i = j"
using upd by (auto simp: bij_betw_def inj_on_iff)
lemma upd_surj: "upd ` {..< n} = {..< n}"
using upd by (auto simp: bij_betw_def)
lemma in_upd_image: "A \<subseteq> {..< n} \<Longrightarrow> i < n \<Longrightarrow> upd i \<in> upd ` A \<longleftrightarrow> i \<in> A"
using inj_on_image_mem_iff[of upd "{..< n}" A i ] upd
by (auto simp: bij_betw_def)
lemma enum_inj: "i \<le> n \<Longrightarrow> j \<le> n \<Longrightarrow> enum i = enum j \<longleftrightarrow> i = j"
using inj_enum by (auto simp: inj_on_iff)
lemma in_enum_image: "A \<subseteq> {.. n} \<Longrightarrow> i \<le> n \<Longrightarrow> enum i \<in> enum ` A \<longleftrightarrow> i \<in> A"
using inj_on_image_mem_iff[OF inj_enum, of A i] by auto
lemma enum_mono: "i \<le> n \<Longrightarrow> j \<le> n \<Longrightarrow> enum i \<le> enum j \<longleftrightarrow> i \<le> j"
by (auto simp: enum_def le_fun_def in_upd_image Ball_def[symmetric])
lemma enum_strict_mono: "i \<le> n \<Longrightarrow> j \<le> n \<Longrightarrow> enum i < enum j \<longleftrightarrow> i < j"
using enum_mono[of i j] enum_inj[of i j] by (auto simp add: le_less)
lemma chain: "a \<in> s \<Longrightarrow> b \<in> s \<Longrightarrow> a \<le> b \<or> b \<le> a"
by (auto simp: s_eq enum_mono)
lemma less: "a \<in> s \<Longrightarrow> b \<in> s \<Longrightarrow> a i < b i \<Longrightarrow> a < b"
using chain[of a b] by (auto simp: less_fun_def le_fun_def not_le[symmetric])
lemma enum_0_bot: "a \<in> s \<Longrightarrow> a = enum 0 \<longleftrightarrow> (\<forall>a'\<in>s. a \<le> a')"
unfolding s_eq by (auto simp: enum_mono Ball_def)
lemma enum_n_top: "a \<in> s \<Longrightarrow> a = enum n \<longleftrightarrow> (\<forall>a'\<in>s. a' \<le> a)"
unfolding s_eq by (auto simp: enum_mono Ball_def)
lemma enum_Suc: "i < n \<Longrightarrow> enum (Suc i) = (enum i)(upd i := Suc (enum i (upd i)))"
by (auto simp: fun_eq_iff enum_def upd_inj)
lemma enum_eq_p: "i \<le> n \<Longrightarrow> n \<le> j \<Longrightarrow> enum i j = p"
by (induct i) (auto simp: enum_Suc enum_0 base_out upd_space not_less[symmetric])
lemma out_eq_p: "a \<in> s \<Longrightarrow> n \<le> j \<Longrightarrow> a j = p"
unfolding s_eq by (auto simp add: enum_eq_p)
lemma s_le_p: "a \<in> s \<Longrightarrow> a j \<le> p"
using out_eq_p[of a j] s_space by (cases "j < n") auto
lemma le_Suc_base: "a \<in> s \<Longrightarrow> a j \<le> Suc (base j)"
unfolding s_eq by (auto simp: enum_def)
lemma base_le: "a \<in> s \<Longrightarrow> base j \<le> a j"
unfolding s_eq by (auto simp: enum_def)
lemma enum_le_p: "i \<le> n \<Longrightarrow> j < n \<Longrightarrow> enum i j \<le> p"
using enum_in[of i] s_space by auto
lemma enum_less: "a \<in> s \<Longrightarrow> i < n \<Longrightarrow> enum i < a \<longleftrightarrow> enum (Suc i) \<le> a"
unfolding s_eq by (auto simp: enum_strict_mono enum_mono)
lemma ksimplex_0:
"n = 0 \<Longrightarrow> s = {(\<lambda>x. p)}"
using s_eq enum_def base_out by auto
lemma replace_0:
assumes "j < n" "a \<in> s" and p: "\<forall>x\<in>s - {a}. x j = 0" and "x \<in> s"
shows "x \<le> a"
proof cases
assume "x \<noteq> a"
have "a j \<noteq> 0"
using assms by (intro one_step[where a=a]) auto
with less[OF `x\<in>s` `a\<in>s`, of j] p[rule_format, of x] `x \<in> s` `x \<noteq> a`
show ?thesis
by auto
qed simp
lemma replace_1:
assumes "j < n" "a \<in> s" and p: "\<forall>x\<in>s - {a}. x j = p" and "x \<in> s"
shows "a \<le> x"
proof cases
assume "x \<noteq> a"
have "a j \<noteq> p"
using assms by (intro one_step[where a=a]) auto
with enum_le_p[of _ j] `j < n` `a\<in>s`
have "a j < p"
by (auto simp: less_le s_eq)
with less[OF `a\<in>s` `x\<in>s`, of j] p[rule_format, of x] `x \<in> s` `x \<noteq> a`
show ?thesis
by auto
qed simp
end
locale kuhn_simplex_pair = s: kuhn_simplex p n b_s u_s s + t: kuhn_simplex p n b_t u_t t
for p n b_s u_s s b_t u_t t
begin
lemma enum_eq:
assumes l: "i \<le> l" "l \<le> j" and "j + d \<le> n"
assumes eq: "s.enum ` {i .. j} = t.enum ` {i + d .. j + d}"
shows "s.enum l = t.enum (l + d)"
using l proof (induct l rule: dec_induct)
case base
then have s: "s.enum i \<in> t.enum ` {i + d .. j + d}" and t: "t.enum (i + d) \<in> s.enum ` {i .. j}"
using eq by auto
from t `i \<le> j` `j + d \<le> n` have "s.enum i \<le> t.enum (i + d)"
by (auto simp: s.enum_mono)
moreover from s `i \<le> j` `j + d \<le> n` have "t.enum (i + d) \<le> s.enum i"
by (auto simp: t.enum_mono)
ultimately show ?case
by auto
next
case (step l)
moreover from step.prems `j + d \<le> n` have
"s.enum l < s.enum (Suc l)"
"t.enum (l + d) < t.enum (Suc l + d)"
by (simp_all add: s.enum_strict_mono t.enum_strict_mono)
moreover have
"s.enum (Suc l) \<in> t.enum ` {i + d .. j + d}"
"t.enum (Suc l + d) \<in> s.enum ` {i .. j}"
using step `j + d \<le> n` eq by (auto simp: s.enum_inj t.enum_inj)
ultimately have "s.enum (Suc l) = t.enum (Suc (l + d))"
using `j + d \<le> n`
by (intro antisym s.enum_less[THEN iffD1] t.enum_less[THEN iffD1])
(auto intro!: s.enum_in t.enum_in)
then show ?case by simp
qed
lemma ksimplex_eq_bot:
assumes a: "a \<in> s" "\<And>a'. a' \<in> s \<Longrightarrow> a \<le> a'"
assumes b: "b \<in> t" "\<And>b'. b' \<in> t \<Longrightarrow> b \<le> b'"
assumes eq: "s - {a} = t - {b}"
shows "s = t"
proof cases
assume "n = 0" with s.ksimplex_0 t.ksimplex_0 show ?thesis by simp
next
assume "n \<noteq> 0"
have "s.enum 0 = (s.enum (Suc 0)) (u_s 0 := s.enum (Suc 0) (u_s 0) - 1)"
"t.enum 0 = (t.enum (Suc 0)) (u_t 0 := t.enum (Suc 0) (u_t 0) - 1)"
using `n \<noteq> 0` by (simp_all add: s.enum_Suc t.enum_Suc)
moreover have e0: "a = s.enum 0" "b = t.enum 0"
using a b by (simp_all add: s.enum_0_bot t.enum_0_bot)
moreover
{ fix j assume "0 < j" "j \<le> n"
moreover have "s - {a} = s.enum ` {Suc 0 .. n}" "t - {b} = t.enum ` {Suc 0 .. n}"
unfolding s.s_eq t.s_eq e0 by (auto simp: s.enum_inj t.enum_inj)
ultimately have "s.enum j = t.enum j"
using enum_eq[of "1" j n 0] eq by auto }
note enum_eq = this
then have "s.enum (Suc 0) = t.enum (Suc 0)"
using `n \<noteq> 0` by auto
moreover
{ fix j assume "Suc j < n"
with enum_eq[of "Suc j"] enum_eq[of "Suc (Suc j)"]
have "u_s (Suc j) = u_t (Suc j)"
using s.enum_Suc[of "Suc j"] t.enum_Suc[of "Suc j"]
by (auto simp: fun_eq_iff split: split_if_asm) }
then have "\<And>j. 0 < j \<Longrightarrow> j < n \<Longrightarrow> u_s j = u_t j"
by (auto simp: gr0_conv_Suc)
with `n \<noteq> 0` have "u_t 0 = u_s 0"
by (intro bij_betw_singleton_eq[OF t.upd s.upd, of 0]) auto
ultimately have "a = b"
by simp
with assms show "s = t"
by auto
qed
lemma ksimplex_eq_top:
assumes a: "a \<in> s" "\<And>a'. a' \<in> s \<Longrightarrow> a' \<le> a"
assumes b: "b \<in> t" "\<And>b'. b' \<in> t \<Longrightarrow> b' \<le> b"
assumes eq: "s - {a} = t - {b}"
shows "s = t"
proof (cases n)
assume "n = 0" with s.ksimplex_0 t.ksimplex_0 show ?thesis by simp
next
case (Suc n')
have "s.enum n = (s.enum n') (u_s n' := Suc (s.enum n' (u_s n')))"
"t.enum n = (t.enum n') (u_t n' := Suc (t.enum n' (u_t n')))"
using Suc by (simp_all add: s.enum_Suc t.enum_Suc)
moreover have en: "a = s.enum n" "b = t.enum n"
using a b by (simp_all add: s.enum_n_top t.enum_n_top)
moreover
{ fix j assume "j < n"
moreover have "s - {a} = s.enum ` {0 .. n'}" "t - {b} = t.enum ` {0 .. n'}"
unfolding s.s_eq t.s_eq en by (auto simp: s.enum_inj t.enum_inj Suc)
ultimately have "s.enum j = t.enum j"
using enum_eq[of "0" j n' 0] eq Suc by auto }
note enum_eq = this
then have "s.enum n' = t.enum n'"
using Suc by auto
moreover
{ fix j assume "j < n'"
with enum_eq[of j] enum_eq[of "Suc j"]
have "u_s j = u_t j"
using s.enum_Suc[of j] t.enum_Suc[of j]
by (auto simp: Suc fun_eq_iff split: split_if_asm) }
then have "\<And>j. j < n' \<Longrightarrow> u_s j = u_t j"
by (auto simp: gr0_conv_Suc)
then have "u_t n' = u_s n'"
by (intro bij_betw_singleton_eq[OF t.upd s.upd, of n']) (auto simp: Suc)
ultimately have "a = b"
by simp
with assms show "s = t"
by auto
qed
end
inductive ksimplex for p n :: nat where
ksimplex: "kuhn_simplex p n base upd s \<Longrightarrow> ksimplex p n s"
lemma finite_ksimplexes: "finite {s. ksimplex p n s}"
proof (rule finite_subset)
{ fix a s assume "ksimplex p n s" "a \<in> s"
then obtain b u where "kuhn_simplex p n b u s" by (auto elim: ksimplex.cases)
then interpret kuhn_simplex p n b u s .
