src/HOL/Library/FSet.thy
author haftmann
Tue Oct 13 09:21:15 2015 +0200 (2015-10-13)
changeset 61424 c3658c18b7bc
parent 60712 3ba16d28449d
child 61585 a9599d3d7610
permissions -rw-r--r--
prod_case as canonical name for product type eliminator
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(*  Title:      HOL/Library/FSet.thy
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    Author:     Ondrej Kuncar, TU Muenchen
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    Author:     Cezary Kaliszyk and Christian Urban
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    Author:     Andrei Popescu, TU Muenchen
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*)
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section \<open>Type of finite sets defined as a subtype of sets\<close>
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theory FSet
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imports Conditionally_Complete_Lattices
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begin
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subsection \<open>Definition of the type\<close>
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typedef 'a fset = "{A :: 'a set. finite A}"  morphisms fset Abs_fset
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by auto
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setup_lifting type_definition_fset
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subsection \<open>Basic operations and type class instantiations\<close>
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(* FIXME transfer and right_total vs. bi_total *)
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instantiation fset :: (finite) finite
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begin
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instance by (standard; transfer; simp)
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end
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instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
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begin
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interpretation lifting_syntax .
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lift_definition bot_fset :: "'a fset" is "{}" parametric empty_transfer by simp 
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lift_definition less_eq_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" is subset_eq parametric subset_transfer 
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  .
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definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
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lemma less_fset_transfer[transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A" 
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  shows "((pcr_fset A) ===> (pcr_fset A) ===> op =) op \<subset> op <"
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  unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover
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lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer
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  by simp
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lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer
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  by simp
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lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer
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  by simp
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instance
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  by (standard; transfer; auto)+
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end
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abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot"
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abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
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abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys"
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abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys"
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abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"
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abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"
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instantiation fset :: (equal) equal
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begin
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definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A"
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instance by intro_classes (auto simp add: equal_fset_def)
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end 
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instantiation fset :: (type) conditionally_complete_lattice
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begin
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interpretation lifting_syntax .
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lemma right_total_Inf_fset_transfer:
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  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
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  shows "(rel_set (rel_set A) ===> rel_set A) 
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    (\<lambda>S. if finite (Inter S \<inter> Collect (Domainp A)) then Inter S \<inter> Collect (Domainp A) else {}) 
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      (\<lambda>S. if finite (Inf S) then Inf S else {})"
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    by transfer_prover
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lemma Inf_fset_transfer:
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  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
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  shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Inf A) then Inf A else {}) 
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    (\<lambda>A. if finite (Inf A) then Inf A else {})"
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  by transfer_prover
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lift_definition Inf_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Inf A) then Inf A else {}" 
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parametric right_total_Inf_fset_transfer Inf_fset_transfer by simp
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lemma Sup_fset_transfer:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Sup A) then Sup A else {})
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  (\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover
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lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}"
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parametric Sup_fset_transfer by simp
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lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"
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by (auto intro: finite_subset)
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lemma transfer_bdd_below[transfer_rule]: "(rel_set (pcr_fset op =) ===> op =) bdd_below bdd_below"
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  by auto
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instance
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proof 
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  fix x z :: "'a fset"
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  fix X :: "'a fset set"
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  {
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    assume "x \<in> X" "bdd_below X" 
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    then show "Inf X |\<subseteq>| x" by transfer auto
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  next
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    assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"
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    then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)
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  next
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    assume "x \<in> X" "bdd_above X"
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    then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
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      by (auto simp: bdd_above_def)
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    then show "x |\<subseteq>| Sup X"
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      by transfer (auto intro!: finite_Sup)
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  next
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    assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
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    then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)
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  }
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qed
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end
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instantiation fset :: (finite) complete_lattice 
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begin
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lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer
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  by simp
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instance
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  by (standard; transfer; auto)
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end
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instantiation fset :: (finite) complete_boolean_algebra
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begin
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lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus 
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  parametric right_total_Compl_transfer Compl_transfer by simp
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instance
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  by (standard, simp_all only: INF_def SUP_def) (transfer, simp add: Compl_partition Diff_eq)+
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end
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abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top"
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abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x"
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declare top_fset.rep_eq[simp]
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subsection \<open>Other operations\<close>
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lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer
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  by simp
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syntax
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  "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")
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translations
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  "{|x, xs|}" == "CONST finsert x {|xs|}"
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  "{|x|}"     == "CONST finsert x {||}"
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lift_definition fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) is Set.member 
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  parametric member_transfer .
