--- a/CONTRIBUTORS Thu Oct 06 11:27:03 2016 +0200
+++ b/CONTRIBUTORS Thu Oct 06 11:27:28 2016 +0200
@@ -6,13 +6,17 @@
Contributions to this Isabelle version
--------------------------------------
-* January 2016: Florian Haftmann
+* January 2016: Florian Haftmann, TUM
Abolition of compound operators INFIMUM and SUPREMUM
for complete lattices.
-* March 2016: Florian Haftmann
+* March 2016: Florian Haftmann, TUM
Abstract factorial rings with unique factorization.
+* March 2016: Florian Haftmann, TUM
+ Reworking of the HOL char type as special case of a
+ finite numeral type.
+
* March 2016: Andreas Lochbihler
Reasoning support for monotonicity, continuity and
admissibility in chain-complete partial orders.
@@ -23,6 +27,11 @@
* June 2016: Andreas Lochbihler
Formalisation of discrete subprobability distributions.
+* June 2016: Florian Haftmann, TUM
+ Improvements to code generation: optional timing measurements,
+ more succint closures for static evaluation, less ambiguities
+ concering Scala implicits.
+
* July 2016: Daniel Stuewe
Height-size proofs in HOL/Data_Structures
--- a/NEWS Thu Oct 06 11:27:03 2016 +0200
+++ b/NEWS Thu Oct 06 11:27:28 2016 +0200
@@ -60,6 +60,9 @@
introduction and elimination rules after each split rule. As a
result the subgoal may be split into several subgoals.
+* Solve direct: option 'solve_direct_strict_warnings' gives explicit
+ warnings for lemma statements with trivial proofs.
+
*** Prover IDE -- Isabelle/Scala/jEdit ***
--- a/src/HOL/Analysis/Analysis.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Analysis.thy Thu Oct 06 11:27:28 2016 +0200
@@ -10,6 +10,7 @@
Bounded_Continuous_Function
Weierstrass_Theorems
Polytope
+ FurtherTopology
Poly_Roots
Conformal_Mappings
Generalised_Binomial_Theorem
--- a/src/HOL/Analysis/Binary_Product_Measure.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Binary_Product_Measure.thy Thu Oct 06 11:27:28 2016 +0200
@@ -686,6 +686,53 @@
show "a \<in> A" and "b \<in> B" by auto
qed
+lemma sets_pair_eq:
+ assumes Ea: "Ea \<subseteq> Pow (space A)" "sets A = sigma_sets (space A) Ea"
+ and Ca: "countable Ca" "Ca \<subseteq> Ea" "\<Union>Ca = space A"
+ and Eb: "Eb \<subseteq> Pow (space B)" "sets B = sigma_sets (space B) Eb"
+ and Cb: "countable Cb" "Cb \<subseteq> Eb" "\<Union>Cb = space B"
+ shows "sets (A \<Otimes>\<^sub>M B) = sets (sigma (space A \<times> space B) { a \<times> b | a b. a \<in> Ea \<and> b \<in> Eb })"
+ (is "_ = sets (sigma ?\<Omega> ?E)")
+proof
+ show "sets (sigma ?\<Omega> ?E) \<subseteq> sets (A \<Otimes>\<^sub>M B)"
+ using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
+ have "?E \<subseteq> Pow ?\<Omega>"
+ using Ea(1) Eb(1) by auto
+ then have E: "a \<in> Ea \<Longrightarrow> b \<in> Eb \<Longrightarrow> a \<times> b \<in> sets (sigma ?\<Omega> ?E)" for a b
+ by auto
+ have "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (Sup {vimage_algebra ?\<Omega> fst A, vimage_algebra ?\<Omega> snd B})"
+ unfolding sets_pair_eq_sets_fst_snd ..
+ also have "vimage_algebra ?\<Omega> fst A = vimage_algebra ?\<Omega> fst (sigma (space A) Ea)"
+ by (intro vimage_algebra_cong[OF refl refl]) (simp add: Ea)
+ also have "\<dots> = sigma ?\<Omega> {fst -` A \<inter> ?\<Omega> |A. A \<in> Ea}"
+ by (intro Ea vimage_algebra_sigma) auto
+ also have "vimage_algebra ?\<Omega> snd B = vimage_algebra ?\<Omega> snd (sigma (space B) Eb)"
+ by (intro vimage_algebra_cong[OF refl refl]) (simp add: Eb)
+ also have "\<dots> = sigma ?\<Omega> {snd -` A \<inter> ?\<Omega> |A. A \<in> Eb}"
+ by (intro Eb vimage_algebra_sigma) auto
+ also have "{sigma ?\<Omega> {fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, sigma ?\<Omega> {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}} =
+ sigma ?\<Omega> ` {{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}"
+ by auto
+ also have "sets (SUP S:{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}. sigma ?\<Omega> S) =
+ sets (sigma ?\<Omega> (\<Union>{{fst -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Ea}, {snd -` Aa \<inter> ?\<Omega> |Aa. Aa \<in> Eb}}))"
+ using Ea(1) Eb(1) by (intro sets_Sup_sigma) auto
+ also have "\<dots> \<subseteq> sets (sigma ?\<Omega> ?E)"
+ proof (subst sigma_le_sets, safe intro!: space_in_measure_of)
+ fix a assume "a \<in> Ea"
+ then have "fst -` a \<inter> ?\<Omega> = (\<Union>b\<in>Cb. a \<times> b)"
+ using Cb(3)[symmetric] Ea(1) by auto
+ then show "fst -` a \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)"
+ using Cb \<open>a \<in> Ea\<close> by (auto intro!: sets.countable_UN' E)
+ next
+ fix b assume "b \<in> Eb"
+ then have "snd -` b \<inter> ?\<Omega> = (\<Union>a\<in>Ca. a \<times> b)"
+ using Ca(3)[symmetric] Eb(1) by auto
+ then show "snd -` b \<inter> ?\<Omega> \<in> sets (sigma ?\<Omega> ?E)"
+ using Ca \<open>b \<in> Eb\<close> by (auto intro!: sets.countable_UN' E)
+ qed
+ finally show "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets (sigma ?\<Omega> ?E)" .
+qed
+
lemma borel_prod:
"(borel \<Otimes>\<^sub>M borel) = (borel :: ('a::second_countable_topology \<times> 'b::second_countable_topology) measure)"
(is "?P = ?B")
--- a/src/HOL/Analysis/Bochner_Integration.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Bochner_Integration.thy Thu Oct 06 11:27:28 2016 +0200
@@ -951,6 +951,10 @@
unfolding integrable.simps
by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)
+lemma integrable_cong_AE_imp:
+ "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
+ using integrable_cong_AE[of f M g] by (auto simp: eq_commute)
+
lemma integral_cong:
"M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g"
by (simp cong: has_bochner_integral_cong cong del: if_weak_cong add: lebesgue_integral_def)
@@ -1682,6 +1686,16 @@
finally show ?thesis .
qed
+lemma nn_integral_eq_integrable:
+ assumes f: "f \<in> M \<rightarrow>\<^sub>M borel" "AE x in M. 0 \<le> f x" and "0 \<le> x"
+ shows "(\<integral>\<^sup>+x. f x \<partial>M) = ennreal x \<longleftrightarrow> (integrable M f \<and> integral\<^sup>L M f = x)"
+proof (safe intro!: nn_integral_eq_integral assms)
+ assume *: "(\<integral>\<^sup>+x. f x \<partial>M) = ennreal x"
+ with integrableI_nn_integral_finite[OF f this] nn_integral_eq_integral[of M f, OF _ f(2)]
+ show "integrable M f" "integral\<^sup>L M f = x"
+ by (simp_all add: * assms integral_nonneg_AE)
+qed
+
lemma
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes integrable[measurable]: "\<And>i. integrable M (f i)"
@@ -2227,6 +2241,27 @@
shows "integrable (count_space UNIV) f \<Longrightarrow> integral\<^sup>L (count_space UNIV) f = (\<Sum>x. f x)"
using sums_integral_count_space_nat by (rule sums_unique)
+lemma integrable_bij_count_space:
+ fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+ assumes g: "bij_betw g A B"
+ shows "integrable (count_space A) (\<lambda>x. f (g x)) \<longleftrightarrow> integrable (count_space B) f"
+ unfolding integrable_iff_bounded by (subst nn_integral_bij_count_space[OF g]) auto
+
+lemma integral_bij_count_space:
+ fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+ assumes g: "bij_betw g A B"
+ shows "integral\<^sup>L (count_space A) (\<lambda>x. f (g x)) = integral\<^sup>L (count_space B) f"
+ using g[THEN bij_betw_imp_funcset]
+ apply (subst distr_bij_count_space[OF g, symmetric])
+ apply (intro integral_distr[symmetric])
+ apply auto
+ done
+
+lemma has_bochner_integral_count_space_nat:
+ fixes f :: "nat \<Rightarrow> _::{banach,second_countable_topology}"
+ shows "has_bochner_integral (count_space UNIV) f x \<Longrightarrow> f sums x"
+ unfolding has_bochner_integral_iff by (auto intro!: sums_integral_count_space_nat)
+
subsection \<open>Point measure\<close>
lemma lebesgue_integral_point_measure_finite:
--- a/src/HOL/Analysis/Borel_Space.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Borel_Space.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1500,6 +1500,9 @@
apply auto
done
+lemma measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel"
+ by simp
+
subsection "Borel space on the extended reals"
lemma borel_measurable_ereal[measurable (raw)]:
@@ -1909,6 +1912,56 @@
shows "mono f \<Longrightarrow> f \<in> borel_measurable borel"
using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def)
+lemma measurable_bdd_below_real[measurable (raw)]:
+ fixes F :: "'a \<Rightarrow> 'i \<Rightarrow> real"
+ assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M borel"
+ shows "Measurable.pred M (\<lambda>x. bdd_below ((\<lambda>i. F i x)`I))"
+proof (subst measurable_cong)
+ show "bdd_below ((\<lambda>i. F i x)`I) \<longleftrightarrow> (\<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i x)" for x
+ by (auto simp: bdd_below_def intro!: bexI[of _ "of_int (floor _)"] intro: order_trans of_int_floor_le)
+ show "Measurable.pred M (\<lambda>w. \<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i w)"
+ using countable_int by measurable
+qed
+
+lemma borel_measurable_cINF_real[measurable (raw)]:
+ fixes F :: "_ \<Rightarrow> _ \<Rightarrow> real"
+ assumes [simp]: "countable I"
+ assumes F[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"
+ shows "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable M"
+proof (rule measurable_piecewise_restrict)
+ let ?\<Omega> = "{x\<in>space M. bdd_below ((\<lambda>i. F i x)`I)}"
+ show "countable {?\<Omega>, - ?\<Omega>}" "space M \<subseteq> \<Union>{?\<Omega>, - ?\<Omega>}" "\<And>X. X \<in> {?\<Omega>, - ?\<Omega>} \<Longrightarrow> X \<inter> space M \<in> sets M"
+ by auto
+ fix X assume "X \<in> {?\<Omega>, - ?\<Omega>}" then show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M X)"
+ proof safe
+ show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M ?\<Omega>)"
+ by (intro borel_measurable_cINF measurable_restrict_space1 F)
+ (auto simp: space_restrict_space)
+ show "(\<lambda>x. INF i:I. F i x) \<in> borel_measurable (restrict_space M (-?\<Omega>))"
+ proof (subst measurable_cong)
+ fix x assume "x \<in> space (restrict_space M (-?\<Omega>))"
+ then have "\<not> (\<forall>i\<in>I. - F i x \<le> y)" for y
+ by (auto simp: space_restrict_space bdd_above_def bdd_above_uminus[symmetric])
+ then show "(INF i:I. F i x) = - (THE x. False)"
+ by (auto simp: space_restrict_space Inf_real_def Sup_real_def Least_def simp del: Set.ball_simps(10))
+ qed simp
+ qed
+qed
+
+lemma borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))"
+proof (safe intro!: borel_eq_sigmaI1[OF borel_Iio])
+ fix x :: real
+ have eq: "{..<x} = space (sigma UNIV (range atLeast)) - {x ..}"
+ by auto
+ show "{..<x} \<in> sets (sigma UNIV (range atLeast))"
+ unfolding eq by (intro sets.compl_sets) auto
+qed auto
+
+lemma borel_measurable_pred_less[measurable (raw)]:
+ fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}"
+ shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> Measurable.pred M (\<lambda>w. f w < g w)"
+ unfolding Measurable.pred_def by (rule borel_measurable_less)
+
no_notation
eucl_less (infix "<e" 50)
--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1975,7 +1975,7 @@
text \<open>So we get the no-retraction theorem.\<close>
-lemma no_retraction_cball:
+theorem no_retraction_cball:
fixes a :: "'a::euclidean_space"
assumes "e > 0"
shows "\<not> (frontier (cball a e) retract_of (cball a e))"
@@ -2001,6 +2001,26 @@
using x assms by auto
qed
+corollary contractible_sphere:
+ fixes a :: "'a::euclidean_space"
+ shows "contractible(sphere a r) \<longleftrightarrow> r \<le> 0"
+proof (cases "0 < r")
+ case True
+ then show ?thesis
+ unfolding contractible_def nullhomotopic_from_sphere_extension
+ using no_retraction_cball [OF True, of a]
+ by (auto simp: retract_of_def retraction_def)
+next
+ case False
+ then show ?thesis
+ unfolding contractible_def nullhomotopic_from_sphere_extension
+ apply (simp add: not_less)
+ apply (rule_tac x=id in exI)
+ apply (auto simp: continuous_on_def)
+ apply (meson dist_not_less_zero le_less less_le_trans)
+ done
+qed
+
subsection\<open>Retractions\<close>
lemma retraction:
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Thu Oct 06 11:27:28 2016 +0200
@@ -7495,6 +7495,11 @@
by (auto simp: closed_segment_commute)
qed
+lemma open_segment_eq_real_ivl:
+ fixes a b::real
+ shows "open_segment a b = (if a \<le> b then {a<..<b} else {b<..<a})"
+by (auto simp: closed_segment_eq_real_ivl open_segment_def split: if_split_asm)
+
lemma closed_segment_real_eq:
fixes u::real shows "closed_segment u v = (\<lambda>x. (v - u) * x + u) ` {0..1}"
by (simp add: add.commute [of u] image_affinity_atLeastAtMost [where c=u] closed_segment_eq_real_ivl)
@@ -11353,6 +11358,81 @@
by (metis connected_segment convex_contains_segment ends_in_segment imageI
is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
+lemma continuous_injective_image_segment_1:
+ fixes f :: "'a::euclidean_space \<Rightarrow> real"
+ assumes contf: "continuous_on (closed_segment a b) f"
+ and injf: "inj_on f (closed_segment a b)"
+ shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
+proof
+ show "closed_segment (f a) (f b) \<subseteq> f ` closed_segment a b"
+ by (metis subset_continuous_image_segment_1 contf)
+ show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
+ proof (cases "a = b")
+ case True
+ then show ?thesis by auto
+ next
+ case False
+ then have fnot: "f a \<noteq> f b"
+ using inj_onD injf by fastforce
+ moreover
+ have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
+ proof (clarsimp simp add: open_segment_def)
+ assume fa: "f a \<in> closed_segment (f c) (f b)"
+ moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
+ by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
+ ultimately have "f a \<in> f ` closed_segment c b"
+ by blast
+ then have a: "a \<in> closed_segment c b"
+ by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
+ have cb: "closed_segment c b \<subseteq> closed_segment a b"
+ by (simp add: closed_segment_subset that)
+ show "f a = f c"
+ proof (rule between_antisym)
+ show "between (f c, f b) (f a)"
+ by (simp add: between_mem_segment fa)
+ show "between (f a, f b) (f c)"
+ by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
+ qed
+ qed
+ moreover
+ have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
+ proof (clarsimp simp add: open_segment_def fnot eq_commute)
+ assume fb: "f b \<in> closed_segment (f a) (f c)"
+ moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
+ by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
+ ultimately have "f b \<in> f ` closed_segment a c"
+ by blast
+ then have b: "b \<in> closed_segment a c"
+ by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
+ have ca: "closed_segment a c \<subseteq> closed_segment a b"
+ by (simp add: closed_segment_subset that)
+ show "f b = f c"
+ proof (rule between_antisym)
+ show "between (f c, f a) (f b)"
+ by (simp add: between_commute between_mem_segment fb)
+ show "between (f b, f a) (f c)"
+ by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
+ qed
+ qed
+ ultimately show ?thesis
+ by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
+ qed
+qed
+
+lemma continuous_injective_image_open_segment_1:
+ fixes f :: "'a::euclidean_space \<Rightarrow> real"
+ assumes contf: "continuous_on (closed_segment a b) f"
+ and injf: "inj_on f (closed_segment a b)"
+ shows "f ` (open_segment a b) = open_segment (f a) (f b)"
+proof -
+ have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
+ by (metis (no_types, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
+ also have "... = open_segment (f a) (f b)"
+ using continuous_injective_image_segment_1 [OF assms]
+ by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
+ finally show ?thesis .
+qed
+
lemma collinear_imp_coplanar:
"collinear s ==> coplanar s"
by (metis collinear_affine_hull coplanar_def insert_absorb2)
--- a/src/HOL/Analysis/Derivative.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Derivative.thy Thu Oct 06 11:27:28 2016 +0200
@@ -2370,6 +2370,16 @@
shows "(\<exists>c. (f has_real_derivative c) F) = (\<exists>D. (f has_derivative D) F)"
by (metis has_field_derivative_def has_real_derivative)
+lemma has_vector_derivative_cong_ev:
+ assumes *: "eventually (\<lambda>x. x \<in> s \<longrightarrow> f x = g x) (nhds x)" "f x = g x"
+ shows "(f has_vector_derivative f') (at x within s) = (g has_vector_derivative f') (at x within s)"
+ unfolding has_vector_derivative_def has_derivative_def
+ using *
+ apply (cases "at x within s \<noteq> bot")
+ apply (intro refl conj_cong filterlim_cong)
+ apply (auto simp: netlimit_within eventually_at_filter elim: eventually_mono)
+ done
+
definition deriv :: "('a \<Rightarrow> 'a::real_normed_field) \<Rightarrow> 'a \<Rightarrow> 'a" where
"deriv f x \<equiv> SOME D. DERIV f x :> D"
--- a/src/HOL/Analysis/Finite_Product_Measure.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Finite_Product_Measure.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1196,4 +1196,14 @@
by (subst emeasure_distr) (auto simp: measurable_pair_iff)
qed simp
+lemma infprod_in_sets[intro]:
+ fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
+ shows "Pi UNIV E \<in> sets (\<Pi>\<^sub>M i\<in>UNIV::nat set. M i)"
+proof -
+ have "Pi UNIV E = (\<Inter>i. prod_emb UNIV M {..i} (\<Pi>\<^sub>E j\<in>{..i}. E j))"
+ using E E[THEN sets.sets_into_space]
+ by (auto simp: prod_emb_def Pi_iff extensional_def)
+ with E show ?thesis by auto
+qed
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/FurtherTopology.thy Thu Oct 06 11:27:28 2016 +0200
@@ -0,0 +1,1891 @@
+section \<open>Extending Continous Maps, etc..\<close>
+
+text\<open>Ported from HOL Light (moretop.ml) by L C Paulson\<close>
+
+theory "FurtherTopology"
+ imports Equivalence_Lebesgue_Henstock_Integration Weierstrass_Theorems Polytope
+
+begin
+
+subsection\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
+
+lemma spheremap_lemma1:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes "subspace S" "subspace T" and dimST: "dim S < dim T"
+ and "S \<subseteq> T"
+ and diff_f: "f differentiable_on sphere 0 1 \<inter> S"
+ shows "f ` (sphere 0 1 \<inter> S) \<noteq> sphere 0 1 \<inter> T"
+proof
+ assume fim: "f ` (sphere 0 1 \<inter> S) = sphere 0 1 \<inter> T"
+ have inS: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> 0\<rbrakk> \<Longrightarrow> (x /\<^sub>R norm x) \<in> S"
+ using subspace_mul \<open>subspace S\<close> by blast
+ have subS01: "(\<lambda>x. x /\<^sub>R norm x) ` (S - {0}) \<subseteq> sphere 0 1 \<inter> S"
+ using \<open>subspace S\<close> subspace_mul by fastforce
+ then have diff_f': "f differentiable_on (\<lambda>x. x /\<^sub>R norm x) ` (S - {0})"
+ by (rule differentiable_on_subset [OF diff_f])
+ define g where "g \<equiv> \<lambda>x. norm x *\<^sub>R f(inverse(norm x) *\<^sub>R x)"
+ have gdiff: "g differentiable_on S - {0}"
+ unfolding g_def
+ by (rule diff_f' derivative_intros differentiable_on_compose [where f=f] | force)+
+ have geq: "g ` (S - {0}) = T - {0}"
+ proof
+ have "g ` (S - {0}) \<subseteq> T"
+ apply (auto simp: g_def subspace_mul [OF \<open>subspace T\<close>])
+ apply (metis (mono_tags, lifting) DiffI subS01 subspace_mul [OF \<open>subspace T\<close>] fim image_subset_iff inf_le2 singletonD)
+ done
+ moreover have "g ` (S - {0}) \<subseteq> UNIV - {0}"
+ proof (clarsimp simp: g_def)
+ fix y
+ assume "y \<in> S" and f0: "f (y /\<^sub>R norm y) = 0"
+ then have "y \<noteq> 0 \<Longrightarrow> y /\<^sub>R norm y \<in> sphere 0 1 \<inter> S"
+ by (auto simp: subspace_mul [OF \<open>subspace S\<close>])
+ then show "y = 0"
+ by (metis fim f0 Int_iff image_iff mem_sphere_0 norm_eq_zero zero_neq_one)
+ qed
+ ultimately show "g ` (S - {0}) \<subseteq> T - {0}"
+ by auto
+ next
+ have *: "sphere 0 1 \<inter> T \<subseteq> f ` (sphere 0 1 \<inter> S)"
+ using fim by (simp add: image_subset_iff)
+ have "x \<in> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
+ if "x \<in> T" "x \<noteq> 0" for x
+ proof -
+ have "x /\<^sub>R norm x \<in> T"
+ using \<open>subspace T\<close> subspace_mul that by blast
+ then show ?thesis
+ using * [THEN subsetD, of "x /\<^sub>R norm x"] that apply clarsimp
+ apply (rule_tac x="norm x *\<^sub>R xa" in image_eqI, simp)
+ apply (metis norm_eq_zero right_inverse scaleR_one scaleR_scaleR)
+ using \<open>subspace S\<close> subspace_mul apply force
+ done
+ qed
+ then have "T - {0} \<subseteq> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
+ by force
+ then show "T - {0} \<subseteq> g ` (S - {0})"
+ by (simp add: g_def)
+ qed
+ define T' where "T' \<equiv> {y. \<forall>x \<in> T. orthogonal x y}"
+ have "subspace T'"
+ by (simp add: subspace_orthogonal_to_vectors T'_def)
+ have dim_eq: "dim T' + dim T = DIM('a)"
+ using dim_subspace_orthogonal_to_vectors [of T UNIV] \<open>subspace T\<close>
+ by (simp add: dim_UNIV T'_def)
+ have "\<exists>v1 v2. v1 \<in> span T \<and> (\<forall>w \<in> span T. orthogonal v2 w) \<and> x = v1 + v2" for x
+ by (force intro: orthogonal_subspace_decomp_exists [of T x])
+ then obtain p1 p2 where p1span: "p1 x \<in> span T"
+ and "\<And>w. w \<in> span T \<Longrightarrow> orthogonal (p2 x) w"
+ and eq: "p1 x + p2 x = x" for x
+ by metis
+ then have p1: "\<And>z. p1 z \<in> T" and ortho: "\<And>w. w \<in> T \<Longrightarrow> orthogonal (p2 x) w" for x
+ using span_eq \<open>subspace T\<close> by blast+
+ then have p2: "\<And>z. p2 z \<in> T'"
+ by (simp add: T'_def orthogonal_commute)
+ have p12_eq: "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p1(x + y) = x \<and> p2(x + y) = y"
+ proof (rule orthogonal_subspace_decomp_unique [OF eq p1span, where T=T'])
+ show "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p2 (x + y) \<in> span T'"
+ using span_eq p2 \<open>subspace T'\<close> by blast
+ show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
+ using T'_def by blast
+ qed (auto simp: span_superset)
+ then have "\<And>c x. p1 (c *\<^sub>R x) = c *\<^sub>R p1 x \<and> p2 (c *\<^sub>R x) = c *\<^sub>R p2 x"
+ by (metis eq \<open>subspace T\<close> \<open>subspace T'\<close> p1 p2 scaleR_add_right subspace_mul)
+ moreover have lin_add: "\<And>x y. p1 (x + y) = p1 x + p1 y \<and> p2 (x + y) = p2 x + p2 y"
+ proof (rule orthogonal_subspace_decomp_unique [OF _ p1span, where T=T'])
+ show "\<And>x y. p1 (x + y) + p2 (x + y) = p1 x + p1 y + (p2 x + p2 y)"
+ by (simp add: add.assoc add.left_commute eq)
+ show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
+ using T'_def by blast
+ qed (auto simp: p1span p2 span_superset subspace_add)
+ ultimately have "linear p1" "linear p2"
+ by unfold_locales auto
+ have "(\<lambda>z. g (p1 z)) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ apply (rule differentiable_on_compose [where f=g])
+ apply (rule linear_imp_differentiable_on [OF \<open>linear p1\<close>])
+ apply (rule differentiable_on_subset [OF gdiff])
+ using p12_eq \<open>S \<subseteq> T\<close> apply auto
+ done
+ then have diff: "(\<lambda>x. g (p1 x) + p2 x) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ by (intro derivative_intros linear_imp_differentiable_on [OF \<open>linear p2\<close>])
+ have "dim {x + y |x y. x \<in> S - {0} \<and> y \<in> T'} \<le> dim {x + y |x y. x \<in> S \<and> y \<in> T'}"
+ by (blast intro: dim_subset)
+ also have "... = dim S + dim T' - dim (S \<inter> T')"
+ using dim_sums_Int [OF \<open>subspace S\<close> \<open>subspace T'\<close>]
+ by (simp add: algebra_simps)
+ also have "... < DIM('a)"
+ using dimST dim_eq by auto
+ finally have neg: "negligible {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ by (rule negligible_lowdim)
+ have "negligible ((\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'})"
+ by (rule negligible_differentiable_image_negligible [OF order_refl neg diff])
+ then have "negligible {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
+ proof (rule negligible_subset)
+ have "\<lbrakk>t' \<in> T'; s \<in> S; s \<noteq> 0\<rbrakk>
+ \<Longrightarrow> g s + t' \<in> (\<lambda>x. g (p1 x) + p2 x) `
+ {x + t' |x t'. x \<in> S \<and> x \<noteq> 0 \<and> t' \<in> T'}" for t' s
+ apply (rule_tac x="s + t'" in image_eqI)
+ using \<open>S \<subseteq> T\<close> p12_eq by auto
+ then show "{x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}
+ \<subseteq> (\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
+ by auto
+ qed
+ moreover have "- T' \<subseteq> {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
+ proof clarsimp
+ fix z assume "z \<notin> T'"
+ show "\<exists>x y. z = x + y \<and> x \<in> g ` (S - {0}) \<and> y \<in> T'"
+ apply (rule_tac x="p1 z" in exI)
+ apply (rule_tac x="p2 z" in exI)
+ apply (simp add: p1 eq p2 geq)
+ by (metis \<open>z \<notin> T'\<close> add.left_neutral eq p2)
+ qed
+ ultimately have "negligible (-T')"
+ using negligible_subset by blast
+ moreover have "negligible T'"
+ using negligible_lowdim
+ by (metis add.commute assms(3) diff_add_inverse2 diff_self_eq_0 dim_eq le_add1 le_antisym linordered_semidom_class.add_diff_inverse not_less0)
+ ultimately have "negligible (-T' \<union> T')"
+ by (metis negligible_Un_eq)
+ then show False
+ using negligible_Un_eq non_negligible_UNIV by simp
+qed
+
+
+lemma spheremap_lemma2:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes ST: "subspace S" "subspace T" "dim S < dim T"
+ and "S \<subseteq> T"
+ and contf: "continuous_on (sphere 0 1 \<inter> S) f"
+ and fim: "f ` (sphere 0 1 \<inter> S) \<subseteq> sphere 0 1 \<inter> T"
+ shows "\<exists>c. homotopic_with (\<lambda>x. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) f (\<lambda>x. c)"
+proof -
+ have [simp]: "\<And>x. \<lbrakk>norm x = 1; x \<in> S\<rbrakk> \<Longrightarrow> norm (f x) = 1"
+ using fim by (simp add: image_subset_iff)
+ have "compact (sphere 0 1 \<inter> S)"
+ by (simp add: \<open>subspace S\<close> closed_subspace compact_Int_closed)
+ then obtain g where pfg: "polynomial_function g" and gim: "g ` (sphere 0 1 \<inter> S) \<subseteq> T"
+ and g12: "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> norm(f x - g x) < 1/2"
+ apply (rule Stone_Weierstrass_polynomial_function_subspace [OF _ contf _ \<open>subspace T\<close>, of "1/2"])
+ using fim apply auto
+ done
+ have gnz: "g x \<noteq> 0" if "x \<in> sphere 0 1 \<inter> S" for x
+ proof -
+ have "norm (f x) = 1"
+ using fim that by (simp add: image_subset_iff)
+ then show ?thesis
+ using g12 [OF that] by auto
+ qed
+ have diffg: "g differentiable_on sphere 0 1 \<inter> S"
+ by (metis pfg differentiable_on_polynomial_function)
+ define h where "h \<equiv> \<lambda>x. inverse(norm(g x)) *\<^sub>R g x"
+ have h: "x \<in> sphere 0 1 \<inter> S \<Longrightarrow> h x \<in> sphere 0 1 \<inter> T" for x
+ unfolding h_def
+ using gnz [of x]
+ by (auto simp: subspace_mul [OF \<open>subspace T\<close>] subsetD [OF gim])
+ have diffh: "h differentiable_on sphere 0 1 \<inter> S"
+ unfolding h_def
+ apply (intro derivative_intros diffg differentiable_on_compose [OF diffg])
+ using gnz apply auto
+ done
+ have homfg: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) f g"
+ proof (rule homotopic_with_linear [OF contf])
+ show "continuous_on (sphere 0 1 \<inter> S) g"
+ using pfg by (simp add: differentiable_imp_continuous_on diffg)
+ next
+ have non0fg: "0 \<notin> closed_segment (f x) (g x)" if "norm x = 1" "x \<in> S" for x
+ proof -
+ have "f x \<in> sphere 0 1"
+ using fim that by (simp add: image_subset_iff)
+ moreover have "norm(f x - g x) < 1/2"
+ apply (rule g12)
+ using that by force
+ ultimately show ?thesis
+ by (auto simp: norm_minus_commute dest: segment_bound)
+ qed
+ show "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> T - {0}"
+ apply (simp add: subset_Diff_insert non0fg)
+ apply (simp add: segment_convex_hull)
+ apply (rule hull_minimal)
+ using fim image_eqI gim apply force
+ apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
+ done
+ qed
+ obtain d where d: "d \<in> (sphere 0 1 \<inter> T) - h ` (sphere 0 1 \<inter> S)"
+ using h spheremap_lemma1 [OF ST \<open>S \<subseteq> T\<close> diffh] by force
+ then have non0hd: "0 \<notin> closed_segment (h x) (- d)" if "norm x = 1" "x \<in> S" for x
+ using midpoint_between [of 0 "h x" "-d"] that h [of x]
+ by (auto simp: between_mem_segment midpoint_def)
+ have conth: "continuous_on (sphere 0 1 \<inter> S) h"
+ using differentiable_imp_continuous_on diffh by blast
+ have hom_hd: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) h (\<lambda>x. -d)"
+ apply (rule homotopic_with_linear [OF conth continuous_on_const])
+ apply (simp add: subset_Diff_insert non0hd)
+ apply (simp add: segment_convex_hull)
+ apply (rule hull_minimal)
+ using h d apply (force simp: subspace_neg [OF \<open>subspace T\<close>])
+ apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
+ done
+ have conT0: "continuous_on (T - {0}) (\<lambda>y. inverse(norm y) *\<^sub>R y)"
+ by (intro continuous_intros) auto
+ have sub0T: "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0}) \<subseteq> sphere 0 1 \<inter> T"
+ by (fastforce simp: assms(2) subspace_mul)
+ obtain c where homhc: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) h (\<lambda>x. c)"
+ apply (rule_tac c="-d" in that)
+ apply (rule homotopic_with_eq)
+ apply (rule homotopic_compose_continuous_left [OF hom_hd conT0 sub0T])
+ using d apply (auto simp: h_def)
+ done
+ show ?thesis
+ apply (rule_tac x=c in exI)
+ apply (rule homotopic_with_trans [OF _ homhc])
+ apply (rule homotopic_with_eq)
+ apply (rule homotopic_compose_continuous_left [OF homfg conT0 sub0T])
+ apply (auto simp: h_def)
+ done
+qed
+
+
+lemma spheremap_lemma3:
+ assumes "bounded S" "convex S" "subspace U" and affSU: "aff_dim S \<le> dim U"
+ obtains T where "subspace T" "T \<subseteq> U" "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"
+ "(rel_frontier S) homeomorphic (sphere 0 1 \<inter> T)"
+proof (cases "S = {}")
+ case True
+ with \<open>subspace U\<close> subspace_0 show ?thesis
+ by (rule_tac T = "{0}" in that) auto
+next
+ case False
+ then obtain a where "a \<in> S"
+ by auto
+ then have affS: "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
+ by (metis hull_inc aff_dim_eq_dim)
+ with affSU have "dim ((\<lambda>x. -a+x) ` S) \<le> dim U"
+ by linarith
+ with choose_subspace_of_subspace
+ obtain T where "subspace T" "T \<subseteq> span U" and dimT: "dim T = dim ((\<lambda>x. -a+x) ` S)" .