from s_space `a \<in> s` out_eq_p[OF `a \<in> s`]
have "a \<in> (\<lambda>f x. if n \<le> x then p else f x) ` ({..< n} \<rightarrow>\<^sub>E {.. p})"
by (auto simp: image_iff subset_eq Pi_iff split: split_if_asm
intro!: bexI[of _ "restrict a {..< n}"]) }
then show "{s. ksimplex p n s} \<subseteq> Pow ((\<lambda>f x. if n \<le> x then p else f x) ` ({..< n} \<rightarrow>\<^sub>E {.. p}))"
by auto
qed (simp add: finite_PiE)
lemma ksimplex_card:
assumes "ksimplex p n s" shows "card s = Suc n"
using assms proof cases
case (ksimplex u b)
then interpret kuhn_simplex p n u b s .
show ?thesis
by (simp add: card_image s_eq inj_enum)
qed
lemma simplex_top_face:
assumes "0 < p" "\<forall>x\<in>s'. x n = p"
shows "ksimplex p n s' \<longleftrightarrow> (\<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> s' = s - {a})"
using assms
proof safe
fix s a assume "ksimplex p (Suc n) s" and a: "a \<in> s" and na: "\<forall>x\<in>s - {a}. x n = p"
then show "ksimplex p n (s - {a})"
proof cases
case (ksimplex base upd)
then interpret kuhn_simplex p "Suc n" base upd "s" .
have "a n < p"
using one_step[of a n p] na `a\<in>s` s_space by (auto simp: less_le)
then have "a = enum 0"
using `a \<in> s` na by (subst enum_0_bot) (auto simp: le_less intro!: less[of a _ n])
then have s_eq: "s - {a} = enum ` Suc ` {.. n}"
using s_eq by (simp add: atMost_Suc_eq_insert_0 insert_ident Zero_notin_Suc in_enum_image subset_eq)
then have "enum 1 \<in> s - {a}"
by auto
then have "upd 0 = n"
using `a n < p` `a = enum 0` na[rule_format, of "enum 1"]
by (auto simp: fun_eq_iff enum_Suc split: split_if_asm)
then have "bij_betw upd (Suc ` {..< n}) {..< n}"
using upd
by (subst notIn_Un_bij_betw3[where b=0])
(auto simp: lessThan_Suc[symmetric] lessThan_Suc_eq_insert_0)
then have "bij_betw (upd\<circ>Suc) {..<n} {..<n}"
by (rule bij_betw_trans[rotated]) (auto simp: bij_betw_def)
have "a n = p - 1"
using enum_Suc[of 0] na[rule_format, OF `enum 1 \<in> s - {a}`] `a = enum 0` by (auto simp: `upd 0 = n`)
show ?thesis
proof (rule ksimplex.intros, default)
show "bij_betw (upd\<circ>Suc) {..< n} {..< n}" by fact
show "base(n := p) \<in> {..<n} \<rightarrow> {..<p}" "\<And>i. n\<le>i \<Longrightarrow> (base(n := p)) i = p"
using base base_out by (auto simp: Pi_iff)
have "\<And>i. Suc ` {..< i} = {..< Suc i} - {0}"
by (auto simp: image_iff Ball_def) arith
then have upd_Suc: "\<And>i. i \<le> n \<Longrightarrow> (upd\<circ>Suc) ` {..< i} = upd ` {..< Suc i} - {n}"
using `upd 0 = n`
by (simp add: image_comp[symmetric] inj_on_image_set_diff[OF inj_upd])
have n_in_upd: "\<And>i. n \<in> upd ` {..< Suc i}"
using `upd 0 = n` by auto
def f' \<equiv> "\<lambda>i j. if j \<in> (upd\<circ>Suc)`{..< i} then Suc ((base(n := p)) j) else (base(n := p)) j"
{ fix x i assume i[arith]: "i \<le> n" then have "enum (Suc i) x = f' i x"
unfolding f'_def enum_def using `a n < p` `a = enum 0` `upd 0 = n` `a n = p - 1`
by (simp add: upd_Suc enum_0 n_in_upd) }
then show "s - {a} = f' ` {.. n}"
unfolding s_eq image_comp by (intro image_cong) auto
qed
qed
next
assume "ksimplex p n s'" and *: "\<forall>x\<in>s'. x n = p"
then show "\<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> s' = s - {a}"
proof cases
case (ksimplex base upd)
then interpret kuhn_simplex p n base upd s' .
def b \<equiv> "base (n := p - 1)"
def u \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> n | Suc i \<Rightarrow> upd i"
have "ksimplex p (Suc n) (s' \<union> {b})"
proof (rule ksimplex.intros, default)
show "b \<in> {..<Suc n} \<rightarrow> {..<p}"
using base `0 < p` unfolding lessThan_Suc b_def by (auto simp: PiE_iff)
show "\<And>i. Suc n \<le> i \<Longrightarrow> b i = p"
using base_out by (auto simp: b_def)
have "bij_betw u (Suc ` {..< n} \<union> {0}) ({..<n} \<union> {u 0})"
using upd
by (intro notIn_Un_bij_betw) (auto simp: u_def bij_betw_def image_comp comp_def inj_on_def)
then show "bij_betw u {..<Suc n} {..<Suc n}"
by (simp add: u_def lessThan_Suc[symmetric] lessThan_Suc_eq_insert_0)
def f' \<equiv> "\<lambda>i j. if j \<in> u`{..< i} then Suc (b j) else b j"
have u_eq: "\<And>i. i \<le> n \<Longrightarrow> u ` {..< Suc i} = upd ` {..< i} \<union> { n }"
by (auto simp: u_def image_iff upd_inj Ball_def split: nat.split) arith
{ fix x have "x \<le> n \<Longrightarrow> n \<notin> upd ` {..<x}"
using upd_space by (simp add: image_iff neq_iff) }
note n_not_upd = this
have *: "f' ` {.. Suc n} = f' ` (Suc ` {.. n} \<union> {0})"
unfolding atMost_Suc_eq_insert_0 by simp
also have "\<dots> = (f' \<circ> Suc) ` {.. n} \<union> {b}"
by (auto simp: f'_def)
also have "(f' \<circ> Suc) ` {.. n} = s'"
using `0 < p` base_out[of n]
unfolding s_eq enum_def[abs_def] f'_def[abs_def] upd_space
by (intro image_cong) (simp_all add: u_eq b_def fun_eq_iff n_not_upd)
finally show "s' \<union> {b} = f' ` {.. Suc n}" ..
qed
moreover have "b \<notin> s'"
using * `0 < p` by (auto simp: b_def)
ultimately show ?thesis by auto
qed
qed
lemma ksimplex_replace_0:
assumes s: "ksimplex p n s" and a: "a \<in> s"
assumes j: "j < n" and p: "\<forall>x\<in>s - {a}. x j = 0"
shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1"
using s
proof cases
case (ksimplex b_s u_s)
{ fix t b assume "ksimplex p n t"
then obtain b_t u_t where "kuhn_simplex p n b_t u_t t"
by (auto elim: ksimplex.cases)
interpret kuhn_simplex_pair p n b_s u_s s b_t u_t t
by intro_locales fact+
assume b: "b \<in> t" "t - {b} = s - {a}"
with a j p s.replace_0[of _ a] t.replace_0[of _ b] have "s = t"
by (intro ksimplex_eq_top[of a b]) auto }
then have "{s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = {s}"
using s `a \<in> s` by auto
then show ?thesis
by simp
qed
lemma ksimplex_replace_1:
assumes s: "ksimplex p n s" and a: "a \<in> s"
assumes j: "j < n" and p: "\<forall>x\<in>s - {a}. x j = p"
shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1"
using s
proof cases
case (ksimplex b_s u_s)
{ fix t b assume "ksimplex p n t"
then obtain b_t u_t where "kuhn_simplex p n b_t u_t t"
by (auto elim: ksimplex.cases)
interpret kuhn_simplex_pair p n b_s u_s s b_t u_t t
by intro_locales fact+
assume b: "b \<in> t" "t - {b} = s - {a}"
with a j p s.replace_1[of _ a] t.replace_1[of _ b] have "s = t"
by (intro ksimplex_eq_bot[of a b]) auto }
then have "{s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = {s}"
using s `a \<in> s` by auto
then show ?thesis
by simp
qed
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))"
by (auto simp add: card_Suc_eq eval_nat_numeral)
lemma ksimplex_replace_2:
assumes s: "ksimplex p n s" and "a \<in> s" and "n \<noteq> 0"
and lb: "\<forall>j<n. \<exists>x\<in>s - {a}. x j \<noteq> 0"
and ub: "\<forall>j<n. \<exists>x\<in>s - {a}. x j \<noteq> p"
shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 2"
using s
proof cases
case (ksimplex base upd)
then interpret kuhn_simplex p n base upd s .
