--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Arcwise_Connected.thy Thu Jan 05 16:03:23 2017 +0000
@@ -0,0 +1,2213 @@
+(* Title: HOL/Analysis/Arcwise_Connected.thy
+ Authors: LC Paulson, based on material from HOL Light
+*)
+
+section \<open>Arcwise-connected sets\<close>
+
+theory Arcwise_Connected
+ imports Path_Connected Ordered_Euclidean_Space "~~/src/HOL/Number_Theory/Primes"
+
+begin
+
+subsection\<open>The Brouwer reduction theorem\<close>
+
+theorem Brouwer_reduction_theorem_gen:
+ fixes S :: "'a::euclidean_space set"
+ assumes "closed S" "\<phi> S"
+ and \<phi>: "\<And>F. \<lbrakk>\<And>n. closed(F n); \<And>n. \<phi>(F n); \<And>n. F(Suc n) \<subseteq> F n\<rbrakk> \<Longrightarrow> \<phi>(\<Inter>range F)"
+ obtains T where "T \<subseteq> S" "closed T" "\<phi> T" "\<And>U. \<lbrakk>U \<subseteq> S; closed U; \<phi> U\<rbrakk> \<Longrightarrow> \<not> (U \<subset> T)"
+proof -
+ obtain B :: "nat \<Rightarrow> 'a set"
+ where "inj B" "\<And>n. open(B n)" and open_cov: "\<And>S. open S \<Longrightarrow> \<exists>K. S = \<Union>(B ` K)"
+ by (metis Setcompr_eq_image that univ_second_countable_sequence)
+ define A where "A \<equiv> rec_nat S (\<lambda>n a. if \<exists>U. U \<subseteq> a \<and> closed U \<and> \<phi> U \<and> U \<inter> (B n) = {}
+ then @U. U \<subseteq> a \<and> closed U \<and> \<phi> U \<and> U \<inter> (B n) = {}
+ else a)"
+ have [simp]: "A 0 = S"
+ by (simp add: A_def)
+ have ASuc: "A(Suc n) = (if \<exists>U. U \<subseteq> A n \<and> closed U \<and> \<phi> U \<and> U \<inter> (B n) = {}
+ then @U. U \<subseteq> A n \<and> closed U \<and> \<phi> U \<and> U \<inter> (B n) = {}
+ else A n)" for n
+ by (auto simp: A_def)
+ have sub: "\<And>n. A(Suc n) \<subseteq> A n"
+ by (auto simp: ASuc dest!: someI_ex)
+ have subS: "A n \<subseteq> S" for n
+ by (induction n) (use sub in auto)
+ have clo: "closed (A n) \<and> \<phi> (A n)" for n
+ by (induction n) (auto simp: assms ASuc dest!: someI_ex)
+ show ?thesis
+ proof
+ show "\<Inter>range A \<subseteq> S"
+ using \<open>\<And>n. A n \<subseteq> S\<close> by blast
+ show "closed (INTER UNIV A)"
+ using clo by blast
+ show "\<phi> (INTER UNIV A)"
+ by (simp add: clo \<phi> sub)
+ show "\<not> U \<subset> INTER UNIV A" if "U \<subseteq> S" "closed U" "\<phi> U" for U
+ proof -
+ have "\<exists>y. x \<notin> A y" if "x \<notin> U" and Usub: "U \<subseteq> (\<Inter>x. A x)" for x
+ proof -
+ obtain e where "e > 0" and e: "ball x e \<subseteq> -U"
+ using \<open>closed U\<close> \<open>x \<notin> U\<close> openE [of "-U"] by blast
+ moreover obtain K where K: "ball x e = UNION K B"
+ using open_cov [of "ball x e"] by auto
+ ultimately have "UNION K B \<subseteq> -U"
+ by blast
+ have "K \<noteq> {}"
+ using \<open>0 < e\<close> \<open>ball x e = UNION K B\<close> by auto
+ then obtain n where "n \<in> K" "x \<in> B n"
+ by (metis K UN_E \<open>0 < e\<close> centre_in_ball)
+ then have "U \<inter> B n = {}"
+ using K e by auto
+ show ?thesis
+ proof (cases "\<exists>U\<subseteq>A n. closed U \<and> \<phi> U \<and> U \<inter> B n = {}")
+ case True
+ then show ?thesis
+ apply (rule_tac x="Suc n" in exI)
+ apply (simp add: ASuc)
+ apply (erule someI2_ex)
+ using \<open>x \<in> B n\<close> by blast
+ next
+ case False
+ then show ?thesis
+ by (meson Inf_lower Usub \<open>U \<inter> B n = {}\<close> \<open>\<phi> U\<close> \<open>closed U\<close> range_eqI subset_trans)
+ qed
+ qed
+ with that show ?thesis
+ by (meson Inter_iff psubsetE rangeI subsetI)
+ qed
+ qed
+qed
+
+corollary Brouwer_reduction_theorem:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S" "\<phi> S" "S \<noteq> {}"
+ and \<phi>: "\<And>F. \<lbrakk>\<And>n. compact(F n); \<And>n. F n \<noteq> {}; \<And>n. \<phi>(F n); \<And>n. F(Suc n) \<subseteq> F n\<rbrakk> \<Longrightarrow> \<phi>(\<Inter>range F)"
+ obtains T where "T \<subseteq> S" "compact T" "T \<noteq> {}" "\<phi> T"
+ "\<And>U. \<lbrakk>U \<subseteq> S; closed U; U \<noteq> {}; \<phi> U\<rbrakk> \<Longrightarrow> \<not> (U \<subset> T)"
+proof (rule Brouwer_reduction_theorem_gen [of S "\<lambda>T. T \<noteq> {} \<and> T \<subseteq> S \<and> \<phi> T"])
+ fix F
+ assume cloF: "\<And>n. closed (F n)"
+ and F: "\<And>n. F n \<noteq> {} \<and> F n \<subseteq> S \<and> \<phi> (F n)" and Fsub: "\<And>n. F (Suc n) \<subseteq> F n"
+ show "INTER UNIV F \<noteq> {} \<and> INTER UNIV F \<subseteq> S \<and> \<phi> (INTER UNIV F)"
+ proof (intro conjI)
+ show "INTER UNIV F \<noteq> {}"
+ apply (rule compact_nest)
+ apply (meson F cloF \<open>compact S\<close> seq_compact_closed_subset seq_compact_eq_compact)
+ apply (simp add: F)
+ by (meson Fsub lift_Suc_antimono_le)
+ show " INTER UNIV F \<subseteq> S"
+ using F by blast
+ show "\<phi> (INTER UNIV F)"
+ by (metis F Fsub \<phi> \<open>compact S\<close> cloF closed_Int_compact inf.orderE)
+ qed
+next
+ show "S \<noteq> {} \<and> S \<subseteq> S \<and> \<phi> S"
+ by (simp add: assms)
+qed (meson assms compact_imp_closed seq_compact_closed_subset seq_compact_eq_compact)+
+
+
+subsection\<open>Arcwise Connections\<close>
+
+subsection\<open>Density of points with dyadic rational coordinates.\<close>
+
+proposition closure_dyadic_rationals:
+ "closure (\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>.
+ { \<Sum>i :: 'a :: euclidean_space \<in> Basis. (f i / 2^k) *\<^sub>R i }) = UNIV"
+proof -
+ have "x \<in> closure (\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>. {\<Sum>i \<in> Basis. (f i / 2^k) *\<^sub>R i})" for x::'a
+ proof (clarsimp simp: closure_approachable)
+ fix e::real
+ assume "e > 0"
+ then obtain k where k: "(1/2)^k < e/DIM('a)"
+ by (meson DIM_positive divide_less_eq_1_pos of_nat_0_less_iff one_less_numeral_iff real_arch_pow_inv semiring_norm(76) zero_less_divide_iff zero_less_numeral)
+ have "dist (\<Sum>i\<in>Basis. (real_of_int \<lfloor>2^k*(x \<bullet> i)\<rfloor> / 2^k) *\<^sub>R i) x =
+ dist (\<Sum>i\<in>Basis. (real_of_int \<lfloor>2^k*(x \<bullet> i)\<rfloor> / 2^k) *\<^sub>R i) (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
+ by (simp add: euclidean_representation)
+ also have "... = norm ((\<Sum>i\<in>Basis. (real_of_int \<lfloor>2^k*(x \<bullet> i)\<rfloor> / 2^k) *\<^sub>R i - (x \<bullet> i) *\<^sub>R i))"
+ by (simp add: dist_norm sum_subtractf)
+ also have "... \<le> DIM('a)*((1/2)^k)"
+ proof (rule sum_norm_bound, simp add: algebra_simps)
+ fix i::'a
+ assume "i \<in> Basis"
+ then have "norm ((real_of_int \<lfloor>x \<bullet> i*2^k\<rfloor> / 2^k) *\<^sub>R i - (x \<bullet> i) *\<^sub>R i) =
+ \<bar>real_of_int \<lfloor>x \<bullet> i*2^k\<rfloor> / 2^k - x \<bullet> i\<bar>"
+ by (simp add: scaleR_left_diff_distrib [symmetric])
+ also have "... \<le> (1/2) ^ k"
+ by (simp add: divide_simps) linarith
+ finally show "norm ((real_of_int \<lfloor>x \<bullet> i*2^k\<rfloor> / 2^k) *\<^sub>R i - (x \<bullet> i) *\<^sub>R i) \<le> (1/2) ^ k" .
+ qed
+ also have "... < DIM('a)*(e/DIM('a))"
+ using DIM_positive k linordered_comm_semiring_strict_class.comm_mult_strict_left_mono of_nat_0_less_iff by blast
+ also have "... = e"
+ by simp
+ finally have "dist (\<Sum>i\<in>Basis. (\<lfloor>2^k*(x \<bullet> i)\<rfloor> / 2^k) *\<^sub>R i) x < e" .
+ then
+ show "\<exists>k. \<exists>f \<in> Basis \<rightarrow> \<int>. dist (\<Sum>b\<in>Basis. (f b / 2^k) *\<^sub>R b) x < e"
+ apply (rule_tac x=k in exI)
+ apply (rule_tac x="\<lambda>i. of_int (floor (2^k*(x \<bullet> i)))" in bexI)
+ apply auto
+ done
+ qed
+ then show ?thesis by auto
+qed
+
+corollary closure_rational_coordinates:
+ "closure (\<Union>f \<in> Basis \<rightarrow> \<rat>. { \<Sum>i :: 'a :: euclidean_space \<in> Basis. f i *\<^sub>R i }) = UNIV"
+proof -
+ have *: "(\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>. { \<Sum>i::'a \<in> Basis. (f i / 2^k) *\<^sub>R i })
+ \<subseteq> (\<Union>f \<in> Basis \<rightarrow> \<rat>. { \<Sum>i \<in> Basis. f i *\<^sub>R i })"
+ proof clarsimp
+ fix k and f :: "'a \<Rightarrow> real"
+ assume f: "f \<in> Basis \<rightarrow> \<int>"
+ show "\<exists>x \<in> Basis \<rightarrow> \<rat>. (\<Sum>i \<in> Basis. (f i / 2^k) *\<^sub>R i) = (\<Sum>i \<in> Basis. x i *\<^sub>R i)"
+ apply (rule_tac x="\<lambda>i. f i / 2^k" in bexI)
+ using Ints_subset_Rats f by auto
+ qed
+ show ?thesis
+ using closure_dyadic_rationals closure_mono [OF *] by blast
+qed
+
+lemma closure_dyadic_rationals_in_convex_set:
+ "\<lbrakk>convex S; interior S \<noteq> {}\<rbrakk>
+ \<Longrightarrow> closure(S \<inter>
+ (\<Union>k. \<Union>f \<in> Basis \<rightarrow> \<int>.
+ { \<Sum>i :: 'a :: euclidean_space \<in> Basis. (f i / 2^k) *\<^sub>R i })) =
+ closure S"
+ by (simp add: closure_dyadic_rationals closure_convex_Int_superset)
+
+lemma closure_rationals_in_convex_set:
+ "\<lbrakk>convex S; interior S \<noteq> {}\<rbrakk>
+ \<Longrightarrow> closure(S \<inter> (\<Union>f \<in> Basis \<rightarrow> \<rat>. { \<Sum>i :: 'a :: euclidean_space \<in> Basis. f i *\<^sub>R i })) =
+ closure S"
+ by (simp add: closure_rational_coordinates closure_convex_Int_superset)
+
+
+text\<open> Every path between distinct points contains an arc, and hence
+path connection is equivalent to arcwise connection for distinct points.
+The proof is based on Whyburn's "Topological Analysis".\<close>
+
+lemma closure_dyadic_rationals_in_convex_set_pos_1:
+ fixes S :: "real set"
+ assumes "convex S" and intnz: "interior S \<noteq> {}" and pos: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> x"
+ shows "closure(S \<inter> (\<Union>k m. {of_nat m / 2^k})) = closure S"
+proof -
+ have "\<exists>m. f 1/2^k = real m / 2^k" if "(f 1) / 2^k \<in> S" "f 1 \<in> \<int>" for k and f :: "real \<Rightarrow> real"
+ using that by (force simp: Ints_def zero_le_divide_iff power_le_zero_eq dest: pos zero_le_imp_eq_int)
+ then have "S \<inter> (\<Union>k m. {real m / 2^k}) = S \<inter>
+ (\<Union>k. \<Union>f\<in>Basis \<rightarrow> \<int>. {\<Sum>i\<in>Basis. (f i / 2^k) *\<^sub>R i})"
+ by force
+ then show ?thesis
+ using closure_dyadic_rationals_in_convex_set [OF \<open>convex S\<close> intnz] by simp
+qed
+
+
+definition dyadics :: "'a::field_char_0 set" where "dyadics \<equiv> \<Union>k m. {of_nat m / 2^k}"
+
+lemma real_in_dyadics [simp]: "real m \<in> dyadics"
+ apply (simp add: dyadics_def)
+ by (metis divide_numeral_1 numeral_One power_0)
+
+lemma nat_neq_4k1: "of_nat m \<noteq> (4 * of_nat k + 1) / (2 * 2^n :: 'a::field_char_0)"
+proof
+ assume "of_nat m = (4 * of_nat k + 1) / (2 * 2^n :: 'a)"
+ then have "of_nat (m * (2 * 2^n)) = (of_nat (Suc (4 * k)) :: 'a)"
+ by (simp add: divide_simps)
+ then have "m * (2 * 2^n) = Suc (4 * k)"
+ using of_nat_eq_iff by blast
+ then have "odd (m * (2 * 2^n))"
+ by simp
+ then show False
+ by simp
+qed
+
+lemma nat_neq_4k3: "of_nat m \<noteq> (4 * of_nat k + 3) / (2 * 2^n :: 'a::field_char_0)"
+proof
+ assume "of_nat m = (4 * of_nat k + 3) / (2 * 2^n :: 'a)"
+ then have "of_nat (m * (2 * 2^n)) = (of_nat (4 * k + 3) :: 'a)"
+ by (simp add: divide_simps)
+ then have "m * (2 * 2^n) = (4 * k) + 3"
+ using of_nat_eq_iff by blast
+ then have "odd (m * (2 * 2^n))"
+ by simp
+ then show False
+ by simp
+qed
+
+lemma iff_4k:
+ assumes "r = real k" "odd k"
+ shows "(4 * real m + r) / (2 * 2^n) = (4 * real m' + r) / (2 * 2 ^ n') \<longleftrightarrow> m=m' \<and> n=n'"
+proof -
+ { assume "(4 * real m + r) / (2 * 2^n) = (4 * real m' + r) / (2 * 2 ^ n')"
+ then have "real ((4 * m + k) * (2 * 2 ^ n')) = real ((4 * m' + k) * (2 * 2^n))"
+ using assms by (auto simp: field_simps)
+ then have "(4 * m + k) * (2 * 2 ^ n') = (4 * m' + k) * (2 * 2^n)"
+ using of_nat_eq_iff by blast
+ then have "(4 * m + k) * (2 ^ n') = (4 * m' + k) * (2^n)"
+ by linarith
+ then obtain "4*m + k = 4*m' + k" "n=n'"
+ apply (rule prime_power_cancel2 [OF two_is_prime_nat])
+ using assms by auto
+ then have "m=m'" "n=n'"
+ by auto
+ }
+ then show ?thesis by blast
+qed
+
+lemma neq_4k1_k43: "(4 * real m + 1) / (2 * 2^n) \<noteq> (4 * real m' + 3) / (2 * 2 ^ n')"
+proof
+ assume "(4 * real m + 1) / (2 * 2^n) = (4 * real m' + 3) / (2 * 2 ^ n')"
+ then have "real (Suc (4 * m) * (2 * 2 ^ n')) = real ((4 * m' + 3) * (2 * 2^n))"
+ by (auto simp: field_simps)
+ then have "Suc (4 * m) * (2 * 2 ^ n') = (4 * m' + 3) * (2 * 2^n)"
+ using of_nat_eq_iff by blast
+ then have "Suc (4 * m) * (2 ^ n') = (4 * m' + 3) * (2^n)"
+ by linarith
+ then have "Suc (4 * m) = (4 * m' + 3)"
+ by (rule prime_power_cancel2 [OF two_is_prime_nat]) auto
+ then have "1 + 2 * m' = 2 * m"
+ using \<open>Suc (4 * m) = 4 * m' + 3\<close> by linarith
+ then show False
+ using even_Suc by presburger
+qed
+
+lemma dyadic_413_cases:
+ obtains "(of_nat m::'a::field_char_0) / 2^k \<in> Nats"
+ | m' k' where "k' < k" "(of_nat m:: 'a) / 2^k = of_nat (4*m' + 1) / 2^Suc k'"
+ | m' k' where "k' < k" "(of_nat m:: 'a) / 2^k = of_nat (4*m' + 3) / 2^Suc k'"
+proof (cases "m>0")
+ case False
+ then have "m=0" by simp
+ with that show ?thesis by auto
+next
+ case True
+ obtain k' m' where m': "odd m'" and k': "m = m' * 2^k'"
+ using prime_power_canonical [OF two_is_prime_nat True] by blast
+ then obtain q r where q: "m' = 4*q + r" and r: "r < 4"
+ by (metis not_add_less2 split_div zero_neq_numeral)
+ show ?thesis
+ proof (cases "k \<le> k'")
+ case True
+ have "(of_nat m:: 'a) / 2^k = of_nat m' * (2 ^ k' / 2^k)"
+ using k' by (simp add: field_simps)
+ also have "... = (of_nat m'::'a) * 2 ^ (k'-k)"
+ using k' True by (simp add: power_diff)
+ also have "... \<in> \<nat>"
+ by (metis Nats_mult of_nat_in_Nats of_nat_numeral of_nat_power)
+ finally show ?thesis by (auto simp: that)
+ next
+ case False
+ then obtain kd where kd: "Suc kd = k - k'"
+ using Suc_diff_Suc not_less by blast
+ have "(of_nat m:: 'a) / 2^k = of_nat m' * (2 ^ k' / 2^k)"
+ using k' by (simp add: field_simps)
+ also have "... = (of_nat m'::'a) / 2 ^ (k-k')"
+ using k' False by (simp add: power_diff)
+ also have "... = ((of_nat r + 4 * of_nat q)::'a) / 2 ^ (k-k')"
+ using q by force
+ finally have meq: "(of_nat m:: 'a) / 2^k = (of_nat r + 4 * of_nat q) / 2 ^ (k - k')" .
