--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy Mon Jul 17 12:31:06 2023 +0100
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy Mon Jul 17 21:49:49 2023 +0100
@@ -22,40 +22,39 @@
lemma retract_of_contractible:
assumes "contractible T" "S retract_of T"
- shows "contractible S"
-using assms
-apply (clarsimp simp add: retract_of_def contractible_def retraction_def homotopic_with image_subset_iff_funcset)
-apply (rule_tac x="r a" in exI)
-apply (rule_tac x="r \<circ> h" in exI)
-apply (intro conjI continuous_intros continuous_on_compose)
-apply (erule continuous_on_subset | force)+
-done
+ shows "contractible S"
+ using assms
+ apply (clarsimp simp add: retract_of_def contractible_def retraction_def homotopic_with image_subset_iff_funcset)
+ apply (rule_tac x="r a" in exI)
+ apply (rule_tac x="r \<circ> h" in exI)
+ apply (intro conjI continuous_intros continuous_on_compose)
+ apply (erule continuous_on_subset | force)+
+ done
lemma retract_of_path_connected:
"\<lbrakk>path_connected T; S retract_of T\<rbrakk> \<Longrightarrow> path_connected S"
by (metis path_connected_continuous_image retract_of_def retraction)
lemma retract_of_simply_connected:
- "\<lbrakk>simply_connected T; S retract_of T\<rbrakk> \<Longrightarrow> simply_connected S"
-apply (simp add: retract_of_def retraction_def, clarify)
-apply (rule simply_connected_retraction_gen)
-apply (force elim!: continuous_on_subset)+
-done
+ "\<lbrakk>simply_connected T; S retract_of T\<rbrakk> \<Longrightarrow> simply_connected S"
+ apply (simp add: retract_of_def retraction_def Pi_iff, clarify)
+ apply (rule simply_connected_retraction_gen)
+ apply (force elim!: continuous_on_subset)+
+ done
lemma retract_of_homotopically_trivial:
assumes ts: "T retract_of S"
- and hom: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
- continuous_on U g; g ` U \<subseteq> S\<rbrakk>
+ and hom: "\<And>f g. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S;
+ continuous_on U g; g \<in> U \<rightarrow> S\<rbrakk>
\<Longrightarrow> homotopic_with_canon (\<lambda>x. True) U S f g"
- and "continuous_on U f" "f ` U \<subseteq> T"
- and "continuous_on U g" "g ` U \<subseteq> T"
+ and "continuous_on U f" "f \<in> U \<rightarrow> T"
+ and "continuous_on U g" "g \<in> U \<rightarrow> T"
shows "homotopic_with_canon (\<lambda>x. True) U T f g"
proof -
- obtain r where "r ` S \<subseteq> S" "continuous_on S r" "\<forall>x\<in>S. r (r x) = r x" "T = r ` S"
+ obtain r where "r \<in> S \<rightarrow> S" "continuous_on S r" "\<forall>x\<in>S. r (r x) = r x" "T = r ` S"
using ts by (auto simp: retract_of_def retraction)
then obtain k where "Retracts S r T k"
- unfolding Retracts_def
- by (metis continuous_on_subset dual_order.trans image_iff image_mono)
+ unfolding Retracts_def using continuous_on_id by blast
then show ?thesis
apply (rule Retracts.homotopically_trivial_retraction_gen)
using assms
@@ -65,16 +64,15 @@
lemma retract_of_homotopically_trivial_null:
assumes ts: "T retract_of S"
- and hom: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
+ and hom: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S\<rbrakk>
\<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. c)"
- and "continuous_on U f" "f ` U \<subseteq> T"
+ and "continuous_on U f" "f \<in> U \<rightarrow> T"
obtains c where "homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)"
proof -
- obtain r where "r ` S \<subseteq> S" "continuous_on S r" "\<forall>x\<in>S. r (r x) = r x" "T = r ` S"
+ obtain r where "r \<in> S \<rightarrow> S" "continuous_on S r" "\<forall>x\<in>S. r (r x) = r x" "T = r ` S"
using ts by (auto simp: retract_of_def retraction)
then obtain k where "Retracts S r T k"
- unfolding Retracts_def
- by (metis continuous_on_subset dual_order.trans image_iff image_mono)
+ unfolding Retracts_def by fastforce
then show ?thesis
apply (rule Retracts.homotopically_trivial_retraction_null_gen)
apply (rule TrueI refl assms that | assumption)+