from `a \<in> s` obtain i where "i \<le> n" "a = enum i"
unfolding s_eq by auto
from `i \<le> n` have "i = 0 \<or> i = n \<or> (0 < i \<and> i < n)"
by linarith
then have "\<exists>!s'. s' \<noteq> s \<and> ksimplex p n s' \<and> (\<exists>b\<in>s'. s - {a} = s'- {b})"
proof (elim disjE conjE)
assume "i = 0"
def rot \<equiv> "\<lambda>i. if i + 1 = n then 0 else i + 1"
let ?upd = "upd \<circ> rot"
have rot: "bij_betw rot {..< n} {..< n}"
by (auto simp: bij_betw_def inj_on_def image_iff Ball_def rot_def)
arith+
from rot upd have "bij_betw ?upd {..<n} {..<n}"
by (rule bij_betw_trans)
def f' \<equiv> "\<lambda>i j. if j \<in> ?upd`{..< i} then Suc (enum (Suc 0) j) else enum (Suc 0) j"
interpret b: kuhn_simplex p n "enum (Suc 0)" "upd \<circ> rot" "f' ` {.. n}"
proof
from `a = enum i` ub `n \<noteq> 0` `i = 0`
obtain i' where "i' \<le> n" "enum i' \<noteq> enum 0" "enum i' (upd 0) \<noteq> p"
unfolding s_eq by (auto intro: upd_space simp: enum_inj)
then have "enum 1 \<le> enum i'" "enum i' (upd 0) < p"
using enum_le_p[of i' "upd 0"] by (auto simp add: enum_inj enum_mono upd_space)
then have "enum 1 (upd 0) < p"
by (auto simp add: le_fun_def intro: le_less_trans)
then show "enum (Suc 0) \<in> {..<n} \<rightarrow> {..<p}"
using base `n \<noteq> 0` by (auto simp add: enum_0 enum_Suc PiE_iff extensional_def upd_space)
{ fix i assume "n \<le> i" then show "enum (Suc 0) i = p"
using `n \<noteq> 0` by (auto simp: enum_eq_p) }
show "bij_betw ?upd {..<n} {..<n}" by fact
qed (simp add: f'_def)
have ks_f': "ksimplex p n (f' ` {.. n})"
by rule unfold_locales
have b_enum: "b.enum = f'" unfolding f'_def b.enum_def[abs_def] ..
with b.inj_enum have inj_f': "inj_on f' {.. n}" by simp
have [simp]: "\<And>j. j < n \<Longrightarrow> rot ` {..< j} = {0 <..< Suc j}"
by (auto simp: rot_def image_iff Ball_def)
arith
{ fix j assume j: "j < n"
from j `n \<noteq> 0` have "f' j = enum (Suc j)"
by (auto simp add: f'_def enum_def upd_inj in_upd_image image_comp[symmetric] fun_eq_iff) }
note f'_eq_enum = this
then have "enum ` Suc ` {..< n} = f' ` {..< n}"
by (force simp: enum_inj)
also have "Suc ` {..< n} = {.. n} - {0}"
by (auto simp: image_iff Ball_def) arith
also have "{..< n} = {.. n} - {n}"
by auto
finally have eq: "s - {a} = f' ` {.. n} - {f' n}"
unfolding s_eq `a = enum i` `i = 0`
by (simp add: inj_on_image_set_diff[OF inj_enum] inj_on_image_set_diff[OF inj_f'])
have "enum 0 < f' 0"
using `n \<noteq> 0` by (simp add: enum_strict_mono f'_eq_enum)
also have "\<dots> < f' n"
using `n \<noteq> 0` b.enum_strict_mono[of 0 n] unfolding b_enum by simp
finally have "a \<noteq> f' n"
using `a = enum i` `i = 0` by auto
{ fix t c assume "ksimplex p n t" "c \<in> t" and eq_sma: "s - {a} = t - {c}"
obtain b u where "kuhn_simplex p n b u t"
using `ksimplex p n t` by (auto elim: ksimplex.cases)
then interpret t: kuhn_simplex p n b u t .
{ fix x assume "x \<in> s" "x \<noteq> a"
then have "x (upd 0) = enum (Suc 0) (upd 0)"
by (auto simp: `a = enum i` `i = 0` s_eq enum_def enum_inj) }
then have eq_upd0: "\<forall>x\<in>t-{c}. x (upd 0) = enum (Suc 0) (upd 0)"
unfolding eq_sma[symmetric] by auto
then have "c (upd 0) \<noteq> enum (Suc 0) (upd 0)"
using `n \<noteq> 0` by (intro t.one_step[OF `c\<in>t` ]) (auto simp: upd_space)
then have "c (upd 0) < enum (Suc 0) (upd 0) \<or> c (upd 0) > enum (Suc 0) (upd 0)"
by auto
then have "t = s \<or> t = f' ` {..n}"
proof (elim disjE conjE)
assume *: "c (upd 0) < enum (Suc 0) (upd 0)"
interpret st: kuhn_simplex_pair p n base upd s b u t ..
{ fix x assume "x \<in> t" with * `c\<in>t` eq_upd0[rule_format, of x] have "c \<le> x"
by (auto simp: le_less intro!: t.less[of _ _ "upd 0"]) }
note top = this
have "s = t"
using `a = enum i` `i = 0` `c \<in> t`
by (intro st.ksimplex_eq_bot[OF _ _ _ _ eq_sma])
(auto simp: s_eq enum_mono t.s_eq t.enum_mono top)
then show ?thesis by simp
next
assume *: "c (upd 0) > enum (Suc 0) (upd 0)"
interpret st: kuhn_simplex_pair p n "enum (Suc 0)" "upd \<circ> rot" "f' ` {.. n}" b u t ..
have eq: "f' ` {..n} - {f' n} = t - {c}"
using eq_sma eq by simp
{ fix x assume "x \<in> t" with * `c\<in>t` eq_upd0[rule_format, of x] have "x \<le> c"
by (auto simp: le_less intro!: t.less[of _ _ "upd 0"]) }
note top = this
have "f' ` {..n} = t"
using `a = enum i` `i = 0` `c \<in> t`
by (intro st.ksimplex_eq_top[OF _ _ _ _ eq])
(auto simp: b.s_eq b.enum_mono t.s_eq t.enum_mono b_enum[symmetric] top)
then show ?thesis by simp
qed }
with ks_f' eq `a \<noteq> f' n` `n \<noteq> 0` show ?thesis
apply (intro ex1I[of _ "f' ` {.. n}"])
apply auto []
apply metis
done
next
assume "i = n"
from `n \<noteq> 0` obtain n' where n': "n = Suc n'"
by (cases n) auto
def rot \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> n' | Suc i \<Rightarrow> i"
let ?upd = "upd \<circ> rot"
have rot: "bij_betw rot {..< n} {..< n}"
by (auto simp: bij_betw_def inj_on_def image_iff Bex_def rot_def n' split: nat.splits)
arith
from rot upd have "bij_betw ?upd {..<n} {..<n}"
by (rule bij_betw_trans)
def b \<equiv> "base (upd n' := base (upd n') - 1)"
def f' \<equiv> "\<lambda>i j. if j \<in> ?upd`{..< i} then Suc (b j) else b j"
interpret b: kuhn_simplex p n b "upd \<circ> rot" "f' ` {.. n}"
proof
{ fix i assume "n \<le> i" then show "b i = p"
using base_out[of i] upd_space[of n'] by (auto simp: b_def n') }
show "b \<in> {..<n} \<rightarrow> {..<p}"
using base `n \<noteq> 0` upd_space[of n']
by (auto simp: b_def PiE_def Pi_iff Ball_def upd_space extensional_def n')
show "bij_betw ?upd {..<n} {..<n}" by fact
qed (simp add: f'_def)
have f': "b.enum = f'" unfolding f'_def b.enum_def[abs_def] ..
have ks_f': "ksimplex p n (b.enum ` {.. n})"
unfolding f' by rule unfold_locales
have "0 < n"
using `n \<noteq> 0` by auto
{ from `a = enum i` `n \<noteq> 0` `i = n` lb upd_space[of n']
obtain i' where "i' \<le> n" "enum i' \<noteq> enum n" "0 < enum i' (upd n')"
unfolding s_eq by (auto simp: enum_inj n')
moreover have "enum i' (upd n') = base (upd n')"
unfolding enum_def using `i' \<le> n` `enum i' \<noteq> enum n` by (auto simp: n' upd_inj enum_inj)
ultimately have "0 < base (upd n')"
by auto }
then have benum1: "b.enum (Suc 0) = base"
unfolding b.enum_Suc[OF `0<n`] b.enum_0 by (auto simp: b_def rot_def)
have [simp]: "\<And>j. Suc j < n \<Longrightarrow> rot ` {..< Suc j} = {n'} \<union> {..< j}"
by (auto simp: rot_def image_iff Ball_def split: nat.splits)
have rot_simps: "\<And>j. rot (Suc j) = j" "rot 0 = n'"
by (simp_all add: rot_def)
{ fix j assume j: "Suc j \<le> n" then have "b.enum (Suc j) = enum j"
by (induct j) (auto simp add: benum1 enum_0 b.enum_Suc enum_Suc rot_simps) }
note b_enum_eq_enum = this
then have "enum ` {..< n} = b.enum ` Suc ` {..< n}"
by (auto simp add: image_comp intro!: image_cong)
also have "Suc ` {..< n} = {.. n} - {0}"
by (auto simp: image_iff Ball_def) arith
also have "{..< n} = {.. n} - {n}"
by auto
finally have eq: "s - {a} = b.enum ` {.. n} - {b.enum 0}"
unfolding s_eq `a = enum i` `i = n`
using inj_on_image_set_diff[OF inj_enum order_refl, of "{n}"]
inj_on_image_set_diff[OF b.inj_enum order_refl, of "{0}"]
by (simp add: comp_def)
have "b.enum 0 \<le> b.enum n"
by (simp add: b.enum_mono)
also have "b.enum n < enum n"
using `n \<noteq> 0` by (simp add: enum_strict_mono b_enum_eq_enum n')
finally have "a \<noteq> b.enum 0"
using `a = enum i` `i = n` by auto
{ fix t c assume "ksimplex p n t" "c \<in> t" and eq_sma: "s - {a} = t - {c}"
obtain b' u where "kuhn_simplex p n b' u t"
using `ksimplex p n t` by (auto elim: ksimplex.cases)
then interpret t: kuhn_simplex p n b' u t .