+ have "r \<noteq> 0" "r \<noteq> 2"
+ using q m' by presburger+
+ with r consider "r = 1" | "r = 3"
+ by linarith
+ then show ?thesis
+ proof cases
+ assume "r = 1"
+ with meq kd that(2) [of kd q] show ?thesis
+ by simp
+ next
+ assume "r = 3"
+ with meq kd that(3) [of kd q] show ?thesis
+ by simp
+ qed
+ qed
+qed
+
+
+lemma dyadics_iff:
+ "(dyadics :: 'a::field_char_0 set) =
+ Nats \<union> (\<Union>k m. {of_nat (4*m + 1) / 2^Suc k}) \<union> (\<Union>k m. {of_nat (4*m + 3) / 2^Suc k})"
+ (is "_ = ?rhs")
+proof
+ show "dyadics \<subseteq> ?rhs"
+ unfolding dyadics_def
+ apply clarify
+ apply (rule dyadic_413_cases, force+)
+ done
+next
+ show "?rhs \<subseteq> dyadics"
+ apply (clarsimp simp: dyadics_def Nats_def simp del: power_Suc)
+ apply (intro conjI subsetI)
+ apply (auto simp del: power_Suc)
+ apply (metis divide_numeral_1 numeral_One power_0)
+ apply (metis of_nat_Suc of_nat_mult of_nat_numeral)
+ by (metis of_nat_add of_nat_mult of_nat_numeral)
+qed
+
+
+function (domintros) dyad_rec :: "[nat \<Rightarrow> 'a, 'a\<Rightarrow>'a, 'a\<Rightarrow>'a, real] \<Rightarrow> 'a" where
+ "dyad_rec b l r (real m) = b m"
+ | "dyad_rec b l r ((4 * real m + 1) / 2 ^ (Suc n)) = l (dyad_rec b l r ((2*m + 1) / 2^n))"
+ | "dyad_rec b l r ((4 * real m + 3) / 2 ^ (Suc n)) = r (dyad_rec b l r ((2*m + 1) / 2^n))"
+ | "x \<notin> dyadics \<Longrightarrow> dyad_rec b l r x = undefined"
+ using iff_4k [of _ 1] iff_4k [of _ 3]
+ apply (simp_all add: nat_neq_4k1 nat_neq_4k3 neq_4k1_k43, atomize_elim)
+ apply (fastforce simp add: dyadics_iff Nats_def field_simps)+
+ done
+
+lemma dyadics_levels: "dyadics = (\<Union>K. \<Union>k<K. \<Union> m. {of_nat m / 2^k})"
+ unfolding dyadics_def by auto
+
+lemma dyad_rec_level_termination:
+ assumes "k < K"
+ shows "dyad_rec_dom(b, l, r, real m / 2^k)"
+ using assms
+proof (induction K arbitrary: k m)
+ case 0
+ then show ?case by auto
+next
+ case (Suc K)
+ then consider "k = K" | "k < K"
+ using less_antisym by blast
+ then show ?case
+ proof cases
+ assume "k = K"
+ show ?case
+ proof (rule dyadic_413_cases [of m k, where 'a=real])
+ show "real m / 2^k \<in> \<nat> \<Longrightarrow> dyad_rec_dom (b, l, r, real m / 2^k)"
+ by (force simp: Nats_def nat_neq_4k1 nat_neq_4k3 intro: dyad_rec.domintros)
+ show ?case if "k' < k" and eq: "real m / 2^k = real (4 * m' + 1) / 2^Suc k'" for m' k'
+ proof -
+ have "dyad_rec_dom (b, l, r, (4 * real m' + 1) / 2^Suc k')"
+ proof (rule dyad_rec.domintros)
+ fix m n
+ assume "(4 * real m' + 1) / (2 * 2 ^ k') = (4 * real m + 1) / (2 * 2^n)"
+ then have "m' = m" "k' = n" using iff_4k [of _ 1]
+ by auto
+ have "dyad_rec_dom (b, l, r, real (2 * m + 1) / 2 ^ k')"
+ using Suc.IH \<open>k = K\<close> \<open>k' < k\<close> by blast
+ then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)"
+ using \<open>k' = n\<close> by (auto simp: algebra_simps)
+ next
+ fix m n
+ assume "(4 * real m' + 1) / (2 * 2 ^ k') = (4 * real m + 3) / (2 * 2^n)"
+ then have "False"
+ by (metis neq_4k1_k43)
+ then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)" ..
+ qed
+ then show ?case by (simp add: eq add_ac)
+ qed
+ show ?case if "k' < k" and eq: "real m / 2^k = real (4 * m' + 3) / 2^Suc k'" for m' k'
+ proof -
+ have "dyad_rec_dom (b, l, r, (4 * real m' + 3) / 2^Suc k')"
+ proof (rule dyad_rec.domintros)
+ fix m n
+ assume "(4 * real m' + 3) / (2 * 2 ^ k') = (4 * real m + 1) / (2 * 2^n)"
+ then have "False"
+ by (metis neq_4k1_k43)
+ then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)" ..
+ next
+ fix m n
+ assume "(4 * real m' + 3) / (2 * 2 ^ k') = (4 * real m + 3) / (2 * 2^n)"
+ then have "m' = m" "k' = n" using iff_4k [of _ 3]
+ by auto
+ have "dyad_rec_dom (b, l, r, real (2 * m + 1) / 2 ^ k')"
+ using Suc.IH \<open>k = K\<close> \<open>k' < k\<close> by blast
+ then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)"
+ using \<open>k' = n\<close> by (auto simp: algebra_simps)
+ qed
+ then show ?case by (simp add: eq add_ac)
+ qed
+ qed
+ next
+ assume "k < K"
+ then show ?case
+ using Suc.IH by blast
+ qed
+qed
+
+
+lemma dyad_rec_termination: "x \<in> dyadics \<Longrightarrow> dyad_rec_dom(b,l,r,x)"
+ by (auto simp: dyadics_levels intro: dyad_rec_level_termination)
+
+lemma dyad_rec_of_nat [simp]: "dyad_rec b l r (real m) = b m"
+ by (simp add: dyad_rec.psimps dyad_rec_termination)
+
+lemma dyad_rec_41 [simp]: "dyad_rec b l r ((4 * real m + 1) / 2 ^ (Suc n)) = l (dyad_rec b l r ((2*m + 1) / 2^n))"
+ apply (rule dyad_rec.psimps)
+ by (metis dyad_rec_level_termination lessI add.commute of_nat_Suc of_nat_mult of_nat_numeral)
+
+
+lemma dyad_rec_43 [simp]: "dyad_rec b l r ((4 * real m + 3) / 2 ^ (Suc n)) = r (dyad_rec b l r ((2*m + 1) / 2^n))"
+ apply (rule dyad_rec.psimps)
+ by (metis dyad_rec_level_termination lessI of_nat_add of_nat_mult of_nat_numeral)
+
+lemma dyad_rec_41_times2:
+ assumes "n > 0"
+ shows "dyad_rec b l r (2 * ((4 * real m + 1) / 2^Suc n)) = l (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
+proof -
+ obtain n' where n': "n = Suc n'"
+ using assms not0_implies_Suc by blast
+ have "dyad_rec b l r (2 * ((4 * real m + 1) / 2^Suc n)) = dyad_rec b l r ((2 * (4 * real m + 1)) / (2 * 2^n))"
+ by auto
+ also have "... = dyad_rec b l r ((4 * real m + 1) / 2^n)"
+ by (subst mult_divide_mult_cancel_left) auto
+ also have "... = l (dyad_rec b l r ((2 * real m + 1) / 2 ^ n'))"
+ by (simp add: add.commute [of 1] n' del: power_Suc)
+ also have "... = l (dyad_rec b l r ((2 * (2 * real m + 1)) / (2 * 2 ^ n')))"
+ by (subst mult_divide_mult_cancel_left) auto
+ also have "... = l (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
+ by (simp add: add.commute n')
+ finally show ?thesis .
+qed
+
+lemma dyad_rec_43_times2:
+ assumes "n > 0"
+ shows "dyad_rec b l r (2 * ((4 * real m + 3) / 2^Suc n)) = r (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
+proof -
+ obtain n' where n': "n = Suc n'"
+ using assms not0_implies_Suc by blast
+ have "dyad_rec b l r (2 * ((4 * real m + 3) / 2^Suc n)) = dyad_rec b l r ((2 * (4 * real m + 3)) / (2 * 2^n))"
+ by auto
+ also have "... = dyad_rec b l r ((4 * real m + 3) / 2^n)"
+ by (subst mult_divide_mult_cancel_left) auto
+ also have "... = r (dyad_rec b l r ((2 * real m + 1) / 2 ^ n'))"
+ by (simp add: n' del: power_Suc)
+ also have "... = r (dyad_rec b l r ((2 * (2 * real m + 1)) / (2 * 2 ^ n')))"
+ by (subst mult_divide_mult_cancel_left) auto
+ also have "... = r (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
+ by (simp add: n')
+ finally show ?thesis .
+qed
+
+definition dyad_rec2
+ where "dyad_rec2 u v lc rc x =
+ dyad_rec (\<lambda>z. (u,v)) (\<lambda>(a,b). (a, lc a b (midpoint a b))) (\<lambda>(a,b). (rc a b (midpoint a b), b)) (2*x)"
+
+abbreviation leftrec where "leftrec u v lc rc x \<equiv> fst (dyad_rec2 u v lc rc x)"
+abbreviation rightrec where "rightrec u v lc rc x \<equiv> snd (dyad_rec2 u v lc rc x)"
+
+lemma leftrec_base: "leftrec u v lc rc (real m / 2) = u"
+ by (simp add: dyad_rec2_def)
+
+lemma leftrec_41: "n > 0 \<Longrightarrow> leftrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) = leftrec u v lc rc ((2 * real m + 1) / 2^n)"
+ apply (simp only: dyad_rec2_def dyad_rec_41_times2)
+ apply (simp add: case_prod_beta)
+ done
+
+lemma leftrec_43: "n > 0 \<Longrightarrow>
+ leftrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) =
+ rc (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n))
+ (midpoint (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n)))"
+ apply (simp only: dyad_rec2_def dyad_rec_43_times2)
+ apply (simp add: case_prod_beta)
+ done
+
+lemma rightrec_base: "rightrec u v lc rc (real m / 2) = v"
+ by (simp add: dyad_rec2_def)
+
+lemma rightrec_41: "n > 0 \<Longrightarrow>
+ rightrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) =
+ lc (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n))
+ (midpoint (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n)))"
+ apply (simp only: dyad_rec2_def dyad_rec_41_times2)
+ apply (simp add: case_prod_beta)
+ done
+
+lemma rightrec_43: "n > 0 \<Longrightarrow> rightrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) = rightrec u v lc rc ((2 * real m + 1) / 2^n)"
+ apply (simp only: dyad_rec2_def dyad_rec_43_times2)
+ apply (simp add: case_prod_beta)
+ done
+
+lemma dyadics_in_open_unit_interval:
+ "{0<..<1} \<inter> (\<Union>k m. {real m / 2^k}) = (\<Union>k. \<Union>m \<in> {0<..<2^k}. {real m / 2^k})"
+ by (auto simp: divide_simps)
+
+
+
+theorem homeomorphic_monotone_image_interval:
+ fixes f :: "real \<Rightarrow> 'a::{real_normed_vector,complete_space}"
+ assumes cont_f: "continuous_on {0..1} f"
+ and conn: "\<And>y. connected ({0..1} \<inter> f -` {y})"
+ and f_1not0: "f 1 \<noteq> f 0"
+ shows "(f ` {0..1}) homeomorphic {0..1::real}"
+proof -
+ have "\<exists>c d. a \<le> c \<and> c \<le> m \<and> m \<le> d \<and> d \<le> b \<and>
+ (\<forall>x \<in> {c..d}. f x = f m) \<and>
+ (\<forall>x \<in> {a..<c}. (f x \<noteq> f m)) \<and>
+ (\<forall>x \<in> {d<..b}. (f x \<noteq> f m)) \<and>
+ (\<forall>x \<in> {a..<c}. \<forall>y \<in> {d<..b}. f x \<noteq> f y)"
+ if m: "m \<in> {a..b}" and ab01: "{a..b} \<subseteq> {0..1}" for a b m
+ proof -
+ have comp: "compact (f -` {f m} \<inter> {0..1})"
+ by (simp add: compact_eq_bounded_closed bounded_Int closed_vimage_Int cont_f)
+ obtain c0 d0 where cd0: "{0..1} \<inter> f -` {f m} = {c0..d0}"
+ using connected_compact_interval_1 [of "{0..1} \<inter> f -` {f m}"] conn comp
+ by (metis Int_commute)
+ with that have "m \<in> cbox c0 d0"
+ by auto
+ obtain c d where cd: "{a..b} \<inter> f -` {f m} = {c..d}"
+ apply (rule_tac c="max a c0" and d="min b d0" in that)
+ using ab01 cd0 by auto
+ then have cdab: "{c..d} \<subseteq> {a..b}"
+ by blast
+ show ?thesis
+ proof (intro exI conjI ballI)
+ show "a \<le> c" "d \<le> b"
+ using cdab cd m by auto
+ show "c \<le> m" "m \<le> d"
+ using cd m by auto
+ show "\<And>x. x \<in> {c..d} \<Longrightarrow> f x = f m"
+ using cd by blast
+ show "f x \<noteq> f m" if "x \<in> {a..<c}" for x
+ using that m cd [THEN equalityD1, THEN subsetD] \<open>c \<le> m\<close> by force