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abbreviation notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
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context
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begin
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interpretation lifting_syntax .
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lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter 
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  parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp
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lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer 
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by (simp add: finite_subset)
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lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer .
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lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image 
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  parametric image_transfer by simp
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lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem .
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lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer 
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by (simp add: Set.bind_def)
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lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer by simp
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lift_definition fBall :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Ball parametric Ball_transfer .
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lift_definition fBex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Bex parametric Bex_transfer .
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lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold .
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subsection \<open>Transferred lemmas from Set.thy\<close>
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lemmas fset_eqI = set_eqI[Transfer.transferred]
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lemmas fset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]
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lemmas fBallI[intro!] = ballI[Transfer.transferred]
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lemmas fbspec[dest?] = bspec[Transfer.transferred]
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lemmas fBallE[elim] = ballE[Transfer.transferred]
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lemmas fBexI[intro] = bexI[Transfer.transferred]
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lemmas rev_fBexI[intro?] = rev_bexI[Transfer.transferred]
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lemmas fBexCI = bexCI[Transfer.transferred]
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lemmas fBexE[elim!] = bexE[Transfer.transferred]
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lemmas fBall_triv[simp] = ball_triv[Transfer.transferred]
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lemmas fBex_triv[simp] = bex_triv[Transfer.transferred]
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lemmas fBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]
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lemmas fBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]
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lemmas fBex_one_point1[simp] = bex_one_point1[Transfer.transferred]
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lemmas fBex_one_point2[simp] = bex_one_point2[Transfer.transferred]
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lemmas fBall_one_point1[simp] = ball_one_point1[Transfer.transferred]
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lemmas fBall_one_point2[simp] = ball_one_point2[Transfer.transferred]
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lemmas fBall_conj_distrib = ball_conj_distrib[Transfer.transferred]
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lemmas fBex_disj_distrib = bex_disj_distrib[Transfer.transferred]
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lemmas fBall_cong = ball_cong[Transfer.transferred]
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lemmas fBex_cong = bex_cong[Transfer.transferred]
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lemmas fsubsetI[intro!] = subsetI[Transfer.transferred]
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lemmas fsubsetD[elim, intro?] = subsetD[Transfer.transferred]
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lemmas rev_fsubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]
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lemmas fsubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]
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lemmas fsubset_eq[no_atp] = subset_eq[Transfer.transferred]
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lemmas contra_fsubsetD[no_atp] = contra_subsetD[Transfer.transferred]
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lemmas fsubset_refl = subset_refl[Transfer.transferred]
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lemmas fsubset_trans = subset_trans[Transfer.transferred]
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lemmas fset_rev_mp = set_rev_mp[Transfer.transferred]
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lemmas fset_mp = set_mp[Transfer.transferred]
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lemmas fsubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]
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lemmas eq_fmem_trans = eq_mem_trans[Transfer.transferred]
