got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
1 (* Title: HOL/Lifting_Option.thy
2 Author: Brian Huffman and Ondrej Kuncar
5 header {* Setup for Lifting/Transfer for the option type *}
11 subsection {* Relator and predicator properties *}
14 option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
16 "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
17 | (Some x, Some y) \<Rightarrow> R x y
18 | _ \<Rightarrow> False)"
20 lemma option_rel_simps[simp]:
21 "option_rel R None None = True"
22 "option_rel R (Some x) None = False"
23 "option_rel R None (Some y) = False"
24 "option_rel R (Some x) (Some y) = R x y"
25 unfolding option_rel_def by simp_all
27 abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
28 "option_pred \<equiv> option_case True"
30 lemma option_rel_eq [relator_eq]:
31 "option_rel (op =) = (op =)"
32 by (simp add: option_rel_def fun_eq_iff split: option.split)
34 lemma option_rel_mono[relator_mono]:
36 shows "(option_rel A) \<le> (option_rel B)"
37 using assms by (auto simp: option_rel_def split: option.splits)
39 lemma option_rel_OO[relator_distr]:
40 "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
41 by (rule ext)+ (auto simp: option_rel_def OO_def split: option.split)
43 lemma Domainp_option[relator_domain]:
44 assumes "Domainp A = P"
45 shows "Domainp (option_rel A) = (option_pred P)"
46 using assms unfolding Domainp_iff[abs_def] option_rel_def[abs_def]
47 by (auto iff: fun_eq_iff split: option.split)
49 lemma reflp_option_rel[reflexivity_rule]:
50 "reflp R \<Longrightarrow> reflp (option_rel R)"
51 unfolding reflp_def split_option_all by simp
53 lemma left_total_option_rel[reflexivity_rule]:
54 "left_total R \<Longrightarrow> left_total (option_rel R)"
55 unfolding left_total_def split_option_all split_option_ex by simp
57 lemma left_unique_option_rel [reflexivity_rule]:
58 "left_unique R \<Longrightarrow> left_unique (option_rel R)"
59 unfolding left_unique_def split_option_all by simp
61 lemma right_total_option_rel [transfer_rule]:
62 "right_total R \<Longrightarrow> right_total (option_rel R)"
63 unfolding right_total_def split_option_all split_option_ex by simp
65 lemma right_unique_option_rel [transfer_rule]:
66 "right_unique R \<Longrightarrow> right_unique (option_rel R)"
67 unfolding right_unique_def split_option_all by simp
69 lemma bi_total_option_rel [transfer_rule]:
70 "bi_total R \<Longrightarrow> bi_total (option_rel R)"
71 unfolding bi_total_def split_option_all split_option_ex by simp
73 lemma bi_unique_option_rel [transfer_rule]:
74 "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
75 unfolding bi_unique_def split_option_all by simp
77 lemma option_invariant_commute [invariant_commute]:
78 "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
79 by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
81 subsection {* Quotient theorem for the Lifting package *}
83 lemma Quotient_option[quot_map]:
84 assumes "Quotient R Abs Rep T"
85 shows "Quotient (option_rel R) (Option.map Abs)
86 (Option.map Rep) (option_rel T)"
87 using assms unfolding Quotient_alt_def option_rel_def
88 by (simp split: option.split)
90 subsection {* Transfer rules for the Transfer package *}
94 interpretation lifting_syntax .
96 lemma None_transfer [transfer_rule]: "(option_rel A) None None"
99 lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
100 unfolding fun_rel_def by simp
102 lemma option_case_transfer [transfer_rule]:
103 "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
104 unfolding fun_rel_def split_option_all by simp
106 lemma option_map_transfer [transfer_rule]:
107 "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
108 unfolding Option.map_def by transfer_prover
110 lemma option_bind_transfer [transfer_rule]:
111 "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
112 Option.bind Option.bind"
113 unfolding fun_rel_def split_option_all by simp