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theory Brouwer_Fixpoint
(* ========================================================================= *)
(* 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 *)
(* ========================================================================= *)
(* Author: John Harrison
Translation from HOL light: Robert Himmelmann, TU Muenchen *)
header {* Results connected with topological dimension. *}
theory Brouwer_Fixpoint
imports Convex_Euclidean_Space
begin
lemma brouwer_compactness_lemma:
assumes "compact s" "continuous_on s f" "¬ (∃x∈s. (f x = (0::real^'n)))"
obtains d where "0 < d" "∀x∈s. d ≤ norm(f x)" proof(cases "s={}") case False
have "continuous_on s (norm o f)" by(rule continuous_on_intros continuous_on_norm assms(2))+
then obtain x where x:"x∈s" "∀y∈s. (norm o f) x ≤ (norm o f) y"
using continuous_attains_inf[OF assms(1), of "norm o f"] and False unfolding o_def by auto
have "(norm o f) x > 0" using assms(3) and x(1) by auto
thus ?thesis apply(rule that) using x(2) unfolding o_def by auto qed(rule that[of 1], auto)
lemma kuhn_labelling_lemma:
assumes "(∀x::real^_. P x --> P (f x))" "∀x. P x --> (∀i. Q i --> 0 ≤ x$i ∧ x$i ≤ 1)"
shows "∃l. (∀x i. l x i ≤ (1::nat)) ∧
(∀x i. P x ∧ Q i ∧ (x$i = 0) --> (l x i = 0)) ∧
(∀x i. P x ∧ Q i ∧ (x$i = 1) --> (l x i = 1)) ∧
(∀x i. P x ∧ Q i ∧ (l x i = 0) --> x$i ≤ f(x)$i) ∧
(∀x i. P x ∧ Q i ∧ (l x i = 1) --> f(x)$i ≤ x$i)" proof-
have and_forall_thm:"!!P Q. (∀x. P x) ∧ (∀x. Q x) <-> (∀x. P x ∧ Q x)" by auto
have *:"∀x y::real. 0 ≤ x ∧ x ≤ 1 ∧ 0 ≤ y ∧ y ≤ 1 --> (x ≠ 1 ∧ x ≤ y ∨ x ≠ 0 ∧ y ≤ x)" by auto
show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
let ?R = "λy. (P x ∧ Q xa ∧ x $ xa = 0 --> y = (0::nat)) ∧
(P x ∧ Q xa ∧ x $ xa = 1 --> y = 1) ∧ (P x ∧ Q xa ∧ y = 0 --> x $ xa ≤ f x $ xa) ∧ (P x ∧ Q xa ∧ y = 1 --> f x $ xa ≤ x $ xa)"
{ assume "P x" "Q xa" hence "0 ≤ f x $ xa ∧ f x $ xa ≤ 1" using assms(2)[rule_format,of "f x" xa]
apply(drule_tac assms(1)[rule_format]) by auto }
hence "?R 0 ∨ ?R 1" by auto thus ?case by auto qed qed
subsection {* The key "counting" observation, somewhat abstracted. *}
lemma setsum_Un_disjoint':assumes "finite A" "finite B" "A ∩ B = {}" "A ∪ 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 kuhn_counting_lemma: assumes "finite faces" "finite simplices"
"∀f∈faces. bnd f --> (card {s ∈ simplices. face f s} = 1)"
"∀f∈faces. ¬ bnd f --> (card {s ∈ simplices. face f s} = 2)"
"∀s∈simplices. compo s --> (card {f ∈ faces. face f s ∧ compo' f} = 1)"
"∀s∈simplices. ¬ compo s --> (card {f ∈ faces. face f s ∧ compo' f} = 0) ∨
(card {f ∈ faces. face f s ∧ compo' f} = 2)"
"odd(card {f ∈ faces. compo' f ∧ bnd f})"
shows "odd(card {s ∈ simplices. compo s})" proof-
have "!!x. {f∈faces. compo' f ∧ bnd f ∧ face f x} ∪ {f∈faces. compo' f ∧ ¬bnd f ∧ face f x} = {f∈faces. compo' f ∧ face f x}"
"!!x. {f ∈ faces. compo' f ∧ bnd f ∧ face f x} ∩ {f ∈ faces. compo' f ∧ ¬ bnd f ∧ face f x} = {}" by auto
hence lem1:"setsum (λs. (card {f ∈ faces. face f s ∧ compo' f})) simplices =
setsum (λs. card {f ∈ {f ∈ faces. compo' f ∧ bnd f}. face f s}) simplices +
setsum (λs. card {f ∈ {f ∈ faces. compo' f ∧ ¬ (bnd f)}. face f s}) simplices"
unfolding setsum_addf[THEN sym] apply- apply(rule setsum_cong2)
using assms(1) by(auto simp add: card_Un_Int, auto simp add:conj_commute)
have lem2:"setsum (λj. card {f ∈ {f ∈ faces. compo' f ∧ bnd f}. face f j}) simplices =
1 * card {f ∈ faces. compo' f ∧ bnd f}"
"setsum (λj. card {f ∈ {f ∈ faces. compo' f ∧ ¬ bnd f}. face f j}) simplices =
2 * card {f ∈ faces. compo' f ∧ ¬ bnd f}"
apply(rule_tac[!] setsum_multicount) using assms by auto
have lem3:"setsum (λs. card {f ∈ faces. face f s ∧ compo' f}) simplices =
setsum (λs. card {f ∈ faces. face f s ∧ compo' f}) {s ∈ simplices. compo s}+
setsum (λs. card {f ∈ faces. face f s ∧ compo' f}) {s ∈ simplices. ¬ compo s}"
apply(rule setsum_Un_disjoint') using assms(2) by auto
have lem4:"setsum (λs. card {f ∈ faces. face f s ∧ compo' f}) {s ∈ simplices. compo s}
= setsum (λs. 1) {s ∈ simplices. compo s}"
apply(rule setsum_cong2) using assms(5) by auto
have lem5: "setsum (λs. card {f ∈ faces. face f s ∧ compo' f}) {s ∈ simplices. ¬ compo s} =
setsum (λs. card {f ∈ faces. face f s ∧ compo' f})
{s ∈ simplices. (¬ compo s) ∧ (card {f ∈ faces. face f s ∧ compo' f} = 0)} +
setsum (λs. card {f ∈ faces. face f s ∧ compo' f})
{s ∈ simplices. (¬ compo s) ∧ (card {f ∈ faces. face f s ∧ compo' f} = 2)}"
apply(rule setsum_Un_disjoint') using assms(2,6) by auto
have *:"int (∑s∈{s ∈ simplices. compo s}. card {f ∈ faces. face f s ∧ compo' f}) =
int (card {f ∈ faces. compo' f ∧ bnd f} + 2 * card {f ∈ faces. compo' f ∧ ¬ bnd f}) -
int (card {s ∈ simplices. ¬ compo s ∧ card {f ∈ faces. face f s ∧ compo' f} = 2} * 2)"
using lem1[unfolded lem3 lem2 lem5] by auto
have even_minus_odd:"!!x y. even x ==> odd (y::int) ==> odd (x - y)" using assms by auto
have odd_minus_even:"!!x y. odd x ==> even (y::int) ==> odd (x - y)" using assms by auto
show ?thesis unfolding even_nat_def unfolding card_eq_setsum and lem4[THEN sym] and *[unfolded card_eq_setsum]
unfolding card_eq_setsum[THEN sym] apply (rule odd_minus_even) unfolding zadd_int[THEN sym] apply(rule odd_plus_even)
apply(rule assms(7)[unfolded even_nat_def]) unfolding int_mult by auto qed
subsection {* The odd/even result for faces of complete vertices, generalized. *}
lemma card_1_exists: "card s = 1 <-> (∃!x. x ∈ s)" unfolding One_nat_def
apply rule apply(drule card_eq_SucD) defer apply(erule ex1E) proof-
fix x assume as:"x ∈ s" "∀y. y ∈ s --> y = x"
have *:"s = insert x {}" apply- apply(rule set_ext,rule) unfolding singleton_iff
apply(rule as(2)[rule_format]) using as(1) by auto
show "card s = Suc 0" unfolding * using card_insert by auto qed auto
lemma card_2_exists: "card s = 2 <-> (∃x∈s. ∃y∈s. x ≠ y ∧ (∀z∈s. (z = x) ∨ (z = y)))" proof
assume "card s = 2" then obtain x y where obt:"s = {x, y}" "x≠y" unfolding numeral_2_eq_2 apply - apply(erule exE conjE|drule card_eq_SucD)+ by auto
show "∃x∈s. ∃y∈s. x ≠ y ∧ (∀z∈s. z = x ∨ z = y)" using obt by auto next
assume "∃x∈s. ∃y∈s. x ≠ y ∧ (∀z∈s. z = x ∨ z = y)" then guess x .. from this(2) guess y ..
with `x∈s` have *:"s = {x, y}" "x≠y" by auto
from this(2) show "card s = 2" unfolding * by auto qed
lemma image_lemma_0: assumes "card {a∈s. f ` (s - {a}) = t - {b}} = n"
shows "card {s'. ∃a∈s. (s' = s - {a}) ∧ (f ` s' = t - {b})} = n" proof-
have *:"{s'. ∃a∈s. (s' = s - {a}) ∧ (f ` s' = t - {b})} = (λa. s - {a}) ` {a∈s. f ` (s - {a}) = t - {b}}" by auto
show ?thesis unfolding * unfolding assms[THEN sym] apply(rule card_image) unfolding inj_on_def
apply(rule,rule,rule) unfolding mem_Collect_eq by auto qed
lemma image_lemma_1: assumes "finite s" "finite t" "card s = card t" "f ` s = t" "b ∈ t"
shows "card {s'. ∃a∈s. s' = s - {a} ∧ f ` s' = t - {b}} = 1" proof-
obtain a where a:"b = f a" "a∈s" using assms(4-5) by auto
have inj:"inj_on f s" apply(rule eq_card_imp_inj_on) using assms(1-4) by auto
have *:"{a ∈ s. f ` (s - {a}) = t - {b}} = {a}" apply(rule set_ext) unfolding singleton_iff
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
show ?thesis apply(rule image_lemma_0) unfolding * by auto qed
lemma image_lemma_2: assumes "finite s" "finite t" "card s = card t" "f ` s ⊆ t" "f ` s ≠ t" "b ∈ t"
shows "(card {s'. ∃a∈s. (s' = s - {a}) ∧ f ` s' = t - {b}} = 0) ∨
(card {s'. ∃a∈s. (s' = s - {a}) ∧ f ` s' = t - {b}} = 2)" proof(cases "{a∈s. f ` (s - {a}) = t - {b}} = {}")
case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by auto
next let ?M = "{a∈s. f ` (s - {a}) = t - {b}}"
case False then obtain a where "a∈?M" by auto hence a:"a∈s" "f ` (s - {a}) = t - {b}" by auto
have "f a ∈ t - {b}" using a and assms by auto
hence "∃c ∈ s - {a}. f a = f c" unfolding image_iff[symmetric] and a by auto
then obtain c where c:"c ∈ s" "a ≠ c" "f a = f c" by auto
hence *:"f ` (s - {c}) = f ` (s - {a})" apply-apply(rule set_ext,rule) proof-
fix x assume "x ∈ f ` (s - {a})" then obtain y where y:"f y = x" "y∈s- {a}" by auto
thus "x ∈ 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
have "c∈?M" unfolding mem_Collect_eq and * using a and c(1) by auto
show ?thesis apply(rule disjI2, rule image_lemma_0) unfolding card_2_exists
apply(rule bexI[OF _ `a∈?M`], rule bexI[OF _ `c∈?M`],rule,rule `a≠c`) proof(rule,unfold mem_Collect_eq,erule conjE)
fix z assume as:"z ∈ s" "f ` (s - {z}) = t - {b}"
have inj:"inj_on f (s - {z})" apply(rule eq_card_imp_inj_on) unfolding as using as(1) and assms by auto
show "z = a ∨ z = c" proof(rule ccontr)
assume "¬ (z = a ∨ z = c)" thus False using inj[unfolded inj_on_def,THEN bspec[where x=a],THEN bspec[where x=c]]
using `a∈s` `c∈s` `f a = f c` `a≠c` by auto qed qed qed
subsection {* Combine this with the basic counting lemma. *}
lemma kuhn_complete_lemma:
assumes "finite simplices"
"∀f s. face f s <-> (∃a∈s. f = s - {a})" "∀s∈simplices. card s = n + 2" "∀s∈simplices. (rl ` s) ⊆ {0..n+1}"
"∀f∈ {f. ∃s∈simplices. face f s}. bnd f --> (card {s∈simplices. face f s} = 1)"
"∀f∈ {f. ∃s∈simplices. face f s}. ¬bnd f --> (card {s∈simplices. face f s} = 2)"
"odd(card {f∈{f. ∃s∈simplices. face f s}. rl ` f = {0..n} ∧ bnd f})"
shows "odd (card {s∈simplices. (rl ` s = {0..n+1})})"
apply(rule kuhn_counting_lemma) defer apply(rule assms)+ prefer 3 apply(rule assms) proof(rule_tac[1-2] ballI impI)+
have *:"{f. ∃s∈simplices. ∃a∈s. f = s - {a}} = (\<Union>s∈simplices. {f. ∃a∈s. (f = s - {a})})" by auto
have **: "∀s∈simplices. card s = n + 2 ∧ finite s" using assms(3) by (auto intro: card_ge_0_finite)
show "finite {f. ∃s∈simplices. face f s}" unfolding assms(2)[rule_format] and *
apply(rule finite_UN_I[OF assms(1)]) using ** by auto
have *:"!! P f s. s∈simplices ==> (f ∈ {f. ∃s∈simplices. ∃a∈s. f = s - {a}}) ∧
(∃a∈s. (f = s - {a})) ∧ P f <-> (∃a∈s. (f = s - {a}) ∧ P f)" by auto
fix s assume s:"s∈simplices" let ?S = "{f ∈ {f. ∃s∈simplices. face f s}. face f s ∧ rl ` f = {0..n}}"
have "{0..n + 1} - {n + 1} = {0..n}" by auto
hence S:"?S = {s'. ∃a∈s. s' = s - {a} ∧ rl ` s' = {0..n + 1} - {n + 1}}" apply- apply(rule set_ext)
unfolding assms(2)[rule_format] mem_Collect_eq and *[OF s, unfolded mem_Collect_eq, where P="λx. rl ` x = {0..n}"] by auto
show "rl ` s = {0..n+1} ==> card ?S = 1" "rl ` s ≠ {0..n+1} ==> card ?S = 0 ∨ card ?S = 2" unfolding S
apply(rule_tac[!] image_lemma_1 image_lemma_2) using ** assms(4) and s by auto qed
subsection {*We use the following notion of ordering rather than pointwise indexing. *}
definition "kle n x y <-> (∃k⊆{1..n::nat}. (∀j. y(j) = x(j) + (if j ∈ k then (1::nat) else 0)))"
lemma kle_refl[intro]: "kle n x x" unfolding kle_def by auto
lemma kle_antisym: "kle n x y ∧ kle n y x <-> (x = y)"
unfolding kle_def apply rule apply(rule ext) by auto
lemma pointwise_minimal_pointwise_maximal: fixes s::"(nat=>nat) set"
assumes "finite s" "s ≠ {}" "∀x∈s. ∀y∈s. (∀j. x j ≤ y j) ∨ (∀j. y j ≤ x j)"
shows "∃a∈s. ∀x∈s. ∀j. a j ≤ x j" "∃a∈s. ∀x∈s. ∀j. x j ≤ a j"
using assms unfolding atomize_conj apply- proof(induct s rule:finite_induct)
fix x and F::"(nat=>nat) set" assume as:"finite F" "x ∉ F"
"[|F ≠ {}; ∀x∈F. ∀y∈F. (∀j. x j ≤ y j) ∨ (∀j. y j ≤ x j)|]
==> (∃a∈F. ∀x∈F. ∀j. a j ≤ x j) ∧ (∃a∈F. ∀x∈F. ∀j. x j ≤ a j)" "insert x F ≠ {}"
"∀xa∈insert x F. ∀y∈insert x F. (∀j. xa j ≤ y j) ∨ (∀j. y j ≤ xa j)"
show "(∃a∈insert x F. ∀x∈insert x F. ∀j. a j ≤ x j) ∧ (∃a∈insert x F. ∀x∈insert x F. ∀j. x j ≤ a j)" proof(cases "F={}")
case True thus ?thesis apply-apply(rule,rule_tac[!] x=x in bexI) by auto next
case False obtain a b where a:"a∈insert x F" "∀x∈F. ∀j. a j ≤ x j" and
b:"b∈insert x F" "∀x∈F. ∀j. x j ≤ b j" using as(3)[OF False] using as(5) by auto
have "∃a∈insert x F. ∀x∈insert x F. ∀j. a j ≤ x j"
using as(5)[rule_format,OF a(1) insertI1] apply- proof(erule disjE)
assume "∀j. a j ≤ x j" thus ?thesis apply(rule_tac x=a in bexI) using a by auto next
assume "∀j. x j ≤ a j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using a apply -
apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed moreover
have "∃b∈insert x F. ∀x∈insert x F. ∀j. x j ≤ b j"
using as(5)[rule_format,OF b(1) insertI1] apply- proof(erule disjE)
assume "∀j. x j ≤ b j" thus ?thesis apply(rule_tac x=b in bexI) using b by auto next
