--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy Tue Nov 17 18:52:30 2009 +0100
@@ -0,0 +1,1983 @@
+
+(* ========================================================================= *)
+(* 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
+ Translated to from HOL light: Robert Himmelmann, TU Muenchen *)
+
+header {* Results connected with topological dimension. *}
+
+theory Brouwer_Fixpoint
+ imports Convex_Euclidean_Space
+begin
+
+declare norm_scaleR[simp]
+
+lemma brouwer_compactness_lemma:
+ assumes "compact s" "continuous_on s f" "\<not> (\<exists>x\<in>s. (f x = (0::real^'n::finite)))"
+ obtains d where "0 < d" "\<forall>x\<in>s. d \<le> norm(f x)" proof(cases "s={}") case False
+ have "continuous_on s (norm \<circ> f)" by(rule continuous_on_intros continuous_on_norm assms(2))+
+ then obtain x where x:"x\<in>s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y"
+ using continuous_attains_inf[OF assms(1), of "norm \<circ> f"] and False unfolding o_def by auto
+ have "(norm \<circ> 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 "(\<forall>x::real^'n. P x \<longrightarrow> P (f x))" "\<forall>x. P x \<longrightarrow> (\<forall>i::'n. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"
+ shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)" proof-
+ have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto
+ have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto
+ show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
+ let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
+ (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
+ { assume "P x" "Q xa" hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1" using assms(2)[rule_format,of "f x" xa]
+ apply(drule_tac assms(1)[rule_format]) by auto }
+ hence "?R 0 \<or> ?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 \<inter> B = {}" "A \<union> B = C"
+ shows "setsum g C = setsum g A + setsum g B"
+ using setsum_Un_disjoint[OF assms(1-3)] and assms(4) by auto
+
+lemma kuhn_counting_lemma: assumes "finite faces" "finite simplices"
+ "\<forall>f\<in>faces. bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 1)"
+ "\<forall>f\<in>faces. \<not> bnd f \<longrightarrow> (card {s \<in> simplices. face f s} = 2)"
+ "\<forall>s\<in>simplices. compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 1)"
+ "\<forall>s\<in>simplices. \<not> compo s \<longrightarrow> (card {f \<in> faces. face f s \<and> compo' f} = 0) \<or>
+ (card {f \<in> faces. face f s \<and> compo' f} = 2)"
+ "odd(card {f \<in> faces. compo' f \<and> bnd f})"
+ shows "odd(card {s \<in> simplices. compo s})" proof-
+ have "\<And>x. {f\<in>faces. compo' f \<and> bnd f \<and> face f x} \<union> {f\<in>faces. compo' f \<and> \<not>bnd f \<and> face f x} = {f\<in>faces. compo' f \<and> face f x}"
+ "\<And>x. {f \<in> faces. compo' f \<and> bnd f \<and> face f x} \<inter> {f \<in> faces. compo' f \<and> \<not> bnd f \<and> face f x} = {}" by auto
+ hence lem1:"setsum (\<lambda>s. (card {f \<in> faces. face f s \<and> compo' f})) simplices =
+ setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f s}) simplices +
+ setsum (\<lambda>s. card {f \<in> {f \<in> faces. compo' f \<and> \<not> (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 (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> bnd f}. face f j}) simplices =
+ 1 * card {f \<in> faces. compo' f \<and> bnd f}"
+ "setsum (\<lambda>j. card {f \<in> {f \<in> faces. compo' f \<and> \<not> bnd f}. face f j}) simplices =
+ 2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}"
+ apply(rule_tac[!] setsum_multicount) using assms by auto
+ have lem3:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) simplices =
+ setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. compo s}+
+ setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s}"
+ apply(rule setsum_Un_disjoint') using assms(2) by auto
+ have lem4:"setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. compo s}
+ = setsum (\<lambda>s. 1) {s \<in> simplices. compo s}"
+ apply(rule setsum_cong2) using assms(5) by auto
+ have lem5: "setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f}) {s \<in> simplices. \<not> compo s} =
+ setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
+ {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 0)} +
+ setsum (\<lambda>s. card {f \<in> faces. face f s \<and> compo' f})
+ {s \<in> simplices. (\<not> compo s) \<and> (card {f \<in> faces. face f s \<and> compo' f} = 2)}"
+ apply(rule setsum_Un_disjoint') using assms(2,6) by auto
+ have *:"int (\<Sum>s\<in>{s \<in> simplices. compo s}. card {f \<in> faces. face f s \<and> compo' f}) =
+ int (card {f \<in> faces. compo' f \<and> bnd f} + 2 * card {f \<in> faces. compo' f \<and> \<not> bnd f}) -
+ int (card {s \<in> simplices. \<not> compo s \<and> card {f \<in> faces. face f s \<and> compo' f} = 2} * 2)"
+ using lem1[unfolded lem3 lem2 lem5] by auto
+ have even_minus_odd:"\<And>x y. even x \<Longrightarrow> odd (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
+ have odd_minus_even:"\<And>x y. odd x \<Longrightarrow> even (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
+ show ?thesis unfolding even_nat_def unfolding card_def and lem4[THEN sym] and *[unfolded card_def]
+ unfolding card_def[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 \<longleftrightarrow> (\<exists>!x. x \<in> s)" unfolding One_nat_def
+ apply rule apply(drule card_eq_SucD) defer apply(erule ex1E) proof-
+ fix x assume as:"x \<in> s" "\<forall>y. y \<in> s \<longrightarrow> 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 \<longleftrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. (z = x) \<or> (z = y)))" proof
+ assume "card s = 2" then obtain x y where obt:"s = {x, y}" "x\<noteq>y" unfolding numeral_2_eq_2 apply - apply(erule exE conjE|drule card_eq_SucD)+ by auto
+ show "\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. z = x \<or> z = y)" using obt by auto next
+ assume "\<exists>x\<in>s. \<exists>y\<in>s. x \<noteq> y \<and> (\<forall>z\<in>s. z = x \<or> z = y)" then guess x .. from this(2) guess y ..
+ with `x\<in>s` have *:"s = {x, y}" "x\<noteq>y" by auto
+ from this(2) show "card s = 2" unfolding * by auto qed
+
+lemma image_lemma_0: assumes "card {a\<in>s. f ` (s - {a}) = t - {b}} = n"
+ shows "card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> (f ` s' = t - {b})} = n" proof-
+ have *:"{s'. \<exists>a\<in>s. (s' = s - {a}) \<and> (f ` s' = t - {b})} = (\<lambda>a. s - {a}) ` {a\<in>s. f ` (s - {a}) = t - {b}}" by auto
+ 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 \<in> t"
+ shows "card {s'. \<exists>a\<in>s. s' = s - {a} \<and> f ` s' = t - {b}} = 1" proof-
+ obtain a where a:"b = f a" "a\<in>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 \<in> 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 \<subseteq> t" "f ` s \<noteq> t" "b \<in> t"
+ shows "(card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 0) \<or>
+ (card {s'. \<exists>a\<in>s. (s' = s - {a}) \<and> f ` s' = t - {b}} = 2)" proof(cases "{a\<in>s. f ` (s - {a}) = t - {b}} = {}")
+ case True thus ?thesis apply-apply(rule disjI1, rule image_lemma_0) using assms(1) by(auto simp add:card_0_eq)
+next let ?M = "{a\<in>s. f ` (s - {a}) = t - {b}}"
+ case False then obtain a where "a\<in>?M" by auto hence a:"a\<in>s" "f ` (s - {a}) = t - {b}" by auto
+ have "f a \<in> t - {b}" using a and assms by auto
+ hence "\<exists>c \<in> s - {a}. f a = f c" unfolding image_iff[symmetric] and a by auto
+ then obtain c where c:"c \<in> s" "a \<noteq> 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 \<in> f ` (s - {a})" then obtain y where y:"f y = x" "y\<in>s- {a}" by auto
+ thus "x \<in> f ` (s - {c})" unfolding image_iff apply(rule_tac x="if y = c then a else y" in bexI) using c a by auto qed auto
+ have "c\<in>?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\<in>?M`], rule bexI[OF _ `c\<in>?M`],rule,rule `a\<noteq>c`) proof(rule,unfold mem_Collect_eq,erule conjE)
+ fix z assume as:"z \<in> 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 \<or> z = c" proof(rule ccontr)
+ assume "\<not> (z = a \<or> z = c)" thus False using inj[unfolded inj_on_def,THEN bspec[where x=a],THEN bspec[where x=c]]
+ using `a\<in>s` `c\<in>s` `f a = f c` `a\<noteq>c` by auto qed qed qed
+
+subsection {* Combine this with the basic counting lemma. *}
+
+lemma kuhn_complete_lemma:
+ assumes "finite simplices"
+ "\<forall>f s. face f s \<longleftrightarrow> (\<exists>a\<in>s. f = s - {a})" "\<forall>s\<in>simplices. card s = n + 2" "\<forall>s\<in>simplices. (rl ` s) \<subseteq> {0..n+1}"
+ "\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. bnd f \<longrightarrow> (card {s\<in>simplices. face f s} = 1)"
+ "\<forall>f\<in> {f. \<exists>s\<in>simplices. face f s}. \<not>bnd f \<longrightarrow> (card {s\<in>simplices. face f s} = 2)"
+ "odd(card {f\<in>{f. \<exists>s\<in>simplices. face f s}. rl ` f = {0..n} \<and> bnd f})"
+ shows "odd (card {s\<in>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. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s - {a}} = (\<Union>s\<in>simplices. {f. \<exists>a\<in>s. (f = s - {a})})" by auto
+ have **: "\<forall>s\<in>simplices. card s = n + 2 \<and> finite s" using assms(3) by (auto intro: card_ge_0_finite)
+ show "finite {f. \<exists>s\<in>simplices. face f s}" unfolding assms(2)[rule_format] and *
+ apply(rule finite_UN_I[OF assms(1)]) using ** by auto
+ have *:"\<And> P f s. s\<in>simplices \<Longrightarrow> (f \<in> {f. \<exists>s\<in>simplices. \<exists>a\<in>s. f = s - {a}}) \<and>
+ (\<exists>a\<in>s. (f = s - {a})) \<and> P f \<longleftrightarrow> (\<exists>a\<in>s. (f = s - {a}) \<and> P f)" by auto
+ fix s assume s:"s\<in>simplices" let ?S = "{f \<in> {f. \<exists>s\<in>simplices. face f s}. face f s \<and> rl ` f = {0..n}}"
+ have "{0..n + 1} - {n + 1} = {0..n}" by auto
+ hence S:"?S = {s'. \<exists>a\<in>s. s' = s - {a} \<and> 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="\<lambda>x. rl ` x = {0..n}"] by auto
+ show "rl ` s = {0..n+1} \<Longrightarrow> card ?S = 1" "rl ` s \<noteq> {0..n+1} \<Longrightarrow> card ?S = 0 \<or> card ?S = 2" unfolding S
+ 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 \<longleftrightarrow> (\<exists>k\<subseteq>{1..n::nat}. (\<forall>j. y(j) = x(j) + (if j \<in> 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 \<and> kle n y x \<longleftrightarrow> (x = y)"
+ unfolding kle_def apply rule apply(rule ext) by auto
+
+lemma pointwise_minimal_pointwise_maximal: fixes s::"(nat\<Rightarrow>nat) set"
+ assumes "finite s" "s \<noteq> {}" "\<forall>x\<in>s. \<forall>y\<in>s. (\<forall>j. x j \<le> y j) \<or> (\<forall>j. y j \<le> x j)"
+ shows "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. a j \<le> x j" "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. x j \<le> a j"
+ using assms unfolding atomize_conj apply- proof(induct s rule:finite_induct)
+ fix x and F::"(nat\<Rightarrow>nat) set" assume as:"finite F" "x \<notin> F"
+ "\<lbrakk>F \<noteq> {}; \<forall>x\<in>F. \<forall>y\<in>F. (\<forall>j. x j \<le> y j) \<or> (\<forall>j. y j \<le> x j)\<rbrakk>
+ \<Longrightarrow> (\<exists>a\<in>F. \<forall>x\<in>F. \<forall>j. a j \<le> x j) \<and> (\<exists>a\<in>F. \<forall>x\<in>F. \<forall>j. x j \<le> a j)" "insert x F \<noteq> {}"
+ "\<forall>xa\<in>insert x F. \<forall>y\<in>insert x F. (\<forall>j. xa j \<le> y j) \<or> (\<forall>j. y j \<le> xa j)"
+ show "(\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. a j \<le> x j) \<and> (\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. x j \<le> a j)" proof(cases "F={}")
+ case True thus ?thesis apply-apply(rule,rule_tac[!] x=x in bexI) by auto next
+ case False obtain a b where a:"a\<in>insert x F" "\<forall>x\<in>F. \<forall>j. a j \<le> x j" and
+ b:"b\<in>insert x F" "\<forall>x\<in>F. \<forall>j. x j \<le> b j" using as(3)[OF False] using as(5) by auto
+ have "\<exists>a\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. a j \<le> x j"
+ using as(5)[rule_format,OF a(1) insertI1] apply- proof(erule disjE)
+ assume "\<forall>j. a j \<le> x j" thus ?thesis apply(rule_tac x=a in bexI) using a by auto next
+ assume "\<forall>j. x j \<le> 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 "\<exists>b\<in>insert x F. \<forall>x\<in>insert x F. \<forall>j. x j \<le> b j"
+ using as(5)[rule_format,OF b(1) insertI1] apply- proof(erule disjE)
+ assume "\<forall>j. x j \<le> b j" thus ?thesis apply(rule_tac x=b in bexI) using b by auto next
+ assume "\<forall>j. b j \<le> 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 \<Longrightarrow> (\<forall>j. x j \<le> y j)" unfolding kle_def by auto
+
+lemma pointwise_antisym: fixes x::"nat \<Rightarrow> nat"
+ shows "(\<forall>j. x j \<le> y j) \<and> (\<forall>j. y j \<le> x j) \<longleftrightarrow> (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 \<or> 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 \<noteq> y"
+ shows "\<forall>j. x j \<le> y j" "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> 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 "\<exists>k. 1 \<le> k \<and> k \<le> n \<and> 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' \<in> k) \<in> k" apply-apply(rule someI_ex) by auto
+ thus ?thesis apply(rule_tac x="SOME k'. k' \<in> k" in exI) using k by auto qed qed
+
+lemma kle_minimal: assumes "finite s" "s \<noteq> {}" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x"
+ shows "\<exists>a\<in>s. \<forall>x\<in>s. kle n a x" proof-
+ have "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. a j \<le> x j" apply(rule pointwise_minimal_pointwise_maximal(1)[OF assms(1-2)])
+ 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\<in>s"
+ show "kle n a x" using assms(3)[rule_format,OF a(1) `x\<in>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\<in>s`] by auto
+ thus ?thesis using kle_refl by auto qed qed(insert a, auto) qed
+
+lemma kle_maximal: assumes "finite s" "s \<noteq> {}" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x"
+ shows "\<exists>a\<in>s. \<forall>x\<in>s. kle n x a" proof-
+ have "\<exists>a\<in>s. \<forall>x\<in>s. \<forall>j. a j \<ge> x j" apply(rule pointwise_minimal_pointwise_maximal(2)[OF assms(1-2)])
+ 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\<in>s"
+ show "kle n x a" using assms(3)[rule_format,OF a(1) `x\<in>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\<in>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 \<noteq> y"
+ shows "1 \<le> card {k\<in>{1..n}. x k < y k}" proof-
+ guess i using kle_strict(2)[OF assms] ..