+ show ?thesis
+ proof (rule that [OF \<open>subspace T\<close>])
+ show "T \<subseteq> U"
+ using span_eq \<open>subspace U\<close> \<open>T \<subseteq> span U\<close> by blast
+ show "aff_dim T = aff_dim S"
+ using dimT \<open>subspace T\<close> affS aff_dim_subspace by fastforce
+ show "rel_frontier S homeomorphic sphere 0 1 \<inter> T"
+ proof -
+ have "aff_dim (ball 0 1 \<inter> T) = aff_dim (T)"
+ by (metis IntI interior_ball \<open>subspace T\<close> aff_dim_convex_Int_nonempty_interior centre_in_ball empty_iff inf_commute subspace_0 subspace_imp_convex zero_less_one)
+ then have affS_eq: "aff_dim S = aff_dim (ball 0 1 \<inter> T)"
+ using \<open>aff_dim T = aff_dim S\<close> by simp
+ have "rel_frontier S homeomorphic rel_frontier(ball 0 1 \<inter> T)"
+ apply (rule homeomorphic_rel_frontiers_convex_bounded_sets [OF \<open>convex S\<close> \<open>bounded S\<close>])
+ apply (simp add: \<open>subspace T\<close> convex_Int subspace_imp_convex)
+ apply (simp add: bounded_Int)
+ apply (rule affS_eq)
+ done
+ also have "... = frontier (ball 0 1) \<inter> T"
+ apply (rule convex_affine_rel_frontier_Int [OF convex_ball])
+ apply (simp add: \<open>subspace T\<close> subspace_imp_affine)
+ using \<open>subspace T\<close> subspace_0 by force
+ also have "... = sphere 0 1 \<inter> T"
+ by auto
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+
+proposition inessential_spheremap_lowdim_gen:
+ fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes "convex S" "bounded S" "convex T" "bounded T"
+ and affST: "aff_dim S < aff_dim T"
+ and contf: "continuous_on (rel_frontier S) f"
+ and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
+ obtains c where "homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (simp add: that)
+next
+ case False
+ then show ?thesis
+ proof (cases "T = {}")
+ case True
+ then show ?thesis
+ using fim that by auto
+ next
+ case False
+ obtain T':: "'a set"
+ where "subspace T'" and affT': "aff_dim T' = aff_dim T"
+ and homT: "rel_frontier T homeomorphic sphere 0 1 \<inter> T'"
+ apply (rule spheremap_lemma3 [OF \<open>bounded T\<close> \<open>convex T\<close> subspace_UNIV, where 'b='a])
+ apply (simp add: dim_UNIV aff_dim_le_DIM)
+ using \<open>T \<noteq> {}\<close> by blast
+ with homeomorphic_imp_homotopy_eqv
+ have relT: "sphere 0 1 \<inter> T' homotopy_eqv rel_frontier T"
+ using homotopy_eqv_sym by blast
+ have "aff_dim S \<le> int (dim T')"
+ using affT' \<open>subspace T'\<close> affST aff_dim_subspace by force
+ with spheremap_lemma3 [OF \<open>bounded S\<close> \<open>convex S\<close> \<open>subspace T'\<close>] \<open>S \<noteq> {}\<close>
+ obtain S':: "'a set" where "subspace S'" "S' \<subseteq> T'"
+ and affS': "aff_dim S' = aff_dim S"
+ and homT: "rel_frontier S homeomorphic sphere 0 1 \<inter> S'"
+ by metis
+ with homeomorphic_imp_homotopy_eqv
+ have relS: "sphere 0 1 \<inter> S' homotopy_eqv rel_frontier S"
+ using homotopy_eqv_sym by blast
+ have dimST': "dim S' < dim T'"
+ by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> affS' affST affT' less_irrefl not_le subspace_dim_equal)
+ have "\<exists>c. homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
+ apply (rule homotopy_eqv_homotopic_triviality_null_imp [OF relT contf fim])
+ apply (rule homotopy_eqv_cohomotopic_triviality_null[OF relS, THEN iffD1, rule_format])
+ apply (metis dimST' \<open>subspace S'\<close> \<open>subspace T'\<close> \<open>S' \<subseteq> T'\<close> spheremap_lemma2, blast)
+ done
+ with that show ?thesis by blast
+ qed
+qed
+
+lemma inessential_spheremap_lowdim:
+ fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes
+ "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
+ obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
+proof (cases "s \<le> 0")
+ case True then show ?thesis
+ by (meson nullhomotopic_into_contractible f contractible_sphere that)
+next
+ case False
+ show ?thesis
+ proof (cases "r \<le> 0")
+ case True then show ?thesis
+ by (meson f nullhomotopic_from_contractible contractible_sphere that)
+ next
+ case False
+ with \<open>~ s \<le> 0\<close> have "r > 0" "s > 0" by auto
+ show ?thesis
+ apply (rule inessential_spheremap_lowdim_gen [of "cball a r" "cball b s" f])
+ using \<open>0 < r\<close> \<open>0 < s\<close> assms(1)
+ apply (simp_all add: f aff_dim_cball)
+ using that by blast
+ qed
+qed
+
+
+
+subsection\<open> Some technical lemmas about extending maps from cell complexes.\<close>
+
+lemma extending_maps_Union_aux:
+ assumes fin: "finite \<F>"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
+using assms
+proof (induction \<F>)
+ case empty show ?case by simp
+next
+ case (insert S \<F>)
+ then obtain f where contf: "continuous_on (S) f" and fim: "f ` S \<subseteq> T" and feq: "\<forall>x \<in> S \<inter> K. f x = h x"
+ by (meson insertI1)
+ obtain g where contg: "continuous_on (\<Union>\<F>) g" and gim: "g ` \<Union>\<F> \<subseteq> T" and geq: "\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x"
+ using insert by auto
+ have fg: "f x = g x" if "x \<in> T" "T \<in> \<F>" "x \<in> S" for x T
+ proof -
+ have "T \<inter> S \<subseteq> K \<or> S = T"
+ using that by (metis (no_types) insert.prems(2) insertCI)
+ then show ?thesis
+ using UnionI feq geq \<open>S \<notin> \<F>\<close> subsetD that by fastforce
+ qed
+ show ?case
+ apply (rule_tac x="\<lambda>x. if x \<in> S then f x else g x" in exI, simp)
+ apply (intro conjI continuous_on_cases)
+ apply (simp_all add: insert closed_Union contf contg)
+ using fim gim feq geq
+ apply (force simp: insert closed_Union contf contg inf_commute intro: fg)+
+ done
+qed
+
+lemma extending_maps_Union:
+ assumes fin: "finite \<F>"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~ X \<subseteq> Y; ~ Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
+ shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
+apply (simp add: Union_maximal_sets [OF fin, symmetric])
+apply (rule extending_maps_Union_aux)
+apply (simp_all add: Union_maximal_sets [OF fin] assms)
+by (metis K psubsetI)
+
+
+lemma extend_map_lemma:
+ assumes "finite \<F>" "\<G> \<subseteq> \<F>" "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"
+ and face: "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> (S \<inter> T) face_of S \<and> (S \<inter> T) face_of T"
+ and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
+ obtains g where "continuous_on (\<Union>\<F>) g" "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
+proof (cases "\<F> - \<G> = {}")
+ case True
+ then have "\<Union>\<F> \<subseteq> \<Union>\<G>"
+ by (simp add: Union_mono)
+ then show ?thesis
+ apply (rule_tac g=f in that)
+ using contf continuous_on_subset apply blast
+ using fim apply blast
+ by simp
+next
+ case False
+ then have "0 \<le> aff_dim T"
+ by (metis aff aff_dim_empty aff_dim_geq aff_dim_negative_iff all_not_in_conv not_less)
+ then obtain i::nat where i: "int i = aff_dim T"
+ by (metis nonneg_eq_int)
+ have Union_empty_eq: "\<Union>{D. D = {} \<and> P D} = {}" for P :: "'a set \<Rightarrow> bool"
+ by auto
+ have extendf: "\<exists>g. continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) g \<and>
+ g ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) \<subseteq> rel_frontier T \<and>
+ (\<forall>x \<in> \<Union>\<G>. g x = f x)"
+ if "i \<le> aff_dim T" for i::nat
+ using that
+ proof (induction i)
+ case 0 then show ?case
+ apply (simp add: Union_empty_eq)
+ apply (rule_tac x=f in exI)
+ apply (intro conjI)
+ using contf continuous_on_subset apply blast
+ using fim apply blast
+ by simp
+ next
+ case (Suc p)
+ with \<open>bounded T\<close> have "rel_frontier T \<noteq> {}"
+ by (auto simp: rel_frontier_eq_empty affine_bounded_eq_lowdim [of T])
+ then obtain t where t: "t \<in> rel_frontier T" by auto
+ have ple: "int p \<le> aff_dim T" using Suc.prems by force
+ obtain h where conth: "continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})) h"
+ and him: "h ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}))
+ \<subseteq> rel_frontier T"
+ and heq: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
+ using Suc.IH [OF ple] by auto
+ let ?Faces = "{D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D \<le> p}"
+ have extendh: "\<exists>g. continuous_on D g \<and>
+ g ` D \<subseteq> rel_frontier T \<and>
+ (\<forall>x \<in> D \<inter> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
+ if D: "D \<in> \<G> \<union> ?Faces" for D
+ proof (cases "D \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})")
+ case True
+ then show ?thesis
+ apply (rule_tac x=h in exI)
+ apply (intro conjI)
+ apply (blast intro: continuous_on_subset [OF conth])
+ using him apply blast
+ by simp
+ next
+ case False
+ note notDsub = False
+ show ?thesis
+ proof (cases "\<exists>a. D = {a}")
+ case True
+ then obtain a where "D = {a}" by auto
+ with notDsub t show ?thesis
+ by (rule_tac x="\<lambda>x. t" in exI) simp
+ next
+ case False
+ have "D \<noteq> {}" using notDsub by auto
+ have Dnotin: "D \<notin> \<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
+ using notDsub by auto
+ then have "D \<notin> \<G>" by simp
+ have "D \<in> ?Faces - {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
+ using Dnotin that by auto
+ then obtain C where "C \<in> \<F>" "D face_of C" and affD: "aff_dim D = int p"
+ by auto
+ then have "bounded D"
+ using face_of_polytope_polytope poly polytope_imp_bounded by blast
+ then have [simp]: "\<not> affine D"
+ using affine_bounded_eq_trivial False \<open>D \<noteq> {}\<close> \<open>bounded D\<close> by blast
+ have "{F. F facet_of D} \<subseteq> {E. E face_of C \<and> aff_dim E < int p}"
+ apply clarify
+ apply (metis \<open>D face_of C\<close> affD eq_iff face_of_trans facet_of_def zle_diff1_eq)
+ done
+ moreover have "polyhedron D"
+ using \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face_of_polytope_polytope poly polytope_imp_polyhedron by auto
+ ultimately have relf_sub: "rel_frontier D \<subseteq> \<Union> {E. E face_of C \<and> aff_dim E < p}"
+ by (simp add: rel_frontier_of_polyhedron Union_mono)
+ then have him_relf: "h ` rel_frontier D \<subseteq> rel_frontier T"
+ using \<open>C \<in> \<F>\<close> him by blast
+ have "convex D"
+ by (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex)
+ have affD_lessT: "aff_dim D < aff_dim T"
+ using Suc.prems affD by linarith
+ have contDh: "continuous_on (rel_frontier D) h"
+ using \<open>C \<in> \<F>\<close> relf_sub by (blast intro: continuous_on_subset [OF conth])
+ then have *: "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)) =
+ (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> rel_frontier T \<and>
+ (\<forall>x\<in>rel_frontier D. g x = h x))"
+ apply (rule nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
+ apply (simp_all add: assms rel_frontier_eq_empty him_relf)
+ done
+ have "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D)
+ (rel_frontier T) h (\<lambda>x. c))"
+ by (metis inessential_spheremap_lowdim_gen
+ [OF \<open>convex D\<close> \<open>bounded D\<close> \<open>convex T\<close> \<open>bounded T\<close> affD_lessT contDh him_relf])
+ then obtain g where contg: "continuous_on UNIV g"
+ and gim: "range g \<subseteq> rel_frontier T"
+ and gh: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> g x = h x"
+ by (metis *)
+ have "D \<inter> E \<subseteq> rel_frontier D"
+ if "E \<in> \<G> \<union> {D. Bex \<F> (op face_of D) \<and> aff_dim D < int p}" for E
+ proof (rule face_of_subset_rel_frontier)
+ show "D \<inter> E face_of D"
+ using that \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face
+ apply auto
+ apply (meson face_of_Int_subface \<open>\<G> \<subseteq> \<F>\<close> face_of_refl_eq poly polytope_imp_convex subsetD)
+ using face_of_Int_subface apply blast
+ done
+ show "D \<inter> E \<noteq> D"
+ using that notDsub by auto
+ qed
+ then show ?thesis
+ apply (rule_tac x=g in exI)
+ apply (intro conjI ballI)
+ using continuous_on_subset contg apply blast
+ using gim apply blast
+ using gh by fastforce
+ qed
+ qed
+ have intle: "i < 1 + int j \<longleftrightarrow> i \<le> int j" for i j
+ by auto
+ have "finite \<G>"
+ using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> rev_finite_subset by blast
+ then have fin: "finite (\<G> \<union> ?Faces)"
+ apply simp
+ apply (rule_tac B = "\<Union>{{D. D face_of C}| C. C \<in> \<F>}" in finite_subset)
+ by (auto simp: \<open>finite \<F>\<close> finite_polytope_faces poly)
+ have clo: "closed S" if "S \<in> \<G> \<union> ?Faces" for S
+ using that \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly polytope_imp_closed by blast
+ have K: "X \<inter> Y \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int p})"
+ if "X \<in> \<G> \<union> ?Faces" "Y \<in> \<G> \<union> ?Faces" "~ Y \<subseteq> X" for X Y
+ proof -
+ have ff: "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ if XY: "X face_of D" "Y face_of E" and DE: "D \<in> \<F>" "E \<in> \<F>" for D E
+ apply (rule face_of_Int_subface [OF _ _ XY])
+ apply (auto simp: face DE)
+ done
+ show ?thesis
+ using that
+ apply auto
+ apply (drule_tac x="X \<inter> Y" in spec, safe)
+ using ff face_of_imp_convex [of X] face_of_imp_convex [of Y]
+ apply (fastforce dest: face_of_aff_dim_lt)
+ by (meson face_of_trans ff)
+ qed
+ obtain g where "continuous_on (\<Union>(\<G> \<union> ?Faces)) g"
+ "g ` \<Union>(\<G> \<union> ?Faces) \<subseteq> rel_frontier T"
+ "(\<forall>x \<in> \<Union>(\<G> \<union> ?Faces) \<inter>
+ \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
+ apply (rule exE [OF extending_maps_Union [OF fin extendh clo K]], blast+)
+ done
+ then show ?case
+ apply (simp add: intle local.heq [symmetric], blast)
+ done
+ qed
+ have eq: "\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i}) = \<Union>\<F>"
+ proof
+ show "\<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) \<subseteq> \<Union>\<F>"
+ apply (rule Union_subsetI)
+ using \<open>\<G> \<subseteq> \<F>\<close> face_of_imp_subset apply force
+ done
+ show "\<Union>\<F> \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < i})"
+ apply (rule Union_mono)
+ using face apply (fastforce simp: aff i)
+ done
+ qed
+ have "int i \<le> aff_dim T" by (simp add: i)
+ then show ?thesis
+ using extendf [of i] unfolding eq by (metis that)
+qed
+
+lemma extend_map_lemma_cofinite0:
+ assumes "finite \<F>"
+ and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and>
+ continuous_on (\<Union>\<F> - C) g \<and> g ` (\<Union>\<F> - C) \<subseteq> T
+ \<and> (\<forall>x \<in> (\<Union>\<F> - C) \<inter> K. g x = h x)"
+ using assms
+proof induction
+ case empty then show ?case
+ by force
+next
+ case (insert X \<F>)
+ then have "closed X" and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
+ and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
+ and pwX: "\<And>Y. Y \<in> \<F> \<and> Y \<noteq> X \<longrightarrow> X \<inter> Y \<subseteq> K \<and> Y \<inter> X \<subseteq> K"
+ and pwF: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) \<F>"
+ by (simp_all add: pairwise_insert)
+ obtain C g where C: "finite C" "disjnt C U" "card C \<le> card \<F>"
+ and contg: "continuous_on (\<Union>\<F> - C) g"
+ and gim: "g ` (\<Union>\<F> - C) \<subseteq> T"
+ and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
+ using insert.IH [OF pwF \<F> clo] by auto
+ obtain a f where "a \<notin> U"
+ and contf: "continuous_on (X - {a}) f"
+ and fim: "f ` (X - {a}) \<subseteq> T"
+ and fh: "(\<forall>x \<in> X \<inter> K. f x = h x)"
+ using insert.prems by (meson insertI1)
+ show ?case
+ proof (intro exI conjI)
+ show "finite (insert a C)"
+ by (simp add: C)
+ show "disjnt (insert a C) U"
+ using C \<open>a \<notin> U\<close> by simp
+ show "card (insert a C) \<le> card (insert X \<F>)"
+ by (simp add: C card_insert_if insert.hyps le_SucI)
+ have "closed (\<Union>\<F>)"
+ using clo insert.hyps by blast
+ have "continuous_on (X - insert a C \<union> (\<Union>\<F> - insert a C)) (\<lambda>x. if x \<in> X then f x else g x)"
+ apply (rule continuous_on_cases_local)
+ apply (simp_all add: closedin_closed)
+ using \<open>closed X\<close> apply blast
+ using \<open>closed (\<Union>\<F>)\<close> apply blast
+ using contf apply (force simp: elim: continuous_on_subset)
+ using contg apply (force simp: elim: continuous_on_subset)
+ using fh gh insert.hyps pwX by fastforce
+ then show "continuous_on (\<Union>insert X \<F> - insert a C) (\<lambda>a. if a \<in> X then f a else g a)"
+ by (blast intro: continuous_on_subset)
+ show "\<forall>x\<in>(\<Union>insert X \<F> - insert a C) \<inter> K. (if x \<in> X then f x else g x) = h x"
+ using gh by (auto simp: fh)
+ show "(\<lambda>a. if a \<in> X then f a else g a) ` (\<Union>insert X \<F> - insert a C) \<subseteq> T"
+ using fim gim by auto force
+ qed
+qed
+
+
+lemma extend_map_lemma_cofinite1:
+assumes "finite \<F>"
+ and \<F>: "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x \<in> X \<inter> K. g x = h x)"
+ and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
+ and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~(X \<subseteq> Y); ~(Y \<subseteq> X)\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
+ obtains C g where "finite C" "disjnt C U" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union>\<F> - C) \<subseteq> T"
+ "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
+proof -
+ let ?\<F> = "{X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
+ have [simp]: "\<Union>?\<F> = \<Union>\<F>"
+ by (simp add: Union_maximal_sets assms)
+ have fin: "finite ?\<F>"
+ by (force intro: finite_subset [OF _ \<open>finite \<F>\<close>])
+ have pw: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) ?\<F>"
+ by (simp add: pairwise_def) (metis K psubsetI)
+ have "card {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y} \<le> card \<F>"
+ by (simp add: \<open>finite \<F>\<close> card_mono)
+ moreover
+ obtain C g where "finite C \<and> disjnt C U \<and> card C \<le> card ?\<F> \<and>
+ continuous_on (\<Union>?\<F> - C) g \<and> g ` (\<Union>?\<F> - C) \<subseteq> T
+ \<and> (\<forall>x \<in> (\<Union>?\<F> - C) \<inter> K. g x = h x)"
+ apply (rule exE [OF extend_map_lemma_cofinite0 [OF fin pw, of U T h]])
+ apply (fastforce intro!: clo \<F>)+
+ done
+ ultimately show ?thesis
+ by (rule_tac C=C and g=g in that) auto
+qed
+
+
+lemma extend_map_lemma_cofinite:
+ assumes "finite \<F>" "\<G> \<subseteq> \<F>" and T: "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
+ and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+ and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+ obtains C g where
+ "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
+proof -
+ define \<H> where "\<H> \<equiv> \<G> \<union> {D. \<exists>C \<in> \<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
+ have "finite \<G>"
+ using assms finite_subset by blast
+ moreover have "finite (\<Union>{{D. D face_of C} |C. C \<in> \<F>})"
+ apply (rule finite_Union)
+ apply (simp add: \<open>finite \<F>\<close>)
+ using finite_polytope_faces poly by auto
+ ultimately have "finite \<H>"
+ apply (simp add: \<H>_def)
+ apply (rule finite_subset [of _ "\<Union> {{D. D face_of C} | C. C \<in> \<F>}"], auto)
+ done
+ have *: "\<And>X Y. \<lbrakk>X \<in> \<H>; Y \<in> \<H>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ unfolding \<H>_def
+ apply (elim UnE bexE CollectE DiffE)
+ using subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] apply (simp_all add: face)
+ apply (meson subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] face face_of_Int_subface face_of_imp_subset face_of_refl poly polytope_imp_convex)+
+ done
+ obtain h where conth: "continuous_on (\<Union>\<H>) h" and him: "h ` (\<Union>\<H>) \<subseteq> rel_frontier T"
+ and hf: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
+ using \<open>finite \<H>\<close>
+ unfolding \<H>_def
+ apply (rule extend_map_lemma [OF _ Un_upper1 T _ _ _ contf fim])
+ using \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly apply fastforce
+ using * apply (auto simp: \<H>_def)
+ done
+ have "bounded (\<Union>\<G>)"
+ using \<open>finite \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> poly polytope_imp_bounded by blast
+ then have "\<Union>\<G> \<noteq> UNIV"
+ by auto
+ then obtain a where a: "a \<notin> \<Union>\<G>"
+ by blast
+ have \<F>: "\<exists>a g. a \<notin> \<Union>\<G> \<and> continuous_on (D - {a}) g \<and>
+ g ` (D - {a}) \<subseteq> rel_frontier T \<and> (\<forall>x \<in> D \<inter> \<Union>\<H>. g x = h x)"
+ if "D \<in> \<F>" for D
+ proof (cases "D \<subseteq> \<Union>\<H>")
+ case True
+ then show ?thesis
+ apply (rule_tac x=a in exI)
+ apply (rule_tac x=h in exI)
+ using him apply (blast intro!: \<open>a \<notin> \<Union>\<G>\<close> continuous_on_subset [OF conth]) +
+ done
+ next
+ case False
+ note D_not_subset = False
+ show ?thesis
+ proof (cases "D \<in> \<G>")
+ case True
+ with D_not_subset show ?thesis
+ by (auto simp: \<H>_def)
+ next
+ case False
+ then have affD: "aff_dim D \<le> aff_dim T"
+ by (simp add: \<open>D \<in> \<F>\<close> aff)
+ show ?thesis
+ proof (cases "rel_interior D = {}")
+ case True
+ with \<open>D \<in> \<F>\<close> poly a show ?thesis
+ by (force simp: rel_interior_eq_empty polytope_imp_convex)
+ next
+ case False
+ then obtain b where brelD: "b \<in> rel_interior D"
+ by blast
+ have "polyhedron D"
+ by (simp add: poly polytope_imp_polyhedron that)
+ have "rel_frontier D retract_of affine hull D - {b}"
+ by (simp add: rel_frontier_retract_of_punctured_affine_hull poly polytope_imp_bounded polytope_imp_convex that brelD)
+ then obtain r where relfD: "rel_frontier D \<subseteq> affine hull D - {b}"
+ and contr: "continuous_on (affine hull D - {b}) r"
+ and rim: "r ` (affine hull D - {b}) \<subseteq> rel_frontier D"
+ and rid: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> r x = x"
+ by (auto simp: retract_of_def retraction_def)
+ show ?thesis
+ proof (intro exI conjI ballI)
+ show "b \<notin> \<Union>\<G>"
+ proof clarify
+ fix E
+ assume "b \<in> E" "E \<in> \<G>"
+ then have "E \<inter> D face_of E \<and> E \<inter> D face_of D"
+ using \<open>\<G> \<subseteq> \<F>\<close> face that by auto
+ with face_of_subset_rel_frontier \<open>E \<in> \<G>\<close> \<open>b \<in> E\<close> brelD rel_interior_subset [of D]
+ D_not_subset rel_frontier_def \<H>_def
+ show False
+ by blast
+ qed
+ have "r ` (D - {b}) \<subseteq> r ` (affine hull D - {b})"
+ by (simp add: Diff_mono hull_subset image_mono)
+ also have "... \<subseteq> rel_frontier D"
+ by (rule rim)
+ also have "... \<subseteq> \<Union>{E. E face_of D \<and> aff_dim E < aff_dim T}"
+ using affD
+ by (force simp: rel_frontier_of_polyhedron [OF \<open>polyhedron D\<close>] facet_of_def)
+ also have "... \<subseteq> \<Union>(\<H>)"
+ using D_not_subset \<H>_def that by fastforce
+ finally have rsub: "r ` (D - {b}) \<subseteq> \<Union>(\<H>)" .
