(* 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 ===> (=)) ===> (=)) (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 ===> (=)) ===> (=)) (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 ===> (=)) ===> rel_set T===> (=)) (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 ===> (=)) ===> rel_set T===> (=)) (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>open\<close>, SOME \<^typ>\<open>'a set \<Rightarrow> bool\<close>)\<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>Alternative construction using \<open>unoverload_type\<close>
(This does not require fiddling with \<open>Sign.add_const_constraint\<close>).\<close>
lemmas dictionary_second_step' = dictionary_first_step[unoverload_type 'a]
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>open\<close>, SOME \<^typ>\<open>'a::topological_space set \<Rightarrow> bool\<close>)\<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