src/HOL/Lifting_Option.thy
 author kuncar Tue Aug 13 15:59:22 2013 +0200 (2013-08-13) changeset 53012 cb82606b8215 child 53026 e1a548c11845 permissions -rw-r--r--
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
```     1 (*  Title:      HOL/Lifting_Option.thy
```
```     2     Author:     Brian Huffman and Ondrej Kuncar
```
```     3 *)
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```     4
```
```     5 header {* Setup for Lifting/Transfer for the option type *}
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```     6
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```     7 theory Lifting_Option
```
```     8 imports Lifting FunDef
```
```     9 begin
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```    10
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```    11 subsection {* Relator and predicator properties *}
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```    12
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```    13 fun
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```    14   option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
```
```    15 where
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```    16   "option_rel R None None = True"
```
```    17 | "option_rel R (Some x) None = False"
```
```    18 | "option_rel R None (Some x) = False"
```
```    19 | "option_rel R (Some x) (Some y) = R x y"
```
```    20
```
```    21 lemma option_rel_unfold:
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```    22   "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
```
```    23     | (Some x, Some y) \<Rightarrow> R x y
```
```    24     | _ \<Rightarrow> False)"
```
```    25   by (cases x) (cases y, simp_all)+
```
```    26
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```    27 fun option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
```
```    28 where
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```    29   "option_pred R None = True"
```
```    30 | "option_pred R (Some x) = R x"
```
```    31
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```    32 lemma option_pred_unfold:
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```    33   "option_pred P x = (case x of None \<Rightarrow> True
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```    34     | Some x \<Rightarrow> P x)"
```
```    35 by (cases x) simp_all
```
```    36
```
```    37 lemma option_rel_eq [relator_eq]:
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```    38   "option_rel (op =) = (op =)"
```
```    39   by (simp add: option_rel_unfold fun_eq_iff split: option.split)
```
```    40
```
```    41 lemma option_rel_mono[relator_mono]:
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```    42   assumes "A \<le> B"
```
```    43   shows "(option_rel A) \<le> (option_rel B)"
```
```    44 using assms by (auto simp: option_rel_unfold split: option.splits)
```
```    45
```
```    46 lemma option_rel_OO[relator_distr]:
```
```    47   "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
```
```    48 by (rule ext)+ (auto simp: option_rel_unfold OO_def split: option.split)
```
```    49
```
```    50 lemma Domainp_option[relator_domain]:
```
```    51   assumes "Domainp A = P"
```
```    52   shows "Domainp (option_rel A) = (option_pred P)"
```
```    53 using assms unfolding Domainp_iff[abs_def] option_rel_unfold[abs_def] option_pred_unfold[abs_def]
```
```    54 by (auto iff: fun_eq_iff split: option.split)
```
```    55
```
```    56 lemma reflp_option_rel[reflexivity_rule]:
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```    57   "reflp R \<Longrightarrow> reflp (option_rel R)"
```
```    58   unfolding reflp_def split_option_all by simp
```
```    59
```
```    60 lemma left_total_option_rel[reflexivity_rule]:
```
```    61   "left_total R \<Longrightarrow> left_total (option_rel R)"
```
```    62   unfolding left_total_def split_option_all split_option_ex by simp
```
```    63
```
```    64 lemma left_unique_option_rel [reflexivity_rule]:
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```    65   "left_unique R \<Longrightarrow> left_unique (option_rel R)"
```
```    66   unfolding left_unique_def split_option_all by simp
```
```    67
```
```    68 lemma right_total_option_rel [transfer_rule]:
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```    69   "right_total R \<Longrightarrow> right_total (option_rel R)"
```
```    70   unfolding right_total_def split_option_all split_option_ex by simp
```
```    71
```
```    72 lemma right_unique_option_rel [transfer_rule]:
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```    73   "right_unique R \<Longrightarrow> right_unique (option_rel R)"
```
```    74   unfolding right_unique_def split_option_all by simp
```
```    75
```
```    76 lemma bi_total_option_rel [transfer_rule]:
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```    77   "bi_total R \<Longrightarrow> bi_total (option_rel R)"
```
```    78   unfolding bi_total_def split_option_all split_option_ex by simp
```
```    79
```
```    80 lemma bi_unique_option_rel [transfer_rule]:
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```    81   "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
```
```    82   unfolding bi_unique_def split_option_all by simp
```
```    83
```
```    84 lemma option_invariant_commute [invariant_commute]:
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```    85   "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
```
```    86   by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
```
```    87
```
```    88 subsection {* Quotient theorem for the Lifting package *}
```
```    89
```
```    90 lemma Quotient_option[quot_map]:
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```    91   assumes "Quotient R Abs Rep T"
```
```    92   shows "Quotient (option_rel R) (Option.map Abs)
```
```    93     (Option.map Rep) (option_rel T)"
```
```    94   using assms unfolding Quotient_alt_def option_rel_unfold
```
```    95   by (simp split: option.split)
```
```    96
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```    97 subsection {* Transfer rules for the Transfer package *}
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```    98
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```    99 context
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```   100 begin
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```   101 interpretation lifting_syntax .
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```   102
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```   103 lemma None_transfer [transfer_rule]: "(option_rel A) None None"
```
```   104   by simp
```
```   105
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```   106 lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
```
```   107   unfolding fun_rel_def by simp
```
```   108
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```   109 lemma option_case_transfer [transfer_rule]:
```
```   110   "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
```
```   111   unfolding fun_rel_def split_option_all by simp
```
```   112
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```   113 lemma option_map_transfer [transfer_rule]:
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```   114   "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
```
```   115   unfolding Option.map_def by transfer_prover
```
```   116
```
```   117 lemma option_bind_transfer [transfer_rule]:
```
```   118   "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
```
```   119     Option.bind Option.bind"
```
```   120   unfolding fun_rel_def split_option_all by simp
```
```   121
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```   122 end
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```   123
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```   124 end
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```   125
```