+ show "continuous_on (D - {b}) (h \<circ> r)"
+ apply (intro conjI \<open>b \<notin> \<Union>\<G>\<close> continuous_on_compose)
+ apply (rule continuous_on_subset [OF contr])
+ apply (simp add: Diff_mono hull_subset)
+ apply (rule continuous_on_subset [OF conth rsub])
+ done
+ show "(h \<circ> r) ` (D - {b}) \<subseteq> rel_frontier T"
+ using brelD him rsub by fastforce
+ show "(h \<circ> r) x = h x" if x: "x \<in> D \<inter> \<Union>\<H>" for x
+ proof -
+ consider A where "x \<in> D" "A \<in> \<G>" "x \<in> A"
+ | A B where "x \<in> D" "A face_of B" "B \<in> \<F>" "B \<notin> \<G>" "aff_dim A < aff_dim T" "x \<in> A"
+ using x by (auto simp: \<H>_def)
+ then have xrel: "x \<in> rel_frontier D"
+ proof cases
+ case 1 show ?thesis
+ proof (rule face_of_subset_rel_frontier [THEN subsetD])
+ show "D \<inter> A face_of D"
+ using \<open>A \<in> \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> face \<open>D \<in> \<F>\<close> by blast
+ show "D \<inter> A \<noteq> D"
+ using \<open>A \<in> \<G>\<close> D_not_subset \<H>_def by blast
+ qed (auto simp: 1)
+ next
+ case 2 show ?thesis
+ proof (rule face_of_subset_rel_frontier [THEN subsetD])
+ show "D \<inter> A face_of D"
+ apply (rule face_of_Int_subface [of D B _ A, THEN conjunct1])
+ apply (simp_all add: 2 \<open>D \<in> \<F>\<close> face)
+ apply (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex face_of_refl)
+ done
+ show "D \<inter> A \<noteq> D"
+ using "2" D_not_subset \<H>_def by blast
+ qed (auto simp: 2)
+ qed
+ show ?thesis
+ by (simp add: rid xrel)
+ qed
+ qed
+ qed
+ qed
+ qed
+ have clo: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
+ by (simp add: poly polytope_imp_closed)
+ obtain C g where "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
+ and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> \<Union>\<H> \<Longrightarrow> g x = h x"
+ proof (rule extend_map_lemma_cofinite1 [OF \<open>finite \<F>\<close> \<F> clo])
+ show "X \<inter> Y \<subseteq> \<Union>\<H>" if XY: "X \<in> \<F>" "Y \<in> \<F>" and "\<not> X \<subseteq> Y" "\<not> Y \<subseteq> X" for X Y
+ proof (cases "X \<in> \<G>")
+ case True
+ then show ?thesis
+ by (auto simp: \<H>_def)
+ next
+ case False
+ have "X \<inter> Y \<noteq> X"
+ using \<open>\<not> X \<subseteq> Y\<close> by blast
+ with XY
+ show ?thesis
+ by (clarsimp simp: \<H>_def)
+ (metis Diff_iff Int_iff aff antisym_conv face face_of_aff_dim_lt face_of_refl
+ not_le poly polytope_imp_convex)
+ qed
+ qed (blast)+
+ with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis
+ apply (rule_tac C=C and g=g in that)
+ apply (auto simp: disjnt_def hf [symmetric] \<H>_def intro!: gh)
+ done
+qed
+
+text\<open>The next two proofs are similar\<close>
+theorem extend_map_cell_complex_to_sphere:
+ assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
+ and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+ obtains g where "continuous_on (\<Union>\<F>) g"
+ "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
+ have "compact S"
+ by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
+ then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+ using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
+ obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
+ and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
+ and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+ and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
+ and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ proof (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly _ face])
+ show "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
+ by (simp add: aff)
+ qed auto
+ obtain h where conth: "continuous_on (\<Union>\<G>) h" and him: "h ` \<Union>\<G> \<subseteq> rel_frontier T" and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
+ proof (rule extend_map_lemma [of \<G> "\<G> \<inter> Pow V" T g])
+ show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
+ by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
+ qed (use \<open>finite \<G>\<close> T polyG affG faceG gim in fastforce)+
+ show ?thesis
+ proof
+ show "continuous_on (\<Union>\<F>) h"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
+ show "h ` \<Union>\<F> \<subseteq> rel_frontier T"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
+ show "h x = f x" if "x \<in> S" for x
+ proof -
+ have "x \<in> \<Union>\<G>"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> that by auto
+ then obtain X where "x \<in> X" "X \<in> \<G>" by blast
+ then have "diameter X < d" "bounded X"
+ by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
+ then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
+ by fastforce
+ have "h x = g x"
+ apply (rule hg)
+ using \<open>X \<in> \<G>\<close> \<open>X \<subseteq> V\<close> \<open>x \<in> X\<close> by blast
+ also have "... = f x"
+ by (simp add: gf that)
+ finally show "h x = f x" .
+ qed
+ qed
+qed
+
+
+theorem extend_map_cell_complex_to_sphere_cofinite:
+ assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
+ and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
+ and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+ and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
+ and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
+ obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
+ "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
+ have "compact S"
+ by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
+ then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+ using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
+ obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
+ and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
+ and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
+ and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
+ and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ by (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly aff face]) auto
+ obtain C h where "finite C" and dis: "disjnt C (\<Union>(\<G> \<inter> Pow V))"
+ and card: "card C \<le> card \<G>" and conth: "continuous_on (\<Union>\<G> - C) h"
+ and him: "h ` (\<Union>\<G> - C) \<subseteq> rel_frontier T"
+ and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
+ proof (rule extend_map_lemma_cofinite [of \<G> "\<G> \<inter> Pow V" T g])
+ show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
+ by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
+ show "g ` \<Union>(\<G> \<inter> Pow V) \<subseteq> rel_frontier T"
+ using gim by force
+ qed (auto intro: \<open>finite \<G>\<close> T polyG affG dest: faceG)
+ have Ssub: "S \<subseteq> \<Union>(\<G> \<inter> Pow V)"
+ proof
+ fix x
+ assume "x \<in> S"
+ then have "x \<in> \<Union>\<G>"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> by auto
+ then obtain X where "x \<in> X" "X \<in> \<G>" by blast
+ then have "diameter X < d" "bounded X"
+ by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
+ then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
+ by fastforce
+ then show "x \<in> \<Union>(\<G> \<inter> Pow V)"
+ using \<open>X \<in> \<G>\<close> \<open>x \<in> X\<close> by blast
+ qed
+ show ?thesis
+ proof
+ show "continuous_on (\<Union>\<F>-C) h"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
+ show "h ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
+ using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
+ show "h x = f x" if "x \<in> S" for x
+ proof -
+ have "h x = g x"
+ apply (rule hg)
+ using Ssub that by blast
+ also have "... = f x"
+ by (simp add: gf that)
+ finally show "h x = f x" .
+ qed
+ show "disjnt C S"
+ using dis Ssub by (meson disjnt_iff subset_eq)
+ qed (intro \<open>finite C\<close>)
+qed
+
+
+
+subsection\<open> Special cases and corollaries involving spheres.\<close>
+
+lemma disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
+ by (auto simp: disjnt_def)
+
+proposition extend_map_affine_to_sphere_cofinite_simple:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact S" "convex U" "bounded U"
+ and aff: "aff_dim T \<le> aff_dim U"
+ and "S \<subseteq> T" and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> rel_frontier U"
+ obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+ "g ` (T - K) \<subseteq> rel_frontier U"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
+ g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
+ if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U" for T
+ proof (cases "S = {}")
+ case True
+ show ?thesis
+ proof (cases "rel_frontier U = {}")
+ case True
+ with \<open>bounded U\<close> have "aff_dim U \<le> 0"
+ using affine_bounded_eq_lowdim rel_frontier_eq_empty by auto
+ with aff have "aff_dim T \<le> 0" by auto
+ then obtain a where "T \<subseteq> {a}"
+ using \<open>affine T\<close> affine_bounded_eq_lowdim affine_bounded_eq_trivial by auto
+ then show ?thesis
+ using \<open>S = {}\<close> fim
+ by (metis Diff_cancel contf disjnt_empty2 finite.emptyI finite_insert finite_subset)
+ next
+ case False
+ then obtain a where "a \<in> rel_frontier U"
+ by auto
+ then show ?thesis
+ using continuous_on_const [of _ a] \<open>S = {}\<close> by force
+ qed
+ next
+ case False
+ have "bounded S"
+ by (simp add: \<open>compact S\<close> compact_imp_bounded)
+ then obtain b where b: "S \<subseteq> cbox (-b) b"
+ using bounded_subset_cbox_symmetric by blast
+ define bbox where "bbox \<equiv> cbox (-(b+One)) (b+One)"
+ have "cbox (-b) b \<subseteq> bbox"
+ by (auto simp: bbox_def algebra_simps intro!: subset_box_imp)
+ with b \<open>S \<subseteq> T\<close> have "S \<subseteq> bbox \<inter> T"
+ by auto
+ then have Ssub: "S \<subseteq> \<Union>{bbox \<inter> T}"
+ by auto
+ then have "aff_dim (bbox \<inter> T) \<le> aff_dim U"
+ by (metis aff aff_dim_subset inf_commute inf_le1 order_trans)
+ obtain K g where K: "finite K" "disjnt K S"
+ and contg: "continuous_on (\<Union>{bbox \<inter> T} - K) g"
+ and gim: "g ` (\<Union>{bbox \<inter> T} - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ proof (rule extend_map_cell_complex_to_sphere_cofinite
+ [OF _ Ssub _ \<open>convex U\<close> \<open>bounded U\<close> _ _ _ contf fim])
+ show "closed S"
+ using \<open>compact S\<close> compact_eq_bounded_closed by auto
+ show poly: "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> polytope X"
+ by (simp add: polytope_Int_polyhedron bbox_def polytope_interval affine_imp_polyhedron \<open>affine T\<close>)
+ show "\<And>X Y. \<lbrakk>X \<in> {bbox \<inter> T}; Y \<in> {bbox \<inter> T}\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
+ by (simp add:poly face_of_refl polytope_imp_convex)
+ show "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> aff_dim X \<le> aff_dim U"
+ by (simp add: \<open>aff_dim (bbox \<inter> T) \<le> aff_dim U\<close>)
+ qed auto
+ define fro where "fro \<equiv> \<lambda>d. frontier(cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
+ obtain d where d12: "1/2 \<le> d" "d \<le> 1" and dd: "disjnt K (fro d)"
+ proof (rule disjoint_family_elem_disjnt [OF _ \<open>finite K\<close>])
+ show "infinite {1/2..1::real}"
+ by (simp add: infinite_Icc)
+ have dis1: "disjnt (fro x) (fro y)" if "x<y" for x y
+ by (auto simp: algebra_simps that subset_box_imp disjnt_Diff1 frontier_def fro_def)
+ then show "disjoint_family_on fro {1/2..1}"
+ by (auto simp: disjoint_family_on_def disjnt_def neq_iff)
+ qed auto
+ define c where "c \<equiv> b + d *\<^sub>R One"
+ have cbsub: "cbox (-b) b \<subseteq> box (-c) c" "cbox (-b) b \<subseteq> cbox (-c) c" "cbox (-c) c \<subseteq> bbox"
+ using d12 by (auto simp: algebra_simps subset_box_imp c_def bbox_def)
+ have clo_cbT: "closed (cbox (- c) c \<inter> T)"
+ by (simp add: affine_closed closed_Int closed_cbox \<open>affine T\<close>)
+ have cpT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> b cbsub(2) \<open>S \<subseteq> T\<close> by fastforce
+ have "closest_point (cbox (- c) c \<inter> T) x \<notin> K" if "x \<in> T" "x \<notin> K" for x
+ proof (cases "x \<in> cbox (-c) c")
+ case True with that show ?thesis
+ by (simp add: closest_point_self)
+ next
+ case False
+ have int_ne: "interior (cbox (-c) c) \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b \<open>cbox (- b) b \<subseteq> box (- c) c\<close> by force
+ have "convex T"
+ by (meson \<open>affine T\<close> affine_imp_convex)
+ then have "x \<in> affine hull (cbox (- c) c \<inter> T)"
+ by (metis Int_commute Int_iff \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> cbsub(1) \<open>x \<in> T\<close> affine_hull_convex_Int_nonempty_interior all_not_in_conv b hull_inc inf.orderE interior_cbox)
+ then have "x \<in> affine hull (cbox (- c) c \<inter> T) - rel_interior (cbox (- c) c \<inter> T)"
+ by (meson DiffI False Int_iff rel_interior_subset subsetCE)
+ then have "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
+ by (rule closest_point_in_rel_frontier [OF clo_cbT cpT_ne])
+ moreover have "(rel_frontier (cbox (- c) c \<inter> T)) \<subseteq> fro d"
+ apply (subst convex_affine_rel_frontier_Int [OF _ \<open>affine T\<close> int_ne])
+ apply (auto simp: fro_def c_def)
+ done
+ ultimately show ?thesis
+ using dd by (force simp: disjnt_def)
+ qed
+ then have cpt_subset: "closest_point (cbox (- c) c \<inter> T) ` (T - K) \<subseteq> \<Union>{bbox \<inter> T} - K"
+ using closest_point_in_set [OF clo_cbT cpT_ne] cbsub(3) by force
+ show ?thesis
+ proof (intro conjI ballI exI)
+ have "continuous_on (T - K) (closest_point (cbox (- c) c \<inter> T))"
+ apply (rule continuous_on_closest_point)
+ using \<open>S \<noteq> {}\<close> cbsub(2) b that
+ by (auto simp: affine_imp_convex convex_Int affine_closed closed_Int closed_cbox \<open>affine T\<close>)
+ then show "continuous_on (T - K) (g \<circ> closest_point (cbox (- c) c \<inter> T))"
+ by (metis continuous_on_compose continuous_on_subset [OF contg cpt_subset])
+ have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> g ` (\<Union>{bbox \<inter> T} - K)"
+ by (metis image_comp image_mono cpt_subset)
+ also have "... \<subseteq> rel_frontier U"
+ by (rule gim)
+ finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> rel_frontier U" .
+ show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x" if "x \<in> S" for x
+ proof -
+ have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = g x"
+ unfolding o_def
+ by (metis IntI \<open>S \<subseteq> T\<close> b cbsub(2) closest_point_self subset_eq that)
+ also have "... = f x"
+ by (simp add: that gf)
+ finally show ?thesis .
+ qed
+ qed (auto simp: K)
+ qed
+ then obtain K g where "finite K" "disjnt K S"
+ and contg: "continuous_on (affine hull T - K) g"
+ and gim: "g ` (affine hull T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (metis aff affine_affine_hull aff_dim_affine_hull
+ order_trans [OF \<open>S \<subseteq> T\<close> hull_subset [of T affine]])
+ then obtain K g where "finite K" "disjnt K S"
+ and contg: "continuous_on (T - K) g"
+ and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (rule_tac K=K and g=g in that) (auto simp: hull_inc elim: continuous_on_subset)
+ then show ?thesis
+ by (rule_tac K="K \<inter> T" and g=g in that) (auto simp: disjnt_iff Diff_Int contg)
+qed
+
+subsection\<open>Extending maps to spheres\<close>
+
+(*Up to extend_map_affine_to_sphere_cofinite_gen*)
+
+lemma closedin_closed_subset:
+ "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
+ \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
+
+lemma extend_map_affine_to_sphere1:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
+ assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
+ and fim: "f ` (U - K) \<subseteq> T"
+ and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
+ obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "K = {}")
+ case True
+ then show ?thesis
+ by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI rev_image_eqI subset_iff that)
+next
+ case False
+ have "S \<subseteq> U"
+ using clo closedin_limpt by blast
+ then have "(U - S) \<inter> K \<noteq> {}"
+ by (metis Diff_triv False Int_Diff K disjnt_def inf.absorb_iff2 inf_commute)
+ then have "\<Union>(components (U - S)) \<inter> K \<noteq> {}"
+ using Union_components by simp
+ then obtain C0 where C0: "C0 \<in> components (U - S)" "C0 \<inter> K \<noteq> {}"
+ by blast
+ have "convex U"
+ by (simp add: affine_imp_convex \<open>affine U\<close>)
+ then have "locally connected U"
+ by (rule convex_imp_locally_connected)
+ have "\<exists>a g. a \<in> C \<and> a \<in> L \<and> continuous_on (S \<union> (C - {a})) g \<and>
+ g ` (S \<union> (C - {a})) \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x)"
+ if C: "C \<in> components (U - S)" and CK: "C \<inter> K \<noteq> {}" for C
+ proof -
+ have "C \<subseteq> U-S" "C \<inter> L \<noteq> {}"
+ by (simp_all add: in_components_subset comps that)
+ then obtain a where a: "a \<in> C" "a \<in> L" by auto
+ have opeUC: "openin (subtopology euclidean U) C"
+ proof (rule openin_trans)
+ show "openin (subtopology euclidean (U-S)) C"
+ by (simp add: \<open>locally connected U\<close> clo locally_diff_closed openin_components_locally_connected [OF _ C])
+ show "openin (subtopology euclidean U) (U - S)"
+ by (simp add: clo openin_diff)
+ qed
+ then obtain d where "C \<subseteq> U" "0 < d" and d: "cball a d \<inter> U \<subseteq> C"
+ using openin_contains_cball by (metis \<open>a \<in> C\<close>)
+ then have "ball a d \<inter> U \<subseteq> C"
+ by auto
+ obtain h k where homhk: "homeomorphism (S \<union> C) (S \<union> C) h k"
+ and subC: "{x. (~ (h x = x \<and> k x = x))} \<subseteq> C"
+ and bou: "bounded {x. (~ (h x = x \<and> k x = x))}"
+ and hin: "\<And>x. x \<in> C \<inter> K \<Longrightarrow> h x \<in> ball a d \<inter> U"
+ proof (rule homeomorphism_grouping_points_exists_gen [of C "ball a d \<inter> U" "C \<inter> K" "S \<union> C"])
+ show "openin (subtopology euclidean C) (ball a d \<inter> U)"
+ by (metis Topology_Euclidean_Space.open_ball \<open>C \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> inf.absorb_iff2 inf.orderE inf_assoc open_openin openin_subtopology)
+ show "openin (subtopology euclidean (affine hull C)) C"
+ by (metis \<open>a \<in> C\<close> \<open>openin (subtopology euclidean U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
+ show "ball a d \<inter> U \<noteq> {}"
+ using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by force
+ show "finite (C \<inter> K)"
+ by (simp add: \<open>finite K\<close>)
+ show "S \<union> C \<subseteq> affine hull C"
+ by (metis \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> opeUC affine_hull_eq affine_hull_openin all_not_in_conv assms(2) sup.bounded_iff)
+ show "connected C"
+ by (metis C in_components_connected)
+ qed auto
+ have a_BU: "a \<in> ball a d \<inter> U"
+ using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
+ have "rel_frontier (cball a d \<inter> U) retract_of (affine hull (cball a d \<inter> U) - {a})"
+ apply (rule rel_frontier_retract_of_punctured_affine_hull)
+ apply (auto simp: \<open>convex U\<close> convex_Int)
+ by (metis \<open>affine U\<close> convex_cball empty_iff interior_cball a_BU rel_interior_convex_Int_affine)
+ moreover have "rel_frontier (cball a d \<inter> U) = frontier (cball a d) \<inter> U"
+ apply (rule convex_affine_rel_frontier_Int)
+ using a_BU by (force simp: \<open>affine U\<close>)+
+ moreover have "affine hull (cball a d \<inter> U) = U"
+ by (metis \<open>convex U\<close> a_BU affine_hull_convex_Int_nonempty_interior affine_hull_eq \<open>affine U\<close> equals0D inf.commute interior_cball)
+ ultimately have "frontier (cball a d) \<inter> U retract_of (U - {a})"
+ by metis
+ then obtain r where contr: "continuous_on (U - {a}) r"
+ and rim: "r ` (U - {a}) \<subseteq> sphere a d" "r ` (U - {a}) \<subseteq> U"
+ and req: "\<And>x. x \<in> sphere a d \<inter> U \<Longrightarrow> r x = x"
+ using \<open>affine U\<close> by (auto simp: retract_of_def retraction_def hull_same)
+ define j where "j \<equiv> \<lambda>x. if x \<in> ball a d then r x else x"
+ have kj: "\<And>x. x \<in> S \<Longrightarrow> k (j x) = x"
+ using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def subC by auto
+ have Uaeq: "U - {a} = (cball a d - {a}) \<inter> U \<union> (U - ball a d)"
+ using \<open>0 < d\<close> by auto
+ have jim: "j ` (S \<union> (C - {a})) \<subseteq> (S \<union> C) - ball a d"
+ proof clarify
+ fix y assume "y \<in> S \<union> (C - {a})"
+ then have "y \<in> U - {a}"
+ using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
+ then have "r y \<in> sphere a d"
+ using rim by auto
+ then show "j y \<in> S \<union> C - ball a d"
+ apply (simp add: j_def)
+ using \<open>r y \<in> sphere a d\<close> \<open>y \<in> U - {a}\<close> \<open>y \<in> S \<union> (C - {a})\<close> d rim by fastforce
+ qed
+ have contj: "continuous_on (U - {a}) j"
+ unfolding j_def Uaeq
+ proof (intro continuous_on_cases_local continuous_on_id, simp_all add: req closedin_closed Uaeq [symmetric])
+ show "\<exists>T. closed T \<and> (cball a d - {a}) \<inter> U = (U - {a}) \<inter> T"
+ apply (rule_tac x="(cball a d) \<inter> U" in exI)
+ using affine_closed \<open>affine U\<close> by blast
+ show "\<exists>T. closed T \<and> U - ball a d = (U - {a}) \<inter> T"
+ apply (rule_tac x="U - ball a d" in exI)
+ using \<open>0 < d\<close> by (force simp: affine_closed \<open>affine U\<close> closed_Diff)
+ show "continuous_on ((cball a d - {a}) \<inter> U) r"
+ by (force intro: continuous_on_subset [OF contr])
+ qed
+ have fT: "x \<in> U - K \<Longrightarrow> f x \<in> T" for x
+ using fim by blast
+ show ?thesis
+ proof (intro conjI exI)
+ show "continuous_on (S \<union> (C - {a})) (f \<circ> k \<circ> j)"
+ proof (intro continuous_on_compose)
+ show "continuous_on (S \<union> (C - {a})) j"
+ apply (rule continuous_on_subset [OF contj])
+ using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by force
+ show "continuous_on (j ` (S \<union> (C - {a}))) k"
+ apply (rule continuous_on_subset [OF homeomorphism_cont2 [OF homhk]])
+ using jim \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def by fastforce
+ show "continuous_on (k ` j ` (S \<union> (C - {a}))) f"
+ proof (clarify intro!: continuous_on_subset [OF contf])
+ fix y assume "y \<in> S \<union> (C - {a})"
+ have ky: "k y \<in> S \<union> C"
+ using homeomorphism_image2 [OF homhk] \<open>y \<in> S \<union> (C - {a})\<close> by blast
+ have jy: "j y \<in> S \<union> C - ball a d"
+ using Un_iff \<open>y \<in> S \<union> (C - {a})\<close> jim by auto
+ show "k (j y) \<in> U - K"
+ apply safe
+ using \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> homeomorphism_image2 [OF homhk] jy apply blast
+ by (metis DiffD1 DiffD2 Int_iff Un_iff \<open>disjnt K S\<close> disjnt_def empty_iff hin homeomorphism_apply2 homeomorphism_image2 homhk imageI jy)
+ qed
+ qed
+ have ST: "\<And>x. x \<in> S \<Longrightarrow> (f \<circ> k \<circ> j) x \<in> T"
+ apply (simp add: kj)
+ apply (metis DiffI \<open>S \<subseteq> U\<close> \<open>disjnt K S\<close> subsetD disjnt_iff fim image_subset_iff)
+ done
+ moreover have "(f \<circ> k \<circ> j) x \<in> T" if "x \<in> C" "x \<noteq> a" "x \<notin> S" for x
+ proof -
+ have rx: "r x \<in> sphere a d"
+ using \<open>C \<subseteq> U\<close> rim that by fastforce
+ have jj: "j x \<in> S \<union> C - ball a d"
+ using jim that by blast
+ have "k (j x) = j x \<longrightarrow> k (j x) \<in> C \<or> j x \<in> C"
+ by (metis Diff_iff Int_iff Un_iff \<open>S \<subseteq> U\<close> subsetD d j_def jj rx sphere_cball that(1))
+ then have "k (j x) \<in> C"
+ using homeomorphism_apply2 [OF homhk, of "j x"] \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> a rx
+ by (metis (mono_tags, lifting) Diff_iff subsetD jj mem_Collect_eq subC)
+ with jj \<open>C \<subseteq> U\<close> show ?thesis
+ apply safe
+ using ST j_def apply fastforce
+ apply (auto simp: not_less intro!: fT)
+ by (metis DiffD1 DiffD2 Int_iff hin homeomorphism_apply2 [OF homhk] jj)
+ qed