{ fix x assume "x \<in> s" "x \<noteq> a"
then have "x (upd n') = enum n' (upd n')"
by (auto simp: `a = enum i` n' `i = n` s_eq enum_def enum_inj in_upd_image) }
then have eq_upd0: "\<forall>x\<in>t-{c}. x (upd n') = enum n' (upd n')"
unfolding eq_sma[symmetric] by auto
then have "c (upd n') \<noteq> enum n' (upd n')"
using `n \<noteq> 0` by (intro t.one_step[OF `c\<in>t` ]) (auto simp: n' upd_space[unfolded n'])
then have "c (upd n') < enum n' (upd n') \<or> c (upd n') > enum n' (upd n')"
by auto
then have "t = s \<or> t = b.enum ` {..n}"
proof (elim disjE conjE)
assume *: "c (upd n') > enum n' (upd n')"
interpret st: kuhn_simplex_pair p n base upd s b' u t ..
{ fix x assume "x \<in> t" with * `c\<in>t` eq_upd0[rule_format, of x] have "x \<le> c"
by (auto simp: le_less intro!: t.less[of _ _ "upd n'"]) }
note top = this
have "s = t"
using `a = enum i` `i = n` `c \<in> t`
by (intro st.ksimplex_eq_top[OF _ _ _ _ eq_sma])
(auto simp: s_eq enum_mono t.s_eq t.enum_mono top)
then show ?thesis by simp
next
assume *: "c (upd n') < enum n' (upd n')"
interpret st: kuhn_simplex_pair p n b "upd \<circ> rot" "f' ` {.. n}" b' u t ..
have eq: "f' ` {..n} - {b.enum 0} = t - {c}"
using eq_sma eq f' by simp
{ fix x assume "x \<in> t" with * `c\<in>t` eq_upd0[rule_format, of x] have "c \<le> x"
by (auto simp: le_less intro!: t.less[of _ _ "upd n'"]) }
note bot = this
have "f' ` {..n} = t"
using `a = enum i` `i = n` `c \<in> t`
by (intro st.ksimplex_eq_bot[OF _ _ _ _ eq])
(auto simp: b.s_eq b.enum_mono t.s_eq t.enum_mono bot)
with f' show ?thesis by simp
qed }
with ks_f' eq `a \<noteq> b.enum 0` `n \<noteq> 0` show ?thesis
apply (intro ex1I[of _ "b.enum ` {.. n}"])
apply auto []
apply metis
done
next
assume i: "0 < i" "i < n"
def i' \<equiv> "i - 1"
with i have "Suc i' < n"
by simp
with i have Suc_i': "Suc i' = i"
by (simp add: i'_def)
let ?upd = "Fun.swap i' i upd"
from i upd have "bij_betw ?upd {..< n} {..< n}"
by (subst bij_betw_swap_iff) (auto simp: i'_def)
def f' \<equiv> "\<lambda>i j. if j \<in> ?upd`{..< i} then Suc (base j) else base j"
interpret b: kuhn_simplex p n base ?upd "f' ` {.. n}"
proof
show "base \<in> {..<n} \<rightarrow> {..<p}" by fact
{ fix i assume "n \<le> i" then show "base i = p" by fact }
show "bij_betw ?upd {..<n} {..<n}" by fact
qed (simp add: f'_def)
have f': "b.enum = f'" unfolding f'_def b.enum_def[abs_def] ..
have ks_f': "ksimplex p n (b.enum ` {.. n})"
unfolding f' by rule unfold_locales
have "{i} \<subseteq> {..n}"
using i by auto
{ fix j assume "j \<le> n"
moreover have "j < i \<or> i = j \<or> i < j" by arith
moreover note i
ultimately have "enum j = b.enum j \<longleftrightarrow> j \<noteq> i"
unfolding enum_def[abs_def] b.enum_def[abs_def]
by (auto simp add: fun_eq_iff swap_image i'_def
in_upd_image inj_on_image_set_diff[OF inj_upd]) }
note enum_eq_benum = this
then have "enum ` ({.. n} - {i}) = b.enum ` ({.. n} - {i})"
by (intro image_cong) auto
then have eq: "s - {a} = b.enum ` {.. n} - {b.enum i}"
unfolding s_eq `a = enum i`
using inj_on_image_set_diff[OF inj_enum order_refl `{i} \<subseteq> {..n}`]
inj_on_image_set_diff[OF b.inj_enum order_refl `{i} \<subseteq> {..n}`]
by (simp add: comp_def)
have "a \<noteq> b.enum i"
using `a = enum i` enum_eq_benum i by auto
{ fix t c assume "ksimplex p n t" "c \<in> t" and eq_sma: "s - {a} = t - {c}"
obtain b' u where "kuhn_simplex p n b' u t"
using `ksimplex p n t` by (auto elim: ksimplex.cases)
then interpret t: kuhn_simplex p n b' u t .
have "enum i' \<in> s - {a}" "enum (i + 1) \<in> s - {a}"
using `a = enum i` i enum_in by (auto simp: enum_inj i'_def)
then obtain l k where
l: "t.enum l = enum i'" "l \<le> n" "t.enum l \<noteq> c" and
k: "t.enum k = enum (i + 1)" "k \<le> n" "t.enum k \<noteq> c"
unfolding eq_sma by (auto simp: t.s_eq)
with i have "t.enum l < t.enum k"
by (simp add: enum_strict_mono i'_def)
with `l \<le> n` `k \<le> n` have "l < k"
by (simp add: t.enum_strict_mono)
{ assume "Suc l = k"
have "enum (Suc (Suc i')) = t.enum (Suc l)"
using i by (simp add: k `Suc l = k` i'_def)
then have False
using `l < k` `k \<le> n` `Suc i' < n`
by (auto simp: t.enum_Suc enum_Suc l upd_inj fun_eq_iff split: split_if_asm)
(metis Suc_lessD n_not_Suc_n upd_inj) }
with `l < k` have "Suc l < k"
by arith
have c_eq: "c = t.enum (Suc l)"
proof (rule ccontr)
assume "c \<noteq> t.enum (Suc l)"
then have "t.enum (Suc l) \<in> s - {a}"
using `l < k` `k \<le> n` by (simp add: t.s_eq eq_sma)
then obtain j where "t.enum (Suc l) = enum j" "j \<le> n" "enum j \<noteq> enum i"
unfolding s_eq `a = enum i` by auto
with i have "t.enum (Suc l) \<le> t.enum l \<or> t.enum k \<le> t.enum (Suc l)"
by (auto simp add: i'_def enum_mono enum_inj l k)
with `Suc l < k` `k \<le> n` show False
by (simp add: t.enum_mono)
qed
{ have "t.enum (Suc (Suc l)) \<in> s - {a}"
unfolding eq_sma c_eq t.s_eq using `Suc l < k` `k \<le> n` by (auto simp: t.enum_inj)
then obtain j where eq: "t.enum (Suc (Suc l)) = enum j" and "j \<le> n" "j \<noteq> i"
by (auto simp: s_eq `a = enum i`)
moreover have "enum i' < t.enum (Suc (Suc l))"
unfolding l(1)[symmetric] using `Suc l < k` `k \<le> n` by (auto simp: t.enum_strict_mono)
ultimately have "i' < j"
using i by (simp add: enum_strict_mono i'_def)
with `j \<noteq> i` `j \<le> n` have "t.enum k \<le> t.enum (Suc (Suc l))"
unfolding i'_def by (simp add: enum_mono k eq)
then have "k \<le> Suc (Suc l)"
using `k \<le> n` `Suc l < k` by (simp add: t.enum_mono) }
with `Suc l < k` have "Suc (Suc l) = k" by simp
then have "enum (Suc (Suc i')) = t.enum (Suc (Suc l))"
using i by (simp add: k i'_def)
also have "\<dots> = (enum i') (u l := Suc (enum i' (u l)), u (Suc l) := Suc (enum i' (u (Suc l))))"
using `Suc l < k` `k \<le> n` by (simp add: t.enum_Suc l t.upd_inj)
finally have "(u l = upd i' \<and> u (Suc l) = upd (Suc i')) \<or>
(u l = upd (Suc i') \<and> u (Suc l) = upd i')"
using `Suc i' < n` by (auto simp: enum_Suc fun_eq_iff split: split_if_asm)
then have "t = s \<or> t = b.enum ` {..n}"
proof (elim disjE conjE)
assume u: "u l = upd i'"
have "c = t.enum (Suc l)" unfolding c_eq ..
also have "t.enum (Suc l) = enum (Suc i')"
using u `l < k` `k \<le> n` `Suc i' < n` by (simp add: enum_Suc t.enum_Suc l)
also have "\<dots> = a"
using `a = enum i` i by (simp add: i'_def)
finally show ?thesis
using eq_sma `a \<in> s` `c \<in> t` by auto
next
assume u: "u l = upd (Suc i')"
def B \<equiv> "b.enum ` {..n}"
have "b.enum i' = enum i'"
using enum_eq_benum[of i'] i by (auto simp add: i'_def gr0_conv_Suc)
have "c = t.enum (Suc l)" unfolding c_eq ..
also have "t.enum (Suc l) = b.enum (Suc i')"
using u `l < k` `k \<le> n` `Suc i' < n`
by (simp_all add: enum_Suc t.enum_Suc l b.enum_Suc `b.enum i' = enum i'` swap_apply1)
(simp add: Suc_i')
also have "\<dots> = b.enum i"
using i by (simp add: i'_def)
finally have "c = b.enum i" .
