src/HOL/Types_To_Sets/Examples/T2_Spaces.thy

+(*  Title:      HOL/Types_To_Sets/Examples/T2_Spaces.thy
+    Author:     Ondřej Kunčar, TU München
+*)
+
+theory T2_Spaces
+  imports Complex_Main "../Types_To_Sets" Prerequisites
+begin
+
+section \<open>The Type-Based Theorem\<close>
+
+text\<open>We relativize a theorem that contains a type class with an associated (overloaded) operation.
+     The key technique is to compile out the overloaded operation by the dictionary construction
+     using the Unoverloading rule.\<close>
+
+text\<open>This is the type-based statement that we want to relativize.\<close>
+thm compact_imp_closed
+text\<open>The type is class a T2 typological space.\<close>
+typ "'a :: t2_space"
+text\<open>The associated operation is the predicate open that determines the open sets in the T2 space.\<close>
+term "open"
+
+section \<open>Definitions and Setup for The Relativization\<close>
+
+text\<open>We gradually define relativization of topological spaces, t2 spaces, compact and closed
+     predicates and prove that they are indeed the relativization of the original predicates.\<close>
+
+definition topological_space_on_with :: "'a set \<Rightarrow> ('a set \<Rightarrow> bool) \<Rightarrow> bool"
+  where "topological_space_on_with A \<equiv> \<lambda>open. open A \<and>
+    (\<forall>S \<subseteq> A. \<forall>T \<subseteq> A. open S \<longrightarrow> open T \<longrightarrow> open (S \<inter> T))
+    \<and> (\<forall>K \<subseteq> Pow A. (\<forall>S\<in>K. open S) \<longrightarrow> open (\<Union>K))"
+
+lemma topological_space_transfer[transfer_rule]:
+  includes lifting_syntax
+  assumes [transfer_rule]: "right_total T" "bi_unique T"
+  shows "((rel_set T ===> op=) ===> op=) (topological_space_on_with (Collect (Domainp T)))
+    class.topological_space"
+  unfolding topological_space_on_with_def[abs_def] class.topological_space_def[abs_def]
+  apply transfer_prover_start
+  apply transfer_step+
+  unfolding Pow_def Ball_Collect[symmetric]
+  by blast
+
+definition t2_space_on_with :: "'a set \<Rightarrow> ('a set \<Rightarrow> bool) \<Rightarrow> bool"
+  where "t2_space_on_with A \<equiv> \<lambda>open. topological_space_on_with A open \<and>
+  (\<forall>x \<in> A. \<forall>y \<in> A. x \<noteq> y \<longrightarrow> (\<exists>U\<subseteq>A. \<exists>V\<subseteq>A. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}))"
+
+lemma t2_space_transfer[transfer_rule]:
+  includes lifting_syntax
+  assumes [transfer_rule]: "right_total T" "bi_unique T"
+  shows "((rel_set T ===> op=) ===> op=) (t2_space_on_with (Collect (Domainp T))) class.t2_space"
+  unfolding t2_space_on_with_def[abs_def] class.t2_space_def[abs_def]
+    class.t2_space_axioms_def[abs_def]
+  apply transfer_prover_start
+  apply transfer_step+
+  unfolding Ball_Collect[symmetric]
+  by blast
+
+definition closed_with :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+  where "closed_with \<equiv> \<lambda>open S. open (- S)"
+
+lemma closed_closed_with: "closed s = closed_with open s"
+  unfolding closed_with_def closed_def[abs_def] ..
+
+definition closed_on_with :: "'a set \<Rightarrow> ('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+  where "closed_on_with A \<equiv> \<lambda>open S. open (-S \<inter> A)"
+
+lemma closed_with_transfer[transfer_rule]:
+  includes lifting_syntax
+  assumes [transfer_rule]: "right_total T" "bi_unique T"
+  shows "((rel_set T ===> op=) ===> rel_set T===> op=) (closed_on_with (Collect (Domainp T)))
+    closed_with"
+  unfolding closed_with_def closed_on_with_def by transfer_prover
+
+definition compact_with :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+  where "compact_with \<equiv> \<lambda>open S. (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
+
+lemma compact_compact_with: "compact s = compact_with open s"
+  unfolding compact_with_def compact_eq_heine_borel[abs_def] ..