@@ -92,25 +90,24 @@
by (metis locally_compact_closedin closedin_retract)
lemma homotopic_into_retract:
- "\<lbrakk>f ` S \<subseteq> T; g ` S \<subseteq> T; T retract_of U; homotopic_with_canon (\<lambda>x. True) S U f g\<rbrakk>
+ "\<lbrakk>f \<in> S \<rightarrow> T; g \<in> S \<rightarrow> T; T retract_of U; homotopic_with_canon (\<lambda>x. True) S U f g\<rbrakk>
\<Longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g"
-apply (subst (asm) homotopic_with_def)
-apply (simp add: homotopic_with retract_of_def retraction_def, clarify)
-apply (rule_tac x="r \<circ> h" in exI)
-apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
-done
+ apply (subst (asm) homotopic_with_def)
+ apply (simp add: homotopic_with retract_of_def retraction_def Pi_iff, clarify)
+ apply (rule_tac x="r \<circ> h" in exI)
+ by (smt (verit, ccfv_SIG) comp_def continuous_on_compose continuous_on_subset image_subset_iff)
lemma retract_of_locally_connected:
assumes "locally connected T" "S retract_of T"
shows "locally connected S"
using assms
- by (auto simp: idempotent_imp_retraction intro!: retraction_openin_vimage_iff elim!: locally_connected_quotient_image retract_ofE)
+ by (metis Abstract_Topology_2.retraction_openin_vimage_iff idempotent_imp_retraction locally_connected_quotient_image retract_ofE)
lemma retract_of_locally_path_connected:
assumes "locally path_connected T" "S retract_of T"
shows "locally path_connected S"
using assms
- by (auto simp: idempotent_imp_retraction intro!: retraction_openin_vimage_iff elim!: locally_path_connected_quotient_image retract_ofE)
+ by (metis Abstract_Topology_2.retraction_openin_vimage_iff idempotent_imp_retraction locally_path_connected_quotient_image retract_ofE)
text \<open>A few simple lemmas about deformation retracts\<close>
@@ -133,7 +130,7 @@
lemma deformation_retract:
fixes S :: "'a::euclidean_space set"
shows "(\<exists>r. homotopic_with_canon (\<lambda>x. True) S S id r \<and> retraction S T r) \<longleftrightarrow>
- T retract_of S \<and> (\<exists>f. homotopic_with_canon (\<lambda>x. True) S S id f \<and> f ` S \<subseteq> T)"
+ T retract_of S \<and> (\<exists>f. homotopic_with_canon (\<lambda>x. True) S S id f \<and> f \<in> S \<rightarrow> T)"
(is "?lhs = ?rhs")
proof
assume ?lhs
@@ -147,7 +144,7 @@
apply (rule homotopic_with_trans, assumption)
apply (rule_tac f = "r \<circ> f" and g="r \<circ> id" in homotopic_with_eq)
apply (rule_tac Y=S in homotopic_with_compose_continuous_left)
- apply (auto simp: homotopic_with_sym)
+ apply (auto simp: homotopic_with_sym Pi_iff)
done
qed
@@ -161,10 +158,10 @@
moreover have "homotopic_with_canon (\<lambda>x. True) S S id (\<lambda>x. a)"
using assms
by (auto simp: contractible_def homotopic_into_contractible image_subset_iff)
- moreover have "(\<lambda>x. a) ` S \<subseteq> {a}"
+ moreover have "(\<lambda>x. a) \<in> S \<rightarrow> {a}"
by (simp add: image_subsetI)
ultimately show ?thesis
- using that deformation_retract by metis
+ by (metis that deformation_retract)
qed
--- a/src/HOL/Analysis/Homotopy.thy Mon Jul 17 12:31:06 2023 +0100
+++ b/src/HOL/Analysis/Homotopy.thy Mon Jul 17 21:49:49 2023 +0100
@@ -2971,7 +2971,7 @@
assumes S: "subspace S"
and T: "subspace T"
and d: "dim S \<le> dim T"
- obtains f where "linear f" "f ` S \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
+ obtains f where "linear f" "f \<in> S \<rightarrow> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
proof -
obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
@@ -3011,7 +3011,7 @@
by (simp add: norm_eq_sqrt_inner)
qed
then show ?thesis
- by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>])
+ by (meson \<open>f ` S \<subseteq> T\<close> \<open>linear f\<close> image_subset_iff_funcset that)
qed
proposition isometries_subspaces:
@@ -3190,36 +3190,36 @@
assumes conth: "continuous_on S h"
and imh: "h ` S = t"
and contk: "continuous_on t k"
- and imk: "k ` t \<subseteq> S"
+ and imk: "k \<in> t \<rightarrow> S"
