src/HOL/Lifting_Option.thy
 author blanchet Mon Jan 20 23:34:26 2014 +0100 (2014-01-20) changeset 55089 181751ad852f parent 53026 e1a548c11845 child 55090 9475b16e520b permissions -rw-r--r--
swapped dependencies of 'Finite_Set' and 'Option' (to move BNF up)
```     1 (*  Title:      HOL/Lifting_Option.thy
```
```     2     Author:     Brian Huffman and Ondrej Kuncar
```
```     3 *)
```
```     4
```
```     5 header {* Setup for Lifting/Transfer for the option type *}
```
```     6
```
```     7 theory Lifting_Option
```
```     8 imports Lifting Option
```
```     9 begin
```
```    10
```
```    11 subsection {* Relator and predicator properties *}
```
```    12
```
```    13 definition
```
```    14   option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
```
```    15 where
```
```    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)"
```
```    19
```
```    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
```
```    26
```
```    27 abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
```
```    28   "option_pred \<equiv> option_case True"
```
```    29
```
```    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)
```
```    33
```
```    34 lemma option_rel_mono[relator_mono]:
```
```    35   assumes "A \<le> B"
```
```    36   shows "(option_rel A) \<le> (option_rel B)"
```
```    37 using assms by (auto simp: option_rel_def split: option.splits)
```
```    38
```
```    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)
```
```    42
```
```    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)
```
```    48
```
```    49 lemma reflp_option_rel[reflexivity_rule]:
```
```    50   "reflp R \<Longrightarrow> reflp (option_rel R)"
```
```    51   unfolding reflp_def split_option_all by simp
```
```    52
```
```    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
```
```    56
```
```    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
```
```    60
```
```    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
```
```    64
```
```    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
```
```    68
```
```    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
```
```    72
```
```    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
```
```    76
```
```    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)
```
```    80
```
```    81 subsection {* Quotient theorem for the Lifting package *}
```
```    82
```
```    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)
```
```    89
```
```    90 subsection {* Transfer rules for the Transfer package *}
```
```    91
```
```    92 context
```
```    93 begin
```
```    94 interpretation lifting_syntax .
```
```    95
```
```    96 lemma None_transfer [transfer_rule]: "(option_rel A) None None"
```
```    97   by simp
```
```    98
```
```    99 lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
```
```   100   unfolding fun_rel_def by simp
```
```   101
```
```   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
```
```   105
```
```   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
```
```   109
```
```   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
```
```   114
```
```   115 end
```
```   116
```
```   117 end
```
```   118
```