author | Andreas Lochbihler |
Wed, 12 Feb 2014 10:59:25 +0100 | |
changeset 55428 | 0ab52bf7b5e6 |
parent 55404 | 5cb95b79a51f |
child 55466 | 786edc984c98 |
permissions | -rw-r--r-- |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
1 |
(* Title: HOL/Lifting_Option.thy |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
2 |
Author: Brian Huffman and Ondrej Kuncar |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
55090
diff
changeset
|
3 |
Author: Andreas Lochbihler, Karlsruhe Institute of Technology |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
4 |
*) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
5 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
6 |
header {* Setup for Lifting/Transfer for the option type *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
7 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
8 |
theory Lifting_Option |
55090
9475b16e520b
technical import to avoid infamous 'duplicate Option.size' error at merge time in 'List.thy'
blanchet
parents:
55089
diff
changeset
|
9 |
imports Lifting Partial_Function |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
10 |
begin |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
11 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
12 |
subsection {* Relator and predicator properties *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
13 |
|
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
14 |
definition |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
15 |
option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
16 |
where |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
17 |
"option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
18 |
| (Some x, Some y) \<Rightarrow> R x y |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
19 |
| _ \<Rightarrow> False)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
20 |
|
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
21 |
lemma option_rel_simps[simp]: |
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
22 |
"option_rel R None None = True" |
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
23 |
"option_rel R (Some x) None = False" |
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
24 |
"option_rel R None (Some y) = False" |
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
25 |
"option_rel R (Some x) (Some y) = R x y" |
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
26 |
unfolding option_rel_def by simp_all |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
27 |
|
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
28 |
abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where |
55404
5cb95b79a51f
transformed 'option' and 'list' into new-style datatypes (but register them as old-style as well)
blanchet
parents:
55129
diff
changeset
|
29 |
"option_pred \<equiv> case_option True" |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
30 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
31 |
lemma option_rel_eq [relator_eq]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
32 |
"option_rel (op =) = (op =)" |
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
33 |
by (simp add: option_rel_def fun_eq_iff split: option.split) |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
34 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
35 |
lemma option_rel_mono[relator_mono]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
36 |
assumes "A \<le> B" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
37 |
shows "(option_rel A) \<le> (option_rel B)" |
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
38 |
using assms by (auto simp: option_rel_def split: option.splits) |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
39 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
40 |
lemma option_rel_OO[relator_distr]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
41 |
"(option_rel A) OO (option_rel B) = option_rel (A OO B)" |
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
42 |
by (rule ext)+ (auto simp: option_rel_def OO_def split: option.split) |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
43 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
44 |
lemma Domainp_option[relator_domain]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
45 |
assumes "Domainp A = P" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
46 |
shows "Domainp (option_rel A) = (option_pred P)" |
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
47 |
using assms unfolding Domainp_iff[abs_def] option_rel_def[abs_def] |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
48 |
by (auto iff: fun_eq_iff split: option.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
49 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
50 |
lemma reflp_option_rel[reflexivity_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
51 |
"reflp R \<Longrightarrow> reflp (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
52 |
unfolding reflp_def split_option_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
53 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
54 |
lemma left_total_option_rel[reflexivity_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
55 |
"left_total R \<Longrightarrow> left_total (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
56 |
unfolding left_total_def split_option_all split_option_ex by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
57 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
58 |
lemma left_unique_option_rel [reflexivity_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
59 |
"left_unique R \<Longrightarrow> left_unique (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
60 |
unfolding left_unique_def split_option_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
61 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
62 |
lemma right_total_option_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
63 |
"right_total R \<Longrightarrow> right_total (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
64 |
unfolding right_total_def split_option_all split_option_ex by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
65 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
66 |
lemma right_unique_option_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
67 |
"right_unique R \<Longrightarrow> right_unique (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
68 |
unfolding right_unique_def split_option_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
69 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
70 |
lemma bi_total_option_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
71 |
"bi_total R \<Longrightarrow> bi_total (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
72 |
unfolding bi_total_def split_option_all split_option_ex by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
73 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
74 |
lemma bi_unique_option_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
75 |
"bi_unique R \<Longrightarrow> bi_unique (option_rel R)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
76 |
unfolding bi_unique_def split_option_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
77 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
78 |
lemma option_invariant_commute [invariant_commute]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
79 |
"option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
80 |
by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
81 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
82 |
subsection {* Quotient theorem for the Lifting package *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
83 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
84 |
lemma Quotient_option[quot_map]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
85 |
assumes "Quotient R Abs Rep T" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
86 |
shows "Quotient (option_rel R) (Option.map Abs) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
87 |
(Option.map Rep) (option_rel T)" |
53026
e1a548c11845
got rid of the dependency of Lifting_* on the function package; use the original rel constants for basic BNFs;
traytel
parents:
53012
diff
changeset
|
88 |
using assms unfolding Quotient_alt_def option_rel_def |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
89 |
by (simp split: option.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
90 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
91 |
subsection {* Transfer rules for the Transfer package *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
92 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
93 |
context |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
94 |
begin |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
95 |
interpretation lifting_syntax . |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
96 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
97 |
lemma None_transfer [transfer_rule]: "(option_rel A) None None" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
98 |
by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
99 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
100 |
lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
101 |
unfolding fun_rel_def by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
102 |
|
55404
5cb95b79a51f
transformed 'option' and 'list' into new-style datatypes (but register them as old-style as well)
blanchet
parents:
55129
diff
changeset
|
103 |
lemma case_option_transfer [transfer_rule]: |
5cb95b79a51f
transformed 'option' and 'list' into new-style datatypes (but register them as old-style as well)
blanchet
parents:
55129
diff
changeset
|
104 |
"(B ===> (A ===> B) ===> option_rel A ===> B) case_option case_option" |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
105 |
unfolding fun_rel_def split_option_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
106 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
107 |
lemma option_map_transfer [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
108 |
"((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
109 |
unfolding Option.map_def by transfer_prover |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
110 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
111 |
lemma option_bind_transfer [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
112 |
"(option_rel A ===> (A ===> option_rel B) ===> option_rel B) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
113 |
Option.bind Option.bind" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
114 |
unfolding fun_rel_def split_option_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
115 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
116 |
end |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
117 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
118 |
end |