+ hence "card {i} \<le> card {k\<in>{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 \<or> kle n z x"
+ "m1 \<le> card {k\<in>{1..n}. x k < y k}"
+ "m2 \<le> card {k\<in>{1..n}. y k < z k}"
+ shows "kle n x z \<and> m1 + m2 \<le> card {k\<in>{1..n}. x k < z k}"
+ apply(rule,rule kle_trans[OF assms(1-3)]) proof-
+ have "\<And>j. x j < y j \<Longrightarrow> x j < z j" apply(rule less_le_trans) using kle_imp_pointwise[OF assms(2)] by auto moreover
+ have "\<And>j. y j < z j \<Longrightarrow> x j < z j" apply(rule le_less_trans) using kle_imp_pointwise[OF assms(1)] by auto ultimately
+ have *:"{k\<in>{1..n}. x k < y k} \<union> {k\<in>{1..n}. y k < z k} = {k\<in>{1..n}. x k < z k}" by auto
+ have **:"{k \<in> {1..n}. x k < y k} \<inter> {k \<in> {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 \<in> {1..n}" "x i < y i" "j \<in> {1..n}" "y j < z j" "\<not> i \<noteq> j"
+ guess kx using assms(1)[unfolded kle_def] .. note kx=this
+ have "x i < y i" using as by auto hence "i \<in> 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 \<in> 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:"\<And>P. finite {x\<in>{1..n::nat}. P x}" by auto
+ have "m1 + m2 \<le> card {k\<in>{1..n}. x k < y k} + card {k\<in>{1..n}. y k < z k}" using assms(4-5) by auto
+ also have "\<dots> \<le> card {k\<in>{1..n}. x k < z k}" unfolding card_Un_Int[OF fin fin] unfolding * ** by auto
+ finally show " m1 + m2 \<le> card {k \<in> {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 \<or> kle n z x" "m \<le> card {k\<in>{1..n}. y(k) < z(k)}"
+ shows "kle n x z \<and> m \<le> card {k\<in>{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 \<or> kle n z x" "m \<le> card {k\<in>{1..n}. x k < y k}"
+ shows "kle n x z \<and> m \<le> card {k\<in>{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" "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x"
+ shows "\<exists>x\<in>s. \<exists>y\<in>s. kle n x y \<and> m \<le> card {k\<in>{1..n}. x k < y k}" proof-
+have "finite s" "s\<noteq>{}" 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\<in>s" "\<forall>x\<in>s. kle n a x" using kle_minimal[of s n] by auto
+ show ?case proof(cases "s \<subseteq> {a}") case False
+ hence "card (s - {a}) = Suc m" "s - {a} \<noteq> {}" using card_Diff_singleton[OF _ a(1)] Suc(4) `finite s` by auto
+ then obtain x b where xb:"x\<in>s - {a}" "b\<in>s - {a}" "kle n x b" "m \<le> card {k \<in> {1..n}. x k < b k}"
+ using Suc(1)[of "s - {a}"] using Suc(5) `finite s` by auto
+ have "1 \<le> card {k \<in> {1..n}. a k < x k}" "m \<le> card {k \<in> {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 \<Longrightarrow> 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 \<le> y j" "y j \<le> x j + 1"
+ using assms[unfolded kle_def] by auto
+
+lemma kle_trans_2: assumes "kle n a b" "kle n b c" "\<forall>j. c j \<le> a j + 1" shows "kle n a c" proof-
+ 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 \<union> kk2" in exI) apply(rule) defer proof
+ fix i show "c i = a i + (if i \<in> kk1 \<union> kk2 then 1 else 0)" proof(cases "i\<in>kk1 \<union> kk2")
+ case True hence "c i \<ge> a i + (if i \<in> kk1 \<union> 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 \<le> a i + (if i \<in> kk1 \<union> 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 *:"\<And>c b x::nat. x \<le> c + 1 \<Longrightarrow> c \<le> b \<Longrightarrow> x \<le> b + 1" by auto
+ fix j show "x j \<le> b j + 1" apply(rule *)using kle_trans_1[OF assms(1),of j] kle_trans_1[OF assms(3), of j] by auto qed
+
+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 *:"\<And>c b x::nat. c \<le> x + 1 \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> x + 1" by auto
+ fix j show "b j \<le> x j + 1" apply(rule *) using kle_trans_1[OF assms(2),of j] kle_trans_1[OF assms(4), of j] by auto qed
+
+lemma kle_adjacent:
+ assumes "\<forall>j. b j = (if j = k then a(j) + 1 else a j)" "kle n a x" "kle n x b"
+ shows "(x = a) \<or> (x = b)" proof(cases "x k = a k")
+ case True show ?thesis apply(rule disjI1,rule ext) proof- fix j
+ have "x j \<le> 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 \<ge> 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 \<Rightarrow> nat) set) \<longleftrightarrow>
+ (card s = n + 1 \<and>
+ (\<forall>x\<in>s. \<forall>j. x(j) \<le> p) \<and>
+ (\<forall>x\<in>s. \<forall>j. j\<notin>{1..n} \<longrightarrow> (x j = p)) \<and>
+ (\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x))"
+
+lemma ksimplexI:"card s = n + 1 \<Longrightarrow> \<forall>x\<in>s. \<forall>j. x j \<le> p \<Longrightarrow> \<forall>x\<in>s. \<forall>j. j \<notin> {1..?n} \<longrightarrow> x j = ?p \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x \<Longrightarrow> ksimplex p n s"
+ unfolding ksimplex_def by auto
+
+lemma ksimplex_eq: "ksimplex p n (s::(nat \<Rightarrow> nat) set) \<longleftrightarrow>
+ (card s = n + 1 \<and> finite s \<and>
+ (\<forall>x\<in>s. \<forall>j. x(j) \<le> p) \<and>
+ (\<forall>x\<in>s. \<forall>j. j\<notin>{1..n} \<longrightarrow> (x j = p)) \<and>
+ (\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> 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 \<in> s" "b \<in> s"
+ "\<forall>x\<in>s. kle n a x \<and> kle n x b" "\<forall>i. b(i) = (if i \<in> {1..n} then a(i) + 1 else a(i))" 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\<noteq>{}" using assm by auto
+ obtain a where a:"a\<in>s" "\<forall>x\<in>s. kle n a x" using `s\<noteq>{}` assm using kle_minimal[of s n] by auto
+ obtain b where b:"b\<in>s" "\<forall>x\<in>s. kle n x b" using `s\<noteq>{}` assm using kle_maximal[of s n] by auto
+ obtain c d where c_d:"c\<in>s" "d\<in>s" "kle n c d" "n \<le> card {k \<in> {1..n}. c k < d k}"
+ using kle_range_induct[of s n n] using assm by auto
+ have "kle n c b \<and> n \<le> card {k \<in> {1..n}. c k < b k}" apply(rule kle_range_combine_r[where y=d]) using c_d a b by auto
+ hence "kle n a b \<and> n \<le> card {k\<in>{1..n}. a(k) < b(k)}" apply-apply(rule kle_range_combine_l[where y=c]) using a `c\<in>s` `b\<in>s` by auto
+ moreover have "card {1..n} \<ge> card {k\<in>{1..n}. a(k) < b(k)}" apply(rule card_mono) by auto
+ ultimately have *:"{k\<in>{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 \<in> {1..n} \<and> a i < b i then a i + 1 else a i)" proof(cases "i \<in> {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\<in>s` by auto
+ moreover have "b i = p" using assm and False `b\<in>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 \<noteq> 0" obtains a b where "a \<in> s" "b \<in> s" "a \<noteq> b"
+ "\<forall>x\<in>s. kle n a x \<and> kle n x b" "\<forall>i. b(i) = (if i \<in> {1..n} then a(i) + 1 else a(i))" proof-
+ obtain a b where ab:"a \<in> s" "b \<in> s" "\<forall>x\<in>s. kle n a x \<and> kle n x b" "\<forall>i. b(i) = (if i \<in> {1..n} then a(i) + 1 else a(i))"
+ apply(rule ksimplex_extrema[OF assms(1)]) by auto
+ have "a \<noteq> 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" "\<forall>x\<in>s. \<forall>j. x j \<le> p" "\<forall>x\<in>s. \<forall>j. j \<notin> {1..?n} \<longrightarrow> x j = p"
+ "\<forall>x\<in>s. \<forall>y\<in>s. kle n x y \<or> kle n y x" using assms unfolding ksimplex_eq by auto
+
+lemma ksimplex_successor:
+ assumes "ksimplex p n s" "a \<in> s"
+ shows "(\<forall>x\<in>s. kle n x a) \<or> (\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. y(j) = (if j = k then a(j) + 1 else a(j)))"
+proof(cases "\<forall>x\<in>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\<in>s" "\<not> kle n b a" "\<forall>x\<in>{x \<in> s. \<not> kle n x a}. kle n b x"
+ using kle_minimal[of "{x\<in>s. \<not> kle n x a}" n] and assm by auto
+ hence **:"1 \<le> card {k\<in>{1..n}. a k < b k}" apply- apply(rule kle_strict_set) using assm(6) and `a\<in>s` by(auto simp add:kle_refl)
+
+ let ?kle1 = "{x \<in> s. \<not> 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 \<in> s" "\<not> kle n c a" "d \<in> s" "\<not> kle n d a" "kle n c d" "card ?kle1 - 1 \<le> card {k \<in> {1..n}. c k < d k}"
+ using kle_range_induct[OF sizekle1, of n] using assm by auto
+
+ let ?kle2 = "{x \<in> s. kle n x a}"
+ have "card ?kle2 > 0" apply(rule ccontr) using assm(6)[rule_format,of a a] and `a\<in>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 \<in> s" "kle n e a" "f \<in> s" "kle n f a" "kle n e f" "card ?kle2 - 1 \<le> card {k \<in> {1..n}. e k < f k}"
+ using kle_range_induct[OF sizekle2, of n] using assm by auto
+
+ have "card {k\<in>{1..n}. a k < b k} = 1" proof(rule ccontr) case goal1
+ hence as:"card {k\<in>{1..n}. a k < b k} \<ge> 2" using ** by auto
+ have *:"finite ?kle2" "finite ?kle1" "?kle2 \<union> ?kle1 = s" "?kle2 \<inter> ?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 "\<dots> = 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 \<and> card {x \<in> s. kle n x a} - 1 \<le> card {k \<in> {1..n}. e k < a k}"
+ apply(rule kle_range_combine_r[where y=f]) using e_f using `a\<in>s` assm(6) by auto
+ moreover have "kle n b d \<and> card {x \<in> s. \<not> kle n x a} - 1 \<le> card {k \<in> {1..n}. b k < d k}"
+ apply(rule kle_range_combine_l[where y=c]) using c_d using assm(6) and b by auto
+ hence "kle n a d \<and> 2 + (card {x \<in> s. \<not> kle n x a} - 1) \<le> card {k \<in> {1..n}. a k < d k}" apply-
+ apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a\<in>s` `d\<in>s` apply- by blast+
+ ultimately have "kle n e d \<and> (card ?kle2 - 1) + (2 + (card ?kle1 - 1)) \<le> card {k\<in>{1..n}. e k < d k}" apply-
+ apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply - by blast+
+ moreover have "card {k \<in> {1..n}. e k < d k} \<le> 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\<in>s` `b\<in>s`] by auto
+ then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format]
+ have kkk:"k\<in>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\<in>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\<noteq>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\<in>s`, auto) qed
+
+lemma ksimplex_predecessor:
+ assumes "ksimplex p n s" "a \<in> s"
+ shows "(\<forall>x\<in>s. kle n a x) \<or> (\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. a(j) = (if j = k then y(j) + 1 else y(j)))"
+proof(cases "\<forall>x\<in>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\<in>s" "\<not> kle n a b" "\<forall>x\<in>{x \<in> s. \<not> kle n a x}. kle n x b"
+ using kle_maximal[of "{x\<in>s. \<not> kle n a x}" n] and assm by auto
+ hence **:"1 \<le> card {k\<in>{1..n}. a k > b k}" apply- apply(rule kle_strict_set) using assm(6) and `a\<in>s` by(auto simp add:kle_refl)
+
+ let ?kle1 = "{x \<in> s. \<not> 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 \<in> s" "\<not> kle n a c" "d \<in> s" "\<not> kle n a d" "kle n d c" "card ?kle1 - 1 \<le> card {k \<in> {1..n}. c k > d k}"
+ using kle_range_induct[OF sizekle1, of n] using assm by auto
+
+ let ?kle2 = "{x \<in> s. kle n a x}"
+ have "card ?kle2 > 0" apply(rule ccontr) using assm(6)[rule_format,of a a] and `a\<in>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 \<in> s" "kle n a e" "f \<in> s" "kle n a f" "kle n f e" "card ?kle2 - 1 \<le> card {k \<in> {1..n}. e k > f k}"
+ using kle_range_induct[OF sizekle2, of n] using assm by auto
+
+ have "card {k\<in>{1..n}. a k > b k} = 1" proof(rule ccontr) case goal1
+ hence as:"card {k\<in>{1..n}. a k > b k} \<ge> 2" using ** by auto
+ have *:"finite ?kle2" "finite ?kle1" "?kle2 \<union> ?kle1 = s" "?kle2 \<inter> ?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 "\<dots> = 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 \<and> card {x \<in> s. kle n a x} - 1 \<le> card {k \<in> {1..n}. e k > a k}"
+ apply(rule kle_range_combine_l[where y=f]) using e_f using `a\<in>s` assm(6) by auto
+ moreover have "kle n d b \<and> card {x \<in> s. \<not> kle n a x} - 1 \<le> card {k \<in> {1..n}. b k > d k}"
+ apply(rule kle_range_combine_r[where y=c]) using c_d using assm(6) and b by auto
+ hence "kle n d a \<and> (card {x \<in> s. \<not> kle n a x} - 1) + 2 \<le> card {k \<in> {1..n}. a k > d k}" apply-
+ apply(rule kle_range_combine[where y=b]) using as and b assm(6) `a\<in>s` `d\<in>s` by blast+
+ ultimately have "kle n d e \<and> (card ?kle1 - 1 + 2) + (card ?kle2 - 1) \<le> card {k\<in>{1..n}. e k > d k}" apply-
+ apply(rule kle_range_combine[where y=a]) using assm(6)[rule_format,OF `e\<in>s` `d\<in>s`] apply - by blast+
+ moreover have "card {k \<in> {1..n}. e k > d k} \<le> 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\<in>s` `b\<in>s`] by auto
+ then guess kk unfolding kle_def .. note kk_raw=this note kk=this[THEN conjunct2,rule_format]
+ have kkk:"k\<in>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\<in>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\<noteq>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\<in>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. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>UNIV - s. f x = d)} = n ^ m" (is "card (?M s) = _")
+ using assms apply - proof(induct m arbitrary: s)
+ have *:"{f. \<forall>x. f x = d} = {\<lambda>x. d}" apply(rule set_ext,rule)unfolding mem_Collect_eq apply(rule,rule ext) by auto
+ 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 = "(\<lambda>(b,g) x. if x = a then b else g x)" have *:"?M (insert a s0) = ?l ` {(b,g). b\<in>t \<and> g\<in>?M s0}"
+ apply(rule set_ext,rule) unfolding mem_Collect_eq image_iff apply(erule conjE)
+ apply(rule_tac x="(x a, \<lambda>y. if y\<in>s0 then x y else d)" in bexI) apply(rule ext) prefer 3 apply rule defer
+ apply(erule bexE,rule) unfolding mem_Collect_eq apply(erule splitE)+ apply(erule conjE)+ proof-
+ fix x xa xb xc y assume as:"x = (\<lambda>(b, g) x. if x = a then b else g x) xa" "xb \<in> UNIV - insert a s0" "xa = (xc, y)" "xc \<in> t"
+ "\<forall>x\<in>s0. y x \<in> t" "\<forall>x\<in>UNIV - s0. y x = d" thus "x xb = d" unfolding as by auto qed auto
+ have inj:"inj_on ?l {(b,g). b\<in>t \<and> g\<in>?M s0}" unfolding inj_on_def apply(rule,rule,rule) unfolding mem_Collect_eq apply(erule splitE conjE)+ proof-
+ 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. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>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. (\<forall>x\<in>s. f x \<in> t) \<and> (\<forall>x\<in>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. \<forall>x. f x = d} = {\<lambda>x. 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\<subseteq>{f. (\<forall>i\<in>{1..n}. f i \<in> {0..p}) \<and> (\<forall>i\<in>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" "\<forall>x\<in>f. x (n + 1) = p"
+ shows "(\<exists>s a. ksimplex p (n + 1) s \<and> a \<in> s \<and> (f = s - {a})) \<longleftrightarrow> ksimplex p n f" (is "?ls = ?rs") proof
+ 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 \<in>s - {a}" "y \<in>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 \<or> 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 \<notin> k" using xyp by auto
+ have "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
+ fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> 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 \<notin> k" using xyp by auto
+ hence "\<not> (\<exists>x\<in>k. x\<notin>{1..n})" apply-apply(rule ccontr) unfolding not_not apply(erule bexE) proof-
+ fix x assume as:"x \<in> k" "x \<notin> {1..n}" have "x \<noteq> 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\<in>s - {a}" "j\<notin>{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 \<equiv> "\<lambda>i. if i = (n + 1) then p - 1 else a i"
+ have "c\<notin>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 \<in> insert c f" thus "x j \<le> 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 \<notin> {1..n + 1} \<longrightarrow> x j = p" apply(cases "x=c") using x ab(1) rs(5) unfolding c_def by auto
+ { fix z assume z:"z \<in> insert c f" hence "kle (n + 1) c z" apply(cases "z = c") (*defer apply(rule kle_Suc)*) proof-
+ case False hence "z\<in>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 \<in> insert c f" show "kle (n + 1) x y \<or> kle (n + 1) y x" proof(cases "x = c \<or> y = c")
+ case False hence **:"x\<in>f" "y\<in>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 \<in> s" "j\<in>{1..n::nat}" "\<forall>x\<in>s - {a}. x j = q" "a0 \<in> s" "a1 \<in> s"
+ "\<forall>i. a1 i = ((if i\<in>{1..n} then a0 i + 1 else a0 i)::nat)"
+ shows "(a = a0) \<or> (a = a1)" proof- have *:"\<And>P A x y. \<forall>x\<in>A. P x \<Longrightarrow> x\<in>A \<Longrightarrow> y\<in>A \<Longrightarrow> P x \<and> P y" by auto
+ 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 \<in> s" "j\<in>{1..n::nat}" "\<forall>x\<in>s - {a}. x j = 0" "a0 \<in> s" "a1 \<in> s"
+ "\<forall>i. a1 i = ((if i\<in>{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 \<in> s" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = p" "a0 \<in> s" "a1 \<in> s"
+ "\<forall>i. a1 i = (if i\<in>{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 \<noteq> a0" hence *:"a0 \<in> 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 \<in> s" "n \<noteq> 0" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = 0"
+ shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1" proof-
+ have *:"\<And>s' a a'. s' - {a'} = s - {a} \<Longrightarrow> a' = a \<Longrightarrow> a' \<in> s' \<Longrightarrow> a \<in> s \<Longrightarrow> (s' = s)" by auto
+ have **:"\<And>s' a'. ksimplex p n s' \<Longrightarrow> a' \<in> s' \<Longrightarrow> s' - {a'} = s - {a} \<Longrightarrow> 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 \<in> s" "n \<noteq> 0" "j\<in>{1..n}" "\<forall>x\<in>s - {a}. x j = p"
+ shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1" proof-
+ have lem:"\<And>a a' s'. s' - {a'} = s - {a} \<Longrightarrow> a' = a \<Longrightarrow> a' \<in> s' \<Longrightarrow> a \<in> s \<Longrightarrow> s' = s" by auto
+ have lem:"\<And>s' a'. ksimplex p n s' \<Longrightarrow> a'\<in>s' \<Longrightarrow> s' - {a'} = s - {a} \<Longrightarrow> 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\<in>{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 \<in> s" "n \<noteq> 0" "~(\<exists>j\<in>{1..n}. \<forall>x\<in>s - {a}. x j = 0)" "~(\<exists>j\<in>{1..n}. \<forall>x\<in>s - {a}. x j = p)"
+ shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 2" (is "card ?A = 2") proof-
+ have lem1:"\<And>a a' s s'. s' - {a'} = s - {a} \<Longrightarrow> a' = a \<Longrightarrow> a' \<in> s' \<Longrightarrow> a \<in> s \<Longrightarrow> s' = s" by auto
+ have lem2:"\<And>a b. a\<in>s \<Longrightarrow> b\<noteq>a \<Longrightarrow> s \<noteq> insert b (s - {a})" proof case goal1
+ hence "a\<in>insert b (s - {a})" by auto hence "a\<in> 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 *:"\<And>P Q. (P \<or> Q) \<Longrightarrow> \<not> P \<Longrightarrow> Q" by auto
+ have "\<exists>x\<in>s. \<not> 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 "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>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\<in>s`
+ def a3 \<equiv> "\<lambda>j. if j = k then a1 j + 1 else a1 j"
+ have "a3 \<notin> s" proof assume "a3\<in>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 \<noteq> a0" "a3 \<noteq> a1" using a0a1 by auto
+ have "a2 \<noteq> a0" using k(2)[THEN spec[where x=k]] by auto
+ have lem3:"\<And>x. x\<in>(s - {a0}) \<Longrightarrow> kle n a2 x" proof(rule ccontr) case goal1 hence as:"x\<in>s" "x\<noteq>a0" by auto
+ have "kle n a2 x \<or> kle n x a2" using ksimplexD(6)[OF assms(1)] and as `a2\<in>s` by auto moreover
+ have "kle n a0 x" using a0a1(4) as by auto
+ ultimately have "x = a0 \<or> 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\<noteq>a0` `a3\<notin>s` `a0\<in>s` by(auto simp add:card_insert_if)
+ fix x assume x:"x \<in> insert a3 (s - {a0})"
+ show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3")
+ fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
+ fix j case True show "x j\<le>p" unfolding True proof(cases "j=k")
+ case False thus "a3 j \<le>p" unfolding True a3_def using `a1\<in>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 \<le> a4 k" using lem3[OF a4(1)[unfolded `a=a0`],THEN kle_imp_pointwise] by auto
+ also have "\<dots> < 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 \<le>p" unfolding a3_def unfolding a0a1(5)[rule_format]
+ using k(1) k(2)assms(5) using * by simp qed qed
+ show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j\<notin>{1..n}"
+ { 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\<in>s` j] by auto qed
+ fix y assume y:"y\<in>insert a3 (s - {a0})"
+ have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a0 \<Longrightarrow> kle n x a3" proof- case goal1
+ guess kk using a0a1(4)[rule_format,OF `x\<in>s`,THEN conjunct2,unfolded kle_def]
+ apply-apply(erule exE,erule conjE) . note kk=this
+ have "k\<notin>kk" proof assume "k\<in>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 \<le> 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 \<or> 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\<noteq>a3` by auto qed qed qed
+ hence "insert a3 (s - {a0}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
+ apply(rule_tac x="a3" in bexI) unfolding `a=a0` using `a3\<notin>s` by auto moreover
+ have "s \<in> ?A" using assms(1,2) by auto ultimately have "?A \<supseteq> {s, insert a3 (s - {a0})}" by auto
+ moreover have "?A \<subseteq> {s, insert a3 (s - {a0})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
+ fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>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 *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}"
+ hence "kle n a2 x" apply-apply(rule lem3) using `a=a0` by auto
+ hence "a2 k \<le> x k" apply(drule_tac kle_imp_pointwise) by auto moreover
+ have "x k \<le> 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 \<or> a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto
+ show "s' \<in> {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\<in>s'" using a' unfolding `a=a0` using a0a1 by auto
+ show "a_max \<in> s" proof(rule ccontr) assume "a_max\<notin>s"
+ hence "a_max = a'" using a' min_max by auto
+ thus False unfolding True using min_max by auto qed qed
+ hence "\<forall>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\<in>{1..n}") by auto qed
+ hence "s' = s" apply-apply(rule lem1[OF a'(2)]) using `a\<in>s` `a'\<in>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 \<in> s - {a0}" unfolding a'(2)[THEN sym,unfolded `a=a0`]
+ unfolding as using min_max(1-3) by auto
+ have "a2 \<noteq> a" unfolding `a=a0` using k(2)[rule_format,of k] by auto
+ hence "a2 \<in> s - {a}" using a2 by auto thus "a2 \<in> s'" unfolding a'(2)[THEN sym] by auto qed
+ hence "\<forall>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\<in>{1..n}") defer apply(cases "x=k")
+ using `k\<in>{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\<notin>s` by auto
+ thus ?thesis by auto qed qed
+ ultimately have *:"?A = {s, insert a3 (s - {a0})}" by blast
+ have "s \<noteq> insert a3 (s - {a0})" using `a3\<notin>s` by auto
+ hence ?thesis unfolding * by auto } moreover
+ { assume "a=a1"
+ have *:"\<And>P Q. (P \<or> Q) \<Longrightarrow> \<not> P \<Longrightarrow> Q" by auto
