(* Title: HOL/Types_To_Sets/Examples/Prerequisites.thy
Author: Ondřej Kunčar, TU München
*)
theory Prerequisites
imports Main
keywords "lemmas_with"::thy_decl
begin
context
fixes Rep Abs A T
assumes type: "type_definition Rep Abs A"
assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
begin
lemma type_definition_Domainp: "Domainp T = (\<lambda>x. x \<in> A)"
proof -
interpret type_definition Rep Abs A by(rule type)
show ?thesis
unfolding Domainp_iff[abs_def] T_def fun_eq_iff
by (metis Abs_inverse Rep)
qed
end
subsection \<open>setting up transfer for local typedef\<close>
lemmas [transfer_rule] = \<comment> \<open>prefer right-total rules\<close>
right_total_All_transfer
right_total_UNIV_transfer
right_total_Ex_transfer
locale local_typedef = fixes S ::"'b set" and s::"'s itself"
assumes Ex_type_definition_S: "\<exists>(Rep::'s \<Rightarrow> 'b) (Abs::'b \<Rightarrow> 's). type_definition Rep Abs S"
begin
definition "rep = fst (SOME (Rep::'s \<Rightarrow> 'b, Abs). type_definition Rep Abs S)"
definition "Abs = snd (SOME (Rep::'s \<Rightarrow> 'b, Abs). type_definition Rep Abs S)"
lemma type_definition_S: "type_definition rep Abs S"
unfolding Abs_def rep_def split_beta'
by (rule someI_ex) (use Ex_type_definition_S in auto)
lemma rep_in_S[simp]: "rep x \<in> S"
and rep_inverse[simp]: "Abs (rep x) = x"
and Abs_inverse[simp]: "y \<in> S \<Longrightarrow> rep (Abs y) = y"
using type_definition_S
unfolding type_definition_def by auto
definition cr_S where "cr_S \<equiv> \<lambda>s b. s = rep b"
lemmas Domainp_cr_S = type_definition_Domainp[OF type_definition_S cr_S_def, transfer_domain_rule]
lemmas right_total_cr_S = typedef_right_total[OF type_definition_S cr_S_def, transfer_rule]
and bi_unique_cr_S = typedef_bi_unique[OF type_definition_S cr_S_def, transfer_rule]
and left_unique_cr_S = typedef_left_unique[OF type_definition_S cr_S_def, transfer_rule]
and right_unique_cr_S = typedef_right_unique[OF type_definition_S cr_S_def, transfer_rule]
lemma cr_S_rep[intro, simp]: "cr_S (rep a) a" by (simp add: cr_S_def)
lemma cr_S_Abs[intro, simp]: "a\<in>S \<Longrightarrow> cr_S a (Abs a)" by (simp add: cr_S_def)
end
subsection \<open>some \<close>
subsection \<open>Tool support\<close>
lemmas subset_iff' = subset_iff[folded Ball_def]
ML \<open>
structure More_Simplifier =
struct
fun asm_full_var_simplify ctxt thm =
let
val ((_, [thm']), ctxt') = Variable.import false [thm] ctxt
in
Simplifier.asm_full_simplify ctxt' thm'
|> singleton (Variable.export ctxt' ctxt)
|> Drule.zero_var_indexes
end
fun var_simplify_only ctxt ths thm =
asm_full_var_simplify (Raw_Simplifier.clear_simpset ctxt addsimps ths) thm
val var_simplified = Attrib.thms >>
(fn ths => Thm.rule_attribute ths
(fn context => var_simplify_only (Context.proof_of context) ths))
val _ = Theory.setup (Attrib.setup \<^binding>\<open>var_simplified\<close> var_simplified "simplified rule (with vars)")
end
\<close>
ML \<open>
val _ = Outer_Syntax.local_theory' \<^command_keyword>\<open>lemmas_with\<close> "note theorems with (the same) attributes"
(Parse.attribs --| \<^keyword>\<open>:\<close> -- Parse_Spec.name_facts -- Parse.for_fixes
>> (fn (((attrs),facts), fixes) =>
#2 oo Specification.theorems_cmd Thm.theoremK
(map (apsnd (map (apsnd (fn xs => attrs@xs)))) facts) fixes))
\<close>
end