src/HOL/Lifting_Option.thy
 author blanchet Fri Jan 24 11:51:45 2014 +0100 (2014-01-24) changeset 55129 26bd1cba3ab5 parent 55090 9475b16e520b child 55404 5cb95b79a51f permissions -rw-r--r--
killed 'More_BNFs' by moving its various bits where they (now) belong
```     1 (*  Title:      HOL/Lifting_Option.thy
```
```     2     Author:     Brian Huffman and Ondrej Kuncar
```
```     3     Author:     Andreas Lochbihler, Karlsruhe Institute of Technology
```
```     4 *)
```
```     5
```
```     6 header {* Setup for Lifting/Transfer for the option type *}
```
```     7
```
```     8 theory Lifting_Option
```
```     9 imports Lifting Partial_Function
```
```    10 begin
```
```    11
```
```    12 subsection {* Relator and predicator properties *}
```
```    13
```
```    14 definition
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```    15   option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
```
```    16 where
```
```    17   "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
```
```    18     | (Some x, Some y) \<Rightarrow> R x y
```
```    19     | _ \<Rightarrow> False)"
```
```    20
```
```    21 lemma option_rel_simps[simp]:
```
```    22   "option_rel R None None = True"
```
```    23   "option_rel R (Some x) None = False"
```
```    24   "option_rel R None (Some y) = False"
```
```    25   "option_rel R (Some x) (Some y) = R x y"
```
```    26   unfolding option_rel_def by simp_all
```
```    27
```
```    28 abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
```
```    29   "option_pred \<equiv> option_case True"
```
```    30
```
```    31 lemma option_rel_eq [relator_eq]:
```
```    32   "option_rel (op =) = (op =)"
```
```    33   by (simp add: option_rel_def fun_eq_iff split: option.split)
```
```    34
```
```    35 lemma option_rel_mono[relator_mono]:
```
```    36   assumes "A \<le> B"
```
```    37   shows "(option_rel A) \<le> (option_rel B)"
```
```    38 using assms by (auto simp: option_rel_def split: option.splits)
```
```    39
```
```    40 lemma option_rel_OO[relator_distr]:
```
```    41   "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
```
```    42 by (rule ext)+ (auto simp: option_rel_def OO_def split: option.split)
```
```    43
```
```    44 lemma Domainp_option[relator_domain]:
```
```    45   assumes "Domainp A = P"
```
```    46   shows "Domainp (option_rel A) = (option_pred P)"
```
```    47 using assms unfolding Domainp_iff[abs_def] option_rel_def[abs_def]
```
```    48 by (auto iff: fun_eq_iff split: option.split)
```
```    49
```
```    50 lemma reflp_option_rel[reflexivity_rule]:
```
```    51   "reflp R \<Longrightarrow> reflp (option_rel R)"
```
```    52   unfolding reflp_def split_option_all by simp
```
```    53
```
```    54 lemma left_total_option_rel[reflexivity_rule]:
```
```    55   "left_total R \<Longrightarrow> left_total (option_rel R)"
```
```    56   unfolding left_total_def split_option_all split_option_ex by simp
```
```    57
```
```    58 lemma left_unique_option_rel [reflexivity_rule]:
```
```    59   "left_unique R \<Longrightarrow> left_unique (option_rel R)"
```
```    60   unfolding left_unique_def split_option_all by simp
```
```    61
```
```    62 lemma right_total_option_rel [transfer_rule]:
```
```    63   "right_total R \<Longrightarrow> right_total (option_rel R)"
```
```    64   unfolding right_total_def split_option_all split_option_ex by simp
```
```    65
```
```    66 lemma right_unique_option_rel [transfer_rule]:
```
```    67   "right_unique R \<Longrightarrow> right_unique (option_rel R)"
```
```    68   unfolding right_unique_def split_option_all by simp
```
```    69
```
```    70 lemma bi_total_option_rel [transfer_rule]:
```
```    71   "bi_total R \<Longrightarrow> bi_total (option_rel R)"
```
```    72   unfolding bi_total_def split_option_all split_option_ex by simp
```
```    73
```
```    74 lemma bi_unique_option_rel [transfer_rule]:
```
```    75   "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
```
```    76   unfolding bi_unique_def split_option_all by simp
```
```    77
```
```    78 lemma option_invariant_commute [invariant_commute]:
```
```    79   "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
```
```    80   by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
```
```    81
```
```    82 subsection {* Quotient theorem for the Lifting package *}
```
```    83
```
```    84 lemma Quotient_option[quot_map]:
```
```    85   assumes "Quotient R Abs Rep T"
```
```    86   shows "Quotient (option_rel R) (Option.map Abs)
```
```    87     (Option.map Rep) (option_rel T)"
```
```    88   using assms unfolding Quotient_alt_def option_rel_def
```
```    89   by (simp split: option.split)
```
```    90
```
```    91 subsection {* Transfer rules for the Transfer package *}
```
```    92
```
```    93 context
```
```    94 begin
```
```    95 interpretation lifting_syntax .
```
```    96
```
```    97 lemma None_transfer [transfer_rule]: "(option_rel A) None None"
```
```    98   by simp
```
```    99
```
```   100 lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
```
```   101   unfolding fun_rel_def by simp
```
```   102
```
```   103 lemma option_case_transfer [transfer_rule]:
```
```   104   "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
```
```   105   unfolding fun_rel_def split_option_all by simp
```
```   106
```
```   107 lemma option_map_transfer [transfer_rule]:
```
```   108   "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
```
```   109   unfolding Option.map_def by transfer_prover
```
```   110
```
```   111 lemma option_bind_transfer [transfer_rule]:
```
```   112   "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
```
```   113     Option.bind Option.bind"
```
```   114   unfolding fun_rel_def split_option_all by simp
```
```   115
```
```   116 end
```
```   117
```
```   118
```
```   119 subsubsection {* BNF setup *}
```
```   120
```
```   121 lemma option_rec_conv_option_case: "option_rec = option_case"
```
```   122 by (simp add: fun_eq_iff split: option.split)
```
```   123
```
```   124 bnf "'a option"
```
```   125   map: Option.map
```
```   126   sets: Option.set
```
```   127   bd: natLeq
```
```   128   wits: None
```
```   129   rel: option_rel
```
```   130 proof -
```
```   131   show "Option.map id = id" by (rule Option.map.id)
```
```   132 next
```
```   133   fix f g
```
```   134   show "Option.map (g \<circ> f) = Option.map g \<circ> Option.map f"
```
```   135     by (auto simp add: fun_eq_iff Option.map_def split: option.split)
```
```   136 next
```
```   137   fix f g x
```
```   138   assume "\<And>z. z \<in> Option.set x \<Longrightarrow> f z = g z"
```
```   139   thus "Option.map f x = Option.map g x"
```
```   140     by (simp cong: Option.map_cong)
```
```   141 next
```
```   142   fix f
```
```   143   show "Option.set \<circ> Option.map f = op ` f \<circ> Option.set"
```
```   144     by fastforce
```
```   145 next
```
```   146   show "card_order natLeq" by (rule natLeq_card_order)
```
```   147 next
```
```   148   show "cinfinite natLeq" by (rule natLeq_cinfinite)
```
```   149 next
```
```   150   fix x
```
```   151   show "|Option.set x| \<le>o natLeq"
```
```   152     by (cases x) (simp_all add: ordLess_imp_ordLeq finite_iff_ordLess_natLeq[symmetric])
```
```   153 next
```
```   154   fix R S
```
```   155   show "option_rel R OO option_rel S \<le> option_rel (R OO S)"
```
```   156     by (auto simp: option_rel_def split: option.splits)
```
```   157 next
```
```   158   fix z
```
```   159   assume "z \<in> Option.set None"
```
```   160   thus False by simp
```
```   161 next
```
```   162   fix R
```
```   163   show "option_rel R =
```
```   164         (Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map fst))\<inverse>\<inverse> OO
```
```   165          Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map snd)"
```
```   166   unfolding option_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff prod.cases
```
```   167   by (auto simp: trans[OF eq_commute option_map_is_None] trans[OF eq_commute option_map_eq_Some]
```
```   168            split: option.splits)
```
```   169 qed
```
```   170
```
```   171 end
```