+ show "f x \<noteq> f m" if "x \<in> {d<..b}" for x
+ using that m cd [THEN equalityD1, THEN subsetD, of x] \<open>m \<le> d\<close> by force
+ show "f x \<noteq> f y" if "x \<in> {a..<c}" "y \<in> {d<..b}" for x y
+ proof (cases "f x = f m \<or> f y = f m")
+ case True
+ then show ?thesis
+ using \<open>\<And>x. x \<in> {a..<c} \<Longrightarrow> f x \<noteq> f m\<close> that by auto
+ next
+ case False
+ have False if "f x = f y"
+ proof -
+ have "x \<le> m" "m \<le> y"
+ using \<open>c \<le> m\<close> \<open>x \<in> {a..<c}\<close> \<open>m \<le> d\<close> \<open>y \<in> {d<..b}\<close> by auto
+ then have "x \<in> ({0..1} \<inter> f -` {f y})" "y \<in> ({0..1} \<inter> f -` {f y})"
+ using \<open>x \<in> {a..<c}\<close> \<open>y \<in> {d<..b}\<close> ab01 by (auto simp: that)
+ then have "m \<in> ({0..1} \<inter> f -` {f y})"
+ by (meson \<open>m \<le> y\<close> \<open>x \<le> m\<close> is_interval_connected_1 conn [of "f y"] is_interval_1)
+ with False show False by auto
+ qed
+ then show ?thesis by auto
+ qed
+ qed
+ qed
+ then obtain leftcut rightcut where LR:
+ "\<And>a b m. \<lbrakk>m \<in> {a..b}; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow>
+ (a \<le> leftcut a b m \<and> leftcut a b m \<le> m \<and> m \<le> rightcut a b m \<and> rightcut a b m \<le> b \<and>
+ (\<forall>x \<in> {leftcut a b m..rightcut a b m}. f x = f m) \<and>
+ (\<forall>x \<in> {a..<leftcut a b m}. f x \<noteq> f m) \<and>
+ (\<forall>x \<in> {rightcut a b m<..b}. f x \<noteq> f m) \<and>
+ (\<forall>x \<in> {a..<leftcut a b m}. \<forall>y \<in> {rightcut a b m<..b}. f x \<noteq> f y))"
+ apply atomize
+ apply (clarsimp simp only: imp_conjL [symmetric] choice_iff choice_iff')
+ apply (rule that, blast)
+ done
+ then have left_right: "\<And>a b m. \<lbrakk>m \<in> {a..b}; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow> a \<le> leftcut a b m \<and> rightcut a b m \<le> b"
+ and left_right_m: "\<And>a b m. \<lbrakk>m \<in> {a..b}; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow> leftcut a b m \<le> m \<and> m \<le> rightcut a b m"
+ by auto
+ have left_neq: "\<lbrakk>a \<le> x; x < leftcut a b m; a \<le> m; m \<le> b; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow> f x \<noteq> f m"
+ and right_neq: "\<lbrakk>rightcut a b m < x; x \<le> b; a \<le> m; m \<le> b; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow> f x \<noteq> f m"
+ and left_right_neq: "\<lbrakk>a \<le> x; x < leftcut a b m; rightcut a b m < y; y \<le> b; a \<le> m; m \<le> b; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow> f x \<noteq> f m"
+ and feqm: "\<lbrakk>leftcut a b m \<le> x; x \<le> rightcut a b m; a \<le> m; m \<le> b; {a..b} \<subseteq> {0..1}\<rbrakk>
+ \<Longrightarrow> f x = f m" for a b m x y
+ by (meson atLeastAtMost_iff greaterThanAtMost_iff atLeastLessThan_iff LR)+
+ have f_eqI: "\<And>a b m x y. \<lbrakk>leftcut a b m \<le> x; x \<le> rightcut a b m; leftcut a b m \<le> y; y \<le> rightcut a b m;
+ a \<le> m; m \<le> b; {a..b} \<subseteq> {0..1}\<rbrakk> \<Longrightarrow> f x = f y"
+ by (metis feqm)
+ define u where "u \<equiv> rightcut 0 1 0"
+ have lc[simp]: "leftcut 0 1 0 = 0" and u01: "0 \<le> u" "u \<le> 1"
+ using LR [of 0 0 1] by (auto simp: u_def)
+ have f0u: "\<And>x. x \<in> {0..u} \<Longrightarrow> f x = f 0"
+ using LR [of 0 0 1] unfolding u_def [symmetric]
+ by (metis \<open>leftcut 0 1 0 = 0\<close> atLeastAtMost_iff order_refl zero_le_one)
+ have fu1: "\<And>x. x \<in> {u<..1} \<Longrightarrow> f x \<noteq> f 0"
+ using LR [of 0 0 1] unfolding u_def [symmetric] by fastforce
+ define v where "v \<equiv> leftcut u 1 1"
+ have rc[simp]: "rightcut u 1 1 = 1" and v01: "u \<le> v" "v \<le> 1"
+ using LR [of 1 u 1] u01 by (auto simp: v_def)
+ have fuv: "\<And>x. x \<in> {u..<v} \<Longrightarrow> f x \<noteq> f 1"
+ using LR [of 1 u 1] u01 v_def by fastforce
+ have f0v: "\<And>x. x \<in> {0..<v} \<Longrightarrow> f x \<noteq> f 1"
+ by (metis f_1not0 atLeastAtMost_iff atLeastLessThan_iff f0u fuv linear)
+ have fv1: "\<And>x. x \<in> {v..1} \<Longrightarrow> f x = f 1"
+ using LR [of 1 u 1] u01 v_def by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl rc)
+ define a where "a \<equiv> leftrec u v leftcut rightcut"
+ define b where "b \<equiv> rightrec u v leftcut rightcut"
+ define c where "c \<equiv> \<lambda>x. midpoint (a x) (b x)"
+ have a_real [simp]: "a (real j) = u" for j
+ using a_def leftrec_base
+ by (metis nonzero_mult_div_cancel_right of_nat_mult of_nat_numeral zero_neq_numeral)
+ have b_real [simp]: "b (real j) = v" for j
+ using b_def rightrec_base
+ by (metis nonzero_mult_div_cancel_right of_nat_mult of_nat_numeral zero_neq_numeral)
+ have a41: "a ((4 * real m + 1) / 2^Suc n) = a ((2 * real m + 1) / 2^n)" if "n > 0" for m n
+ using that a_def leftrec_41 by blast
+ have b41: "b ((4 * real m + 1) / 2^Suc n) =
+ leftcut (a ((2 * real m + 1) / 2^n))
+ (b ((2 * real m + 1) / 2^n))
+ (c ((2 * real m + 1) / 2^n))" if "n > 0" for m n
+ using that a_def b_def c_def rightrec_41 by blast
+ have a43: "a ((4 * real m + 3) / 2^Suc n) =
+ rightcut (a ((2 * real m + 1) / 2^n))
+ (b ((2 * real m + 1) / 2^n))
+ (c ((2 * real m + 1) / 2^n))" if "n > 0" for m n
+ using that a_def b_def c_def leftrec_43 by blast
+ have b43: "b ((4 * real m + 3) / 2^Suc n) = b ((2 * real m + 1) / 2^n)" if "n > 0" for m n
+ using that b_def rightrec_43 by blast
+ have uabv: "u \<le> a (real m / 2 ^ n) \<and> a (real m / 2 ^ n) \<le> b (real m / 2 ^ n) \<and> b (real m / 2 ^ n) \<le> v" for m n
+ proof (induction n arbitrary: m)
+ case 0
+ then show ?case by (simp add: v01)
+ next
+ case (Suc n p)
+ show ?case
+ proof (cases "even p")
+ case True
+ then obtain m where "p = 2*m" by (metis evenE)
+ then show ?thesis
+ by (simp add: Suc.IH)
+ next
+ case False
+ then obtain m where m: "p = 2*m + 1" by (metis oddE)
+ show ?thesis
+ proof (cases n)
+ case 0
+ then show ?thesis
+ by (simp add: a_def b_def leftrec_base rightrec_base v01)
+ next
+ case (Suc n')
+ then have "n > 0" by simp
+ have a_le_c: "a (real m / 2^n) \<le> c (real m / 2^n)" for m
+ unfolding c_def by (metis Suc.IH ge_midpoint_1)
+ have c_le_b: "c (real m / 2^n) \<le> b (real m / 2^n)" for m
+ unfolding c_def by (metis Suc.IH le_midpoint_1)
+ have c_ge_u: "c (real m / 2^n) \<ge> u" for m
+ using Suc.IH a_le_c order_trans by blast
+ have c_le_v: "c (real m / 2^n) \<le> v" for m
+ using Suc.IH c_le_b order_trans by blast
+ have a_ge_0: "0 \<le> a (real m / 2^n)" for m
+ using Suc.IH order_trans u01(1) by blast
+ have b_le_1: "b (real m / 2^n) \<le> 1" for m
+ using Suc.IH order_trans v01(2) by blast
+ have left_le: "leftcut (a ((real m) / 2^n)) (b ((real m) / 2^n)) (c ((real m) / 2^n)) \<le> c ((real m) / 2^n)" for m
+ by (simp add: LR a_ge_0 a_le_c b_le_1 c_le_b)
+ have right_ge: "rightcut (a ((real m) / 2^n)) (b ((real m) / 2^n)) (c ((real m) / 2^n)) \<ge> c ((real m) / 2^n)" for m
+ by (simp add: LR a_ge_0 a_le_c b_le_1 c_le_b)
+ show ?thesis
+ proof (cases "even m")
+ case True
+ then obtain r where r: "m = 2*r" by (metis evenE)
+ show ?thesis
+ using a_le_c [of "m+1"] c_le_b [of "m+1"] a_ge_0 [of "m+1"] b_le_1 [of "m+1"]
+ Suc.IH [of "m+1"]
+ apply (simp add: r m add.commute [of 1] \<open>n > 0\<close> a41 b41 del: power_Suc)
+ apply (auto simp: left_right [THEN conjunct1])
+ using order_trans [OF left_le c_le_v]
+ by (metis (no_types, hide_lams) add.commute mult_2 of_nat_Suc of_nat_add)
+ next
+ case False
+ then obtain r where r: "m = 2*r + 1" by (metis oddE)
+ show ?thesis
+ using a_le_c [of "m"] c_le_b [of "m"] a_ge_0 [of "m"] b_le_1 [of "m"]
+ Suc.IH [of "m+1"]
+ apply (simp add: r m add.commute [of 3] \<open>n > 0\<close> a43 b43 del: power_Suc)
+ apply (auto simp: add.commute left_right [THEN conjunct2])
+ using order_trans [OF c_ge_u right_ge]
+ apply (metis (no_types, hide_lams) mult_2 numeral_One of_nat_add of_nat_numeral)
+ apply (metis Suc.IH mult_2 of_nat_1 of_nat_add)
+ done
+ qed
+ qed
+ qed
+ qed
+ have a_ge_0 [simp]: "0 \<le> a(m / 2^n)" and b_le_1 [simp]: "b(m / 2^n) \<le> 1" for m::nat and n
+ using uabv order_trans u01 v01 by blast+
+ then have b_ge_0 [simp]: "0 \<le> b(m / 2^n)" and a_le_1 [simp]: "a(m / 2^n) \<le> 1" for m::nat and n
+ using uabv order_trans by blast+
+ have alec [simp]: "a(m / 2^n) \<le> c(m / 2^n)" and cleb [simp]: "c(m / 2^n) \<le> b(m / 2^n)" for m::nat and n
+ by (auto simp: c_def ge_midpoint_1 le_midpoint_1 uabv)
+ have c_ge_0 [simp]: "0 \<le> c(m / 2^n)" and c_le_1 [simp]: "c(m / 2^n) \<le> 1" for m::nat and n
+ using a_ge_0 alec order_trans apply blast
+ by (meson b_le_1 cleb order_trans)
+ have "\<lbrakk>d = m-n; odd j; \<bar>real i / 2^m - real j / 2^n\<bar> < 1/2 ^ n\<rbrakk>
+ \<Longrightarrow> (a(j / 2^n)) \<le> (c(i / 2^m)) \<and> (c(i / 2^m)) \<le> (b(j / 2^n))" for d i j m n
+ proof (induction d arbitrary: j n rule: less_induct)
+ case (less d j n)
+ show ?case
+ proof (cases "m \<le> n")
+ case True
+ have "\<bar>2^n\<bar> * \<bar>real i / 2^m - real j / 2^n\<bar> = 0"
+ proof (rule Ints_nonzero_abs_less1)
+ have "(real i * 2^n - real j * 2^m) / 2^m = (real i * 2^n) / 2^m - (real j * 2^m) / 2^m"
+ using diff_divide_distrib by blast
+ also have "... = (real i * 2 ^ (n-m)) - (real j)"
+ using True by (auto simp: power_diff field_simps)
+ also have "... \<in> \<int>"
+ by simp
+ finally have "(real i * 2^n - real j * 2^m) / 2^m \<in> \<int>" .
+ with True Ints_abs show "\<bar>2^n\<bar> * \<bar>real i / 2^m - real j / 2^n\<bar> \<in> \<int>"
+ by (fastforce simp: divide_simps)
+ show "\<bar>\<bar>2^n\<bar> * \<bar>real i / 2^m - real j / 2^n\<bar>\<bar> < 1"
+ using less.prems by (auto simp: divide_simps)
+ qed
+ then have "real i / 2^m = real j / 2^n"
+ by auto
+ then show ?thesis
+ by auto
+ next
+ case False
+ then have "n < m" by auto
+ obtain k where k: "j = Suc (2*k)"
+ using \<open>odd j\<close> oddE by fastforce
+ show ?thesis
+ proof (cases "n > 0")
+ case False
+ then have "a (real j / 2^n) = u"
+ by simp
+ also have "... \<le> c (real i / 2^m)"
+ using alec uabv by (blast intro: order_trans)
+ finally have ac: "a (real j / 2^n) \<le> c (real i / 2^m)" .