+ ultimately show "(f \<circ> k \<circ> j) ` (S \<union> (C - {a})) \<subseteq> T"
+ by force
+ show "\<forall>x\<in>S. (f \<circ> k \<circ> j) x = f x" using kj by simp
+ qed (auto simp: a)
+ qed
+ then obtain a h where
+ ah: "\<And>C. \<lbrakk>C \<in> components (U - S); C \<inter> K \<noteq> {}\<rbrakk>
+ \<Longrightarrow> a C \<in> C \<and> a C \<in> L \<and> continuous_on (S \<union> (C - {a C})) (h C) \<and>
+ h C ` (S \<union> (C - {a C})) \<subseteq> T \<and> (\<forall>x \<in> S. h C x = f x)"
+ using that by metis
+ define F where "F \<equiv> {C \<in> components (U - S). C \<inter> K \<noteq> {}}"
+ define G where "G \<equiv> {C \<in> components (U - S). C \<inter> K = {}}"
+ define UF where "UF \<equiv> (\<Union>C\<in>F. C - {a C})"
+ have "C0 \<in> F"
+ by (auto simp: F_def C0)
+ have "finite F"
+ proof (subst finite_image_iff [of "\<lambda>C. C \<inter> K" F, symmetric])
+ show "inj_on (\<lambda>C. C \<inter> K) F"
+ unfolding F_def inj_on_def
+ using components_nonoverlap by blast
+ show "finite ((\<lambda>C. C \<inter> K) ` F)"
+ unfolding F_def
+ by (rule finite_subset [of _ "Pow K"]) (auto simp: \<open>finite K\<close>)
+ qed
+ obtain g where contg: "continuous_on (S \<union> UF) g"
+ and gh: "\<And>x i. \<lbrakk>i \<in> F; x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i}))\<rbrakk>
+ \<Longrightarrow> g x = h i x"
+ proof (rule pasting_lemma_exists_closed [OF \<open>finite F\<close>, of "S \<union> UF" "\<lambda>C. S \<union> (C - {a C})" h])
+ show "S \<union> UF \<subseteq> (\<Union>C\<in>F. S \<union> (C - {a C}))"
+ using \<open>C0 \<in> F\<close> by (force simp: UF_def)
+ show "closedin (subtopology euclidean (S \<union> UF)) (S \<union> (C - {a C}))"
+ if "C \<in> F" for C
+ proof (rule closedin_closed_subset [of U "S \<union> C"])
+ show "closedin (subtopology euclidean U) (S \<union> C)"
+ apply (rule closedin_Un_complement_component [OF \<open>locally connected U\<close> clo])
+ using F_def that by blast
+ next
+ have "x = a C'" if "C' \<in> F" "x \<in> C'" "x \<notin> U" for x C'
+ proof -
+ have "\<forall>A. x \<in> \<Union>A \<or> C' \<notin> A"
+ using \<open>x \<in> C'\<close> by blast
+ with that show "x = a C'"
+ by (metis (lifting) DiffD1 F_def Union_components mem_Collect_eq)
+ qed
+ then show "S \<union> UF \<subseteq> U"
+ using \<open>S \<subseteq> U\<close> by (force simp: UF_def)
+ next
+ show "S \<union> (C - {a C}) = (S \<union> C) \<inter> (S \<union> UF)"
+ using F_def UF_def components_nonoverlap that by auto
+ qed
+ next
+ show "continuous_on (S \<union> (C' - {a C'})) (h C')" if "C' \<in> F" for C'
+ using ah F_def that by blast
+ show "\<And>i j x. \<lbrakk>i \<in> F; j \<in> F;
+ x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i})) \<inter> (S \<union> (j - {a j}))\<rbrakk>
+ \<Longrightarrow> h i x = h j x"
+ using components_eq by (fastforce simp: components_eq F_def ah)
+ qed blast
+ have SU': "S \<union> \<Union>G \<union> (S \<union> UF) \<subseteq> U"
+ using \<open>S \<subseteq> U\<close> in_components_subset by (auto simp: F_def G_def UF_def)
+ have clo1: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> \<Union>G)"
+ proof (rule closedin_closed_subset [OF _ SU'])
+ have *: "\<And>C. C \<in> F \<Longrightarrow> openin (subtopology euclidean U) C"
+ unfolding F_def
+ by clarify (metis (no_types, lifting) \<open>locally connected U\<close> clo closedin_def locally_diff_closed openin_components_locally_connected openin_trans topspace_euclidean_subtopology)
+ show "closedin (subtopology euclidean U) (U - UF)"
+ unfolding UF_def
+ by (force intro: openin_delete *)
+ show "S \<union> \<Union>G = (U - UF) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
+ using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
+ apply (metis Diff_iff UnionI Union_components)
+ apply (metis DiffD1 UnionI Union_components)
+ by (metis (no_types, lifting) IntI components_nonoverlap empty_iff)
+ qed
+ have clo2: "closedin (subtopology euclidean (S \<union> \<Union>G \<union> (S \<union> UF))) (S \<union> UF)"
+ proof (rule closedin_closed_subset [OF _ SU'])
+ show "closedin (subtopology euclidean U) (\<Union>C\<in>F. S \<union> C)"
+ apply (rule closedin_Union)
+ apply (simp add: \<open>finite F\<close>)
+ using F_def \<open>locally connected U\<close> clo closedin_Un_complement_component by blast
+ show "S \<union> UF = (\<Union>C\<in>F. S \<union> C) \<inter> (S \<union> \<Union>G \<union> (S \<union> UF))"
+ using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def)
+ using C0 apply blast
+ by (metis components_nonoverlap disjnt_def disjnt_iff)
+ qed
+ have SUG: "S \<union> \<Union>G \<subseteq> U - K"
+ using \<open>S \<subseteq> U\<close> K apply (auto simp: G_def disjnt_iff)
+ by (meson Diff_iff subsetD in_components_subset)
+ then have contf': "continuous_on (S \<union> \<Union>G) f"
+ by (rule continuous_on_subset [OF contf])
+ have contg': "continuous_on (S \<union> UF) g"
+ apply (rule continuous_on_subset [OF contg])
+ using \<open>S \<subseteq> U\<close> by (auto simp: F_def G_def)
+ have "\<And>x. \<lbrakk>S \<subseteq> U; x \<in> S\<rbrakk> \<Longrightarrow> f x = g x"
+ by (subst gh) (auto simp: ah C0 intro: \<open>C0 \<in> F\<close>)
+ then have f_eq_g: "\<And>x. x \<in> S \<union> UF \<and> x \<in> S \<union> \<Union>G \<Longrightarrow> f x = g x"
+ using \<open>S \<subseteq> U\<close> apply (auto simp: F_def G_def UF_def dest: in_components_subset)
+ using components_eq by blast
+ have cont: "continuous_on (S \<union> \<Union>G \<union> (S \<union> UF)) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
+ by (blast intro: continuous_on_cases_local [OF clo1 clo2 contf' contg' f_eq_g, of "\<lambda>x. x \<in> S \<union> \<Union>G"])
+ show ?thesis
+ proof
+ have UF: "\<Union>F - L \<subseteq> UF"
+ unfolding F_def UF_def using ah by blast
+ have "U - S - L = \<Union>(components (U - S)) - L"
+ by simp
+ also have "... = \<Union>F \<union> \<Union>G - L"
+ unfolding F_def G_def by blast
+ also have "... \<subseteq> UF \<union> \<Union>G"
+ using UF by blast
+ finally have "U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)"
+ by blast
+ then show "continuous_on (U - L) (\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x)"
+ by (rule continuous_on_subset [OF cont])
+ have "((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> ((U - L) \<inter> (-S \<inter> UF))"
+ using \<open>U - L \<subseteq> S \<union> \<Union>G \<union> (S \<union> UF)\<close> by auto
+ moreover have "g ` ((U - L) \<inter> (-S \<inter> UF)) \<subseteq> T"
+ proof -
+ have "g x \<in> T" if "x \<in> U" "x \<notin> L" "x \<notin> S" "C \<in> F" "x \<in> C" "x \<noteq> a C" for x C
+ proof (subst gh)
+ show "x \<in> (S \<union> UF) \<inter> (S \<union> (C - {a C}))"
+ using that by (auto simp: UF_def)
+ show "h C x \<in> T"
+ using ah that by (fastforce simp add: F_def)
+ qed (rule that)
+ then show ?thesis
+ by (force simp: UF_def)
+ qed
+ ultimately have "g ` ((U - L) \<inter> {x. x \<notin> S \<and> (\<forall>xa\<in>G. x \<notin> xa)}) \<subseteq> T"
+ using image_mono order_trans by blast
+ moreover have "f ` ((U - L) \<inter> (S \<union> \<Union>G)) \<subseteq> T"
+ using fim SUG by blast
+ ultimately show "(\<lambda>x. if x \<in> S \<union> \<Union>G then f x else g x) ` (U - L) \<subseteq> T"
+ by force
+ show "\<And>x. x \<in> S \<Longrightarrow> (if x \<in> S \<union> \<Union>G then f x else g x) = f x"
+ by (simp add: F_def G_def)
+ qed
+qed
+
+
+lemma extend_map_affine_to_sphere2:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
+ and affTU: "aff_dim T \<le> aff_dim U"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> rel_frontier U"
+ and ovlap: "\<And>C. C \<in> components(T - S) \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S"
+ "continuous_on (T - K) g" "g ` (T - K) \<subseteq> rel_frontier U"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof -
+ obtain K g where K: "finite K" "K \<subseteq> T" "disjnt K S"
+ and contg: "continuous_on (T - K) g"
+ and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ using assms extend_map_affine_to_sphere_cofinite_simple by metis
+ have "(\<exists>y C. C \<in> components (T - S) \<and> x \<in> C \<and> y \<in> C \<and> y \<in> L)" if "x \<in> K" for x
+ proof -
+ have "x \<in> T-S"
+ using \<open>K \<subseteq> T\<close> \<open>disjnt K S\<close> disjnt_def that by fastforce
+ then obtain C where "C \<in> components(T - S)" "x \<in> C"
+ by (metis UnionE Union_components)
+ with ovlap [of C] show ?thesis
+ by blast
+ qed
+ then obtain \<xi> where \<xi>: "\<And>x. x \<in> K \<Longrightarrow> \<exists>C. C \<in> components (T - S) \<and> x \<in> C \<and> \<xi> x \<in> C \<and> \<xi> x \<in> L"
+ by metis
+ obtain h where conth: "continuous_on (T - \<xi> ` K) h"
+ and him: "h ` (T - \<xi> ` K) \<subseteq> rel_frontier U"
+ and hg: "\<And>x. x \<in> S \<Longrightarrow> h x = g x"
+ proof (rule extend_map_affine_to_sphere1 [OF \<open>finite K\<close> \<open>affine T\<close> contg gim, of S "\<xi> ` K"])
+ show cloTS: "closedin (subtopology euclidean T) S"
+ by (simp add: \<open>compact S\<close> \<open>S \<subseteq> T\<close> closed_subset compact_imp_closed)
+ show "\<And>C. \<lbrakk>C \<in> components (T - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> \<xi> ` K \<noteq> {}"
+ using \<xi> components_eq by blast
+ qed (use K in auto)
+ show ?thesis
+ proof
+ show *: "\<xi> ` K \<subseteq> L"
+ using \<xi> by blast
+ show "finite (\<xi> ` K)"
+ by (simp add: K)
+ show "\<xi> ` K \<subseteq> T"
+ by clarify (meson \<xi> Diff_iff contra_subsetD in_components_subset)
+ show "continuous_on (T - \<xi> ` K) h"
+ by (rule conth)
+ show "disjnt (\<xi> ` K) S"
+ using K
+ apply (auto simp: disjnt_def)
+ by (metis \<xi> DiffD2 UnionI Union_components)
+ qed (simp_all add: him hg gf)
+qed
+
+
+proposition extend_map_affine_to_sphere_cofinite_gen:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes SUT: "compact S" "convex U" "bounded U" "affine T" "S \<subseteq> T"
+ and aff: "aff_dim T \<le> aff_dim U"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> rel_frontier U"
+ and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+ "g ` (T - K) \<subseteq> rel_frontier U"
+ "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "S = {}")
+ case True
+ show ?thesis
+ proof (cases "rel_frontier U = {}")
+ case True
+ with aff have "aff_dim T \<le> 0"
+ apply (simp add: rel_frontier_eq_empty)
+ using affine_bounded_eq_lowdim \<open>bounded U\<close> order_trans by auto
+ with aff_dim_geq [of T] consider "aff_dim T = -1" | "aff_dim T = 0"
+ by linarith
+ then show ?thesis
+ proof cases
+ assume "aff_dim T = -1"
+ then have "T = {}"
+ by (simp add: aff_dim_empty)
+ then show ?thesis
+ by (rule_tac K="{}" in that) auto
+ next
+ assume "aff_dim T = 0"
+ then obtain a where "T = {a}"
+ using aff_dim_eq_0 by blast
+ then have "a \<in> L"
+ using dis [of "{a}"] \<open>S = {}\<close> by (auto simp: in_components_self)
+ with \<open>S = {}\<close> \<open>T = {a}\<close> show ?thesis
+ by (rule_tac K="{a}" and g=f in that) auto
+ qed
+ next
+ case False
+ then obtain y where "y \<in> rel_frontier U"
+ by auto
+ with \<open>S = {}\<close> show ?thesis
+ by (rule_tac K="{}" and g="\<lambda>x. y" in that) (auto simp: continuous_on_const)
+ qed
+next
+ case False
+ have "bounded S"
+ by (simp add: assms compact_imp_bounded)
+ then obtain b where b: "S \<subseteq> cbox (-b) b"
+ using bounded_subset_cbox_symmetric by blast
+ define LU where "LU \<equiv> L \<union> (\<Union> {C \<in> components (T - S). ~bounded C} - cbox (-(b+One)) (b+One))"
+ obtain K g where "finite K" "K \<subseteq> LU" "K \<subseteq> T" "disjnt K S"
+ and contg: "continuous_on (T - K) g"
+ and gim: "g ` (T - K) \<subseteq> rel_frontier U"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ proof (rule extend_map_affine_to_sphere2 [OF SUT aff contf fim])
+ show "C \<inter> LU \<noteq> {}" if "C \<in> components (T - S)" for C
+ proof (cases "bounded C")
+ case True
+ with dis that show ?thesis
+ unfolding LU_def by fastforce
+ next
+ case False
+ then have "\<not> bounded (\<Union>{C \<in> components (T - S). \<not> bounded C})"
+ by (metis (no_types, lifting) Sup_upper bounded_subset mem_Collect_eq that)
+ then show ?thesis
+ apply (clarsimp simp: LU_def Int_Un_distrib Diff_Int_distrib Int_UN_distrib)
+ by (metis (no_types, lifting) False Sup_upper bounded_cbox bounded_subset inf.orderE mem_Collect_eq that)
+ qed
+ qed blast
+ have *: False if "x \<in> cbox (- b - m *\<^sub>R One) (b + m *\<^sub>R One)"
+ "x \<notin> box (- b - n *\<^sub>R One) (b + n *\<^sub>R One)"
+ "0 \<le> m" "m < n" "n \<le> 1" for m n x
+ using that by (auto simp: mem_box algebra_simps)
+ have "disjoint_family_on (\<lambda>d. frontier (cbox (- b - d *\<^sub>R One) (b + d *\<^sub>R One))) {1 / 2..1}"
+ by (auto simp: disjoint_family_on_def neq_iff frontier_def dest: *)
+ then obtain d where d12: "1/2 \<le> d" "d \<le> 1"
+ and ddis: "disjnt K (frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One)))"
+ using disjoint_family_elem_disjnt [of "{1/2..1::real}" K "\<lambda>d. frontier (cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"]
+ by (auto simp: \<open>finite K\<close>)
+ define c where "c \<equiv> b + d *\<^sub>R One"
+ have cbsub: "cbox (-b) b \<subseteq> box (-c) c"
+ "cbox (-b) b \<subseteq> cbox (-c) c"
+ "cbox (-c) c \<subseteq> cbox (-(b+One)) (b+One)"
+ using d12 by (simp_all add: subset_box c_def inner_diff_left inner_left_distrib)
+ have clo_cT: "closed (cbox (- c) c \<inter> T)"
+ using affine_closed \<open>affine T\<close> by blast
+ have cT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub by fastforce
+ have S_sub_cc: "S \<subseteq> cbox (- c) c"
+ using \<open>cbox (- b) b \<subseteq> cbox (- c) c\<close> b by auto
+ show ?thesis
+ proof
+ show "finite (K \<inter> cbox (-(b+One)) (b+One))"
+ using \<open>finite K\<close> by blast
+ show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> L"
+ using \<open>K \<subseteq> LU\<close> by (auto simp: LU_def)
+ show "K \<inter> cbox (- (b + One)) (b + One) \<subseteq> T"
+ using \<open>K \<subseteq> T\<close> by auto
+ show "disjnt (K \<inter> cbox (- (b + One)) (b + One)) S"
+ using \<open>disjnt K S\<close> by (simp add: disjnt_def disjoint_eq_subset_Compl inf.coboundedI1)
+ have cloTK: "closest_point (cbox (- c) c \<inter> T) x \<in> T - K"
+ if "x \<in> T" and Knot: "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
+ proof (cases "x \<in> cbox (- c) c")
+ case True
+ with \<open>x \<in> T\<close> show ?thesis
+ using cbsub(3) Knot by (force simp: closest_point_self)
+ next
+ case False
+ have clo_in_rf: "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
+ proof (intro closest_point_in_rel_frontier [OF clo_cT cT_ne] DiffI notI)
+ have "T \<inter> interior (cbox (- c) c) \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
+ then show "x \<in> affine hull (cbox (- c) c \<inter> T)"
+ by (simp add: Int_commute affine_hull_affine_Int_nonempty_interior \<open>affine T\<close> hull_inc that(1))
+ next
+ show "False" if "x \<in> rel_interior (cbox (- c) c \<inter> T)"
+ proof -
+ have "interior (cbox (- c) c) \<inter> T \<noteq> {}"
+ using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b cbsub(1) by fastforce
+ then have "affine hull (T \<inter> cbox (- c) c) = T"
+ using affine_hull_convex_Int_nonempty_interior [of T "cbox (- c) c"]
+ by (simp add: affine_imp_convex \<open>affine T\<close> inf_commute)
+ then show ?thesis
+ by (meson subsetD le_inf_iff rel_interior_subset that False)
+ qed
+ qed
+ have "closest_point (cbox (- c) c \<inter> T) x \<notin> K"
+ proof
+ assume inK: "closest_point (cbox (- c) c \<inter> T) x \<in> K"
+ have "\<And>x. x \<in> K \<Longrightarrow> x \<notin> frontier (cbox (- (b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
+ by (metis ddis disjnt_iff)
+ then show False
+ by (metis DiffI Int_iff \<open>affine T\<close> cT_ne c_def clo_cT clo_in_rf closest_point_in_set
+ convex_affine_rel_frontier_Int convex_box(1) empty_iff frontier_cbox inK interior_cbox)
+ qed
+ then show ?thesis
+ using cT_ne clo_cT closest_point_in_set by blast
+ qed
+ show "continuous_on (T - K \<inter> cbox (- (b + One)) (b + One)) (g \<circ> closest_point (cbox (-c) c \<inter> T))"
+ apply (intro continuous_on_compose continuous_on_closest_point continuous_on_subset [OF contg])
+ apply (simp_all add: clo_cT affine_imp_convex \<open>affine T\<close> convex_Int cT_ne)
+ using cloTK by blast
+ have "g (closest_point (cbox (- c) c \<inter> T) x) \<in> rel_frontier U"
+ if "x \<in> T" "x \<in> K \<longrightarrow> x \<notin> cbox (- b - One) (b + One)" for x
+ apply (rule gim [THEN subsetD])
+ using that cloTK by blast
+ then show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K \<inter> cbox (- (b + One)) (b + One))
+ \<subseteq> rel_frontier U"
+ by force
+ show "\<And>x. x \<in> S \<Longrightarrow> (g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x"
+ by simp (metis (mono_tags, lifting) IntI \<open>S \<subseteq> T\<close> cT_ne clo_cT closest_point_refl gf subsetD S_sub_cc)
+ qed
+qed
+
+
+corollary extend_map_affine_to_sphere_cofinite:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes SUT: "compact S" "affine T" "S \<subseteq> T"
+ and aff: "aff_dim T \<le> DIM('b)" and "0 \<le> r"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> sphere a r"
+ and dis: "\<And>C. \<lbrakk>C \<in> components(T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
+ "g ` (T - K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+proof (cases "r = 0")
+ case True
+ with fim show ?thesis
+ by (rule_tac K="{}" and g = "\<lambda>x. a" in that) (auto simp: continuous_on_const)
+next
+ case False
+ with assms have "0 < r" by auto
+ then have "aff_dim T \<le> aff_dim (cball a r)"
+ by (simp add: aff aff_dim_cball)
+ then show ?thesis
+ apply (rule extend_map_affine_to_sphere_cofinite_gen
+ [OF \<open>compact S\<close> convex_cball bounded_cball \<open>affine T\<close> \<open>S \<subseteq> T\<close> _ contf])
+ using fim apply (auto simp: assms False that dest: dis)
+ done
+qed
+
+corollary extend_map_UNIV_to_sphere_cofinite:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
+ and SUT: "compact S"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> sphere a r"
+ and dis: "\<And>C. \<lbrakk>C \<in> components(- S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
+ obtains K g where "finite K" "K \<subseteq> L" "disjnt K S" "continuous_on (- K) g"
+ "g ` (- K) \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+apply (rule extend_map_affine_to_sphere_cofinite
+ [OF \<open>compact S\<close> affine_UNIV subset_UNIV _ \<open>0 \<le> r\<close> contf fim dis])
+ apply (auto simp: assms that Compl_eq_Diff_UNIV [symmetric])
+done
+
+corollary extend_map_UNIV_to_sphere_no_bounded_component:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes aff: "DIM('a) \<le> DIM('b)" and "0 \<le> r"
+ and SUT: "compact S"
+ and contf: "continuous_on S f"
+ and fim: "f ` S \<subseteq> sphere a r"
+ and dis: "\<And>C. C \<in> components(- S) \<Longrightarrow> \<not> bounded C"
+ obtains g where "continuous_on UNIV g" "g ` UNIV \<subseteq> sphere a r" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+apply (rule extend_map_UNIV_to_sphere_cofinite [OF aff \<open>0 \<le> r\<close> \<open>compact S\<close> contf fim, of "{}"])
+ apply (auto simp: that dest: dis)
+done
+
+theorem Borsuk_separation_theorem_gen:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S"
+ shows "(\<forall>c \<in> components(- S). ~bounded c) \<longleftrightarrow>
+ (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
+ (is "?lhs = ?rhs")
+proof
+ assume L [rule_format]: ?lhs
+ show ?rhs
+ proof clarify
+ fix f :: "'a \<Rightarrow> 'a"
+ assume contf: "continuous_on S f" and fim: "f ` S \<subseteq> sphere 0 1"
+ obtain g where contg: "continuous_on UNIV g" and gim: "range g \<subseteq> sphere 0 1"
+ and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
+ by (rule extend_map_UNIV_to_sphere_no_bounded_component [OF _ _ \<open>compact S\<close> contf fim L]) auto
+ then show "\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)"
+ using nullhomotopic_from_contractible [OF contg gim]
+ by (metis assms compact_imp_closed contf empty_iff fim homotopic_with_equal nullhomotopic_into_sphere_extension)
+ qed
+next
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ unfolding components_def
+ proof clarify
+ fix a
+ assume "a \<notin> S" and a: "bounded (connected_component_set (- S) a)"
+ have cont: "continuous_on S (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a))"
+ apply (intro continuous_intros)
+ using \<open>a \<notin> S\<close> by auto
+ have im: "(\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a)) ` S \<subseteq> sphere 0 1"
+ by clarsimp (metis \<open>a \<notin> S\<close> eq_iff_diff_eq_0 left_inverse norm_eq_zero)
+ show False
+ using R cont im Borsuk_map_essential_bounded_component [OF \<open>compact S\<close> \<open>a \<notin> S\<close>] a by blast
+ qed
+qed
+
+
+corollary Borsuk_separation_theorem:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S" and 2: "2 \<le> DIM('a)"
+ shows "connected(- S) \<longleftrightarrow>
+ (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S (sphere 0 1) f (\<lambda>x. c)))"
+ (is "?lhs = ?rhs")
+proof
+ assume L: ?lhs
+ show ?rhs
+ proof (cases "S = {}")
+ case True
+ then show ?thesis by auto
+ next
+ case False
+ then have "(\<forall>c\<in>components (- S). \<not> bounded c)"
+ by (metis L assms(1) bounded_empty cobounded_imp_unbounded compact_imp_bounded in_components_maximal order_refl)
+ then show ?thesis
+ by (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>])
+ qed
+next
+ assume R: ?rhs
+ then show ?lhs
+ apply (simp add: Borsuk_separation_theorem_gen [OF \<open>compact S\<close>, symmetric])
+ apply (auto simp: components_def connected_iff_eq_connected_component_set)
+ using connected_component_in apply fastforce
+ using cobounded_unique_unbounded_component [OF _ 2, of "-S"] \<open>compact S\<close> compact_eq_bounded_closed by fastforce
+qed
+
+
+lemma homotopy_eqv_separation:
+ fixes S :: "'a::euclidean_space set" and T :: "'a set"
+ assumes "S homotopy_eqv T" and "compact S" and "compact T"
+ shows "connected(- S) \<longleftrightarrow> connected(- T)"
+proof -
+ consider "DIM('a) = 1" | "2 \<le> DIM('a)"
+ by (metis DIM_ge_Suc0 One_nat_def Suc_1 dual_order.antisym not_less_eq_eq)
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ using bounded_connected_Compl_1 compact_imp_bounded homotopy_eqv_empty1 homotopy_eqv_empty2 assms by metis
+ next
+ case 2
+ with assms show ?thesis
+ by (simp add: Borsuk_separation_theorem homotopy_eqv_cohomotopic_triviality_null)
+ qed
+qed
+
+lemma Jordan_Brouwer_separation:
+ fixes S :: "'a::euclidean_space set" and a::'a
+ assumes hom: "S homeomorphic sphere a r" and "0 < r"
+ shows "\<not> connected(- S)"
+proof -
+ have "- sphere a r \<inter> ball a r \<noteq> {}"
+ using \<open>0 < r\<close> by (simp add: Int_absorb1 subset_eq)
+ moreover
+ have eq: "- sphere a r - ball a r = - cball a r"
+ by auto
+ have "- cball a r \<noteq> {}"
+ proof -
+ have "frontier (cball a r) \<noteq> {}"
+ using \<open>0 < r\<close> by auto
+ then show ?thesis
+ by (metis frontier_complement frontier_empty)
+ qed
+ with eq have "- sphere a r - ball a r \<noteq> {}"
+ by auto
+ moreover
+ have "connected (- S) = connected (- sphere a r)"
+ proof (rule homotopy_eqv_separation)
+ show "S homotopy_eqv sphere a r"
+ using hom homeomorphic_imp_homotopy_eqv by blast
+ show "compact (sphere a r)"
+ by simp
+ then show " compact S"
+ using hom homeomorphic_compactness by blast
+ qed
+ ultimately show ?thesis
+ using connected_Int_frontier [of "- sphere a r" "ball a r"] by (auto simp: \<open>0 < r\<close>)
+qed
+
+
+lemma Jordan_Brouwer_frontier:
+ fixes S :: "'a::euclidean_space set" and a::'a