assume "∀j. b j ≤ x j" thus ?thesis apply(rule_tac x=x in bexI) apply(rule,rule) using b apply -
apply(erule_tac x=xa in ballE) apply(erule_tac x=j in allE)+ by auto qed
ultimately show ?thesis by auto qed qed(auto)
lemma kle_imp_pointwise: "kle n x y ==> (∀j. x j ≤ y j)" unfolding kle_def by auto
lemma pointwise_antisym: fixes x::"nat => nat"
shows "(∀j. x j ≤ y j) ∧ (∀j. y j ≤ x j) <-> (x = y)"
apply(rule, rule ext,erule conjE) apply(erule_tac x=xa in allE)+ by auto
lemma kle_trans: assumes "kle n x y" "kle n y z" "kle n x z ∨ kle n z x" shows "kle n x z"
using assms apply- apply(erule disjE) apply assumption proof- case goal1
hence "x=z" apply- apply(rule ext) apply(drule kle_imp_pointwise)+
apply(erule_tac x=xa in allE)+ by auto thus ?case by auto qed
lemma kle_strict: assumes "kle n x y" "x ≠ y"
shows "∀j. x j ≤ y j" "∃k. 1 ≤ k ∧ k ≤ n ∧ x(k) < y(k)"
apply(rule kle_imp_pointwise[OF assms(1)]) proof-
guess k using assms(1)[unfolded kle_def] .. note k = this
show "∃k. 1 ≤ k ∧ k ≤ n ∧ x(k) < y(k)" proof(cases "k={}")
case True hence "x=y" apply-apply(rule ext) using k by auto
thus ?thesis using assms(2) by auto next
case False hence "(SOME k'. k' ∈ k) ∈ k" apply-apply(rule someI_ex) by auto
thus ?thesis apply(rule_tac x="SOME k'. k' ∈ k" in exI) using k by auto qed qed
lemma kle_minimal: assumes "finite s" "s ≠ {}" "∀x∈s. ∀y∈s. kle n x y ∨ kle n y x"
shows "∃a∈s. ∀x∈s. kle n a x" proof-
have "∃a∈s. ∀x∈s. ∀j. a j ≤ x j" apply(rule pointwise_minimal_pointwise_maximal(1)[OF assms(1-2)])
apply(rule,rule) apply(drule_tac assms(3)[rule_format],assumption) using kle_imp_pointwise by auto
then guess a .. note a=this show ?thesis apply(rule_tac x=a in bexI) proof fix x assume "x∈s"
show "kle n a x" using assms(3)[rule_format,OF a(1) `x∈s`] apply- proof(erule disjE)
assume "kle n x a" hence "x = a" apply- unfolding pointwise_antisym[THEN sym]
apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x∈s`] by auto
thus ?thesis using kle_refl by auto qed qed(insert a, auto) qed
lemma kle_maximal: assumes "finite s" "s ≠ {}" "∀x∈s. ∀y∈s. kle n x y ∨ kle n y x"
shows "∃a∈s. ∀x∈s. kle n x a" proof-
have "∃a∈s. ∀x∈s. ∀j. a j ≥ x j" apply(rule pointwise_minimal_pointwise_maximal(2)[OF assms(1-2)])
apply(rule,rule) apply(drule_tac assms(3)[rule_format],assumption) using kle_imp_pointwise by auto
then guess a .. note a=this show ?thesis apply(rule_tac x=a in bexI) proof fix x assume "x∈s"
show "kle n x a" using assms(3)[rule_format,OF a(1) `x∈s`] apply- proof(erule disjE)
assume "kle n a x" hence "x = a" apply- unfolding pointwise_antisym[THEN sym]
apply(drule kle_imp_pointwise) using a(2)[rule_format,OF `x∈s`] by auto
thus ?thesis using kle_refl by auto qed qed(insert a, auto) qed
lemma kle_strict_set: assumes "kle n x y" "x ≠ y"
shows "1 ≤ card {k∈{1..n}. x k < y k}" proof-
guess i using kle_strict(2)[OF assms] ..
hence "card {i} ≤ card {k∈{1..n}. x k < y k}" apply- apply(rule card_mono) by auto
thus ?thesis by auto qed
lemma kle_range_combine:
assumes "kle n x y" "kle n y z" "kle n x z ∨ kle n z x"
"m1 ≤ card {k∈{1..n}. x k < y k}"
"m2 ≤ card {k∈{1..n}. y k < z k}"
shows "kle n x z ∧ m1 + m2 ≤ card {k∈{1..n}. x k < z k}"
apply(rule,rule kle_trans[OF assms(1-3)]) proof-
have "!!j. x j < y j ==> x j < z j" apply(rule less_le_trans) using kle_imp_pointwise[OF assms(2)] by auto moreover
have "!!j. y j < z j ==> x j < z j" apply(rule le_less_trans) using kle_imp_pointwise[OF assms(1)] by auto ultimately
have *:"{k∈{1..n}. x k < y k} ∪ {k∈{1..n}. y k < z k} = {k∈{1..n}. x k < z k}" by auto
have **:"{k ∈ {1..n}. x k < y k} ∩ {k ∈ {1..n}. y k < z k} = {}" unfolding disjoint_iff_not_equal
apply(rule,rule,unfold mem_Collect_eq,rule ccontr) apply(erule conjE)+ proof-
fix i j assume as:"i ∈ {1..n}" "x i < y i" "j ∈ {1..n}" "y j < z j" "¬ i ≠ j"
guess kx using assms(1)[unfolded kle_def] .. note kx=this
have "x i < y i" using as by auto hence "i ∈ kx" using as(1) kx apply(rule_tac ccontr) by auto
hence "x i + 1 = y i" using kx by auto moreover
guess ky using assms(2)[unfolded kle_def] .. note ky=this
have "y i < z i" using as by auto hence "i ∈ ky" using as(1) ky apply(rule_tac ccontr) by auto
hence "y i + 1 = z i" using ky by auto ultimately
have "z i = x i + 2" by auto
thus False using assms(3) unfolding kle_def by(auto simp add: split_if_eq1) qed
have fin:"!!P. finite {x∈{1..n::nat}. P x}" by auto
have "m1 + m2 ≤ card {k∈{1..n}. x k < y k} + card {k∈{1..n}. y k < z k}" using assms(4-5) by auto
also have "… ≤ card {k∈{1..n}. x k < z k}" unfolding card_Un_Int[OF fin fin] unfolding * ** by auto
finally show " m1 + m2 ≤ card {k ∈ {1..n}. x k < z k}" by auto qed
lemma kle_range_combine_l:
assumes "kle n x y" "kle n y z" "kle n x z ∨ kle n z x" "m ≤ card {k∈{1..n}. y(k) < z(k)}"
shows "kle n x z ∧ m ≤ card {k∈{1..n}. x(k) < z(k)}"
using kle_range_combine[OF assms(1-3) _ assms(4), of 0] by auto
lemma kle_range_combine_r:
assumes "kle n x y" "kle n y z" "kle n x z ∨ kle n z x" "m ≤ card {k∈{1..n}. x k < y k}"
shows "kle n x z ∧ m ≤ card {k∈{1..n}. x(k) < z(k)}"
using kle_range_combine[OF assms(1-3) assms(4), of 0] by auto
lemma kle_range_induct:
assumes "card s = Suc m" "∀x∈s. ∀y∈s. kle n x y ∨ kle n y x"
shows "∃x∈s. ∃y∈s. kle n x y ∧ m ≤ card {k∈{1..n}. x k < y k}" proof-
have "finite s" "s≠{}" using assms(1) by (auto intro: card_ge_0_finite)
thus ?thesis using assms apply- proof(induct m arbitrary: s)
case 0 thus ?case using kle_refl by auto next
case (Suc m) then obtain a where a:"a∈s" "∀x∈s. kle n a x" using kle_minimal[of s n] by auto
show ?case proof(cases "s ⊆ {a}") case False
hence "card (s - {a}) = Suc m" "s - {a} ≠ {}" using card_Diff_singleton[OF _ a(1)] Suc(4) `finite s` by auto
then obtain x b where xb:"x∈s - {a}" "b∈s - {a}" "kle n x b" "m ≤ card {k ∈ {1..n}. x k < b k}"
using Suc(1)[of "s - {a}"] using Suc(5) `finite s` by auto
have "1 ≤ card {k ∈ {1..n}. a k < x k}" "m ≤ card {k ∈ {1..n}. x k < b k}"
apply(rule kle_strict_set) apply(rule a(2)[rule_format]) using a and xb by auto
thus ?thesis apply(rule_tac x=a in bexI, rule_tac x=b in bexI)
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
case True hence "s = {a}" using Suc(3) by auto hence "card s = 1" by auto
hence False using Suc(4) `finite s` by auto thus ?thesis by auto qed qed qed
lemma kle_Suc: "kle n x y ==> kle (n + 1) x y"
unfolding kle_def apply(erule exE) apply(rule_tac x=k in exI) by auto
lemma kle_trans_1: assumes "kle n x y" shows "x j ≤ y j" "y j ≤ x j + 1"
using assms[unfolded kle_def] by auto
lemma kle_trans_2: assumes "kle n a b" "kle n b c" "∀j. c j ≤ a j + 1" shows "kle n a c" proof-
guess kk1 using assms(1)[unfolded kle_def] .. note kk1=this
guess kk2 using assms(2)[unfolded kle_def] .. note kk2=this
show ?thesis unfolding kle_def apply(rule_tac x="kk1 ∪ kk2" in exI) apply(rule) defer proof
fix i show "c i = a i + (if i ∈ kk1 ∪ kk2 then 1 else 0)" proof(cases "i∈kk1 ∪ kk2")
case True hence "c i ≥ a i + (if i ∈ kk1 ∪ kk2 then 1 else 0)"
unfolding kk1[THEN conjunct2,rule_format,of i] kk2[THEN conjunct2,rule_format,of i] by auto
moreover have "c i ≤ a i + (if i ∈ kk1 ∪ kk2 then 1 else 0)" using True assms(3) by auto
ultimately show ?thesis by auto next
case False thus ?thesis using kk1 kk2 by auto qed qed(insert kk1 kk2, auto) qed
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"
apply(rule kle_trans_2[OF assms(2,4)]) proof have *:"!!c b x::nat. x ≤ c + 1 ==> c ≤ b ==> x ≤ b + 1" by auto
fix j show "x j ≤ 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
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"
apply(rule kle_trans_2[OF assms(3,1)]) proof have *:"!!c b x::nat. c ≤ x + 1 ==> b ≤ c ==> b ≤ x + 1" by auto
fix j show "b j ≤ 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
lemma kle_adjacent:
assumes "∀j. b j = (if j = k then a(j) + 1 else a j)" "kle n a x" "kle n x b"
shows "(x = a) ∨ (x = b)" proof(cases "x k = a k")
case True show ?thesis apply(rule disjI1,rule ext) proof- fix j
have "x j ≤ a j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]]
unfolding assms(1)[rule_format] apply-apply(cases "j=k") using True by auto
thus "x j = a j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]] by auto qed next
case False show ?thesis apply(rule disjI2,rule ext) proof- fix j
have "x j ≥ b j" using kle_imp_pointwise[OF assms(2),THEN spec[where x=j]]
unfolding assms(1)[rule_format] apply-apply(cases "j=k") using False by auto
thus "x j = b j" using kle_imp_pointwise[OF assms(3),THEN spec[where x=j]] by auto qed qed
subsection {* kuhn's notion of a simplex (a reformulation to avoid so much indexing). *}
definition "ksimplex p n (s::(nat => nat) set) <->
(card s = n + 1 ∧
(∀x∈s. ∀j. x(j) ≤ p) ∧
(∀x∈s. ∀j. j∉{1..n} --> (x j = p)) ∧
(∀x∈s. ∀y∈s. kle n x y ∨ kle n y x))"
lemma ksimplexI:"card s = n + 1 ==> ∀x∈s. ∀j. x j ≤ p ==> ∀x∈s. ∀j. j ∉ {1..n} --> x j = p ==> ∀x∈s. ∀y∈s. kle n x y ∨ kle n y x ==> ksimplex p n s"
unfolding ksimplex_def by auto
lemma ksimplex_eq: "ksimplex p n (s::(nat => nat) set) <->
(card s = n + 1 ∧ finite s ∧
(∀x∈s. ∀j. x(j) ≤ p) ∧
(∀x∈s. ∀j. j∉{1..n} --> (x j = p)) ∧
(∀x∈s. ∀y∈s. kle n x y ∨ kle n y x))"
unfolding ksimplex_def by (auto intro: card_ge_0_finite)
lemma ksimplex_extrema: assumes "ksimplex p n s" obtains a b where "a ∈ s" "b ∈ s"
"∀x∈s. kle n a x ∧ kle n x b" "∀i. b(i) = (if i ∈ {1..n} then a(i) + 1 else a(i))" proof(cases "n=0")
case True obtain x where *:"s = {x}" using assms[unfolded ksimplex_eq True,THEN conjunct1]
unfolding add_0_left card_1_exists by auto
show ?thesis apply(rule that[of x x]) unfolding * True by auto next
note assm = assms[unfolded ksimplex_eq]
case False have "s≠{}" using assm by auto
obtain a where a:"a∈s" "∀x∈s. kle n a x" using `s≠{}` assm using kle_minimal[of s n] by auto
obtain b where b:"b∈s" "∀x∈s. kle n x b" using `s≠{}` assm using kle_maximal[of s n] by auto
obtain c d where c_d:"c∈s" "d∈s" "kle n c d" "n ≤ card {k ∈ {1..n}. c k < d k}"
using kle_range_induct[of s n n] using assm by auto
have "kle n c b ∧ n ≤ card {k ∈ {1..n}. c k < b k}" apply(rule kle_range_combine_r[where y=d]) using c_d a b by auto
hence "kle n a b ∧ n ≤ card {k∈{1..n}. a(k) < b(k)}" apply-apply(rule kle_range_combine_l[where y=c]) using a `c∈s` `b∈s` by auto
moreover have "card {1..n} ≥ card {k∈{1..n}. a(k) < b(k)}" apply(rule card_mono) by auto
ultimately have *:"{k∈{1 .. n}. a k < b k} = {1..n}" apply- apply(rule card_subset_eq) by auto
show ?thesis apply(rule that[OF a(1) b(1)]) defer apply(subst *[THEN sym]) unfolding mem_Collect_eq proof
guess k using a(2)[rule_format,OF b(1),unfolded kle_def] .. note k=this
fix i show "b i = (if i ∈ {1..n} ∧ a i < b i then a i + 1 else a i)" proof(cases "i ∈ {1..n}")
case True thus ?thesis unfolding k[THEN conjunct2,rule_format] by auto next
case False have "a i = p" using assm and False `a∈s` by auto
moreover have "b i = p" using assm and False `b∈s` by auto
ultimately show ?thesis by auto qed qed(insert a(2) b(2) assm,auto) qed
lemma ksimplex_extrema_strong:
assumes "ksimplex p n s" "n ≠ 0" obtains a b where "a ∈ s" "b ∈ s" "a ≠ b"
"∀x∈s. kle n a x ∧ kle n x b" "∀i. b(i) = (if i ∈ {1..n} then a(i) + 1 else a(i))" proof-
obtain a b where ab:"a ∈ s" "b ∈ s" "∀x∈s. kle n a x ∧ kle n x b" "∀i. b(i) = (if i ∈ {1..n} then a(i) + 1 else a(i))"
apply(rule ksimplex_extrema[OF assms(1)]) by auto
have "a ≠ b" apply(rule ccontr) unfolding not_not apply(drule cong[of _ _ 1 1]) using ab(4) assms(2) by auto
thus ?thesis apply(rule_tac that[of a b]) using ab by auto qed
lemma ksimplexD:
assumes "ksimplex p n s"
shows "card s = n + 1" "finite s" "card s = n + 1" "∀x∈s. ∀j. x j ≤ p" "∀x∈s. ∀j. j ∉ {1..n} --> x j = p"
"∀x∈s. ∀y∈s. kle n x y ∨ kle n y x" using assms unfolding ksimplex_eq by auto
lemma ksimplex_successor:
assumes "ksimplex p n s" "a ∈ s"
shows "(∀x∈s. kle n x a) ∨ (∃y∈s. ∃k∈{1..n}. ∀j. y(j) = (if j = k then a(j) + 1 else a(j)))"
proof(cases "∀x∈s. kle n x a") case True thus ?thesis by auto next note assm = ksimplexD[OF assms(1)]
case False then obtain b where b:"b∈s" "¬ kle n b a" "∀x∈{x ∈ s. ¬ kle n x a}. kle n b x"
using kle_minimal[of "{x∈s. ¬ kle n x a}" n] and assm by auto