then have "t - {c} = B - {c}" "c \<in> B"
unfolding eq_sma[symmetric] eq B_def using i by auto
with `c \<in> t` have "t = B"
by auto
then show ?thesis
by (simp add: B_def)
qed }
with ks_f' eq `a \<noteq> b.enum i` `n \<noteq> 0` `i \<le> n` show ?thesis
apply (intro ex1I[of _ "b.enum ` {.. n}"])
apply auto []
apply metis
done
qed
then show ?thesis
using s `a \<in> s` by (simp add: card_2_exists Ex1_def) metis
qed
text {* Hence another step towards concreteness. *}
lemma kuhn_simplex_lemma:
assumes "\<forall>s. ksimplex p (Suc n) s \<longrightarrow> rl ` s \<subseteq> {.. Suc n}"
and "odd (card {f. \<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> (f = s - {a}) \<and>
rl ` f = {..n} \<and> ((\<exists>j\<le>n. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>f. x j = p))})"
shows "odd (card {s. ksimplex p (Suc n) s \<and> rl ` s = {..Suc n}})"
proof (rule kuhn_complete_lemma[OF finite_ksimplexes refl, unfolded mem_Collect_eq,
where bnd="\<lambda>f. (\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = p)"],
safe del: notI)
have *: "\<And>x y. x = y \<Longrightarrow> odd (card x) \<Longrightarrow> odd (card y)"
by auto
show "odd (card {f. (\<exists>s\<in>{s. ksimplex p (Suc n) s}. \<exists>a\<in>s. f = s - {a}) \<and>
rl ` f = {..n} \<and> ((\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = p))})"
apply (rule *[OF _ assms(2)])
apply (auto simp: atLeast0AtMost)
done
next
fix s assume s: "ksimplex p (Suc n) s"
then show "card s = n + 2"
by (simp add: ksimplex_card)
fix a assume a: "a \<in> s" then show "rl a \<le> Suc n"
using assms(1) s by (auto simp: subset_eq)
let ?S = "{t. ksimplex p (Suc n) t \<and> (\<exists>b\<in>t. s - {a} = t - {b})}"
{ fix j assume j: "j \<le> n" "\<forall>x\<in>s - {a}. x j = 0"
with s a show "card ?S = 1"
using ksimplex_replace_0[of p "n + 1" s a j]
by (subst eq_commute) simp }
{ fix j assume j: "j \<le> n" "\<forall>x\<in>s - {a}. x j = p"
with s a show "card ?S = 1"
using ksimplex_replace_1[of p "n + 1" s a j]
by (subst eq_commute) simp }
{ assume "card ?S \<noteq> 2" "\<not> (\<exists>j\<in>{..n}. \<forall>x\<in>s - {a}. x j = p)"
with s a show "\<exists>j\<in>{..n}. \<forall>x\<in>s - {a}. x j = 0"
using ksimplex_replace_2[of p "n + 1" s a]
by (subst (asm) eq_commute) auto }
qed
subsection {* Reduced labelling *}
definition reduced :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> nat" where "reduced n x = (LEAST k. k = n \<or> x k \<noteq> 0)"
lemma reduced_labelling:
shows "reduced n x \<le> n"
and "\<forall>i<reduced n x. x i = 0"
and "reduced n x = n \<or> x (reduced n x) \<noteq> 0"
proof -
show "reduced n x \<le> n"
unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) auto
show "\<forall>i<reduced n x. x i = 0"
unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) fastforce+
show "reduced n x = n \<or> x (reduced n x) \<noteq> 0"
unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) fastforce+
qed
lemma reduced_labelling_unique:
"r \<le> n \<Longrightarrow> \<forall>i<r. x i = 0 \<Longrightarrow> r = n \<or> x r \<noteq> 0 \<Longrightarrow> reduced n x = r"
unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) (metis le_less not_le)+
lemma reduced_labelling_zero: "j < n \<Longrightarrow> x j = 0 \<Longrightarrow> reduced n x \<noteq> j"
using reduced_labelling[of n x] by auto
lemma reduce_labelling_zero[simp]: "reduced 0 x = 0"
by (rule reduced_labelling_unique) auto
lemma reduced_labelling_nonzero: "j < n \<Longrightarrow> x j \<noteq> 0 \<Longrightarrow> reduced n x \<le> j"
using reduced_labelling[of n x] by (elim allE[where x=j]) auto
lemma reduced_labelling_Suc: "reduced (Suc n) x \<noteq> Suc n \<Longrightarrow> reduced (Suc n) x = reduced n x"
using reduced_labelling[of "Suc n" x]
by (intro reduced_labelling_unique[symmetric]) auto
lemma complete_face_top:
assumes "\<forall>x\<in>f. \<forall>j\<le>n. x j = 0 \<longrightarrow> lab x j = 0"
and "\<forall>x\<in>f. \<forall>j\<le>n. x j = p \<longrightarrow> lab x j = 1"
and eq: "(reduced (Suc n) \<circ> lab) ` f = {..n}"
shows "((\<exists>j\<le>n. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>f. x j = p)) \<longleftrightarrow> (\<forall>x\<in>f. x n = p)"
proof (safe del: disjCI)
fix x j assume j: "j \<le> n" "\<forall>x\<in>f. x j = 0"
{ fix x assume "x \<in> f" with assms j have "reduced (Suc n) (lab x) \<noteq> j"
by (intro reduced_labelling_zero) auto }
moreover have "j \<in> (reduced (Suc n) \<circ> lab) ` f"
using j eq by auto
ultimately show "x n = p"
by force
next
fix x j assume j: "j \<le> n" "\<forall>x\<in>f. x j = p" and x: "x \<in> f"
have "j = n"
proof (rule ccontr)
assume "\<not> ?thesis"
{ fix x assume "x \<in> f"
with assms j have "reduced (Suc n) (lab x) \<le> j"
by (intro reduced_labelling_nonzero) auto
then have "reduced (Suc n) (lab x) \<noteq> n"
using `j \<noteq> n` `j \<le> n` by simp }
moreover
have "n \<in> (reduced (Suc n) \<circ> lab) ` f"
using eq by auto
ultimately show False
by force
qed
moreover have "j \<in> (reduced (Suc n) \<circ> lab) ` f"
using j eq by auto
ultimately show "x n = p"
using j x by auto
qed auto
text {* Hence we get just about the nice induction. *}
lemma kuhn_induction:
assumes "0 < p"
and lab_0: "\<forall>x. \<forall>j\<le>n. (\<forall>j. x j \<le> p) \<and> x j = 0 \<longrightarrow> lab x j = 0"
and lab_1: "\<forall>x. \<forall>j\<le>n. (\<forall>j. x j \<le> p) \<and> x j = p \<longrightarrow> lab x j = 1"
and odd: "odd (card {s. ksimplex p n s \<and> (reduced n\<circ>lab) ` s = {..n}})"
shows "odd (card {s. ksimplex p (Suc n) s \<and> (reduced (Suc n)\<circ>lab) ` s = {..Suc n}})"
proof -
let ?rl = "reduced (Suc n) \<circ> lab" and ?ext = "\<lambda>f v. \<exists>j\<le>n. \<forall>x\<in>f. x j = v"
let ?ext = "\<lambda>s. (\<exists>j\<le>n. \<forall>x\<in>s. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>s. x j = p)"
have "\<forall>s. ksimplex p (Suc n) s \<longrightarrow> ?rl ` s \<subseteq> {..Suc n}"
by (simp add: reduced_labelling subset_eq)
moreover
have "{s. ksimplex p n s \<and> (reduced n \<circ> lab) ` s = {..n}} =
{f. \<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> f = s - {a} \<and> ?rl ` f = {..n} \<and> ?ext f}"
proof (intro set_eqI, safe del: disjCI equalityI disjE)
fix s assume s: "ksimplex p n s" and rl: "(reduced n \<circ> lab) ` s = {..n}"
from s obtain u b where "kuhn_simplex p n u b s" by (auto elim: ksimplex.cases)
then interpret kuhn_simplex p n u b s .
have all_eq_p: "\<forall>x\<in>s. x n = p"
by (auto simp: out_eq_p)
moreover
{ fix x assume "x \<in> s"
with lab_1[rule_format, of n x] all_eq_p s_le_p[of x]
have "?rl x \<le> n"
by (auto intro!: reduced_labelling_nonzero)
then have "?rl x = reduced n (lab x)"
by (auto intro!: reduced_labelling_Suc) }
then have "?rl ` s = {..n}"
using rl by (simp cong: image_cong)
moreover
obtain t a where "ksimplex p (Suc n) t" "a \<in> t" "s = t - {a}"
using s unfolding simplex_top_face[OF `0 < p` all_eq_p] by auto
ultimately
show "\<exists>t a. ksimplex p (Suc n) t \<and> a \<in> t \<and> s = t - {a} \<and> ?rl ` s = {..n} \<and> ?ext s"
by auto
next
fix x s a assume s: "ksimplex p (Suc n) s" and rl: "?rl ` (s - {a}) = {.. n}"
and a: "a \<in> s" and "?ext (s - {a})"
from s obtain u b where "kuhn_simplex p (Suc n) u b s" by (auto elim: ksimplex.cases)
then interpret kuhn_simplex p "Suc n" u b s .