+ have "\<exists>x\<in>s. \<not> 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 "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>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\<in>s`
+ def a3 \<equiv> "\<lambda>j. if j = k then a0 j - 1 else a0 j"
+ have "a2 \<noteq> a1" using k(2)[THEN spec[where x=k]] by auto
+ have lem3:"\<And>x. x\<in>(s - {a1}) \<Longrightarrow> kle n x a2" proof(rule ccontr) case goal1 hence as:"x\<in>s" "x\<noteq>a1" by auto
+ have "kle n a2 x \<or> kle n x a2" using ksimplexD(6)[OF assms(1)] and as `a2\<in>s` by auto moreover
+ have "kle n x a1" using a0a1(4) as by auto
+ ultimately have "x = a2 \<or> 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 \<noteq> 0" proof-
+ guess a4 using assms(4)[unfolded bex_simps ball_simps,rule_format,OF `k\<in>{1..n}`] .. note a4=this
+ have "a4 k \<le> 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:"\<forall>j. a0 j = (if j=k then a3 j + 1 else a3 j)" unfolding a3_def by simp
+ have "\<not> 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 \<notin> s" using a0a1(4) by auto
+ hence "a3 \<noteq> a1" "a3 \<noteq> 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\<noteq>a0` `a3\<notin>s` `a1\<in>s` by(auto simp add:card_insert_if)
+ fix x assume x:"x \<in> insert a3 (s - {a1})"
+ show "\<forall>j. x j \<le> p" proof(rule,cases "x = a3")
+ fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
+ fix j case True show "x j\<le>p" unfolding True proof(cases "j=k")
+ case False thus "a3 j \<le>p" unfolding True a3_def using `a0\<in>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 \<le> a0 k" unfolding lem4[rule_format] by auto
+ also have "\<dots> \<le> p" using ksimplexD(4)[OF assms(1),rule_format,of a0 k] a0a1 by auto
+ finally show "a3 j \<le> p" unfolding True by auto qed qed
+ show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a3") fix j::nat assume j:"j\<notin>{1..n}"
+ { 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\<in>s` j] by auto qed
+ fix y assume y:"y\<in>insert a3 (s - {a1})"
+ have lem4:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a1 \<Longrightarrow> kle n a3 x" proof- case goal1 hence *:"x\<in>s - {a1}" by auto
+ 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\<in>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 \<or> 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\<noteq>a3` by auto qed qed qed
+ hence "insert a3 (s - {a1}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
+ apply(rule_tac x="a3" in bexI) unfolding `a=a1` using `a3\<notin>s` by auto moreover
+ have "s \<in> ?A" using assms(1,2) by auto ultimately have "?A \<supseteq> {s, insert a3 (s - {a1})}" by auto
+ moreover have "?A \<subseteq> {s, insert a3 (s - {a1})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
+ fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>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 *:"\<forall>x\<in>s' - {a'}. x k = a2 k" unfolding a' proof fix x assume x:"x\<in>s-{a}"
+ hence "kle n x a2" apply-apply(rule lem3) using `a=a1` by auto
+ hence "x k \<le> a2 k" apply(drule_tac kle_imp_pointwise) by auto moreover
+ { have "a2 k \<le> a0 k" using k(2)[rule_format,of k] unfolding a0a1(5)[rule_format] using k(1) by simp
+ also have "\<dots> \<le> x k" using a0a1(4)[rule_format,of x,THEN conjunct1,THEN kle_imp_pointwise] x by auto
+ finally have "a2 k \<le> x k" . } ultimately show "x k = a2 k" by auto qed
+ have **:"a'=a_min \<or> a'=a_max" apply(rule ksimplex_fix_plane[OF a'(1) k(1) *]) using min_max by auto
+ have "a2 \<noteq> 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 \<in> s' - {a'}" unfolding a' using a2 unfolding `a=a1` by auto
+ show "s' \<in> {s, insert a3 (s - {a1})}" proof(cases "a'=a_min")
+ case True have "a_max \<in> 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:"\<forall>i. a_max i = a2 i" by auto
+ have *:"\<forall>j. a2 j = (if j\<in>{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\<in>{1..n}",case_tac[!] "j=k") by auto qed
+ have "\<forall>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\<in>{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 \<in> s - {a1}" using min_max(1,3) unfolding a'(2)[THEN sym,unfolded `a=a1`] as by auto
+ thus "a_min \<in> s" by auto have "a0 \<in> s - {a1}" using a0a1(1-3) by auto thus "a0 \<in> s'"
+ unfolding a'(2)[THEN sym,unfolded `a=a1`] by auto qed
+ hence "\<forall>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\<in>s` `a'\<in>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 \<noteq> insert a3 (s - {a1})" using `a3\<notin>s` by auto
+ hence ?thesis unfolding * by auto } moreover
+ { assume as:"a\<noteq>a0" "a\<noteq>a1" have "\<not> (\<forall>x\<in>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 "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. a j = (if j = k then y j + 1 else y j)" using ksimplex_predecessor[OF assms(1-2)] by blast
+ then guess u .. from this(2) guess k .. note k = this[rule_format] note u = `u\<in>s`
+ have "\<not> (\<forall>x\<in>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 "\<exists>y\<in>s. \<exists>k\<in>{1..n}. \<forall>j. y j = (if j = k then a j + 1 else a j)" using ksimplex_successor[OF assms(1-2)] by blast
+ then guess v .. from this(2) guess l .. note l = this[rule_format] note v = `v\<in>s`
+ def a' \<equiv> "\<lambda>j. if j = l then u j + 1 else u j"
+ have kl:"k \<noteq> l" proof assume "k=l" have *:"\<And>P. (if P then (1::nat) else 0) \<noteq> 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'\<noteq>a" apply-apply rule unfolding expand_fun_eq unfolding a'_def k(2)
+ apply(erule_tac x=l in allE) by auto
+ have "a' \<notin> s" apply(rule) apply(drule ksimplexD(6)[OF assms(1),rule_format,OF `a\<in>s`]) proof(cases "kle n a a'")
+ 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:"\<And>x. kle n u x \<Longrightarrow> kle n x v \<Longrightarrow> (x = u) \<or> (x = a) \<or> (x = a') \<or> (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 \<noteq> 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 \<noteq> 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 \<noteq> u l" "x k \<noteq> 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\<noteq>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:"\<And>x. x\<in>s \<Longrightarrow> kle n v x \<Longrightarrow> 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\<in>s` by auto
+ have lem4:"\<And>x. x\<in>s \<Longrightarrow> kle n x u \<Longrightarrow> kle n x a'" apply(rule kle_between_l[of _ u _ v])
+ prefer 4 apply(rule kle_trans[OF _ uv]) defer apply(rule ksimplexD(6)[OF assms(1),rule_format])
+ using kle_uv `v\<in>s` by auto
+ have lem5:"\<And>x. x\<in>s \<Longrightarrow> x\<noteq>a \<Longrightarrow> kle n x a' \<or> kle n a' x" proof- case goal1 thus ?case
+ proof(cases "kle n v x \<or> 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\<in>s` `v\<in>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'\<noteq>a` `a'\<notin>s` `a\<in>s` by(auto simp add:card_insert_if)
+ fix x assume x:"x \<in> insert a' (s - {a})"
+ show "\<forall>j. x j \<le> p" proof(rule,cases "x = a'")
+ fix j case False thus "x j\<le>p" using x ksimplexD(4)[OF assms(1)] by auto next
+ fix j case True show "x j\<le>p" unfolding True proof(cases "j=l")
+ case False thus "a' j \<le>p" unfolding True a'_def using `u\<in>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\<noteq>l` by auto
+ have "u l + 1 \<le> p" unfolding *[THEN sym] using ksimplexD(4)[OF assms(1)] using `v\<in>s` by auto
+ thus "a' j \<le>p" unfolding a'_def True by auto qed qed
+ show "\<forall>j. j \<notin> {1..n} \<longrightarrow> x j = p" proof(rule,rule,cases "x=a'") fix j::nat assume j:"j\<notin>{1..n}"
+ { 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\<in>s` j] by auto qed
+ fix y assume y:"y\<in>insert a' (s - {a})"
+ show "kle n x y \<or> 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\<noteq>a'` by auto qed qed qed
+ hence "insert a' (s - {a}) \<in> ?A" unfolding mem_Collect_eq apply-apply(rule,assumption)
+ apply(rule_tac x="a'" in bexI) using aa' `a'\<notin>s` by auto moreover
+ have "s \<in> ?A" using assms(1,2) by auto ultimately have "?A \<supseteq> {s, insert a' (s - {a})}" by auto
+ moreover have "?A \<subseteq> {s, insert a' (s - {a})}" apply(rule) unfolding mem_Collect_eq proof(erule conjE)
+ fix s' assume as:"ksimplex p n s'" and "\<exists>b\<in>s'. s' - {b} = s - {a}"
+ from this(2) guess a'' .. note a''=this
+ have "u\<noteq>v" unfolding expand_fun_eq unfolding l(2) k(2) by auto
+ hence uv':"\<not> kle n v u" using uv using kle_antisym by auto
+ have "u\<noteq>a" "v\<noteq>a" unfolding expand_fun_eq k(2) l(2) by auto
+ hence uvs':"u\<in>s'" "v\<in>s'" using `u\<in>s` `v\<in>s` using a'' by auto
+ have lem6:"a \<in> s' \<or> a' \<in> s'" proof(cases "\<forall>x\<in>s'. kle n x u \<or> 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' \<or> w = a" using uxv[of w] uvs' w by auto thus ?thesis using w by auto next
+ case True have "\<not> (\<forall>x\<in>s'. kle n x u)" unfolding ball_simps apply(rule_tac x=v in bexI)
+ using uv `u\<noteq>v` unfolding kle_antisym[of n u v,THEN sym] using `v\<in>s'` by auto
+ hence "\<exists>y\<in>s'. \<exists>k\<in>{1..n}. \<forall>j. y j = (if j = k then u j + 1 else u j)" using ksimplex_successor[OF as `u\<in>s'`] by blast
+ then guess w .. note w=this from this(2) guess kk .. note kk=this[rule_format]
+ have "\<not> 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\<noteq>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\<noteq>l` by auto next
+ case False hence "kk\<noteq>k" using `k\<noteq>l` by auto thus False using *[THEN kle_imp_pointwise, unfolded l(2) kk k(2)]
+ apply(erule_tac x=k in allE) using `k\<noteq>l` by auto qed
+ thus ?thesis by auto qed
+ thus "s' \<in> {s, insert a' (s - {a})}" proof(cases "a\<in>s'")
+ case True hence "s' = s" apply-apply(rule lem1[OF a''(2)]) using a'' `a\<in>s` by auto
+ thus ?thesis by auto next case False hence "a'\<in>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'\<notin>s` by auto
+ thus ?thesis by auto qed qed
+ ultimately have *:"?A = {s, insert a' (s - {a})}" by blast
+ have "s \<noteq> insert a' (s - {a})" using `a'\<notin>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 "\<forall>s. ksimplex p (n + 1) s \<longrightarrow> (rl ` s \<subseteq>{0..n+1})"
+ "odd (card{f. \<exists>s a. ksimplex p (n + 1) s \<and> a \<in> s \<and> (f = s - {a}) \<and>
+ (rl ` f = {0 .. n}) \<and> ((\<exists>j\<in>{1..n+1}.\<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n+1}.\<forall>x\<in>f. x j = p))})"
+ shows "odd(card {s\<in>{s. ksimplex p (n + 1) s}. rl ` s = {0..n+1} })" proof-
+ have *:"\<And>x y. x = y \<Longrightarrow> odd (card x) \<Longrightarrow> odd (card y)" by auto
+ have *:"odd(card {f\<in>{f. \<exists>s\<in>{s. ksimplex p (n + 1) s}. (\<exists>a\<in>s. f = s - {a})}.
+ (rl ` f = {0..n}) \<and>
+ ((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or>
+ (\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p))})" 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\<in>s" "f = s - {a}"
+ let ?S = "{s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})}"
+ have S:"?S = {s'. ksimplex p (n + 1) s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})}" unfolding as by blast
+ { fix j assume j:"j \<in> {1..n + 1}" "\<forall>x\<in>f. x j = 0" thus "card {s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})} = 1" unfolding S
+ apply-apply(rule ksimplex_replace_0) apply(rule as)+ unfolding as by auto }
+ { fix j assume j:"j \<in> {1..n + 1}" "\<forall>x\<in>f. x j = p" thus "card {s. ksimplex p (n + 1) s \<and> (\<exists>a\<in>s. f = s - {a})} = 1" unfolding S
+ apply-apply(rule ksimplex_replace_1) apply(rule as)+ unfolding as by auto }
+ show "\<not> ((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p)) \<Longrightarrow> card ?S = 2"
+ 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\<Rightarrow>nat) =
+ (SOME k. k \<le> n \<and> (\<forall>i. 1\<le>i \<and> i<k+1 \<longrightarrow> label x i = 0) \<and> (k = n \<or> label x (k + 1) \<noteq> (0::nat)))"
+
+lemma reduced_labelling: shows "reduced label n x \<le> n" (is ?t1) and
+ "\<forall>i. 1\<le>i \<and> i < reduced label n x + 1 \<longrightarrow> (label x i = 0)" (is ?t2)
+ "(reduced label n x = n) \<or> (label x (reduced label n x + 1) \<noteq> 0)" (is ?t3) proof-
+ have num_WOP:"\<And>P k. P (k::nat) \<Longrightarrow> \<exists>n. P n \<and> (\<forall>m<n. \<not> P m)"
+ apply(drule ex_has_least_nat[where m="\<lambda>x. x"]) apply(erule exE,rule_tac x=x in exI) by auto
+ have *:"n \<le> n \<and> (label x (n + 1) \<noteq> 0 \<or> n = n)" by auto
+ then guess N apply(drule_tac num_WOP[of "\<lambda>j. j\<le>n \<and> (label x (j+1) \<noteq> 0 \<or> n = j)"]) apply(erule exE) . note N=this
+ have N':"N \<le> n" "\<forall>i. 1 \<le> i \<and> i < N + 1 \<longrightarrow> label x i = 0" "N = n \<or> label x (N + 1) \<noteq> 0" defer proof(rule,rule)
+ fix i assume i:"1\<le>i \<and> i<N+1" thus "label x i = 0" using N[THEN conjunct2,THEN spec[where x="i - 1"]] using N by auto qed(insert N, auto)
+ 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 \<Rightarrow> nat"
+ assumes "r \<le> n" "\<forall>i. 1 \<le> i \<and> i < r + 1 \<longrightarrow> (label x i = 0)" "(r = n) \<or> (label x (r + 1) \<noteq> 0)"
+ 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\<in>{1..n}" "label x j = 0" shows "reduced label n x \<noteq> j - 1"
+ using reduced_labelling[of label n x] using assms by fastsimp
+
+lemma reduced_labelling_1: assumes "j\<in>{1..n}" "label x j \<noteq> 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 \<noteq> 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 "\<forall>x\<in>f. \<forall>j\<in>{1..n+1}. x j = 0 \<longrightarrow> lab x j = 0"
+ "\<forall>x\<in>f. \<forall>j\<in>{1..n+1}. x j = p \<longrightarrow> lab x j = 1"
+ shows "((reduced lab (n + 1)) ` f = {0..n}) \<and> ((\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n+1}. \<forall>x\<in>f. x j = p)) \<longleftrightarrow>
+ ((reduced lab (n + 1)) ` f = {0..n}) \<and> (\<forall>x\<in>f. x (n + 1) = p)" (is "?l = ?r") proof
+ assume ?l (is "?as \<and> (?a \<or> ?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\<in>f"
+ have "reduced lab (n+1) x \<noteq> j - 1" using j apply-apply(rule reduced_labelling_0) defer apply(rule assms(1)[rule_format]) using x by auto }
+ moreover have "j - 1 \<in> {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 "\<forall>x\<in>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\<in>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 \<in> {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 "\<forall>x\<in>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" "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = 0) \<longrightarrow> (lab x j = 0)"
+ "\<forall>x. \<forall>j\<in>{1..n+1}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
+ "odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})"
+ shows "odd (card {s. ksimplex p (n+1) s \<and>((reduced lab (n+1)) ` s = {0..n+1})})" proof-
+ have *:"\<And>s t. odd (card s) \<Longrightarrow> s = t \<Longrightarrow> odd (card t)" "\<And>s f. (\<And>x. f x \<le> n +1 ) \<Longrightarrow> f ` s \<subseteq> {0..n+1}" by auto
+ 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 *:"\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = 0 \<longrightarrow> lab x j = 0" "\<forall>x\<in>f. \<forall>j\<in>{1..n + 1}. x j = p \<longrightarrow> lab x j = 1"
+ using assms(2-3) using as(1)[unfolded ksimplex_def] by auto
+ have allp:"\<forall>x\<in>f. x (n + 1) = p" using assms(2) using as(1)[unfolded ksimplex_def] by auto
+ { fix x assume "x\<in>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 "\<exists>s a. ksimplex p (n + 1) s \<and>
+ a \<in> s \<and> f = s - {a} \<and> reduced lab (n + 1) ` f = {0..n} \<and> ((\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p))" (is ?ex)
+ 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:"\<exists>s a. ksimplex p (n + 1) s \<and>
+ a \<in> s \<and> f = s - {a} \<and> reduced lab (n + 1) ` f = {0..n} \<and> ((\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = p))" (is ?ex)
+ then guess s .. then guess a apply-apply(erule exE,(erule conjE)+) . note sa=this
+ { fix x assume "x\<in>f" hence "reduced lab (n + 1) x \<in> 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 *:"\<forall>x\<in>f. x (n + 1) = p" proof(cases "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>f. x j = 0")
+ case True then guess j .. hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab (n + 1) x \<noteq> j - 1" apply-apply(rule reduced_labelling_0) apply assumption
+ apply(rule assms(2)[rule_format]) using sa(1)[unfolded ksimplex_def] unfolding sa by auto moreover
+ have "j - 1 \<in> {0..n}" using `j\<in>{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 "\<exists>j\<in>{1..n + 1}. \<forall>x\<in>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\<in>{1..n}" using j by auto
+ hence "\<And>x. x\<in>f \<Longrightarrow> reduced lab n x < j" apply(rule reduced_labelling_1) proof- fix x assume "x\<in>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 \<noteq> 0" by auto qed
+ moreover have "j\<in>{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" "\<forall>x. \<forall>j\<in>{1..Suc n}. (\<forall>j. x j \<le> p) \<and> (x j = 0) \<longrightarrow> (lab x j = 0)"
+ "\<forall>x. \<forall>j\<in>{1..Suc n}. (\<forall>j. x j \<le> p) \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
+ "odd (card {f. ksimplex p n f \<and> ((reduced lab n) ` f = {0..n})})"
+ shows "odd (card {s. ksimplex p (Suc n) s \<and>((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 \<longleftrightarrow> s = {(\<lambda>x. 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 = (\<lambda>x. 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" "\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n \<and> (x j = 0) \<longrightarrow> (lab x j = 0)"
+ "\<forall>x j. (\<forall>j. x(j) \<le> p) \<and> 1 \<le> j \<and> j \<le> n \<and> (x j = p) \<longrightarrow> (lab x j = 1)"
+ shows " odd (card {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})})" using assms proof(induct n)
+ let ?M = "\<lambda>n. {s. ksimplex p n s \<and> ((reduced lab n) ` s = {0..n})}"
+ { case 0 have *:"?M 0 = {{(\<lambda>x. 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)"
+ "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (label x i = (0::nat)) \<or> (label x i = 1))"
+ "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = 0) \<longrightarrow> (label x i = 0))"
+ "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. (x i = p) \<longrightarrow> (label x i = 1))"
+ obtains q where "\<forall>i\<in>{1..n}. q i < p"
+ "\<forall>i\<in>{1..n}. \<exists>r s. (\<forall>j\<in>{1..n}. q(j) \<le> r(j) \<and> r(j) \<le> q(j) + 1) \<and>
+ (\<forall>j\<in>{1..n}. q(j) \<le> s(j) \<and> s(j) \<le> q(j) + 1) \<and>
+ ~(label r i = label s i)" proof-
+ let ?A = "{s. ksimplex p n s \<and> reduced label n ` s = {0..n}}" have "n\<noteq>0" using assms by auto
+ have conjD:"\<And>P Q. P \<and> Q \<Longrightarrow> P" "\<And>P Q. P \<and> Q \<Longrightarrow> Q" by auto
+ have "odd (card ?A)" apply(rule kuhn_combinatorial[of p n label]) using assms by auto
+ hence "card ?A \<noteq> 0" apply-apply(rule ccontr) by auto hence "?A \<noteq> {}" unfolding card_eq_0_iff by auto
+ then obtain s where "s\<in>?A" by auto note s=conjD[OF this[unfolded mem_Collect_eq]]
+ guess a b apply(rule ksimplex_extrema_strong[OF s(1) `n\<noteq>0`]) . note ab=this
+ show ?thesis apply(rule that[of a]) proof(rule_tac[!] ballI) fix i assume "i\<in>{1..n}"
+ hence "a i + 1 \<le> 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 \<in> reduced label n ` s" using s by auto then guess u unfolding image_iff .. note u=this
+ from goal2 have "i - 1 \<in> 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 \<noteq> 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\<in>{1..n}" show "a j \<le> u j \<and> u j \<le> a j + 1" "a j \<le> v j \<and> v j \<le> a j + 1"
+ 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 "(\<forall>x::nat\<Rightarrow>real. P x \<longrightarrow> P (f x))" "\<forall>x. P x \<longrightarrow> (\<forall>i::nat. Q i \<longrightarrow> 0 \<le> x i \<and> x i \<le> 1)"
+ shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (x i = 0) \<longrightarrow> (l x i = 0)) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (x i = 1) \<longrightarrow> (l x i = 1)) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x i \<le> f(x) i) \<and>
+ (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x) i \<le> x i)" proof-
+ have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto
+ have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto
+ show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
+ let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x xa = 0 \<longrightarrow> y = (0::nat)) \<and>
+ (P x \<and> Q xa \<and> x xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x xa \<le> (f x) xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> (f x) xa \<le> x xa)"
+ { assume "P x" "Q xa" hence "0 \<le> (f x) xa \<and> (f x) xa \<le> 1" using assms(2)[rule_format,of "f x" xa]
+ apply(drule_tac assms(1)[rule_format]) by auto }
+ hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed
+
+lemma brouwer_cube: fixes f::"real^'n::finite \<Rightarrow> real^'n::finite"
+ assumes "continuous_on {0..1} f" "f ` {0..1} \<subseteq> {0..1}"
+ shows "\<exists>x\<in>{0..1}. f x = x" apply(rule ccontr) proof-
+ def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
+ assume "\<not> (\<exists>x\<in>{0..1}. f x = x)" hence *:"\<not> (\<exists>x\<in>{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 *:"\<forall>x. x \<in> {0..1} \<longrightarrow> f x \<in> {0..1}" "\<forall>x. x \<in> {0..1::real^'n} \<longrightarrow> (\<forall>i. True \<longrightarrow> 0 \<le> x $ i \<and> x $ i \<le> 1)"
+ 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:"\<forall>x\<in>{0..1}.\<forall>y\<in>{0..1}.\<forall>i. label x i \<noteq> label y i
+ \<longrightarrow> abs(f x $ i - x $ i) \<le> norm(f y - f x) + norm(y - x)" proof(rule,rule,rule,rule)
+ fix x y assume xy:"x\<in>{0..1::real^'n}" "y\<in>{0..1::real^'n}" fix i::'n assume i:"label x i \<noteq> label y i"
+ have *:"\<And>x y fx fy::real. (x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy)
+ \<Longrightarrow> abs(fx - x) \<le> abs(fy - fx) + abs(y - x)" by auto
+ have "\<bar>(f x - x) $ i\<bar> \<le> 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 \<le> f x $ i" apply(rule label(4)[rule_format]) using xy lx by auto
+ show "f y $ i \<le> y $ i" apply(rule label(5)[rule_format]) using xy ly by auto next
+ assume "label x i \<noteq> 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 \<le> x $ i" apply(rule label(5)[rule_format]) using xy l by auto
+ show "y $ i \<le> f y $ i" apply(rule label(4)[rule_format]) using xy l by auto qed
+ also have "\<dots> \<le> norm (f y - f x) + norm (y - x)" apply(rule add_mono) by(rule component_le_norm)+
+ finally show "\<bar>f x $ i - x $ i\<bar> \<le> norm (f y - f x) + norm (y - x)" unfolding vector_minus_component . qed
+ have "\<exists>e>0. \<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. \<forall>z\<in>{0..1}. \<forall>i.