+
+definition compact_on_with :: "'a set \<Rightarrow> ('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+  where "compact_on_with A \<equiv> \<lambda>open S. (\<forall>C\<subseteq>Pow A. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
+    (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
+
+lemma compact_on_with_subset_trans: "(\<forall>C\<subseteq>Pow A. (\<forall>c\<in>C. open' c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
+  (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)) =
+  ((\<forall>C\<subseteq>Pow A. (\<forall>c\<in>C. open' c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>Pow A. D\<subseteq>C \<and> finite D \<and> S \<subseteq> \<Union>D)))"
+  by (meson subset_trans)
+
+lemma compact_with_transfer[transfer_rule]:
+  includes lifting_syntax
+  assumes [transfer_rule]: "right_total T" "bi_unique T"
+  shows "((rel_set T ===> op=) ===> rel_set T===> op=) (compact_on_with (Collect (Domainp T)))
+    compact_with"
+  unfolding compact_with_def compact_on_with_def
+  apply transfer_prover_start
+  apply transfer_step+
+  unfolding compact_on_with_subset_trans
+  unfolding Pow_def Ball_Collect[symmetric] Ball_def Bex_def mem_Collect_eq
+  by blast
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name "open"}, SOME @{typ "'a set \<Rightarrow> bool"})\<close>
+
+text\<open>The aforementioned development can be automated. The main part is already automated
+     by the transfer_prover.\<close>
+
+section \<open>The Relativization to The Set-Based Theorem\<close>
+
+text\<open>The first step of the dictionary construction.\<close>
+lemmas dictionary_first_step = compact_imp_closed[unfolded compact_compact_with closed_closed_with]
+thm dictionary_first_step
+
+text\<open>Internalization of the type class t2_space.\<close>
+lemmas internalized_sort = dictionary_first_step[internalize_sort "'a::t2_space"]
+thm internalized_sort
+
+text\<open>We unoverload the overloaded constant open and thus finish compiling out of it.\<close>
+lemmas dictionary_second_step =  internalized_sort[unoverload "open :: 'a set \<Rightarrow> bool"]
+text\<open>The theorem with internalized type classes and compiled out operations is the starting point
+     for the original relativization algorithm.\<close>
+thm dictionary_second_step
+
+
+text\<open>This is the set-based variant of the theorem compact_imp_closed.\<close>
+lemma compact_imp_closed_set_based:
+  assumes "(A::'a set) \<noteq> {}"
+  shows "\<forall>open. t2_space_on_with A open \<longrightarrow> (\<forall>S\<subseteq>A. compact_on_with A open S \<longrightarrow>
+    closed_on_with A open S)"
+proof -
+  {
+    text\<open>We define the type 'b to be isomorphic to A.\<close>
+    assume T: "\<exists>(Rep :: 'b \<Rightarrow> 'a) Abs. type_definition Rep Abs A"
+    from T obtain rep :: "'b \<Rightarrow> 'a" and abs :: "'a \<Rightarrow> 'b" where t: "type_definition rep abs A"
+      by auto
+
+    text\<open>Setup for the Transfer tool.\<close>
+    define cr_b where "cr_b == \<lambda>r a. r = rep a"
+    note type_definition_Domainp[OF t cr_b_def, transfer_domain_rule]
+    note typedef_right_total[OF t cr_b_def, transfer_rule]
+    note typedef_bi_unique[OF t cr_b_def, transfer_rule]
+
+    have ?thesis
+      text\<open>Relativization by the Transfer tool.\<close>
+      using dictionary_second_step[where 'a = 'b, untransferred, simplified]
+      by blast
+
+  } note * = this[cancel_type_definition, OF assms]
+
+  show ?thesis by (rule *)
+qed
+
+setup \<open>Sign.add_const_constraint
+  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"})\<close>
+
+text\<open>The Final Result. We can compare the type-based and the set-based statement.\<close>
+thm compact_imp_closed compact_imp_closed_set_based
+
+declare [[show_sorts]]
+text\<open>The Final Result. This time with explicitly shown type-class annotations.\<close>
+thm compact_imp_closed compact_imp_closed_set_based
+
+end