have all_eq_p: "\<forall>x\<in>s. x (Suc n) = p"
by (auto simp: out_eq_p)
{ fix x assume "x \<in> s - {a}"
then have "?rl x \<in> ?rl ` (s - {a})"
by auto
then have "?rl x \<le> n"
unfolding rl by auto
then have "?rl x = reduced n (lab x)"
by (auto intro!: reduced_labelling_Suc) }
then show rl': "(reduced n\<circ>lab) ` (s - {a}) = {..n}"
unfolding rl[symmetric] by (intro image_cong) auto
from `?ext (s - {a})`
have all_eq_p: "\<forall>x\<in>s - {a}. x n = p"
proof (elim disjE exE conjE)
fix j assume "j \<le> n" "\<forall>x\<in>s - {a}. x j = 0"
with lab_0[rule_format, of j] all_eq_p s_le_p
have "\<And>x. x \<in> s - {a} \<Longrightarrow> reduced (Suc n) (lab x) \<noteq> j"
by (intro reduced_labelling_zero) auto
moreover have "j \<in> ?rl ` (s - {a})"
using `j \<le> n` unfolding rl by auto
ultimately show ?thesis
by force
next
fix j assume "j \<le> n" and eq_p: "\<forall>x\<in>s - {a}. x j = p"
show ?thesis
proof cases
assume "j = n" with eq_p show ?thesis by simp
next
assume "j \<noteq> n"
{ fix x assume x: "x \<in> s - {a}"
have "reduced n (lab x) \<le> j"
proof (rule reduced_labelling_nonzero)
show "lab x j \<noteq> 0"
using lab_1[rule_format, of j x] x s_le_p[of x] eq_p `j \<le> n` by auto
show "j < n"
using `j \<le> n` `j \<noteq> n` by simp
qed
then have "reduced n (lab x) \<noteq> n"
using `j \<le> n` `j \<noteq> n` by simp }
moreover have "n \<in> (reduced n\<circ>lab) ` (s - {a})"
unfolding rl' by auto
ultimately show ?thesis
by force
qed
qed
show "ksimplex p n (s - {a})"
unfolding simplex_top_face[OF `0 < p` all_eq_p] using s a by auto
qed
ultimately show ?thesis
using assms by (intro kuhn_simplex_lemma) auto
qed
text {* And so we get the final combinatorial result. *}
lemma ksimplex_0: "ksimplex p 0 s \<longleftrightarrow> s = {(\<lambda>x. p)}"
proof
assume "ksimplex p 0 s" then show "s = {(\<lambda>x. p)}"
by (blast dest: kuhn_simplex.ksimplex_0 elim: ksimplex.cases)
next
assume s: "s = {(\<lambda>x. p)}"
show "ksimplex p 0 s"
proof (intro ksimplex, unfold_locales)
show "(\<lambda>_. p) \<in> {..<0::nat} \<rightarrow> {..<p}" by auto
show "bij_betw id {..<0} {..<0}"
by simp
qed (auto simp: s)
qed
lemma kuhn_combinatorial:
assumes "0 < p"
and "\<forall>x j. (\<forall>j. x j \<le> p) \<and> j < n \<and> x j = 0 \<longrightarrow> lab x j = 0"
and "\<forall>x j. (\<forall>j. x j \<le> p) \<and> j < n \<and> x j = p \<longrightarrow> lab x j = 1"
shows "odd (card {s. ksimplex p n s \<and> (reduced n\<circ>lab) ` s = {..n}})"
(is "odd (card (?M n))")
using assms
proof (induct n)
case 0 then show ?case
by (simp add: ksimplex_0 cong: conj_cong)
next
case (Suc n)
then have "odd (card (?M n))"
by force
with Suc show ?case
using kuhn_induction[of p n] by (auto simp: comp_def)
qed
lemma kuhn_lemma:
fixes n p :: nat
assumes "0 < p"
and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. label x i = (0::nat) \<or> label x i = 1)"
and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow> label x i = 0)"
and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow> label x i = 1)"
obtains q where "\<forall>i<n. q i < p"
and "\<forall>i<n. \<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and> (\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and> label r i \<noteq> label s i"
proof -
let ?rl = "reduced n\<circ>label"
let ?A = "{s. ksimplex p n s \<and> ?rl ` s = {..n}}"
have "odd (card ?A)"
using assms by (intro kuhn_combinatorial[of p n label]) auto
then have "?A \<noteq> {}"
by (intro notI) simp
then obtain s b u where "kuhn_simplex p n b u s" and rl: "?rl ` s = {..n}"
by (auto elim: ksimplex.cases)
interpret kuhn_simplex p n b u s by fact
show ?thesis
proof (intro that[of b] allI impI)
fix i assume "i < n" then show "b i < p"
using base by auto
next
case goal2
then have "i \<in> {.. n}" "Suc i \<in> {.. n}"
by auto
then obtain u v where u: "u \<in> s" "Suc i = ?rl u" and v: "v \<in> s" "i = ?rl v"
unfolding rl[symmetric] by blast
have "label u i \<noteq> label v i"
using reduced_labelling [of n "label u"] reduced_labelling [of n "label v"]
u(2)[symmetric] v(2)[symmetric] `i < n`
by auto
moreover
{ fix j assume "j < n"
then have "b j \<le> u j" "u j \<le> b j + 1" "b j \<le> v j" "v j \<le> b j + 1"
using base_le[OF `u\<in>s`] le_Suc_base[OF `u\<in>s`] base_le[OF `v\<in>s`] le_Suc_base[OF `v\<in>s`] by auto }
ultimately show ?case
by blast
qed
qed
subsection {* The main result for the unit cube *}
lemma kuhn_labelling_lemma':
assumes "(\<forall>x::nat\<Rightarrow>real. P x \<longrightarrow> P (f x))"
and "\<forall>x. P x \<longrightarrow> (\<forall>i::nat. Q i \<longrightarrow> 0 \<le> x i \<and> x i \<le> 1)"
shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
(\<forall>x i. P x \<and> Q i \<and> x i = 0 \<longrightarrow> l x i = 0) \<and>
(\<forall>x i. P x \<and> Q i \<and> x i = 1 \<longrightarrow> l x i = 1) \<and>
(\<forall>x i. P x \<and> Q i \<and> l x i = 0 \<longrightarrow> x i \<le> f x i) \<and>
(\<forall>x i. P x \<and> Q i \<and> l x i = 1 \<longrightarrow> f x i \<le> x i)"
proof -
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
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
show ?thesis
unfolding and_forall_thm
apply (subst choice_iff[symmetric])+
apply rule
apply rule
proof -
case goal1
let ?R = "\<lambda>y::nat.
(P x \<and> Q xa \<and> x xa = 0 \<longrightarrow> y = 0) \<and>
(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)"
{
assume "P x" and "Q xa"
then have "0 \<le> f x xa \<and> f x xa \<le> 1"
using assms(2)[rule_format,of "f x" xa]
apply (drule_tac assms(1)[rule_format])
apply auto
done
}
then have "?R 0 \<or> ?R 1"
by auto
then show ?case
by auto
qed
qed
definition unit_cube :: "'a::euclidean_space set"
where "unit_cube = {x. \<forall>i\<in>Basis. 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1}"
lemma mem_unit_cube: "x \<in> unit_cube \<longleftrightarrow> (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
unfolding unit_cube_def by simp
lemma bounded_unit_cube: "bounded unit_cube"
unfolding bounded_def
proof (intro exI ballI)
fix y :: 'a assume y: "y \<in> unit_cube"
have "dist 0 y = norm y" by (rule dist_0_norm)
also have "\<dots> = norm (\<Sum>i\<in>Basis. (y \<bullet> i) *\<^sub>R i)" unfolding euclidean_representation ..
also have "\<dots> \<le> (\<Sum>i\<in>Basis. norm ((y \<bullet> i) *\<^sub>R i))" by (rule norm_setsum)
also have "\<dots> \<le> (\<Sum>i::'a\<in>Basis. 1)"
by (rule setsum_mono, simp add: y [unfolded mem_unit_cube])
finally show "dist 0 y \<le> (\<Sum>i::'a\<in>Basis. 1)" .
qed
lemma closed_unit_cube: "closed unit_cube"
unfolding unit_cube_def Collect_ball_eq Collect_conj_eq
by (rule closed_INT, auto intro!: closed_Collect_le)
lemma compact_unit_cube: "compact unit_cube" (is "compact ?C")
unfolding compact_eq_seq_compact_metric
using bounded_unit_cube closed_unit_cube
by (rule bounded_closed_imp_seq_compact)
lemma brouwer_cube:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "continuous_on unit_cube f"
and "f ` unit_cube \<subseteq> unit_cube"
shows "\<exists>x\<in>unit_cube. f x = x"
proof (rule ccontr)
def n \<equiv> "DIM('a)"
have n: "1 \<le> n" "0 < n" "n \<noteq> 0"
unfolding n_def by (auto simp add: Suc_le_eq DIM_positive)
assume "\<not> ?thesis"
then have *: "\<not> (\<exists>x\<in>unit_cube. f x - x = 0)"
by auto
obtain d where
d: "d > 0" "\<And>x. x \<in> unit_cube \<Longrightarrow> d \<le> norm (f x - x)"
apply (rule brouwer_compactness_lemma[OF compact_unit_cube _ *])
apply (rule continuous_intros assms)+
apply blast
done
have *: "\<forall>x. x \<in> unit_cube \<longrightarrow> f x \<in> unit_cube"
"\<forall>x. x \<in> (unit_cube::'a set) \<longrightarrow> (\<forall>i\<in>Basis. True \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
using assms(2)[unfolded image_subset_iff Ball_def]
unfolding mem_unit_cube
by auto
obtain label :: "'a \<Rightarrow> 'a \<Rightarrow> nat" where
"\<forall>x. \<forall>i\<in>Basis. label x i \<le> 1"
"\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> True \<and> x \<bullet> i = 0 \<longrightarrow> label x i = 0"
"\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> True \<and> x \<bullet> i = 1 \<longrightarrow> label x i = 1"
"\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> True \<and> label x i = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i"
"\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> True \<and> label x i = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i"
using kuhn_labelling_lemma[OF *] by blast
note label = this [rule_format]
have lem1: "\<forall>x\<in>unit_cube. \<forall>y\<in>unit_cube. \<forall>i\<in>Basis. label x i \<noteq> label y i \<longrightarrow>
abs (f x \<bullet> i - x \<bullet> i) \<le> norm (f y - f x) + norm (y - x)"
proof safe
fix x y :: 'a
assume x: "x \<in> unit_cube"
assume y: "y \<in> unit_cube"
fix i
assume i: "label x i \<noteq> label y i" "i \<in> Basis"
have *: "\<And>x y fx fy :: real. x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy \<Longrightarrow>
abs (fx - x) \<le> abs (fy - fx) + abs (y - x)" by auto
have "\<bar>(f x - x) \<bullet> i\<bar> \<le> abs ((f y - f x)\<bullet>i) + abs ((y - x)\<bullet>i)"
unfolding inner_simps
apply (rule *)
apply (cases "label x i = 0")
apply (rule disjI1)
apply rule
prefer 3
apply (rule disjI2)
apply rule
proof -
assume lx: "label x i = 0"
then have ly: "label y i = 1"
using i label(1)[of i y]
by auto
show "x \<bullet> i \<le> f x \<bullet> i"
apply (rule label(4)[rule_format])
using x y lx i(2)
apply auto
done
show "f y \<bullet> i \<le> y \<bullet> i"
apply (rule label(5)[rule_format])
using x y ly i(2)
apply auto
done
next
assume "label x i \<noteq> 0"
then have l: "label x i = 1" "label y i = 0"
using i label(1)[of i x] label(1)[of i y]
by auto
show "f x \<bullet> i \<le> x \<bullet> i"
apply (rule label(5)[rule_format])
using x y l i(2)
apply auto
done
show "y \<bullet> i \<le> f y \<bullet> i"
apply (rule label(4)[rule_format])
using x y l i(2)
apply auto
done
qed
also have "\<dots> \<le> norm (f y - f x) + norm (y - x)"
apply (rule add_mono)
apply (rule Basis_le_norm[OF i(2)])+
done
finally show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)"
unfolding inner_simps .
