--- a/src/HOL/Library/Quotient_Option.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_Option.thy Tue Aug 13 15:59:22 2013 +0200
@@ -1,5 +1,5 @@
(* Title: HOL/Library/Quotient_Option.thy
- Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
+ Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the option type *}
@@ -8,31 +8,7 @@
imports Main Quotient_Syntax
begin
-subsection {* Relator for option type *}
-
-fun
- option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
-where
- "option_rel R None None = True"
-| "option_rel R (Some x) None = False"
-| "option_rel R None (Some x) = False"
-| "option_rel R (Some x) (Some y) = R x y"
-
-lemma option_rel_unfold:
- "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
- | (Some x, Some y) \<Rightarrow> R x y
- | _ \<Rightarrow> False)"
- by (cases x) (cases y, simp_all)+
-
-fun option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
-where
- "option_pred R None = True"
-| "option_pred R (Some x) = R x"
-
-lemma option_pred_unfold:
- "option_pred P x = (case x of None \<Rightarrow> True
- | Some x \<Rightarrow> P x)"
-by (cases x) simp_all
+subsection {* Rules for the Quotient package *}
lemma option_rel_map1:
"option_rel R (Option.map f x) y \<longleftrightarrow> option_rel (\<lambda>x. R (f x)) x y"
@@ -46,37 +22,10 @@
"Option.map id = id"
by (simp add: id_def Option.map.identity fun_eq_iff)
-lemma option_rel_eq [id_simps, relator_eq]:
+lemma option_rel_eq [id_simps]:
"option_rel (op =) = (op =)"
by (simp add: option_rel_unfold fun_eq_iff split: option.split)
-lemma option_rel_mono[relator_mono]:
- assumes "A \<le> B"
- shows "(option_rel A) \<le> (option_rel B)"
-using assms by (auto simp: option_rel_unfold split: option.splits)
-
-lemma option_rel_OO[relator_distr]:
- "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
-by (rule ext)+ (auto simp: option_rel_unfold OO_def split: option.split)
-
-lemma Domainp_option[relator_domain]:
- assumes "Domainp A = P"
- shows "Domainp (option_rel A) = (option_pred P)"
-using assms unfolding Domainp_iff[abs_def] option_rel_unfold[abs_def] option_pred_unfold[abs_def]
-by (auto iff: fun_eq_iff split: option.split)
-
-lemma reflp_option_rel[reflexivity_rule]:
- "reflp R \<Longrightarrow> reflp (option_rel R)"
- unfolding reflp_def split_option_all by simp
-
-lemma left_total_option_rel[reflexivity_rule]:
- "left_total R \<Longrightarrow> left_total (option_rel R)"
- unfolding left_total_def split_option_all split_option_ex by simp
-
-lemma left_unique_option_rel [reflexivity_rule]:
- "left_unique R \<Longrightarrow> left_unique (option_rel R)"
- unfolding left_unique_def split_option_all by simp
-
lemma option_symp:
"symp R \<Longrightarrow> symp (option_rel R)"
unfolding symp_def split_option_all option_rel.simps by fast
@@ -89,65 +38,6 @@
"equivp R \<Longrightarrow> equivp (option_rel R)"
by (blast intro: equivpI reflp_option_rel option_symp option_transp elim: equivpE)
-lemma right_total_option_rel [transfer_rule]:
- "right_total R \<Longrightarrow> right_total (option_rel R)"
- unfolding right_total_def split_option_all split_option_ex by simp
-
-lemma right_unique_option_rel [transfer_rule]:
- "right_unique R \<Longrightarrow> right_unique (option_rel R)"
- unfolding right_unique_def split_option_all by simp
-
-lemma bi_total_option_rel [transfer_rule]:
- "bi_total R \<Longrightarrow> bi_total (option_rel R)"
- unfolding bi_total_def split_option_all split_option_ex by simp
-
-lemma bi_unique_option_rel [transfer_rule]:
- "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
- unfolding bi_unique_def split_option_all by simp
-
-subsection {* Transfer rules for transfer package *}
-
-lemma None_transfer [transfer_rule]: "(option_rel A) None None"
- by simp
-
-lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
- unfolding fun_rel_def by simp
-
-lemma option_case_transfer [transfer_rule]:
- "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
- unfolding fun_rel_def split_option_all by simp
-
-lemma option_map_transfer [transfer_rule]:
- "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
- unfolding Option.map_def by transfer_prover
-
-lemma option_bind_transfer [transfer_rule]:
- "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
- Option.bind Option.bind"
- unfolding fun_rel_def split_option_all by simp
-
-subsection {* Setup for lifting package *}
-
-lemma Quotient_option[quot_map]:
- assumes "Quotient R Abs Rep T"
- shows "Quotient (option_rel R) (Option.map Abs)
- (Option.map Rep) (option_rel T)"
- using assms unfolding Quotient_alt_def option_rel_unfold
- by (simp split: option.split)
-
-lemma option_invariant_commute [invariant_commute]:
- "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
- apply (simp add: fun_eq_iff Lifting.invariant_def)
- apply (intro allI)
- apply (case_tac x rule: option.exhaust)
- apply (case_tac xa rule: option.exhaust)
- apply auto[2]
- apply (case_tac xa rule: option.exhaust)
- apply auto
-done
-
-subsection {* Rules for quotient package *}
-
lemma option_quotient [quot_thm]:
assumes "Quotient3 R Abs Rep"
shows "Quotient3 (option_rel R) (Option.map Abs) (Option.map Rep)"