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lemmas fsubset_antisym[intro!] = subset_antisym[Transfer.transferred]
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lemmas fequalityD1 = equalityD1[Transfer.transferred]
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lemmas fequalityD2 = equalityD2[Transfer.transferred]
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lemmas fequalityE = equalityE[Transfer.transferred]
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lemmas fequalityCE[elim] = equalityCE[Transfer.transferred]
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lemmas eqfset_imp_iff = eqset_imp_iff[Transfer.transferred]
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lemmas eqfelem_imp_iff = eqelem_imp_iff[Transfer.transferred]
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lemmas fempty_iff[simp] = empty_iff[Transfer.transferred]
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lemmas fempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]
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lemmas equalsffemptyI = equals0I[Transfer.transferred]
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lemmas equalsffemptyD = equals0D[Transfer.transferred]
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lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]
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lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]
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lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]
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lemmas fPowI = PowI[Transfer.transferred]
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lemmas fPowD = PowD[Transfer.transferred]
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lemmas fPow_bottom = Pow_bottom[Transfer.transferred]
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lemmas fPow_top = Pow_top[Transfer.transferred]
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lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]
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lemmas finter_iff[simp] = Int_iff[Transfer.transferred]
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lemmas finterI[intro!] = IntI[Transfer.transferred]
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lemmas finterD1 = IntD1[Transfer.transferred]
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lemmas finterD2 = IntD2[Transfer.transferred]
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lemmas finterE[elim!] = IntE[Transfer.transferred]
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lemmas funion_iff[simp] = Un_iff[Transfer.transferred]
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lemmas funionI1[elim?] = UnI1[Transfer.transferred]
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lemmas funionI2[elim?] = UnI2[Transfer.transferred]
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lemmas funionCI[intro!] = UnCI[Transfer.transferred]
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lemmas funionE[elim!] = UnE[Transfer.transferred]
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lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]
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lemmas fminusI[intro!] = DiffI[Transfer.transferred]
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lemmas fminusD1 = DiffD1[Transfer.transferred]
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lemmas fminusD2 = DiffD2[Transfer.transferred]
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lemmas fminusE[elim!] = DiffE[Transfer.transferred]
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lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]
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lemmas finsertI1 = insertI1[Transfer.transferred]
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lemmas finsertI2 = insertI2[Transfer.transferred]
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lemmas finsertE[elim!] = insertE[Transfer.transferred]
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lemmas finsertCI[intro!] = insertCI[Transfer.transferred]
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lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]
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lemmas finsert_ident = insert_ident[Transfer.transferred]
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lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
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lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
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lemmas fsingleton_iff = singleton_iff[Transfer.transferred]
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lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]
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lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
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lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
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   288
lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
kuncar@53953
   289
lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
kuncar@53953
   290
lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
kuncar@53953
   291
lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
kuncar@53953
   292
lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
kuncar@53953
   293
lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
kuncar@53953
   294
lemmas fimageI = imageI[Transfer.transferred]
kuncar@53953
   295
lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
kuncar@53953
   296