qed
have "\<exists>e>0. \<forall>x\<in>unit_cube. \<forall>y\<in>unit_cube. \<forall>z\<in>unit_cube. \<forall>i\<in>Basis.
norm (x - z) < e \<and> norm (y - z) < e \<and> label x i \<noteq> label y i \<longrightarrow>
abs ((f(z) - z)\<bullet>i) < d / (real n)"
proof -
have d': "d / real n / 8 > 0"
using d(1) by (simp add: n_def DIM_positive)
have *: "uniformly_continuous_on unit_cube f"
by (rule compact_uniformly_continuous[OF assms(1) compact_unit_cube])
obtain e where e:
"e > 0"
"\<And>x x'. x \<in> unit_cube \<Longrightarrow>
x' \<in> unit_cube \<Longrightarrow>
norm (x' - x) < e \<Longrightarrow>
norm (f x' - f x) < d / real n / 8"
using *[unfolded uniformly_continuous_on_def,rule_format,OF d']
unfolding dist_norm
by blast
show ?thesis
apply (rule_tac x="min (e/2) (d/real n/8)" in exI)
apply safe
proof -
show "0 < min (e / 2) (d / real n / 8)"
using d' e by auto
fix x y z i
assume as:
"x \<in> unit_cube" "y \<in> unit_cube" "z \<in> unit_cube"
"norm (x - z) < min (e / 2) (d / real n / 8)"
"norm (y - z) < min (e / 2) (d / real n / 8)"
"label x i \<noteq> label y i"
assume i: "i \<in> Basis"
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>
n1 < d4 \<Longrightarrow> n2 < 2 * d4 \<Longrightarrow> n3 < d4 \<Longrightarrow> n4 < d4 \<Longrightarrow>
(8 * d4 = d) \<Longrightarrow> abs(fz - z) < d"
by auto
show "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
unfolding inner_simps
proof (rule *)
show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y -f x) + norm (y - x)"
apply (rule lem1[rule_format])
using as i
apply auto
done
show "\<bar>f x \<bullet> i - f z \<bullet> i\<bar> \<le> norm (f x - f z)" "\<bar>x \<bullet> i - z \<bullet> i\<bar> \<le> norm (x - z)"
unfolding inner_diff_left[symmetric]
by (rule Basis_le_norm[OF i])+
have tria: "norm (y - x) \<le> norm (y - z) + norm (x - z)"
using dist_triangle[of y x z, unfolded dist_norm]
unfolding norm_minus_commute
by auto
also have "\<dots> < e / 2 + e / 2"
apply (rule add_strict_mono)
using as(4,5)
apply auto
done
finally show "norm (f y - f x) < d / real n / 8"
apply -
apply (rule e(2))
using as
apply auto
done
have "norm (y - z) + norm (x - z) < d / real n / 8 + d / real n / 8"
apply (rule add_strict_mono)
using as
apply auto
done
then show "norm (y - x) < 2 * (d / real n / 8)"
using tria
by auto
show "norm (f x - f z) < d / real n / 8"
apply (rule e(2))
using as e(1)
apply auto
done
qed (insert as, auto)
qed
qed
then
obtain e where e:
"e > 0"
"\<And>x y z i. x \<in> unit_cube \<Longrightarrow>
y \<in> unit_cube \<Longrightarrow>
z \<in> unit_cube \<Longrightarrow>
i \<in> Basis \<Longrightarrow>
norm (x - z) < e \<and> norm (y - z) < e \<and> label x i \<noteq> label y i \<Longrightarrow>
\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
by blast
obtain p :: nat where p: "1 + real n / e \<le> real p"
using real_arch_simple ..
have "1 + real n / e > 0"
using e(1) n by (simp add: add_pos_pos)
then have "p > 0"
using p by auto
obtain b :: "nat \<Rightarrow> 'a" where b: "bij_betw b {..< n} Basis"
by atomize_elim (auto simp: n_def intro!: finite_same_card_bij)
def b' \<equiv> "inv_into {..< n} b"
then have b': "bij_betw b' Basis {..< n}"
using bij_betw_inv_into[OF b] by auto
then have b'_Basis: "\<And>i. i \<in> Basis \<Longrightarrow> b' i \<in> {..< n}"
unfolding bij_betw_def by (auto simp: set_eq_iff)
have bb'[simp]:"\<And>i. i \<in> Basis \<Longrightarrow> b (b' i) = i"
unfolding b'_def
using b
by (auto simp: f_inv_into_f bij_betw_def)
have b'b[simp]:"\<And>i. i < n \<Longrightarrow> b' (b i) = i"
unfolding b'_def
using b
by (auto simp: inv_into_f_eq bij_betw_def)
have *: "\<And>x :: nat. x = 0 \<or> x = 1 \<longleftrightarrow> x \<le> 1"
by auto
have b'': "\<And>j. j < n \<Longrightarrow> b j \<in> Basis"
using b unfolding bij_betw_def by auto
have q1: "0 < p" "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow>
(\<forall>i<n. (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0 \<or>
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
unfolding *
using `p > 0` `n > 0`
using label(1)[OF b'']
by auto
{ fix x :: "nat \<Rightarrow> nat" and i assume "\<forall>i<n. x i \<le> p" "i < n" "x i = p \<or> x i = 0"
then have "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> (unit_cube::'a set)"
using b'_Basis
by (auto simp add: mem_unit_cube inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) }
note cube = this
have q2: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow>
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)"
unfolding o_def using cube `p > 0` by (intro allI impI label(2)) (auto simp add: b'')
have q3: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow>
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
using cube `p > 0` unfolding o_def by (intro allI impI label(3)) (auto simp add: b'')
obtain q where q:
"\<forall>i<n. q i < p"
"\<forall>i<n.
\<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and>
(\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and>
(label (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i) \<circ> b) i \<noteq>
(label (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i) \<circ> b) i"
by (rule kuhn_lemma[OF q1 q2 q3])
def z \<equiv> "(\<Sum>i\<in>Basis. (real (q (b' i)) / real p) *\<^sub>R i)::'a"
have "\<exists>i\<in>Basis. d / real n \<le> abs ((f z - z)\<bullet>i)"
proof (rule ccontr)
have "\<forall>i\<in>Basis. q (b' i) \<in> {0..p}"
using q(1) b'
by (auto intro: less_imp_le simp: bij_betw_def)
then have "z \<in> unit_cube"
unfolding z_def mem_unit_cube
using b'_Basis
by (auto simp add: bij_betw_def zero_le_divide_iff divide_le_eq_1)
then have d_fz_z: "d \<le> norm (f z - z)"
by (rule d)
assume "\<not> ?thesis"
then have as: "\<forall>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar> < d / real n"
using `n > 0`
by (auto simp add: not_le inner_diff)
have "norm (f z - z) \<le> (\<Sum>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar>)"
unfolding inner_diff_left[symmetric]
by (rule norm_le_l1)
also have "\<dots> < (\<Sum>(i::'a) \<in> Basis. d / real n)"
apply (rule setsum_strict_mono)
using as
apply auto
done
also have "\<dots> = d"
using DIM_positive[where 'a='a]
by (auto simp: real_eq_of_nat n_def)
finally show False
using d_fz_z by auto
qed
then obtain i where i: "i \<in> Basis" "d / real n \<le> \<bar>(f z - z) \<bullet> i\<bar>" ..
have *: "b' i < n"
using i and b'[unfolded bij_betw_def]
by auto
obtain r s where rs:
"\<And>j. j < n \<Longrightarrow> q j \<le> r j \<and> r j \<le> q j + 1"
"\<And>j. j < n \<Longrightarrow> q j \<le> s j \<and> s j \<le> q j + 1"
"(label (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i) \<circ> b) (b' i) \<noteq>
(label (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i) \<circ> b) (b' i)"
using q(2)[rule_format,OF *] by blast
have b'_im: "\<And>i. i \<in> Basis \<Longrightarrow> b' i < n"
using b' unfolding bij_betw_def by auto
def r' \<equiv> "(\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i)::'a"
have "\<And>i. i \<in> Basis \<Longrightarrow> r (b' i) \<le> p"
apply (rule order_trans)
apply (rule rs(1)[OF b'_im,THEN conjunct2])
using q(1)[rule_format,OF b'_im]
apply (auto simp add: Suc_le_eq)
done
then have "r' \<in> unit_cube"
unfolding r'_def mem_unit_cube
using b'_Basis
by (auto simp add: bij_betw_def zero_le_divide_iff divide_le_eq_1)
def s' \<equiv> "(\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i)::'a"
have "\<And>i. i \<in> Basis \<Longrightarrow> s (b' i) \<le> p"
apply (rule order_trans)
apply (rule rs(2)[OF b'_im, THEN conjunct2])
using q(1)[rule_format,OF b'_im]
apply (auto simp add: Suc_le_eq)
done
then have "s' \<in> unit_cube"
unfolding s'_def mem_unit_cube
using b'_Basis
by (auto simp add: bij_betw_def zero_le_divide_iff divide_le_eq_1)
have "z \<in> unit_cube"
unfolding z_def mem_unit_cube
using b'_Basis q(1)[rule_format,OF b'_im] `p > 0`
by (auto simp add: bij_betw_def zero_le_divide_iff divide_le_eq_1 less_imp_le)
have *: "\<And>x. 1 + real x = real (Suc x)"
by auto
{
have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
apply (rule setsum_mono)
using rs(1)[OF b'_im]
apply (auto simp add:* field_simps)
done
also have "\<dots> < e * real p"
using p `e > 0` `p > 0`
by (auto simp add: field_simps n_def real_of_nat_def)
finally have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" .