+ norm(x - z) < e \<and> norm(y - z) < e \<and> label x i \<noteq> label y i \<longrightarrow> abs((f(z) - z)$i) < d / (real n)" proof-
+ 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 vector_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 \<in> {0..1}" "y \<in> {0..1}" "z \<in> {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 \<noteq> label y i"
+ have *:"\<And>z fz x fx n1 n2 n3 n4 d4 d::real. abs(fx - x) \<le> n1 + n2 \<Longrightarrow> abs(fx - fz) \<le> n3 \<Longrightarrow> abs(x - z) \<le> n4 \<Longrightarrow>
+ n1 < d4 \<Longrightarrow> n2 < 2 * d4 \<Longrightarrow> n3 < d4 \<Longrightarrow> n4 < d4 \<Longrightarrow> (8 * d4 = d) \<Longrightarrow> abs(fz - z) < d" by auto
+ show "\<bar>(f z - z) $ i\<bar> < d / real n" unfolding vector_minus_component proof(rule *)
+ show "\<bar>f x $ i - x $ i\<bar> \<le> norm (f y -f x) + norm (y - x)" apply(rule lem1[rule_format]) using as by auto
+ show "\<bar>f x $ i - f z $ i\<bar> \<le> norm (f x - f z)" "\<bar>x $ i - z $ i\<bar> \<le> norm (x - z)"
+ unfolding vector_minus_component[THEN sym] by(rule component_le_norm)+
+ have tria:"norm (y - x) \<le> norm (y - z) + norm (x - z)" using dist_triangle[of y x z,unfolded vector_dist_norm]
+ unfolding norm_minus_commute by auto
+ also have "\<dots> < 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' \<equiv> "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]:"\<And>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]:"\<And>i. i\<in>{1..n} \<Longrightarrow> b' (b i) = i" unfolding b'_def apply(rule inv_into_f_eq)
+ using b unfolding n_def bij_betw_def by auto
+ have *:"\<And>x::nat. x=0 \<or> x=1 \<longleftrightarrow> x\<le>1" by auto
+ have q1:"0 < p" "0 < n" "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow>
+ (\<forall>i\<in>{1..n}. (label (\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0 \<or> (label (\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1)"
+ unfolding * using `p>0` `n>0` using label(1) by auto
+ have q2:"\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = 0 \<longrightarrow> (label (\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 0)"
+ "\<forall>x. (\<forall>i\<in>{1..n}. x i \<le> p) \<longrightarrow> (\<forall>i\<in>{1..n}. x i = p \<longrightarrow> (label (\<chi> i. real (x (b' i)) / real p) \<circ> b) i = 1)"
+ apply(rule,rule,rule,rule) defer proof(rule,rule,rule,rule) fix x i
+ assume as:"\<forall>i\<in>{1..n}. x i \<le> p" "i \<in> {1..n}"
+ { assume "x i = p \<or> x i = 0"
+ have "(\<chi> i. real (x (b' i)) / real p) \<in> {0..1}" unfolding mem_interval Cart_lambda_beta proof(rule,rule)
+ fix j::'n have j:"b' j \<in> {1..n}" using b' unfolding n_def bij_betw_def by auto
+ show "0 $ j \<le> 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 \<le> 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 (\<chi> i. real (x (b' i)) / real p) \<circ> 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 (\<chi> i. real (x (b' i)) / real p) \<circ> 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 \<equiv> "\<chi> i. real (q (b' i)) / real p"
+ have "\<exists>i. d / real n \<le> abs((f z - z)$i)" proof(rule ccontr)
+ have "\<forall>i. q (b' i) \<in> {0..<p}" using q(1) b'[unfolded bij_betw_def] by auto
+ hence "\<forall>i. q (b' i) \<in> {0..p}" apply-apply(rule,erule_tac x=i in allE) by auto
+ hence "z\<in>{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 \<le> norm (f z - z)" apply(drule_tac d) .
+ case goal1 hence as:"\<forall>i. \<bar>f z $ i - z $ i\<bar> < d / real n" using `n>0` by(auto simp add:not_le)
+ have "norm (f z - z) \<le> (\<Sum>i\<in>UNIV. \<bar>f z $ i - z $ i\<bar>)" unfolding vector_minus_component[THEN sym] by(rule norm_le_l1)
+ also have "\<dots> < (\<Sum>(i::'n)\<in>UNIV. d / real n)" apply(rule setsum_strict_mono) using as by auto
+ also have "\<dots> = 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 \<in> {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:"\<And>i. b' i \<in> {1..n}" using b' unfolding bij_betw_def by auto
+ def r' \<equiv> "\<chi> i. real (r (b' i)) / real p"
+ have "\<And>i. r (b' i) \<le> p" apply(rule order_trans) apply(rule rs(1)[OF b'_im,THEN conjunct2])
+ using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
+ hence "r' \<in> {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' \<equiv> "\<chi> i. real (s (b' i)) / real p"
+ have "\<And>i. s (b' i) \<le> p" apply(rule order_trans) apply(rule rs(2)[OF b'_im,THEN conjunct2])
+ using q(1)[rule_format,OF b'_im] by(auto simp add: Suc_le_eq)
+ hence "s' \<in> {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\<in>{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 *:"\<And>x. 1 + real x = real (Suc x)" by auto
+ { have "(\<Sum>i\<in>UNIV. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'n)\<in>UNIV. 1)"
+ apply(rule setsum_mono) using rs(1)[OF b'_im] by(auto simp add:* field_simps)
+ also have "\<dots> < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
+ by(auto simp add:field_simps)
+ finally have "(\<Sum>i\<in>UNIV. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" . } moreover
+ { have "(\<Sum>i\<in>UNIV. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'n)\<in>UNIV. 1)"
+ apply(rule setsum_mono) using rs(2)[OF b'_im] by(auto simp add:* field_simps)
+ also have "\<dots> < e * real p" using p `e>0` `p>0` unfolding n_def real_of_nat_def
+ by(auto simp add:field_simps)
+ finally have "(\<Sum>i\<in>UNIV. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < 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 "\<bar>(f z - z) $ i\<bar> < d / real n" apply-apply(rule e(2)[OF `r'\<in>{0..1}` `s'\<in>{0..1}` `z\<in>{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::finite\<Rightarrow>real^'n) \<longleftrightarrow>
+ t \<subseteq> s \<and> continuous_on s r \<and> (r ` s \<subseteq> t) \<and> (\<forall>x\<in>t. r x = x)"
+
+definition retract_of (infixl "retract'_of" 12) where
+ "(t retract_of s) \<longleftrightarrow> (\<exists>r. retraction s t r)"
+
+lemma retraction_idempotent: "retraction s t r \<Longrightarrow> x \<in> s \<Longrightarrow> r(r x) = r x"
+ unfolding retraction_def by auto
+
+subsection {*preservation of fixpoints under (more general notion of) retraction. *}
+
+lemma invertible_fixpoint_property: fixes s::"(real^'n::finite) set" and t::"(real^'m::finite) set"
+ assumes "continuous_on t i" "i ` t \<subseteq> s" "continuous_on s r" "r ` s \<subseteq> t" "\<forall>y\<in>t. r (i y) = y"
+ "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)" "continuous_on t g" "g ` t \<subseteq> t"
+ obtains y where "y\<in>t" "g y = y" proof-
+ have "\<exists>x\<in>s. (i \<circ> g \<circ> 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) \<in> t" using assms(4,8) by auto
+ have "r ((i \<circ> g \<circ> 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::finite) set" and t::"(real^'m::finite) set" assumes "s homeomorphic t"
+ shows "(\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)) \<longleftrightarrow>
+ (\<forall>g. continuous_on t g \<and> g ` t \<subseteq> t \<longrightarrow> (\<exists>y\<in>t. g y = y))" proof-
+ 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" "\<forall>f. continuous_on s f \<and> f ` s \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. f x = x)" "continuous_on t g" "g ` t \<subseteq> t"
+ obtains y where "y \<in> 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::finite \<Rightarrow> real^'n::finite"
+ assumes "compact s" "convex s" "interior s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
+ obtains x where "x \<in> s" "f x = x" proof-
+ have *:"interior {0..1::real^'n} \<noteq> {}" 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 "\<forall>f. continuous_on {0..1} f \<and> f ` {0..1} \<subseteq> {0..1} \<longrightarrow> (\<exists>x\<in>{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::finite \<Rightarrow> real^'n::finite"
+ assumes "0 < e" "continuous_on (cball a e) f" "f ` (cball a e) \<subseteq> (cball a e)"
+ obtains x where "x \<in> 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 HOMEOMORPHIC_CONVEX_COMPACT theorem, just using
+ a scaling and translation to put the set inside the unit cube. *}
+
+lemma brouwer: fixes f::"real^'n::finite \<Rightarrow> real^'n::finite"
+ assumes "compact s" "convex s" "s \<noteq> {}" "continuous_on s f" "f ` s \<subseteq> s"
+ obtains x where "x \<in> s" "f x = x" proof-
+ have "\<exists>e>0. s \<subseteq> cball 0 e" using compact_imp_bounded[OF assms(1)] unfolding bounded_pos
+ apply(erule_tac exE,rule_tac x=b in exI) by(auto simp add: vector_dist_norm)
+ then guess e apply-apply(erule exE,(erule conjE)+) . note e=this
+ have "\<exists>x\<in> cball 0 e. (f \<circ> closest_point s) x = x"
+ apply(rule_tac brouwer_ball[OF e(1), of 0 "f \<circ> closest_point s"]) apply(rule continuous_on_compose )
+ apply(rule continuous_on_closest_point[OF assms(2) compact_imp_closed[OF assms(1)] assms(3)])
+ apply(rule continuous_on_subset[OF assms(4)])
+ 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:vector_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 "\<not> (frontier(cball a e) retract_of (cball a e))" proof case goal1
+ have *:"\<And>xa. a - (2 *\<^sub>R a - xa) = -(a - xa)" using scaleR_left_distrib[of 1 1 a] by auto
+ guess x apply(rule retract_fixpoint_property[OF goal1, of "\<lambda>x. 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 vector_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:group_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 = (\<lambda> (a,b) (u,v) (x::real^'n::finite).
+ (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
+
+lemma interval_bij_affine:
+ "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
+ (\<chi> 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:group_simps field_simps add_divide_distrib[THEN sym])
+
+lemma continuous_interval_bij:
+ "continuous (at x) (interval_bij (a,b::real^'n::finite) (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 \<ge> 0" "b \<ge> 0" shows "0 \<le> a / (b::real)"
+ apply(cases "b=0") defer apply(rule divide_nonneg_pos) using assms by auto
+
+lemma in_interval_interval_bij: assumes "x \<in> {a..b}" "{u..v} \<noteq> {}"
+ shows "interval_bij (a,b) (u,v) x \<in> {u..v::real^'n::finite}"
+ unfolding interval_bij_def split_conv mem_interval Cart_lambda_beta proof(rule,rule)
+ fix i::'n have "{a..b} \<noteq> {}" using assms by auto
+ hence *:"a$i \<le> b$i" "u$i \<le> v$i" using assms(2) unfolding interval_eq_empty not_ex apply-
+ apply(erule_tac[!] x=i in allE)+ by auto
+ have x:"a$i\<le>x$i" "x$i\<le>b$i" using assms(1)[unfolded mem_interval] by auto
+ have "0 \<le> (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 \<le> 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) \<le> 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) \<le> v $ i" using * by auto qed
+
+lemma interval_bij_bij: assumes "\<forall>i. a$i < b$i \<and> 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
+
+subsection {*Fashoda meet theorem. *}
+
+lemma infnorm_2: "infnorm (x::real^2) = max (abs(x$1)) (abs(x$2))"
+ unfolding infnorm_def UNIV_2 apply(rule Sup_eq) by auto
+
+lemma infnorm_eq_1_2: "infnorm (x::real^2) = 1 \<longleftrightarrow>
+ (abs(x$1) \<le> 1 \<and> abs(x$2) \<le> 1 \<and> (x$1 = -1 \<or> x$1 = 1 \<or> x$2 = -1 \<or> x$2 = 1))"
+ unfolding infnorm_2 by auto
+
+lemma infnorm_eq_1_imp: assumes "infnorm (x::real^2) = 1" shows "abs(x$1) \<le> 1" "abs(x$2) \<le> 1"
+ using assms unfolding infnorm_eq_1_2 by auto
+
+lemma fashoda_unit: fixes f g::"real^1 \<Rightarrow> real^2"
+ assumes "f ` {- 1..1} \<subseteq> {- 1..1}" "g ` {- 1..1} \<subseteq> {- 1..1}"
+ "continuous_on {- 1..1} f" "continuous_on {- 1..1} g"
+ "f (- 1)$1 = - 1" "f 1$1 = 1" "g (- 1) $2 = -1" "g 1 $2 = 1"
+ shows "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. f s = g t" proof(rule ccontr)
+ case goal1 note as = this[unfolded bex_simps,rule_format]
+ def sqprojection \<equiv> "\<lambda>z::real^2. (inverse (infnorm z)) *\<^sub>R z"
+ def negatex \<equiv> "\<lambda>x::real^2. (vector [-(x$1), x$2])::real^2"
+ have lem1:"\<forall>z::real^2. infnorm(negatex z) = infnorm z"
+ unfolding negatex_def infnorm_2 vector_2 by auto
+ have lem2:"\<forall>z. z\<noteq>0 \<longrightarrow> infnorm(sqprojection z) = 1" unfolding sqprojection_def
+ unfolding infnorm_mul[unfolded smult_conv_scaleR] unfolding abs_inverse real_abs_infnorm
+ unfolding infnorm_eq_0[THEN sym] by auto
+ let ?F = "(\<lambda>w::real^2. (f \<circ> vec1 \<circ> (\<lambda>x. x$1)) w - (g \<circ> vec1 \<circ> (\<lambda>x. x$2)) w)"
+ have *:"\<And>i. vec1 ` (\<lambda>x::real^2. x $ i) ` {- 1..1} = {- 1..1::real^1}"
+ apply(rule set_ext) unfolding image_iff Bex_def mem_interval apply rule defer
+ apply(rule_tac x="dest_vec1 x" in exI) apply rule apply(rule_tac x="vec (dest_vec1 x)" in exI)
+ by(auto simp add: dest_vec1_def[THEN sym])
+ { fix x assume "x \<in> (\<lambda>w. (f \<circ> vec1 \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> vec1 \<circ> (\<lambda>x. x $ 2)) w) ` {- 1..1::real^2}"
+ then guess w unfolding image_iff .. note w = this
+ hence "x \<noteq> 0" using as[of "vec1 (w$1)" "vec1 (w$2)"] unfolding mem_interval by auto} note x0=this
+ have 21:"\<And>i::2. i\<noteq>1 \<Longrightarrow> i=2" using UNIV_2 by auto
+ have 1:"{- 1<..<1::real^2} \<noteq> {}" unfolding interval_eq_empty by auto
+ have 2:"continuous_on {- 1..1} (negatex \<circ> sqprojection \<circ> ?F)" apply(rule continuous_on_intros continuous_on_component continuous_on_vec1)+
+ prefer 2 apply(rule continuous_on_intros continuous_on_component continuous_on_vec1)+ unfolding *
+ apply(rule assms)+ apply(rule continuous_on_compose,subst sqprojection_def)
+ apply(rule continuous_on_mul ) apply(rule continuous_at_imp_continuous_on,rule) apply(rule continuous_at_inv[unfolded o_def])
+ apply(rule continuous_at_infnorm) unfolding infnorm_eq_0 defer apply(rule continuous_on_id) apply(rule linear_continuous_on) proof-
+ show "bounded_linear negatex" apply(rule bounded_linearI') unfolding Cart_eq proof(rule_tac[!] allI) fix i::2 and x y::"real^2" and c::real
+ show "negatex (x + y) $ i = (negatex x + negatex y) $ i" "negatex (c *s x) $ i = (c *s negatex x) $ i"
+ apply-apply(case_tac[!] "i\<noteq>1") prefer 3 apply(drule_tac[1-2] 21)
+ unfolding negatex_def by(auto simp add:vector_2 ) qed qed(insert x0, auto)
+ have 3:"(negatex \<circ> sqprojection \<circ> ?F) ` {- 1..1} \<subseteq> {- 1..1}" unfolding subset_eq apply rule proof-
+ case goal1 then guess y unfolding image_iff .. note y=this have "?F y \<noteq> 0" apply(rule x0) using y(1) by auto
+ hence *:"infnorm (sqprojection (?F y)) = 1" unfolding y o_def apply- by(rule lem2[rule_format])
+ have "infnorm x = 1" unfolding *[THEN sym] y o_def by(rule lem1[rule_format])
+ thus "x\<in>{- 1..1}" unfolding mem_interval infnorm_2 apply- apply rule
+ proof-case goal1 thus ?case apply(cases "i=1") defer apply(drule 21) by auto qed qed
+ guess x apply(rule brouwer_weak[of "{- 1..1::real^2}" "negatex \<circ> sqprojection \<circ> ?F"])
+ apply(rule compact_interval convex_interval)+ unfolding interior_closed_interval
+ apply(rule 1 2 3)+ . note x=this
+ have "?F x \<noteq> 0" apply(rule x0) using x(1) by auto
+ hence *:"infnorm (sqprojection (?F x)) = 1" unfolding o_def by(rule lem2[rule_format])
+ have nx:"infnorm x = 1" apply(subst x(2)[THEN sym]) unfolding *[THEN sym] o_def by(rule lem1[rule_format])
+ have "\<forall>x i. x \<noteq> 0 \<longrightarrow> (0 < (sqprojection x)$i \<longleftrightarrow> 0 < x$i)" "\<forall>x i. x \<noteq> 0 \<longrightarrow> ((sqprojection x)$i < 0 \<longleftrightarrow> x$i < 0)"
+ apply- apply(rule_tac[!] allI impI)+ proof- fix x::"real^2" and i::2 assume x:"x\<noteq>0"
+ have "inverse (infnorm x) > 0" using x[unfolded infnorm_pos_lt[THEN sym]] by auto
+ thus "(0 < sqprojection x $ i) = (0 < x $ i)" "(sqprojection x $ i < 0) = (x $ i < 0)"
+ unfolding sqprojection_def vector_component_simps Cart_nth.scaleR real_scaleR_def
+ unfolding zero_less_mult_iff mult_less_0_iff by(auto simp add:field_simps) qed
+ note lem3 = this[rule_format]
+ have x1:"vec1 (x $ 1) \<in> {- 1..1::real^1}" "vec1 (x $ 2) \<in> {- 1..1::real^1}" using x(1) unfolding mem_interval by auto
+ hence nz:"f (vec1 (x $ 1)) - g (vec1 (x $ 2)) \<noteq> 0" unfolding right_minus_eq apply-apply(rule as) by auto
+ have "x $ 1 = -1 \<or> x $ 1 = 1 \<or> x $ 2 = -1 \<or> x $ 2 = 1" using nx unfolding infnorm_eq_1_2 by auto
+ thus False proof- fix P Q R S
+ presume "P \<or> Q \<or> R \<or> S" "P\<Longrightarrow>False" "Q\<Longrightarrow>False" "R\<Longrightarrow>False" "S\<Longrightarrow>False" thus False by auto
+ next assume as:"x$1 = 1" hence "vec1 (x$1) = 1" unfolding Cart_eq by auto
+ hence *:"f (vec1 (x $ 1)) $ 1 = 1" using assms(6) by auto
+ have "sqprojection (f (vec1 (x$1)) - g (vec1 (x$2))) $ 1 < 0"
+ using x(2)[unfolded o_def Cart_eq,THEN spec[where x=1]]
+ unfolding as negatex_def vector_2 by auto moreover
+ from x1 have "g (vec1 (x $ 2)) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
+ ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval
+ apply(erule_tac x=1 in allE) by auto
+ next assume as:"x$1 = -1" hence "vec1 (x$1) = - 1" unfolding Cart_eq by auto
+ hence *:"f (vec1 (x $ 1)) $ 1 = - 1" using assms(5) by auto
+ have "sqprojection (f (vec1 (x$1)) - g (vec1 (x$2))) $ 1 > 0"
+ using x(2)[unfolded o_def Cart_eq,THEN spec[where x=1]]