and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y"
begin
lemma homotopically_trivial_retraction_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
- and Q: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
+ assumes P: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
+ and Q: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
and Qeq: "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S; P f;
- continuous_on U g; g ` U \<subseteq> S; P g\<rbrakk>
+ and hom: "\<And>f g. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S; P f;
+ continuous_on U g; g \<in> U \<rightarrow> S; P g\<rbrakk>
\<Longrightarrow> homotopic_with_canon P U S f g"
- and contf: "continuous_on U f" and imf: "f ` U \<subseteq> t" and Qf: "Q f"
- and contg: "continuous_on U g" and img: "g ` U \<subseteq> t" and Qg: "Q g"
+ and contf: "continuous_on U f" and imf: "f \<in> U \<rightarrow> t" and Qf: "Q f"
+ and contg: "continuous_on U g" and img: "g \<in> U \<rightarrow> t" and Qg: "Q g"
shows "homotopic_with_canon Q U t f g"
proof -
have "continuous_on U (k \<circ> f)"
- using contf continuous_on_compose continuous_on_subset contk imf by blast
+ by (meson contf continuous_on_compose continuous_on_subset contk funcset_image imf)
moreover have "(k \<circ> f) ` U \<subseteq> S"
using imf imk by fastforce
moreover have "P (k \<circ> f)"
by (simp add: P Qf contf imf)
moreover have "continuous_on U (k \<circ> g)"
- using contg continuous_on_compose continuous_on_subset contk img by blast
+ by (meson contg continuous_on_compose continuous_on_subset contk funcset_image img)
moreover have "(k \<circ> g) ` U \<subseteq> S"
using img imk by fastforce
moreover have "P (k \<circ> g)"
by (simp add: P Qg contg img)
ultimately have "homotopic_with_canon P U S (k \<circ> f) (k \<circ> g)"
- by (rule hom)
+ by (simp add: hom image_subset_iff)
then have "homotopic_with_canon Q U t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
using Q conth imh by force+
@@ -3228,23 +3228,23 @@
show "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
using Qeq topspace_euclidean_subtopology by blast
show "\<And>x. x \<in> U \<Longrightarrow> f x = h (k (f x))" "\<And>x. x \<in> U \<Longrightarrow> g x = h (k (g x))"
- using idhk imf img by auto
+ using idhk imf img by fastforce+
qed
qed
lemma homotopically_trivial_retraction_null_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
- and Q: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
+ assumes P: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
+ and Q: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
and Qeq: "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S; P f\<rbrakk>
+ and hom: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S; P f\<rbrakk>
\<Longrightarrow> \<exists>c. homotopic_with_canon P U S f (\<lambda>x. c)"
- and contf: "continuous_on U f" and imf:"f ` U \<subseteq> t" and Qf: "Q f"
+ and contf: "continuous_on U f" and imf:"f \<in> U \<rightarrow> t" and Qf: "Q f"
obtains c where "homotopic_with_canon Q U t f (\<lambda>x. c)"
proof -
have feq: "\<And>x. x \<in> U \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
have "continuous_on U (k \<circ> f)"
- using contf continuous_on_compose continuous_on_subset contk imf by blast
- moreover have "(k \<circ> f) ` U \<subseteq> S"
+ by (meson contf continuous_on_compose continuous_on_subset contk funcset_image imf)
+ moreover have "(k \<circ> f) \<in> U \<rightarrow> S"
using imf imk by fastforce
moreover have "P (k \<circ> f)"
by (simp add: P Qf contf imf)
@@ -3265,32 +3265,32 @@
qed
lemma cohomotopically_trivial_retraction_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> U; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
- and Q: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
+ assumes P: "\<And>f. \<lbrakk>continuous_on t f; f \<in> t \<rightarrow> U; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