+ have "c (real i / 2^m) \<le> v"
+ using cleb uabv by (blast intro: order_trans)
+ also have "... = b (real j / 2^n)"
+ using False by simp
+ finally show ?thesis
+ by (auto simp: ac)
+ next
+ case True show ?thesis
+ proof (cases "real i / 2^m" "real j / 2^n" rule: linorder_cases)
+ case less
+ moreover have "real (4 * k + 1) / 2 ^ Suc n + 1 / (2 ^ Suc n) = real j / 2 ^ n"
+ using k by (force simp: divide_simps)
+ moreover have "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 2 / (2 ^ Suc n)"
+ using less.prems by simp
+ ultimately have closer: "\<bar>real i / 2 ^ m - real (4 * k + 1) / 2 ^ Suc n\<bar> < 1 / (2 ^ Suc n)"
+ using less.prems by linarith
+ have *: "a (real (4 * k + 1) / 2 ^ Suc n) \<le> c (real i / 2 ^ m) \<and>
+ c (real i / 2 ^ m) \<le> b (real (4 * k + 1) / 2 ^ Suc n)"
+ apply (rule less.IH [OF _ refl])
+ using closer \<open>n < m\<close> \<open>d = m - n\<close> apply (auto simp: divide_simps \<open>n < m\<close> diff_less_mono2)
+ done
+ show ?thesis
+ using LR [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
+ using alec [of "2*k+1"] cleb [of "2*k+1"] a_ge_0 [of "2*k+1"] b_le_1 [of "2*k+1"]
+ using k a41 b41 * \<open>0 < n\<close>
+ apply (simp add: add.commute)
+ done
+ next
+ case equal then show ?thesis by simp
+ next
+ case greater
+ moreover have "real (4 * k + 3) / 2 ^ Suc n - 1 / (2 ^ Suc n) = real j / 2 ^ n"
+ using k by (force simp: divide_simps)
+ moreover have "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 2 * 1 / (2 ^ Suc n)"
+ using less.prems by simp
+ ultimately have closer: "\<bar>real i / 2 ^ m - real (4 * k + 3) / 2 ^ Suc n\<bar> < 1 / (2 ^ Suc n)"
+ using less.prems by linarith
+ have *: "a (real (4 * k + 3) / 2 ^ Suc n) \<le> c (real i / 2 ^ m) \<and>
+ c (real i / 2 ^ m) \<le> b (real (4 * k + 3) / 2 ^ Suc n)"
+ apply (rule less.IH [OF _ refl])
+ using closer \<open>n < m\<close> \<open>d = m - n\<close> apply (auto simp: divide_simps \<open>n < m\<close> diff_less_mono2)
+ done
+ show ?thesis
+ using LR [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
+ using alec [of "2*k+1"] cleb [of "2*k+1"] a_ge_0 [of "2*k+1"] b_le_1 [of "2*k+1"]
+ using k a43 b43 * \<open>0 < n\<close>
+ apply (simp add: add.commute)
+ done
+ qed
+ qed
+ qed
+ qed
+ then have aj_le_ci: "a (real j / 2 ^ n) \<le> c (real i / 2 ^ m)"
+ and ci_le_bj: "c (real i / 2 ^ m) \<le> b (real j / 2 ^ n)" if "odd j" "\<bar>real i / 2^m - real j / 2^n\<bar> < 1/2 ^ n" for i j m n
+ using that by blast+
+ have close_ab: "odd m \<Longrightarrow> \<bar>a (real m / 2 ^ n) - b (real m / 2 ^ n)\<bar> \<le> 2 / 2^n" for m n
+ proof (induction n arbitrary: m)
+ case 0
+ with u01 v01 show ?case by auto
+ next
+ case (Suc n m)
+ with oddE obtain k where k: "m = Suc (2*k)" by fastforce
+ show ?case
+ proof (cases "n > 0")
+ case False
+ with u01 v01 show ?thesis
+ by (simp add: a_def b_def leftrec_base rightrec_base)
+ next
+ case True
+ show ?thesis
+ proof (cases "even k")
+ case True
+ then obtain j where j: "k = 2*j" by (metis evenE)
+ have "\<bar>a ((2 * real j + 1) / 2 ^ n) - (b ((2 * real j + 1) / 2 ^ n))\<bar> \<le> 2/2 ^ n"
+ proof -
+ have "odd (Suc k)"
+ using True by auto
+ then show ?thesis
+ by (metis (no_types) Groups.add_ac(2) Suc.IH j of_nat_Suc of_nat_mult of_nat_numeral)
+ qed
+ moreover have "a ((2 * real j + 1) / 2 ^ n) \<le>
+ leftcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n))"
+ using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
+ by (auto simp: add.commute left_right)
+ moreover have "leftcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n)) \<le>
+ c ((2 * real j + 1) / 2 ^ n)"
+ using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
+ by (auto simp: add.commute left_right_m)
+ ultimately have "\<bar>a ((2 * real j + 1) / 2 ^ n) -
+ leftcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n))\<bar>
+ \<le> 2/2 ^ Suc n"
+ by (simp add: c_def midpoint_def)
+ with j k \<open>n > 0\<close> show ?thesis
+ by (simp add: add.commute [of 1] a41 b41 del: power_Suc)
+ next
+ case False
+ then obtain j where j: "k = 2*j + 1" by (metis oddE)
+ have "\<bar>a ((2 * real j + 1) / 2 ^ n) - (b ((2 * real j + 1) / 2 ^ n))\<bar> \<le> 2/2 ^ n"
+ using Suc.IH [OF False] j by (auto simp: algebra_simps)
+ moreover have "c ((2 * real j + 1) / 2 ^ n) \<le>
+ rightcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n))"
+ using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
+ by (auto simp: add.commute left_right_m)
+ moreover have "rightcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n)) \<le>
+ b ((2 * real j + 1) / 2 ^ n)"
+ using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
+ by (auto simp: add.commute left_right)
+ ultimately have "\<bar>rightcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n)) -
+ b ((2 * real j + 1) / 2 ^ n)\<bar> \<le> 2/2 ^ Suc n"
+ by (simp add: c_def midpoint_def)
+ with j k \<open>n > 0\<close> show ?thesis
+ by (simp add: add.commute [of 3] a43 b43 del: power_Suc)
+ qed
+ qed
+ qed
+ have m1_to_3: "4 * real k - 1 = real (4 * (k-1)) + 3" if "0 < k" for k
+ using that by auto
+ have fb_eq_fa: "\<lbrakk>0 < j; 2*j < 2 ^ n\<rbrakk> \<Longrightarrow> f(b((2 * real j - 1) / 2^n)) = f(a((2 * real j + 1) / 2^n))" for n j
+ proof (induction n arbitrary: j)
+ case 0
+ then show ?case by auto
+ next
+ case (Suc n j) show ?case
+ proof (cases "n > 0")
+ case False
+ with Suc.prems show ?thesis by auto
+ next
+ case True
+ show ?thesis proof (cases "even j")
+ case True
+ then obtain k where k: "j = 2*k" by (metis evenE)
+ with \<open>0 < j\<close> have "k > 0" "2 * k < 2 ^ n"
+ using Suc.prems(2) k by auto
+ with k \<open>0 < n\<close> Suc.IH [of k] show ?thesis
+ apply (simp add: m1_to_3 a41 b43 del: power_Suc)
+ apply (subst of_nat_diff, auto)
+ done
+ next
+ case False
+ then obtain k where k: "j = 2*k + 1" by (metis oddE)
+ have "f (leftcut (a ((2 * k + 1) / 2^n)) (b ((2 * k + 1) / 2^n)) (c ((2 * k + 1) / 2^n)))
+ = f (c ((2 * k + 1) / 2^n))"
+ "f (c ((2 * k + 1) / 2^n))
+ = f (rightcut (a ((2 * k + 1) / 2^n)) (b ((2 * k + 1) / 2^n)) (c ((2 * k + 1) / 2^n)))"
+ using alec [of "2*k+1" n] cleb [of "2*k+1" n] a_ge_0 [of "2*k+1" n] b_le_1 [of "2*k+1" n] k
+ using left_right_m [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
+ apply (auto simp: add.commute feqm [OF order_refl] feqm [OF _ order_refl, symmetric])
+ done
+ then
+ show ?thesis
+ by (simp add: k add.commute [of 1] add.commute [of 3] a43 b41\<open>0 < n\<close> del: power_Suc)
+ qed
+ qed
+ qed
+ have f_eq_fc: "\<lbrakk>0 < j; j < 2 ^ n\<rbrakk>
+ \<Longrightarrow> f(b((2*j - 1) / 2 ^ (Suc n))) = f(c(j / 2^n)) \<and>
+ f(a((2*j + 1) / 2 ^ (Suc n))) = f(c(j / 2^n))" for n and j::nat
+ proof (induction n arbitrary: j)
+ case 0
+ then show ?case by auto
+ next
+ case (Suc n)
+ show ?case
+ proof (cases "even j")
+ case True
+ then obtain k where k: "j = 2*k" by (metis evenE)
+ then have less2n: "k < 2 ^ n"
+ using Suc.prems(2) by auto
+ have "0 < k" using \<open>0 < j\<close> k by linarith
+ then have m1_to_3: "real (4 * k - Suc 0) = real (4 * (k-1)) + 3"
+ by auto
+ then show ?thesis
+ using Suc.IH [of k] k \<open>0 < k\<close>
+ apply (simp add: less2n add.commute [of 1] m1_to_3 a41 b43 del: power_Suc)
+ apply (auto simp: of_nat_diff)
+ done
+ next
+ case False
+ then obtain k where k: "j = 2*k + 1" by (metis oddE)
+ with Suc.prems have "k < 2^n" by auto
+ show ?thesis
+ using alec [of "2*k+1" "Suc n"] cleb [of "2*k+1" "Suc n"] a_ge_0 [of "2*k+1" "Suc n"] b_le_1 [of "2*k+1" "Suc n"] k
+ using left_right_m [of "c((2*k + 1) / 2 ^ Suc n)" "a((2*k + 1) / 2 ^ Suc n)" "b((2*k + 1) / 2 ^ Suc n)"]
+ apply (simp add: add.commute [of 1] add.commute [of 3] m1_to_3 b41 a43 del: power_Suc)
+ apply (force intro: feqm)
+ done
+ qed
+ qed
+ define D01 where "D01 \<equiv> {0<..<1} \<inter> (\<Union>k m. {real m / 2^k})"
+ have cloD01 [simp]: "closure D01 = {0..1}"
+ unfolding D01_def
+ by (subst closure_dyadic_rationals_in_convex_set_pos_1) auto
+ have "uniformly_continuous_on D01 (f \<circ> c)"
+ proof (clarsimp simp: uniformly_continuous_on_def)
+ fix e::real
+ assume "0 < e"
+ have ucontf: "uniformly_continuous_on {0..1} f"
+ by (simp add: compact_uniformly_continuous [OF cont_f])
+ then obtain d where "0 < d" and d: "\<And>x x'. \<lbrakk>x \<in> {0..1}; x' \<in> {0..1}; norm (x' - x) < d\<rbrakk> \<Longrightarrow> norm (f x' - f x) < e/2"
+ unfolding uniformly_continuous_on_def dist_norm
+ by (metis \<open>0 < e\<close> less_divide_eq_numeral1(1) mult_zero_left)
+ obtain n where n: "1/2^n < min d 1"
+ by (metis \<open>0 < d\<close> divide_less_eq_1 less_numeral_extra(1) min_def one_less_numeral_iff power_one_over real_arch_pow_inv semiring_norm(76) zero_less_numeral)
+ with gr0I have "n > 0"
+ by (force simp: divide_simps)
+ show "\<exists>d>0. \<forall>x\<in>D01. \<forall>x'\<in>D01. dist x' x < d \<longrightarrow> dist (f (c x')) (f (c x)) < e"
+ proof (intro exI ballI impI conjI)
+ show "(0::real) < 1/2^n" by auto
+ next
+ have dist_fc_close: "dist (f(c(real i / 2^m))) (f(c(real j / 2^n))) < e/2"
+ if i: "0 < i" "i < 2 ^ m" and j: "0 < j" "j < 2 ^ n" and clo: "abs(i / 2^m - j / 2^n) < 1/2 ^ n" for i j m
+ proof -
+ have abs3: "\<bar>x - a\<bar> < e \<Longrightarrow> x = a \<or> \<bar>x - (a - e/2)\<bar> < e/2 \<or> \<bar>x - (a + e/2)\<bar> < e/2" for x a e::real
+ by linarith
+ consider "i / 2 ^ m = j / 2 ^ n"
+ | "\<bar>i / 2 ^ m - (2 * j - 1) / 2 ^ Suc n\<bar> < 1/2 ^ Suc n"
+ | "\<bar>i / 2 ^ m - (2 * j + 1) / 2 ^ Suc n\<bar> < 1/2 ^ Suc n"
+ using abs3 [OF clo] j by (auto simp: field_simps of_nat_diff)
+ then show ?thesis
+ proof cases
+ case 1 with \<open>0 < e\<close> show ?thesis by auto
+ next
+ case 2
+ have *: "abs(a - b) \<le> 1/2 ^ n \<and> 1/2 ^ n < d \<and> a \<le> c \<and> c \<le> b \<Longrightarrow> b - c < d" for a b c
+ by auto
+ have "norm (c (real i / 2 ^ m) - b (real (2 * j - 1) / 2 ^ Suc n)) < d"
+ using 2 j n close_ab [of "2*j-1" "Suc n"]
+ using b_ge_0 [of "2*j-1" "Suc n"] b_le_1 [of "2*j-1" "Suc n"]
+ using aj_le_ci [of "2*j-1" i m "Suc n"]
+ using ci_le_bj [of "2*j-1" i m "Suc n"]
+ apply (simp add: divide_simps of_nat_diff del: power_Suc)
+ apply (auto simp: divide_simps intro!: *)
+ done
+ moreover have "f(c(j / 2^n)) = f(b ((2*j - 1) / 2 ^ (Suc n)))"
+ using f_eq_fc [OF j] by metis
+ ultimately show ?thesis
+ by (metis dist_norm atLeastAtMost_iff b_ge_0 b_le_1 c_ge_0 c_le_1 d)
+ next
+ case 3
+ have *: "abs(a - b) \<le> 1/2 ^ n \<and> 1/2 ^ n < d \<and> a \<le> c \<and> c \<le> b \<Longrightarrow> c - a < d" for a b c
+ by auto
+ have "norm (c (real i / 2 ^ m) - a (real (2 * j + 1) / 2 ^ Suc n)) < d"
+ using 3 j n close_ab [of "2*j+1" "Suc n"]
+ using b_ge_0 [of "2*j+1" "Suc n"] b_le_1 [of "2*j+1" "Suc n"]
+ using aj_le_ci [of "2*j+1" i m "Suc n"]
+ using ci_le_bj [of "2*j+1" i m "Suc n"]
+ apply (simp add: divide_simps of_nat_diff del: power_Suc)
+ apply (auto simp: divide_simps intro!: *)
+ done
+ moreover have "f(c(j / 2^n)) = f(a ((2*j + 1) / 2 ^ (Suc n)))"
+ using f_eq_fc [OF j] by metis
+ ultimately show ?thesis
+ by (metis dist_norm a_ge_0 atLeastAtMost_iff a_ge_0 a_le_1 c_ge_0 c_le_1 d)
+ qed
+ qed
+ show "dist (f (c x')) (f (c x)) < e"
+ if "x \<in> D01" "x' \<in> D01" "dist x' x < 1/2^n" for x x'
+ using that unfolding D01_def dyadics_in_open_unit_interval
+ proof clarsimp
+ fix i k::nat and m p
+ assume i: "0 < i" "i < 2 ^ m" and k: "0<k" "k < 2 ^ p"
+ assume clo: "dist (real k / 2 ^ p) (real i / 2 ^ m) < 1/2 ^ n"
+ obtain j::nat where "0 < j" "j < 2 ^ n"
+ and clo_ij: "abs(i / 2^m - j / 2^n) < 1/2 ^ n"
+ and clo_kj: "abs(k / 2^p - j / 2^n) < 1/2 ^ n"
+ proof -
+ have "max (2^n * i / 2^m) (2^n * k / 2^p) \<ge> 0"
+ by (auto simp: le_max_iff_disj)
+ then obtain j where "floor (max (2^n*i / 2^m) (2^n*k / 2^p)) = int j"
+ using zero_le_floor zero_le_imp_eq_int by blast
+ then have j_le: "real j \<le> max (2^n * i / 2^m) (2^n * k / 2^p)"
+ and less_j1: "max (2^n * i / 2^m) (2^n * k / 2^p) < real j + 1"
+ using floor_correct [of "max (2^n * i / 2^m) (2^n * k / 2^p)"] by linarith+
+ show thesis
+ proof (cases "j = 0")
+ case True
+ show thesis
+ proof
+ show "(1::nat) < 2 ^ n"
+ apply (subst one_less_power)
+ using \<open>n > 0\<close> by auto
+ show "\<bar>real i / 2 ^ m - real 1/2 ^ n\<bar> < 1/2 ^ n"
+ using i less_j1 by (simp add: dist_norm field_simps True)
+ show "\<bar>real k / 2 ^ p - real 1/2 ^ n\<bar> < 1/2 ^ n"
+ using k less_j1 by (simp add: dist_norm field_simps True)
+ qed simp
+ next
+ case False
+ have 1: "real j * 2 ^ m < real i * 2 ^ n"
+ if j: "real j * 2 ^ p \<le> real k * 2 ^ n" and k: "real k * 2 ^ m < real i * 2 ^ p"
+ for i k m p
+ proof -
+ have "real j * 2 ^ p * 2 ^ m \<le> real k * 2 ^ n * 2 ^ m"
+ using j by simp
+ moreover have "real k * 2 ^ m * 2 ^ n < real i * 2 ^ p * 2 ^ n"
+ using k by simp
+ ultimately have "real j * 2 ^ p * 2 ^ m < real i * 2 ^ p * 2 ^ n"
+ by (simp only: mult_ac)
+ then show ?thesis
+ by simp
+ qed
+ have 2: "real j * 2 ^ m < 2 ^ m + real i * 2 ^ n"
+ if j: "real j * 2 ^ p \<le> real k * 2 ^ n" and k: "real k * (2 ^ m * 2 ^ n) < 2 ^ m * 2 ^ p + real i * (2 ^ n * 2 ^ p)"
+ for i k m p
+ proof -
+ have "real j * 2 ^ p * 2 ^ m \<le> real k * (2 ^ m * 2 ^ n)"
+ using j by simp
+ also have "... < 2 ^ m * 2 ^ p + real i * (2 ^ n * 2 ^ p)"
+ by (rule k)
+ finally have "(real j * 2 ^ m) * 2 ^ p < (2 ^ m + real i * 2 ^ n) * 2 ^ p"
+ by (simp add: algebra_simps)
+ then show ?thesis
+ by simp
+ qed
+ have 3: "real j * 2 ^ p < 2 ^ p + real k * 2 ^ n"
+ if j: "real j * 2 ^ m \<le> real i * 2 ^ n" and i: "real i * 2 ^ p \<le> real k * 2 ^ m"
+ proof -
+ have "real j * 2 ^ m * 2 ^ p \<le> real i * 2 ^ n * 2 ^ p"
+ using j by simp
+ moreover have "real i * 2 ^ p * 2 ^ n \<le> real k * 2 ^ m * 2 ^ n"
+ using i by simp
+ ultimately have "real j * 2 ^ m * 2 ^ p \<le> real k * 2 ^ m * 2 ^ n"
+ by (simp only: mult_ac)
+ then have "real j * 2 ^ p \<le> real k * 2 ^ n"
+ by simp
+ also have "... < 2 ^ p + real k * 2 ^ n"
+ by auto
+ finally show ?thesis by simp
+ qed
+ show ?thesis
+ proof
+ have "real j < 2 ^ n"
+ using j_le i k
+ apply (auto simp: le_max_iff_disj simp del: real_of_nat_less_numeral_power_cancel_iff elim!: le_less_trans)
+ apply (auto simp: field_simps)
+ done
+ then show "j < 2 ^ n"
+ by auto
+ show "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 1/2 ^ n"
+ using clo less_j1 j_le
+ apply (auto simp: le_max_iff_disj divide_simps dist_norm)
+ apply (auto simp: algebra_simps abs_if split: if_split_asm dest: 1 2)
+ done
+ show "\<bar>real k / 2 ^ p - real j / 2 ^ n\<bar> < 1/2 ^ n"
+ using clo less_j1 j_le
+ apply (auto simp: le_max_iff_disj divide_simps dist_norm)
+ apply (auto simp: algebra_simps not_less abs_if split: if_split_asm dest: 3 2)
+ done
+ qed (use False in simp)
+ qed
+ qed
+ show "dist (f (c (real k / 2 ^ p))) (f (c (real i / 2 ^ m))) < e"
+ proof (rule dist_triangle_half_l)
+ show "dist (f (c (real k / 2 ^ p))) (f(c(j / 2^n))) < e/2"
+ apply (rule dist_fc_close)
+ using \<open>0 < j\<close> \<open>j < 2 ^ n\<close> k clo_kj by auto
+ show "dist (f (c (real i / 2 ^ m))) (f (c (real j / 2 ^ n))) < e/2"