+ unfolding as negatex_def vector_2 by auto moreover
+ from x1 have "g (vec1 (x $ 2)) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
+ ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval
+ apply(erule_tac x=1 in allE) by auto
+ next assume as:"x$2 = 1" hence "vec1 (x$2) = 1" unfolding Cart_eq by auto
+ hence *:"g (vec1 (x $ 2)) $ 2 = 1" using assms(8) by auto
+ have "sqprojection (f (vec1 (x$1)) - g (vec1 (x$2))) $ 2 > 0"
+ using x(2)[unfolded o_def Cart_eq,THEN spec[where x=2]]
+ unfolding as negatex_def vector_2 by auto moreover
+ from x1 have "f (vec1 (x $ 1)) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
+ ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval
+ apply(erule_tac x=2 in allE) by auto
+ next assume as:"x$2 = -1" hence "vec1 (x$2) = - 1" unfolding Cart_eq by auto
+ hence *:"g (vec1 (x $ 2)) $ 2 = - 1" using assms(7) by auto
+ have "sqprojection (f (vec1 (x$1)) - g (vec1 (x$2))) $ 2 < 0"
+ using x(2)[unfolded o_def Cart_eq,THEN spec[where x=2]]
+ unfolding as negatex_def vector_2 by auto moreover
+ from x1 have "f (vec1 (x $ 1)) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
+ ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval
+ apply(erule_tac x=2 in allE) by auto qed(auto) qed
+
+lemma fashoda_unit_path: fixes f ::"real^1 \<Rightarrow> real^2" and g ::"real^1 \<Rightarrow> real^2"
+ assumes "path f" "path g" "path_image f \<subseteq> {- 1..1}" "path_image g \<subseteq> {- 1..1}"
+ "(pathstart f)$1 = -1" "(pathfinish f)$1 = 1" "(pathstart g)$2 = -1" "(pathfinish g)$2 = 1"
+ obtains z where "z \<in> path_image f" "z \<in> path_image g" proof-
+ note assms=assms[unfolded path_def pathstart_def pathfinish_def path_image_def]
+ def iscale \<equiv> "\<lambda>z::real^1. inverse 2 *\<^sub>R (z + 1)"
+ have isc:"iscale ` {- 1..1} \<subseteq> {0..1}" unfolding iscale_def by(auto simp add:dest_vec1_add dest_vec1_neg)
+ have "\<exists>s\<in>{- 1..1}. \<exists>t\<in>{- 1..1}. (f \<circ> iscale) s = (g \<circ> iscale) t" proof(rule fashoda_unit)
+ show "(f \<circ> iscale) ` {- 1..1} \<subseteq> {- 1..1}" "(g \<circ> iscale) ` {- 1..1} \<subseteq> {- 1..1}"
+ using isc and assms(3-4) unfolding image_compose by auto
+ have *:"continuous_on {- 1..1} iscale" unfolding iscale_def by(rule continuous_on_intros)+
+ show "continuous_on {- 1..1} (f \<circ> iscale)" "continuous_on {- 1..1} (g \<circ> iscale)"
+ apply-apply(rule_tac[!] continuous_on_compose[OF *]) apply(rule_tac[!] continuous_on_subset[OF _ isc])
+ by(rule assms)+ have *:"(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" unfolding Cart_eq by auto
+ show "(f \<circ> iscale) (- 1) $ 1 = - 1" "(f \<circ> iscale) 1 $ 1 = 1" "(g \<circ> iscale) (- 1) $ 2 = -1" "(g \<circ> iscale) 1 $ 2 = 1"
+ unfolding o_def iscale_def using assms by(auto simp add:*) qed
+ then guess s .. from this(2) guess t .. note st=this
+ show thesis apply(rule_tac z="f (iscale s)" in that)
+ using st `s\<in>{- 1..1}` unfolding o_def path_image_def image_iff apply-
+ apply(rule_tac x="iscale s" in bexI) prefer 3 apply(rule_tac x="iscale t" in bexI)
+ using isc[unfolded subset_eq, rule_format] by auto qed
+
+lemma fashoda: fixes b::"real^2"
+ assumes "path f" "path g" "path_image f \<subseteq> {a..b}" "path_image g \<subseteq> {a..b}"
+ "(pathstart f)$1 = a$1" "(pathfinish f)$1 = b$1"
+ "(pathstart g)$2 = a$2" "(pathfinish g)$2 = b$2"
+ obtains z where "z \<in> path_image f" "z \<in> path_image g" proof-
+ fix P Q S presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" "Q \<Longrightarrow> thesis" "S \<Longrightarrow> thesis" thus thesis by auto
+next have "{a..b} \<noteq> {}" using assms(3) using path_image_nonempty by auto
+ hence "a \<le> b" unfolding interval_eq_empty vector_less_eq_def by(auto simp add: not_less)
+ thus "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" unfolding vector_less_eq_def forall_2 by auto
+next assume as:"a$1 = b$1" have "\<exists>z\<in>path_image g. z$2 = (pathstart f)$2" apply(rule connected_ivt_component)
+ apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
+ unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"]
+ unfolding pathstart_def by(auto simp add: vector_less_eq_def) then guess z .. note z=this
+ have "z \<in> {a..b}" using z(1) assms(4) unfolding path_image_def by blast
+ hence "z = f 0" unfolding Cart_eq forall_2 unfolding z(2) pathstart_def
+ using assms(3)[unfolded path_image_def subset_eq mem_interval,rule_format,of "f 0" 1]
+ unfolding mem_interval apply(erule_tac x=1 in allE) using as by auto
+ thus thesis apply-apply(rule that[OF _ z(1)]) unfolding path_image_def by auto
+next assume as:"a$2 = b$2" have "\<exists>z\<in>path_image f. z$1 = (pathstart g)$1" apply(rule connected_ivt_component)
+ apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
+ unfolding assms using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"]
+ unfolding pathstart_def by(auto simp add: vector_less_eq_def) then guess z .. note z=this
+ have "z \<in> {a..b}" using z(1) assms(3) unfolding path_image_def by blast
+ hence "z = g 0" unfolding Cart_eq forall_2 unfolding z(2) pathstart_def
+ using assms(4)[unfolded path_image_def subset_eq mem_interval,rule_format,of "g 0" 2]
+ unfolding mem_interval apply(erule_tac x=2 in allE) using as by auto
+ thus thesis apply-apply(rule that[OF z(1)]) unfolding path_image_def by auto
+next assume as:"a $ 1 < b $ 1 \<and> a $ 2 < b $ 2"
+ have int_nem:"{- 1..1::real^2} \<noteq> {}" unfolding interval_eq_empty by auto
+ guess z apply(rule fashoda_unit_path[of "interval_bij (a,b) (- 1,1) \<circ> f" "interval_bij (a,b) (- 1,1) \<circ> g"])
+ unfolding path_def path_image_def pathstart_def pathfinish_def
+ apply(rule_tac[1-2] continuous_on_compose) apply(rule assms[unfolded path_def] continuous_on_interval_bij)+
+ unfolding subset_eq apply(rule_tac[1-2] ballI)
+ proof- fix x assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}"
+ then guess y unfolding image_iff .. note y=this
+ show "x \<in> {- 1..1}" unfolding y o_def apply(rule in_interval_interval_bij)
+ using y(1) using assms(3)[unfolded path_image_def subset_eq] int_nem by auto
+ next fix x assume "x \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}"
+ then guess y unfolding image_iff .. note y=this
+ show "x \<in> {- 1..1}" unfolding y o_def apply(rule in_interval_interval_bij)
+ using y(1) using assms(4)[unfolded path_image_def subset_eq] int_nem by auto
+ next show "(interval_bij (a, b) (- 1, 1) \<circ> f) 0 $ 1 = -1"
+ "(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1"
+ "(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1"
+ "(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1" unfolding interval_bij_def Cart_lambda_beta vector_component_simps o_def split_conv
+ unfolding assms[unfolded pathstart_def pathfinish_def] using as by auto qed note z=this
+ from z(1) guess zf unfolding image_iff .. note zf=this
+ from z(2) guess zg unfolding image_iff .. note zg=this
+ have *:"\<forall>i. (- 1) $ i < (1::real^2) $ i \<and> a $ i < b $ i" unfolding forall_2 using as by auto
+ show thesis apply(rule_tac z="interval_bij (- 1,1) (a,b) z" in that)
+ apply(subst zf) defer apply(subst zg) unfolding o_def interval_bij_bij[OF *] path_image_def
+ using zf(1) zg(1) by auto qed
+
+subsection {*Some slightly ad hoc lemmas I use below*}
+
+lemma segment_vertical: fixes a::"real^2" assumes "a$1 = b$1"
+ shows "x \<in> closed_segment a b \<longleftrightarrow> (x$1 = a$1 \<and> x$1 = b$1 \<and>
+ (a$2 \<le> x$2 \<and> x$2 \<le> b$2 \<or> b$2 \<le> x$2 \<and> x$2 \<le> a$2))" (is "_ = ?R")
+proof-
+ let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
+ { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
+ unfolding Cart_eq forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+ { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
+ { fix b a assume "b + u * a > a + u * b"
+ hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
+ hence "b \<ge> a" apply(drule_tac mult_less_imp_less_left) using u by auto
+ hence "u * a \<le> u * b" apply-apply(rule mult_left_mono[OF _ u(3)])
+ using u(3-4) by(auto simp add:field_simps) } note * = this
+ { fix a b assume "u * b > u * a" hence "(1 - u) * a \<le> (1 - u) * b" apply-apply(rule mult_left_mono)
+ apply(drule mult_less_imp_less_left) using u by auto
+ hence "a + u * b \<le> b + u * a" by(auto simp add:field_simps) } note ** = this
+ thus ?R unfolding u assms using u by(auto simp add:field_simps not_le intro:* **) }
+ { assume ?R thus ?L proof(cases "x$2 = b$2")
+ case True thus ?L apply(rule_tac x="(x$2 - a$2) / (b$2 - a$2)" in exI) unfolding assms True
+ using `?R` by(auto simp add:field_simps)
+ next case False thus ?L apply(rule_tac x="1 - (x$2 - b$2) / (a$2 - b$2)" in exI) unfolding assms using `?R`
+ by(auto simp add:field_simps)
+ qed } qed
+
+lemma segment_horizontal: fixes a::"real^2" assumes "a$2 = b$2"
+ shows "x \<in> closed_segment a b \<longleftrightarrow> (x$2 = a$2 \<and> x$2 = b$2 \<and>
+ (a$1 \<le> x$1 \<and> x$1 \<le> b$1 \<or> b$1 \<le> x$1 \<and> x$1 \<le> a$1))" (is "_ = ?R")
+proof-
+ let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
+ { presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
+ unfolding Cart_eq forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+ { assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
+ { fix b a assume "b + u * a > a + u * b"
+ hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
+ hence "b \<ge> a" apply(drule_tac mult_less_imp_less_left) using u by auto
+ hence "u * a \<le> u * b" apply-apply(rule mult_left_mono[OF _ u(3)])
+ using u(3-4) by(auto simp add:field_simps) } note * = this
+ { fix a b assume "u * b > u * a" hence "(1 - u) * a \<le> (1 - u) * b" apply-apply(rule mult_left_mono)
+ apply(drule mult_less_imp_less_left) using u by auto
+ hence "a + u * b \<le> b + u * a" by(auto simp add:field_simps) } note ** = this
+ thus ?R unfolding u assms using u by(auto simp add:field_simps not_le intro:* **) }
+ { assume ?R thus ?L proof(cases "x$1 = b$1")
+ case True thus ?L apply(rule_tac x="(x$1 - a$1) / (b$1 - a$1)" in exI) unfolding assms True
+ using `?R` by(auto simp add:field_simps)
+ next case False thus ?L apply(rule_tac x="1 - (x$1 - b$1) / (a$1 - b$1)" in exI) unfolding assms using `?R`
+ by(auto simp add:field_simps)
+ qed } qed
+
+subsection {*useful Fashoda corollary pointed out to me by Tom Hales. *}
+
+lemma fashoda_interlace: fixes a::"real^2"
+ assumes "path f" "path g"
+ "path_image f \<subseteq> {a..b}" "path_image g \<subseteq> {a..b}"
+ "(pathstart f)$2 = a$2" "(pathfinish f)$2 = a$2"
+ "(pathstart g)$2 = a$2" "(pathfinish g)$2 = a$2"
+ "(pathstart f)$1 < (pathstart g)$1" "(pathstart g)$1 < (pathfinish f)$1"
+ "(pathfinish f)$1 < (pathfinish g)$1"
+ obtains z where "z \<in> path_image f" "z \<in> path_image g"
+proof-
+ have "{a..b} \<noteq> {}" using path_image_nonempty using assms(3) by auto
+ note ab=this[unfolded interval_eq_empty not_ex forall_2 not_less]
+ have "pathstart f \<in> {a..b}" "pathfinish f \<in> {a..b}" "pathstart g \<in> {a..b}" "pathfinish g \<in> {a..b}"
+ using pathstart_in_path_image pathfinish_in_path_image using assms(3-4) by auto
+ note startfin = this[unfolded mem_interval forall_2]
+ let ?P1 = "linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2]) +++
+ linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f) +++ f +++
+ linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2]) +++
+ linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2])"
+ let ?P2 = "linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g) +++ g +++
+ linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1]) +++
+ linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1]) +++
+ linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3])"
+ let ?a = "vector[a$1 - 2, a$2 - 3]"
+ let ?b = "vector[b$1 + 2, b$2 + 3]"
+ have P1P2:"path_image ?P1 = path_image (linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2])) \<union>
+ path_image (linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f)) \<union> path_image f \<union>
+ path_image (linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2])) \<union>
+ path_image (linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2]))"
+ "path_image ?P2 = path_image(linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g)) \<union> path_image g \<union>
+ path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) \<union>
+ path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union>
+ path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
+ by(auto simp add: pathstart_join pathfinish_join path_image_join path_image_linepath path_join path_linepath)
+ have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:vector_less_eq_def forall_2 vector_2)
+ guess z apply(rule fashoda[of ?P1 ?P2 ?a ?b])
+ unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2 proof-
+ show "path ?P1" "path ?P2" using assms by(auto simp add: pathstart_join pathfinish_join path_join)
+ have "path_image ?P1 \<subseteq> {?a .. ?b}" unfolding P1P2 path_image_linepath apply(rule Un_least)+ defer 3
+ apply(rule_tac[1-4] convex_interval(1)[unfolded convex_contains_segment,rule_format])
+ unfolding mem_interval forall_2 vector_2 using ab startfin abab assms(3)
+ using assms(9-) unfolding assms by(auto simp add:field_simps)
+ thus "path_image ?P1 \<subseteq> {?a .. ?b}" .
+ have "path_image ?P2 \<subseteq> {?a .. ?b}" unfolding P1P2 path_image_linepath apply(rule Un_least)+ defer 2
+ apply(rule_tac[1-4] convex_interval(1)[unfolded convex_contains_segment,rule_format])
+ unfolding mem_interval forall_2 vector_2 using ab startfin abab assms(4)
+ using assms(9-) unfolding assms by(auto simp add:field_simps)
+ thus "path_image ?P2 \<subseteq> {?a .. ?b}" .
+ show "a $ 1 - 2 = a $ 1 - 2" "b $ 1 + 2 = b $ 1 + 2" "pathstart g $ 2 - 3 = a $ 2 - 3" "b $ 2 + 3 = b $ 2 + 3"
+ by(auto simp add: assms)
+ qed note z=this[unfolded P1P2 path_image_linepath]
+ show thesis apply(rule that[of z]) proof-
+ have "(z \<in> closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) \<or>
+ z \<in> closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f)) \<or>
+ z \<in> closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) \<or>
+ z \<in> closed_segment (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]) \<Longrightarrow>
+ (((z \<in> closed_segment (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g)) \<or>
+ z \<in> closed_segment (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1])) \<or>
+ z \<in> closed_segment (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1])) \<or>
+ z \<in> closed_segment (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]) \<Longrightarrow> False"
+ apply(simp only: segment_vertical segment_horizontal vector_2) proof- case goal1 note as=this
+ have "pathfinish f \<in> {a..b}" using assms(3) pathfinish_in_path_image[of f] by auto
+ hence "1 + b $ 1 \<le> pathfinish f $ 1 \<Longrightarrow> False" unfolding mem_interval forall_2 by auto
+ hence "z$1 \<noteq> pathfinish f$1" using as(2) using assms ab by(auto simp add:field_simps)
+ moreover have "pathstart f \<in> {a..b}" using assms(3) pathstart_in_path_image[of f] by auto
+ hence "1 + b $ 1 \<le> pathstart f $ 1 \<Longrightarrow> False" unfolding mem_interval forall_2 by auto
+ hence "z$1 \<noteq> pathstart f$1" using as(2) using assms ab by(auto simp add:field_simps)
+ ultimately have *:"z$2 = a$2 - 2" using goal1(1) by auto
+ have "z$1 \<noteq> pathfinish g$1" using as(2) using assms ab by(auto simp add:field_simps *)
+ moreover have "pathstart g \<in> {a..b}" using assms(4) pathstart_in_path_image[of g] by auto
+ note this[unfolded mem_interval forall_2]
+ hence "z$1 \<noteq> pathstart g$1" using as(1) using assms ab by(auto simp add:field_simps *)
+ ultimately have "a $ 2 - 1 \<le> z $ 2 \<and> z $ 2 \<le> b $ 2 + 3 \<or> b $ 2 + 3 \<le> z $ 2 \<and> z $ 2 \<le> a $ 2 - 1"
+ using as(2) unfolding * assms by(auto simp add:field_simps)
+ thus False unfolding * using ab by auto
+ qed hence "z \<in> path_image f \<or> z \<in> path_image g" using z unfolding Un_iff by blast
+ hence z':"z\<in>{a..b}" using assms(3-4) by auto
+ have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow> (z = pathstart f \<or> z = pathfinish f)"
+ unfolding Cart_eq forall_2 assms by auto
+ with z' show "z\<in>path_image f" using z(1) unfolding Un_iff mem_interval forall_2 apply-
+ apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
+ have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow> (z = pathstart g \<or> z = pathfinish g)"
+ unfolding Cart_eq forall_2 assms by auto
+ with z' show "z\<in>path_image g" using z(2) unfolding Un_iff mem_interval forall_2 apply-
+ apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
+ qed qed
+
+(** The Following still needs to be translated. Maybe I will do that later.
+
+(* ------------------------------------------------------------------------- *)
+(* Complement in dimension N >= 2 of set homeomorphic to any interval in *)
+(* any dimension is (path-)connected. This naively generalizes the argument *)
+(* in Ryuji Maehara's paper "The Jordan curve theorem via the Brouwer *)
+(* fixed point theorem", American Mathematical Monthly 1984. *)
+(* ------------------------------------------------------------------------- *)
+
+let RETRACTION_INJECTIVE_IMAGE_INTERVAL = prove
+ (`!p:real^M->real^N a b.