hence **:"1 ≤ card {k∈{1..n}. a k < b k}" apply- apply(rule kle_strict_set) using assm(6) and `a∈s` by(auto simp add:kle_refl)
let ?kle1 = "{x ∈ s. ¬ kle n x a}" have "card ?kle1 > 0" apply(rule ccontr) using assm(2) and False by auto
hence sizekle1: "card ?kle1 = Suc (card ?kle1 - 1)" using assm(2) by auto
obtain c d where c_d: "c ∈ s" "¬ kle n c a" "d ∈ s" "¬ kle n d a" "kle n c d" "card ?kle1 - 1 ≤ card {k ∈ {1..n}. c k < d k}"
using kle_range_induct[OF sizekle1, of n] using assm by auto
let ?kle2 = "{x ∈ s. kle n x a}"
have "card ?kle2 > 0" apply(rule ccontr) using assm(6)[rule_format,of a a] and `a∈s` and assm(2) by auto
hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)" using assm(2) by auto
obtain e f where e_f: "e ∈ s" "kle n e a" "f ∈ s" "kle n f a" "kle n e f" "card ?kle2 - 1 ≤ card {k ∈ {1..n}. e k < f k}"
using kle_range_induct[OF sizekle2, of n] using assm by auto
have "card {k∈{1..n}. a k < b k} = 1" proof(rule ccontr) case goal1
hence as:"card {k∈{1..n}. a k < b k} ≥ 2" using ** by auto
have *:"finite ?kle2" "finite ?kle1" "?kle2 ∪ ?kle1 = s" "?kle2 ∩ ?kle1 = {}" using assm(2) by auto
have "(card ?kle2 - 1) + 2 + (card ?kle1 - 1) = card ?kle2 + card ?kle1" using sizekle1 sizekle2 by auto
also have "… = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto
finally have n:"(card ?kle2 - 1) + (2 + (card ?kle1 - 1)) = n + 1" by auto
have "kle n e a ∧ card {x ∈ s. kle n x a} - 1 ≤ card {k ∈ {1..n}. e k < a k}"
apply(rule kle_range_combine_r[where y=f]) using e_f using `a∈s` assm(6) by auto
moreover have "kle n b d ∧ card {x ∈ s. ¬ kle n x a} - 1 ≤ card {k ∈ {1..n}. b k < d k}"
apply(rule kle_range_combine_l[where y=c]) using c_d using assm(6) and b by auto
hence "kle n a d ∧ 2 + (card {x ∈ s. ¬ kle n x a} - 1) ≤ card {k ∈ {1..n}. a k < d k}" apply-
apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a∈s` `d∈s` apply- by blast+
ultimately have "kle n e d ∧ (card ?kle2 - 1) + (2 + (card ?kle1 - 1)) ≤ card {k∈{1..n}. e k < d k}" apply-
apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e∈s` `d∈s`] apply - by blast+
moreover have "card {k ∈ {1..n}. e k < d k} ≤ card {1..n}" apply(rule card_mono) by auto
ultimately show False unfolding n by auto qed
then guess k unfolding card_1_exists .. note k=this[unfolded mem_Collect_eq]
show ?thesis apply(rule disjI2) apply(rule_tac x=b in bexI,rule_tac x=k in bexI) proof
fix j::nat have "kle n a b" using b and assm(6)[rule_format, OF `a∈s` `b∈s`] by auto
then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format]
have kkk:"k∈kk" apply(rule ccontr) using k(1) unfolding kk by auto
show "b j = (if j = k then a j + 1 else a j)" proof(cases "j∈kk")
case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto
thus ?thesis unfolding kk using kkk by auto next
case False hence "j≠k" using k(2)[rule_format, of j k] using kk_raw kkk by auto
thus ?thesis unfolding kk using kkk using False by auto qed qed(insert k(1) `b∈s`, auto) qed
lemma ksimplex_predecessor:
assumes "ksimplex p n s" "a ∈ s"
shows "(∀x∈s. kle n a x) ∨ (∃y∈s. ∃k∈{1..n}. ∀j. a(j) = (if j = k then y(j) + 1 else y(j)))"
proof(cases "∀x∈s. kle n a x") case True thus ?thesis by auto next note assm = ksimplexD[OF assms(1)]
case False then obtain b where b:"b∈s" "¬ kle n a b" "∀x∈{x ∈ s. ¬ kle n a x}. kle n x b"
using kle_maximal[of "{x∈s. ¬ kle n a x}" n] and assm by auto
hence **:"1 ≤ card {k∈{1..n}. a k > b k}" apply- apply(rule kle_strict_set) using assm(6) and `a∈s` by(auto simp add:kle_refl)
let ?kle1 = "{x ∈ s. ¬ kle n a x}" have "card ?kle1 > 0" apply(rule ccontr) using assm(2) and False by auto
hence sizekle1:"card ?kle1 = Suc (card ?kle1 - 1)" using assm(2) by auto
obtain c d where c_d: "c ∈ s" "¬ kle n a c" "d ∈ s" "¬ kle n a d" "kle n d c" "card ?kle1 - 1 ≤ card {k ∈ {1..n}. c k > d k}"
using kle_range_induct[OF sizekle1, of n] using assm by auto
let ?kle2 = "{x ∈ s. kle n a x}"
have "card ?kle2 > 0" apply(rule ccontr) using assm(6)[rule_format,of a a] and `a∈s` and assm(2) by auto
hence sizekle2:"card ?kle2 = Suc (card ?kle2 - 1)" using assm(2) by auto
obtain e f where e_f: "e ∈ s" "kle n a e" "f ∈ s" "kle n a f" "kle n f e" "card ?kle2 - 1 ≤ card {k ∈ {1..n}. e k > f k}"
using kle_range_induct[OF sizekle2, of n] using assm by auto
have "card {k∈{1..n}. a k > b k} = 1" proof(rule ccontr) case goal1
hence as:"card {k∈{1..n}. a k > b k} ≥ 2" using ** by auto
have *:"finite ?kle2" "finite ?kle1" "?kle2 ∪ ?kle1 = s" "?kle2 ∩ ?kle1 = {}" using assm(2) by auto
have "(card ?kle2 - 1) + 2 + (card ?kle1 - 1) = card ?kle2 + card ?kle1" using sizekle1 sizekle2 by auto
also have "… = n + 1" unfolding card_Un_Int[OF *(1-2)] *(3-) using assm(3) by auto
finally have n:"(card ?kle1 - 1) + 2 + (card ?kle2 - 1) = n + 1" by auto
have "kle n a e ∧ card {x ∈ s. kle n a x} - 1 ≤ card {k ∈ {1..n}. e k > a k}"
apply(rule kle_range_combine_l[where y=f]) using e_f using `a∈s` assm(6) by auto
moreover have "kle n d b ∧ card {x ∈ s. ¬ kle n a x} - 1 ≤ card {k ∈ {1..n}. b k > d k}"
apply(rule kle_range_combine_r[where y=c]) using c_d using assm(6) and b by auto
hence "kle n d a ∧ (card {x ∈ s. ¬ kle n a x} - 1) + 2 ≤ card {k ∈ {1..n}. a k > d k}" apply-
apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a∈s` `d∈s` by blast+
ultimately have "kle n d e ∧ (card ?kle1 - 1 + 2) + (card ?kle2 - 1) ≤ card {k∈{1..n}. e k > d k}" apply-
apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e∈s` `d∈s`] apply - by blast+
moreover have "card {k ∈ {1..n}. e k > d k} ≤ card {1..n}" apply(rule card_mono) by auto
ultimately show False unfolding n by auto qed
then guess k unfolding card_1_exists .. note k=this[unfolded mem_Collect_eq]
show ?thesis apply(rule disjI2) apply(rule_tac x=b in bexI,rule_tac x=k in bexI) proof
fix j::nat have "kle n b a" using b and assm(6)[rule_format, OF `a∈s` `b∈s`] by auto
then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format]
have kkk:"k∈kk" apply(rule ccontr) using k(1) unfolding kk by auto
show "a j = (if j = k then b j + 1 else b j)" proof(cases "j∈kk")
case True hence "j=k" apply-apply(rule k(2)[rule_format]) using kk_raw kkk by auto
thus ?thesis unfolding kk using kkk by auto next
case False hence "j≠k" using k(2)[rule_format, of j k] using kk_raw kkk by auto
thus ?thesis unfolding kk using kkk using False by auto qed qed(insert k(1) `b∈s`, auto) qed
subsection {* The lemmas about simplices that we need. *}
lemma card_funspace': assumes "finite s" "finite t" "card s = m" "card t = n"
shows "card {f. (∀x∈s. f x ∈ t) ∧ (∀x∈UNIV - s. f x = d)} = n ^ m" (is "card (?M s) = _")
using assms apply - proof(induct m arbitrary: s)
have *:"{f. ∀x. f x = d} = {λx. d}" apply(rule set_ext,rule)unfolding mem_Collect_eq apply(rule,rule ext) by auto
case 0 thus ?case by(auto simp add: *) next
case (Suc m) guess a using card_eq_SucD[OF Suc(4)] .. then guess s0
apply(erule_tac exE) apply(erule conjE)+ . note as0 = this
have **:"card s0 = m" using as0 using Suc(2) Suc(4) by auto
let ?l = "(λ(b,g) x. if x = a then b else g x)" have *:"?M (insert a s0) = ?l ` {(b,g). b∈t ∧ g∈?M s0}"
apply(rule set_ext,rule) unfolding mem_Collect_eq image_iff apply(erule conjE)
apply(rule_tac x="(x a, λy. if y∈s0 then x y else d)" in bexI) apply(rule ext) prefer 3 apply rule defer
apply(erule bexE,rule) unfolding mem_Collect_eq apply(erule splitE)+ apply(erule conjE)+ proof-
fix x xa xb xc y assume as:"x = (λ(b, g) x. if x = a then b else g x) xa" "xb ∈ UNIV - insert a s0" "xa = (xc, y)" "xc ∈ t"
"∀x∈s0. y x ∈ t" "∀x∈UNIV - s0. y x = d" thus "x xb = d" unfolding as by auto qed auto
have inj:"inj_on ?l {(b,g). b∈t ∧ g∈?M s0}" unfolding inj_on_def apply(rule,rule,rule) unfolding mem_Collect_eq apply(erule splitE conjE)+ proof-
case goal1 note as = this(1,4-)[unfolded goal1 split_conv]
have "xa = xb" using as(1)[THEN cong[of _ _ a]] by auto
moreover have "ya = yb" proof(rule ext) fix x show "ya x = yb x" proof(cases "x = a")
case False thus ?thesis using as(1)[THEN cong[of _ _ x x]] by auto next
case True thus ?thesis using as(5,7) using as0(2) by auto qed qed
ultimately show ?case unfolding goal1 by auto qed
have "finite s0" using `finite s` unfolding as0 by simp
show ?case unfolding as0 * card_image[OF inj] using assms
unfolding SetCompr_Sigma_eq apply-
unfolding card_cartesian_product
using Suc(1)[OF `finite s0` `finite t` ** `card t = n`] by auto
qed
lemma card_funspace: assumes "finite s" "finite t"
shows "card {f. (∀x∈s. f x ∈ t) ∧ (∀x∈UNIV - s. f x = d)} = (card t) ^ (card s)"
using assms by (auto intro: card_funspace')
lemma finite_funspace: assumes "finite s" "finite t"
shows "finite {f. (∀x∈s. f x ∈ t) ∧ (∀x∈UNIV - s. f x = d)}" (is "finite ?S")
proof (cases "card t > 0")
case True
have "card ?S = (card t) ^ (card s)"
using assms by (auto intro!: card_funspace)
thus ?thesis using True by (auto intro: card_ge_0_finite)
next
case False hence "t = {}" using `finite t` by auto
show ?thesis
proof (cases "s = {}")
have *:"{f. ∀x. f x = d} = {λx. d}" by (auto intro: ext)
case True thus ?thesis using `t = {}` by (auto simp: *)
next
case False thus ?thesis using `t = {}` by simp
qed
qed
lemma finite_simplices: "finite {s. ksimplex p n s}"
apply(rule finite_subset[of _ "{s. s⊆{f. (∀i∈{1..n}. f i ∈ {0..p}) ∧ (∀i∈UNIV-{1..n}. f i = p)}}"])
unfolding ksimplex_def defer apply(rule finite_Collect_subsets) apply(rule finite_funspace) by auto
lemma simplex_top_face: assumes "0<p" "∀x∈f. x (n + 1) = p"
shows "(∃s a. ksimplex p (n + 1) s ∧ a ∈ s ∧ (f = s - {a})) <-> ksimplex p n f" (is "?ls = ?rs") proof
assume ?ls then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
show ?rs unfolding ksimplex_def sa(3) apply(rule) defer apply rule defer apply(rule,rule,rule,rule) defer apply(rule,rule) proof-
fix x y assume as:"x ∈s - {a}" "y ∈s - {a}" have xyp:"x (n + 1) = y (n + 1)"
using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]]
using as(2)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]] by auto
show "kle n x y ∨ kle n y x" proof(cases "kle (n + 1) x y")
case True then guess k unfolding kle_def .. note k=this hence *:"n+1 ∉ k" using xyp by auto
have "¬ (∃x∈k. x∉{1..n})" apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
fix x assume as:"x ∈ k" "x ∉ {1..n}" have "x ≠ n+1" using as and * by auto
thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed
thus ?thesis apply-apply(rule disjI1) unfolding kle_def using k apply(rule_tac x=k in exI) by auto next
case False hence "kle (n + 1) y x" using ksimplexD(6)[OF sa(1),rule_format, of x y] using as by auto
then guess k unfolding kle_def .. note k=this hence *:"n+1 ∉ k" using xyp by auto
hence "¬ (∃x∈k. x∉{1..n})" apply-apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
fix x assume as:"x ∈ k" "x ∉ {1..n}" have "x ≠ n+1" using as and * by auto
thus False using as and k[THEN conjunct1,unfolded subset_eq] by auto qed
thus ?thesis apply-apply(rule disjI2) unfolding kle_def using k apply(rule_tac x=k in exI) by auto qed next
fix x j assume as:"x∈s - {a}" "j∉{1..n}"
thus "x j = p" using as(1)[unfolded sa(3)[THEN sym], THEN assms(2)[rule_format]]
apply(cases "j = n+1") using sa(1)[unfolded ksimplex_def] by auto qed(insert sa ksimplexD[OF sa(1)], auto) next
assume ?rs note rs=ksimplexD[OF this] guess a b apply(rule ksimplex_extrema[OF `?rs`]) . note ab = this
def c ≡ "λi. if i = (n + 1) then p - 1 else a i"
have "c∉f" apply(rule ccontr) unfolding not_not apply(drule assms(2)[rule_format]) unfolding c_def using assms(1) by auto
thus ?ls apply(rule_tac x="insert c f" in exI,rule_tac x=c in exI) unfolding ksimplex_def conj_assoc
apply(rule conjI) defer apply(rule conjI) defer apply(rule conjI) defer apply(rule conjI) defer
proof(rule_tac[3-5] ballI allI)+
fix x j assume x:"x ∈ insert c f" thus "x j ≤ p" proof (cases "x=c")
case True show ?thesis unfolding True c_def apply(cases "j=n+1") using ab(1) and rs(4) by auto
qed(insert x rs(4), auto simp add:c_def)
show "j ∉ {1..n + 1} --> x j = p" apply(cases "x=c") using x ab(1) rs(5) unfolding c_def by auto
{ fix z assume z:"z ∈ insert c f" hence "kle (n + 1) c z" apply(cases "z = c") (*defer apply(rule kle_Suc)*) proof-
case False hence "z∈f" using z by auto
then guess k apply(drule_tac ab(3)[THEN bspec[where x=z], THEN conjunct1]) unfolding kle_def apply(erule exE) .