}
moreover
{
have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
apply (rule setsum_mono)
using rs(2)[OF b'_im]
apply (auto simp add:* field_simps)
done
also have "\<dots> < e * real p"
using p `e > 0` `p > 0`
by (auto simp add: field_simps n_def real_of_nat_def)
finally have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" .
}
ultimately
have "norm (r' - z) < e" and "norm (s' - z) < e"
unfolding r'_def s'_def z_def
using `p > 0`
apply (rule_tac[!] le_less_trans[OF norm_le_l1])
apply (auto simp add: field_simps setsum_divide_distrib[symmetric] inner_diff_left)
done
then have "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
using rs(3) i
unfolding r'_def[symmetric] s'_def[symmetric] o_def bb'
by (intro e(2)[OF `r'\<in>unit_cube` `s'\<in>unit_cube` `z\<in>unit_cube`]) auto
then show False
using i by auto
qed
subsection {* Retractions *}
definition "retraction s t r \<longleftrightarrow> t \<subseteq> s \<and> continuous_on s r \<and> r ` s \<subseteq> t \<and> (\<forall>x\<in>t. r x = x)"
definition retract_of (infixl "retract'_of" 12)
where "(t retract_of s) \<longleftrightarrow> (\<exists>r. retraction s t r)"
lemma retraction_idempotent: "retraction s t r \<Longrightarrow> x \<in> s \<Longrightarrow> r (r x) = r x"
unfolding retraction_def by auto
subsection {* Preservation of fixpoints under (more general notion of) retraction *}
lemma invertible_fixpoint_property:
fixes s :: "'a::euclidean_space set"
and t :: "'b::euclidean_space set"
assumes "continuous_on t i"
and "i ` t \<subseteq> s"
and "continuous_on s r"
and "r ` s \<subseteq> t"
and "\<forall>y\<in>t. r (i y) = y"
and "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)"
and "continuous_on t g"
and "g ` t \<subseteq> t"
obtains y where "y \<in> t" and "g y = y"
proof -
have "\<exists>x\<in>s. (i \<circ> g \<circ> r) x = x"
apply (rule assms(6)[rule_format])
apply rule
apply (rule continuous_on_compose assms)+
apply ((rule continuous_on_subset)?, rule assms)+
using assms(2,4,8)
apply auto
apply blast
done
then obtain x where x: "x \<in> s" "(i \<circ> g \<circ> r) x = x" ..
then have *: "g (r x) \<in> t"
using assms(4,8) by auto
have "r ((i \<circ> g \<circ> r) x) = r x"
using x by auto
then show ?thesis
apply (rule_tac that[of "r x"])
using x
unfolding o_def
unfolding assms(5)[rule_format,OF *]
using assms(4)
apply auto
done
qed
lemma homeomorphic_fixpoint_property:
fixes s :: "'a::euclidean_space set"
and t :: "'b::euclidean_space set"
assumes "s homeomorphic t"
shows "(\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)) \<longleftrightarrow>
(\<forall>g. continuous_on t g \<and> g ` t \<subseteq> t \<longrightarrow> (\<exists>y\<in>t. g y = y))"
proof -
obtain r i where
"\<forall>x\<in>s. i (r x) = x"
"r ` s = t"
"continuous_on s r"
"\<forall>y\<in>t. r (i y) = y"
"i ` t = s"
"continuous_on t i"
using assms
unfolding homeomorphic_def homeomorphism_def
by blast
then show ?thesis
apply -
apply rule
apply (rule_tac[!] allI impI)+
apply (rule_tac g=g in invertible_fixpoint_property[of t i s r])
prefer 10
apply (rule_tac g=f in invertible_fixpoint_property[of s r t i])
apply auto
done
qed
lemma retract_fixpoint_property:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
and s :: "'a set"
assumes "t retract_of s"
and "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)"
and "continuous_on t g"
and "g ` t \<subseteq> t"
obtains y where "y \<in> t" and "g y = y"
proof -
obtain h where "retraction s t h"
using assms(1) unfolding retract_of_def ..
then show ?thesis
unfolding retraction_def
apply -
apply (rule invertible_fixpoint_property[OF continuous_on_id _ _ _ _ assms(2), of t h g])
prefer 7
apply (rule_tac y = y in that)
using assms
apply auto
done
qed
subsection {* The Brouwer theorem for any set with nonempty interior *}
lemma convex_unit_cube: "convex unit_cube"
apply (rule is_interval_convex)
apply (clarsimp simp add: is_interval_def mem_unit_cube)
apply (drule (1) bspec)+
apply auto
done
lemma brouwer_weak:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "compact s"
and "convex s"
and "interior s \<noteq> {}"
and "continuous_on s f"
and "f ` s \<subseteq> s"
obtains x where "x \<in> s" and "f x = x"
proof -
let ?U = "unit_cube :: 'a set"
have "\<Sum>Basis /\<^sub>R 2 \<in> interior ?U"
proof (rule interiorI)
let ?I = "(\<Inter>i\<in>Basis. {x::'a. 0 < x \<bullet> i} \<inter> {x. x \<bullet> i < 1})"
show "open ?I"
by (intro open_INT finite_Basis ballI open_Int, auto intro: open_Collect_less)
show "\<Sum>Basis /\<^sub>R 2 \<in> ?I"
by simp
show "?I \<subseteq> unit_cube"
unfolding unit_cube_def by force
qed
then have *: "interior ?U \<noteq> {}" by fast
have *: "?U homeomorphic s"
using homeomorphic_convex_compact[OF convex_unit_cube compact_unit_cube * assms(2,1,3)] .
have "\<forall>f. continuous_on ?U f \<and> f ` ?U \<subseteq> ?U \<longrightarrow>
(\<exists>x\<in>?U. f x = x)"
using brouwer_cube by auto
then show ?thesis
unfolding homeomorphic_fixpoint_property[OF *]
apply (erule_tac x=f in allE)
apply (erule impE)
defer
apply (erule bexE)
apply (rule_tac x=y in that)
using assms
apply auto
done
qed
text {* And in particular for a closed ball. *}
lemma brouwer_ball:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "e > 0"
and "continuous_on (cball a e) f"
and "f ` cball a e \<subseteq> cball a e"
obtains x where "x \<in> cball a e" and "f x = x"
using brouwer_weak[OF compact_cball convex_cball, of a e f]
unfolding interior_cball ball_eq_empty
using assms by auto
text {*Still more general form; could derive this directly without using the
rather involved @{text "HOMEOMORPHIC_CONVEX_COMPACT"} theorem, just using
a scaling and translation to put the set inside the unit cube. *}
lemma brouwer:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "compact s"
and "convex s"
and "s \<noteq> {}"
and "continuous_on s f"
and "f ` s \<subseteq> s"
obtains x where "x \<in> s" and "f x = x"
proof -
have "\<exists>e>0. s \<subseteq> cball 0 e"
using compact_imp_bounded[OF assms(1)]
unfolding bounded_pos
apply (erule_tac exE)
apply (rule_tac x=b in exI)
apply (auto simp add: dist_norm)
done
then obtain e where e: "e > 0" "s \<subseteq> cball 0 e"
by blast
have "\<exists>x\<in> cball 0 e. (f \<circ> closest_point s) x = x"
apply (rule_tac brouwer_ball[OF e(1), of 0 "f \<circ> closest_point s"])
apply (rule continuous_on_compose )
apply (rule continuous_on_closest_point[OF assms(2) compact_imp_closed[OF assms(1)] assms(3)])
apply (rule continuous_on_subset[OF assms(4)])
apply (insert closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)])
defer
using assms(5)[unfolded subset_eq]
using e(2)[unfolded subset_eq mem_cball]
apply (auto simp add: dist_norm)
done
then obtain x where x: "x \<in> cball 0 e" "(f \<circ> closest_point s) x = x" ..
have *: "closest_point s x = x"
apply (rule closest_point_self)
apply (rule assms(5)[unfolded subset_eq,THEN bspec[where x="x"], unfolded image_iff])
apply (rule_tac x="closest_point s x" in bexI)
using x
unfolding o_def
using closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3), of x]
apply auto
done
show thesis
apply (rule_tac x="closest_point s x" in that)
unfolding x(2)[unfolded o_def]
apply (rule closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)])
using *
apply auto
done
qed
text {*So we get the no-retraction theorem. *}
lemma no_retraction_cball:
fixes a :: "'a::euclidean_space"
assumes "e > 0"
shows "\<not> (frontier (cball a e) retract_of (cball a e))"
proof
case goal1
have *: "\<And>xa. a - (2 *\<^sub>R a - xa) = - (a - xa)"
using scaleR_left_distrib[of 1 1 a] by auto
obtain x where x:
"x \<in> {x. norm (a - x) = e}"
"2 *\<^sub>R a - x = x"
apply (rule retract_fixpoint_property[OF goal1, of "\<lambda>x. scaleR 2 a - x"])
apply rule
apply rule
apply (erule conjE)
apply (rule brouwer_ball[OF assms])
apply assumption+
apply (rule_tac x=x in bexI)
apply assumption+
apply (rule continuous_intros)+
unfolding frontier_cball subset_eq Ball_def image_iff
apply rule
apply rule
apply (erule bexE)
unfolding dist_norm
apply (simp add: * norm_minus_commute)
apply blast
done
then have "scaleR 2 a = scaleR 1 x + scaleR 1 x"
by (auto simp add: algebra_simps)
then have "a = x"
unfolding scaleR_left_distrib[symmetric]
by auto
then show False
using x assms by auto
qed
end