+ ~(interval[a,b] = {}) /\
+ p continuous_on interval[a,b] /\
+ (!x y. x IN interval[a,b] /\ y IN interval[a,b] /\ p x = p y ==> x = y)
+ ==> ?f. f continuous_on (:real^N) /\
+ IMAGE f (:real^N) SUBSET (IMAGE p (interval[a,b])) /\
+ (!x. x IN (IMAGE p (interval[a,b])) ==> f x = x)`,
+ REPEAT STRIP_TAC THEN
+ FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
+ DISCH_THEN(X_CHOOSE_TAC `q:real^N->real^M`) THEN
+ SUBGOAL_THEN `(q:real^N->real^M) continuous_on
+ (IMAGE p (interval[a:real^M,b]))`
+ ASSUME_TAC THENL
+ [MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[COMPACT_INTERVAL];
+ ALL_TAC] THEN
+ MP_TAC(ISPECL [`q:real^N->real^M`;
+ `IMAGE (p:real^M->real^N)
+ (interval[a,b])`;
+ `a:real^M`; `b:real^M`]
+ TIETZE_CLOSED_INTERVAL) THEN
+ ASM_SIMP_TAC[COMPACT_INTERVAL; COMPACT_CONTINUOUS_IMAGE;
+ COMPACT_IMP_CLOSED] THEN
+ ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
+ DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^M` STRIP_ASSUME_TAC) THEN
+ EXISTS_TAC `(p:real^M->real^N) o (r:real^N->real^M)` THEN
+ REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_THM; IN_UNIV] THEN
+ CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
+ MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN
+ FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ]
+ CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[]);;
+
+let UNBOUNDED_PATH_COMPONENTS_COMPLEMENT_HOMEOMORPHIC_INTERVAL = prove
+ (`!s:real^N->bool a b:real^M.
+ s homeomorphic (interval[a,b])
+ ==> !x. ~(x IN s) ==> ~bounded(path_component((:real^N) DIFF s) x)`,
+ REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN
+ REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
+ MAP_EVERY X_GEN_TAC [`p':real^N->real^M`; `p:real^M->real^N`] THEN
+ DISCH_TAC THEN
+ SUBGOAL_THEN
+ `!x y. x IN interval[a,b] /\ y IN interval[a,b] /\
+ (p:real^M->real^N) x = p y ==> x = y`
+ ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
+ FIRST_X_ASSUM(MP_TAC o funpow 4 CONJUNCT2) THEN
+ DISCH_THEN(CONJUNCTS_THEN2 (SUBST1_TAC o SYM) ASSUME_TAC) THEN
+ ASM_CASES_TAC `interval[a:real^M,b] = {}` THEN
+ ASM_REWRITE_TAC[IMAGE_CLAUSES; DIFF_EMPTY; PATH_COMPONENT_UNIV;
+ NOT_BOUNDED_UNIV] THEN
+ ABBREV_TAC `s = (:real^N) DIFF (IMAGE p (interval[a:real^M,b]))` THEN
+ X_GEN_TAC `c:real^N` THEN REPEAT STRIP_TAC THEN
+ SUBGOAL_THEN `(c:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
+ SUBGOAL_THEN `bounded((path_component s c) UNION
+ (IMAGE (p:real^M->real^N) (interval[a,b])))`
+ MP_TAC THENL
+ [ASM_SIMP_TAC[BOUNDED_UNION; COMPACT_IMP_BOUNDED; COMPACT_IMP_BOUNDED;
+ COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
+ ALL_TAC] THEN
+ DISCH_THEN(MP_TAC o SPEC `c:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN
+ REWRITE_TAC[UNION_SUBSET] THEN
+ DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
+ MP_TAC(ISPECL [`p:real^M->real^N`; `a:real^M`; `b:real^M`]
+ RETRACTION_INJECTIVE_IMAGE_INTERVAL) THEN
+ ASM_REWRITE_TAC[SUBSET; IN_UNIV] THEN
+ DISCH_THEN(X_CHOOSE_THEN `r:real^N->real^N` MP_TAC) THEN
+ DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
+ (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
+ REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN DISCH_TAC THEN
+ ABBREV_TAC `q = \z:real^N. if z IN path_component s c then r(z) else z` THEN
+ SUBGOAL_THEN
+ `(q:real^N->real^N) continuous_on
+ (closure(path_component s c) UNION ((:real^N) DIFF (path_component s c)))`
+ MP_TAC THENL
+ [EXPAND_TAC "q" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES THEN
+ REWRITE_TAC[CLOSED_CLOSURE; CONTINUOUS_ON_ID; GSYM OPEN_CLOSED] THEN
+ REPEAT CONJ_TAC THENL
+ [MATCH_MP_TAC OPEN_PATH_COMPONENT THEN EXPAND_TAC "s" THEN
+ ASM_SIMP_TAC[GSYM CLOSED_OPEN; COMPACT_IMP_CLOSED;
+ COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
+ ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV];
+ ALL_TAC] THEN
+ X_GEN_TAC `z:real^N` THEN
+ REWRITE_TAC[SET_RULE `~(z IN (s DIFF t) /\ z IN t)`] THEN
+ STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
+ MP_TAC(ISPECL
+ [`path_component s (z:real^N)`; `path_component s (c:real^N)`]
+ OPEN_INTER_CLOSURE_EQ_EMPTY) THEN
+ ASM_REWRITE_TAC[GSYM DISJOINT; PATH_COMPONENT_DISJOINT] THEN ANTS_TAC THENL
+ [MATCH_MP_TAC OPEN_PATH_COMPONENT THEN EXPAND_TAC "s" THEN
+ ASM_SIMP_TAC[GSYM CLOSED_OPEN; COMPACT_IMP_CLOSED;
+ COMPACT_CONTINUOUS_IMAGE; COMPACT_INTERVAL];
+ REWRITE_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
+ DISCH_THEN(MP_TAC o SPEC `z:real^N`) THEN ASM_REWRITE_TAC[] THEN
+ GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [IN] THEN
+ REWRITE_TAC[PATH_COMPONENT_REFL_EQ] THEN ASM SET_TAC[]];
+ ALL_TAC] THEN
+ SUBGOAL_THEN
+ `closure(path_component s c) UNION ((:real^N) DIFF (path_component s c)) =
+ (:real^N)`
+ SUBST1_TAC THENL
+ [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> t UNION (UNIV DIFF s) = UNIV`) THEN
+ REWRITE_TAC[CLOSURE_SUBSET];
+ DISCH_TAC] THEN
+ MP_TAC(ISPECL
+ [`(\x. &2 % c - x) o
+ (\x. c + B / norm(x - c) % (x - c)) o (q:real^N->real^N)`;
+ `cball(c:real^N,B)`]
+ BROUWER) THEN
+ REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; COMPACT_CBALL; CONVEX_CBALL] THEN
+ ASM_SIMP_TAC[CBALL_EQ_EMPTY; REAL_LT_IMP_LE; REAL_NOT_LT] THEN
+ SUBGOAL_THEN `!x. ~((q:real^N->real^N) x = c)` ASSUME_TAC THENL
+ [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "q" THEN
+ REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN COND_CASES_TAC THEN
+ ASM SET_TAC[PATH_COMPONENT_REFL_EQ];
+ ALL_TAC] THEN
+ REPEAT CONJ_TAC THENL
+ [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
+ SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN
+ MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL
+ [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_UNIV]; ALL_TAC] THEN
+ MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
+ MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
+ SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST] THEN
+ REWRITE_TAC[o_DEF; real_div; LIFT_CMUL] THEN
+ MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN
+ MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
+ ASM_REWRITE_TAC[FORALL_IN_IMAGE; NORM_EQ_0; VECTOR_SUB_EQ] THEN
+ SUBGOAL_THEN
+ `(\x:real^N. lift(norm(x - c))) = (lift o norm) o (\x. x - c)`
+ SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
+ MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
+ ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST;
+ CONTINUOUS_ON_LIFT_NORM];
+ REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_CBALL; o_THM; dist] THEN
+ X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
+ REWRITE_TAC[VECTOR_ARITH `c - (&2 % c - (c + x)) = x`] THEN
+ REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
+ ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
+ ASM_REAL_ARITH_TAC;
+ REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(c /\ b) <=> c ==> ~b`] THEN
+ REWRITE_TAC[IN_CBALL; o_THM; dist] THEN
+ X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
+ REWRITE_TAC[VECTOR_ARITH `&2 % c - (c + x') = x <=> --x' = x - c`] THEN
+ ASM_CASES_TAC `(x:real^N) IN path_component s c` THENL
+ [MATCH_MP_TAC(NORM_ARITH `norm(y) < B /\ norm(x) = B ==> ~(--x = y)`) THEN
+ REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
+ ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
+ ASM_SIMP_TAC[REAL_ARITH `&0 < B ==> abs B = B`] THEN
+ UNDISCH_TAC `path_component s c SUBSET ball(c:real^N,B)` THEN
+ REWRITE_TAC[SUBSET; IN_BALL] THEN ASM_MESON_TAC[dist; NORM_SUB];
+ EXPAND_TAC "q" THEN REWRITE_TAC[] THEN ASM_REWRITE_TAC[] THEN
+ REWRITE_TAC[VECTOR_ARITH `--(c % x) = x <=> (&1 + c) % x = vec 0`] THEN
+ ASM_REWRITE_TAC[DE_MORGAN_THM; VECTOR_SUB_EQ; VECTOR_MUL_EQ_0] THEN
+ SUBGOAL_THEN `~(x:real^N = c)` ASSUME_TAC THENL
+ [ASM_MESON_TAC[PATH_COMPONENT_REFL; IN]; ALL_TAC] THEN
+ ASM_REWRITE_TAC[] THEN
+ MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`) THEN
+ ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]]);;
+
+let PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_INTERVAL = prove
+ (`!s:real^N->bool a b:real^M.
+ 2 <= dimindex(:N) /\ s homeomorphic interval[a,b]
+ ==> path_connected((:real^N) DIFF s)`,
+ REPEAT STRIP_TAC THEN REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN
+ FIRST_ASSUM(MP_TAC o MATCH_MP
+ UNBOUNDED_PATH_COMPONENTS_COMPLEMENT_HOMEOMORPHIC_INTERVAL) THEN
+ ASM_REWRITE_TAC[SET_RULE `~(x IN s) <=> x IN (UNIV DIFF s)`] THEN
+ ABBREV_TAC `t = (:real^N) DIFF s` THEN
+ DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
+ STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_COMPACTNESS) THEN
+ REWRITE_TAC[COMPACT_INTERVAL] THEN
+ DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
+ REWRITE_TAC[BOUNDED_POS; LEFT_IMP_EXISTS_THM] THEN
+ X_GEN_TAC `B:real` THEN STRIP_TAC THEN
+ SUBGOAL_THEN `(?u:real^N. u IN path_component t x /\ B < norm(u)) /\
+ (?v:real^N. v IN path_component t y /\ B < norm(v))`
+ STRIP_ASSUME_TAC THENL
+ [ASM_MESON_TAC[BOUNDED_POS; REAL_NOT_LE]; ALL_TAC] THEN
+ MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `u:real^N` THEN
+ CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
+ MATCH_MP_TAC PATH_COMPONENT_SYM THEN
+ MATCH_MP_TAC PATH_COMPONENT_TRANS THEN EXISTS_TAC `v:real^N` THEN
+ CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
+ MATCH_MP_TAC PATH_COMPONENT_OF_SUBSET THEN
+ EXISTS_TAC `(:real^N) DIFF cball(vec 0,B)` THEN CONJ_TAC THENL
+ [EXPAND_TAC "t" THEN MATCH_MP_TAC(SET_RULE
+ `s SUBSET t ==> (u DIFF t) SUBSET (u DIFF s)`) THEN
+ ASM_REWRITE_TAC[SUBSET; IN_CBALL_0];
+ MP_TAC(ISPEC `cball(vec 0:real^N,B)`
+ PATH_CONNECTED_COMPLEMENT_BOUNDED_CONVEX) THEN
+ ASM_REWRITE_TAC[BOUNDED_CBALL; CONVEX_CBALL] THEN
+ REWRITE_TAC[PATH_CONNECTED_IFF_PATH_COMPONENT] THEN
+ DISCH_THEN MATCH_MP_TAC THEN
+ ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; IN_CBALL_0; REAL_NOT_LE]]);;
+
+(* ------------------------------------------------------------------------- *)
+(* In particular, apply all these to the special case of an arc. *)
+(* ------------------------------------------------------------------------- *)
+
+let RETRACTION_ARC = prove
+ (`!p. arc p
+ ==> ?f. f continuous_on (:real^N) /\
+ IMAGE f (:real^N) SUBSET path_image p /\
+ (!x. x IN path_image p ==> f x = x)`,
+ REWRITE_TAC[arc; path; path_image] THEN REPEAT STRIP_TAC THEN
+ MATCH_MP_TAC RETRACTION_INJECTIVE_IMAGE_INTERVAL THEN
+ ASM_REWRITE_TAC[INTERVAL_EQ_EMPTY_1; DROP_VEC; REAL_NOT_LT; REAL_POS]);;
+
+let PATH_CONNECTED_ARC_COMPLEMENT = prove
+ (`!p. 2 <= dimindex(:N) /\ arc p
+ ==> path_connected((:real^N) DIFF path_image p)`,
+ REWRITE_TAC[arc; path] THEN REPEAT STRIP_TAC THEN SIMP_TAC[path_image] THEN
+ MP_TAC(ISPECL [`path_image p:real^N->bool`; `vec 0:real^1`; `vec 1:real^1`]
+ PATH_CONNECTED_COMPLEMENT_HOMEOMORPHIC_INTERVAL) THEN
+ ASM_REWRITE_TAC[path_image] THEN DISCH_THEN MATCH_MP_TAC THEN
+ ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
+ MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN
+ EXISTS_TAC `p:real^1->real^N` THEN ASM_REWRITE_TAC[COMPACT_INTERVAL]);;
+
+let CONNECTED_ARC_COMPLEMENT = prove
+ (`!p. 2 <= dimindex(:N) /\ arc p
+ ==> connected((:real^N) DIFF path_image p)`,
+ SIMP_TAC[PATH_CONNECTED_ARC_COMPLEMENT; PATH_CONNECTED_IMP_CONNECTED]);; *)
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Tue Nov 17 18:52:30 2009 +0100
@@ -0,0 +1,1332 @@
+(* Title: HOL/Library/Convex_Euclidean_Space.thy
+ Author: John Harrison
+ Translated to from HOL light: Robert Himmelmann, TU Muenchen *)
+
+header {* Multivariate calculus in Euclidean space. *}
+
+theory Derivative
+ imports Brouwer_Fixpoint RealVector
+begin
+
+
+(* Because I do not want to type this all the time *)
+lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
+
+subsection {* Derivatives *}
+
+text {* The definition is slightly tricky since we make it work over
+ nets of a particular form. This lets us prove theorems generally and use
+ "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
+
+definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
+(infixl "(has'_derivative)" 12) where
+ "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
+ (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
+
+lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
+ unfolding has_derivative_def by auto
+
+lemma FDERIV_conv_has_derivative:"FDERIV f (x::'a::{real_normed_vector,perfect_space}) :> f' = (f has_derivative f') (at x)" (is "?l = ?r") proof
+ assume ?l note as = this[unfolded fderiv_def]
+ show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
+ fix e::real assume "e>0"
+ guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
+ thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
+ dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
+ apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
+ unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed next
+ assume ?r note as = this[unfolded has_derivative_def]
+ show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
+ fix e::real assume "e>0"
+ guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
+ thus "\<exists>s>0. \<forall>xa. xa \<noteq> 0 \<and> dist xa 0 < s \<longrightarrow> dist (norm (f (x + xa) - f x - f' xa) / norm xa) 0 < e" apply-
+ apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
+ unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed qed
+
+subsection {* These are the only cases we'll care about, probably. *}
+
+lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
+ bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
+ unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
+
+lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
+ bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
+ apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto
+
+subsection {* More explicit epsilon-delta forms. *}
+
+lemma has_derivative_within':
+ "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
+ (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
+ \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
+ unfolding has_derivative_within Lim_within vector_dist_norm
+ unfolding diff_0_right norm_mul by(simp add: group_simps)
+
+lemma has_derivative_at':
+ "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
+ (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
+ \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
+ apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto
+
+lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
+ unfolding has_derivative_within' has_derivative_at' by meson
+
+lemma has_derivative_within_open:
+ "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
+ unfolding has_derivative_within has_derivative_at using Lim_within_open by auto
+
+subsection {* Derivatives on real = Derivatives on real^1 *}
+
+lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps)
+
+lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
+ shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
+ { assume ?l guess K using linear_bounded[OF `?l`] ..
+ hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)
+ unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }
+ thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
+ unfolding vec1_dest_vec1_simps by auto qed
+
+lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
+ "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
+ = (f has_derivative f') (at x within s)"
+ unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
+ unfolding o_def Lim_within Ball_def unfolding forall_vec1
+ unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
+
+lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
+ "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
+ using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto
+
+lemma bounded_linear_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real"
+ shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
+ unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
+ unfolding vec1_dest_vec1_simps by auto
+
+lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"
+ shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
+ unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
+ unfolding vec1_dest_vec1_simps by auto
+
+lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows
+ "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
+ unfolding has_derivative_at unfolding bounded_linear_vec1[unfolded linear_conv_bounded_linear]
+ unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto
+
+lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
+ "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"
+ unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def
+ unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
+
+lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
+ "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
+ using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV)
+
+lemma derivative_is_linear: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite" shows
+ "(f has_derivative f') net \<Longrightarrow> linear f'"
+ unfolding has_derivative_def and linear_conv_bounded_linear by auto
+
+
+subsection {* Combining theorems. *}
+
+lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
+ unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
+ unfolding diff by(simp add: Lim_const)
+
+lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
+ apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
+
+lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
+ unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)
+
+lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)" proof
+ guess K using pos_bounded ..