+ assumes S: "S homeomorphic sphere a r" and T: "T \<in> components(- S)" and 2: "2 \<le> DIM('a)"
+ shows "frontier T = S"
+proof (cases r rule: linorder_cases)
+ assume "r < 0"
+ with S T show ?thesis by auto
+next
+ assume "r = 0"
+ with S T card_eq_SucD obtain b where "S = {b}"
+ by (auto simp: homeomorphic_finite [of "{a}" S])
+ have "components (- {b}) = { -{b}}"
+ using T \<open>S = {b}\<close> by (auto simp: components_eq_sing_iff connected_punctured_universe 2)
+ with T show ?thesis
+ by (metis \<open>S = {b}\<close> cball_trivial frontier_cball frontier_complement singletonD sphere_trivial)
+next
+ assume "r > 0"
+ have "compact S"
+ using homeomorphic_compactness compact_sphere S by blast
+ show ?thesis
+ proof (rule frontier_minimal_separating_closed)
+ show "closed S"
+ using \<open>compact S\<close> compact_eq_bounded_closed by blast
+ show "\<not> connected (- S)"
+ using Jordan_Brouwer_separation S \<open>0 < r\<close> by blast
+ obtain f g where hom: "homeomorphism S (sphere a r) f g"
+ using S by (auto simp: homeomorphic_def)
+ show "connected (- T)" if "closed T" "T \<subset> S" for T
+ proof -
+ have "f ` T \<subseteq> sphere a r"
+ using \<open>T \<subset> S\<close> hom homeomorphism_image1 by blast
+ moreover have "f ` T \<noteq> sphere a r"
+ using \<open>T \<subset> S\<close> hom
+ by (metis homeomorphism_image2 homeomorphism_of_subsets order_refl psubsetE)
+ ultimately have "f ` T \<subset> sphere a r" by blast
+ then have "connected (- f ` T)"
+ by (rule psubset_sphere_Compl_connected [OF _ \<open>0 < r\<close> 2])
+ moreover have "compact T"
+ using \<open>compact S\<close> bounded_subset compact_eq_bounded_closed that by blast
+ moreover then have "compact (f ` T)"
+ by (meson compact_continuous_image continuous_on_subset hom homeomorphism_def psubsetE \<open>T \<subset> S\<close>)
+ moreover have "T homotopy_eqv f ` T"
+ by (meson \<open>f ` T \<subseteq> sphere a r\<close> dual_order.strict_implies_order hom homeomorphic_def homeomorphic_imp_homotopy_eqv homeomorphism_of_subsets \<open>T \<subset> S\<close>)
+ ultimately show ?thesis
+ using homotopy_eqv_separation [of T "f`T"] by blast
+ qed
+ qed (rule T)
+qed
+
+lemma Jordan_Brouwer_nonseparation:
+ fixes S :: "'a::euclidean_space set" and a::'a
+ assumes S: "S homeomorphic sphere a r" and "T \<subset> S" and 2: "2 \<le> DIM('a)"
+ shows "connected(- T)"
+proof -
+ have *: "connected(C \<union> (S - T))" if "C \<in> components(- S)" for C
+ proof (rule connected_intermediate_closure)
+ show "connected C"
+ using in_components_connected that by auto
+ have "S = frontier C"
+ using "2" Jordan_Brouwer_frontier S that by blast
+ with closure_subset show "C \<union> (S - T) \<subseteq> closure C"
+ by (auto simp: frontier_def)
+ qed auto
+ have "components(- S) \<noteq> {}"
+ by (metis S bounded_empty cobounded_imp_unbounded compact_eq_bounded_closed compact_sphere
+ components_eq_empty homeomorphic_compactness)
+ then have "- T = (\<Union>C \<in> components(- S). C \<union> (S - T))"
+ using Union_components [of "-S"] \<open>T \<subset> S\<close> by auto
+ then show ?thesis
+ apply (rule ssubst)
+ apply (rule connected_Union)
+ using \<open>T \<subset> S\<close> apply (auto simp: *)
+ done
+qed
+
+end
--- a/src/HOL/Analysis/Lebesgue_Measure.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy Thu Oct 06 11:27:28 2016 +0200
@@ -11,6 +11,24 @@
imports Finite_Product_Measure Bochner_Integration Caratheodory Complete_Measure Summation_Tests Regularity
begin
+lemma measure_eqI_lessThan:
+ fixes M N :: "real measure"
+ assumes sets: "sets M = sets borel" "sets N = sets borel"
+ assumes fin: "\<And>x. emeasure M {x <..} < \<infinity>"
+ assumes "\<And>x. emeasure M {x <..} = emeasure N {x <..}"
+ shows "M = N"
+proof (rule measure_eqI_generator_eq_countable)
+ let ?LT = "\<lambda>a::real. {a <..}" let ?E = "range ?LT"
+ show "Int_stable ?E"
+ by (auto simp: Int_stable_def lessThan_Int_lessThan)
+
+ show "?E \<subseteq> Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
+ unfolding sets borel_Ioi by auto
+
+ show "?LT`Rats \<subseteq> ?E" "(\<Union>i\<in>Rats. ?LT i) = UNIV" "\<And>a. a \<in> ?LT`Rats \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
+ using fin by (auto intro: Rats_no_bot_less simp: less_top)
+qed (auto intro: assms countable_rat)
+
subsection \<open>Every right continuous and nondecreasing function gives rise to a measure\<close>
definition interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
--- a/src/HOL/Analysis/Measurable.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Measurable.thy Thu Oct 06 11:27:28 2016 +0200
@@ -618,6 +618,24 @@
shows "Measurable.pred M ((R ^^ n) T)"
by (induct n) (auto intro: assms)
+lemma measurable_compose_countable_restrict:
+ assumes P: "countable {i. P i}"
+ and f: "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"
+ and Q: "\<And>i. P i \<Longrightarrow> pred M (Q i)"
+ shows "pred M (\<lambda>x. P (f x) \<and> Q (f x) x)"
+proof -
+ have P_f: "{x \<in> space M. P (f x)} \<in> sets M"
+ unfolding pred_def[symmetric] by (rule measurable_compose[OF f]) simp
+ have "pred (restrict_space M {x\<in>space M. P (f x)}) (\<lambda>x. Q (f x) x)"
+ proof (rule measurable_compose_countable'[where g=f, OF _ _ P])
+ show "f \<in> restrict_space M {x\<in>space M. P (f x)} \<rightarrow>\<^sub>M count_space {i. P i}"
+ by (rule measurable_count_space_extend[OF subset_UNIV])
+ (auto simp: space_restrict_space intro!: measurable_restrict_space1 f)
+ qed (auto intro!: measurable_restrict_space1 Q)
+ then show ?thesis
+ unfolding pred_restrict_space[OF P_f] by (simp cong: measurable_cong)
+qed
+
hide_const (open) pred
end
--- a/src/HOL/Analysis/Measure_Space.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Measure_Space.thy Thu Oct 06 11:27:28 2016 +0200
@@ -838,6 +838,38 @@
qed
qed
+lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"
+ by (rule measure_eqI) (simp_all add: space_empty_iff)
+
+lemma measure_eqI_generator_eq_countable:
+ fixes M N :: "'a measure" and E :: "'a set set" and A :: "'a set set"
+ assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
+ and sets: "sets M = sigma_sets \<Omega> E" "sets N = sigma_sets \<Omega> E"
+ and A: "A \<subseteq> E" "(\<Union>A) = \<Omega>" "countable A" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
+ shows "M = N"
+proof cases
+ assume "\<Omega> = {}"
+ have *: "sigma_sets \<Omega> E = sets (sigma \<Omega> E)"
+ using E(2) by simp
+ have "space M = \<Omega>" "space N = \<Omega>"
+ using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of)
+ then show "M = N"
+ unfolding \<open>\<Omega> = {}\<close> by (auto dest: space_empty)
+next
+ assume "\<Omega> \<noteq> {}" with \<open>\<Union>A = \<Omega>\<close> have "A \<noteq> {}" by auto
+ from this \<open>countable A\<close> have rng: "range (from_nat_into A) = A"
+ by (rule range_from_nat_into)
+ show "M = N"
+ proof (rule measure_eqI_generator_eq[OF E sets])
+ show "range (from_nat_into A) \<subseteq> E"
+ unfolding rng using \<open>A \<subseteq> E\<close> .
+ show "(\<Union>i. from_nat_into A i) = \<Omega>"
+ unfolding rng using \<open>\<Union>A = \<Omega>\<close> .
+ show "emeasure M (from_nat_into A i) \<noteq> \<infinity>" for i
+ using rng by (intro A) auto
+ qed
+qed
+
lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
proof (intro measure_eqI emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
@@ -1097,6 +1129,9 @@
"(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)"
by auto
+lemma AE_cong_strong: "M = N \<Longrightarrow> (\<And>x. x \<in> space N =simp=> P x = Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in N. Q x)"
+ by (auto simp: simp_implies_def)
+
lemma AE_all_countable:
"(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
proof
@@ -2135,9 +2170,6 @@
qed simp
qed (simp add: emeasure_notin_sets)
-lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"
- by (rule measure_eqI) (simp_all add: space_empty_iff)
-
lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
--- a/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1692,6 +1692,16 @@
by (simp add: ** nn_integral_suminf from_nat_into)
qed
+lemma of_bool_Bex_eq_nn_integral:
+ assumes unique: "\<And>x y. x \<in> X \<Longrightarrow> y \<in> X \<Longrightarrow> P x \<Longrightarrow> P y \<Longrightarrow> x = y"
+ shows "of_bool (\<exists>y\<in>X. P y) = (\<integral>\<^sup>+y. of_bool (P y) \<partial>count_space X)"
+proof cases
+ assume "\<exists>y\<in>X. P y"
+ then obtain y where "P y" "y \<in> X" by auto
+ then show ?thesis
+ by (subst nn_integral_count_space'[where A="{y}"]) (auto dest: unique)
+qed (auto cong: nn_integral_cong_simp)
+
lemma emeasure_UN_countable:
assumes sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I[simp]: "countable I"
assumes disj: "disjoint_family_on X I"
--- a/src/HOL/Analysis/Path_Connected.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Path_Connected.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1883,6 +1883,10 @@
finally show ?thesis .
qed
+corollary connected_punctured_universe:
+ "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(- {a::'N})"
+ by (simp add: path_connected_punctured_universe path_connected_imp_connected)
+
lemma path_connected_sphere:
assumes "2 \<le> DIM('a::euclidean_space)"
shows "path_connected {x::'a. norm (x - a) = r}"
@@ -2104,6 +2108,32 @@
thus ?case by (metis Diff_insert)
qed
+lemma psubset_sphere_Compl_connected:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "S \<subset> sphere a r" and "0 < r" and 2: "2 \<le> DIM('a)"
+ shows "connected(- S)"
+proof -
+ have "S \<subseteq> sphere a r"
+ using S by blast
+ obtain b where "dist a b = r" and "b \<notin> S"
+ using S mem_sphere by blast
+ have CS: "- S = {x. dist a x \<le> r \<and> (x \<notin> S)} \<union> {x. r \<le> dist a x \<and> (x \<notin> S)}"
+ by (auto simp: )
+ have "{x. dist a x \<le> r \<and> x \<notin> S} \<inter> {x. r \<le> dist a x \<and> x \<notin> S} \<noteq> {}"
+ using \<open>b \<notin> S\<close> \<open>dist a b = r\<close> by blast
+ moreover have "connected {x. dist a x \<le> r \<and> x \<notin> S}"
+ apply (rule connected_intermediate_closure [of "ball a r"])
+ using assms by auto
+ moreover
+ have "connected {x. r \<le> dist a x \<and> x \<notin> S}"
+ apply (rule connected_intermediate_closure [of "- cball a r"])
+ using assms apply (auto intro: connected_complement_bounded_convex)
+ apply (metis ComplI interior_cball interior_closure mem_ball not_less)
+ done
+ ultimately show ?thesis
+ by (simp add: CS connected_Un)
+qed
+
subsection\<open>Relations between components and path components\<close>
lemma open_connected_component:
@@ -2505,9 +2535,9 @@
{ fix y
assume y1: "y \<in> closure (connected_component_set S x)"
and y2: "y \<notin> interior (connected_component_set S x)"
- have 1: "y \<in> closure S"
+ have "y \<in> closure S"
using y1 closure_mono connected_component_subset by blast
- have "z \<in> interior (connected_component_set S x)"
+ moreover have "z \<in> interior (connected_component_set S x)"
if "0 < e" "ball y e \<subseteq> interior S" "dist y z < e" for e z
proof -
have "ball y e \<subseteq> connected_component_set S y"
@@ -2516,12 +2546,12 @@
done
then show ?thesis
using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
- by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in subsetD
- dist_commute mem_Collect_eq mem_ball mem_interior \<open>0 < e\<close> y2)
+ by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \<open>0 < e\<close> y2)
qed
- then have 2: "y \<notin> interior S"
+ then have "y \<notin> interior S"
using y2 by (force simp: open_contains_ball_eq [OF open_interior])
- note 1 2
+ ultimately have "y \<in> frontier S"
+ by (auto simp: frontier_def)
}
then show ?thesis by (auto simp: frontier_def)
qed
@@ -2565,6 +2595,49 @@
by (rule order_trans [OF frontier_Union_subset_closure])
(auto simp: closure_subset_eq)
+lemma frontier_of_components_subset:
+ fixes S :: "'a::real_normed_vector set"
+ shows "C \<in> components S \<Longrightarrow> frontier C \<subseteq> frontier S"
+ by (metis Path_Connected.frontier_of_connected_component_subset components_iff)
+
+lemma frontier_of_components_closed_complement:
+ fixes S :: "'a::real_normed_vector set"
+ shows "\<lbrakk>closed S; C \<in> components (- S)\<rbrakk> \<Longrightarrow> frontier C \<subseteq> S"
+ using frontier_complement frontier_of_components_subset frontier_subset_eq by blast
+
+lemma frontier_minimal_separating_closed:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "closed S"
+ and nconn: "~ connected(- S)"
+ and C: "C \<in> components (- S)"
+ and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected(- T)"
+ shows "frontier C = S"
+proof (rule ccontr)
+ assume "frontier C \<noteq> S"
+ then have "frontier C \<subset> S"
+ using frontier_of_components_closed_complement [OF \<open>closed S\<close> C] by blast
+ then have "connected(- (frontier C))"
+ by (simp add: conn)
+ have "\<not> connected(- (frontier C))"
+ unfolding connected_def not_not
+ proof (intro exI conjI)
+ show "open C"
+ using C \<open>closed S\<close> open_components by blast
+ show "open (- closure C)"
+ by blast
+ show "C \<inter> - closure C \<inter> - frontier C = {}"
+ using closure_subset by blast
+ show "C \<inter> - frontier C \<noteq> {}"
+ using C \<open>open C\<close> components_eq frontier_disjoint_eq by fastforce
+ show "- frontier C \<subseteq> C \<union> - closure C"
+ by (simp add: \<open>open C\<close> closed_Compl frontier_closures)
+ then show "- closure C \<inter> - frontier C \<noteq> {}"
+ by (metis (no_types, lifting) C Compl_subset_Compl_iff \<open>frontier C \<subset> S\<close> compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb)
+ qed
+ then show False
+ using \<open>connected (- frontier C)\<close> by blast
+qed
+
lemma connected_component_UNIV [simp]:
fixes x :: "'a::real_normed_vector"
shows "connected_component_set UNIV x = UNIV"
@@ -6140,6 +6213,51 @@
apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)
+lemma homotopy_eqv_homotopic_triviality_null_imp:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ and f: "continuous_on U f" "f ` U \<subseteq> T"
+ and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
+ \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c)"
+ shows "\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)"
+proof -
+ obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+ and k: "continuous_on T k" "k ` T \<subseteq> S"
+ and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+ "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+ using assms by (auto simp: homotopy_eqv_def)
+ obtain c::'a where "homotopic_with (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
+ apply (rule exE [OF homSU [of "k \<circ> f"]])
+ apply (intro continuous_on_compose h)
+ using k f apply (force elim!: continuous_on_subset)+
+ done
+ then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
+ apply (rule homotopic_with_compose_continuous_left [where Y=S])
+ using h by auto
+ moreover have "homotopic_with (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
+ apply (rule homotopic_with_compose_continuous_right [where X=T])
+ apply (simp add: hom homotopic_with_symD)
+ using f apply auto
+ done
+ ultimately show ?thesis
+ using homotopic_with_trans by (fastforce simp add: o_def)
+qed
+
+lemma homotopy_eqv_homotopic_triviality_null:
+ fixes S :: "'a::real_normed_vector set"
+ and T :: "'b::real_normed_vector set"
+ and U :: "'c::real_normed_vector set"
+ assumes "S homotopy_eqv T"
+ shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
+ (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
+ \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)))"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
+by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_sym)
+
lemma homotopy_eqv_contractible_sets:
fixes S :: "'a::real_normed_vector set"
and T :: "'b::real_normed_vector set"
--- a/src/HOL/Analysis/Sigma_Algebra.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Analysis/Sigma_Algebra.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1244,6 +1244,10 @@
lemma (in algebra) Int_stable: "Int_stable M"
unfolding Int_stable_def by auto
+lemma Int_stableI_image:
+ "(\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. A i \<inter> A j = A k) \<Longrightarrow> Int_stable (A ` I)"
+ by (auto simp: Int_stable_def image_def)
+
lemma Int_stableI:
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
unfolding Int_stable_def by auto
@@ -1574,6 +1578,9 @@
using assms
by(simp_all add: sets_measure_of_conv space_measure_of_conv)
+lemma space_in_measure_of[simp]: "\<Omega> \<in> sets (measure_of \<Omega> M \<mu>)"
+ by (subst sets_measure_of_conv) (auto simp: sigma_sets_top)
+
lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
using space_closed by (auto intro!: sigma_sets_eq)
@@ -2259,4 +2266,7 @@
by (rule measurable_restrict_countable[OF X])
(auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)
+lemma measurable_count_space_extend: "A \<subseteq> B \<Longrightarrow> f \<in> space M \<rightarrow> A \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M count_space B \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M count_space A"
+ by (auto simp: measurable_def)
+
end
--- a/src/HOL/Cardinals/Ordinal_Arithmetic.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Cardinals/Ordinal_Arithmetic.thy Thu Oct 06 11:27:28 2016 +0200
@@ -1660,7 +1660,7 @@
thus "((x, y), (s.max_fun_diff (rev_curr f m) (rev_curr g m), m)) \<in> s *o t"
using rst.max_fun_diff_in[OF diff1] rs.max_fun_diff_in[OF diff2] diff1 diff2
rst.max_fun_diff_max[OF diff1, of y] rs.max_fun_diff_le_eq[OF _ diff2, of x]
- unfolding oprod_def m_def rev_curr_def fun_eq_iff by auto (metis s.in_notinI)
+ unfolding oprod_def m_def rev_curr_def fun_eq_iff by (auto intro: s.in_notinI)
qed
qed
--- a/src/HOL/Data_Structures/Balance.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Data_Structures/Balance.thy Thu Oct 06 11:27:28 2016 +0200
@@ -4,10 +4,138 @@
theory Balance
imports
+ Complex_Main
"~~/src/HOL/Library/Tree"
- "~~/src/HOL/Library/Log_Nat"
begin
+(* mv *)
+
+text \<open>The lemmas about \<open>floor\<close> and \<open>ceiling\<close> of \<open>log 2\<close> should be generalized
+from 2 to \<open>n\<close> and should be made executable. \<close>
+
+lemma floor_log_nat: fixes b n k :: nat
+assumes "b \<ge> 2" "b^n \<le> k" "k < b^(n+1)"
+shows "floor (log b (real k)) = int(n)"
+proof -
+ have "k \<ge> 1"
+ using assms(1,2) one_le_power[of b n] by linarith
+ show ?thesis
+ proof(rule floor_eq2)
+ show "int n \<le> log b k"
+ using assms(1,2) \<open>k \<ge> 1\<close>
+ by(simp add: powr_realpow le_log_iff of_nat_power[symmetric] del: of_nat_power)
+ next
+ have "real k < b powr (real(n + 1))" using assms(1,3)
+ by (simp only: powr_realpow) (metis of_nat_less_iff of_nat_power)
+ thus "log b k < real_of_int (int n) + 1"
+ using assms(1) \<open>k \<ge> 1\<close> by(simp add: log_less_iff add_ac)
+ qed
+qed
+
+lemma ceil_log_nat: fixes b n k :: nat
+assumes "b \<ge> 2" "b^n < k" "k \<le> b^(n+1)"
+shows "ceiling (log b (real k)) = int(n)+1"
+proof(rule ceiling_eq)
+ show "int n < log b k"
+ using assms(1,2)
+ by(simp add: powr_realpow less_log_iff of_nat_power[symmetric] del: of_nat_power)
+next
+ have "real k \<le> b powr (real(n + 1))"
+ using assms(1,3)
+ by (simp only: powr_realpow) (metis of_nat_le_iff of_nat_power)
+ thus "log b k \<le> real_of_int (int n) + 1"
+ using assms(1,2) by(simp add: log_le_iff add_ac)
+qed
+
+lemma ex_power_ivl1: fixes b k :: nat assumes "b \<ge> 2"
+shows "k \<ge> 1 \<Longrightarrow> \<exists>n. b^n \<le> k \<and> k < b^(n+1)" (is "_ \<Longrightarrow> \<exists>n. ?P k n")
+proof(induction k)
+ case 0 thus ?case by simp
+next
+ case (Suc k)
+ show ?case
+ proof cases
+ assume "k=0"
+ hence "?P (Suc k) 0"
+ using assms by simp
+ thus ?case ..
+ next
+ assume "k\<noteq>0"
+ with Suc obtain n where IH: "?P k n" by auto
+ show ?case
+ proof (cases "k = b^(n+1) - 1")
+ case True
+ hence "?P (Suc k) (n+1)" using assms
+ by (simp add: not_less_eq_eq[symmetric])
+ thus ?thesis ..
+ next
+ case False
+ hence "?P (Suc k) n" using IH by auto
+ thus ?thesis ..
+ qed
+ qed
+qed
+
+lemma ex_power_ivl2: fixes b k :: nat assumes "b \<ge> 2" "(k::nat) \<ge> 2"
+shows "\<exists>n. b^n < k \<and> k \<le> b^(n+1)"
+proof -
+ have "1 \<le> k - 1"
+ using assms(2) by arith
+ from ex_power_ivl1[OF assms(1) this]
+ obtain n where "b ^ n \<le> k - 1 \<and> k - 1 < b ^ (n + 1)" ..
+ hence "b^n < k \<and> k \<le> b^(n+1)"
+ using assms by auto
+ thus ?thesis ..
+qed
+
+lemma ceil_log2_div2: assumes "n \<ge> 2"
+shows "ceiling(log 2 (real n)) = ceiling(log 2 ((n-1) div 2 + 1)) + 1"
+proof cases
+ assume "n=2"
+ thus ?thesis by simp
+next
+ let ?m = "(n-1) div 2 + 1"
+ assume "n\<noteq>2"
+ hence "2 \<le> ?m"
+ using assms by arith
+ then obtain i where i: "2 ^ i < ?m" "?m \<le> 2 ^ (i + 1)"
+ using ex_power_ivl2[of 2 ?m] by auto
+ have "n \<le> 2*?m"
+ by arith
+ also have "2*?m \<le> 2 ^ ((i+1)+1)"
+ using i(2) by simp
+ finally have *: "n \<le> \<dots>" .
+ have "2^(i+1) < n"
+ using i(1) by (auto simp add: less_Suc_eq_0_disj)
+ from ceil_log_nat[OF _ this *] ceil_log_nat[OF _ i]
+ show ?thesis by simp
+qed
+
+lemma floor_log2_div2: fixes n :: nat assumes "n \<ge> 2"
+shows "floor(log 2 n) = floor(log 2 (n div 2)) + 1"
+proof cases
+ assume "n=2"
+ thus ?thesis by simp
+next
+ let ?m = "n div 2"
+ assume "n\<noteq>2"
+ hence "1 \<le> ?m"
+ using assms by arith
+ then obtain i where i: "2 ^ i \<le> ?m" "?m < 2 ^ (i + 1)"
+ using ex_power_ivl1[of 2 ?m] by auto
+ have "2^(i+1) \<le> 2*?m"
+ using i(1) by simp
+ also have "2*?m \<le> n"
+ by arith
+ finally have *: "2^(i+1) \<le> \<dots>" .
+ have "n < 2^(i+1+1)"
+ using i(2) by simp
+ from floor_log_nat[OF _ * this] floor_log_nat[OF _ i]
+ show ?thesis by simp
+qed
+
+(* end of mv *)
+
fun bal :: "'a list \<Rightarrow> nat \<Rightarrow> 'a tree * 'a list" where
"bal xs n = (if n=0 then (Leaf,xs) else
(let m = n div 2;
@@ -28,8 +156,8 @@
"n > 0 \<Longrightarrow>
bal xs n =
(let m = n div 2;
- (l, ys) = Balance.bal xs m;
- (r, zs) = Balance.bal (tl ys) (n-1-m)
+ (l, ys) = bal xs m;
+ (r, zs) = bal (tl ys) (n-1-m)
in (Node l (hd ys) r, zs))"
by(simp_all add: bal.simps)
@@ -78,39 +206,22 @@
using bal_inorder[of xs "length xs"]
by (metis balance_list_def order_refl prod.collapse take_all)
-lemma bal_height: "bal xs n = (t,ys) \<Longrightarrow> height t = floorlog 2 n"
+corollary inorder_balance_tree[simp]: "inorder(balance_tree t) = inorder t"
+by(simp add: balance_tree_def inorder_balance_list)
+
+corollary size_balance_list[simp]: "size(balance_list xs) = length xs"
+by (metis inorder_balance_list length_inorder)
+
+corollary size_balance_tree[simp]: "size(balance_tree t) = size t"
+by(simp add: balance_tree_def inorder_balance_list)
+
+lemma min_height_bal:
+ "bal xs n = (t,ys) \<Longrightarrow> min_height t = nat(floor(log 2 (n + 1)))"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n) show ?case
proof cases
assume "n = 0" thus ?thesis
- using "1.prems" by (simp add: floorlog_def bal_simps)
- next
- assume [arith]: "n \<noteq> 0"
- from "1.prems" obtain l r xs' where
- b1: "bal xs (n div 2) = (l,xs')" and
- b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
- t: "t = \<langle>l, hd xs', r\<rangle>"
- by(auto simp: bal_simps Let_def split: prod.splits)
- let ?log1 = "floorlog 2 (n div 2)"
- let ?log2 = "floorlog 2 (n - 1 - n div 2)"
- have IH1: "height l = ?log1" using "1.IH"(1) b1 by simp
- have IH2: "height r = ?log2" using "1.IH"(2) b1 b2 by simp
- have "n div 2 \<ge> n - 1 - n div 2" by arith
- hence le: "?log2 \<le> ?log1" by(simp add:floorlog_mono)
- have "height t = max ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
- also have "\<dots> = ?log1 + 1" using le by (simp add: max_absorb1)
- also have "\<dots> = floorlog 2 n" by (simp add: compute_floorlog)
- finally show ?thesis .