+ apply (rule dist_fc_close)
+ using \<open>0 < j\<close> \<open>j < 2 ^ n\<close> i clo_ij by auto
+ qed
+ qed
+ qed
+ qed
+ then obtain h where ucont_h: "uniformly_continuous_on {0..1} h"
+ and fc_eq: "\<And>x. x \<in> D01 \<Longrightarrow> (f \<circ> c) x = h x"
+ proof (rule uniformly_continuous_on_extension_on_closure [of D01 "f \<circ> c"])
+ qed (use closure_subset [of D01] in \<open>auto intro!: that\<close>)
+ then have cont_h: "continuous_on {0..1} h"
+ using uniformly_continuous_imp_continuous by blast
+ have h_eq: "h (real k / 2 ^ m) = f (c (real k / 2 ^ m))" if "0 < k" "k < 2^m" for k m
+ using fc_eq that by (force simp: D01_def)
+ have "h ` {0..1} = f ` {0..1}"
+ proof
+ have "h ` (closure D01) \<subseteq> f ` {0..1}"
+ proof (rule image_closure_subset)
+ show "continuous_on (closure D01) h"
+ using cont_h by simp
+ show "closed (f ` {0..1})"
+ using compact_continuous_image [OF cont_f] compact_imp_closed by blast
+ show "h ` D01 \<subseteq> f ` {0..1}"
+ by (force simp: dyadics_in_open_unit_interval D01_def h_eq)
+ qed
+ with cloD01 show "h ` {0..1} \<subseteq> f ` {0..1}" by simp
+ have a12 [simp]: "a (1/2) = u"
+ by (metis a_def leftrec_base numeral_One of_nat_numeral)
+ have b12 [simp]: "b (1/2) = v"
+ by (metis b_def rightrec_base numeral_One of_nat_numeral)
+ have "f ` {0..1} \<subseteq> closure(h ` D01)"
+ proof (clarsimp simp: closure_approachable dyadics_in_open_unit_interval D01_def)
+ fix x e::real
+ assume "0 \<le> x" "x \<le> 1" "0 < e"
+ have ucont_f: "uniformly_continuous_on {0..1} f"
+ using compact_uniformly_continuous cont_f by blast
+ then obtain \<delta> where "\<delta> > 0"
+ and \<delta>: "\<And>x x'. \<lbrakk>x \<in> {0..1}; x' \<in> {0..1}; dist x' x < \<delta>\<rbrakk> \<Longrightarrow> norm (f x' - f x) < e"
+ using \<open>0 < e\<close> by (auto simp: uniformly_continuous_on_def dist_norm)
+ have *: "\<exists>m::nat. \<exists>y. odd m \<and> 0 < m \<and> m < 2 ^ n \<and> y \<in> {a(m / 2^n) .. b(m / 2^n)} \<and> f y = f x"
+ if "n \<noteq> 0" for n
+ using that
+ proof (induction n)
+ case 0 then show ?case by auto
+ next
+ case (Suc n)
+ show ?case
+ proof (cases "n=0")
+ case True
+ consider "x \<in> {0..u}" | "x \<in> {u..v}" | "x \<in> {v..1}"
+ using \<open>0 \<le> x\<close> \<open>x \<le> 1\<close> by force
+ then have "\<exists>y\<ge>a (real 1/2). y \<le> b (real 1/2) \<and> f y = f x"
+ proof cases
+ case 1
+ then show ?thesis
+ apply (rule_tac x=u in exI)
+ using uabv [of 1 1] f0u [of u] f0u [of x] by auto
+ next
+ case 2
+ then show ?thesis
+ by (rule_tac x=x in exI) auto
+ next
+ case 3
+ then show ?thesis
+ apply (rule_tac x=v in exI)
+ using uabv [of 1 1] fv1 [of v] fv1 [of x] by auto
+ qed
+ with \<open>n=0\<close> show ?thesis
+ by (rule_tac x=1 in exI) auto
+ next
+ case False
+ with Suc obtain m y
+ where "odd m" "0 < m" and mless: "m < 2 ^ n"
+ and y: "y \<in> {a (real m / 2 ^ n)..b (real m / 2 ^ n)}" and feq: "f y = f x"
+ by metis
+ then obtain j where j: "m = 2*j + 1" by (metis oddE)
+ consider "y \<in> {a((2*j + 1) / 2^n) .. b((4*j + 1) / 2 ^ (Suc n))}"
+ | "y \<in> {b((4*j + 1) / 2 ^ (Suc n)) .. a((4*j + 3) / 2 ^ (Suc n))}"
+ | "y \<in> {a((4*j + 3) / 2 ^ (Suc n)) .. b((2*j + 1) / 2^n)}"
+ using y j by force
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ apply (rule_tac x="4*j + 1" in exI)
+ apply (rule_tac x=y in exI)
+ using mless j \<open>n \<noteq> 0\<close>
+ apply (simp add: feq a41 b41 add.commute [of 1] del: power_Suc)
+ apply (simp add: algebra_simps)
+ done
+ next
+ case 2
+ show ?thesis
+ apply (rule_tac x="4*j + 1" in exI)
+ apply (rule_tac x="b((4*j + 1) / 2 ^ (Suc n))" in exI)
+ using mless \<open>n \<noteq> 0\<close> 2 j
+ using alec [of "2*j+1" n] cleb [of "2*j+1" n] a_ge_0 [of "2*j+1" n] b_le_1 [of "2*j+1" n]
+ using left_right [of "c((2*j + 1) / 2^n)" "a((2*j + 1) / 2^n)" "b((2*j + 1) / 2^n)"]
+ apply (simp add: a41 b41 a43 b43 add.commute [of 1] add.commute [of 3] del: power_Suc)
+ apply (auto simp: feq [symmetric] intro: f_eqI)
+ done
+ next
+ case 3
+ then show ?thesis
+ apply (rule_tac x="4*j + 3" in exI)
+ apply (rule_tac x=y in exI)
+ using mless j \<open>n \<noteq> 0\<close>
+ apply (simp add: feq a43 b43 del: power_Suc)
+ apply (simp add: algebra_simps)
+ done
+ qed
+ qed
+ qed
+ obtain n where n: "1/2^n < min (\<delta> / 2) 1"
+ by (metis \<open>0 < \<delta>\<close> divide_less_eq_1 less_numeral_extra(1) min_less_iff_conj one_less_numeral_iff power_one_over real_arch_pow_inv semiring_norm(76) zero_less_divide_iff zero_less_numeral)
+ with gr0I have "n \<noteq> 0"
+ by fastforce
+ with * obtain m::nat and y
+ where "odd m" "0 < m" and mless: "m < 2 ^ n"
+ and y: "y \<in> {a(m / 2^n) .. b(m / 2^n)}" and feq: "f x = f y"
+ by metis
+ then have "0 \<le> y" "y \<le> 1"
+ by (metis atLeastAtMost_iff a_ge_0 b_le_1 order.trans)+
+ moreover have "y < \<delta> + c (real m / 2 ^ n)" "c (real m / 2 ^ n) < \<delta> + y"
+ using y apply simp_all
+ using alec [of m n] cleb [of m n] n real_sum_of_halves close_ab [OF \<open>odd m\<close>, of n]
+ by linarith+
+ moreover note \<open>0 < m\<close> mless \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>
+ ultimately show "\<exists>k. \<exists>m\<in>{0<..<2 ^ k}. dist (h (real m / 2 ^ k)) (f x) < e"
+ apply (rule_tac x=n in exI)
+ apply (rule_tac x=m in bexI)
+ apply (auto simp: dist_norm h_eq feq \<delta>)
+ done
+ qed
+ also have "... \<subseteq> h ` {0..1}"
+ apply (rule closure_minimal)
+ using compact_continuous_image [OF cont_h] compact_imp_closed by (auto simp: D01_def)
+ finally show "f ` {0..1} \<subseteq> h ` {0..1}" .
+ qed
+ moreover have "inj_on h {0..1}"
+ proof -
+ have "u < v"
+ by (metis atLeastAtMost_iff f0u f_1not0 fv1 order.not_eq_order_implies_strict u01(1) u01(2) v01(1))
+ have f_not_fu: "\<And>x. \<lbrakk>u < x; x \<le> v\<rbrakk> \<Longrightarrow> f x \<noteq> f u"
+ by (metis atLeastAtMost_iff f0u fu1 greaterThanAtMost_iff order_refl order_trans u01(1) v01(2))
+ have f_not_fv: "\<And>x. \<lbrakk>u \<le> x; x < v\<rbrakk> \<Longrightarrow> f x \<noteq> f v"
+ by (metis atLeastAtMost_iff order_refl order_trans v01(2) atLeastLessThan_iff fuv fv1)
+ have a_less_b:
+ "a(j / 2^n) < b(j / 2^n) \<and>
+ (\<forall>x. a(j / 2^n) < x \<longrightarrow> x \<le> b(j / 2^n) \<longrightarrow> f x \<noteq> f(a(j / 2^n))) \<and>
+ (\<forall>x. a(j / 2^n) \<le> x \<longrightarrow> x < b(j / 2^n) \<longrightarrow> f x \<noteq> f(b(j / 2^n)))" for n and j::nat
+ proof (induction n arbitrary: j)
+ case 0 then show ?case
+ by (simp add: \<open>u < v\<close> f_not_fu f_not_fv)
+ next
+ case (Suc n j) show ?case
+ proof (cases "n > 0")
+ case False then show ?thesis
+ by (auto simp: a_def b_def leftrec_base rightrec_base \<open>u < v\<close> f_not_fu f_not_fv)
+ next
+ case True show ?thesis
+ proof (cases "even j")
+ case True
+ with \<open>0 < n\<close> Suc.IH show ?thesis
+ by (auto elim!: evenE)
+ next
+ case False
+ then obtain k where k: "j = 2*k + 1" by (metis oddE)
+ then show ?thesis
+ proof (cases "even k")
+ case True
+ then obtain m where m: "k = 2*m" by (metis evenE)
+ have fleft: "f (leftcut (a ((2*m + 1) / 2^n)) (b ((2*m + 1) / 2^n)) (c ((2*m + 1) / 2^n))) =
+ f (c((2*m + 1) / 2^n))"
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
+ by (auto intro: f_eqI)
+ show ?thesis
+ proof (intro conjI impI notI allI)
+ have False if "b (real j / 2 ^ Suc n) \<le> a (real j / 2 ^ Suc n)"
+ proof -
+ have "f (c ((1 + real m * 2) / 2 ^ n)) = f (a ((1 + real m * 2) / 2 ^ n))"
+ using k m \<open>0 < n\<close> fleft that a41 [of n m] b41 [of n m]
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ using left_right [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
+ by (auto simp: algebra_simps)
+ moreover have "a (real (1 + m * 2) / 2 ^ n) < c (real (1 + m * 2) / 2 ^ n)"
+ using Suc.IH [of "1 + m * 2"] by (simp add: c_def midpoint_def)
+ moreover have "c (real (1 + m * 2) / 2 ^ n) \<le> b (real (1 + m * 2) / 2 ^ n)"
+ using cleb by blast
+ ultimately show ?thesis
+ using Suc.IH [of "1 + m * 2"] by force
+ qed
+ then show "a (real j / 2 ^ Suc n) < b (real j / 2 ^ Suc n)" by force
+ next
+ fix x
+ assume "a (real j / 2 ^ Suc n) < x" "x \<le> b (real j / 2 ^ Suc n)" "f x = f (a (real j / 2 ^ Suc n))"
+ then show False
+ using Suc.IH [of "1 + m * 2", THEN conjunct2, THEN conjunct1]
+ using k m \<open>0 < n\<close> a41 [of n m] b41 [of n m]
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
+ by (auto simp: algebra_simps)
+ next
+ fix x
+ assume "a (real j / 2 ^ Suc n) \<le> x" "x < b (real j / 2 ^ Suc n)" "f x = f (b (real j / 2 ^ Suc n))"
+ then show False
+ using k m \<open>0 < n\<close> a41 [of n m] b41 [of n m] fleft left_neq
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ by (auto simp: algebra_simps)
+ qed
+ next
+ case False
+ with oddE obtain m where m: "k = Suc (2*m)" by fastforce
+ have fright: "f (rightcut (a ((2*m + 1) / 2^n)) (b ((2*m + 1) / 2^n)) (c ((2*m + 1) / 2^n))) = f (c((2*m + 1) / 2^n))"
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
+ by (auto intro: f_eqI [OF _ order_refl])
+ show ?thesis
+ proof (intro conjI impI notI allI)
+ have False if "b (real j / 2 ^ Suc n) \<le> a (real j / 2 ^ Suc n)"
+ proof -
+ have "f (c ((1 + real m * 2) / 2 ^ n)) = f (b ((1 + real m * 2) / 2 ^ n))"
+ using k m \<open>0 < n\<close> fright that a43 [of n m] b43 [of n m]
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ using left_right [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
+ by (auto simp: algebra_simps)
+ moreover have "a (real (1 + m * 2) / 2 ^ n) \<le> c (real (1 + m * 2) / 2 ^ n)"
+ using alec by blast
+ moreover have "c (real (1 + m * 2) / 2 ^ n) < b (real (1 + m * 2) / 2 ^ n)"
+ using Suc.IH [of "1 + m * 2"] by (simp add: c_def midpoint_def)
+ ultimately show ?thesis
+ using Suc.IH [of "1 + m * 2"] by force
+ qed
+ then show "a (real j / 2 ^ Suc n) < b (real j / 2 ^ Suc n)" by force
+ next
+ fix x
+ assume "a (real j / 2 ^ Suc n) < x" "x \<le> b (real j / 2 ^ Suc n)" "f x = f (a (real j / 2 ^ Suc n))"
+ then show False
+ using k m \<open>0 < n\<close> a43 [of n m] b43 [of n m] fright right_neq
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ by (auto simp: algebra_simps)
+ next
+ fix x
+ assume "a (real j / 2 ^ Suc n) \<le> x" "x < b (real j / 2 ^ Suc n)" "f x = f (b (real j / 2 ^ Suc n))"
+ then show False
+ using Suc.IH [of "1 + m * 2", THEN conjunct2, THEN conjunct2]
+ using k m \<open>0 < n\<close> a43 [of n m] b43 [of n m]
+ using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
+ using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
+ by (auto simp: algebra_simps fright simp del: power_Suc)
+ qed
+ qed
+ qed
+ qed
+ qed
+ have c_gt_0 [simp]: "0 < c(m / 2^n)" and c_less_1 [simp]: "c(m / 2^n) < 1" for m::nat and n
+ using a_less_b [of m n] apply (simp_all add: c_def midpoint_def)
+ using a_ge_0 [of m n] b_le_1 [of m n] apply linarith+
+ done
+ have approx: "\<exists>j n. odd j \<and> n \<noteq> 0 \<and>
+ real i / 2^m \<le> real j / 2^n \<and>
+ real j / 2^n \<le> real k / 2^p \<and>
+ \<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 1/2^n \<and>
+ \<bar>real k / 2 ^ p - real j / 2 ^ n\<bar> < 1/2^n"
+ if "0 < i" "i < 2 ^ m" "0 < k" "k < 2 ^ p" "i / 2^m < k / 2^p" "m + p = N" for N m p i k
+ using that
+ proof (induction N arbitrary: m p i k rule: less_induct)
+ case (less N)
+ then consider "i / 2^m \<le> 1/2" "1/2 \<le> k / 2^p" | "k / 2^p < 1/2" | "k / 2^p \<ge> 1/2" "1/2 < i / 2^m"
+ by linarith
+ then show ?case
+ proof cases
+ case 1
+ with less.prems show ?thesis
+ by (rule_tac x=1 in exI)+ (fastforce simp: divide_simps)
+ next
+ case 2 show ?thesis
+ proof (cases m)
+ case 0 with less.prems show ?thesis
+ by auto
+ next
+ case (Suc m') show ?thesis
+ proof (cases p)
+ case 0 with less.prems show ?thesis by auto
+ next
+ case (Suc p')
+ have False if "real i * 2 ^ p' < real k * 2 ^ m'" "k < 2 ^ p'" "2 ^ m' \<le> i"
+ proof -
+ have "real k * 2 ^ m' < 2 ^ p' * 2 ^ m'"
+ using that by simp
+ then have "real i * 2 ^ p' < 2 ^ p' * 2 ^ m'"
+ using that by linarith
+ with that show ?thesis by simp
+ qed
+ then show ?thesis
+ using less.IH [of "m'+p'" i m' k p'] less.prems \<open>m = Suc m'\<close> 2 Suc
+ apply atomize
+ apply (force simp: divide_simps)
+ done
+ qed
+ qed
+ next
+ case 3 show ?thesis
+ proof (cases m)
+ case 0 with less.prems show ?thesis
+ by auto
+ next
+ case (Suc m') show ?thesis
+ proof (cases p)
+ case 0 with less.prems show ?thesis by auto
+ next
+ case (Suc p')
+ then show ?thesis
+ using less.IH [of "m'+p'" "i - 2^m'" m' "k - 2 ^ p'" p'] less.prems \<open>m = Suc m'\<close> Suc 3
+ apply atomize
+ apply (auto simp: field_simps of_nat_diff)
+ apply (rule_tac x="2 ^ n + j" in exI, simp)
+ apply (rule_tac x="Suc n" in exI)
+ apply (auto simp: field_simps)
+ done
+ qed
+ qed
+ qed
+ qed
+ have clec: "c(real i / 2^m) \<le> c(real j / 2^n)"
+ if i: "0 < i" "i < 2 ^ m" and j: "0 < j" "j < 2 ^ n" and ij: "i / 2^m < j / 2^n" for m i n j
+ proof -
+ obtain j' n' where "odd j'" "n' \<noteq> 0"
+ and i_le_j: "real i / 2 ^ m \<le> real j' / 2 ^ n'"
+ and j_le_j: "real j' / 2 ^ n' \<le> real j / 2 ^ n"
+ and clo_ij: "\<bar>real i / 2 ^ m - real j' / 2 ^ n'\<bar> < 1/2 ^ n'"
+ and clo_jj: "\<bar>real j / 2 ^ n - real j' / 2 ^ n'\<bar> < 1/2 ^ n'"
+ using approx [of i m j n "m+n"] that i j ij by auto
+ with oddE obtain q where q: "j' = Suc (2*q)" by fastforce
+ have "c (real i / 2 ^ m) \<le> c((2*q + 1) / 2^n')"
+ proof (cases "i / 2^m = (2*q + 1) / 2^n'")
+ case True then show ?thesis by simp
+ next
+ case False
+ with i_le_j q have less: "i / 2^m < (2*q + 1) / 2^n'"
+ by auto
+ have *: "\<lbrakk>i < q; abs(i - q) < s*2; q = r + s\<rbrakk> \<Longrightarrow> abs(i - r) < s" for i q s r::real
+ by auto
+ have "c(i / 2^m) \<le> b(real(4 * q + 1) / 2 ^ (Suc n'))"
+ apply (rule ci_le_bj, force)
+ apply (rule * [OF less])
+ using i_le_j clo_ij q apply (auto simp: divide_simps)
+ done
+ then show ?thesis
+ using alec [of "2*q+1" n'] cleb [of "2*q+1" n'] a_ge_0 [of "2*q+1" n'] b_le_1 [of "2*q+1" n'] b41 [of n' q] \<open>n' \<noteq> 0\<close>
+ using left_right_m [of "c((2*q + 1) / 2^n')" "a((2*q + 1) / 2^n')" "b((2*q + 1) / 2^n')"]
+ by (auto simp: algebra_simps)
+ qed
+ also have "... \<le> c(real j / 2^n)"
+ proof (cases "j / 2^n = (2*q + 1) / 2^n'")
+ case True
+ then show ?thesis by simp
+ next
+ case False
+ with j_le_j q have less: "(2*q + 1) / 2^n' < j / 2^n"
+ by auto
+ have *: "\<lbrakk>q < i; abs(i - q) < s*2; r = q + s\<rbrakk> \<Longrightarrow> abs(i - r) < s" for i q s r::real
+ by auto
+ have "a(real(4*q + 3) / 2 ^ (Suc n')) \<le> c(j / 2^n)"
+ apply (rule aj_le_ci, force)
+ apply (rule * [OF less])
+ using j_le_j clo_jj q apply (auto simp: divide_simps)
+ done
+ then show ?thesis
+ using alec [of "2*q+1" n'] cleb [of "2*q+1" n'] a_ge_0 [of "2*q+1" n'] b_le_1 [of "2*q+1" n'] a43 [of n' q] \<open>n' \<noteq> 0\<close>
+ using left_right_m [of "c((2*q + 1) / 2^n')" "a((2*q + 1) / 2^n')" "b((2*q + 1) / 2^n')"]
+ by (auto simp: algebra_simps)
+ qed
+ finally show ?thesis .