+ thus "\<exists>K. \<forall>x. norm ((c::real) *\<^sub>R f x) \<le> norm x * K" apply(rule_tac x="abs c * K" in exI) proof
+ fix x case goal1
+ hence "abs c * norm (f x) \<le> abs c * (norm x * K)" apply-apply(erule conjE,erule_tac x=x in allE)
+ apply(rule mult_left_mono) by auto
+ thus ?case by(auto simp add:field_simps)
+ qed qed(auto simp add: scaleR.add_right add scaleR)
+
+lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net"
+ unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
+ using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
+ unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto
+
+lemma has_derivative_cmul_eq: assumes "c \<noteq> 0"
+ shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
+ apply(rule) defer apply(rule has_derivative_cmul,assumption)
+ apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto
+
+lemma has_derivative_neg:
+ "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
+ apply(drule has_derivative_cmul[where c="-1"]) by auto
+
+lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
+ apply(rule, drule_tac[!] has_derivative_neg) by auto
+
+lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
+ shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
+ note as = assms[unfolded has_derivative_def]
+ show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
+ using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
+ by(auto simp add:group_simps scaleR_right_diff_distrib scaleR_right_distrib) qed
+
+lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
+ apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
+
+lemma has_derivative_sub:
+ "(f has_derivative f') net \<Longrightarrow> (g has_derivative g') net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
+ apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:group_simps)
+
+lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
+ shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
+ apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1))
+proof- fix x F assume as:"finite F" "x \<notin> F" "x\<in>s" "((\<lambda>x. \<Sum>a\<in>F. f a x) has_derivative (\<lambda>h. \<Sum>a\<in>F. f' a h)) net"
+ thus "((\<lambda>xa. \<Sum>a\<in>insert x F. f a xa) has_derivative (\<lambda>h. \<Sum>a\<in>insert x F. f' a h)) net"
+ unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
+qed(auto intro!: has_derivative_const)
+
+lemma has_derivative_setsum_numseg:
+ "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
+ ((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
+ apply(rule has_derivative_setsum) by auto
+
+subsection {* somewhat different results for derivative of scalar multiplier. *}
+
+lemma has_derivative_vmul_component: fixes c::"real^'a::finite \<Rightarrow> real^'b::finite" and v::"real^'c::finite"
+ assumes "(c has_derivative c') net"
+ shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" proof-
+ have *:"\<And>y. (c y $ k *\<^sub>R v - (c (netlimit net) $ k *\<^sub>R v + c' (y - netlimit net) $ k *\<^sub>R v)) =
+ (c y $ k - (c (netlimit net) $ k + c' (y - netlimit net) $ k)) *\<^sub>R v"
+ unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
+ show ?thesis unfolding has_derivative_def and * and linear_conv_bounded_linear[symmetric]
+ apply(rule,rule linear_vmul_component[of c' k v, unfolded smult_conv_scaleR]) defer
+ apply(subst vector_smult_lzero[THEN sym, of v]) unfolding scaleR_scaleR smult_conv_scaleR apply(rule Lim_vmul)
+ using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
+ apply(rule,assumption,rule disjI2,rule,rule) proof-
+ have *:"\<And>x. x - vec 0 = (x::real^'n)" by auto
+ have **:"\<And>d x. d * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k)) = (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $k" by(auto simp add:field_simps)
+ fix e assume "\<not> trivial_limit net" "0 < (e::real)"
+ then obtain A where A:"A\<in>Rep_net net" "\<forall>x\<in>A. dist ((1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e"
+ using assms[unfolded has_derivative_def Lim] unfolding eventually_def by auto
+ show "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k))) 0 < e) net"
+ unfolding eventually_def apply(rule_tac x=A in bexI) apply rule proof-
+ case goal1 thus ?case apply -apply(drule A(2)[rule_format]) unfolding vector_dist_norm vec1_vec apply(rule le_less_trans) prefer 2 apply assumption unfolding * ** and norm_vec1[unfolded vec1_vec]
+ using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto
+ qed(insert A, auto) qed(insert assms[unfolded has_derivative_def], auto simp add:linear_conv_bounded_linear) qed
+
+lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real" and v::"real^'a::finite"
+ assumes "(c has_derivative c') (at x within s)"
+ shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)" proof-
+ have *:"\<And>c. (\<lambda>x. (vec1 \<circ> c \<circ> dest_vec1) x $ 1 *\<^sub>R v) = (\<lambda>x. (c x) *\<^sub>R v) \<circ> dest_vec1" unfolding o_def by auto
+ show ?thesis using has_derivative_vmul_component[of "vec1 \<circ> c \<circ> dest_vec1" "vec1 \<circ> c' \<circ> dest_vec1" "at (vec1 x) within vec1 ` s" 1 v]
+ unfolding * and has_derivative_within_vec1_dest_vec1 unfolding has_derivative_within_dest_vec1 using assms by auto qed
+
+lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real" and v::"real^'a::finite"
+ assumes "(c has_derivative c') (at x)"
+ shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
+ using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)
+
+lemma has_derivative_lift_dot:
+ assumes "(f has_derivative f') net"
+ shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
+ show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
+ apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
+ apply(drule Lim_inner[where a=v]) unfolding o_def
+ by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
+
+lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
+ has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul
+ bounded_linear.has_derivative has_derivative_lift_dot
+
+subsection {* limit transformation for derivatives. *}
+
+lemma has_derivative_transform_within:
+ assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x within s)"
+ shows "(g has_derivative f') (at x within s)"
+ using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
+ apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
+ apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
+
+lemma has_derivative_transform_at:
+ assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
+ shows "(g has_derivative f') (at x)"
+ apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)])
+ using assms(2-3) unfolding within_UNIV by auto
+
+lemma has_derivative_transform_within_open:
+ assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
+ shows "(g has_derivative f') (at x)"
+ using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
+ apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
+ apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
+
+subsection {* differentiability. *}
+
+definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
+ "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
+
+definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
+ "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
+
+lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
+ unfolding differentiable_def by auto
+
+lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
+ unfolding differentiable_def using has_derivative_at_within by blast
+
+lemma differentiable_within_open: assumes "a \<in> s" "open s" shows
+ "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
+ unfolding differentiable_def has_derivative_within_open[OF assms] by auto
+
+lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n::finite) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
+ unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
+
+lemma differentiable_on_eq_differentiable_at: "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
+ unfolding differentiable_on_def by(auto simp add: differentiable_within_open)
+
+lemma differentiable_transform_within:
+ assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable (at x within s)"
+ shows "g differentiable (at x within s)"
+ using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])
+
+lemma differentiable_transform_at:
+ assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
+ shows "g differentiable at x"
+ using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto
+
+subsection {* Frechet derivative and Jacobian matrix. *}
+
+definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
+
+lemma frechet_derivative_works:
+ "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
+ unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
+
+lemma linear_frechet_derivative: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
+ unfolding frechet_derivative_works has_derivative_def unfolding linear_conv_bounded_linear by auto
+
+definition "jacobian f net = matrix(frechet_derivative f net)"
+
+lemma jacobian_works: "(f::(real^'a::finite) \<Rightarrow> (real^'b::finite)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
+ apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
+ apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
+
+subsection {* Differentiability implies continuity. *}
+
+lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+ shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
+ unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
+ apply(rule_tac x="min d 1" in exI) apply rule defer apply(rule,erule_tac x=x in ballE) unfolding vector_dist_norm diff_0_right norm_mul
+ by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
+
+lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)"
+ shows "continuous (at x within s) f" proof-
+ from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
+ then interpret bounded_linear f' by auto
+ have *:"\<And>xa. x\<noteq>xa \<Longrightarrow> (f' \<circ> (\<lambda>y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \<circ> (\<lambda>y. y - x)) x + 0) = f xa - f x"
+ using zero by auto
+ have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
+ apply(rule continuous_within_compose) apply(rule continuous_intros)+
+ by(rule linear_continuous_within[OF f'[THEN conjunct1]])
+ show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
+ apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and vector_dist_norm
+ apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
+ by(auto simp add:zero * elim!:allE) qed
+
+lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
+ by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
+
+lemma differentiable_imp_continuous_on: "f differentiable_on s \<Longrightarrow> continuous_on s f"
+ unfolding differentiable_on_def continuous_on_eq_continuous_within
+ using differentiable_imp_continuous_within by blast
+
+lemma has_derivative_within_subset:
+ "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
+ unfolding has_derivative_within using Lim_within_subset by blast
+
+lemma differentiable_within_subset:
+ "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
+ unfolding differentiable_def using has_derivative_within_subset by blast
+
+lemma differentiable_on_subset: "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
+ unfolding differentiable_on_def using differentiable_within_subset by blast
+
+lemma differentiable_on_empty: "f differentiable_on {}"
+ unfolding differentiable_on_def by auto
+
+subsection {* Several results are easier using a "multiplied-out" variant. *)
+(* (I got this idea from Dieudonne's proof of the chain rule). *}
+
+lemma has_derivative_within_alt:
+ "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
+ (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm(f(y) - f(x) - f'(y - x)) \<le> e * norm(y - x))" (is "?lhs \<longleftrightarrow> ?rhs")
+proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
+ unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
+ apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
+ fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
+ dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
+ then interpret bounded_linear f' by auto
+ show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
+ case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
+ case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
+ unfolding vector_dist_norm diff_0_right norm_mul using as(3)
+ using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded vector_dist_norm]
+ by(auto simp add:linear_0 linear_sub group_simps)
+ thus ?thesis by(auto simp add:group_simps) qed qed next
+ assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
+ apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
+ apply(erule conjE,rule,assumption,rule,rule) unfolding vector_dist_norm diff_0_right norm_scaleR
+ apply(erule_tac x=xa in ballE,erule impE) proof-
+ fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
+ "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
+ thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
+ apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:group_simps) qed auto qed
+
+lemma has_derivative_at_alt:
+ "(f has_derivative f') (at (x::real^'n::finite)) \<longleftrightarrow> bounded_linear f' \<and>
+ (\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm(f y - f x - f'(y - x)) \<le> e * norm(y - x))"
+ using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
+
+subsection {* The chain rule. *}
+
+lemma diff_chain_within:
+ assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
+ shows "((g o f) has_derivative (g' o f'))(at x within s)"
+ unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
+ apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
+ apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
+ note assms = assms[unfolded has_derivative_within_alt]
+ fix e::real assume "0<e"
+ guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
+ guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
+ have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
+ guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
+ have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
+ guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
+ guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this
+
+ def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
+ def d \<equiv> "(min d0 (de / (B1 + 1))) / 2" have "de * 2 / (B1 + 1) > de / (B1 + 1)" apply(rule divide_strict_right_mono) using B1 de by auto
+ hence d:"d>0" "d < d1" "d < d2" "d < (de / (B1 + 1))" unfolding d_def using d0 d1 d2 de B1 by(auto intro!: divide_pos_pos simp add:min_less_iff_disj not_less)
+
+ show "\<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" apply(rule_tac x=d in exI)
+ proof(rule,rule `d>0`,rule,rule)
+ fix y assume as:"y \<in> s" "norm (y - x) < d"
+ hence 1:"norm (f y - f x - f' (y - x)) \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto
+
+ have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
+ using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:group_simps)
+ also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" apply(rule add_left_mono) using B1 by(auto simp add:group_simps)
+ also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" apply(rule add_right_mono) using d1 d2 d as by auto
+ also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
+ also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
+ finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto
+
+ hence "norm (f y - f x) \<le> d * (1 + B1)" apply- apply(rule order_trans,assumption,rule mult_right_mono) using as B1 by auto
+ also have "\<dots> < de" using d B1 by(auto simp add:field_simps)
+ finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
+ apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto
+ also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto
+ also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono) using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
+ finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
+
+ interpret g': bounded_linear g' using assms(2) by auto
+ interpret f': bounded_linear f' using assms(1) by auto
+ have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
+ by(auto simp add:group_simps f'.diff g'.diff g'.add)
+ also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:group_simps)
+ also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto
+ also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
+ finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
+
+ have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)" using 5 4 by auto
+ thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) by assumption qed qed
+
+lemma diff_chain_at:
+ "(f has_derivative f') (at x) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow> ((g o f) has_derivative (g' o f')) (at x)"
+ using diff_chain_within[of f f' x UNIV g g'] using has_derivative_within_subset[of g g' "f x" UNIV "range f"] unfolding within_UNIV by auto
+
+subsection {* Composition rules stated just for differentiability. *}
+
+lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
+ unfolding differentiable_def using has_derivative_const by auto
+
+lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
+ unfolding differentiable_def using has_derivative_id by auto
+
+lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
+ unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
+
+lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
+ unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
+
+lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
+ \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
+ unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
+ apply(rule has_derivative_add) by auto
+
+lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
+ \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
+ unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
+ apply(rule has_derivative_sub) by auto
+
+lemma differentiable_setsum: fixes f::"'a \<Rightarrow> (real^'n::finite \<Rightarrow>real^'n)"
+ assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
+ shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
+ guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
+ thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed
+
+lemma differentiable_setsum_numseg: fixes f::"_ \<Rightarrow> (real^'n::finite \<Rightarrow>real^'n)"
+ shows "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> (f i) differentiable net \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) {m::nat..n}) differentiable net"
+ apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
+
+lemma differentiable_chain_at:
+ "f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
+ unfolding differentiable_def by(meson diff_chain_at)
+
+lemma differentiable_chain_within:
+ "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
+ \<Longrightarrow> (g o f) differentiable (at x within s)"
+ unfolding differentiable_def by(meson diff_chain_within)
+
+subsection {* Uniqueness of derivative. *)
+(* *)
+(* The general result is a bit messy because we need approachability of the *)
+(* limit point from any direction. But OK for nontrivial intervals etc. *}
+
+lemma frechet_derivative_unique_within: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
+ "(\<forall>i::'a::finite. \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)" shows "f' = f''" proof-
+ note as = assms(1,2)[unfolded has_derivative_def]
+ then interpret f': bounded_linear f' by auto from as interpret f'': bounded_linear f'' by auto
+ have "x islimpt s" unfolding islimpt_approachable proof(rule,rule)
+ guess a using UNIV_witness[where 'a='a] ..
+ fix e::real assume "0<e" guess d using assms(3)[rule_format,OF`e>0`,of a] ..
+ thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x="x + d*\<^sub>R basis a" in bexI)
+ using basis_nonzero[of a] norm_basis[of a] unfolding vector_dist_norm by auto qed
+ hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
+ show ?thesis apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
+ apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule ccontr)
+ fix i::'a def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
+ assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
+ guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
+ guess c using assms(3)[rule_format,OF d[THEN conjunct1],of i] .. note c=this
+ have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
+ unfolding scaleR_right_distrib by auto
+ also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"
+ unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
+ also have "\<dots> = e" unfolding e_def norm_mul using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by(auto simp add:group_simps)
+ finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"] using norm_basis[of i] unfolding vector_dist_norm
+ unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib by auto qed qed
+
+lemma frechet_derivative_unique_at: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
+ apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+
+ apply(rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto
+
+lemma "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
+ unfolding continuous_at Lim_at unfolding dist_nz by auto
+
+lemma frechet_derivative_unique_within_closed_interval: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" (is "x\<in>?I") and
+ "(f has_derivative f' ) (at x within {a..b})" and
+ "(f has_derivative f'') (at x within {a..b})"
+ shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule)
+ fix e::real and i::'a assume "e>0"
+ thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}" proof(cases "x$i=a$i")
+ case True thus ?thesis apply(rule_tac x="(min (b$i - a$i) e) / 2" in exI)
+ using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
+ unfolding mem_interval by(auto simp add:field_simps) next
+ note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
+ case False moreover have "a $ i < x $ i" using False * by auto
+ moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
+ also have "\<dots> = a$i + x$i" by auto also have "\<dots> \<le> 2 * x$i" using * by auto
+ finally have "a $ i * 2 + min (x $ i - a $ i) e \<le> x $ i * 2" by auto }
+ moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
+ hence "x $ i * 2 \<le> b $ i * 2 + min (x $ i - a $ i) e" using * by auto
+ ultimately show ?thesis apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
+ using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
+ unfolding mem_interval by(auto simp add:field_simps) qed qed
+
+lemma frechet_derivative_unique_within_open_interval: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
+ "(f has_derivative f'') (at x within {a<..<b})"
+ shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(2-3))+ proof(rule,rule,rule)
+ fix e::real and i::'a assume "e>0"
+ note * = assms(1)[unfolded mem_interval,THEN spec[where x=i]]
+ have "a $ i < x $ i" using * by auto
+ moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
+ also have "\<dots> = a$i + x$i" by auto also have "\<dots> < 2 * x$i" using * by auto
+ finally have "a $ i * 2 + min (x $ i - a $ i) e < x $ i * 2" by auto }
+ moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
+ hence "x $ i * 2 < b $ i * 2 + min (x $ i - a $ i) e" using * by auto
+ ultimately show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a<..<b}"
+ apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
+ using `e>0` and assms(1) unfolding mem_interval by(auto simp add:field_simps) qed
+
+lemma frechet_derivative_at: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
+ apply(rule frechet_derivative_unique_at[of f],assumption)
+ unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
+
+lemma frechet_derivative_within_closed_interval: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
+ shows "frechet_derivative f (at x within {a.. b}) = f'"
+ apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
+ apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
+ unfolding differentiable_def using assms(3) by auto
+
+subsection {* Component of the differential must be zero if it exists at a local *)
+(* maximum or minimum for that corresponding component. *}
+
+lemma differential_zero_maxmin_component: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
+ "f differentiable (at x)" shows "jacobian f (at x) $ k = 0" proof(rule ccontr)
+ def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
+ then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
+ hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
+ note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
+ guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
+ guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
+ { fix c assume "abs c \<le> d"
+ hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
+ have "\<bar>(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j))" by(rule component_le_norm)
+ also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto
+ finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
+ hence "\<bar>f (x + c *\<^sub>R basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
+ unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric]
+ unfolding dot_rmult dot_basis unfolding smult_conv_scaleR by simp } note * = this
+ have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
+ unfolding mem_ball vector_dist_norm using norm_basis[of j] d by auto
+ hence **:"((f (x - d *\<^sub>R basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R basis j))$k \<le> (f x)$k) \<or>
+ ((f (x - d *\<^sub>R basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R basis j))$k \<ge> (f x)$k)" using assms(2) by auto
+ have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
+ show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"])
+ using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
+ unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding group_simps by auto qed
+
+subsection {* In particular if we have a mapping into R^1. *}
+
+lemma differential_zero_maxmin: fixes f::"real^'a::finite \<Rightarrow> real"
+ assumes "x \<in> s" "open s" "(f has_derivative f') (at x)"
+ "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
+ shows "f' = (\<lambda>v. 0)" proof-
+ note deriv = assms(3)[unfolded has_derivative_at_vec1]
+ obtain e where e:"e>0" "ball x e \<subseteq> s" using assms(2)[unfolded open_contains_ball] and assms(1) by auto
+ hence **:"(jacobian (vec1 \<circ> f) (at x)) $ 1 = 0" using differential_zero_maxmin_component[of e x "\<lambda>x. vec1 (f x)" 1]
+ unfolding dest_vec1_def[THEN sym] vec1_dest_vec1 using assms(4) and assms(3)[unfolded has_derivative_at_vec1 o_def]
+ unfolding differentiable_def o_def by auto
+ have *:"jacobian (vec1 \<circ> f) (at x) = matrix (vec1 \<circ> f')" unfolding jacobian_def and frechet_derivative_at[OF deriv] ..
+ have "vec1 \<circ> f' = (\<lambda>x. 0)" apply(rule) unfolding matrix_works[OF derivative_is_linear[OF deriv],THEN sym]
+ unfolding Cart_eq matrix_vector_mul_component using **[unfolded *] by auto
+ thus ?thesis apply-apply(rule,subst vec1_eq[THEN sym]) unfolding o_def apply(drule fun_cong) by auto qed
+
+subsection {* The traditional Rolle theorem in one dimension. *}
+
+lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
+ unfolding vector_less_eq_def by auto
+lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
+ unfolding vector_less_def by auto
+
+lemma rolle: fixes f::"real\<Rightarrow>real"
+ assumes "a < b" "f a = f b" "continuous_on {a..b} f"
+ "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
+ have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))" proof-
+ have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
+ guess d using continuous_attains_sup[OF compact_real_interval * assms(3)] .. note d=this
+ guess c using continuous_attains_inf[OF compact_real_interval * assms(3)] .. note c=this
+ show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
+ case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
+ apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
+ case False hence "f d = f c" using d c assms(2) by auto
+ hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d" using c d apply- apply(erule_tac x=x in ballE)+ by auto
+ thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
+ then guess x .. note x=this
+ hence "f' x \<circ> dest_vec1 = (\<lambda>v. 0)" apply(rule_tac differential_zero_maxmin[of "vec1 x" "vec1 ` {a<..<b}" "f \<circ> dest_vec1" "(f' x) \<circ> dest_vec1"])
+ unfolding vec1_interval defer apply(rule open_interval)
+ apply(rule assms(4)[unfolded has_derivative_at_dest_vec1[THEN sym],THEN bspec[where x=x]],assumption)
+ unfolding o_def apply(erule disjE,rule disjI2) by(auto simp add: vector_less_def dest_vec1_def)
+ thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
+ apply(drule_tac x="vec1 v" in fun_cong) unfolding vec1_dest_vec1 using x(1) by auto qed
+
+subsection {* One-dimensional mean value theorem. *}
+
+lemma mvt: fixes f::"real \<Rightarrow> real"
+ assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
+ have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
+ apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
+ apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
+ fix x assume x:"x \<in> {a<..<b}"
+ show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
+ by(rule has_derivative_intros assms(3)[rule_format,OF x]
+ has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+
+ qed(insert assms(1), auto simp add:field_simps)
+ then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed
+
+lemma mvt_simple: fixes f::"real \<Rightarrow> real"
+ assumes "a<b" "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
+ shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
+ apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
+ unfolding differentiable_on_def differentiable_def defer proof
+ fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)" unfolding has_derivative_within_open[OF x open_interval_real,THEN sym]
+ apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)
+
+lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
+ assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
+ shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b")
+ interpret bounded_linear "f' b" using assms(2) assms(1) by auto
+ case True thus ?thesis apply(rule_tac x=a in bexI)
+ using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
+ unfolding True using zero by auto next
+ case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed
+
+subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
+
+lemma inner_eq_dot: fixes a::"real^'n::finite"
+ shows "a \<bullet> b = inner a b" unfolding inner_vector_def dot_def by auto
+
+lemma mvt_general: fixes f::"real\<Rightarrow>real^'n::finite"
+ assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))" proof-
+ have "\<exists>x\<in>{a<..<b}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
+ apply(rule mvt) apply(rule assms(1))unfolding inner_eq_dot apply(rule continuous_on_inner continuous_on_intros assms(2))+
+ unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
+ then guess x .. note x=this
+ show ?thesis proof(cases "f a = f b")
+ case False have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add:class_semiring.semiring_rules)
+ also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding norm_pow_2 ..