- qed
-qed
-
-lemma bal_min_height:
- "bal xs n = (t,ys) \<Longrightarrow> min_height t = floorlog 2 (n + 1) - 1"
-proof(induction xs n arbitrary: t ys rule: bal.induct)
- case (1 xs n) show ?case
- proof cases
- assume "n = 0" thus ?thesis
- using "1.prems" by (simp add: floorlog_def bal_simps)
+ using "1.prems" by (simp add: bal_simps)
next
assume [arith]: "n \<noteq> 0"
from "1.prems" obtain l r xs' where
@@ -118,54 +229,78 @@
b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
by(auto simp: bal_simps Let_def split: prod.splits)
- let ?log1 = "floorlog 2 (n div 2 + 1) - 1"
- let ?log2 = "floorlog 2 (n - 1 - n div 2 + 1) - 1"
- let ?log2' = "floorlog 2 (n - n div 2) - 1"
- have "n - 1 - n div 2 + 1 = n - n div 2" by arith
- hence IH2: "min_height r = ?log2'" using "1.IH"(2) b1 b2 by simp
+ let ?log1 = "nat (floor(log 2 (n div 2 + 1)))"
+ let ?log2 = "nat (floor(log 2 (n - 1 - n div 2 + 1)))"
have IH1: "min_height l = ?log1" using "1.IH"(1) b1 by simp
- have *: "floorlog 2 (n - n div 2) \<ge> 1" by (simp add: floorlog_def)
- have "n div 2 + 1 \<ge> n - n div 2" by arith
- with * have le: "?log2' \<le> ?log1" by(simp add: floorlog_mono diff_le_mono)
- have "min_height t = min ?log1 ?log2' + 1" by (simp add: t IH1 IH2)
- also have "\<dots> = ?log2' + 1" using le by (simp add: min_absorb2)
- also have "\<dots> = floorlog 2 (n - n div 2)" by(simp add: floorlog_def)
- also have "n - n div 2 = (n+1) div 2" by arith
- also have "floorlog 2 \<dots> = floorlog 2 (n+1) - 1"
- by (simp add: compute_floorlog)
+ have IH2: "min_height r = ?log2" using "1.IH"(2) b1 b2 by simp
+ have "(n+1) div 2 \<ge> 1" by arith
+ hence 0: "log 2 ((n+1) div 2) \<ge> 0" by simp
+ have "n - 1 - n div 2 + 1 \<le> n div 2 + 1" by arith
+ hence le: "?log2 \<le> ?log1"
+ by(simp add: nat_mono floor_mono)
+ have "min_height t = min ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
+ also have "\<dots> = ?log2 + 1" using le by (simp add: min_absorb2)
+ also have "n - 1 - n div 2 + 1 = (n+1) div 2" by linarith
+ also have "nat (floor(log 2 ((n+1) div 2))) + 1
+ = nat (floor(log 2 ((n+1) div 2) + 1))"
+ using 0 by linarith
+ also have "\<dots> = nat (floor(log 2 (n + 1)))"
+ using floor_log2_div2[of "n+1"] by (simp add: log_mult)
+ finally show ?thesis .
+ qed
+qed
+
+lemma height_bal:
+ "bal xs n = (t,ys) \<Longrightarrow> height t = nat \<lceil>log 2 (n + 1)\<rceil>"
+proof(induction xs n arbitrary: t ys rule: bal.induct)
+ case (1 xs n) show ?case
+ proof cases
+ assume "n = 0" thus ?thesis
+ using "1.prems" by (simp add: bal_simps)
+ next
+ assume [arith]: "n \<noteq> 0"
+ from "1.prems" obtain l r xs' where
+ b1: "bal xs (n div 2) = (l,xs')" and
+ b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
+ t: "t = \<langle>l, hd xs', r\<rangle>"
+ by(auto simp: bal_simps Let_def split: prod.splits)
+ let ?log1 = "nat \<lceil>log 2 (n div 2 + 1)\<rceil>"
+ let ?log2 = "nat \<lceil>log 2 (n - 1 - n div 2 + 1)\<rceil>"
+ have IH1: "height l = ?log1" using "1.IH"(1) b1 by simp
+ have IH2: "height r = ?log2" using "1.IH"(2) b1 b2 by simp
+ have 0: "log 2 (n div 2 + 1) \<ge> 0" by auto
+ have "n - 1 - n div 2 + 1 \<le> n div 2 + 1" by arith
+ hence le: "?log2 \<le> ?log1"
+ by(simp add: nat_mono ceiling_mono del: nat_ceiling_le_eq)
+ have "height t = max ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
+ also have "\<dots> = ?log1 + 1" using le by (simp add: max_absorb1)
+ also have "\<dots> = nat \<lceil>log 2 (n div 2 + 1) + 1\<rceil>" using 0 by linarith
+ also have "\<dots> = nat \<lceil>log 2 (n + 1)\<rceil>"
+ using ceil_log2_div2[of "n+1"] by (simp)
finally show ?thesis .
qed
qed
lemma balanced_bal:
assumes "bal xs n = (t,ys)" shows "balanced t"
-proof -
- have "floorlog 2 n \<le> floorlog 2 (n+1)" by (rule floorlog_mono) auto
- thus ?thesis unfolding balanced_def
- using bal_height[OF assms] bal_min_height[OF assms] by linarith
-qed
+unfolding balanced_def
+using height_bal[OF assms] min_height_bal[OF assms]
+by linarith
-corollary size_balance_list[simp]: "size(balance_list xs) = length xs"
-by (metis inorder_balance_list length_inorder)
+lemma height_balance_list:
+ "height (balance_list xs) = nat \<lceil>log 2 (length xs + 1)\<rceil>"
+by (metis balance_list_def height_bal prod.collapse)
+
+corollary height_balance_tree:
+ "height (balance_tree t) = nat(ceiling(log 2 (size t + 1)))"
+by(simp add: balance_tree_def height_balance_list)
corollary balanced_balance_list[simp]: "balanced (balance_list xs)"
by (metis balance_list_def balanced_bal prod.collapse)
-lemma height_balance_list: "height(balance_list xs) = floorlog 2 (length xs)"
-by (metis bal_height balance_list_def prod.collapse)
-
-lemma inorder_balance_tree[simp]: "inorder(balance_tree t) = inorder t"
-by(simp add: balance_tree_def inorder_balance_list)
-
-lemma size_balance_tree[simp]: "size(balance_tree t) = size t"
-by(simp add: balance_tree_def inorder_balance_list)
-
corollary balanced_balance_tree[simp]: "balanced (balance_tree t)"
by (simp add: balance_tree_def)
-lemma height_balance_tree: "height(balance_tree t) = floorlog 2 (size t)"
-by(simp add: balance_tree_def height_balance_list)
-
lemma wbalanced_bal: "bal xs n = (t,ys) \<Longrightarrow> wbalanced t"
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n)
--- a/src/HOL/Divides.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Divides.thy Thu Oct 06 11:27:28 2016 +0200
@@ -542,6 +542,10 @@
"even a \<longleftrightarrow> a mod 2 = 0"
by (fact dvd_eq_mod_eq_0)
+lemma odd_iff_mod_2_eq_one:
+ "odd a \<longleftrightarrow> a mod 2 = 1"
+ by (auto simp add: even_iff_mod_2_eq_zero)
+
lemma even_succ_div_two [simp]:
"even a \<Longrightarrow> (a + 1) div 2 = a div 2"
by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
--- a/src/HOL/Int.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Int.thy Thu Oct 06 11:27:28 2016 +0200
@@ -983,6 +983,20 @@
end
+lemma transfer_rule_of_int:
+ fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
+ assumes [transfer_rule]: "R 0 0" "R 1 1"
+ "rel_fun R (rel_fun R R) plus plus"
+ "rel_fun R R uminus uminus"
+ shows "rel_fun HOL.eq R of_int of_int"
+proof -
+ note transfer_rule_of_nat [transfer_rule]
+ have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat"
+ by transfer_prover
+ show ?thesis
+ by (unfold of_int_of_nat [abs_def]) transfer_prover
+qed
+
lemma nat_mult_distrib:
fixes z z' :: int
assumes "0 \<le> z"
--- a/src/HOL/Library/Countable_Set.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Library/Countable_Set.thy Thu Oct 06 11:27:28 2016 +0200
@@ -284,6 +284,9 @@
lemma countable_Collect_finite: "countable (Collect (finite::'a::countable set\<Rightarrow>bool))"
by (simp add: Collect_finite_eq_lists)
+lemma countable_int: "countable \<int>"
+ unfolding Ints_def by auto
+
lemma countable_rat: "countable \<rat>"
unfolding Rats_def by auto
--- a/src/HOL/Library/Extended_Nonnegative_Real.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Library/Extended_Nonnegative_Real.thy Thu Oct 06 11:27:28 2016 +0200
@@ -220,6 +220,11 @@
shows "summable f \<Longrightarrow> finite I \<Longrightarrow> \<forall>m\<in>- I. 0 \<le> f m \<Longrightarrow> setsum f I \<le> suminf f"
by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
+lemma suminf_eq_SUP_real:
+ assumes X: "summable X" "\<And>i. 0 \<le> X i" shows "suminf X = (SUP i. \<Sum>n<i. X n::real)"
+ by (intro LIMSEQ_unique[OF summable_LIMSEQ] X LIMSEQ_incseq_SUP)
+ (auto intro!: bdd_aboveI2[where M="\<Sum>i. X i"] setsum_le_suminf X monoI setsum_mono3)
+
subsection \<open>Defining the extended non-negative reals\<close>
text \<open>Basic definitions and type class setup\<close>
--- a/src/HOL/Library/Multiset.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Library/Multiset.thy Thu Oct 06 11:27:28 2016 +0200
@@ -545,6 +545,8 @@
interpretation subset_mset: ordered_cancel_comm_monoid_diff "op +" 0 "op \<le>#" "op <#" "op -"
by standard (simp, fact mset_subset_eq_exists_conv)
+declare subset_mset.add_diff_assoc[simp] subset_mset.add_diff_assoc2[simp]
+
lemma mset_subset_eq_mono_add_right_cancel: "(A::'a multiset) + C \<subseteq># B + C \<longleftrightarrow> A \<subseteq># B"
by (fact subset_mset.add_le_cancel_right)
@@ -2649,7 +2651,7 @@
using K N trans True by (meson that transE)
ultimately show ?thesis
by (rule_tac x = I in exI, rule_tac x = J in exI, rule_tac x = "(K - {#a#}) + K'" in exI)
- (use z y N in \<open>auto simp: subset_mset.add_diff_assoc dest: in_diffD\<close>)
+ (use z y N in \<open>auto simp del: subset_mset.add_diff_assoc2 dest: in_diffD\<close>)
next
case False
then have "a \<in># I" by (metis N(2) union_iff union_single_eq_member z)
@@ -2658,7 +2660,7 @@
ultimately show ?thesis
by (rule_tac x = "I - {#a#}" in exI, rule_tac x = "add_mset a J" in exI,
rule_tac x = "K + K'" in exI)
- (use z y N False K in \<open>auto simp: subset_mset.diff_add_assoc2\<close>)
+ (use z y N False K in \<open>auto simp: add.assoc\<close>)
qed
qed
--- a/src/HOL/Probability/Central_Limit_Theorem.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Central_Limit_Theorem.thy Thu Oct 06 11:27:28 2016 +0200
@@ -8,7 +8,7 @@
imports Levy
begin
-theorem (in prob_space) central_limit_theorem:
+theorem (in prob_space) central_limit_theorem_zero_mean:
fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real"
and \<mu> :: "real measure"
and \<sigma> :: real
@@ -114,4 +114,31 @@
qed (auto intro!: real_dist_normal_dist simp: S_def)
qed
+theorem (in prob_space) central_limit_theorem:
+ fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real"
+ and \<mu> :: "real measure"
+ and \<sigma> :: real
+ and S :: "nat \<Rightarrow> 'a \<Rightarrow> real"
+ assumes X_indep: "indep_vars (\<lambda>i. borel) X UNIV"
+ and X_integrable: "\<And>n. integrable M (X n)"
+ and X_mean: "\<And>n. expectation (X n) = m"
+ and \<sigma>_pos: "\<sigma> > 0"
+ and X_square_integrable: "\<And>n. integrable M (\<lambda>x. (X n x)\<^sup>2)"
+ and X_variance: "\<And>n. variance (X n) = \<sigma>\<^sup>2"
+ and X_distrib: "\<And>n. distr M borel (X n) = \<mu>"
+ defines "X' i x \<equiv> X i x - m"
+ shows "weak_conv_m (\<lambda>n. distr M borel (\<lambda>x. (\<Sum>i<n. X' i x) / sqrt (n*\<sigma>\<^sup>2))) std_normal_distribution"
+proof (intro central_limit_theorem_zero_mean)
+ show "indep_vars (\<lambda>i. borel) X' UNIV"
+ unfolding X'_def[abs_def] using X_indep by (rule indep_vars_compose2) auto
+ show "integrable M (X' n)" "expectation (X' n) = 0" for n
+ using X_integrable X_mean by (auto simp: X'_def[abs_def] prob_space)
+ show "\<sigma> > 0" "integrable M (\<lambda>x. (X' n x)\<^sup>2)" "variance (X' n) = \<sigma>\<^sup>2" for n
+ using \<open>0 < \<sigma>\<close> X_integrable X_mean X_square_integrable X_variance unfolding X'_def
+ by (auto simp: prob_space power2_diff)
+ show "distr M borel (X' n) = distr \<mu> borel (\<lambda>x. x - m)" for n
+ unfolding X_distrib[of n, symmetric] using X_integrable
+ by (subst distr_distr) (auto simp: X'_def[abs_def] comp_def)
+qed
+
end
--- a/src/HOL/Probability/Giry_Monad.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Giry_Monad.thy Thu Oct 06 11:27:28 2016 +0200
@@ -28,6 +28,9 @@
show "subprob_space M" by standard fact+
qed
+lemma (in subprob_space) emeasure_subprob_space_less_top: "emeasure M A \<noteq> top"
+ using emeasure_finite[of A] .
+
lemma prob_space_imp_subprob_space:
"prob_space M \<Longrightarrow> subprob_space M"
by (rule subprob_spaceI) (simp_all add: prob_space.emeasure_space_1 prob_space.not_empty)
@@ -245,6 +248,43 @@
by (auto dest: subprob_space_kernel sets_kernel)
qed
+lemma measurable_subprob_algebra_generated:
+ assumes eq: "sets N = sigma_sets \<Omega> G" and "Int_stable G" "G \<subseteq> Pow \<Omega>"
+ assumes subsp: "\<And>a. a \<in> space M \<Longrightarrow> subprob_space (K a)"
+ assumes sets: "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"
+ assumes "\<And>A. A \<in> G \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
+ assumes \<Omega>: "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M"
+ shows "K \<in> measurable M (subprob_algebra N)"
+proof (rule measurable_subprob_algebra)
+ fix a assume "a \<in> space M" then show "subprob_space (K a)" "sets (K a) = sets N" by fact+
+next
+ interpret G: sigma_algebra \<Omega> "sigma_sets \<Omega> G"
+ using \<open>G \<subseteq> Pow \<Omega>\<close> by (rule sigma_algebra_sigma_sets)
+ fix A assume "A \<in> sets N" with assms(2,3) show "(\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
+ unfolding \<open>sets N = sigma_sets \<Omega> G\<close>
+ proof (induction rule: sigma_sets_induct_disjoint)
+ case (basic A) then show ?case by fact
+ next
+ case empty then show ?case by simp
+ next
+ case (compl A)
+ have "(\<lambda>a. emeasure (K a) (\<Omega> - A)) \<in> borel_measurable M \<longleftrightarrow>
+ (\<lambda>a. emeasure (K a) \<Omega> - emeasure (K a) A) \<in> borel_measurable M"
+ using G.top G.sets_into_space sets eq compl subprob_space.emeasure_subprob_space_less_top[OF subsp]
+ by (intro measurable_cong emeasure_Diff) auto
+ with compl \<Omega> show ?case
+ by simp
+ next
+ case (union F)
+ moreover have "(\<lambda>a. emeasure (K a) (\<Union>i. F i)) \<in> borel_measurable M \<longleftrightarrow>
+ (\<lambda>a. \<Sum>i. emeasure (K a) (F i)) \<in> borel_measurable M"
+ using sets union eq
+ by (intro measurable_cong suminf_emeasure[symmetric]) auto
+ ultimately show ?case
+ by auto
+ qed
+qed
+
lemma space_subprob_algebra_empty_iff:
"space (subprob_algebra N) = {} \<longleftrightarrow> space N = {}"
proof
@@ -1080,7 +1120,7 @@
shows "space (bind M f) = space N"
using assms by (intro sets_eq_imp_space_eq sets_bind)
-lemma bind_cong:
+lemma bind_cong_All:
assumes "\<forall>x \<in> space M. f x = g x"
shows "bind M f = bind M g"
proof (cases "space M = {}")
@@ -1090,6 +1130,10 @@
with \<open>space M \<noteq> {}\<close> and assms show ?thesis by (simp add: bind_nonempty cong: distr_cong)
qed (simp add: bind_empty)
+lemma bind_cong:
+ "M = N \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> bind M f = bind N g"
+ using bind_cong_All[of M f g] by auto
+
lemma bind_nonempty':
assumes "f \<in> measurable M (subprob_algebra N)" "x \<in> space M"
shows "bind M f = join (distr M (subprob_algebra N) f)"
@@ -1121,8 +1165,8 @@
qed (simp add: bind_empty space_empty[of M] nn_integral_count_space)
lemma AE_bind:
+ assumes N[measurable]: "N \<in> measurable M (subprob_algebra B)"
assumes P[measurable]: "Measurable.pred B P"
- assumes N[measurable]: "N \<in> measurable M (subprob_algebra B)"
shows "(AE x in M \<bind> N. P x) \<longleftrightarrow> (AE x in M. AE y in N x. P y)"
proof cases
assume M: "space M = {}" show ?thesis
@@ -1454,7 +1498,7 @@
also from Mh have "\<And>x. x \<in> space M \<Longrightarrow> h x \<in> measurable M' N" by measurable
hence "do {x \<leftarrow> M; do {y \<leftarrow> f x; return N (h x y)} \<bind> g} =
do {x \<leftarrow> M; y \<leftarrow> f x; return N (h x y) \<bind> g}"
- apply (intro ballI bind_cong bind_assoc)
+ apply (intro ballI bind_cong refl bind_assoc)
apply (subst measurable_cong_sets[OF sets_kernel[OF Mf] refl], simp)
apply (rule measurable_compose[OF _ return_measurable], auto intro: Mg)
done
@@ -1522,4 +1566,216 @@
"null_measure M \<in> space (subprob_algebra M) \<longleftrightarrow> space M \<noteq> {}"
by(simp add: space_subprob_algebra subprob_space_null_measure_iff)
+subsection \<open>Giry monad on probability spaces\<close>
+
+definition prob_algebra :: "'a measure \<Rightarrow> 'a measure measure" where
+ "prob_algebra K = restrict_space (subprob_algebra K) {M. prob_space M}"
+
+lemma space_prob_algebra: "space (prob_algebra M) = {N. sets N = sets M \<and> prob_space N}"
+ unfolding prob_algebra_def by (auto simp: space_subprob_algebra space_restrict_space prob_space_imp_subprob_space)
+
+lemma measurable_measure_prob_algebra[measurable]:
+ "a \<in> sets A \<Longrightarrow> (\<lambda>M. Sigma_Algebra.measure M a) \<in> prob_algebra A \<rightarrow>\<^sub>M borel"
+ unfolding prob_algebra_def by (intro measurable_restrict_space1 measurable_measure_subprob_algebra)
+
+lemma measurable_prob_algebraD:
+ "f \<in> N \<rightarrow>\<^sub>M prob_algebra M \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M subprob_algebra M"
+ unfolding prob_algebra_def measurable_restrict_space2_iff by auto
+
+lemma measure_measurable_prob_algebra2:
+ "Sigma (space M) A \<in> sets (M \<Otimes>\<^sub>M N) \<Longrightarrow> L \<in> M \<rightarrow>\<^sub>M prob_algebra N \<Longrightarrow>
+ (\<lambda>x. Sigma_Algebra.measure (L x) (A x)) \<in> borel_measurable M"
+ using measure_measurable_subprob_algebra2[of M A N L] by (auto intro: measurable_prob_algebraD)
+
+lemma measurable_prob_algebraI:
+ "(\<And>x. x \<in> space N \<Longrightarrow> prob_space (f x)) \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M subprob_algebra M \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M prob_algebra M"
+ unfolding prob_algebra_def by (intro measurable_restrict_space2) auto
+
+lemma measurable_distr_prob_space:
+ assumes f: "f \<in> M \<rightarrow>\<^sub>M N"
+ shows "(\<lambda>M'. distr M' N f) \<in> prob_algebra M \<rightarrow>\<^sub>M prob_algebra N"
+ unfolding prob_algebra_def measurable_restrict_space2_iff
+proof (intro conjI measurable_restrict_space1 measurable_distr f)
+ show "(\<lambda>M'. distr M' N f) \<in> space (restrict_space (subprob_algebra M) (Collect prob_space)) \<rightarrow> Collect prob_space"
+ using f by (auto simp: space_restrict_space space_subprob_algebra intro!: prob_space.prob_space_distr)
+qed
+
+lemma measurable_return_prob_space[measurable]: "return N \<in> N \<rightarrow>\<^sub>M prob_algebra N"
+ by (rule measurable_prob_algebraI) (auto simp: prob_space_return)
+
+lemma measurable_distr_prob_space2[measurable (raw)]:
+ assumes f: "g \<in> L \<rightarrow>\<^sub>M prob_algebra M" "(\<lambda>(x, y). f x y) \<in> L \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N"
+ shows "(\<lambda>x. distr (g x) N (f x)) \<in> L \<rightarrow>\<^sub>M prob_algebra N"
+ unfolding prob_algebra_def measurable_restrict_space2_iff
+proof (intro conjI measurable_restrict_space1 measurable_distr2[where M=M] f measurable_prob_algebraD)
+ show "(\<lambda>x. distr (g x) N (f x)) \<in> space L \<rightarrow> Collect prob_space"
+ using f subprob_measurableD[OF measurable_prob_algebraD[OF f(1)]]
+ by (auto simp: measurable_restrict_space2_iff prob_algebra_def
+ intro!: prob_space.prob_space_distr)
+qed
+
+lemma measurable_bind_prob_space:
+ assumes f: "f \<in> M \<rightarrow>\<^sub>M prob_algebra N" and g: "g \<in> N \<rightarrow>\<^sub>M prob_algebra R"
+ shows "(\<lambda>x. bind (f x) g) \<in> M \<rightarrow>\<^sub>M prob_algebra R"
+ unfolding prob_algebra_def measurable_restrict_space2_iff
+proof (intro conjI measurable_restrict_space1 measurable_bind2[where N=N] f g measurable_prob_algebraD)
+ show "(\<lambda>x. f x \<bind> g) \<in> space M \<rightarrow> Collect prob_space"
+ using g f subprob_measurableD[OF measurable_prob_algebraD[OF f]]
+ by (auto simp: measurable_restrict_space2_iff prob_algebra_def
+ intro!: prob_space.prob_space_bind[where S=R] AE_I2)
+qed
+
+lemma measurable_bind_prob_space2[measurable (raw)]:
+ assumes f: "f \<in> M \<rightarrow>\<^sub>M prob_algebra N" and g: "(\<lambda>(x, y). g x y) \<in> (M \<Otimes>\<^sub>M N) \<rightarrow>\<^sub>M prob_algebra R"
+ shows "(\<lambda>x. bind (f x) (g x)) \<in> M \<rightarrow>\<^sub>M prob_algebra R"
+ unfolding prob_algebra_def measurable_restrict_space2_iff
+proof (intro conjI measurable_restrict_space1 measurable_bind[where N=N] f g measurable_prob_algebraD)
+ show "(\<lambda>x. f x \<bind> g x) \<in> space M \<rightarrow> Collect prob_space"
+ using g f subprob_measurableD[OF measurable_prob_algebraD[OF f]]
+ using measurable_space[OF g]
+ by (auto simp: measurable_restrict_space2_iff prob_algebra_def space_pair_measure Pi_iff
+ intro!: prob_space.prob_space_bind[where S=R] AE_I2)
+qed (insert g, simp)
+
+
+lemma measurable_prob_algebra_generated:
+ assumes eq: "sets N = sigma_sets \<Omega> G" and "Int_stable G" "G \<subseteq> Pow \<Omega>"
+ assumes subsp: "\<And>a. a \<in> space M \<Longrightarrow> prob_space (K a)"
+ assumes sets: "\<And>a. a \<in> space M \<Longrightarrow> sets (K a) = sets N"
+ assumes "\<And>A. A \<in> G \<Longrightarrow> (\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
+ shows "K \<in> measurable M (prob_algebra N)"
+ unfolding measurable_restrict_space2_iff prob_algebra_def
+proof
+ show "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
+ proof (rule measurable_subprob_algebra_generated[OF assms(1,2,3) _ assms(5,6)])
+ fix a assume "a \<in> space M" then show "subprob_space (K a)"
+ using subsp[of a] by (intro prob_space_imp_subprob_space)
+ next
+ have "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M \<longleftrightarrow> (\<lambda>a. 1::ennreal) \<in> borel_measurable M"
+ using sets_eq_imp_space_eq[of "sigma \<Omega> G" N] \<open>G \<subseteq> Pow \<Omega>\<close> eq sets_eq_imp_space_eq[OF sets]
+ prob_space.emeasure_space_1[OF subsp]
+ by (intro measurable_cong) auto
+ then show "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M" by simp
+ qed
+qed (insert subsp, auto)
+
+lemma in_space_prob_algebra:
+ "x \<in> space (prob_algebra M) \<Longrightarrow> emeasure x (space M) = 1"
+ unfolding prob_algebra_def space_restrict_space space_subprob_algebra
+ by (auto dest!: prob_space.emeasure_space_1 sets_eq_imp_space_eq)
+
+lemma prob_space_pair:
+ assumes "prob_space M" "prob_space N" shows "prob_space (M \<Otimes>\<^sub>M N)"
+proof -
+ interpret M: prob_space M by fact
+ interpret N: prob_space N by fact
+ interpret P: pair_prob_space M N proof qed
+ show ?thesis
+ by unfold_locales
+qed
+
+lemma measurable_pair_prob[measurable]:
+ "f \<in> M \<rightarrow>\<^sub>M prob_algebra N \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M prob_algebra L \<Longrightarrow> (\<lambda>x. f x \<Otimes>\<^sub>M g x) \<in> M \<rightarrow>\<^sub>M prob_algebra (N \<Otimes>\<^sub>M L)"
+ unfolding prob_algebra_def measurable_restrict_space2_iff
+ by (auto intro!: measurable_pair_measure prob_space_pair)
+
+lemma emeasure_bind_prob_algebra:
+ assumes A: "A \<in> space (prob_algebra N)"
+ assumes B: "B \<in> N \<rightarrow>\<^sub>M prob_algebra L"
+ assumes X: "X \<in> sets L"
+ shows "emeasure (bind A B) X = (\<integral>\<^sup>+x. emeasure (B x) X \<partial>A)"
+ using A B
+ by (intro emeasure_bind[OF _ _ X])
+ (auto simp: space_prob_algebra measurable_prob_algebraD cong: measurable_cong_sets intro!: prob_space.not_empty)
+
+lemma prob_space_bind':
+ assumes A: "A \<in> space (prob_algebra M)" and B: "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "prob_space (A \<bind> B)"
+ using measurable_bind_prob_space[OF measurable_const, OF A B, THEN measurable_space, of undefined "count_space UNIV"]
+ by (simp add: space_prob_algebra)
+
+lemma sets_bind':
+ assumes A: "A \<in> space (prob_algebra M)" and B: "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "sets (A \<bind> B) = sets N"
+ using measurable_bind_prob_space[OF measurable_const, OF A B, THEN measurable_space, of undefined "count_space UNIV"]
+ by (simp add: space_prob_algebra)
+
+lemma bind_cong_AE':
+ assumes M: "M \<in> space (prob_algebra L)"
+ and f: "f \<in> L \<rightarrow>\<^sub>M prob_algebra N" and g: "g \<in> L \<rightarrow>\<^sub>M prob_algebra N"
+ and ae: "AE x in M. f x = g x"
+ shows "bind M f = bind M g"
+proof (rule measure_eqI)
+ show "sets (M \<bind> f) = sets (M \<bind> g)"
+ unfolding sets_bind'[OF M f] sets_bind'[OF M g] ..