thus "kle (n + 1) c z" unfolding kle_def apply(rule_tac x="insert (n + 1) k" in exI) unfolding c_def
using ab using rs(5)[rule_format,OF ab(1),of "n + 1"] assms(1) by auto qed auto } note * = this
fix y assume y:"y ∈ insert c f" show "kle (n + 1) x y ∨ kle (n + 1) y x" proof(cases "x = c ∨ y = c")
case False hence **:"x∈f" "y∈f" using x y by auto
show ?thesis using rs(6)[rule_format,OF **] by(auto dest: kle_Suc) qed(insert * x y, auto)
qed(insert rs, auto) qed
lemma ksimplex_fix_plane:
assumes "a ∈ s" "j∈{1..n::nat}" "∀x∈s - {a}. x j = q" "a0 ∈ s" "a1 ∈ s"
"∀i. a1 i = ((if i∈{1..n} then a0 i + 1 else a0 i)::nat)"
shows "(a = a0) ∨ (a = a1)" proof- have *:"!!P A x y. ∀x∈A. P x ==> x∈A ==> y∈A ==> P x ∧ P y" by auto
show ?thesis apply(rule ccontr) using *[OF assms(3), of a0 a1] unfolding assms(6)[THEN spec[where x=j]]
using assms(1-2,4-5) by auto qed
lemma ksimplex_fix_plane_0:
assumes "a ∈ s" "j∈{1..n::nat}" "∀x∈s - {a}. x j = 0" "a0 ∈ s" "a1 ∈ s"
"∀i. a1 i = ((if i∈{1..n} then a0 i + 1 else a0 i)::nat)"
shows "a = a1" apply(rule ccontr) using ksimplex_fix_plane[OF assms]
using assms(3)[THEN bspec[where x=a1]] using assms(2,5)
unfolding assms(6)[THEN spec[where x=j]] by simp
lemma ksimplex_fix_plane_p:
assumes "ksimplex p n s" "a ∈ s" "j∈{1..n}" "∀x∈s - {a}. x j = p" "a0 ∈ s" "a1 ∈ s"
"∀i. a1 i = (if i∈{1..n} then a0 i + 1 else a0 i)"
shows "a = a0" proof(rule ccontr) note s = ksimplexD[OF assms(1),rule_format]
assume as:"a ≠ a0" hence *:"a0 ∈ s - {a}" using assms(5) by auto
hence "a1 = a" using ksimplex_fix_plane[OF assms(2-)] by auto
thus False using as using assms(3,5) and assms(7)[rule_format,of j]
unfolding assms(4)[rule_format,OF *] using s(4)[OF assms(6), of j] by auto qed
lemma ksimplex_replace_0:
assumes "ksimplex p n s" "a ∈ s" "n ≠ 0" "j∈{1..n}" "∀x∈s - {a}. x j = 0"
shows "card {s'. ksimplex p n s' ∧ (∃b∈s'. s' - {b} = s - {a})} = 1" proof-
have *:"!!s' a a'. s' - {a'} = s - {a} ==> a' = a ==> a' ∈ s' ==> a ∈ s ==> (s' = s)" by auto
have **:"!!s' a'. ksimplex p n s' ==> a' ∈ s' ==> s' - {a'} = s - {a} ==> s' = s" proof- case goal1
guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note exta = this[rule_format]
have a:"a = a1" apply(rule ksimplex_fix_plane_0[OF assms(2,4-5)]) using exta(1-2,5) by auto moreover
guess b0 b1 apply(rule ksimplex_extrema_strong[OF goal1(1) assms(3)]) . note extb = this[rule_format]
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
have "b0 = a0" unfolding kle_antisym[THEN sym, of b0 a0 n] using exta extb using goal1(3) unfolding a a' by blast
hence "b1 = a1" apply-apply(rule ext) unfolding exta(5) extb(5) by auto ultimately
show "s' = s" apply-apply(rule *[of _ a1 b1]) using exta(1-2) extb(1-2) goal1 by auto qed
show ?thesis unfolding card_1_exists apply-apply(rule ex1I[of _ s])
unfolding mem_Collect_eq defer apply(erule conjE bexE)+ apply(rule_tac a'=b in **) using assms(1,2) by auto qed
lemma ksimplex_replace_1:
assumes "ksimplex p n s" "a ∈ s" "n ≠ 0" "j∈{1..n}" "∀x∈s - {a}. x j = p"
shows "card {s'. ksimplex p n s' ∧ (∃b∈s'. s' - {b} = s - {a})} = 1" proof-
have lem:"!!a a' s'. s' - {a'} = s - {a} ==> a' = a ==> a' ∈ s' ==> a ∈ s ==> s' = s" by auto
have lem:"!!s' a'. ksimplex p n s' ==> a'∈s' ==> s' - {a'} = s - {a} ==> s' = s" proof- case goal1
guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note exta = this[rule_format]
have a:"a = a0" apply(rule ksimplex_fix_plane_p[OF assms(1-2,4-5) exta(1,2)]) unfolding exta by auto moreover
guess b0 b1 apply(rule ksimplex_extrema_strong[OF goal1(1) assms(3)]) . note extb = this[rule_format]
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
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
have "a0 = b0" apply(rule ext) proof- case goal1 show "a0 x = b0 x"
using *[THEN cong, of x x] unfolding exta extb apply-apply(cases "x∈{1..n}") by auto qed
ultimately show "s' = s" apply-apply(rule lem[OF goal1(3) _ goal1(2) assms(2)]) by auto qed
show ?thesis unfolding card_1_exists apply(rule ex1I[of _ s]) unfolding mem_Collect_eq apply(rule,rule assms(1))
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
lemma ksimplex_replace_2:
assumes "ksimplex p n s" "a ∈ s" "n ≠ 0" "~(∃j∈{1..n}. ∀x∈s - {a}. x j = 0)" "~(∃j∈{1..n}. ∀x∈s - {a}. x j = p)"
shows "card {s'. ksimplex p n s' ∧ (∃b∈s'. s' - {b} = s - {a})} = 2" (is "card ?A = 2") proof-
have lem1:"!!a a' s s'. s' - {a'} = s - {a} ==> a' = a ==> a' ∈ s' ==> a ∈ s ==> s' = s" by auto
have lem2:"!!a b. a∈s ==> b≠a ==> s ≠ insert b (s - {a})" proof case goal1
hence "a∈insert b (s - {a})" by auto hence "a∈ s - {a}" unfolding insert_iff using goal1 by auto
thus False by auto qed
guess a0 a1 apply(rule ksimplex_extrema_strong[OF assms(1,3)]) . note a0a1=this
{ assume "a=a0"
have *:"!!P Q. (P ∨ Q) ==> ¬ P ==> Q" by auto
have "∃x∈s. ¬ kle n x a0" apply(rule_tac x=a1 in bexI) proof assume as:"kle n a1 a0"
show False using kle_imp_pointwise[OF as,THEN spec[where x=1]] unfolding a0a1(5)[THEN spec[where x=1]]
using assms(3) by auto qed(insert a0a1,auto)
hence "∃y∈s. ∃k∈{1..n}. ∀j. y j = (if j = k then a0 j + 1 else a0 j)"
apply(rule_tac *[OF ksimplex_successor[OF assms(1-2),unfolded `a=a0`]]) by auto
then guess a2 .. from this(2) guess k .. note k=this note a2=`a2∈s`
def a3 ≡ "λj. if j = k then a1 j + 1 else a1 j"
have "a3 ∉ s" proof assume "a3∈s" hence "kle n a3 a1" using a0a1(4) by auto
thus False apply(drule_tac kle_imp_pointwise) unfolding a3_def
apply(erule_tac x=k in allE) by auto qed
hence "a3 ≠ a0" "a3 ≠ a1" using a0a1 by auto
have "a2 ≠ a0" using k(2)[THEN spec[where x=k]] by auto
have lem3:"!!x. x∈(s - {a0}) ==> kle n a2 x" proof(rule ccontr) case goal1 hence as:"x∈s" "x≠a0" by auto
have "kle n a2 x ∨ kle n x a2" using ksimplexD(6)[OF assms(1)] and as `a2∈s` by auto moreover
have "kle n a0 x" using a0a1(4) as by auto
ultimately have "x = a0 ∨ x = a2" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto
hence "x = a2" using as by auto thus False using goal1(2) using kle_refl by auto qed
let ?s = "insert a3 (s - {a0})" have "ksimplex p n ?s" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI)
show "card ?s = n + 1" using ksimplexD(2-3)[OF assms(1)]
using `a3≠a0` `a3∉s` `a0∈s` by(auto simp add:card_insert_if)
fix x assume x:"x ∈ insert a3 (s - {a0})"
show "∀j. x j ≤ p" proof(rule,cases "x = a3")
fix j case False thus "x j≤p" using x ksimplexD(4)[OF assms(1)] by auto next
fix j case True show "x j≤p" unfolding True proof(cases "j=k")
case False thus "a3 j ≤p" unfolding True a3_def using `a1∈s` ksimplexD(4)[OF assms(1)] by auto next
guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this
have "a2 k ≤ a4 k" using lem3[OF a4(1)[unfolded `a=a0`],THEN kle_imp_pointwise] by auto
also have "… < p" using ksimplexD(4)[OF assms(1),rule_format,of a4 k] using a4 by auto
finally have *:"a0 k + 1 < p" unfolding k(2)[rule_format] by auto
case True thus "a3 j ≤p" unfolding a3_def unfolding a0a1(5)[rule_format]
using k(1) k(2)assms(5) using * by simp qed qed
show "∀j. j ∉ {1..n} --> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j∉{1..n}"
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
case True show "x j = p" unfolding True a3_def using j k(1)
using ksimplexD(5)[OF assms(1),rule_format,OF `a1∈s` j] by auto qed
fix y assume y:"y∈insert a3 (s - {a0})"
have lem4:"!!x. x∈s ==> x≠a0 ==> kle n x a3" proof- case goal1
guess kk using a0a1(4)[rule_format,OF `x∈s`,THEN conjunct2,unfolded kle_def]
apply-apply(erule exE,erule conjE) . note kk=this
have "k∉kk" proof assume "k∈kk"
hence "a1 k = x k + 1" using kk by auto
hence "a0 k = x k" unfolding a0a1(5)[rule_format] using k(1) by auto
hence "a2 k = x k + 1" unfolding k(2)[rule_format] by auto moreover
have "a2 k ≤ x k" using lem3[of x,THEN kle_imp_pointwise] goal1 by auto
ultimately show False by auto qed
thus ?case unfolding kle_def apply(rule_tac x="insert k kk" in exI) using kk(1)
unfolding a3_def kle_def kk(2)[rule_format] using k(1) by auto qed
show "kle n x y ∨ kle n y x" proof(cases "y=a3")
case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI1,rule lem4)
using x by auto next
case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True
apply(rule disjI2,rule lem4) using y False by auto next
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format])
using x y `y≠a3` by auto qed qed qed
hence "insert a3 (s - {a0}) ∈ ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
apply(rule_tac x="a3" in bexI) unfolding `a=a0` using `a3∉s` by auto moreover
have "s ∈ ?A" using assms(1,2) by auto ultimately have "?A ⊇ {s, insert a3 (s - {a0})}" by auto
moreover have "?A ⊆ {s, insert a3 (s - {a0})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
fix s' assume as:"ksimplex p n s'" and "∃b∈s'. s' - {b} = s - {a}"
from this(2) guess a' .. note a'=this
guess a_min a_max apply(rule ksimplex_extrema_strong[OF as assms(3)]) . note min_max=this
have *:"∀x∈s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x∈s-{a}"
hence "kle n a2 x" apply-apply(rule lem3) using `a=a0` by auto
hence "a2 k ≤ x k" apply(drule_tac kle_imp_pointwise) by auto moreover
have "x k ≤ a2 k" unfolding k(2)[rule_format] using a0a1(4)[rule_format,of x,THEN conjunct1]
unfolding kle_def using x by auto ultimately show "x k = a2 k" by auto qed
have **:"a'=a_min ∨ a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto
show "s' ∈ {s, insert a3 (s - {a0})}" proof(cases "a'=a_min")
case True have "a_max = a1" unfolding kle_antisym[THEN sym,of a_max a1 n] apply(rule)
apply(rule a0a1(4)[rule_format,THEN conjunct2]) defer proof(rule min_max(4)[rule_format,THEN conjunct2])
show "a1∈s'" using a' unfolding `a=a0` using a0a1 by auto
show "a_max ∈ s" proof(rule ccontr) assume "a_max∉s"
hence "a_max = a'" using a' min_max by auto
thus False unfolding True using min_max by auto qed qed
hence "∀i. a_max i = a1 i" by auto
hence "a' = a" unfolding True `a=a0` apply-apply(subst expand_fun_eq,rule)
apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format]
proof- case goal1 thus ?case apply(cases "x∈{1..n}") by auto qed
hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a∈s` `a'∈s'` by auto
thus ?thesis by auto next
case False hence as:"a' = a_max" using ** by auto
have "a_min = a2" unfolding kle_antisym[THEN sym, of _ _ n] apply rule
apply(rule min_max(4)[rule_format,THEN conjunct1]) defer proof(rule lem3)
show "a_min ∈ s - {a0}" unfolding a'(2)[THEN sym,unfolded `a=a0`]
unfolding as using min_max(1-3) by auto
have "a2 ≠ a" unfolding `a=a0` using k(2)[rule_format,of k] by auto
hence "a2 ∈ s - {a}" using a2 by auto thus "a2 ∈ s'" unfolding a'(2)[THEN sym] by auto qed
hence "∀i. a_min i = a2 i" by auto
hence "a' = a3" unfolding as `a=a0` apply-apply(subst expand_fun_eq,rule)
apply(erule_tac x=x in allE) unfolding a0a1(5)[rule_format] min_max(5)[rule_format]
unfolding a3_def k(2)[rule_format] unfolding a0a1(5)[rule_format] proof- case goal1
show ?case unfolding goal1 apply(cases "x∈{1..n}") defer apply(cases "x=k")