lemmas fimageE[elim!] = imageE[Transfer.transferred]
kuncar@53953
   297
lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
kuncar@53953
   298
lemmas fimage_funion = image_Un[Transfer.transferred]
kuncar@53953
   299
lemmas fimage_iff = image_iff[Transfer.transferred]
kuncar@53964
   300
lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
kuncar@53964
   301
lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
kuncar@53953
   302
lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
kuncar@53953
   303
lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
kuncar@53953
   304
lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
kuncar@53964
   305
lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
kuncar@53964
   306
lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
kuncar@53964
   307
lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
kuncar@53964
   308
lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
kuncar@53964
   309
lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
kuncar@53964
   310
lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
kuncar@53964
   311
lemmas pfsubsetD = psubsetD[Transfer.transferred]
kuncar@53964
   312
lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
kuncar@53964
   313
lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
kuncar@53964
   314
lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
kuncar@53953
   315
lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
kuncar@53953
   316
lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
kuncar@53964
   317
lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
kuncar@53964
   318
lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
kuncar@53964
   319
lemmas fsubset_finsert = subset_insert[Transfer.transferred]
kuncar@53953
   320
lemmas funion_upper1 = Un_upper1[Transfer.transferred]
kuncar@53953
   321
lemmas funion_upper2 = Un_upper2[Transfer.transferred]
kuncar@53953
   322
lemmas funion_least = Un_least[Transfer.transferred]
kuncar@53953
   323
lemmas finter_lower1 = Int_lower1[Transfer.transferred]
kuncar@53953
   324
lemmas finter_lower2 = Int_lower2[Transfer.transferred]
kuncar@53953
   325
lemmas finter_greatest = Int_greatest[Transfer.transferred]
kuncar@53964
   326
lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
kuncar@53964
   327
lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
kuncar@53964
   328
lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
kuncar@53964
   329
lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
kuncar@53953
   330
lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
kuncar@53953
   331
lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
kuncar@53953
   332
lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
kuncar@53953
   333
lemmas finsert_absorb = insert_absorb[Transfer.transferred]
kuncar@53953
   334
lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
kuncar@53953
   335
lemmas finsert_commute = insert_commute[Transfer.transferred]
kuncar@53964
   336
lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
kuncar@53953
   337
lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
kuncar@53953
   338
lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
kuncar@53953
   339
lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
kuncar@53953
   340
lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
kuncar@53953
   341
lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
kuncar@53953
   342
lemmas fimage_constant = image_constant[Transfer.transferred]
kuncar@53953
   343
lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
kuncar@53953
   344
lemmas fimage_fimage = image_image[Transfer.transferred]
kuncar@53953
   345
lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
kuncar@53953
   346
lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
kuncar@53953
   347
lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
kuncar@53953
   348
lemmas fimage_cong = image_cong[Transfer.transferred]
kuncar@53964
   349
lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
kuncar@53964
   350
lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
kuncar@53953
   351
lemmas finter_absorb = Int_absorb[Transfer.transferred]
kuncar@53953
   352
lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
kuncar@53953
   353
lemmas finter_commute = Int_commute[Transfer.transferred]
kuncar@53953
   354
lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
kuncar@53953
   355
lemmas finter_assoc = Int_assoc[Transfer.transferred]
kuncar@53953
   356
lemmas finter_ac = Int_ac[Transfer.transferred]
kuncar@53953
   357
lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
kuncar@53953
   358
lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
kuncar@53953
   359
lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
kuncar@53953
   360
lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
kuncar@53953
   361
lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
kuncar@53953
   362
lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
kuncar@53953
   363
lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
kuncar@53964
   364
lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
kuncar@53953
   365
lemmas funion_absorb = Un_absorb[Transfer.transferred]
kuncar@53953
   366
lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
kuncar@53953
   367
lemmas funion_commute = Un_commute[Transfer.transferred]
kuncar@53953
   368
lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
kuncar@53953
   369
lemmas funion_assoc = Un_assoc[Transfer.transferred]
kuncar@53953
   370
lemmas funion_ac = Un_ac[Transfer.transferred]
kuncar@53953
   371
lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
kuncar@53953
   372
lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
kuncar@53953
   373
lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
kuncar@53953
   374
lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
kuncar@53953
   375
lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
kuncar@53953
   376
lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
kuncar@53953
   377
lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
kuncar@53953
   378
lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
kuncar@53953
   379
lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
kuncar@53953
   380
lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
kuncar@53953
   381
lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
kuncar@53953
   382
lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
kuncar@53953
   383
lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
kuncar@53953
   384
lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
kuncar@53953
   385
lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
kuncar@53964
   386
lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
kuncar@53953
   387
lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
kuncar@53964
   388
lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
kuncar@53953
   389
lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
kuncar@53953
   390
lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
kuncar@53953
   391
lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
kuncar@53953
   392
lemmas fBall_funion = ball_Un[Transfer.transferred]
kuncar@53953
   393
lemmas fBex_funion = bex_Un[Transfer.transferred]
kuncar@53953
   394
lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
kuncar@53953
   395
lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
kuncar@53953
   396
lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
kuncar@53953
   397
lemmas fminus_triv = Diff_triv[Transfer.transferred]
kuncar@53953
   398
lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
kuncar@53953
   399
lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
kuncar@53953
   400
lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
kuncar@53953
   401
lemmas fminus_finsert = Diff_insert[Transfer.transferred]
kuncar@53953
   402
lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
kuncar@53953
   403
lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
kuncar@53953
   404
lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
kuncar@53953
   405
lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
kuncar@53953
   406
lemmas finsert_fminus = insert_Diff[Transfer.transferred]
kuncar@53953
   407
lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
kuncar@53953
   408
lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
kuncar@53953
   409
lemmas fminus_partition = Diff_partition[Transfer.transferred]
kuncar@53953
   410
lemmas double_fminus = double_diff[Transfer.transferred]
kuncar@53953
   411
lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
kuncar@53953
   412
lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
kuncar@53953
   413
lemmas fminus_funion = Diff_Un[Transfer.transferred]
kuncar@53953
   414
lemmas fminus_finter = Diff_Int[Transfer.transferred]
kuncar@53953
   415
lemmas funion_fminus = Un_Diff[Transfer.transferred]
kuncar@53953
   416
lemmas finter_fminus = Int_Diff[Transfer.transferred]
kuncar@53953
   417
lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
kuncar@53953
   418
lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
kuncar@53953
   419
lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
kuncar@53953
   420
lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
kuncar@53953
   421
lemmas fPow_finsert = Pow_insert[Transfer.transferred]
kuncar@53964
   422
lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
kuncar@53953
   423
lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
kuncar@53964
   424
lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
kuncar@53964
   425
lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
kuncar@53964
   426
lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
kuncar@53953
   427
lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
kuncar@53953
   428
lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
kuncar@53953
   429
lemmas fimage_mono = image_mono[Transfer.transferred]
kuncar@53953
   430
lemmas fPow_mono = Pow_mono[Transfer.transferred]
kuncar@53953
   431
lemmas finsert_mono = insert_mono[Transfer.transferred]
kuncar@53953
   432
lemmas funion_mono = Un_mono[Transfer.transferred]
kuncar@53953
   433
lemmas finter_mono = Int_mono[Transfer.transferred]
kuncar@53953
   434
lemmas fminus_mono = Diff_mono[Transfer.transferred]
kuncar@53953
   435
lemmas fin_mono = in_mono[Transfer.transferred]
kuncar@53953
   436
lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
kuncar@53953
   437
lemmas fLeast_mono = Least_mono[Transfer.transferred]
kuncar@53953
   438
lemmas fbind_fbind = bind_bind[Transfer.transferred]
kuncar@53953
   439
lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
kuncar@53953
   440
lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
kuncar@53953
   441
lemmas fbind_const = bind_const[Transfer.transferred]
kuncar@53953
   442
lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
kuncar@53953
   443
lemmas fequalityI = equalityI[Transfer.transferred]
kuncar@53953
   444
blanchet@55129
   445
wenzelm@60500
   446
subsection \<open>Additional lemmas\<close>
kuncar@53953
   447
wenzelm@60500
   448
subsubsection \<open>@{text fsingleton}\<close>
kuncar@53953
   449
kuncar@53953
   450
lemmas fsingletonE = fsingletonD [elim_format]
kuncar@53953
   451
blanchet@55129
   452
wenzelm@60500
   453
