enhanced simplifier solver for preconditions of rewrite rule, can now deal with conjunctions
(* Title: HOL/Lifting_Option.thy
Author: Brian Huffman and Ondrej Kuncar
Author: Andreas Lochbihler, Karlsruhe Institute of Technology
*)
header {* Setup for Lifting/Transfer for the option type *}
theory Lifting_Option
imports Lifting Partial_Function
begin
subsection {* Relator and predicator properties *}
lemma rel_option_iff:
"rel_option R x y = (case (x, y) of (None, None) \<Rightarrow> True
| (Some x, Some y) \<Rightarrow> R x y
| _ \<Rightarrow> False)"
by (auto split: prod.split option.split)
abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
"option_pred \<equiv> case_option True"
lemma rel_option_eq [relator_eq]:
"rel_option (op =) = (op =)"
by (simp add: rel_option_iff fun_eq_iff split: option.split)
lemma rel_option_mono[relator_mono]:
assumes "A \<le> B"
shows "(rel_option A) \<le> (rel_option B)"
using assms by (auto simp: rel_option_iff split: option.splits)
lemma rel_option_OO[relator_distr]:
"(rel_option A) OO (rel_option B) = rel_option (A OO B)"
by (rule ext)+ (auto simp: rel_option_iff OO_def split: option.split)
lemma Domainp_option[relator_domain]:
assumes "Domainp A = P"
shows "Domainp (rel_option A) = (option_pred P)"
using assms unfolding Domainp_iff[abs_def] rel_option_iff[abs_def]
by (auto iff: fun_eq_iff split: option.split)
lemma left_total_rel_option[reflexivity_rule]:
"left_total R \<Longrightarrow> left_total (rel_option R)"
unfolding left_total_def split_option_all split_option_ex by simp
lemma left_unique_rel_option [reflexivity_rule]:
"left_unique R \<Longrightarrow> left_unique (rel_option R)"
unfolding left_unique_def split_option_all by simp
lemma right_total_rel_option [transfer_rule]:
"right_total R \<Longrightarrow> right_total (rel_option R)"
unfolding right_total_def split_option_all split_option_ex by simp
lemma right_unique_rel_option [transfer_rule]:
"right_unique R \<Longrightarrow> right_unique (rel_option R)"
unfolding right_unique_def split_option_all by simp
lemma bi_total_rel_option [transfer_rule]:
"bi_total R \<Longrightarrow> bi_total (rel_option R)"
unfolding bi_total_def split_option_all split_option_ex by simp
lemma bi_unique_rel_option [transfer_rule]:
"bi_unique R \<Longrightarrow> bi_unique (rel_option R)"
unfolding bi_unique_def split_option_all by simp
lemma option_invariant_commute [invariant_commute]:
"rel_option (Lifting.invariant P) = Lifting.invariant (option_pred P)"
by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
subsection {* Quotient theorem for the Lifting package *}
lemma Quotient_option[quot_map]:
assumes "Quotient R Abs Rep T"
shows "Quotient (rel_option R) (map_option Abs)
(map_option Rep) (rel_option T)"
using assms unfolding Quotient_alt_def rel_option_iff
by (simp split: option.split)
subsection {* Transfer rules for the Transfer package *}
context
begin
interpretation lifting_syntax .
lemma None_transfer [transfer_rule]: "(rel_option A) None None"
by (rule option.rel_inject)
lemma Some_transfer [transfer_rule]: "(A ===> rel_option A) Some Some"
unfolding rel_fun_def by simp
lemma case_option_transfer [transfer_rule]:
"(B ===> (A ===> B) ===> rel_option A ===> B) case_option case_option"
unfolding rel_fun_def split_option_all by simp
lemma map_option_transfer [transfer_rule]:
"((A ===> B) ===> rel_option A ===> rel_option B) map_option map_option"
unfolding map_option_case[abs_def] by transfer_prover
lemma option_bind_transfer [transfer_rule]:
"(rel_option A ===> (A ===> rel_option B) ===> rel_option B)
Option.bind Option.bind"
unfolding rel_fun_def split_option_all by simp
end
end