+ show "A \<in> sets (M \<bind> f) \<Longrightarrow> emeasure (M \<bind> f) A = emeasure (M \<bind> g) A" for A
+ unfolding sets_bind'[OF M f]
+ using emeasure_bind_prob_algebra[OF M f, of A] emeasure_bind_prob_algebra[OF M g, of A] ae
+ by (auto intro: nn_integral_cong_AE)
+qed
+
+lemma density_discrete:
+ "countable A \<Longrightarrow> sets N = Set.Pow A \<Longrightarrow> (\<And>x. f x \<ge> 0) \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x = emeasure N {x}) \<Longrightarrow>
+ density (count_space A) f = N"
+ by (rule measure_eqI_countable[of _ A]) (auto simp: emeasure_density)
+
+lemma distr_density_discrete:
+ fixes f'
+ assumes "countable A"
+ assumes "f' \<in> borel_measurable M"
+ assumes "g \<in> measurable M (count_space A)"
+ defines "f \<equiv> \<lambda>x. \<integral>\<^sup>+t. (if g t = x then 1 else 0) * f' t \<partial>M"
+ assumes "\<And>x. x \<in> space M \<Longrightarrow> g x \<in> A"
+ shows "density (count_space A) (\<lambda>x. f x) = distr (density M f') (count_space A) g"
+proof (rule density_discrete)
+ fix x assume x: "x \<in> A"
+ have "f x = \<integral>\<^sup>+t. indicator (g -` {x} \<inter> space M) t * f' t \<partial>M" (is "_ = ?I") unfolding f_def
+ by (intro nn_integral_cong) (simp split: split_indicator)
+ also from x have in_sets: "g -` {x} \<inter> space M \<in> sets M"
+ by (intro measurable_sets[OF assms(3)]) simp
+ have "?I = emeasure (density M f') (g -` {x} \<inter> space M)" unfolding f_def
+ by (subst emeasure_density[OF assms(2) in_sets], subst mult.commute) (rule refl)
+ also from assms(3) x have "... = emeasure (distr (density M f') (count_space A) g) {x}"
+ by (subst emeasure_distr) simp_all
+ finally show "f x = emeasure (distr (density M f') (count_space A) g) {x}" .
+qed (insert assms, auto)
+
+lemma bind_cong_AE:
+ assumes "M = N"
+ assumes f: "f \<in> measurable N (subprob_algebra B)"
+ assumes g: "g \<in> measurable N (subprob_algebra B)"
+ assumes ae: "AE x in N. f x = g x"
+ shows "bind M f = bind N g"
+proof cases
+ assume "space N = {}" then show ?thesis
+ using `M = N` by (simp add: bind_empty)
+next
+ assume "space N \<noteq> {}"
+ show ?thesis unfolding `M = N`
+ proof (rule measure_eqI)
+ have *: "sets (N \<bind> f) = sets B"
+ using sets_bind[OF sets_kernel[OF f] `space N \<noteq> {}`] by simp
+ then show "sets (N \<bind> f) = sets (N \<bind> g)"
+ using sets_bind[OF sets_kernel[OF g] `space N \<noteq> {}`] by auto
+ fix A assume "A \<in> sets (N \<bind> f)"
+ then have "A \<in> sets B"
+ unfolding * .
+ with ae f g `space N \<noteq> {}` show "emeasure (N \<bind> f) A = emeasure (N \<bind> g) A"
+ by (subst (1 2) emeasure_bind[where N=B]) (auto intro!: nn_integral_cong_AE)
+ qed
+qed
+
+lemma bind_cong_strong: "M = N \<Longrightarrow> (\<And>x. x\<in>space M =simp=> f x = g x) \<Longrightarrow> bind M f = bind N g"
+ by (auto simp: simp_implies_def intro!: bind_cong)
+
+lemma sets_bind_measurable:
+ assumes f: "f \<in> measurable M (subprob_algebra B)"
+ assumes M: "space M \<noteq> {}"
+ shows "sets (M \<bind> f) = sets B"
+ using M by (intro sets_bind[OF sets_kernel[OF f]]) auto
+
+lemma space_bind_measurable:
+ assumes f: "f \<in> measurable M (subprob_algebra B)"
+ assumes M: "space M \<noteq> {}"
+ shows "space (M \<bind> f) = space B"
+ using M by (intro space_bind[OF sets_kernel[OF f]]) auto
+
end
--- a/src/HOL/Probability/Infinite_Product_Measure.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy Thu Oct 06 11:27:28 2016 +0200
@@ -63,6 +63,21 @@
using emeasure_PiM_emb[of "{}" "\<lambda>_. {}"] by (simp add: *)
qed
+lemma prob_space_PiM:
+ assumes M: "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)" shows "prob_space (PiM I M)"
+proof -
+ let ?M = "\<lambda>i. if i \<in> I then M i else count_space {undefined}"
+ interpret M': prob_space "?M i" for i
+ using M by (cases "i \<in> I") (auto intro!: prob_spaceI)
+ interpret product_prob_space ?M I
+ by unfold_locales
+ have "prob_space (\<Pi>\<^sub>M i\<in>I. ?M i)"
+ by unfold_locales
+ also have "(\<Pi>\<^sub>M i\<in>I. ?M i) = (\<Pi>\<^sub>M i\<in>I. M i)"
+ by (intro PiM_cong) auto
+ finally show ?thesis .
+qed
+
lemma (in product_prob_space) emeasure_PiM_Collect:
assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
shows "emeasure (Pi\<^sub>M I M) {x\<in>space (Pi\<^sub>M I M). \<forall>i\<in>J. x i \<in> X i} = (\<Prod> i\<in>J. emeasure (M i) (X i))"
@@ -123,6 +138,107 @@
apply simp_all
done
+lemma emeasure_PiM_emb:
+ assumes M: "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"
+ assumes J: "J \<subseteq> I" "finite J" and A: "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)"
+ shows "emeasure (Pi\<^sub>M I M) (prod_emb I M J (Pi\<^sub>E J A)) = (\<Prod>i\<in>J. emeasure (M i) (A i))"
+proof -
+ let ?M = "\<lambda>i. if i \<in> I then M i else count_space {undefined}"
+ interpret M': prob_space "?M i" for i
+ using M by (cases "i \<in> I") (auto intro!: prob_spaceI)
+ interpret P: product_prob_space ?M I
+ by unfold_locales
+ have "emeasure (Pi\<^sub>M I M) (prod_emb I M J (Pi\<^sub>E J A)) = emeasure (Pi\<^sub>M I ?M) (P.emb I J (Pi\<^sub>E J A))"
+ by (auto simp: prod_emb_def PiE_iff intro!: arg_cong2[where f=emeasure] PiM_cong)
+ also have "\<dots> = (\<Prod>i\<in>J. emeasure (M i) (A i))"
+ using J A by (subst P.emeasure_PiM_emb[OF J]) (auto intro!: setprod.cong)
+ finally show ?thesis .
+qed
+
+lemma distr_pair_PiM_eq_PiM:
+ fixes i' :: "'i" and I :: "'i set" and M :: "'i \<Rightarrow> 'a measure"
+ assumes M: "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)" "prob_space (M i')"
+ shows "distr (M i' \<Otimes>\<^sub>M (\<Pi>\<^sub>M i\<in>I. M i)) (\<Pi>\<^sub>M i\<in>insert i' I. M i) (\<lambda>(x, X). X(i' := x)) =
+ (\<Pi>\<^sub>M i\<in>insert i' I. M i)" (is "?L = _")
+proof (rule measure_eqI_PiM_infinite[symmetric, OF refl])
+ interpret M': prob_space "M i'" by fact
+ interpret I: prob_space "(\<Pi>\<^sub>M i\<in>I. M i)"
+ using M by (intro prob_space_PiM) auto
+ interpret I': prob_space "(\<Pi>\<^sub>M i\<in>insert i' I. M i)"
+ using M by (intro prob_space_PiM) auto
+ show "finite_measure (\<Pi>\<^sub>M i\<in>insert i' I. M i)"
+ by unfold_locales
+ fix J A assume J: "finite J" "J \<subseteq> insert i' I" and A: "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)"
+ let ?X = "prod_emb (insert i' I) M J (Pi\<^sub>E J A)"
+ have "Pi\<^sub>M (insert i' I) M ?X = (\<Prod>i\<in>J. M i (A i))"
+ using M J A by (intro emeasure_PiM_emb) auto
+ also have "\<dots> = M i' (if i' \<in> J then (A i') else space (M i')) * (\<Prod>i\<in>J-{i'}. M i (A i))"
+ using setprod.insert_remove[of J "\<lambda>i. M i (A i)" i'] J M'.emeasure_space_1
+ by (cases "i' \<in> J") (auto simp: insert_absorb)
+ also have "(\<Prod>i\<in>J-{i'}. M i (A i)) = Pi\<^sub>M I M (prod_emb I M (J - {i'}) (Pi\<^sub>E (J - {i'}) A))"
+ using M J A by (intro emeasure_PiM_emb[symmetric]) auto
+ also have "M i' (if i' \<in> J then (A i') else space (M i')) * \<dots> =
+ (M i' \<Otimes>\<^sub>M Pi\<^sub>M I M) ((if i' \<in> J then (A i') else space (M i')) \<times> prod_emb I M (J - {i'}) (Pi\<^sub>E (J - {i'}) A))"
+ using J A by (intro I.emeasure_pair_measure_Times[symmetric] sets_PiM_I) auto
+ also have "((if i' \<in> J then (A i') else space (M i')) \<times> prod_emb I M (J - {i'}) (Pi\<^sub>E (J - {i'}) A)) =
+ (\<lambda>(x, X). X(i' := x)) -` ?X \<inter> space (M i' \<Otimes>\<^sub>M Pi\<^sub>M I M)"
+ using A[of i', THEN sets.sets_into_space] unfolding set_eq_iff
+ by (simp add: prod_emb_def space_pair_measure space_PiM PiE_fun_upd ac_simps cong: conj_cong)
+ (auto simp add: Pi_iff Ball_def all_conj_distrib)
+ finally show "Pi\<^sub>M (insert i' I) M ?X = ?L ?X"
+ using J A by (simp add: emeasure_distr)
+qed simp
+
+lemma distr_PiM_reindex:
+ assumes M: "\<And>i. i \<in> K \<Longrightarrow> prob_space (M i)"
+ assumes f: "inj_on f I" "f \<in> I \<rightarrow> K"
+ shows "distr (Pi\<^sub>M K M) (\<Pi>\<^sub>M i\<in>I. M (f i)) (\<lambda>\<omega>. \<lambda>n\<in>I. \<omega> (f n)) = (\<Pi>\<^sub>M i\<in>I. M (f i))"
+ (is "distr ?K ?I ?t = ?I")
+proof (rule measure_eqI_PiM_infinite[symmetric, OF refl])
+ interpret prob_space ?I
+ using f M by (intro prob_space_PiM) auto
+ show "finite_measure ?I"
+ by unfold_locales
+ fix A J assume J: "finite J" "J \<subseteq> I" and A: "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M (f i))"
+ have [simp]: "i \<in> J \<Longrightarrow> the_inv_into I f (f i) = i" for i
+ using J f by (intro the_inv_into_f_f) auto
+ have "?I (prod_emb I (\<lambda>i. M (f i)) J (Pi\<^sub>E J A)) = (\<Prod>j\<in>J. M (f j) (A j))"
+ using f J A by (intro emeasure_PiM_emb M) auto
+ also have "\<dots> = (\<Prod>j\<in>f`J. M j (A (the_inv_into I f j)))"
+ using f J by (subst setprod.reindex) (auto intro!: setprod.cong intro: inj_on_subset simp: the_inv_into_f_f)
+ also have "\<dots> = ?K (prod_emb K M (f`J) (\<Pi>\<^sub>E j\<in>f`J. A (the_inv_into I f j)))"
+ using f J A by (intro emeasure_PiM_emb[symmetric] M) (auto simp: the_inv_into_f_f)
+ also have "prod_emb K M (f`J) (\<Pi>\<^sub>E j\<in>f`J. A (the_inv_into I f j)) = ?t -` prod_emb I (\<lambda>i. M (f i)) J (Pi\<^sub>E J A) \<inter> space ?K"
+ using f J A by (auto simp: prod_emb_def space_PiM Pi_iff PiE_iff Int_absorb1)
+ also have "?K \<dots> = distr ?K ?I ?t (prod_emb I (\<lambda>i. M (f i)) J (Pi\<^sub>E J A))"
+ using f J A by (intro emeasure_distr[symmetric] sets_PiM_I) (auto simp: Pi_iff)
+ finally show "?I (prod_emb I (\<lambda>i. M (f i)) J (Pi\<^sub>E J A)) = distr ?K ?I ?t (prod_emb I (\<lambda>i. M (f i)) J (Pi\<^sub>E J A))" .
+qed simp
+
+lemma distr_PiM_component:
+ assumes M: "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"
+ assumes "i \<in> I"
+ shows "distr (Pi\<^sub>M I M) (M i) (\<lambda>\<omega>. \<omega> i) = M i"
+proof -
+ have *: "(\<lambda>\<omega>. \<omega> i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E i'\<in>{i}. A)" for A
+ by (auto simp: prod_emb_def space_PiM)
+ show ?thesis
+ apply (intro measure_eqI)
+ apply (auto simp add: emeasure_distr \<open>i\<in>I\<close> * emeasure_PiM_emb M)
+ apply (subst emeasure_PiM_emb)
+ apply (simp_all add: M \<open>i\<in>I\<close>)
+ done
+qed
+
+lemma AE_PiM_component:
+ "(\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)) \<Longrightarrow> i \<in> I \<Longrightarrow> AE x in M i. P x \<Longrightarrow> AE x in PiM I M. P (x i)"
+ using AE_distrD[of "\<lambda>x. x i" "PiM I M" "M i"]
+ by (subst (asm) distr_PiM_component[of I _ i]) (auto intro: AE_distrD[of "\<lambda>x. x i" _ _ P])
+
+lemma decseq_emb_PiE:
+ "incseq J \<Longrightarrow> decseq (\<lambda>i. prod_emb I M (J i) (\<Pi>\<^sub>E j\<in>J i. X j))"
+ by (fastforce simp: decseq_def prod_emb_def incseq_def Pi_iff)
+
subsection \<open>Sequence space\<close>
definition comb_seq :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a)" where
--- a/src/HOL/Probability/Information.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Information.thy Thu Oct 06 11:27:28 2016 +0200
@@ -389,10 +389,6 @@
done
qed
-lemma integrable_cong_AE_imp:
- "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
- using integrable_cong_AE[of f M g] by (auto simp: eq_commute)
-
lemma (in information_space) finite_entropy_integrable:
"finite_entropy S X Px \<Longrightarrow> integrable S (\<lambda>x. Px x * log b (Px x))"
unfolding finite_entropy_def by auto
--- a/src/HOL/Probability/Probability_Mass_Function.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Probability_Mass_Function.thy Thu Oct 06 11:27:28 2016 +0200
@@ -246,7 +246,7 @@
shows "finite (set_pmf M) \<Longrightarrow> integrable M f"
by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top)
-lemma integral_measure_pmf:
+lemma integral_measure_pmf_real:
assumes [simp]: "finite A" and "\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A"
shows "(\<integral>x. f x \<partial>measure_pmf M) = (\<Sum>a\<in>A. f a * pmf M a)"
proof -
@@ -572,9 +572,9 @@
lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
unfolding pair_pmf_def pmf_bind pmf_return
- apply (subst integral_measure_pmf[where A="{b}"])
+ apply (subst integral_measure_pmf_real[where A="{b}"])
apply (auto simp: indicator_eq_0_iff)
- apply (subst integral_measure_pmf[where A="{a}"])
+ apply (subst integral_measure_pmf_real[where A="{a}"])
apply (auto simp: indicator_eq_0_iff setsum_nonneg_eq_0_iff pmf_nonneg)
done
@@ -658,7 +658,10 @@
done
lemma pmf_map_outside: "x \<notin> f ` set_pmf M \<Longrightarrow> pmf (map_pmf f M) x = 0"
-unfolding pmf_eq_0_set_pmf by simp
+ unfolding pmf_eq_0_set_pmf by simp
+
+lemma measurable_set_pmf[measurable]: "Measurable.pred (count_space UNIV) (\<lambda>x. x \<in> set_pmf M)"
+ by simp
subsection \<open> PMFs as function \<close>
@@ -742,6 +745,107 @@
lemma nn_integral_measure_pmf: "(\<integral>\<^sup>+ x. f x \<partial>measure_pmf p) = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg)
+lemma integral_measure_pmf:
+ fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+ assumes A: "finite A"
+ shows "(\<And>a. a \<in> set_pmf M \<Longrightarrow> f a \<noteq> 0 \<Longrightarrow> a \<in> A) \<Longrightarrow> (LINT x|M. f x) = (\<Sum>a\<in>A. pmf M a *\<^sub>R f a)"
+ unfolding measure_pmf_eq_density
+ apply (simp add: integral_density)
+ apply (subst lebesgue_integral_count_space_finite_support)
+ apply (auto intro!: finite_subset[OF _ \<open>finite A\<close>] setsum.mono_neutral_left simp: pmf_eq_0_set_pmf)
+ done
+
+lemma continuous_on_LINT_pmf: -- \<open>This is dominated convergence!?\<close>
+ fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::{banach, second_countable_topology}"
+ assumes f: "\<And>i. i \<in> set_pmf M \<Longrightarrow> continuous_on A (f i)"
+ and bnd: "\<And>a i. a \<in> A \<Longrightarrow> i \<in> set_pmf M \<Longrightarrow> norm (f i a) \<le> B"
+ shows "continuous_on A (\<lambda>a. LINT i|M. f i a)"
+proof cases
+ assume "finite M" with f show ?thesis
+ using integral_measure_pmf[OF \<open>finite M\<close>]
+ by (subst integral_measure_pmf[OF \<open>finite M\<close>])
+ (auto intro!: continuous_on_setsum continuous_on_scaleR continuous_on_const)
+next
+ assume "infinite M"
+ let ?f = "\<lambda>i x. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) x"
+
+ show ?thesis
+ proof (rule uniform_limit_theorem)
+ show "\<forall>\<^sub>F n in sequentially. continuous_on A (\<lambda>a. \<Sum>i<n. ?f i a)"
+ by (intro always_eventually allI continuous_on_setsum continuous_on_scaleR continuous_on_const f
+ from_nat_into set_pmf_not_empty)
+ show "uniform_limit A (\<lambda>n a. \<Sum>i<n. ?f i a) (\<lambda>a. LINT i|M. f i a) sequentially"
+ proof (subst uniform_limit_cong[where g="\<lambda>n a. \<Sum>i<n. ?f i a"])
+ fix a assume "a \<in> A"
+ have 1: "(LINT i|M. f i a) = (LINT i|map_pmf (to_nat_on M) M. f (from_nat_into M i) a)"
+ by (auto intro!: integral_cong_AE AE_pmfI)
+ have 2: "\<dots> = (LINT i|count_space UNIV. pmf (map_pmf (to_nat_on M) M) i *\<^sub>R f (from_nat_into M i) a)"
+ by (simp add: measure_pmf_eq_density integral_density)
+ have "(\<lambda>n. ?f n a) sums (LINT i|M. f i a)"
+ unfolding 1 2
+ proof (intro sums_integral_count_space_nat)
+ have A: "integrable M (\<lambda>i. f i a)"
+ using \<open>a\<in>A\<close> by (auto intro!: measure_pmf.integrable_const_bound AE_pmfI bnd)
+ have "integrable (map_pmf (to_nat_on M) M) (\<lambda>i. f (from_nat_into M i) a)"
+ by (auto simp add: map_pmf_rep_eq integrable_distr_eq intro!: AE_pmfI integrable_cong_AE_imp[OF A])
+ then show "integrable (count_space UNIV) (\<lambda>n. ?f n a)"
+ by (simp add: measure_pmf_eq_density integrable_density)
+ qed
+ then show "(LINT i|M. f i a) = (\<Sum> n. ?f n a)"
+ by (simp add: sums_unique)
+ next
+ show "uniform_limit A (\<lambda>n a. \<Sum>i<n. ?f i a) (\<lambda>a. (\<Sum> n. ?f n a)) sequentially"
+ proof (rule weierstrass_m_test)
+ fix n a assume "a\<in>A"
+ then show "norm (?f n a) \<le> pmf (map_pmf (to_nat_on M) M) n * B"
+ using bnd by (auto intro!: mult_mono simp: from_nat_into set_pmf_not_empty)
+ next
+ have "integrable (map_pmf (to_nat_on M) M) (\<lambda>n. B)"
+ by auto
+ then show "summable (\<lambda>n. pmf (map_pmf (to_nat_on (set_pmf M)) M) n * B)"
+ by (simp add: measure_pmf_eq_density integrable_density integrable_count_space_nat_iff summable_rabs_cancel)
+ qed
+ qed simp
+ qed simp
+qed
+
+lemma continuous_on_LBINT:
+ fixes f :: "real \<Rightarrow> real"
+ assumes f: "\<And>b. a \<le> b \<Longrightarrow> set_integrable lborel {a..b} f"
+ shows "continuous_on UNIV (\<lambda>b. LBINT x:{a..b}. f x)"
+proof (subst set_borel_integral_eq_integral)
+ { fix b :: real assume "a \<le> b"
+ from f[OF this] have "continuous_on {a..b} (\<lambda>b. integral {a..b} f)"
+ by (intro indefinite_integral_continuous set_borel_integral_eq_integral) }
+ note * = this
+
+ have "continuous_on (\<Union>b\<in>{a..}. {a <..< b}) (\<lambda>b. integral {a..b} f)"
+ proof (intro continuous_on_open_UN)
+ show "b \<in> {a..} \<Longrightarrow> continuous_on {a<..<b} (\<lambda>b. integral {a..b} f)" for b
+ using *[of b] by (rule continuous_on_subset) auto
+ qed simp
+ also have "(\<Union>b\<in>{a..}. {a <..< b}) = {a <..}"
+ by (auto simp: lt_ex gt_ex less_imp_le) (simp add: Bex_def less_imp_le gt_ex cong: rev_conj_cong)
+ finally have "continuous_on {a+1 ..} (\<lambda>b. integral {a..b} f)"
+ by (rule continuous_on_subset) auto
+ moreover have "continuous_on {a..a+1} (\<lambda>b. integral {a..b} f)"
+ by (rule *) simp
+ moreover
+ have "x \<le> a \<Longrightarrow> {a..x} = (if a = x then {a} else {})" for x
+ by auto
+ then have "continuous_on {..a} (\<lambda>b. integral {a..b} f)"
+ by (subst continuous_on_cong[OF refl, where g="\<lambda>x. 0"]) (auto intro!: continuous_on_const)
+ ultimately have "continuous_on ({..a} \<union> {a..a+1} \<union> {a+1 ..}) (\<lambda>b. integral {a..b} f)"
+ by (intro continuous_on_closed_Un) auto
+ also have "{..a} \<union> {a..a+1} \<union> {a+1 ..} = UNIV"
+ by auto
+ finally show "continuous_on UNIV (\<lambda>b. integral {a..b} f)"
+ by auto
+next
+ show "set_integrable lborel {a..b} f" for b
+ using f by (cases "a \<le> b") auto
+qed
+
locale pmf_as_function
begin
--- a/src/HOL/Probability/Probability_Measure.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Probability_Measure.thy Thu Oct 06 11:27:28 2016 +0200
@@ -508,8 +508,6 @@
"distributed M N X f \<longleftrightarrow>
distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> measurable M N"
-term distributed
-
lemma
assumes "distributed M N X f"
shows distributed_distr_eq_density: "distr M N X = density N f"
--- a/src/HOL/Probability/Stream_Space.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Probability/Stream_Space.thy Thu Oct 06 11:27:28 2016 +0200
@@ -109,6 +109,10 @@
shows "(\<lambda>x. stake n (f x) @- g x) \<in> measurable N (stream_space M)"
using f by (induction n arbitrary: f) simp_all
+lemma measurable_case_stream_replace[measurable (raw)]:
+ "(\<lambda>x. f x (shd (g x)) (stl (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. case_stream (f x) (g x)) \<in> measurable M N"
+ unfolding stream.case_eq_if .