using `k∈{1..n}` by auto qed
hence "s' = insert a3 (s - {a0})" apply-apply(rule lem1) defer apply assumption
apply(rule a'(1)) unfolding a' `a=a0` using `a3∉s` by auto
thus ?thesis by auto qed qed
ultimately have *:"?A = {s, insert a3 (s - {a0})}" by blast
have "s ≠ insert a3 (s - {a0})" using `a3∉s` by auto
hence ?thesis unfolding * by auto } moreover
{ assume "a=a1"
have *:"!!P Q. (P ∨ Q) ==> ¬ P ==> Q" by auto
have "∃x∈s. ¬ kle n a1 x" apply(rule_tac x=a0 in bexI) proof assume as:"kle n a1 a0"
show False using kle_imp_pointwise[OF as,THEN spec[where x=1]] unfolding a0a1(5)[THEN spec[where x=1]]
using assms(3) by auto qed(insert a0a1,auto)
hence "∃y∈s. ∃k∈{1..n}. ∀j. a1 j = (if j = k then y j + 1 else y j)"
apply(rule_tac *[OF ksimplex_predecessor[OF assms(1-2),unfolded `a=a1`]]) by auto
then guess a2 .. from this(2) guess k .. note k=this note a2=`a2∈s`
def a3 ≡ "λj. if j = k then a0 j - 1 else a0 j"
have "a2 ≠ a1" using k(2)[THEN spec[where x=k]] by auto
have lem3:"!!x. x∈(s - {a1}) ==> kle n x a2" proof(rule ccontr) case goal1 hence as:"x∈s" "x≠a1" by auto
have "kle n a2 x ∨ kle n x a2" using ksimplexD(6)[OF assms(1)] and as `a2∈s` by auto moreover
have "kle n x a1" using a0a1(4) as by auto
ultimately have "x = a2 ∨ x = a1" apply-apply(rule kle_adjacent[OF k(2)]) using goal1(2) by auto
hence "x = a2" using as by auto thus False using goal1(2) using kle_refl by auto qed
have "a0 k ≠ 0" proof-
guess a4 using assms(4)[unfolded bex_simps ball_simps,rule_format,OF `k∈{1..n}`] .. note a4=this
have "a4 k ≤ a2 k" using lem3[OF a4(1)[unfolded `a=a1`],THEN kle_imp_pointwise] by auto
moreover have "a4 k > 0" using a4 by auto ultimately have "a2 k > 0" by auto
hence "a1 k > 1" unfolding k(2)[rule_format] by simp
thus ?thesis unfolding a0a1(5)[rule_format] using k(1) by simp qed
hence lem4:"∀j. a0 j = (if j=k then a3 j + 1 else a3 j)" unfolding a3_def by simp
have "¬ kle n a0 a3" apply(rule ccontr) unfolding not_not apply(drule kle_imp_pointwise)
unfolding lem4[rule_format] apply(erule_tac x=k in allE) by auto
hence "a3 ∉ s" using a0a1(4) by auto
hence "a3 ≠ a1" "a3 ≠ a0" using a0a1 by auto
let ?s = "insert a3 (s - {a1})" have "ksimplex p n ?s" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI)
show "card ?s = n+1" using ksimplexD(2-3)[OF assms(1)]
using `a3≠a0` `a3∉s` `a1∈s` by(auto simp add:card_insert_if)
fix x assume x:"x ∈ insert a3 (s - {a1})"
show "∀j. x j ≤ p" proof(rule,cases "x = a3")
fix j case False thus "x j≤p" using x ksimplexD(4)[OF assms(1)] by auto next
fix j case True show "x j≤p" unfolding True proof(cases "j=k")
case False thus "a3 j ≤p" unfolding True a3_def using `a0∈s` ksimplexD(4)[OF assms(1)] by auto next
guess a4 using assms(5)[unfolded bex_simps ball_simps,rule_format,OF k(1)] .. note a4=this
case True have "a3 k ≤ a0 k" unfolding lem4[rule_format] by auto
also have "… ≤ p" using ksimplexD(4)[OF assms(1),rule_format,of a0 k] a0a1 by auto
finally show "a3 j ≤ p" unfolding True by auto qed qed
show "∀j. j ∉ {1..n} --> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j∉{1..n}"
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
case True show "x j = p" unfolding True a3_def using j k(1)
using ksimplexD(5)[OF assms(1),rule_format,OF `a0∈s` j] by auto qed
fix y assume y:"y∈insert a3 (s - {a1})"
have lem4:"!!x. x∈s ==> x≠a1 ==> kle n a3 x" proof- case goal1 hence *:"x∈s - {a1}" by auto
have "kle n a3 a2" proof- have "kle n a0 a1" using a0a1 by auto then guess kk unfolding kle_def ..
thus ?thesis unfolding kle_def apply(rule_tac x=kk in exI) unfolding lem4[rule_format] k(2)[rule_format]
apply(rule)defer proof(rule) case goal1 thus ?case apply-apply(erule conjE)
apply(erule_tac[!] x=j in allE) apply(cases "j∈kk") apply(case_tac[!] "j=k") by auto qed auto qed moreover
have "kle n a3 a0" unfolding kle_def lem4[rule_format] apply(rule_tac x="{k}" in exI) using k(1) by auto
ultimately show ?case apply-apply(rule kle_between_l[of _ a0 _ a2]) using lem3[OF *]
using a0a1(4)[rule_format,OF goal1(1)] by auto qed
show "kle n x y ∨ kle n y x" proof(cases "y=a3")
case True show ?thesis unfolding True apply(cases "x=a3") defer apply(rule disjI2,rule lem4)
using x by auto next
case False show ?thesis proof(cases "x=a3") case True show ?thesis unfolding True
apply(rule disjI1,rule lem4) using y False by auto next
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format])
using x y `y≠a3` by auto qed qed qed
hence "insert a3 (s - {a1}) ∈ ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
apply(rule_tac x="a3" in bexI) unfolding `a=a1` using `a3∉s` by auto moreover
have "s ∈ ?A" using assms(1,2) by auto ultimately have "?A ⊇ {s, insert a3 (s - {a1})}" by auto
moreover have "?A ⊆ {s, insert a3 (s - {a1})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
fix s' assume as:"ksimplex p n s'" and "∃b∈s'. s' - {b} = s - {a}"
from this(2) guess a' .. note a'=this
guess a_min a_max apply(rule ksimplex_extrema_strong[OF as assms(3)]) . note min_max=this
have *:"∀x∈s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x∈s-{a}"
hence "kle n x a2" apply-apply(rule lem3) using `a=a1` by auto
hence "x k ≤ a2 k" apply(drule_tac kle_imp_pointwise) by auto moreover
{ have "a2 k ≤ a0 k" using k(2)[rule_format,of k] unfolding a0a1(5)[rule_format] using k(1) by simp
also have "… ≤ x k" using a0a1(4)[rule_format,of x,THEN conjunct1,THEN kle_imp_pointwise] x by auto
finally have "a2 k ≤ x k" . } ultimately show "x k = a2 k" by auto qed
have **:"a'=a_min ∨ a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto
have "a2 ≠ a1" proof assume as:"a2 = a1"
show False using k(2) unfolding as apply(erule_tac x=k in allE) by auto qed
hence a2':"a2 ∈ s' - {a'}" unfolding a' using a2 unfolding `a=a1` by auto
show "s' ∈ {s, insert a3 (s - {a1})}" proof(cases "a'=a_min")
case True have "a_max ∈ s - {a1}" using min_max unfolding a'(2)[unfolded `a=a1`,THEN sym] True by auto
hence "a_max = a2" unfolding kle_antisym[THEN sym,of a_max a2 n] apply-apply(rule)
apply(rule lem3,assumption) apply(rule min_max(4)[rule_format,THEN conjunct2]) using a2' by auto
hence a_max:"∀i. a_max i = a2 i" by auto
have *:"∀j. a2 j = (if j∈{1..n} then a3 j + 1 else a3 j)"
using k(2) unfolding lem4[rule_format] a0a1(5)[rule_format] apply-apply(rule,erule_tac x=j in allE)
proof- case goal1 thus ?case apply(cases "j∈{1..n}",case_tac[!] "j=k") by auto qed
have "∀i. a_min i = a3 i" using a_max apply-apply(rule,erule_tac x=i in allE)
unfolding min_max(5)[rule_format] *[rule_format] proof- case goal1
thus ?case apply(cases "i∈{1..n}") by auto qed hence "a_min = a3" unfolding expand_fun_eq .
hence "s' = insert a3 (s - {a1})" using a' unfolding `a=a1` True by auto thus ?thesis by auto next
case False hence as:"a'=a_max" using ** by auto
have "a_min = a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
apply(rule min_max(4)[rule_format,THEN conjunct1]) defer apply(rule a0a1(4)[rule_format,THEN conjunct1]) proof-
have "a_min ∈ s - {a1}" using min_max(1,3) unfolding a'(2)[THEN sym,unfolded `a=a1`] as by auto
thus "a_min ∈ s" by auto have "a0 ∈ s - {a1}" using a0a1(1-3) by auto thus "a0 ∈ s'"
unfolding a'(2)[THEN sym,unfolded `a=a1`] by auto qed
hence "∀i. a_max i = a1 i" unfolding a0a1(5)[rule_format] min_max(5)[rule_format] by auto
hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a∈s` `a'∈s'` unfolding as `a=a1` unfolding expand_fun_eq by auto
thus ?thesis by auto qed qed
ultimately have *:"?A = {s, insert a3 (s - {a1})}" by blast
have "s ≠ insert a3 (s - {a1})" using `a3∉s` by auto
hence ?thesis unfolding * by auto } moreover
{ assume as:"a≠a0" "a≠a1" have "¬ (∀x∈s. kle n a x)" proof case goal1
have "a=a0" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
using goal1 a0a1 assms(2) by auto thus False using as by auto qed
hence "∃y∈s. ∃k∈{1..n}. ∀j. a j = (if j = k then y j + 1 else y j)" using ksimplex_predecessor[OF assms(1-2)] by blast
then guess u .. from this(2) guess k .. note k = this[rule_format] note u = `u∈s`
have "¬ (∀x∈s. kle n x a)" proof case goal1
have "a=a1" unfolding kle_antisym[THEN sym,of _ _ n] apply(rule)
using goal1 a0a1 assms(2) by auto thus False using as by auto qed
hence "∃y∈s. ∃k∈{1..n}. ∀j. y j = (if j = k then a j + 1 else a j)" using ksimplex_successor[OF assms(1-2)] by blast
then guess v .. from this(2) guess l .. note l = this[rule_format] note v = `v∈s`
def a' ≡ "λj. if j = l then u j + 1 else u j"
have kl:"k ≠ l" proof assume "k=l" have *:"!!P. (if P then (1::nat) else 0) ≠ 2" by auto
thus False using ksimplexD(6)[OF assms(1),rule_format,OF u v] unfolding kle_def
unfolding l(2) k(2) `k=l` apply-apply(erule disjE)apply(erule_tac[!] exE conjE)+
apply(erule_tac[!] x=l in allE)+ by(auto simp add: *) qed
hence aa':"a'≠a" apply-apply rule unfolding expand_fun_eq unfolding a'_def k(2)
apply(erule_tac x=l in allE) by auto
have "a' ∉ s" apply(rule) apply(drule ksimplexD(6)[OF assms(1),rule_format,OF `a∈s`]) proof(cases "kle n a a'")
case goal2 hence "kle n a' a" by auto thus False apply(drule_tac kle_imp_pointwise)
apply(erule_tac x=l in allE) unfolding a'_def k(2) using kl by auto next
case True thus False apply(drule_tac kle_imp_pointwise)
apply(erule_tac x=k in allE) unfolding a'_def k(2) using kl by auto qed
have kle_uv:"kle n u a" "kle n u a'" "kle n a v" "kle n a' v" unfolding kle_def apply-
apply(rule_tac[1] x="{k}" in exI,rule_tac[2] x="{l}" in exI)
apply(rule_tac[3] x="{l}" in exI,rule_tac[4] x="{k}" in exI)
unfolding l(2) k(2) a'_def using l(1) k(1) by auto
have uxv:"!!x. kle n u x ==> kle n x v ==> (x = u) ∨ (x = a) ∨ (x = a') ∨ (x = v)"
proof- case goal1 thus ?case proof(cases "x k = u k", case_tac[!] "x l = u l")
assume as:"x l = u l" "x k = u k"
have "x = u" unfolding expand_fun_eq
using goal1(2)[THEN kle_imp_pointwise,unfolded l(2)] unfolding k(2) apply-
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
assume as:"x l ≠ u l" "x k = u k"
have "x = a'" unfolding expand_fun_eq unfolding a'_def
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
assume as:"x l = u l" "x k ≠ u k"
have "x = a" unfolding expand_fun_eq
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as by auto qed thus ?case by auto next
assume as:"x l ≠ u l" "x k ≠ u k"
have "x = v" unfolding expand_fun_eq
using goal1(2)[THEN kle_imp_pointwise] unfolding l(2) k(2) apply-
using goal1(1)[THEN kle_imp_pointwise] apply-apply rule apply(erule_tac x=xa in allE)+ proof- case goal1
thus ?case apply(cases "x=l") apply(case_tac[!] "x=k") using as `k≠l` by auto qed thus ?case by auto qed qed
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
have lem3:"!!x. x∈s ==> kle n v x ==> kle n a' x" apply(rule kle_between_r[of _ u _ v])
prefer 3 apply(rule kle_trans[OF uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format])
using kle_uv `u∈s` by auto
have lem4:"!!x. x∈s ==> kle n x u ==> kle n x a'" apply(rule kle_between_l[of _ u _ v])
prefer 4 apply(rule kle_trans[OF _ uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format])