+ qed
+ have "x = y" if "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" "h x = h y" for x y
+ using that
+ proof (induction x y rule: linorder_class.linorder_less_wlog)
+ case (less x1 x2)
+ obtain m n where m: "0 < m" "m < 2 ^ n"
+ and x12: "x1 < m / 2^n" "m / 2^n < x2"
+ and neq: "h x1 \<noteq> h (real m / 2^n)"
+ proof -
+ have "(x1 + x2) / 2 \<in> closure D01"
+ using cloD01 less.hyps less.prems by auto
+ with less obtain y where "y \<in> D01" and dist_y: "dist y ((x1 + x2) / 2) < (x2 - x1) / 64"
+ unfolding closure_approachable
+ by (metis diff_gt_0_iff_gt less_divide_eq_numeral1(1) mult_zero_left)
+ obtain m n where m: "0 < m" "m < 2 ^ n"
+ and clo: "\<bar>real m / 2 ^ n - (x1 + x2) / 2\<bar> < (x2 - x1) / 64"
+ and n: "1/2^n < (x2 - x1) / 128"
+ proof -
+ have "min 1 ((x2 - x1) / 128) > 0" "1/2 < (1::real)"
+ using less by auto
+ then obtain N where N: "1/2^N < min 1 ((x2 - x1) / 128)"
+ by (metis power_one_over real_arch_pow_inv)
+ then have "N > 0"
+ using less_divide_eq_1 by force
+ obtain p q where p: "p < 2 ^ q" "p \<noteq> 0" and yeq: "y = real p / 2 ^ q"
+ using \<open>y \<in> D01\<close> by (auto simp: zero_less_divide_iff D01_def)
+ show ?thesis
+ proof
+ show "0 < 2^N * p"
+ using p by auto
+ show "2 ^ N * p < 2 ^ (N+q)"
+ by (simp add: p power_add)
+ have "\<bar>real (2 ^ N * p) / 2 ^ (N + q) - (x1 + x2) / 2\<bar> = \<bar>real p / 2 ^ q - (x1 + x2) / 2\<bar>"
+ by (simp add: power_add)
+ also have "... = \<bar>y - (x1 + x2) / 2\<bar>"
+ by (simp add: yeq)
+ also have "... < (x2 - x1) / 64"
+ using dist_y by (simp add: dist_norm)
+ finally show "\<bar>real (2 ^ N * p) / 2 ^ (N + q) - (x1 + x2) / 2\<bar> < (x2 - x1) / 64" .
+ have "(1::real) / 2 ^ (N + q) \<le> 1/2^N"
+ by (simp add: field_simps)
+ also have "... < (x2 - x1) / 128"
+ using N by force
+ finally show "1/2 ^ (N + q) < (x2 - x1) / 128" .
+ qed
+ qed
+ obtain m' n' m'' n'' where "0 < m'" "m' < 2 ^ n'" "x1 < m' / 2^n'" "m' / 2^n' < x2"
+ and "0 < m''" "m'' < 2 ^ n''" "x1 < m'' / 2^n''" "m'' / 2^n'' < x2"
+ and neq: "h (real m'' / 2^n'') \<noteq> h (real m' / 2^n')"
+ proof
+ show "0 < Suc (2*m)"
+ by simp
+ show m21: "Suc (2*m) < 2 ^ Suc n"
+ using m by auto
+ show "x1 < real (Suc (2 * m)) / 2 ^ Suc n"
+ using clo by (simp add: field_simps abs_if split: if_split_asm)
+ show "real (Suc (2 * m)) / 2 ^ Suc n < x2"
+ using n clo by (simp add: field_simps abs_if split: if_split_asm)
+ show "0 < 4*m + 3"
+ by simp
+ have "m+1 \<le> 2 ^ n"
+ using m by simp
+ then have "4 * (m+1) \<le> 4 * (2 ^ n)"
+ by simp
+ then show m43: "4*m + 3 < 2 ^ (n+2)"
+ by (simp add: algebra_simps)
+ show "x1 < real (4 * m + 3) / 2 ^ (n + 2)"
+ using clo by (simp add: field_simps abs_if split: if_split_asm)
+ show "real (4 * m + 3) / 2 ^ (n + 2) < x2"
+ using n clo by (simp add: field_simps abs_if split: if_split_asm)
+ have c_fold: "midpoint (a ((2 * real m + 1) / 2 ^ Suc n)) (b ((2 * real m + 1) / 2 ^ Suc n)) = c ((2 * real m + 1) / 2 ^ Suc n)"
+ by (simp add: c_def)
+ define R where "R \<equiv> rightcut (a ((2 * real m + 1) / 2 ^ Suc n)) (b ((2 * real m + 1) / 2 ^ Suc n)) (c ((2 * real m + 1) / 2 ^ Suc n))"
+ have "R < b ((2 * real m + 1) / 2 ^ Suc n)"
+ unfolding R_def using a_less_b [of "4*m + 3" "n+2"] a43 [of "Suc n" m] b43 [of "Suc n" m]
+ by simp
+ then have Rless: "R < midpoint R (b ((2 * real m + 1) / 2 ^ Suc n))"
+ by (simp add: midpoint_def)
+ have midR_le: "midpoint R (b ((2 * real m + 1) / 2 ^ Suc n)) \<le> b ((2 * real m + 1) / (2 * 2 ^ n))"
+ using \<open>R < b ((2 * real m + 1) / 2 ^ Suc n)\<close>
+ by (simp add: midpoint_def)
+ have "(real (Suc (2 * m)) / 2 ^ Suc n) \<in> D01" "real (4 * m + 3) / 2 ^ (n + 2) \<in> D01"
+ by (simp_all add: D01_def m21 m43 del: power_Suc of_nat_Suc of_nat_add add_2_eq_Suc') blast+
+ then show "h (real (4 * m + 3) / 2 ^ (n + 2)) \<noteq> h (real (Suc (2 * m)) / 2 ^ Suc n)"
+ using a_less_b [of "4*m + 3" "n+2", THEN conjunct1]
+ using a43 [of "Suc n" m] b43 [of "Suc n" m]
+ using alec [of "2*m+1" "Suc n"] cleb [of "2*m+1" "Suc n"] a_ge_0 [of "2*m+1" "Suc n"] b_le_1 [of "2*m+1" "Suc n"]
+ apply (simp add: fc_eq [symmetric] c_def del: power_Suc)
+ apply (simp only: add.commute [of 1] c_fold R_def [symmetric])
+ apply (rule right_neq)
+ using Rless apply (simp add: R_def)
+ apply (rule midR_le, auto)
+ done
+ qed
+ then show ?thesis by (metis that)
+ qed
+ have m_div: "0 < m / 2^n" "m / 2^n < 1"
+ using m by (auto simp: divide_simps)
+ have closure0m: "{0..m / 2^n} = closure ({0<..< m / 2^n} \<inter> (\<Union>k m. {real m / 2 ^ k}))"
+ by (subst closure_dyadic_rationals_in_convex_set_pos_1, simp_all add: not_le m)
+ have closurem1: "{m / 2^n .. 1} = closure ({m / 2^n <..< 1} \<inter> (\<Union>k m. {real m / 2 ^ k}))"
+ apply (subst closure_dyadic_rationals_in_convex_set_pos_1; simp add: not_le m)
+ using \<open>0 < real m / 2 ^ n\<close> by linarith
+ have cont_h': "continuous_on (closure ({u<..<v} \<inter> (\<Union>k m. {real m / 2 ^ k}))) h"
+ if "0 \<le> u" "v \<le> 1" for u v
+ apply (rule continuous_on_subset [OF cont_h])
+ apply (rule closure_minimal [OF subsetI])
+ using that apply auto
+ done
+ have closed_f': "closed (f ` {u..v})" if "0 \<le> u" "v \<le> 1" for u v
+ by (metis compact_continuous_image cont_f compact_interval atLeastatMost_subset_iff
+ compact_imp_closed continuous_on_subset that)
+ have less_2I: "\<And>k i. real i / 2 ^ k < 1 \<Longrightarrow> i < 2 ^ k"
+ by simp
+ have "h ` ({0<..<m / 2 ^ n} \<inter> (\<Union>q p. {real p / 2 ^ q})) \<subseteq> f ` {0..c (m / 2 ^ n)}"
+ proof clarsimp
+ fix p q
+ assume p: "0 < real p / 2 ^ q" "real p / 2 ^ q < real m / 2 ^ n"
+ then have [simp]: "0 < p" "p < 2 ^ q"
+ apply (simp add: divide_simps)
+ apply (blast intro: p less_2I m_div less_trans)
+ done
+ have "f (c (real p / 2 ^ q)) \<in> f ` {0..c (real m / 2 ^ n)}"
+ by (auto simp: clec p m)
+ then show "h (real p / 2 ^ q) \<in> f ` {0..c (real m / 2 ^ n)}"
+ by (simp add: h_eq)
+ qed
+ then have "h ` {0 .. m / 2^n} \<subseteq> f ` {0 .. c(m / 2^n)}"
+ apply (subst closure0m)
+ apply (rule image_closure_subset [OF cont_h' closed_f'])
+ using m_div apply auto
+ done
+ then have hx1: "h x1 \<in> f ` {0 .. c(m / 2^n)}"
+ using x12 less.prems(1) by auto
+ then obtain t1 where t1: "h x1 = f t1" "0 \<le> t1" "t1 \<le> c (m / 2 ^ n)"
+ by auto
+ have "h ` ({m / 2 ^ n<..<1} \<inter> (\<Union>q p. {real p / 2 ^ q})) \<subseteq> f ` {c (m / 2 ^ n)..1}"
+ proof clarsimp
+ fix p q
+ assume p: "real m / 2 ^ n < real p / 2 ^ q" and [simp]: "p < 2 ^ q"
+ then have [simp]: "0 < p"
+ using gr_zeroI m_div by fastforce
+ have "f (c (real p / 2 ^ q)) \<in> f ` {c (m / 2 ^ n)..1}"
+ by (auto simp: clec p m)
+ then show "h (real p / 2 ^ q) \<in> f ` {c (real m / 2 ^ n)..1}"
+ by (simp add: h_eq)
+ qed
+ then have "h ` {m / 2^n .. 1} \<subseteq> f ` {c(m / 2^n) .. 1}"
+ apply (subst closurem1)
+ apply (rule image_closure_subset [OF cont_h' closed_f'])
+ using m apply auto
+ done
+ then have hx2: "h x2 \<in> f ` {c(m / 2^n)..1}"
+ using x12 less.prems by auto
+ then obtain t2 where t2: "h x2 = f t2" "c (m / 2 ^ n) \<le> t2" "t2 \<le> 1"
+ by auto
+ with t1 less neq have False
+ using conn [of "h x2", unfolded is_interval_connected_1 [symmetric] is_interval_1, rule_format, of t1 t2 "c(m / 2^n)"]
+ by (simp add: h_eq m)
+ then show ?case by blast
+ qed auto
+ then show ?thesis
+ by (auto simp: inj_on_def)
+ qed
+ ultimately have "{0..1::real} homeomorphic f ` {0..1}"
+ using homeomorphic_compact [OF _ cont_h] by blast
+ then show ?thesis
+ using homeomorphic_sym by blast
+qed
+
+
+theorem path_contains_arc:
+ fixes p :: "real \<Rightarrow> 'a::{complete_space,real_normed_vector}"
+ assumes "path p" and a: "pathstart p = a" and b: "pathfinish p = b" and "a \<noteq> b"
+ obtains q where "arc q" "path_image q \<subseteq> path_image p" "pathstart q = a" "pathfinish q = b"
+proof -
+ have ucont_p: "uniformly_continuous_on {0..1} p"
+ using \<open>path p\<close> unfolding path_def
+ by (metis compact_Icc compact_uniformly_continuous)
+ define \<phi> where "\<phi> \<equiv> \<lambda>S. S \<subseteq> {0..1} \<and> 0 \<in> S \<and> 1 \<in> S \<and>
+ (\<forall>x \<in> S. \<forall>y \<in> S. open_segment x y \<inter> S = {} \<longrightarrow> p x = p y)"
+ obtain T where "closed T" "\<phi> T" and T: "\<And>U. \<lbrakk>closed U; \<phi> U\<rbrakk> \<Longrightarrow> \<not> (U \<subset> T)"
+ proof (rule Brouwer_reduction_theorem_gen [of "{0..1}" \<phi>])
+ have *: "{x<..<y} \<inter> {0..1} = {x<..<y}" if "0 \<le> x" "y \<le> 1" "x \<le> y" for x y::real
+ using that by auto
+ show "\<phi> {0..1}"
+ by (auto simp: \<phi>_def open_segment_eq_real_ivl *)
+ show "\<phi> (INTER UNIV F)"
+ if "\<And>n. closed (F n)" and \<phi>: "\<And>n. \<phi> (F n)" and Fsub: "\<And>n. F (Suc n) \<subseteq> F n" for F
+ proof -
+ have F01: "\<And>n. F n \<subseteq> {0..1} \<and> 0 \<in> F n \<and> 1 \<in> F n"
+ and peq: "\<And>n x y. \<lbrakk>x \<in> F n; y \<in> F n; open_segment x y \<inter> F n = {}\<rbrakk> \<Longrightarrow> p x = p y"
+ by (metis \<phi> \<phi>_def)+
+ have pqF: False if "\<forall>u. x \<in> F u" "\<forall>x. y \<in> F x" "open_segment x y \<inter> (\<Inter>x. F x) = {}" and neg: "p x \<noteq> p y"
+ for x y
+ using that
+ proof (induction x y rule: linorder_class.linorder_less_wlog)
+ case (less x y)
+ have xy: "x \<in> {0..1}" "y \<in> {0..1}"
+ by (metis less.prems subsetCE F01)+
+ have "norm(p x - p y) / 2 > 0"
+ using less by auto
+ then obtain e where "e > 0"
+ and e: "\<And>u v. \<lbrakk>u \<in> {0..1}; v \<in> {0..1}; dist v u < e\<rbrakk> \<Longrightarrow> dist (p v) (p u) < norm(p x - p y) / 2"
+ by (metis uniformly_continuous_onE [OF ucont_p])
+ have minxy: "min e (y - x) < (y - x) * (3 / 2)"
+ by (subst min_less_iff_disj) (simp add: less)
+ obtain w z where "w < z" and w: "w \<in> {x<..<y}" and z: "z \<in> {x<..<y}"
+ and wxe: "norm(w - x) < e" and zye: "norm(z - y) < e"
+ apply (rule_tac w = "x + (min e (y - x) / 3)" and z = "y - (min e (y - x) / 3)" in that)
+ using minxy \<open>0 < e\<close> less by simp_all
+ have Fclo: "\<And>T. T \<in> range F \<Longrightarrow> closed T"
+ by (metis \<open>\<And>n. closed (F n)\<close> image_iff)
+ have eq: "{w..z} \<inter> INTER UNIV F = {}"
+ using less w z apply (auto simp: open_segment_eq_real_ivl)
+ by (metis (no_types, hide_lams) INT_I IntI empty_iff greaterThanLessThan_iff not_le order.trans)
+ then obtain K where "finite K" and K: "{w..z} \<inter> (\<Inter> (F ` K)) = {}"
+ by (metis finite_subset_image compact_imp_fip [OF compact_interval Fclo])
+ then have "K \<noteq> {}"
+ using \<open>w < z\<close> \<open>{w..z} \<inter> INTER K F = {}\<close> by auto
+ define n where "n \<equiv> Max K"
+ have "n \<in> K" unfolding n_def by (metis \<open>K \<noteq> {}\<close> \<open>finite K\<close> Max_in)
+ have "F n \<subseteq> \<Inter> (F ` K)"
+ unfolding n_def by (metis Fsub Max_ge \<open>K \<noteq> {}\<close> \<open>finite K\<close> cINF_greatest lift_Suc_antimono_le)
+ with K have wzF_null: "{w..z} \<inter> F n = {}"
+ by (metis disjoint_iff_not_equal subset_eq)
+ obtain u where u: "u \<in> F n" "u \<in> {x..w}" "({u..w} - {u}) \<inter> F n = {}"
+ proof (cases "w \<in> F n")
+ case True
+ then show ?thesis
+ by (metis wzF_null \<open>w < z\<close> atLeastAtMost_iff disjoint_iff_not_equal less_eq_real_def)
+ next
+ case False
+ obtain u where "u \<in> F n" "u \<in> {x..w}" "{u<..<w} \<inter> F n = {}"
+ proof (rule segment_to_point_exists [of "F n \<inter> {x..w}" w])
+ show "closed (F n \<inter> {x..w})"
+ by (metis \<open>\<And>n. closed (F n)\<close> closed_Int closed_real_atLeastAtMost)
+ show "F n \<inter> {x..w} \<noteq> {}"
+ by (metis atLeastAtMost_iff disjoint_iff_not_equal greaterThanLessThan_iff less.prems(1) less_eq_real_def w)
+ qed (auto simp: open_segment_eq_real_ivl intro!: that)
+ with False show thesis