+
lemma measurable_ev_at[measurable]:
assumes [measurable]: "Measurable.pred (stream_space M) P"
shows "Measurable.pred (stream_space M) (ev_at P n)"
@@ -442,4 +446,212 @@
by (cases "xs = []") (auto simp: * space_stream_space del: in_listsD)
qed (auto simp: * ae sets_M del: in_listsD intro!: streams_sets)
+primrec scylinder :: "'a set \<Rightarrow> 'a set list \<Rightarrow> 'a stream set"
+where
+ "scylinder S [] = streams S"
+| "scylinder S (A # As) = {\<omega>\<in>streams S. shd \<omega> \<in> A \<and> stl \<omega> \<in> scylinder S As}"
+
+lemma scylinder_streams: "scylinder S xs \<subseteq> streams S"
+ by (induction xs) auto
+
+lemma sets_scylinder: "(\<forall>x\<in>set xs. x \<in> sets S) \<Longrightarrow> scylinder (space S) xs \<in> sets (stream_space S)"
+ by (induction xs) (auto simp: space_stream_space[symmetric])
+
+lemma stream_space_eq_scylinder:
+ assumes P: "prob_space M" "prob_space N"
+ assumes "Int_stable G" and S: "sets S = sets (sigma (space S) G)"
+ and C: "countable C" "C \<subseteq> G" "\<Union>C = space S" and G: "G \<subseteq> Pow (space S)"
+ assumes sets_M: "sets M = sets (stream_space S)"
+ assumes sets_N: "sets N = sets (stream_space S)"
+ assumes *: "\<And>xs. xs \<noteq> [] \<Longrightarrow> xs \<in> lists G \<Longrightarrow> emeasure M (scylinder (space S) xs) = emeasure N (scylinder (space S) xs)"
+ shows "M = N"
+proof (rule measure_eqI_generator_eq)
+ interpret M: prob_space M by fact
+ interpret N: prob_space N by fact
+
+ let ?G = "scylinder (space S) ` lists G"
+ show sc_Pow: "?G \<subseteq> Pow (streams (space S))"
+ using scylinder_streams by auto
+
+ have "sets (stream_space S) = sets (sigma (streams (space S)) ?G)"
+ (is "?S = sets ?R")
+ proof (rule antisym)
+ let ?V = "\<lambda>i. vimage_algebra (streams (space S)) (\<lambda>s. s !! i) S"
+ show "?S \<subseteq> sets ?R"
+ unfolding sets_stream_space_eq
+ proof (safe intro!: sets_Sup_in_sets del: subsetI equalityI)
+ fix i :: nat
+ show "space (?V i) = space ?R"
+ using scylinder_streams by (subst space_measure_of) (auto simp: )
+ { fix A assume "A \<in> G"
+ then have "scylinder (space S) (replicate i (space S) @ [A]) = (\<lambda>s. s !! i) -` A \<inter> streams (space S)"
+ by (induction i) (auto simp add: streams_shd streams_stl cong: conj_cong)
+ also have "scylinder (space S) (replicate i (space S) @ [A]) = (\<Union>xs\<in>{xs\<in>lists C. length xs = i}. scylinder (space S) (xs @ [A]))"
+ apply (induction i)
+ apply auto []
+ apply (simp add: length_Suc_conv set_eq_iff ex_simps(1,2)[symmetric] cong: conj_cong del: ex_simps(1,2))
+ apply rule
+ subgoal for i x
+ apply (cases x)
+ apply (subst (2) C(3)[symmetric])
+ apply (simp del: ex_simps(1,2) add: ex_simps(1,2)[symmetric] ac_simps Bex_def)
+ apply auto
+ done
+ done
+ finally have "(\<lambda>s. s !! i) -` A \<inter> streams (space S) = (\<Union>xs\<in>{xs\<in>lists C. length xs = i}. scylinder (space S) (xs @ [A]))"
+ ..
+ also have "\<dots> \<in> ?R"
+ using C(2) \<open>A\<in>G\<close>
+ by (intro sets.countable_UN' countable_Collect countable_lists C)
+ (auto intro!: in_measure_of[OF sc_Pow] imageI)
+ finally have "(\<lambda>s. s !! i) -` A \<inter> streams (space S) \<in> ?R" . }
+ then show "sets (?V i) \<subseteq> ?R"
+ apply (subst vimage_algebra_cong[OF refl refl S])
+ apply (subst vimage_algebra_sigma[OF G])
+ apply (simp add: streams_iff_snth) []
+ apply (subst sigma_le_sets)
+ apply auto
+ done
+ qed
+ have "G \<subseteq> sets S"
+ unfolding S using G by auto
+ with C(2) show "sets ?R \<subseteq> ?S"
+ unfolding sigma_le_sets[OF sc_Pow] by (auto intro!: sets_scylinder)
+ qed
+ then show "sets M = sigma_sets (streams (space S)) (scylinder (space S) ` lists G)"
+ "sets N = sigma_sets (streams (space S)) (scylinder (space S) ` lists G)"
+ unfolding sets_M sets_N by (simp_all add: sc_Pow)
+
+ show "Int_stable ?G"
+ proof (rule Int_stableI_image)
+ fix xs ys assume "xs \<in> lists G" "ys \<in> lists G"
+ then show "\<exists>zs\<in>lists G. scylinder (space S) xs \<inter> scylinder (space S) ys = scylinder (space S) zs"
+ proof (induction xs arbitrary: ys)
+ case Nil then show ?case
+ by (auto simp add: Int_absorb1 scylinder_streams)
+ next
+ case xs: (Cons x xs)
+ show ?case
+ proof (cases ys)
+ case Nil with xs.hyps show ?thesis
+ by (auto simp add: Int_absorb2 scylinder_streams intro!: bexI[of _ "x#xs"])
+ next
+ case ys: (Cons y ys')
+ with xs.IH[of ys'] xs.prems obtain zs where
+ "zs \<in> lists G" and eq: "scylinder (space S) xs \<inter> scylinder (space S) ys' = scylinder (space S) zs"
+ by auto
+ show ?thesis
+ proof (intro bexI[of _ "(x \<inter> y)#zs"])
+ show "x \<inter> y # zs \<in> lists G"
+ using \<open>zs\<in>lists G\<close> \<open>x\<in>G\<close> \<open>ys\<in>lists G\<close> ys \<open>Int_stable G\<close>[THEN Int_stableD, of x y] by auto
+ show "scylinder (space S) (x # xs) \<inter> scylinder (space S) ys = scylinder (space S) (x \<inter> y # zs)"
+ by (auto simp add: eq[symmetric] ys)
+ qed
+ qed
+ qed
+ qed
+
+ show "range (\<lambda>_::nat. streams (space S)) \<subseteq> scylinder (space S) ` lists G"
+ "(\<Union>i. streams (space S)) = streams (space S)"
+ "emeasure M (streams (space S)) \<noteq> \<infinity>"
+ by (auto intro!: image_eqI[of _ _ "[]"])
+
+ fix X assume "X \<in> scylinder (space S) ` lists G"
+ then obtain xs where xs: "xs \<in> lists G" and eq: "X = scylinder (space S) xs"
+ by auto
+ then show "emeasure M X = emeasure N X"
+ proof (cases "xs = []")
+ assume "xs = []" then show ?thesis
+ unfolding eq
+ using sets_M[THEN sets_eq_imp_space_eq] sets_N[THEN sets_eq_imp_space_eq]
+ M.emeasure_space_1 N.emeasure_space_1
+ by (simp add: space_stream_space[symmetric])
+ next
+ assume "xs \<noteq> []" with xs show ?thesis
+ unfolding eq by (intro *)
+ qed
+qed
+
+lemma stream_space_coinduct:
+ fixes R :: "'a stream measure \<Rightarrow> 'a stream measure \<Rightarrow> bool"
+ assumes "R A B"
+ assumes R: "\<And>A B. R A B \<Longrightarrow> \<exists>K\<in>space (prob_algebra M).
+ \<exists>A'\<in>M \<rightarrow>\<^sub>M prob_algebra (stream_space M). \<exists>B'\<in>M \<rightarrow>\<^sub>M prob_algebra (stream_space M).
+ (AE y in K. R (A' y) (B' y) \<or> A' y = B' y) \<and>
+ A = do { y \<leftarrow> K; \<omega> \<leftarrow> A' y; return (stream_space M) (y ## \<omega>) } \<and>
+ B = do { y \<leftarrow> K; \<omega> \<leftarrow> B' y; return (stream_space M) (y ## \<omega>) }"
+ shows "A = B"
+proof (rule stream_space_eq_scylinder)
+ let ?step = "\<lambda>K L. do { y \<leftarrow> K; \<omega> \<leftarrow> L y; return (stream_space M) (y ## \<omega>) }"
+ { fix K A A' assume K: "K \<in> space (prob_algebra M)"
+ and A'[measurable]: "A' \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)" and A_eq: "A = ?step K A'"
+ have ps: "prob_space A"
+ unfolding A_eq by (rule prob_space_bind'[OF K]) measurable
+ have "sets A = sets (stream_space M)"
+ unfolding A_eq by (rule sets_bind'[OF K]) measurable
+ note ps this }
+ note ** = this
+
+ { fix A B assume "R A B"
+ obtain K A' B' where K: "K \<in> space (prob_algebra M)"
+ and A': "A' \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)" "A = ?step K A'"
+ and B': "B' \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)" "B = ?step K B'"
+ using R[OF \<open>R A B\<close>] by blast
+ have "prob_space A" "prob_space B" "sets A = sets (stream_space M)" "sets B = sets (stream_space M)"
+ using **[OF K A'] **[OF K B'] by auto }
+ note R_D = this
+
+ show "prob_space A" "prob_space B" "sets A = sets (stream_space M)" "sets B = sets (stream_space M)"
+ using R_D[OF \<open>R A B\<close>] by auto
+
+ show "Int_stable (sets M)" "sets M = sets (sigma (space M) (sets M))" "countable {space M}"
+ "{space M} \<subseteq> sets M" "\<Union>{space M} = space M" "sets M \<subseteq> Pow (space M)"
+ using sets.space_closed[of M] by (auto simp: Int_stable_def)
+
+ { fix A As L K assume K[measurable]: "K \<in> space (prob_algebra M)" and A: "A \<in> sets M" "As \<in> lists (sets M)"
+ and L[measurable]: "L \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)"
+ from A have [measurable]: "\<forall>x\<in>set (A # As). x \<in> sets M" "\<forall>x\<in>set As. x \<in> sets M"
+ by auto
+ have [simp]: "space K = space M" "sets K = sets M"
+ using K by (auto simp: space_prob_algebra intro!: sets_eq_imp_space_eq)
+ have [simp]: "x \<in> space M \<Longrightarrow> sets (L x) = sets (stream_space M)" for x
+ using measurable_space[OF L] by (auto simp: space_prob_algebra)
+ note sets_scylinder[measurable]
+ have *: "indicator (scylinder (space M) (A # As)) (x ## \<omega>) =
+ (indicator A x * indicator (scylinder (space M) As) \<omega> :: ennreal)" for \<omega> x
+ using scylinder_streams[of "space M" As] \<open>A \<in> sets M\<close>[THEN sets.sets_into_space]
+ by (auto split: split_indicator)
+ have "emeasure (?step K L) (scylinder (space M) (A # As)) = (\<integral>\<^sup>+y. L y (scylinder (space M) As) * indicator A y \<partial>K)"
+ apply (subst emeasure_bind_prob_algebra[OF K])
+ apply measurable
+ apply (rule nn_integral_cong)
+ apply (subst emeasure_bind_prob_algebra[OF L[THEN measurable_space]])
+ apply (simp_all add: ac_simps * nn_integral_cmult_indicator del: scylinder.simps)
+ apply measurable
+ done }
+ note emeasure_step = this
+
+ fix Xs assume "Xs \<in> lists (sets M)"
+ from this \<open>R A B\<close> show "emeasure A (scylinder (space M) Xs) = emeasure B (scylinder (space M) Xs)"
+ proof (induction Xs arbitrary: A B)
+ case (Cons X Xs)
+ obtain K A' B' where K: "K \<in> space (prob_algebra M)"
+ and A'[measurable]: "A' \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)" and A: "A = ?step K A'"
+ and B'[measurable]: "B' \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)" and B: "B = ?step K B'"
+ and AE_R: "AE x in K. R (A' x) (B' x) \<or> A' x = B' x"
+ using R[OF \<open>R A B\<close>] by blast
+
+ show ?case
+ unfolding A B emeasure_step[OF K Cons.hyps A'] emeasure_step[OF K Cons.hyps B']
+ apply (rule nn_integral_cong_AE)
+ using AE_R by eventually_elim (auto simp add: Cons.IH)
+ next
+ case Nil
+ note R_D[OF this]
+ from this(1,2)[THEN prob_space.emeasure_space_1] this(3,4)[THEN sets_eq_imp_space_eq]
+ show ?case
+ by (simp add: space_stream_space)
+ qed
+qed
+
end
--- a/src/HOL/ROOT Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/ROOT Thu Oct 06 11:27:28 2016 +0200
@@ -627,6 +627,7 @@
Code_Timing
Perm_Fragments
Argo_Examples
+ Word_Type
theories [skip_proofs = false]
Meson_Test
document_files "root.bib" "root.tex"
--- a/src/HOL/Topological_Spaces.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Topological_Spaces.thy Thu Oct 06 11:27:28 2016 +0200
@@ -699,7 +699,7 @@
assumes "filterlim f (nhds c) F"
assumes "eventually (\<lambda>x. f x \<in> A - {c}) F"
shows "filterlim f (at c within A) F"
- using assms by (simp add: filterlim_at)
+ using assms by (simp add: filterlim_at)
lemma filterlim_atI:
assumes "filterlim f (nhds c) F"
@@ -1644,6 +1644,10 @@
unfolding continuous_on_def
by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
+lemma continuous_on_strong_cong:
+ "s = t \<Longrightarrow> (\<And>x. x \<in> t =simp=> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
+ unfolding simp_implies_def by (rule continuous_on_cong)
+
lemma continuous_on_topological:
"continuous_on s f \<longleftrightarrow>
(\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
--- a/src/HOL/Transfer.thy Thu Oct 06 11:27:03 2016 +0200
+++ b/src/HOL/Transfer.thy Thu Oct 06 11:27:28 2016 +0200
@@ -602,4 +602,14 @@
end
+
+subsection \<open>@{const of_nat}\<close>
+
+lemma transfer_rule_of_nat:
+ fixes R :: "'a::semiring_1 \<Rightarrow> 'b::semiring_1 \<Rightarrow> bool"
+ assumes [transfer_rule]: "R 0 0" "R 1 1"
+ "rel_fun R (rel_fun R R) plus plus"
+ shows "rel_fun HOL.eq R of_nat of_nat"
+ by (unfold of_nat_def [abs_def]) transfer_prover
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Word_Type.thy Thu Oct 06 11:27:28 2016 +0200
@@ -0,0 +1,354 @@
+(* Author: Florian Haftmann, TUM
+*)
+
+section \<open>Proof of concept for algebraically founded bit word types\<close>
+
+theory Word_Type
+ imports
+ Main
+ "~~/src/HOL/Library/Type_Length"
+begin
+
+subsection \<open>Compact syntax for types with a length\<close>
+
+syntax "_type_length" :: "type \<Rightarrow> nat" ("(1LENGTH/(1'(_')))")
+
+translations "LENGTH('a)" \<rightharpoonup>
+ "CONST len_of (CONST Pure.type :: 'a itself)"
+
+print_translation \<open>
+ let
+ fun len_of_itself_tr' ctxt [Const (@{const_syntax Pure.type}, Type (_, [T]))] =
+ Syntax.const @{syntax_const "_type_length"} $ Syntax_Phases.term_of_typ ctxt T
+ in [(@{const_syntax len_of}, len_of_itself_tr')] end
+\<close>
+
+
+subsection \<open>Truncating bit representations of numeric types\<close>
+
+class semiring_bits = semiring_div_parity +
+ assumes semiring_bits: "(1 + 2 * a) mod of_nat (2 * n) = 1 + 2 * (a mod of_nat n)"
+
+context semiring_bits
+begin
+
+definition bits :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+ where bits_eq_mod: "bits n a = a mod of_nat (2 ^ n)"
+
+lemma bits_bits [simp]:
+ "bits n (bits n a) = bits n a"
+ by (simp add: bits_eq_mod)
+
+lemma bits_0 [simp]:
+ "bits 0 a = 0"
+ by (simp add: bits_eq_mod)
+
+lemma bits_Suc [simp]:
+ "bits (Suc n) a = bits n (a div 2) * 2 + a mod 2"
+proof -
+ define b and c
+ where "b = a div 2" and "c = a mod 2"
+ then have a: "a = b * 2 + c"
+ and "c = 0 \<or> c = 1"
+ by (simp_all add: mod_div_equality parity)
+ from \<open>c = 0 \<or> c = 1\<close>
+ have "bits (Suc n) (b * 2 + c) = bits n b * 2 + c"
+ proof
+ assume "c = 0"
+ moreover have "(2 * b) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n)"
+ by (simp add: mod_mult_mult1)
+ ultimately show ?thesis
+ by (simp add: bits_eq_mod ac_simps)
+ next
+ assume "c = 1"
+ with semiring_bits [of b "2 ^ n"] show ?thesis
+ by (simp add: bits_eq_mod ac_simps)
+ qed
+ with a show ?thesis
+ by (simp add: b_def c_def)
+qed
+
+lemma bits_of_0 [simp]:
+ "bits n 0 = 0"
+ by (simp add: bits_eq_mod)
+
+lemma bits_plus:
+ "bits n (bits n a + bits n b) = bits n (a + b)"
+ by (simp add: bits_eq_mod mod_add_eq [symmetric])
+
+lemma bits_of_1_eq_0_iff [simp]:
+ "bits n 1 = 0 \<longleftrightarrow> n = 0"
+ by (induct n) simp_all
+
+end
+
+instance nat :: semiring_bits
+ by standard (simp add: mod_Suc Suc_double_not_eq_double)
+
+instance int :: semiring_bits
+ by standard (simp add: pos_zmod_mult_2)
+
+lemma bits_uminus:
+ fixes k :: int
+ shows "bits n (- (bits n k)) = bits n (- k)"
+ by (simp add: bits_eq_mod mod_minus_eq [symmetric])
+
+lemma bits_minus:
+ fixes k l :: int
+ shows "bits n (bits n k - bits n l) = bits n (k - l)"
+ by (simp add: bits_eq_mod mod_diff_eq [symmetric])
+
+lemma bits_nonnegative [simp]:
+ fixes k :: int
+ shows "bits n k \<ge> 0"
+ by (simp add: bits_eq_mod)
+
+definition signed_bits :: "nat \<Rightarrow> int \<Rightarrow> int"
+ where signed_bits_eq_bits:
+ "signed_bits n k = bits (Suc n) (k + 2 ^ n) - 2 ^ n"
+
+lemma signed_bits_eq_bits':
+ assumes "n > 0"
+ shows "signed_bits (n - Suc 0) k = bits n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)"
+ using assms by (simp add: signed_bits_eq_bits)
+
+lemma signed_bits_0 [simp]:
+ "signed_bits 0 k = - (k mod 2)"
+proof (cases "even k")
+ case True
+ then have "odd (k + 1)"
+ by simp
+ then have "(k + 1) mod 2 = 1"
+ by (simp add: even_iff_mod_2_eq_zero)
+ with True show ?thesis
+ by (simp add: signed_bits_eq_bits)
+next
+ case False
+ then show ?thesis
+ by (simp add: signed_bits_eq_bits odd_iff_mod_2_eq_one)
+qed
+
+lemma signed_bits_Suc [simp]:
+ "signed_bits (Suc n) k = signed_bits n (k div 2) * 2 + k mod 2"
+ using zero_not_eq_two by (simp add: signed_bits_eq_bits algebra_simps)
+
+lemma signed_bits_of_0 [simp]:
+ "signed_bits n 0 = 0"
+ by (simp add: signed_bits_eq_bits bits_eq_mod)
+
+lemma signed_bits_of_minus_1 [simp]:
+ "signed_bits n (- 1) = - 1"
+ by (induct n) simp_all
+
+lemma signed_bits_eq_iff_bits_eq:
+ assumes "n > 0"
+ shows "signed_bits (n - Suc 0) k = signed_bits (n - Suc 0) l \<longleftrightarrow> bits n k = bits n l" (is "?P \<longleftrightarrow> ?Q")
+proof -
+ from assms obtain m where m: "n = Suc m"
+ by (cases n) auto
+ show ?thesis
+ proof
+ assume ?Q
+ have "bits (Suc m) (k + 2 ^ m) =
+ bits (Suc m) (bits (Suc m) k + bits (Suc m) (2 ^ m))"
+ by (simp only: bits_plus)
+ also have "\<dots> =
+ bits (Suc m) (bits (Suc m) l + bits (Suc m) (2 ^ m))"
+ by (simp only: \<open>?Q\<close> m [symmetric])
+ also have "\<dots> = bits (Suc m) (l + 2 ^ m)"
+ by (simp only: bits_plus)
+ finally show ?P
+ by (simp only: signed_bits_eq_bits m) simp
+ next
+ assume ?P
+ with assms have "(k + 2 ^ (n - Suc 0)) mod 2 ^ n = (l + 2 ^ (n - Suc 0)) mod 2 ^ n"
+ by (simp add: signed_bits_eq_bits' bits_eq_mod)
+ then have "(i + (k + 2 ^ (n - Suc 0))) mod 2 ^ n = (i + (l + 2 ^ (n - Suc 0))) mod 2 ^ n" for i
+ by (metis mod_add_eq)
+ then have "k mod 2 ^ n = l mod 2 ^ n"
+ by (metis add_diff_cancel_right' uminus_add_conv_diff)
+ then show ?Q
+ by (simp add: bits_eq_mod)
+ qed
+qed
+
+
+subsection \<open>Bit strings as quotient type\<close>
+
+subsubsection \<open>Basic properties\<close>
+
+quotient_type (overloaded) 'a word = int / "\<lambda>k l. bits LENGTH('a) k = bits LENGTH('a::len0) l"
+ by (auto intro!: equivpI reflpI sympI transpI)
+
+instantiation word :: (len0) "{semiring_numeral, comm_semiring_0, comm_ring}"
+begin
+
+lift_definition zero_word :: "'a word"
+ is 0
+ .
+
+lift_definition one_word :: "'a word"
+ is 1
+ .
+
+lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
+ is plus
+ by (subst bits_plus [symmetric]) (simp add: bits_plus)
+
+lift_definition uminus_word :: "'a word \<Rightarrow> 'a word"
+ is uminus
+ by (subst bits_uminus [symmetric]) (simp add: bits_uminus)
+
+lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
+ is minus
+ by (subst bits_minus [symmetric]) (simp add: bits_minus)
+
+lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
+ is times
+ by (auto simp add: bits_eq_mod intro: mod_mult_cong)
+
+instance
+ by standard (transfer; simp add: algebra_simps)+
+
+end
+
+instance word :: (len) comm_ring_1
+ by standard (transfer; simp)+
+
+
+subsubsection \<open>Conversions\<close>
+
+lemma [transfer_rule]:
+ "rel_fun HOL.eq pcr_word int of_nat"
+proof -
+ note transfer_rule_of_nat [transfer_rule]
+ show ?thesis by transfer_prover
+qed
+
+lemma [transfer_rule]:
+ "rel_fun HOL.eq pcr_word (\<lambda>k. k) of_int"
+proof -
+ note transfer_rule_of_int [transfer_rule]
+ have "rel_fun HOL.eq pcr_word (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> 'a word)"
+ by transfer_prover
+ then show ?thesis by (simp add: id_def)
+qed
+
+context semiring_1
+begin
+
+lift_definition unsigned :: "'b::len0 word \<Rightarrow> 'a"
+ is "of_nat \<circ> nat \<circ> bits LENGTH('b)"
+ by simp
+
+lemma unsigned_0 [simp]:
+ "unsigned 0 = 0"
+ by transfer simp
+
+end
+
+context semiring_char_0
+begin
+
+lemma word_eq_iff_unsigned:
+ "a = b \<longleftrightarrow> unsigned a = unsigned b"
+ by safe (transfer; simp add: eq_nat_nat_iff)
+
+end
+
+context ring_1
+begin
+
+lift_definition signed :: "'b::len word \<Rightarrow> 'a"
+ is "of_int \<circ> signed_bits (LENGTH('b) - 1)"
+ by (simp add: signed_bits_eq_iff_bits_eq [symmetric])
+
+lemma signed_0 [simp]:
+ "signed 0 = 0"
+ by transfer simp
+
+end
+
+lemma unsigned_of_nat [simp]:
+ "unsigned (of_nat n :: 'a word) = bits LENGTH('a::len) n"
+ by transfer (simp add: nat_eq_iff bits_eq_mod zmod_int)
+
+lemma of_nat_unsigned [simp]:
+ "of_nat (unsigned a) = a"
+ by transfer simp
+
+lemma of_int_unsigned [simp]:
+ "of_int (unsigned a) = a"
+ by transfer simp
+
+context ring_char_0
+begin
+
+lemma word_eq_iff_signed:
+ "a = b \<longleftrightarrow> signed a = signed b"
+ by safe (transfer; auto simp add: signed_bits_eq_iff_bits_eq)
+
+end
+
+lemma signed_of_int [simp]:
+ "signed (of_int k :: 'a word) = signed_bits (LENGTH('a::len) - 1) k"
+ by transfer simp
+
+lemma of_int_signed [simp]:
+ "of_int (signed a) = a"
+ by transfer (simp add: signed_bits_eq_bits bits_eq_mod zdiff_zmod_left)
+
+
+subsubsection \<open>Properties\<close>
+
+
+subsubsection \<open>Division\<close>
+
+instantiation word :: (len0) modulo
+begin
+
+lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
+ is "\<lambda>a b. bits LENGTH('a) a div bits LENGTH('a) b"
+ by simp
+
+lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
+ is "\<lambda>a b. bits LENGTH('a) a mod bits LENGTH('a) b"
+ by simp
+
+instance ..
+
+end
+
+
+subsubsection \<open>Orderings\<close>
+
+instantiation word :: (len0) linorder
+begin
+
+lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
+ is "\<lambda>a b. bits LENGTH('a) a \<le> bits LENGTH('a) b"
+ by simp
+
+lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
+ is "\<lambda>a b. bits LENGTH('a) a < bits LENGTH('a) b"
+ by simp
+
+instance
+ by standard (transfer; auto)+
+
+end
+
+context linordered_semidom
+begin
+
+lemma word_less_eq_iff_unsigned:
+ "a \<le> b \<longleftrightarrow> unsigned a \<le> unsigned b"
+ by (transfer fixing: less_eq) (simp add: nat_le_eq_zle)
+
+lemma word_less_iff_unsigned:
+ "a < b \<longleftrightarrow> unsigned a < unsigned b"
+ by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF bits_nonnegative])
+
+end
+
+end
--- a/src/Tools/solve_direct.ML Thu Oct 06 11:27:03 2016 +0200
+++ b/src/Tools/solve_direct.ML Thu Oct 06 11:27:28 2016 +0200
@@ -14,7 +14,8 @@
val someN: string
val noneN: string
val unknownN: string
- val max_solutions: int Unsynchronized.ref
+ val max_solutions: int Config.T
+ val strict_warnings: bool Config.T
val solve_direct: Proof.state -> bool * (string * string list)
end;
@@ -32,7 +33,8 @@
(* preferences *)
-val max_solutions = Unsynchronized.ref 5;
+val max_solutions = Attrib.setup_config_int @{binding solve_direct_max_solutions} (K 5);
+val strict_warnings = Attrib.setup_config_bool @{binding solve_direct_strict_warnings} (K false);
(* solve_direct command *)
@@ -44,7 +46,8 @@
val crits = [(true, Find_Theorems.Solves)];
fun find g =
- snd (Find_Theorems.find_theorems find_ctxt (SOME g) (SOME (! max_solutions)) false crits);
+ snd (Find_Theorems.find_theorems find_ctxt (SOME g)
+ (SOME (Config.get find_ctxt max_solutions)) false crits);
fun prt_result (goal, results) =
let
@@ -76,8 +79,15 @@
(case try seek_against_goal () of
SOME (SOME results) =>
(someN,
- let val msg = Pretty.string_of (Pretty.chunks (message results))
- in if mode = Auto_Try then [msg] else (writeln msg; []) end)
+ let
+ val msg = Pretty.string_of (Pretty.chunks (message results))
+ in
+ if Config.get ctxt strict_warnings
+ then (warning msg; [])
+ else if mode = Auto_Try
+ then [msg]
+ else (writeln msg; [])
+ end)
| SOME NONE =>
(if mode = Normal then writeln "No proof found"
else ();