using kle_uv `v∈s` by auto
have lem5:"!!x. x∈s ==> x≠a ==> kle n x a' ∨ kle n a' x" proof- case goal1 thus ?case
proof(cases "kle n v x ∨ kle n x u") case True thus ?thesis using goal1 by(auto intro:lem3 lem4) next
case False hence *:"kle n u x" "kle n x v" using ksimplexD(6)[OF assms(1)] using goal1 `u∈s` `v∈s` by auto
show ?thesis using uxv[OF *] using kle_uv using goal1 by auto qed qed
have "ksimplex p n (insert a' (s - {a}))" apply(rule ksimplexI) proof(rule_tac[2-] ballI,rule_tac[4] ballI)
show "card (insert a' (s - {a})) = n + 1" using ksimplexD(2-3)[OF assms(1)]
using `a'≠a` `a'∉s` `a∈s` by(auto simp add:card_insert_if)
fix x assume x:"x ∈ insert a' (s - {a})"
show "∀j. x j ≤ p" proof(rule,cases "x = a'")
fix j case False thus "x j≤p" using x ksimplexD(4)[OF assms(1)] by auto next
fix j case True show "x j≤p" unfolding True proof(cases "j=l")
case False thus "a' j ≤p" unfolding True a'_def using `u∈s` ksimplexD(4)[OF assms(1)] by auto next
case True have *:"a l = u l" "v l = a l + 1" using k(2)[of l] l(2)[of l] `k≠l` by auto
have "u l + 1 ≤ p" unfolding *[THEN sym] using ksimplexD(4)[OF assms(1)] using `v∈s` by auto
thus "a' j ≤p" unfolding a'_def True by auto qed qed
show "∀j. j ∉ {1..n} --> x j = p" proof(rule,rule,cases "x=a'") fix j::nat assume j:"j∉{1..n}"
{ case False thus "x j = p" using j x ksimplexD(5)[OF assms(1)] by auto }
case True show "x j = p" unfolding True a'_def using j l(1)
using ksimplexD(5)[OF assms(1),rule_format,OF `u∈s` j] by auto qed
fix y assume y:"y∈insert a' (s - {a})"
show "kle n x y ∨ kle n y x" proof(cases "y=a'")
case True show ?thesis unfolding True apply(cases "x=a'") defer apply(rule lem5) using x by auto next
case False show ?thesis proof(cases "x=a'") case True show ?thesis unfolding True
using lem5[of y] using y by auto next
case False thus ?thesis apply(rule_tac ksimplexD(6)[OF assms(1),rule_format])
using x y `y≠a'` by auto qed qed qed
hence "insert a' (s - {a}) ∈ ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
apply(rule_tac x="a'" in bexI) using aa' `a'∉s` by auto moreover
have "s ∈ ?A" using assms(1,2) by auto ultimately have "?A ⊇ {s, insert a' (s - {a})}" by auto
moreover have "?A ⊆ {s, insert a' (s - {a})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
fix s' assume as:"ksimplex p n s'" and "∃b∈s'. s' - {b} = s - {a}"
from this(2) guess a'' .. note a''=this
have "u≠v" unfolding expand_fun_eq unfolding l(2) k(2) by auto
hence uv':"¬ kle n v u" using uv using kle_antisym by auto
have "u≠a" "v≠a" unfolding expand_fun_eq k(2) l(2) by auto
hence uvs':"u∈s'" "v∈s'" using `u∈s` `v∈s` using a'' by auto
have lem6:"a ∈ s' ∨ a' ∈ s'" proof(cases "∀x∈s'. kle n x u ∨ kle n v x")
case False then guess w unfolding ball_simps .. note w=this
hence "kle n u w" "kle n w v" using ksimplexD(6)[OF as] uvs' by auto
hence "w = a' ∨ w = a" using uxv[of w] uvs' w by auto thus ?thesis using w by auto next
case True have "¬ (∀x∈s'. kle n x u)" unfolding ball_simps apply(rule_tac x=v in bexI)
using uv `u≠v` unfolding kle_antisym[of n u v,THEN sym] using `v∈s'` by auto
hence "∃y∈s'. ∃k∈{1..n}. ∀j. y j = (if j = k then u j + 1 else u j)" using ksimplex_successor[OF as `u∈s'`] by blast
then guess w .. note w=this from this(2) guess kk .. note kk=this[rule_format]
have "¬ kle n w u" apply-apply(rule,drule kle_imp_pointwise)
apply(erule_tac x=kk in allE) unfolding kk by auto
hence *:"kle n v w" using True[rule_format,OF w(1)] by auto
hence False proof(cases "kk≠l") case True thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)]
apply(erule_tac x=l in allE) using `k≠l` by auto next
case False hence "kk≠k" using `k≠l` by auto thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)]
apply(erule_tac x=k in allE) using `k≠l` by auto qed
thus ?thesis by auto qed
thus "s' ∈ {s, insert a' (s - {a})}" proof(cases "a∈s'")
case True hence "s' = s" apply-apply(rule lem1[OF a''(2)]) using a'' `a∈s` by auto
thus ?thesis by auto next case False hence "a'∈s'" using lem6 by auto
hence "s' = insert a' (s - {a})" apply-apply(rule lem1[of _ a'' _ a'])
unfolding a''(2)[THEN sym] using a'' using `a'∉s` by auto
thus ?thesis by auto qed qed
ultimately have *:"?A = {s, insert a' (s - {a})}" by blast
have "s ≠ insert a' (s - {a})" using `a'∉s` by auto
hence ?thesis unfolding * by auto }
ultimately show ?thesis by auto qed
subsection {* Hence another step towards concreteness. *}
lemma kuhn_simplex_lemma:
assumes "∀s. ksimplex p (n + 1) s --> (rl ` s ⊆{0..n+1})"
"odd (card{f. ∃s a. ksimplex p (n + 1) s ∧ a ∈ s ∧ (f = s - {a}) ∧
(rl ` f = {0 .. n}) ∧ ((∃j∈{1..n+1}.∀x∈f. x j = 0) ∨ (∃j∈{1..n+1}.∀x∈f. x j = p))})"
shows "odd(card {s∈{s. ksimplex p (n + 1) s}. rl ` s = {0..n+1} })" proof-
have *:"!!x y. x = y ==> odd (card x) ==> odd (card y)" by auto
have *:"odd(card {f∈{f. ∃s∈{s. ksimplex p (n + 1) s}. (∃a∈s. f = s - {a})}.
(rl ` f = {0..n}) ∧
((∃j∈{1..n+1}. ∀x∈f. x j = 0) ∨
(∃j∈{1..n+1}. ∀x∈f. x j = p))})" apply(rule *[OF _ assms(2)]) by auto
show ?thesis apply(rule kuhn_complete_lemma[OF finite_simplices]) prefer 6 apply(rule *) apply(rule,rule,rule)
apply(subst ksimplex_def) defer apply(rule,rule assms(1)[rule_format]) unfolding mem_Collect_eq apply assumption
apply default+ unfolding mem_Collect_eq apply(erule disjE bexE)+ defer apply(erule disjE bexE)+ defer
apply default+ unfolding mem_Collect_eq apply(erule disjE bexE)+ unfolding mem_Collect_eq proof-
fix f s a assume as:"ksimplex p (n + 1) s" "a∈s" "f = s - {a}"
let ?S = "{s. ksimplex p (n + 1) s ∧ (∃a∈s. f = s - {a})}"
have S:"?S = {s'. ksimplex p (n + 1) s' ∧ (∃b∈s'. s' - {b} = s - {a})}" unfolding as by blast
{ fix j assume j:"j ∈ {1..n + 1}" "∀x∈f. x j = 0" thus "card {s. ksimplex p (n + 1) s ∧ (∃a∈s. f = s - {a})} = 1" unfolding S
apply-apply(rule ksimplex_replace_0) apply(rule as)+ unfolding as by auto }
{ fix j assume j:"j ∈ {1..n + 1}" "∀x∈f. x j = p" thus "card {s. ksimplex p (n + 1) s ∧ (∃a∈s. f = s - {a})} = 1" unfolding S
apply-apply(rule ksimplex_replace_1) apply(rule as)+ unfolding as by auto }
show "¬ ((∃j∈{1..n+1}. ∀x∈f. x j = 0) ∨ (∃j∈{1..n+1}. ∀x∈f. x j = p)) ==> card ?S = 2"
unfolding S apply(rule ksimplex_replace_2) apply(rule as)+ unfolding as by auto qed auto qed
subsection {* Reduced labelling. *}
definition "reduced label (n::nat) (x::nat=>nat) =
(SOME k. k ≤ n ∧ (∀i. 1≤i ∧ i<k+1 --> label x i = 0) ∧ (k = n ∨ label x (k + 1) ≠ (0::nat)))"
lemma reduced_labelling: shows "reduced label n x ≤ n" (is ?t1) and
"∀i. 1≤i ∧ i < reduced label n x + 1 --> (label x i = 0)" (is ?t2)
"(reduced label n x = n) ∨ (label x (reduced label n x + 1) ≠ 0)" (is ?t3) proof-
have num_WOP:"!!P k. P (k::nat) ==> ∃n. P n ∧ (∀m<n. ¬ P m)"
apply(drule ex_has_least_nat[where m="λx. x"]) apply(erule exE,rule_tac x=x in exI) by auto
have *:"n ≤ n ∧ (label x (n + 1) ≠ 0 ∨ n = n)" by auto
then guess N apply(drule_tac num_WOP[of "λj. j≤n ∧ (label x (j+1) ≠ 0 ∨ n = j)"]) apply(erule exE) . note N=this
have N':"N ≤ n" "∀i. 1 ≤ i ∧ i < N + 1 --> label x i = 0" "N = n ∨ label x (N + 1) ≠ 0" defer proof(rule,rule)
fix i assume i:"1≤i ∧ 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)
show ?t1 ?t2 ?t3 unfolding reduced_def apply(rule_tac[!] someI2_ex) using N' by(auto intro!: exI[where x=N]) qed
lemma reduced_labelling_unique: fixes x::"nat => nat"
assumes "r ≤ n" "∀i. 1 ≤ i ∧ i < r + 1 --> (label x i = 0)" "(r = n) ∨ (label x (r + 1) ≠ 0)"
shows "reduced label n x = r" apply(rule le_antisym) apply(rule_tac[!] ccontr) unfolding not_le
using reduced_labelling[of label n x] using assms by auto
lemma reduced_labelling_0: assumes "j∈{1..n}" "label x j = 0" shows "reduced label n x ≠ j - 1"
using reduced_labelling[of label n x] using assms by fastsimp
lemma reduced_labelling_1: assumes "j∈{1..n}" "label x j ≠ 0" shows "reduced label n x < j"
using assms and reduced_labelling[of label n x] apply(erule_tac x=j in allE) by auto
lemma reduced_labelling_Suc:
assumes "reduced lab (n + 1) x ≠ n + 1" shows "reduced lab (n + 1) x = reduced lab n x"
apply(subst eq_commute) apply(rule reduced_labelling_unique)
using reduced_labelling[of lab "n+1" x] and assms by auto
lemma complete_face_top:
assumes "∀x∈f. ∀j∈{1..n+1}. x j = 0 --> lab x j = 0"
"∀x∈f. ∀j∈{1..n+1}. x j = p --> lab x j = 1"
shows "((reduced lab (n + 1)) ` f = {0..n}) ∧ ((∃j∈{1..n+1}. ∀x∈f. x j = 0) ∨ (∃j∈{1..n+1}. ∀x∈f. x j = p)) <->
((reduced lab (n + 1)) ` f = {0..n}) ∧ (∀x∈f. x (n + 1) = p)" (is "?l = ?r") proof
assume ?l (is "?as ∧ (?a ∨ ?b)") thus ?r apply-apply(rule,erule conjE,assumption) proof(cases ?a)
case True then guess j .. note j=this {fix x assume x:"x∈f"
have "reduced lab (n+1) x ≠ j - 1" using j apply-apply(rule reduced_labelling_0) defer apply(rule assms(1)[rule_format]) using x by auto }
moreover have "j - 1 ∈ {0..n}" using j by auto
then guess y unfolding `?l`[THEN conjunct1,THEN sym] and image_iff .. note y = this
ultimately have False by auto thus "∀x∈f. x (n + 1) = p" by auto next
case False hence ?b using `?l` by blast then guess j .. note j=this {fix x assume x:"x∈f"
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
have "j = n + 1" proof(rule ccontr) case goal1 hence "j < n + 1" using j by auto moreover
have "n ∈ {0..n}" by auto then guess y unfolding `?l`[THEN conjunct1,THEN sym] image_iff ..
ultimately show False using *[of y] by auto qed
thus "∀x∈f. x (n + 1) = p" using j by auto qed qed(auto)
subsection {* Hence we get just about the nice induction. *}
lemma kuhn_induction:
assumes "0 < p" "∀x. ∀j∈{1..n+1}. (∀j. x j ≤ p) ∧ (x j = 0) --> (lab x j = 0)"
"∀x. ∀j∈{1..n+1}. (∀j. x j ≤ p) ∧ (x j = p) --> (lab x j = 1)"
"odd (card {f. ksimplex p n f ∧ ((reduced lab n) ` f = {0..n})})"
shows "odd (card {s. ksimplex p (n+1) s ∧((reduced lab (n+1)) ` s = {0..n+1})})" proof-
have *:"!!s t. odd (card s) ==> s = t ==> odd (card t)" "!!s f. (!!x. f x ≤ n +1 ) ==> f ` s ⊆ {0..n+1}" by auto
show ?thesis apply(rule kuhn_simplex_lemma[unfolded mem_Collect_eq]) apply(rule,rule,rule *,rule reduced_labelling)
apply(rule *(1)[OF assms(4)]) apply(rule set_ext) unfolding mem_Collect_eq apply(rule,erule conjE) defer apply(rule) proof-(*(rule,rule)*)
fix f assume as:"ksimplex p n f" "reduced lab n ` f = {0..n}"
have *:"∀x∈f. ∀j∈{1..n + 1}. x j = 0 --> lab x j = 0" "∀x∈f. ∀j∈{1..n + 1}. x j = p --> lab x j = 1"
using assms(2-3) using as(1)[unfolded ksimplex_def] by auto
have allp:"∀x∈f. x (n + 1) = p" using assms(2) using as(1)[unfolded ksimplex_def] by auto
{ fix x assume "x∈f" hence "reduced lab (n + 1) x < n + 1" apply-apply(rule reduced_labelling_1)
defer using assms(3) using as(1)[unfolded ksimplex_def] by auto
hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc) using reduced_labelling(1) by auto }
hence "reduced lab (n + 1) ` f = {0..n}" unfolding as(2)[THEN sym] apply- apply(rule set_ext) unfolding image_iff by auto
moreover guess s using as(1)[unfolded simplex_top_face[OF assms(1) allp,THEN sym]] .. then guess a ..