+ apply (auto simp: disjoint_iff_not_equal intro!: that)
+ by (metis greaterThanLessThan_iff less_eq_real_def)
+ qed
+ obtain v where v: "v \<in> F n" "v \<in> {z..y}" "({z..v} - {v}) \<inter> F n = {}"
+ proof (cases "z \<in> F n")
+ case True
+ have "z \<in> {w..z}"
+ using \<open>w < z\<close> by auto
+ then show ?thesis
+ by (metis wzF_null Int_iff True empty_iff)
+ next
+ case False
+ show ?thesis
+ proof (rule segment_to_point_exists [of "F n \<inter> {z..y}" z])
+ show "closed (F n \<inter> {z..y})"
+ by (metis \<open>\<And>n. closed (F n)\<close> closed_Int closed_atLeastAtMost)
+ show "F n \<inter> {z..y} \<noteq> {}"
+ by (metis atLeastAtMost_iff disjoint_iff_not_equal greaterThanLessThan_iff less.prems(2) less_eq_real_def z)
+ show "\<And>b. \<lbrakk>b \<in> F n \<inter> {z..y}; open_segment z b \<inter> (F n \<inter> {z..y}) = {}\<rbrakk> \<Longrightarrow> thesis"
+ apply (rule that)
+ apply (auto simp: open_segment_eq_real_ivl)
+ by (metis DiffI Int_iff atLeastAtMost_diff_ends atLeastAtMost_iff atLeastatMost_empty_iff empty_iff insert_iff False)
+ qed
+ qed
+ obtain u v where "u \<in> {0..1}" "v \<in> {0..1}" "norm(u - x) < e" "norm(v - y) < e" "p u = p v"
+ proof
+ show "u \<in> {0..1}" "v \<in> {0..1}"
+ by (metis F01 \<open>u \<in> F n\<close> \<open>v \<in> F n\<close> subsetD)+
+ show "norm(u - x) < e" "norm (v - y) < e"
+ using \<open>u \<in> {x..w}\<close> \<open>v \<in> {z..y}\<close> atLeastAtMost_iff real_norm_def wxe zye by auto
+ show "p u = p v"
+ proof (rule peq)
+ show "u \<in> F n" "v \<in> F n"
+ by (auto simp: u v)
+ have "False" if "\<xi> \<in> F n" "u < \<xi>" "\<xi> < v" for \<xi>
+ proof -
+ have "\<xi> \<notin> {z..v}"
+ by (metis DiffI disjoint_iff_not_equal less_irrefl singletonD that v(3))
+ moreover have "\<xi> \<notin> {w..z} \<inter> F n"
+ by (metis equals0D wzF_null)
+ ultimately have "\<xi> \<in> {u..w}"
+ using that by auto
+ then show ?thesis
+ by (metis DiffI disjoint_iff_not_equal less_eq_real_def not_le singletonD that u(3))
+ qed
+ moreover
+ have "\<lbrakk>\<xi> \<in> F n; v < \<xi>; \<xi> < u\<rbrakk> \<Longrightarrow> False" for \<xi>
+ using \<open>u \<in> {x..w}\<close> \<open>v \<in> {z..y}\<close> \<open>w < z\<close> by simp
+ ultimately
+ show "open_segment u v \<inter> F n = {}"
+ by (force simp: open_segment_eq_real_ivl)
+ qed
+ qed
+ then show ?case
+ using e [of x u] e [of y v] xy
+ apply (simp add: open_segment_eq_real_ivl dist_norm del: divide_const_simps)
+ by (metis dist_norm dist_triangle_half_r less_irrefl)
+ qed (auto simp: open_segment_commute)
+ show ?thesis
+ unfolding \<phi>_def by (metis (no_types, hide_lams) INT_I Inf_lower2 rangeI that F01 subsetCE pqF)
+ qed
+ show "closed {0..1::real}" by auto
+ qed (meson \<phi>_def)
+ then have "T \<subseteq> {0..1}" "0 \<in> T" "1 \<in> T"
+ and peq: "\<And>x y. \<lbrakk>x \<in> T; y \<in> T; open_segment x y \<inter> T = {}\<rbrakk> \<Longrightarrow> p x = p y"
+ unfolding \<phi>_def by metis+
+ then have "T \<noteq> {}" by auto
+ define h where "h \<equiv> \<lambda>x. p(@y. y \<in> T \<and> open_segment x y \<inter> T = {})"
+ have "p y = p z" if "y \<in> T" "z \<in> T" and xyT: "open_segment x y \<inter> T = {}" and xzT: "open_segment x z \<inter> T = {}"
+ for x y z
+ proof (cases "x \<in> T")
+ case True
+ with that show ?thesis by (metis \<open>\<phi> T\<close> \<phi>_def)
+ next
+ case False
+ have "insert x (open_segment x y \<union> open_segment x z) \<inter> T = {}"
+ by (metis False Int_Un_distrib2 Int_insert_left Un_empty_right xyT xzT)
+ moreover have "open_segment y z \<inter> T \<subseteq> insert x (open_segment x y \<union> open_segment x z) \<inter> T"
+ apply auto
+ by (metis greaterThanLessThan_iff less_eq_real_def less_le_trans linorder_neqE_linordered_idom open_segment_eq_real_ivl)
+ ultimately have "open_segment y z \<inter> T = {}"
+ by blast
+ with that peq show ?thesis by metis
+ qed
+ then have h_eq_p_gen: "h x = p y" if "y \<in> T" "open_segment x y \<inter> T = {}" for x y
+ using that unfolding h_def
+ by (metis (mono_tags, lifting) some_eq_ex)
+ then have h_eq_p: "\<And>x. x \<in> T \<Longrightarrow> h x = p x"
+ by simp
+ have disjoint: "\<And>x. \<exists>y. y \<in> T \<and> open_segment x y \<inter> T = {}"
+ by (meson \<open>T \<noteq> {}\<close> \<open>closed T\<close> segment_to_point_exists)
+ have heq: "h x = h x'" if "open_segment x x' \<inter> T = {}" for x x'
+ proof (cases "x \<in> T \<or> x' \<in> T")
+ case True
+ then show ?thesis
+ by (metis h_eq_p h_eq_p_gen open_segment_commute that)
+ next
+ case False
+ obtain y y' where "y \<in> T" "open_segment x y \<inter> T = {}" "h x = p y"
+ "y' \<in> T" "open_segment x' y' \<inter> T = {}" "h x' = p y'"
+ by (meson disjoint h_eq_p_gen)
+ moreover have "open_segment y y' \<subseteq> (insert x (insert x' (open_segment x y \<union> open_segment x' y' \<union> open_segment x x')))"
+ by (auto simp: open_segment_eq_real_ivl)
+ ultimately show ?thesis
+ using False that by (fastforce simp add: h_eq_p intro!: peq)
+ qed
+ have "h ` {0..1} homeomorphic {0..1::real}"
+ proof (rule homeomorphic_monotone_image_interval)
+ show "continuous_on {0..1} h"
+ proof (clarsimp simp add: continuous_on_iff)
+ fix u \<epsilon>::real
+ assume "0 < \<epsilon>" "0 \<le> u" "u \<le> 1"
+ then obtain \<delta> where "\<delta> > 0" and \<delta>: "\<And>v. v \<in> {0..1} \<Longrightarrow> dist v u < \<delta> \<longrightarrow> dist (p v) (p u) < \<epsilon> / 2"
+ using ucont_p [unfolded uniformly_continuous_on_def]
+ by (metis atLeastAtMost_iff half_gt_zero_iff)
+ then have "dist (h v) (h u) < \<epsilon>" if "v \<in> {0..1}" "dist v u < \<delta>" for v
+ proof (cases "open_segment u v \<inter> T = {}")
+ case True
+ then show ?thesis
+ using \<open>0 < \<epsilon>\<close> heq by auto
+ next
+ case False
+ have uvT: "closed (closed_segment u v \<inter> T)" "closed_segment u v \<inter> T \<noteq> {}"
+ using False open_closed_segment by (auto simp: \<open>closed T\<close> closed_Int)
+ obtain w where "w \<in> T" and w: "w \<in> closed_segment u v" "open_segment u w \<inter> T = {}"
+ apply (rule segment_to_point_exists [OF uvT, of u])
+ by (metis IntD1 Int_commute Int_left_commute ends_in_segment(1) inf.orderE subset_oc_segment)
+ then have puw: "dist (p u) (p w) < \<epsilon> / 2"
+ by (metis (no_types) \<open>T \<subseteq> {0..1}\<close> \<open>dist v u < \<delta>\<close> \<delta> dist_commute dist_in_closed_segment le_less_trans subsetCE)
+ obtain z where "z \<in> T" and z: "z \<in> closed_segment u v" "open_segment v z \<inter> T = {}"
+ apply (rule segment_to_point_exists [OF uvT, of v])
+ by (metis IntD2 Int_commute Int_left_commute ends_in_segment(2) inf.orderE subset_oc_segment)
+ then have "dist (p u) (p z) < \<epsilon> / 2"
+ by (metis \<open>T \<subseteq> {0..1}\<close> \<open>dist v u < \<delta>\<close> \<delta> dist_commute dist_in_closed_segment le_less_trans subsetCE)
+ then show ?thesis
+ using puw by (metis (no_types) \<open>w \<in> T\<close> \<open>z \<in> T\<close> dist_commute dist_triangle_half_l h_eq_p_gen w(2) z(2))
+ qed
+ with \<open>0 < \<delta>\<close> show "\<exists>\<delta>>0. \<forall>v\<in>{0..1}. dist v u < \<delta> \<longrightarrow> dist (h v) (h u) < \<epsilon>" by blast
+ qed
+ show "connected ({0..1} \<inter> h -` {z})" for z
+ proof (clarsimp simp add: connected_iff_connected_component)
+ fix u v
+ assume huv_eq: "h v = h u" and uv: "0 \<le> u" "u \<le> 1" "0 \<le> v" "v \<le> 1"
+ have "\<exists>T. connected T \<and> T \<subseteq> {0..1} \<and> T \<subseteq> h -` {h u} \<and> u \<in> T \<and> v \<in> T"
+ proof (intro exI conjI)
+ show "connected (closed_segment u v)"
+ by simp
+ show "closed_segment u v \<subseteq> {0..1}"
+ by (simp add: uv closed_segment_eq_real_ivl)
+ have pxy: "p x = p y"
+ if "T \<subseteq> {0..1}" "0 \<in> T" "1 \<in> T" "x \<in> T" "y \<in> T"
+ and disjT: "open_segment x y \<inter> (T - open_segment u v) = {}"
+ and xynot: "x \<notin> open_segment u v" "y \<notin> open_segment u v"
+ for x y
+ proof (cases "open_segment x y \<inter> open_segment u v = {}")
+ case True
+ then show ?thesis
+ by (metis Diff_Int_distrib Diff_empty peq disjT \<open>x \<in> T\<close> \<open>y \<in> T\<close>)
+ next
+ case False
+ then have "open_segment x u \<union> open_segment y v \<subseteq> open_segment x y - open_segment u v \<or>
+ open_segment y u \<union> open_segment x v \<subseteq> open_segment x y - open_segment u v" (is "?xuyv \<or> ?yuxv")
+ using xynot by (fastforce simp add: open_segment_eq_real_ivl not_le not_less split: if_split_asm)
+ then show "p x = p y"
+ proof
+ assume "?xuyv"
+ then have "open_segment x u \<inter> T = {}" "open_segment y v \<inter> T = {}"
+ using disjT by auto
+ then have "h x = h y"
+ using heq huv_eq by auto
+ then show ?thesis
+ using h_eq_p \<open>x \<in> T\<close> \<open>y \<in> T\<close> by auto
+ next
+ assume "?yuxv"
+ then have "open_segment y u \<inter> T = {}" "open_segment x v \<inter> T = {}"
+ using disjT by auto
+ then have "h x = h y"
+ using heq [of y u] heq [of x v] huv_eq by auto
+ then show ?thesis
+ using h_eq_p \<open>x \<in> T\<close> \<open>y \<in> T\<close> by auto
+ qed
+ qed
+ have "\<not> T - open_segment u v \<subset> T"
+ proof (rule T)
+ show "closed (T - open_segment u v)"
+ by (simp add: closed_Diff [OF \<open>closed T\<close>] open_segment_eq_real_ivl)
+ have "0 \<notin> open_segment u v" "1 \<notin> open_segment u v"
+ using open_segment_eq_real_ivl uv by auto
+ then show "\<phi> (T - open_segment u v)"
+ using \<open>T \<subseteq> {0..1}\<close> \<open>0 \<in> T\<close> \<open>1 \<in> T\<close>
+ by (auto simp: \<phi>_def) (meson peq pxy)
+ qed
+ then have "open_segment u v \<inter> T = {}"
+ by blast
+ then show "closed_segment u v \<subseteq> h -` {h u}"
+ by (force intro: heq simp: open_segment_eq_real_ivl closed_segment_eq_real_ivl split: if_split_asm)+
+ qed auto
+ then show "connected_component ({0..1} \<inter> h -` {h u}) u v"
+ by (simp add: connected_component_def)
+ qed
+ show "h 1 \<noteq> h 0"
+ by (metis \<open>\<phi> T\<close> \<phi>_def a \<open>a \<noteq> b\<close> b h_eq_p pathfinish_def pathstart_def)
+ qed
+ then obtain f and g :: "real \<Rightarrow> 'a"
+ where gfeq: "(\<forall>x\<in>h ` {0..1}. (g(f x) = x))" and fhim: "f ` h ` {0..1} = {0..1}" and contf: "continuous_on (h ` {0..1}) f"
+ and fgeq: "(\<forall>y\<in>{0..1}. (f(g y) = y))" and pag: "path_image g = h ` {0..1}" and contg: "continuous_on {0..1} g"
+ by (auto simp: homeomorphic_def homeomorphism_def path_image_def)
+ then have "arc g"
+ by (metis arc_def path_def inj_on_def)
+ obtain u v where "u \<in> {0..1}" "a = g u" "v \<in> {0..1}" "b = g v"
+ by (metis (mono_tags, hide_lams) \<open>\<phi> T\<close> \<phi>_def a b fhim gfeq h_eq_p imageI path_image_def pathfinish_def pathfinish_in_path_image pathstart_def pathstart_in_path_image)
+ then have "a \<in> path_image g" "b \<in> path_image g"
+ using path_image_def by blast+
+ have ph: "path_image h \<subseteq> path_image p"
+ by (metis image_mono image_subset_iff path_image_def disjoint h_eq_p_gen \<open>T \<subseteq> {0..1}\<close>)
+ show ?thesis
+ proof
+ show "pathstart (subpath u v g) = a" "pathfinish (subpath u v g) = b"
+ by (simp_all add: \<open>a = g u\<close> \<open>b = g v\<close>)
+ show "path_image (subpath u v g) \<subseteq> path_image p"
+ by (metis \<open>arc g\<close> \<open>u \<in> {0..1}\<close> \<open>v \<in> {0..1}\<close> arc_imp_path order_trans pag path_image_def path_image_subpath_subset ph)
+ show "arc (subpath u v g)"
+ using \<open>arc g\<close> \<open>a = g u\<close> \<open>b = g v\<close> \<open>u \<in> {0..1}\<close> \<open>v \<in> {0..1}\<close> arc_subpath_arc \<open>a \<noteq> b\<close> by blast
+ qed
+qed
+
+
+corollary path_connected_arcwise:
+ fixes S :: "'a::{complete_space,real_normed_vector} set"