+ and Q: "\<And>f. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> U; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f g. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f;
- continuous_on S g; g ` S \<subseteq> U; P g\<rbrakk>
+ and hom: "\<And>f g. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> U; P f;
+ continuous_on S g; g \<in> S \<rightarrow> U; P g\<rbrakk>
\<Longrightarrow> homotopic_with_canon P S U f g"
- and contf: "continuous_on t f" and imf: "f ` t \<subseteq> U" and Qf: "Q f"
- and contg: "continuous_on t g" and img: "g ` t \<subseteq> U" and Qg: "Q g"
+ and contf: "continuous_on t f" and imf: "f \<in> t \<rightarrow> U" and Qf: "Q f"
+ and contg: "continuous_on t g" and img: "g \<in> t \<rightarrow> U" and Qg: "Q g"
shows "homotopic_with_canon Q t U f g"
proof -
have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
have geq: "\<And>x. x \<in> t \<Longrightarrow> (g \<circ> h \<circ> k) x = g x" using idhk img by auto
have "continuous_on S (f \<circ> h)"
using contf conth continuous_on_compose imh by blast
- moreover have "(f \<circ> h) ` S \<subseteq> U"
+ moreover have "(f \<circ> h) \<in> S \<rightarrow> U"
using imf imh by fastforce
moreover have "P (f \<circ> h)"
by (simp add: P Qf contf imf)
moreover have "continuous_on S (g \<circ> h)"
using contg continuous_on_compose continuous_on_subset conth imh by blast
- moreover have "(g \<circ> h) ` S \<subseteq> U"
+ moreover have "(g \<circ> h) \<in> S \<rightarrow> U"
using img imh by fastforce
moreover have "P (g \<circ> h)"
by (simp add: P Qg contg img)
ultimately have "homotopic_with_canon P S U (f \<circ> h) (g \<circ> h)"
- by (rule hom)
+ by (simp add: hom)
then have "homotopic_with_canon Q t U (f \<circ> h \<circ> k) (g \<circ> h \<circ> k)"
apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
using Q contk imk by force+
@@ -3303,18 +3303,18 @@
qed
lemma cohomotopically_trivial_retraction_null_gen:
- assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> U; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
- and Q: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
+ assumes P: "\<And>f. \<lbrakk>continuous_on t f; f \<in> t \<rightarrow> U; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
+ and Q: "\<And>f. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> U; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
- and hom: "\<And>f g. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f\<rbrakk>
+ and hom: "\<And>f g. \<lbrakk>continuous_on S f; f \<in> S \<rightarrow> U; P f\<rbrakk>
\<Longrightarrow> \<exists>c. homotopic_with_canon P S U f (\<lambda>x. c)"
- and contf: "continuous_on t f" and imf: "f ` t \<subseteq> U" and Qf: "Q f"
+ and contf: "continuous_on t f" and imf: "f \<in> t \<rightarrow> U" and Qf: "Q f"
obtains c where "homotopic_with_canon Q t U f (\<lambda>x. c)"
proof -
have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
have "continuous_on S (f \<circ> h)"
using contf conth continuous_on_compose imh by blast
- moreover have "(f \<circ> h) ` S \<subseteq> U"
+ moreover have "(f \<circ> h) \<in> S \<rightarrow> U"
using imf imh by fastforce
moreover have "P (f \<circ> h)"
by (simp add: P Qf contf imf)
@@ -3335,12 +3335,12 @@
lemma simply_connected_retraction_gen:
shows "\<lbrakk>simply_connected S; continuous_on S h; h ` S = T;
- continuous_on T k; k ` T \<subseteq> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk>
+ continuous_on T k; k \<in> T \<rightarrow> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk>
\<Longrightarrow> simply_connected T"
apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify)
apply (rule Retracts.homotopically_trivial_retraction_gen
[of S h _ k _ "\<lambda>p. pathfinish p = pathstart p" "\<lambda>p. pathfinish p = pathstart p"])
-apply (simp_all add: Retracts_def pathfinish_def pathstart_def)
+apply (simp_all add: Retracts_def pathfinish_def pathstart_def image_subset_iff_funcset)
done
lemma homeomorphic_simply_connected: