src/HOL/Lifting_Option.thy
author blanchet
Fri, 24 Jan 2014 11:51:45 +0100
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(*  Title:      HOL/Lifting_Option.thy
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    Author:     Brian Huffman and Ondrej Kuncar
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    Author:     Andreas Lochbihler, Karlsruhe Institute of Technology
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*)
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header {* Setup for Lifting/Transfer for the option type *}
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theory Lifting_Option
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imports Lifting Partial_Function
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begin
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subsection {* Relator and predicator properties *}
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definition
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  option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
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where
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  "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
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    | (Some x, Some y) \<Rightarrow> R x y
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    | _ \<Rightarrow> False)"
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lemma option_rel_simps[simp]:
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  "option_rel R None None = True"
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  "option_rel R (Some x) None = False"
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  "option_rel R None (Some y) = False"
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  "option_rel R (Some x) (Some y) = R x y"
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  unfolding option_rel_def by simp_all
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abbreviation (input) option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool" where
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  "option_pred \<equiv> option_case True"
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lemma option_rel_eq [relator_eq]:
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  "option_rel (op =) = (op =)"
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  by (simp add: option_rel_def fun_eq_iff split: option.split)
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lemma option_rel_mono[relator_mono]:
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  assumes "A \<le> B"
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  shows "(option_rel A) \<le> (option_rel B)"
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using assms by (auto simp: option_rel_def split: option.splits)
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lemma option_rel_OO[relator_distr]:
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  "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
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by (rule ext)+ (auto simp: option_rel_def OO_def split: option.split)
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lemma Domainp_option[relator_domain]:
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  assumes "Domainp A = P"
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  shows "Domainp (option_rel A) = (option_pred P)"
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using assms unfolding Domainp_iff[abs_def] option_rel_def[abs_def]
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by (auto iff: fun_eq_iff split: option.split)
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lemma reflp_option_rel[reflexivity_rule]:
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  "reflp R \<Longrightarrow> reflp (option_rel R)"
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  unfolding reflp_def split_option_all by simp
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lemma left_total_option_rel[reflexivity_rule]:
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  "left_total R \<Longrightarrow> left_total (option_rel R)"
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  unfolding left_total_def split_option_all split_option_ex by simp
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lemma left_unique_option_rel [reflexivity_rule]:
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  "left_unique R \<Longrightarrow> left_unique (option_rel R)"
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  unfolding left_unique_def split_option_all by simp
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lemma right_total_option_rel [transfer_rule]:
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  "right_total R \<Longrightarrow> right_total (option_rel R)"
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  unfolding right_total_def split_option_all split_option_ex by simp
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lemma right_unique_option_rel [transfer_rule]:
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  "right_unique R \<Longrightarrow> right_unique (option_rel R)"
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  unfolding right_unique_def split_option_all by simp
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lemma bi_total_option_rel [transfer_rule]:
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  "bi_total R \<Longrightarrow> bi_total (option_rel R)"
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  unfolding bi_total_def split_option_all split_option_ex by simp
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lemma bi_unique_option_rel [transfer_rule]:
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  "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
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  unfolding bi_unique_def split_option_all by simp
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lemma option_invariant_commute [invariant_commute]:
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  "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
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  by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
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subsection {* Quotient theorem for the Lifting package *}
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lemma Quotient_option[quot_map]:
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  assumes "Quotient R Abs Rep T"
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  shows "Quotient (option_rel R) (Option.map Abs)
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    (Option.map Rep) (option_rel T)"
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  using assms unfolding Quotient_alt_def option_rel_def
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  by (simp split: option.split)
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subsection {* Transfer rules for the Transfer package *}
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context
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begin
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interpretation lifting_syntax .
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lemma None_transfer [transfer_rule]: "(option_rel A) None None"
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  by simp
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lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
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  unfolding fun_rel_def by simp
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lemma option_case_transfer [transfer_rule]:
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  "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
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  unfolding fun_rel_def split_option_all by simp
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lemma option_map_transfer [transfer_rule]:
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  "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
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  unfolding Option.map_def by transfer_prover
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lemma option_bind_transfer [transfer_rule]:
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  "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
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    Option.bind Option.bind"
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  unfolding fun_rel_def split_option_all by simp
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end
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subsubsection {* BNF setup *}
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lemma option_rec_conv_option_case: "option_rec = option_case"
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by (simp add: fun_eq_iff split: option.split)
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bnf "'a option"
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  map: Option.map
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  sets: Option.set
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  bd: natLeq
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  wits: None
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  rel: option_rel
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proof -
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  show "Option.map id = id" by (rule Option.map.id)
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next
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  fix f g
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  show "Option.map (g \<circ> f) = Option.map g \<circ> Option.map f"
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    by (auto simp add: fun_eq_iff Option.map_def split: option.split)
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next
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  fix f g x
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  assume "\<And>z. z \<in> Option.set x \<Longrightarrow> f z = g z"
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  thus "Option.map f x = Option.map g x"
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    by (simp cong: Option.map_cong)
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   141
next
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   142
  fix f
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   143
  show "Option.set \<circ> Option.map f = op ` f \<circ> Option.set"
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   144
    by fastforce
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   145
next
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   146
  show "card_order natLeq" by (rule natLeq_card_order)
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   147
next
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   148
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
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   149
next
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   150
  fix x
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   151
  show "|Option.set x| \<le>o natLeq"
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   152
    by (cases x) (simp_all add: ordLess_imp_ordLeq finite_iff_ordLess_natLeq[symmetric])
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   153
next
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   154
  fix R S
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   155
  show "option_rel R OO option_rel S \<le> option_rel (R OO S)"
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   156
    by (auto simp: option_rel_def split: option.splits)
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   157
next
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   158
  fix z
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   159
  assume "z \<in> Option.set None"
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   160
  thus False by simp
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   161
next
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   162
  fix R
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   163
  show "option_rel R =
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   164
        (Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map fst))\<inverse>\<inverse> OO
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   165
         Grp {x. Option.set x \<subseteq> Collect (split R)} (Option.map snd)"
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   166
  unfolding option_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff prod.cases
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   167
  by (auto simp: trans[OF eq_commute option_map_is_None] trans[OF eq_commute option_map_eq_Some]
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   168
           split: option.splits)
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   169
qed
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   170
53012
cb82606b8215 move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff changeset
   171
end