ultimately show "∃s a. ksimplex p (n + 1) s ∧
a ∈ s ∧ f = s - {a} ∧ reduced lab (n + 1) ` f = {0..n} ∧ ((∃j∈{1..n + 1}. ∀x∈f. x j = 0) ∨ (∃j∈{1..n + 1}. ∀x∈f. x j = p))" (is ?ex)
apply(rule_tac x=s in exI,rule_tac x=a in exI) unfolding complete_face_top[OF *] using allp as(1) by auto
next fix f assume as:"∃s a. ksimplex p (n + 1) s ∧
a ∈ s ∧ f = s - {a} ∧ reduced lab (n + 1) ` f = {0..n} ∧ ((∃j∈{1..n + 1}. ∀x∈f. x j = 0) ∨ (∃j∈{1..n + 1}. ∀x∈f. x j = p))" (is ?ex)
then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
{ fix x assume "x∈f" hence "reduced lab (n + 1) x ∈ reduced lab (n + 1) ` f" by auto
hence "reduced lab (n + 1) x < n + 1" using sa(4) by auto
hence "reduced lab (n + 1) x = reduced lab n x" apply-apply(rule reduced_labelling_Suc)
using reduced_labelling(1) by auto }
thus part1:"reduced lab n ` f = {0..n}" unfolding sa(4)[THEN sym] apply-apply(rule set_ext) unfolding image_iff by auto
have *:"∀x∈f. x (n + 1) = p" proof(cases "∃j∈{1..n + 1}. ∀x∈f. x j = 0")
case True then guess j .. hence "!!x. x∈f ==> reduced lab (n + 1) x ≠ j - 1" apply-apply(rule reduced_labelling_0) apply assumption
apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover
have "j - 1 ∈ {0..n}" using `j∈{1..n+1}` by auto
ultimately have False unfolding sa(4)[THEN sym] unfolding image_iff by fastsimp thus ?thesis by auto next
case False hence "∃j∈{1..n + 1}. ∀x∈f. x j = p" using sa(5) by fastsimp then guess j .. note j=this
thus ?thesis proof(cases "j = n+1")
case False hence *:"j∈{1..n}" using j by auto
hence "!!x. x∈f ==> reduced lab n x < j" apply(rule reduced_labelling_1) proof- fix x assume "x∈f"
hence "lab x j = 1" apply-apply(rule assms(3)[rule_format,OF j(1)])
using sa(1)[unfolded ksimplex_def] using j unfolding sa by auto thus "lab x j ≠ 0" by auto qed
moreover have "j∈{0..n}" using * by auto
ultimately have False unfolding part1[THEN sym] using * unfolding image_iff by auto thus ?thesis by auto qed auto qed
thus "ksimplex p n f" using as unfolding simplex_top_face[OF assms(1) *,THEN sym] by auto qed qed
lemma kuhn_induction_Suc:
assumes "0 < p" "∀x. ∀j∈{1..Suc n}. (∀j. x j ≤ p) ∧ (x j = 0) --> (lab x j = 0)"
"∀x. ∀j∈{1..Suc n}. (∀j. x j ≤ p) ∧ (x j = p) --> (lab x j = 1)"
"odd (card {f. ksimplex p n f ∧ ((reduced lab n) ` f = {0..n})})"
shows "odd (card {s. ksimplex p (Suc n) s ∧((reduced lab (Suc n)) ` s = {0..Suc n})})"
using assms unfolding Suc_eq_plus1 by(rule kuhn_induction)
subsection {* And so we get the final combinatorial result. *}
lemma ksimplex_0: "ksimplex p 0 s <-> s = {(λx. p)}" (is "?l = ?r") proof
assume l:?l guess a using ksimplexD(3)[OF l, unfolded add_0] unfolding card_1_exists .. note a=this
have "a = (λx. p)" using ksimplexD(5)[OF l, rule_format, OF a(1)] by(rule,auto) thus ?r using a by auto next
assume r:?r show ?l unfolding r ksimplex_eq by auto qed
lemma reduce_labelling_0[simp]: "reduced lab 0 x = 0" apply(rule reduced_labelling_unique) by auto
lemma kuhn_combinatorial:
assumes "0 < p" "∀x j. (∀j. x(j) ≤ p) ∧ 1 ≤ j ∧ j ≤ n ∧ (x j = 0) --> (lab x j = 0)"
"∀x j. (∀j. x(j) ≤ p) ∧ 1 ≤ j ∧ j ≤ n ∧ (x j = p) --> (lab x j = 1)"
shows " odd (card {s. ksimplex p n s ∧ ((reduced lab n) ` s = {0..n})})" using assms proof(induct n)
let ?M = "λn. {s. ksimplex p n s ∧ ((reduced lab n) ` s = {0..n})}"
{ case 0 have *:"?M 0 = {{(λx. p)}}" unfolding ksimplex_0 by auto show ?case unfolding * by auto }
case (Suc n) have "odd (card (?M n))" apply(rule Suc(1)[OF Suc(2)]) using Suc(3-) by auto
thus ?case apply-apply(rule kuhn_induction_Suc) using Suc(2-) by auto qed
lemma kuhn_lemma: assumes "0 < (p::nat)" "0 < (n::nat)"
"∀x. (∀i∈{1..n}. x i ≤ p) --> (∀i∈{1..n}. (label x i = (0::nat)) ∨ (label x i = 1))"
"∀x. (∀i∈{1..n}. x i ≤ p) --> (∀i∈{1..n}. (x i = 0) --> (label x i = 0))"
"∀x. (∀i∈{1..n}. x i ≤ p) --> (∀i∈{1..n}. (x i = p) --> (label x i = 1))"
obtains q where "∀i∈{1..n}. q i < p"
"∀i∈{1..n}. ∃r s. (∀j∈{1..n}. q(j) ≤ r(j) ∧ r(j) ≤ q(j) + 1) ∧
(∀j∈{1..n}. q(j) ≤ s(j) ∧ s(j) ≤ q(j) + 1) ∧
~(label r i = label s i)" proof-
let ?A = "{s. ksimplex p n s ∧ reduced label n ` s = {0..n}}" have "n≠0" using assms by auto
have conjD:"!!P Q. P ∧ Q ==> P" "!!P Q. P ∧ Q ==> Q" by auto
have "odd (card ?A)" apply(rule kuhn_combinatorial[of p n label]) using assms by auto
hence "card ?A ≠ 0" apply-apply(rule ccontr) by auto hence "?A ≠ {}" unfolding card_eq_0_iff by auto
then obtain s where "s∈?A" by auto note s=conjD[OF this[unfolded mem_Collect_eq]]
guess a b apply(rule ksimplex_extrema_strong[OF s(1) `n≠0`]) . note ab=this
show ?thesis apply(rule that[of a]) proof(rule_tac[!] ballI) fix i assume "i∈{1..n}"
hence "a i + 1 ≤ p" apply-apply(rule order_trans[of _ "b i"]) apply(subst ab(5)[THEN spec[where x=i]])
using s(1)[unfolded ksimplex_def] defer apply- apply(erule conjE)+ apply(drule_tac bspec[OF _ ab(2)])+ by auto
thus "a i < p" by auto
case goal2 hence "i ∈ reduced label n ` s" using s by auto then guess u unfolding image_iff .. note u=this
from goal2 have "i - 1 ∈ reduced label n ` s" using s by auto then guess v unfolding image_iff .. note v=this
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)
show "label u i ≠ label v i" using reduced_labelling[of label n u] reduced_labelling[of label n v]
unfolding u(2)[THEN sym] v(2)[THEN sym] using goal2 by auto
fix j assume j:"j∈{1..n}" show "a j ≤ u j ∧ u j ≤ a j + 1" "a j ≤ v j ∧ v j ≤ a j + 1"
using conjD[OF ab(4)[rule_format, OF u(1)]] and conjD[OF ab(4)[rule_format, OF v(1)]] apply-
apply(drule_tac[!] kle_imp_pointwise)+ apply(erule_tac[!] x=j in allE)+ unfolding ab(5)[rule_format] using j
by auto qed qed qed
subsection {* The main result for the unit cube. *}
lemma kuhn_labelling_lemma':
assumes "(∀x::nat=>real. P x --> P (f x))" "∀x. P x --> (∀i::nat. Q i --> 0 ≤ x i ∧ x i ≤ 1)"
shows "∃l. (∀x i. l x i ≤ (1::nat)) ∧
(∀x i. P x ∧ Q i ∧ (x i = 0) --> (l x i = 0)) ∧
(∀x i. P x ∧ Q i ∧ (x i = 1) --> (l x i = 1)) ∧
(∀x i. P x ∧ Q i ∧ (l x i = 0) --> x i ≤ f(x) i) ∧
(∀x i. P x ∧ Q i ∧ (l x i = 1) --> f(x) i ≤ x i)" proof-
have and_forall_thm:"!!P Q. (∀x. P x) ∧ (∀x. Q x) <-> (∀x. P x ∧ Q x)" by auto
have *:"∀x y::real. 0 ≤ x ∧ x ≤ 1 ∧ 0 ≤ y ∧ y ≤ 1 --> (x ≠ 1 ∧ x ≤ y ∨ x ≠ 0 ∧ y ≤ x)" by auto
show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
let ?R = "λy. (P x ∧ Q xa ∧ x xa = 0 --> y = (0::nat)) ∧
(P x ∧ Q xa ∧ x xa = 1 --> y = 1) ∧ (P x ∧ Q xa ∧ y = 0 --> x xa ≤ (f x) xa) ∧ (P x ∧ Q xa ∧ y = 1 --> (f x) xa ≤ x xa)"
{ assume "P x" "Q xa" hence "0 ≤ (f x) xa ∧ (f x) xa ≤ 1" using assms(2)[rule_format,of "f x" xa]
apply(drule_tac assms(1)[rule_format]) by auto }
hence "?R 0 ∨ ?R 1" by auto thus ?case by auto qed qed
lemma brouwer_cube: fixes f::"real^'n => real^'n"
assumes "continuous_on {0..1} f" "f ` {0..1} ⊆ {0..1}"
shows "∃x∈{0..1}. f x = x" apply(rule ccontr) proof-
def n ≡ "CARD('n)" have n:"1 ≤ n" "0 < n" "n ≠ 0" unfolding n_def by auto
assume "¬ (∃x∈{0..1}. f x = x)" hence *:"¬ (∃x∈{0..1}. f x - x = 0)" by auto
guess d apply(rule brouwer_compactness_lemma[OF compact_interval _ *])
apply(rule continuous_on_intros assms)+ . note d=this[rule_format]
have *:"∀x. x ∈ {0..1} --> f x ∈ {0..1}" "∀x. x ∈ {0..1::real^'n} --> (∀i. True --> 0 ≤ x $ i ∧ x $ i ≤ 1)"
using assms(2)[unfolded image_subset_iff Ball_def] unfolding mem_interval by auto
guess label using kuhn_labelling_lemma[OF *] apply-apply(erule exE,(erule conjE)+) . note label = this[rule_format]
have lem1:"∀x∈{0..1}.∀y∈{0..1}.∀i. label x i ≠ label y i
--> abs(f x $ i - x $ i) ≤ norm(f y - f x) + norm(y - x)" proof(rule,rule,rule,rule)
fix x y assume xy:"x∈{0..1::real^'n}" "y∈{0..1::real^'n}" fix i::'n assume i:"label x i ≠ label y i"
have *:"!!x y fx fy::real. (x ≤ fx ∧ fy ≤ y ∨ fx ≤ x ∧ y ≤ fy)
==> abs(fx - x) ≤ abs(fy - fx) + abs(y - x)" by auto
have "¦(f x - x) $ i¦ ≤ abs((f y - f x)$i) + abs((y - x)$i)" unfolding vector_minus_component
apply(rule *) apply(cases "label x i = 0") apply(rule disjI1,rule) prefer 3 proof(rule disjI2,rule)
assume lx:"label x i = 0" hence ly:"label y i = 1" using i label(1)[of y i] by auto
show "x $ i ≤ f x $ i" apply(rule label(4)[rule_format]) using xy lx by auto
show "f y $ i ≤ y $ i" apply(rule label(5)[rule_format]) using xy ly by auto next
assume "label x i ≠ 0" hence l:"label x i = 1" "label y i = 0"
using i label(1)[of x i] label(1)[of y i] by auto
show "f x $ i ≤ x $ i" apply(rule label(5)[rule_format]) using xy l by auto
show "y $ i ≤ f y $ i" apply(rule label(4)[rule_format]) using xy l by auto qed
also have "… ≤ norm (f y - f x) + norm (y - x)" apply(rule add_mono) by(rule component_le_norm)+
finally show "¦f x $ i - x $ i¦ ≤ norm (f y - f x) + norm (y - x)" unfolding vector_minus_component . qed
have "∃e>0. ∀x∈{0..1}. ∀y∈{0..1}. ∀z∈{0..1}. ∀i.
norm(x - z) < e ∧ norm(y - z) < e ∧ label x i ≠ label y i --> abs((f(z) - z)$i) < d / (real n)" proof-
have d':"d / real n / 8 > 0" apply(rule divide_pos_pos)+ using d(1) unfolding n_def by auto
have *:"uniformly_continuous_on {0..1} f" by(rule compact_uniformly_continuous[OF assms(1) compact_interval])
guess e using *[unfolded uniformly_continuous_on_def,rule_format,OF d'] apply-apply(erule exE,(erule conjE)+) .