+ shows "path_connected S \<longleftrightarrow>
+ (\<forall>x \<in> S. \<forall>y \<in> S. x \<noteq> y \<longrightarrow> (\<exists>g. arc g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y))"
+ (is "?lhs = ?rhs")
+proof (intro iffI impI ballI)
+ fix x y
+ assume "path_connected S" "x \<in> S" "y \<in> S" "x \<noteq> y"
+ then obtain p where p: "path p" "path_image p \<subseteq> S" "pathstart p = x" "pathfinish p = y"
+ by (force simp: path_connected_def)
+ then show "\<exists>g. arc g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
+ by (metis \<open>x \<noteq> y\<close> order_trans path_contains_arc)
+next
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ unfolding path_connected_def
+ proof (intro ballI)
+ fix x y
+ assume "x \<in> S" "y \<in> S"
+ show "\<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
+ proof (cases "x = y")
+ case True with \<open>x \<in> S\<close> path_component_def path_component_refl show ?thesis
+ by blast
+ next
+ case False with R [OF \<open>x \<in> S\<close> \<open>y \<in> S\<close>] show ?thesis
+ by (auto intro: arc_imp_path)
+ qed
+ qed
+qed
+
+
+corollary arc_connected_trans:
+ fixes g :: "real \<Rightarrow> 'a::{complete_space,real_normed_vector}"
+ assumes "arc g" "arc h" "pathfinish g = pathstart h" "pathstart g \<noteq> pathfinish h"
+ obtains i where "arc i" "path_image i \<subseteq> path_image g \<union> path_image h"
+ "pathstart i = pathstart g" "pathfinish i = pathfinish h"
+ by (metis (no_types, hide_lams) arc_imp_path assms path_contains_arc path_image_join path_join pathfinish_join pathstart_join)
+
+
+
+
+subsection\<open>Accessibility of frontier points\<close>
+
+lemma dense_accessible_frontier_points:
+ fixes S :: "'a::{complete_space,real_normed_vector} set"
+ assumes "open S" and opeSV: "openin (subtopology euclidean (frontier S)) V" and "V \<noteq> {}"
+ obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g \<in> S" "pathfinish g \<in> V"
+proof -
+ obtain z where "z \<in> V"
+ using \<open>V \<noteq> {}\<close> by auto
+ then obtain r where "r > 0" and r: "ball z r \<inter> frontier S \<subseteq> V"
+ by (metis openin_contains_ball opeSV)
+ then have "z \<in> frontier S"
+ using \<open>z \<in> V\<close> opeSV openin_contains_ball by blast
+ then have "z \<in> closure S" "z \<notin> S"
+ by (simp_all add: frontier_def assms interior_open)
+ with \<open>r > 0\<close> have "infinite (S \<inter> ball z r)"
+ by (auto simp: closure_def islimpt_eq_infinite_ball)
+ then obtain y where "y \<in> S" and y: "y \<in> ball z r"
+ using infinite_imp_nonempty by force
+ then have "y \<notin> frontier S"
+ by (meson \<open>open S\<close> disjoint_iff_not_equal frontier_disjoint_eq)
+ have "y \<noteq> z"
+ using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by blast
+ have "path_connected(ball z r)"
+ by (simp add: convex_imp_path_connected)
+ with y \<open>r > 0\<close> obtain g where "arc g" and pig: "path_image g \<subseteq> ball z r"
+ and g: "pathstart g = y" "pathfinish g = z"
+ using \<open>y \<noteq> z\<close> by (force simp: path_connected_arcwise)
+ have "compact (g -` frontier S \<inter> {0..1})"
+ apply (simp add: compact_eq_bounded_closed bounded_Int bounded_closed_interval)
+ apply (rule closed_vimage_Int)
+ using \<open>arc g\<close> apply (auto simp: arc_def path_def)
+ done
+ moreover have "g -` frontier S \<inter> {0..1} \<noteq> {}"
+ proof -
+ have "\<exists>r. r \<in> g -` frontier S \<and> r \<in> {0..1}"
+ by (metis \<open>z \<in> frontier S\<close> g(2) imageE path_image_def pathfinish_in_path_image vimageI2)
+ then show ?thesis
+ by blast
+ qed
+ ultimately obtain t where gt: "g t \<in> frontier S" and "0 \<le> t" "t \<le> 1"
+ and t: "\<And>u. \<lbrakk>g u \<in> frontier S; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> t \<le> u"
+ by (force simp: dest!: compact_attains_inf)
+ moreover have "t \<noteq> 0"
+ by (metis \<open>y \<notin> frontier S\<close> g(1) gt pathstart_def)
+ ultimately have t01: "0 < t" "t \<le> 1"
+ by auto
+ have "V \<subseteq> frontier S"
+ using opeSV openin_contains_ball by blast
+ show ?thesis
+ proof
+ show "arc (subpath 0 t g)"
+ by (simp add: \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> \<open>arc g\<close> \<open>t \<noteq> 0\<close> arc_subpath_arc)
+ have "g 0 \<in> S"
+ by (metis \<open>y \<in> S\<close> g(1) pathstart_def)
+ then show "pathstart (subpath 0 t g) \<in> S"
+ by auto
+ have "g t \<in> V"
+ by (metis IntI atLeastAtMost_iff gt image_eqI path_image_def pig r subsetCE \<open>0 \<le> t\<close> \<open>t \<le> 1\<close>)
+ then show "pathfinish (subpath 0 t g) \<in> V"
+ by auto
+ then have "inj_on (subpath 0 t g) {0..1}"
+ using t01
+ apply (clarsimp simp: inj_on_def subpath_def)
+ apply (drule inj_onD [OF arc_imp_inj_on [OF \<open>arc g\<close>]])
+ using mult_le_one apply auto
+ done
+ then have "subpath 0 t g ` {0..<1} \<subseteq> subpath 0 t g ` {0..1} - {subpath 0 t g 1}"
+ by (force simp: dest: inj_onD)
+ moreover have False if "subpath 0 t g ` ({0..<1}) - S \<noteq> {}"
+ proof -
+ have contg: "continuous_on {0..1} g"
+ using \<open>arc g\<close> by (auto simp: arc_def path_def)
+ have "subpath 0 t g ` {0..<1} \<inter> frontier S \<noteq> {}"
+ proof (rule connected_Int_frontier [OF _ _ that])
+ show "connected (subpath 0 t g ` {0..<1})"
+ apply (rule connected_continuous_image)
+ apply (simp add: subpath_def)
+ apply (intro continuous_intros continuous_on_compose2 [OF contg])
+ apply (auto simp: \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> mult_le_one)
+ done
+ show "subpath 0 t g ` {0..<1} \<inter> S \<noteq> {}"
+ using \<open>y \<in> S\<close> g(1) by (force simp: subpath_def image_def pathstart_def)
+ qed
+ then obtain x where "x \<in> subpath 0 t g ` {0..<1}" "x \<in> frontier S"
+ by blast
+ with t01 \<open>0 \<le> t\<close> mult_le_one t show False
+ by (fastforce simp: subpath_def)
+ qed
+ then have "subpath 0 t g ` {0..1} - {subpath 0 t g 1} \<subseteq> S"
+ using subsetD by fastforce
+ ultimately show "subpath 0 t g ` {0..<1} \<subseteq> S"
+ by auto
+ qed
+qed
+
+
+lemma dense_accessible_frontier_points_connected:
+ fixes S :: "'a::{complete_space,real_normed_vector} set"
+ assumes "open S" "connected S" "x \<in> S" "V \<noteq> {}"
+ and ope: "openin (subtopology euclidean (frontier S)) V"
+ obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g = x" "pathfinish g \<in> V"
+proof -
+ have "V \<subseteq> frontier S"
+ using ope openin_imp_subset by blast
+ with \<open>open S\<close> \<open>x \<in> S\<close> have "x \<notin> V"
+ using interior_open by (auto simp: frontier_def)
+ obtain g where "arc g" and g: "g ` {0..<1} \<subseteq> S" "pathstart g \<in> S" "pathfinish g \<in> V"
+ by (metis dense_accessible_frontier_points [OF \<open>open S\<close> ope \<open>V \<noteq> {}\<close>])
+ then have "path_connected S"
+ by (simp add: assms connected_open_path_connected)
+ with \<open>pathstart g \<in> S\<close> \<open>x \<in> S\<close> have "path_component S x (pathstart g)"
+ by (simp add: path_connected_component)
+ then obtain f where "path f" and f: "path_image f \<subseteq> S" "pathstart f = x" "pathfinish f = pathstart g"
+ by (auto simp: path_component_def)
+ then have "path (f +++ g)"
+ by (simp add: \<open>arc g\<close> arc_imp_path)
+ then obtain h where "arc h"
+ and h: "path_image h \<subseteq> path_image (f +++ g)" "pathstart h = x" "pathfinish h = pathfinish g"
+ apply (rule path_contains_arc [of "f +++ g" x "pathfinish g"])
+ using f \<open>x \<notin> V\<close> \<open>pathfinish g \<in> V\<close> by auto
+ have "h ` {0..1} - {h 1} \<subseteq> S"
+ using f g h apply (clarsimp simp: path_image_join)
+ apply (simp add: path_image_def pathfinish_def subset_iff image_def Bex_def)
+ by (metis le_less)
+ then have "h ` {0..<1} \<subseteq> S"
+ using \<open>arc h\<close> by (force simp: arc_def dest: inj_onD)
+ then show thesis
+ apply (rule that [OF \<open>arc h\<close>])
+ using h \<open>pathfinish g \<in> V\<close> by auto
+qed
+
+lemma dense_access_fp_aux:
+ fixes S :: "'a::{complete_space,real_normed_vector} set"
+ assumes S: "open S" "connected S"
+ and opeSU: "openin (subtopology euclidean (frontier S)) U"
+ and opeSV: "openin (subtopology euclidean (frontier S)) V"
+ and "V \<noteq> {}" "\<not> U \<subseteq> V"
+ obtains g where "arc g" "pathstart g \<in> U" "pathfinish g \<in> V" "g ` {0<..<1} \<subseteq> S"
+proof -
+ have "S \<noteq> {}"
+ using opeSV \<open>V \<noteq> {}\<close> by (metis frontier_empty openin_subtopology_empty)
+ then obtain x where "x \<in> S" by auto
+ obtain g where "arc g" and g: "g ` {0..<1} \<subseteq> S" "pathstart g = x" "pathfinish g \<in> V"
+ using dense_accessible_frontier_points_connected [OF S \<open>x \<in> S\<close> \<open>V \<noteq> {}\<close> opeSV] by blast
+ obtain h where "arc h" and h: "h ` {0..<1} \<subseteq> S" "pathstart h = x" "pathfinish h \<in> U - {pathfinish g}"
+ proof (rule dense_accessible_frontier_points_connected [OF S \<open>x \<in> S\<close>])
+ show "U - {pathfinish g} \<noteq> {}"
+ using \<open>pathfinish g \<in> V\<close> \<open>\<not> U \<subseteq> V\<close> by blast
+ show "openin (subtopology euclidean (frontier S)) (U - {pathfinish g})"
+ by (simp add: opeSU openin_delete)
+ qed auto
+ obtain \<gamma> where "arc \<gamma>"
+ and \<gamma>: "path_image \<gamma> \<subseteq> path_image (reversepath h +++ g)"
+ "pathstart \<gamma> = pathfinish h" "pathfinish \<gamma> = pathfinish g"
+ proof (rule path_contains_arc [of "(reversepath h +++ g)" "pathfinish h" "pathfinish g"])
+ show "path (reversepath h +++ g)"
+ by (simp add: \<open>arc g\<close> \<open>arc h\<close> \<open>pathstart g = x\<close> \<open>pathstart h = x\<close> arc_imp_path)
+ show "pathstart (reversepath h +++ g) = pathfinish h"
+ "pathfinish (reversepath h +++ g) = pathfinish g"
+ by auto
+ show "pathfinish h \<noteq> pathfinish g"
+ using \<open>pathfinish h \<in> U - {pathfinish g}\<close> by auto
+ qed auto
+ show ?thesis
+ proof
+ show "arc \<gamma>" "pathstart \<gamma> \<in> U" "pathfinish \<gamma> \<in> V"
+ using \<gamma> \<open>arc \<gamma>\<close> \<open>pathfinish h \<in> U - {pathfinish g}\<close> \<open>pathfinish g \<in> V\<close> by auto
+ have "\<gamma> ` {0..1} - {\<gamma> 0, \<gamma> 1} \<subseteq> S"
+ using \<gamma> g h
+ apply (simp add: path_image_join)
+ apply (simp add: path_image_def pathstart_def pathfinish_def subset_iff image_def Bex_def)
+ by (metis linorder_neqE_linordered_idom not_less)
+ then show "\<gamma> ` {0<..<1} \<subseteq> S"
+ using \<open>arc h\<close> \<open>arc \<gamma>\<close>
+ by (metis arc_imp_simple_path path_image_def pathfinish_def pathstart_def simple_path_endless)
+ qed
+qed
+
+lemma dense_accessible_frontier_point_pairs:
+ fixes S :: "'a::{complete_space,real_normed_vector} set"
+ assumes S: "open S" "connected S"
+ and opeSU: "openin (subtopology euclidean (frontier S)) U"
+ and opeSV: "openin (subtopology euclidean (frontier S)) V"
+ and "U \<noteq> {}" "V \<noteq> {}" "U \<noteq> V"
+ obtains g where "arc g" "pathstart g \<in> U" "pathfinish g \<in> V" "g ` {0<..<1} \<subseteq> S"
+proof -
+ consider "\<not> U \<subseteq> V" | "\<not> V \<subseteq> U"
+ using \<open>U \<noteq> V\<close> by blast
+ then show ?thesis
+ proof cases
+ case 1 then show ?thesis
+ using assms dense_access_fp_aux [OF S opeSU opeSV] that by blast
+ next
+ case 2
+ obtain g where "arc g" and g: "pathstart g \<in> V" "pathfinish g \<in> U" "g ` {0<..<1} \<subseteq> S"
+ using assms dense_access_fp_aux [OF S opeSV opeSU] "2" by blast
+ show ?thesis
+ proof
+ show "arc (reversepath g)"
+ by (simp add: \<open>arc g\<close> arc_reversepath)
+ show "pathstart (reversepath g) \<in> U" "pathfinish (reversepath g) \<in> V"
+ using g by auto
+ show "reversepath g ` {0<..<1} \<subseteq> S"
+ using g by (auto simp: reversepath_def)
+ qed
+ qed
+qed
+
+end