+ also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x by auto
+ also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
+ finally show ?thesis using False x(1) by(auto simp add: real_mult_left_cancel) next
+ case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed
+
+subsection {* Still more general bound theorem. *}
+
+lemma differentiable_bound: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "convex s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
+ shows "norm(f x - f y) \<le> B * norm(x - y)" proof-
+ let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
+ have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
+ using assms(1)[unfolded convex_alt,rule_format,OF x y] unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib by(auto simp add:group_simps)
+ hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
+ unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
+ unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
+ apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
+ have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)" proof rule case goal1
+ let ?u = "x + u *\<^sub>R (y - x)"
+ have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
+ apply(rule diff_chain_within) apply(rule has_derivative_intros)+
+ apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
+ thus ?case unfolding has_derivative_within_open[OF goal1 open_interval_real] by auto qed
+ guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
+ have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
+ have "norm (f' x y) \<le> onorm (f' x) * norm y"
+ using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
+ also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
+ using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
+ finally show ?case by simp qed
+ have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
+ by(auto simp add:norm_minus_commute)
+ also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
+ also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
+ finally show ?thesis by(auto simp add:norm_minus_commute) qed
+
+lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
+ shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
+ have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 unfolding norm_vec1 by auto
+ hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1)
+ have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
+
+ have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
+ have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
+ show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) qed
+
+lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
+ assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
+ shows "norm(f x - f y) \<le> B * norm(x - y)"
+ using differentiable_bound[of "vec1 ` s" "vec1 \<circ> f \<circ> dest_vec1" "\<lambda>x. vec1 \<circ> (f' (dest_vec1 x)) \<circ> dest_vec1" B "vec1 x" "vec1 y"]
+ unfolding Ball_def forall_vec1 unfolding has_derivative_within_vec1_dest_vec1 image_iff
+ unfolding convex_vec1 unfolding o_def vec1_dest_vec1_simps onorm_vec1 using assms by auto
+
+subsection {* In particular. *}
+
+lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
+ assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
+ shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
+ case False then obtain x where "x\<in>s" by auto
+ have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
+ thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
+ unfolding onorm_vec1[of "\<lambda>x. 0", THEN sym] onorm_const norm_vec1 by auto qed
+ thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto
+
+lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
+ assumes "convex s" "a \<in> s" "f a = c" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
+ shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
+
+subsection {* Differentiability of inverse function (most basic form). *}
+
+lemma has_derivative_inverse_basic: fixes f::"real^'b::finite \<Rightarrow> real^'c::finite"
+ assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
+ "open t" "y \<in> t" "\<forall>z\<in>t. f(g z) = z"
+ shows "(g has_derivative g') (at y)" proof-
+ interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto
+ interpret g': bounded_linear g' using assms by auto
+ guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
+(* have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
+ have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)" proof(rule,rule) case goal1
+ have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
+ guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
+ guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
+ guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
+ guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
+ thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
+ fix z assume as:"norm (z - y) < d" hence "z\<in>t" using d2 d unfolding vector_dist_norm by auto
+ have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
+ unfolding g'.diff f'.diff unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
+ unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
+ also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format])
+ also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
+ apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
+ apply(rule d1[THEN conjunct2, unfolded vector_dist_norm,rule_format]) using as d C d0 by auto
+ also have "\<dots> \<le> e * norm (g z - g y)" using C by(auto simp add:field_simps)
+ finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)" by simp qed auto qed
+ have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
+ have "B>0" unfolding B_def using C by auto
+ have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
+ have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub)
+ also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)" apply(rule add_left_mono) using d and goal1 by auto
+ also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
+ finally show ?case unfolding B_def by(auto simp add:field_simps) qed
+ show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
+ hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
+ guess d' using lem1[rule_format,OF *] .. note d'=this
+ guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
+ show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
+ hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
+ also have "\<dots> \<le> e * norm(z - y)" unfolding mult_frac_num pos_divide_le_eq[OF `B>0`]
+ using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
+ finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
+
+subsection {* Simply rewrite that based on the domain point x. *}
+
+lemma has_derivative_inverse_basic_x: fixes f::"real^'b::finite \<Rightarrow> real^'c::finite"
+ assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
+ "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
+ shows "(g has_derivative g') (at (f(x)))"
+ apply(rule has_derivative_inverse_basic) using assms by auto
+
+subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}
+
+lemma has_derivative_inverse_dieudonne: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
+ (**) "x\<in>s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
+ shows "(g has_derivative g') (at (f x))"
+ apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
+ using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)] by auto
+
+subsection {* Here's the simplest way of not assuming much about g. *}
+
+lemma has_derivative_inverse: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
+ "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
+ shows "(g has_derivative g') (at (f x))" proof-
+ { fix y assume "y\<in>interior (f ` s)"
+ then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
+ have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
+ show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
+ apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
+ apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption) qed
+
+subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
+
+lemma brouwer_surjective: fixes f::"real^'n::finite \<Rightarrow> real^'n"
+ assumes "compact t" "convex t" "t \<noteq> {}" "continuous_on t f"
+ "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
+ shows "\<exists>y\<in>t. f y = x" proof-
+ have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:group_simps)
+ show ?thesis unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
+ apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed
+
+lemma brouwer_surjective_cball: fixes f::"real^'n::finite \<Rightarrow> real^'n"
+ assumes "0 < e" "continuous_on (cball a e) f"
+ "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
+ shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
+ unfolding cball_eq_empty using assms by auto
+
+text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
+
+lemma sussmann_open_mapping: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
+ assumes "open s" "continuous_on s f" "x \<in> s"
+ "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
+ (**) "t \<subseteq> s" "x \<in> interior t"
+ shows "f x \<in> interior (f ` t)" proof-
+ interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
+ interpret g':bounded_linear g' using assms by auto
+ guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
+ guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
+ guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
+ have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
+ guess e using real_lbound_gt_zero[OF *] .. note e=this
+ have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
+ apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
+ prefer 3 apply(rule,rule) proof-
+ show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
+ apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
+ apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
+ apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
+ fix y z assume as:"y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
+ have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and vector_dist_norm by auto
+ also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
+ also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball vector_dist_norm] using B by auto
+ also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
+ finally have "z\<in>cball x e1" unfolding mem_cball by force
+ thus "z \<in> s" using e1 assms(7) by auto qed next
+ fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
+ have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
+ also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using as(2)[unfolded mem_cball vector_dist_norm] and B unfolding norm_minus_commute by auto
+ also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
+ finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
+ have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
+ have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le> norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
+ using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:group_simps)
+ also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding group_simps ** by auto
+ also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball vector_dist_norm] by auto
+ also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
+ also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
+ also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball vector_dist_norm] unfolding norm_minus_commute by auto
+ finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball vector_dist_norm by auto
+ qed(insert e, auto) note lem = this
+ show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
+ apply(rule,rule divide_pos_pos) prefer 3 proof
+ fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
+ guess z using lem[rule_format,OF *] .. note z=this
+ hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
+ also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball vector_dist_norm norm_minus_commute using B by auto
+ also have "\<dots> \<le> e1" using e B unfolding less_divide_eq by auto
+ finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
+ unfolding mem_cball vector_dist_norm by auto
+ thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed
+
+text {* Hence the following eccentric variant of the inverse function theorem. *)
+(* This has no continuity assumptions, but we do need the inverse function. *)
+(* We could put f' o g = I but this happens to fit with the minimal linear *)
+(* algebra theory I've set up so far. *}
+
+lemma has_derivative_inverse_strong: fixes f::"real^'n::finite \<Rightarrow> real^'n"
+ assumes "open s" "x \<in> s" "continuous_on s f"
+ "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
+ shows "(g has_derivative g') (at (f x))" proof-
+ have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
+ hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
+ apply- apply(rule right_inverse_linear) using assms(6) by auto
+ moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
+ using linear_inverse_left by auto
+ moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)" apply(rule,rule,rule,rule sussmann_open_mapping )
+ apply(rule assms ling)+ by auto
+ have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
+ fix e::real assume "e>0"
+ hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
+ by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
+ then guess d unfolding mem_interior .. note d=this
+ show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
+ apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
+ hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
+ by(auto simp add:dist_commute)
+ hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
+ thus "dist (g y) (g (f x)) < e" using assms(4)[rule_format,OF `x\<in>s`] by(auto simp add:dist_commute) qed qed
+ moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
+ apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
+ moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
+ hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
+ thus ?case using assms(4) by auto qed
+ ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed
+
+subsection {* A rewrite based on the other domain. *}
+
+lemma has_derivative_inverse_strong_x: fixes f::"real^'n::finite \<Rightarrow> real^'n"
+ assumes "open s" "g y \<in> s" "continuous_on s f"
+ "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
+ shows "(g has_derivative g') (at y)"
+ using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
+
+subsection {* On a region. *}
+
+lemma has_derivative_inverse_on: fixes f::"real^'n::finite \<Rightarrow> real^'n"
+ assumes "open s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)" "\<forall>x\<in>s. g(f x) = x" "f'(x) o g'(x) = id" "x\<in>s"
+ shows "(g has_derivative g'(x)) (at (f x))"
+ apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
+ unfolding continuous_on_eq_continuous_at[OF assms(1)]
+ apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto
+
+subsection {* Invertible derivative continous at a point implies local injectivity. *)
+(* It's only for this we need continuity of the derivative, except of course *)
+(* if we want the fact that the inverse derivative is also continuous. So if *)
+(* we know for some other reason that the inverse function exists, it's OK. *}
+
+lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
+ using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:group_simps)
+
+lemma has_derivative_locally_injective: fixes f::"real^'n::finite \<Rightarrow> real^'m::finite"
+ assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
+ "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
+ "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
+ obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" proof-
+ interpret bounded_linear g' using assms by auto
+ note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
+ have "g' (f' a 1) = 1" using f'g' by auto
+ hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
+ def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
+ guess d1 using assms(6)[rule_format,OF *] .. note d1=this
+ from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
+ obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
+ guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
+ guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
+ show ?thesis proof show "a\<in>ball a d" using d by auto
+ show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
+ fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
+ def ph \<equiv> "\<lambda>w. w - g'(f w - f x)" have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
+ unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:group_simps)
+ have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
+ apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
+ apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "u\<in>s" using d d2 by auto
+ have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)" unfolding o_def and diff using f'g' by auto
+ show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
+ unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
+ apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
+ apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
+ by(auto intro!: has_derivative_intros derivative_linear)
+ have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
+ apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
+ have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
+ unfolding linear_conv_bounded_linear by(rule assms(3) **)+
+ also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono)
+ using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
+ using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:group_simps)
+ also have "\<dots> \<le> 1/2" unfolding k_def by auto
+ finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
+ moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
+ unfolding ph_def using diff unfolding as by auto
+ ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed
+
+subsection {* Uniformly convergent sequence of derivatives. *}
+
+lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
+ assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
+ shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)" proof(default)+
+ fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
+ show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
+ apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)]) apply(rule_tac[!] ballI) proof-
+ fix x assume "x\<in>s" show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
+ by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
+ { fix h have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
+ using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"] unfolding norm_minus_commute by(auto simp add:group_simps)
+ also have "\<dots> \<le> e * norm h+ e * norm h" using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h] assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
+ by(auto simp add:field_simps)
+ finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
+ thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e" apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
+ unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed
+
+lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
+ assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)" "0 < e"
+ shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)" proof(rule,rule)
+ case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
+ guess N using assms(3)[rule_format,OF *(2)] ..
+ thus ?case apply(rule_tac x=N in exI) apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *]) using assms by auto qed
+
+lemma has_derivative_sequence: fixes f::"nat\<Rightarrow>real^'m::finite\<Rightarrow>real^'n::finite"
+ assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
+ "x0 \<in> s" "((\<lambda>n. f n x0) ---> l) sequentially"
+ shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)" proof-
+ have lem1:"\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
+ apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
+ have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
+ fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
+ case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
+ case False show ?thesis unfolding Cauchy_def proof(rule,rule)
+ fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
+ guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
+ guess N using lem1[rule_format,OF *(2)] .. note N = this
+ show " \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e" apply(rule_tac x="max M N" in exI) proof(default+)
+ fix m n assume as:"max M N \<le>m" "max M N\<le>n"
+ have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
+ unfolding vector_dist_norm by(rule norm_triangle_sub)
+ also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2" using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False by auto
+ also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_right_mono) using as and M[rule_format] unfolding vector_dist_norm by auto
+ finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
+ then guess g .. note g = this
+ have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)" proof(rule,rule)
+ fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
+ show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply(rule_tac x=N in exI) proof(default+)
+ fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
+ have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
+ unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
+ fix m assume "N\<le>m" thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
+ using N[rule_format, of n m x y] and as by(auto simp add:group_simps) qed
+ thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
+ apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\<lambda>m. (f n x - f n y) - (f m x - f m y)"])
+ apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
+ show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
+ apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
+ have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
+ fix u and e::real assume "e>0" show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e" proof(cases "u=0")
+ case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
+ show ?thesis apply(rule_tac x=N in exI) unfolding True
+ using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
+ case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
+ guess N using assms(3)[rule_format,OF *] .. note N=this
+ show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
+ show ?case unfolding vector_dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
+ by (auto simp add:field_simps) qed qed qed
+ show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
+ fix x' y z::"real^'m" and c::real
+ note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
+ show "g' x (c *s x') = c *s g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
+ apply(rule lem3[rule_format]) unfolding smult_conv_scaleR
+ unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
+ apply(rule Lim_cmul) by(rule lem3[rule_format])
+ show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
+ apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
+ apply(rule Lim_add) by(rule lem3[rule_format])+ qed
+ show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)" proof(rule,rule) case goal1
+ have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
+ guess N2 using lem2[rule_format,OF *] .. note N2=this
+ guess d1 using assms(2)[unfolded has_derivative_within_alt, rule_format,OF `x\<in>s`, of "max N1 N2",THEN conjunct2,rule_format,OF *] .. note d1=this
+ show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
+ fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
+ have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)" apply(subst norm_minus_cancel[THEN sym])
+ using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
+ have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)" using d1 and as by auto ultimately
+ have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
+ using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
+ by (auto simp add:group_simps) moreover
+ have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)" using N1 `x\<in>s` by auto
+ ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
+ using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"] by(auto simp add:group_simps)
+ qed qed qed qed
+
+subsection {* Can choose to line up antiderivatives if we want. *}
+
+lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
+ assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm h"
+ shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}")
+ case False then obtain a where "a\<in>s" by auto have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
+ show ?thesis apply(rule *) apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
+ apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
+ apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto
+
+lemma has_antiderivative_limit: fixes g'::"real^'m::finite \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
+ assumes "convex s" "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> e * norm(h))"
+ shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof-
+ have *:"\<forall>n. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm(h))"
+ apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
+ guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
+ show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
+ fix e::real assume "0<e" guess N using reals_Archimedean[OF `e>0`] .. note N=this
+ show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h" apply(rule_tac x=N in exI) proof(default+) case goal1
+ have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
+ using goal1(1) by(auto simp add:field_simps)
+ show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
+ apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed
+
+subsection {* Differentiation of a series. *}
+
+definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
+(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
+
+lemma has_derivative_series: fixes f::"nat \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
+ assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
+ "x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
+ shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) sums_seq (g x)) k \<and> (g has_derivative g'(x)) (at x within s)"
+ unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
+ apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
+ using assms(4-5) unfolding sums_seq_def by auto
+
+subsection {* Derivative with composed bilinear function. *}
+
+lemma has_derivative_bilinear_within: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite" and f::"real^'q::finite \<Rightarrow> real^'m"
+ assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" proof-
+ have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
+ using assms(2) unfolding differentiable_def by auto moreover
+ interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
+ interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
+ interpret h:bounded_bilinear h using assms by auto
+ have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
+ apply(rule Lim_linear[of "\<lambda>y. y - x" 0 "at x within s" f']) using Lim_sub[OF Lim_within_id Lim_const, of x x s]
+ unfolding id_def using assms(1) unfolding has_derivative_def by auto
+ hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
+ using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
+ have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
+ + h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)"
+ apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2) unfolding has_derivative_within by auto
+ guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
+ guess C using f'.pos_bounded .. note C=this
+ guess D using g'.pos_bounded .. note D=this
+ have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
+ have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" unfolding Lim_within proof(rule,rule) case goal1
+ hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
+ thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
+ fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
+ have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
+ also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
+ apply(rule pordered_semiring_class.mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
+ also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
+ also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
+ using as(3)[unfolded vector_dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
+ finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
+ unfolding vector_dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
+ have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))" unfolding linear_linear linear_def
+ unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR
+ unfolding h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right by auto
+ thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within
+ unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
+ h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
+ scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed
+
+lemma has_derivative_bilinear_at: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite" and f::"real^'p::finite \<Rightarrow> real^'m"
+ assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
+ using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto
+
+subsection {* Considering derivative R(^1)->R^n as a vector. *}
+
+definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('b) \<Rightarrow> (real net \<Rightarrow> bool)"
+(infixl "has'_vector'_derivative" 12) where
+ "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
+
+definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
+
+lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
+ shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
+proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
+ then interpret bounded_linear f' by auto
+ thus ?r unfolding vector_derivative_def has_vector_derivative_def
+ apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
+ using f' unfolding scaleR[THEN sym] by auto
+next assume ?r thus ?l unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed
+
+lemma vector_derivative_unique_at: fixes f::"real\<Rightarrow>real^'n::finite"
+ assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof-
+ have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1" apply(rule frechet_derivative_unique_at)
+ using assms[unfolded has_vector_derivative_def] unfolding has_derivative_at_dest_vec1[THEN sym] by auto
+ show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
+ hence "((\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1) (vec1 1) = ((\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1) (vec1 1)" using * by auto
+ ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
+
+lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> real^'n::finite"
+ assumes "a < b" "x \<in> {a..b}"
+ "(f has_vector_derivative f') (at x within {a..b})"
+ "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
+ have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1"
+ apply(rule frechet_derivative_unique_within_closed_interval[of "vec1 a" "vec1 b"])
+ using assms(3-)[unfolded has_vector_derivative_def]
+ unfolding has_derivative_within_dest_vec1[THEN sym] vec1_interval using assms(1-2) by auto
+ show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
+ hence "((\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1) (vec1 1) = ((\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1) (vec1 1)" using * by auto
+ ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
+
+lemma vector_derivative_at: fixes f::"real \<Rightarrow> real^'a::finite" shows
+ "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
+ apply(rule vector_derivative_unique_at) defer apply assumption
+ unfolding vector_derivative_works[THEN sym] differentiable_def
+ unfolding has_vector_derivative_def by auto
+
+lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> real^'a::finite"
+ assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
+ shows "vector_derivative f (at x within {a..b}) = f'"
+ apply(rule vector_derivative_unique_within_closed_interval)
+ using vector_derivative_works[unfolded differentiable_def]
+ using assms by(auto simp add:has_vector_derivative_def)
+
+lemma has_vector_derivative_within_subset: fixes f::"real \<Rightarrow> real^'a::finite" shows
+ "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
+ unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
+
+lemma has_vector_derivative_const: fixes c::"real^'n::finite" shows
+ "((\<lambda>x. c) has_vector_derivative 0) net"
+ unfolding has_vector_derivative_def using has_derivative_const by auto
+
+lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
+ unfolding has_vector_derivative_def using has_derivative_id by auto
+
+lemma has_vector_derivative_cmul: fixes f::"real \<Rightarrow> real^'a::finite"
+ shows "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
+ unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:group_simps)
+
+lemma has_vector_derivative_cmul_eq: fixes f::"real \<Rightarrow> real^'a::finite" assumes "c \<noteq> 0"
+ shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
+ apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
+ apply(rule has_vector_derivative_cmul) using assms by auto
+
+lemma has_vector_derivative_neg:
+ "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
+ unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
+
+lemma has_vector_derivative_add:
+ assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
+ shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
+ using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
+ unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
+
+lemma has_vector_derivative_sub:
+ assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
+ shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
+ using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
+ unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
+
+lemma has_vector_derivative_bilinear_within: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite"
+ assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof-
+ interpret bounded_bilinear h using assms by auto
+ show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def has_derivative_within_dest_vec1[THEN sym]], where h=h]
+ unfolding o_def vec1_dest_vec1 has_vector_derivative_def
+ unfolding has_derivative_within_dest_vec1[unfolded o_def, where f="\<lambda>x. h (f x) (g x)" and f'="\<lambda>d. h (f x) (d *\<^sub>R g') + h (d *\<^sub>R f') (g x)"]
+ using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed
+
+lemma has_vector_derivative_bilinear_at: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite"
+ assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
+ apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto
+
+lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
+ unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto
+
+lemma has_vector_derivative_transform_within:
+ assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x within s)"
+ shows "(g has_vector_derivative f') (at x within s)"
+ using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within)
+
+lemma has_vector_derivative_transform_at:
+ assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
+ shows "(g has_vector_derivative f') (at x)"
+ using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)
+
+lemma has_vector_derivative_transform_within_open:
+ assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_vector_derivative f') (at x)"
+ shows "(g has_vector_derivative f') (at x)"
+ using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open)
+
+lemma vector_diff_chain_at:
+ assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
+ shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
+ using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
+ unfolding o_def scaleR.scaleR_left by auto
+
+lemma vector_diff_chain_within:
+ assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
+ shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
+ using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
+ unfolding o_def scaleR.scaleR_left by auto
+
+end
+