note e=this[rule_format,unfolded dist_norm]
show ?thesis apply(rule_tac x="min (e/2) (d/real n/8)" in exI) apply(rule) defer
apply(rule,rule,rule,rule,rule) apply(erule conjE)+ proof-
show "0 < min (e / 2) (d / real n / 8)" using d' e by auto
fix x y z i assume as:"x ∈ {0..1}" "y ∈ {0..1}" "z ∈ {0..1}" "norm (x - z) < min (e / 2) (d / real n / 8)"
"norm (y - z) < min (e / 2) (d / real n / 8)" "label x i ≠ label y i"
have *:"!!z fz x fx n1 n2 n3 n4 d4 d::real. abs(fx - x) ≤ n1 + n2 ==> abs(fx - fz) ≤ n3 ==> abs(x - z) ≤ n4 ==>
n1 < d4 ==> n2 < 2 * d4 ==> n3 < d4 ==> n4 < d4 ==> (8 * d4 = d) ==> abs(fz - z) < d" by auto
show "¦(f z - z) $ i¦ < d / real n" unfolding vector_minus_component proof(rule *)
show "¦f x $ i - x $ i¦ ≤ norm (f y -f x) + norm (y - x)" apply(rule lem1[rule_format]) using as by auto
show "¦f x $ i - f z $ i¦ ≤ norm (f x - f z)" "¦x $ i - z $ i¦ ≤ norm (x - z)"
unfolding vector_minus_component[THEN sym] by(rule component_le_norm)+
have tria:"norm (y - x) ≤ norm (y - z) + norm (x - z)" using dist_triangle[of y x z,unfolded dist_norm]
unfolding norm_minus_commute by auto
also have "… < e / 2 + e / 2" apply(rule add_strict_mono) using as(4,5) by auto
finally show "norm (f y - f x) < d / real n / 8" apply- apply(rule e(2)) using as by auto
have "norm (y - z) + norm (x - z) < d / real n / 8 + d / real n / 8" apply(rule add_strict_mono) using as by auto
thus "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) by auto qed(insert as, auto) qed qed
then guess e apply-apply(erule exE,(erule conjE)+) . note e=this[rule_format]
guess p using real_arch_simple[of "1 + real n / e"] .. note p=this
have "1 + real n / e > 0" apply(rule add_pos_pos) defer apply(rule divide_pos_pos) using e(1) n by auto
hence "p > 0" using p by auto
guess b using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note b=this
def b' ≡ "inv_into {1..n} b"
have b':"bij_betw b' UNIV {1..n}" using bij_betw_inv_into[OF b] unfolding b'_def n_def by auto
have bb'[simp]:"!!i. b (b' i) = i" unfolding b'_def apply(rule f_inv_into_f) unfolding n_def using b
unfolding bij_betw_def by auto
have b'b[simp]:"!!i. i∈{1..n} ==> b' (b i) = i" unfolding b'_def apply(rule inv_into_f_eq)
using b unfolding n_def bij_betw_def by auto
have *:"!!x::nat. x=0 ∨ x=1 <-> x≤1" by auto
have q1:"0 < p" "0 < n" "∀x. (∀i∈{1..n}. x i ≤ p) -->
(∀i∈{1..n}. (label (χ i. real (x (b' i)) / real p) o b) i = 0 ∨ (label (χ i. real (x (b' i)) / real p) o b) i = 1)"
unfolding * using `p>0` `n>0` using label(1) by auto
have q2:"∀x. (∀i∈{1..n}. x i ≤ p) --> (∀i∈{1..n}. x i = 0 --> (label (χ i. real (x (b' i)) / real p) o b) i = 0)"
"∀x. (∀i∈{1..n}. x i ≤ p) --> (∀i∈{1..n}. x i = p --> (label (χ i. real (x (b' i)) / real p) o b) i = 1)"
apply(rule,rule,rule,rule) defer proof(rule,rule,rule,rule) fix x i
assume as:"∀i∈{1..n}. x i ≤ p" "i ∈ {1..n}"
{ assume "x i = p ∨ x i = 0"
have "(χ i. real (x (b' i)) / real p) ∈ {0..1}" unfolding mem_interval Cart_lambda_beta proof(rule,rule)
fix j::'n have j:"b' j ∈ {1..n}" using b' unfolding n_def bij_betw_def by auto
show "0 $ j ≤ real (x (b' j)) / real p" unfolding zero_index
apply(rule divide_nonneg_pos) using `p>0` using as(1)[rule_format,OF j] by auto
show "real (x (b' j)) / real p ≤ 1 $ j" unfolding one_index divide_le_eq_1
using as(1)[rule_format,OF j] by auto qed } note cube=this
{ assume "x i = p" thus "(label (χ i. real (x (b' i)) / real p) o b) i = 1" unfolding o_def
apply-apply(rule label(3)) using cube using as `p>0` by auto }
{ assume "x i = 0" thus "(label (χ i. real (x (b' i)) / real p) o b) i = 0" unfolding o_def
apply-apply(rule label(2)) using cube using as `p>0` by auto } qed
guess q apply(rule kuhn_lemma[OF q1 q2]) . note q=this
def z ≡ "χ i. real (q (b' i)) / real p"
have "∃i. d / real n ≤ abs((f z - z)$i)" proof(rule ccontr)
have "∀i. q (b' i) ∈ {0..<p}" using q(1) b'[unfolded bij_betw_def] by auto
hence "∀i. q (b' i) ∈ {0..p}" apply-apply(rule,erule_tac x=i in allE) by auto
hence "z∈{0..1}" unfolding z_def mem_interval unfolding one_index zero_index Cart_lambda_beta
apply-apply(rule,rule) apply(rule divide_nonneg_pos) using `p>0` unfolding divide_le_eq_1 by auto
hence d_fz_z:"d ≤ norm (f z - z)" apply(drule_tac d) .
case goal1 hence as:"∀i. ¦f z $ i - z $ i¦ < d / real n" using `n>0` by(auto simp add:not_le)
have "norm (f z - z) ≤ (∑i∈UNIV. ¦f z $ i - z $ i¦)" unfolding vector_minus_component[THEN sym] by(rule norm_le_l1)
also have "… < (∑(i::'n)∈UNIV. d / real n)" apply(rule setsum_strict_mono) using as by auto
also have "… = d" unfolding real_eq_of_nat n_def using n by auto
finally show False using d_fz_z by auto qed then guess i .. note i=this
have *:"b' i ∈ {1..n}" using b'[unfolded bij_betw_def] by auto
guess r using q(2)[rule_format,OF *] .. then guess s apply-apply(erule exE,(erule conjE)+) . note rs=this[rule_format]
have b'_im:"!!i. b' i ∈ {1..n}" using b' unfolding bij_betw_def by auto
def r' ≡ "χ i. real (r (b' i)) / real p"
have "!!i. r (b' i) ≤ p" apply(rule order_trans) apply(rule rs(1)[OF b'_im,THEN conjunct2])
using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
hence "r' ∈ {0..1::real^'n}" unfolding r'_def mem_interval Cart_lambda_beta one_index zero_index
apply-apply(rule,rule,rule divide_nonneg_pos)
using rs(1)[OF b'_im] q(1)[rule_format,OF b'_im] `p>0` by auto
def s' ≡ "χ i. real (s (b' i)) / real p"
have "!!i. s (b' i) ≤ p" apply(rule order_trans) apply(rule rs(2)[OF b'_im,THEN conjunct2])
using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
hence "s' ∈ {0..1::real^'n}" unfolding s'_def mem_interval Cart_lambda_beta one_index zero_index
apply-apply(rule,rule,rule divide_nonneg_pos)
using rs(1)[OF b'_im] q(1)[rule_format,OF b'_im] `p>0` by auto
have "z∈{0..1}" unfolding z_def mem_interval Cart_lambda_beta one_index zero_index
apply(rule,rule,rule divide_nonneg_pos) using q(1)[rule_format,OF b'_im] `p>0` by(auto intro:less_imp_le)
have *:"!!x. 1 + real x = real (Suc x)" by auto
{ have "(∑i∈UNIV. ¦real (r (b' i)) - real (q (b' i))¦) ≤ (∑(i::'n)∈UNIV. 1)"
apply(rule setsum_mono) using rs(1)[OF b'_im] by(auto simp add:* field_simps)
also have "… < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
by(auto simp add:field_simps)
finally have "(∑i∈UNIV. ¦real (r (b' i)) - real (q (b' i))¦) < e * real p" . } moreover
{ have "(∑i∈UNIV. ¦real (s (b' i)) - real (q (b' i))¦) ≤ (∑(i::'n)∈UNIV. 1)"
apply(rule setsum_mono) using rs(2)[OF b'_im] by(auto simp add:* field_simps)
also have "… < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
by(auto simp add:field_simps)
finally have "(∑i∈UNIV. ¦real (s (b' i)) - real (q (b' i))¦) < e * real p" . } ultimately
have "norm (r' - z) < e" "norm (s' - z) < e" unfolding r'_def s'_def z_def apply-
apply(rule_tac[!] le_less_trans[OF norm_le_l1]) using `p>0`
by(auto simp add:field_simps setsum_divide_distrib[THEN sym])
hence "¦(f z - z) $ i¦ < d / real n" apply-apply(rule e(2)[OF `r'∈{0..1}` `s'∈{0..1}` `z∈{0..1}`])
using rs(3) unfolding r'_def[symmetric] s'_def[symmetric] o_def bb' by auto
thus False using i by auto qed
subsection {* Retractions. *}
definition "retraction s t (r::real^'n=>real^'n) <->
t ⊆ s ∧ continuous_on s r ∧ (r ` s ⊆ t) ∧ (∀x∈t. r x = x)"
definition retract_of (infixl "retract'_of" 12) where
"(t retract_of s) <-> (∃r. retraction s t r)"
lemma retraction_idempotent: "retraction s t r ==> x ∈ s ==> 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::"(real^'n) set" and t::"(real^'m) set"
assumes "continuous_on t i" "i ` t ⊆ s" "continuous_on s r" "r ` s ⊆ t" "∀y∈t. r (i y) = y"
"∀f. continuous_on s f ∧ f ` s ⊆ s --> (∃x∈s. f x = x)" "continuous_on t g" "g ` t ⊆ t"
obtains y where "y∈t" "g y = y" proof-
have "∃x∈s. (i o g o r) x = x" apply(rule assms(6)[rule_format],rule)
apply(rule continuous_on_compose assms)+ apply((rule continuous_on_subset)?,rule assms)+
using assms(2,4,8) unfolding image_compose by(auto,blast)
then guess x .. note x = this hence *:"g (r x) ∈ t" using assms(4,8) by auto
have "r ((i o g o r) x) = r x" using x by auto
thus ?thesis apply(rule_tac that[of "r x"]) using x unfolding o_def
unfolding assms(5)[rule_format,OF *] using assms(4) by auto qed
lemma homeomorphic_fixpoint_property:
fixes s::"(real^'n) set" and t::"(real^'m) set" assumes "s homeomorphic t"
shows "(∀f. continuous_on s f ∧ f ` s ⊆ s --> (∃x∈s. f x = x)) <->
(∀g. continuous_on t g ∧ g ` t ⊆ t --> (∃y∈t. g y = y))" proof-
guess r using assms[unfolded homeomorphic_def homeomorphism_def] .. then guess i ..
thus ?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]) by auto qed
lemma retract_fixpoint_property:
assumes "t retract_of s" "∀f. continuous_on s f ∧ f ` s ⊆ s --> (∃x∈s. f x = x)" "continuous_on t g" "g ` t ⊆ t"
obtains y where "y ∈ t" "g y = y" proof- guess h using assms(1) unfolding retract_of_def ..
thus ?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 by auto qed
subsection {*So the Brouwer theorem for any set with nonempty interior. *}
lemma brouwer_weak: fixes f::"real^'n => real^'n"
assumes "compact s" "convex s" "interior s ≠ {}" "continuous_on s f" "f ` s ⊆ s"
obtains x where "x ∈ s" "f x = x" proof-
have *:"interior {0..1::real^'n} ≠ {}" unfolding interior_closed_interval interval_eq_empty by auto
have *:"{0..1::real^'n} homeomorphic s" using homeomorphic_convex_compact[OF convex_interval(1) compact_interval * assms(2,1,3)] .
have "∀f. continuous_on {0..1} f ∧ f ` {0..1} ⊆ {0..1} --> (∃x∈{0..1::real^'n}. f x = x)" using brouwer_cube by auto
thus ?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 by auto qed
subsection {* And in particular for a closed ball. *}
lemma brouwer_ball: fixes f::"real^'n => real^'n"
assumes "0 < e" "continuous_on (cball a e) f" "f ` (cball a e) ⊆ (cball a e)"
obtains x where "x ∈ cball a e" "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::"real^'n => real^'n"
assumes "compact s" "convex s" "s ≠ {}" "continuous_on s f" "f ` s ⊆ s"
obtains x where "x ∈ s" "f x = x" proof-
have "∃e>0. s ⊆ cball 0 e" using compact_imp_bounded[OF assms(1)] unfolding bounded_pos
apply(erule_tac exE,rule_tac x=b in exI) by(auto simp add: dist_norm)
then guess e apply-apply(erule exE,(erule conjE)+) . note e=this
have "∃x∈ cball 0 e. (f o closest_point s) x = x"
apply(rule_tac brouwer_ball[OF e(1), of 0 "f o 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)])
using closest_point_in_set[OF compact_imp_closed[OF assms(1)] assms(3)] apply - defer
using assms(5)[unfolded subset_eq] using e(2)[unfolded subset_eq mem_cball] by(auto simp add: dist_norm)
then guess x .. note x=this
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] by auto
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 * by auto qed
text {*So we get the no-retraction theorem. *}
lemma no_retraction_cball: assumes "0 < e"
shows "¬ (frontier(cball a e) retract_of (cball a e))" proof case goal1
have *:"!!xa. a - (2 *\<^sub>R a - xa) = -(a - xa)" using scaleR_left_distrib[of 1 1 a] by auto
guess x apply(rule retract_fixpoint_property[OF goal1, of "λx. scaleR 2 a - x"])
apply(rule,rule,erule conjE) apply(rule brouwer_ball[OF assms]) apply assumption+
apply(rule_tac x=x in bexI) apply assumption+ apply(rule continuous_on_intros)+
unfolding frontier_cball subset_eq Ball_def image_iff apply(rule,rule,erule bexE)
unfolding dist_norm apply(simp add: * norm_minus_commute) . note x = this
hence "scaleR 2 a = scaleR 1 x + scaleR 1 x" by(auto simp add:algebra_simps)
hence "a = x" unfolding scaleR_left_distrib[THEN sym] by auto
thus False using x using assms by auto qed
subsection {*Bijections between intervals. *}
definition "interval_bij = (λ (a,b) (u,v) (x::real^'n).
(χ i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
lemma interval_bij_affine:
"interval_bij (a,b) (u,v) = (λx. (χ i. (v$i - u$i) / (b$i - a$i) * x$i) +
(χ i. u$i - (v$i - u$i) / (b$i - a$i) * a$i))"
apply rule unfolding Cart_eq interval_bij_def vector_component_simps
by(auto simp add: field_simps add_divide_distrib[THEN sym])
lemma continuous_interval_bij:
"continuous (at x) (interval_bij (a,b::real^'n) (u,v))"
unfolding interval_bij_affine apply(rule continuous_intros)
apply(rule linear_continuous_at) unfolding linear_conv_bounded_linear[THEN sym]
unfolding linear_def unfolding Cart_eq unfolding Cart_lambda_beta defer
apply(rule continuous_intros) by(auto simp add:field_simps add_divide_distrib[THEN sym])
lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a,b) (u,v))"
apply(rule continuous_at_imp_continuous_on) by(rule, rule continuous_interval_bij)
(** move this **)
lemma divide_nonneg_nonneg:assumes "a ≥ 0" "b ≥ 0" shows "0 ≤ a / (b::real)"
apply(cases "b=0") defer apply(rule divide_nonneg_pos) using assms by auto
lemma in_interval_interval_bij: assumes "x ∈ {a..b}" "{u..v} ≠ {}"
shows "interval_bij (a,b) (u,v) x ∈ {u..v::real^'n}"
unfolding interval_bij_def split_conv mem_interval Cart_lambda_beta proof(rule,rule)
fix i::'n have "{a..b} ≠ {}" using assms by auto
hence *:"a$i ≤ b$i" "u$i ≤ v$i" using assms(2) unfolding interval_eq_empty not_ex apply-
apply(erule_tac[!] x=i in allE)+ by auto
have x:"a$i≤x$i" "x$i≤b$i" using assms(1)[unfolded mem_interval] by auto
have "0 ≤ (x $ i - a $ i) / (b $ i - a $ i) * (v $ i - u $ i)"
apply(rule mult_nonneg_nonneg) apply(rule divide_nonneg_nonneg)
using * x by(auto simp add:field_simps)
thus "u $ i ≤ u $ i + (x $ i - a $ i) / (b $ i - a $ i) * (v $ i - u $ i)" using * by auto
have "((x $ i - a $ i) / (b $ i - a $ i)) * (v $ i - u $ i) ≤ 1 * (v $ i - u $ i)"
apply(rule mult_right_mono) unfolding divide_le_eq_1 using * x by auto
thus "u $ i + (x $ i - a $ i) / (b $ i - a $ i) * (v $ i - u $ i) ≤ v $ i" using * by auto qed
lemma interval_bij_bij: assumes "∀i. a$i < b$i ∧ u$i < v$i"
shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
unfolding interval_bij_def split_conv Cart_eq Cart_lambda_beta
apply(rule,insert assms,erule_tac x=i in allE) by auto
end