subsubsection \<open>@{text femepty}\<close>
kuncar@53953
   454
kuncar@53953
   455
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
kuncar@53953
   456
by transfer auto
kuncar@53953
   457
kuncar@53953
   458
(* FIXME, transferred doesn't work here *)
kuncar@53953
   459
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
kuncar@53953
   460
  by simp
kuncar@53953
   461
blanchet@55129
   462
wenzelm@60500
   463
subsubsection \<open>@{text fset}\<close>
kuncar@53953
   464
kuncar@53963
   465
lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
kuncar@53953
   466
kuncar@53953
   467
lemma finite_fset [simp]: 
kuncar@53953
   468
  shows "finite (fset S)"
kuncar@53953
   469
  by transfer simp
kuncar@53953
   470
kuncar@53963
   471
lemmas fset_cong = fset_inject
kuncar@53953
   472
kuncar@53953
   473
lemma filter_fset [simp]:
kuncar@53953
   474
  shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
kuncar@53953
   475
  by transfer auto
kuncar@53953
   476
kuncar@53963
   477
lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
kuncar@53963
   478
kuncar@53963
   479
lemmas inter_fset[simp] = inf_fset.rep_eq
kuncar@53953
   480
kuncar@53963
   481
lemmas union_fset[simp] = sup_fset.rep_eq
kuncar@53953
   482
kuncar@53963
   483
lemmas minus_fset[simp] = minus_fset.rep_eq
kuncar@53953
   484
blanchet@55129
   485
wenzelm@60500
   486
subsubsection \<open>@{text filter_fset}\<close>
kuncar@53953
   487
kuncar@53953
   488
lemma subset_ffilter: 
kuncar@53953
   489
  "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
kuncar@53953
   490
  by transfer auto
kuncar@53953
   491
kuncar@53953
   492
lemma eq_ffilter: 
kuncar@53953
   493
  "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
kuncar@53953
   494
  by transfer auto
kuncar@53953
   495
kuncar@53964
   496
lemma pfsubset_ffilter:
kuncar@53953
   497
  "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow> 
kuncar@53953
   498
    ffilter P A |\<subset>| ffilter Q A"
kuncar@53953
   499
  unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
kuncar@53953
   500
blanchet@55129
   501
wenzelm@60500
   502
subsubsection \<open>@{text finsert}\<close>
kuncar@53953
   503
kuncar@53953
   504
(* FIXME, transferred doesn't work here *)
kuncar@53953
   505
lemma set_finsert:
kuncar@53953
   506
  assumes "x |\<in>| A"
kuncar@53953
   507
  obtains B where "A = finsert x B" and "x |\<notin>| B"
kuncar@53953
   508
using assms by transfer (metis Set.set_insert finite_insert)
kuncar@53953
   509
kuncar@53953
   510
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
kuncar@53953
   511
  by (rule_tac x = "A |-| {|a|}" in exI, blast)
kuncar@53953
   512
blanchet@55129
   513
wenzelm@60500
   514
subsubsection \<open>@{text fimage}\<close>
kuncar@53953
   515
kuncar@53953
   516
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
kuncar@53953
   517
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
kuncar@53953
   518
blanchet@55129
   519
wenzelm@60500
   520
subsubsection \<open>bounded quantification\<close>
kuncar@53953
   521
kuncar@53953
   522
lemma bex_simps [simp, no_atp]:
kuncar@53953
   523
  "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" 
kuncar@53953
   524
  "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
kuncar@53953
   525
  "\<And>P. fBex {||} P = False" 
kuncar@53953
   526
  "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
kuncar@53953
   527
  "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
kuncar@53953
   528
  "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
kuncar@53953
   529
by auto
kuncar@53953
   530
kuncar@53953
   531
lemma ball_simps [simp, no_atp]:
kuncar@53953
   532
  "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
kuncar@53953
   533
  "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
kuncar@53953
   534
  "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
kuncar@53953
   535
  "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
kuncar@53953
   536
  "\<And>P. fBall {||} P = True"
kuncar@53953
   537
  "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
kuncar@53953
   538
  "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
kuncar@53953
   539
  "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
kuncar@53953
   540
by auto
kuncar@53953
   541
kuncar@53953
   542
lemma atomize_fBall:
kuncar@53953
   543
    "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
kuncar@53953
   544
apply (simp only: atomize_all atomize_imp)
kuncar@53953
   545
apply (rule equal_intr_rule)
kuncar@53953
   546
by (transfer, simp)+
kuncar@53953
   547
kuncar@53963
   548
end
kuncar@53963
   549
blanchet@55129
   550
wenzelm@60500
   551
subsubsection \<open>@{text fcard}\<close>
kuncar@53963
   552
kuncar@53964
   553
(* FIXME: improve transferred to handle bounded meta quantification *)
kuncar@53964
   554
kuncar@53963
   555
lemma fcard_fempty:
kuncar@53963
   556
  "fcard {||} = 0"
kuncar@53963
   557
  by transfer (rule card_empty)
kuncar@53963
   558
kuncar@53963
   559
lemma fcard_finsert_disjoint:
kuncar@53963
   560
  "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
kuncar@53963
   561
  by transfer (rule card_insert_disjoint)
kuncar@53963
   562
kuncar@53963
   563
lemma fcard_finsert_if:
kuncar@53963
   564
  "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
kuncar@53963
   565
  by transfer (rule card_insert_if)
kuncar@53963
   566
kuncar@53963
   567
lemma card_0_eq [simp, no_atp]:
kuncar@53963
   568
  "fcard A = 0 \<longleftrightarrow> A = {||}"
kuncar@53963
   569
  by transfer (rule card_0_eq)
kuncar@53963
   570
kuncar@53963
   571
lemma fcard_Suc_fminus1:
kuncar@53963
   572
  "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
kuncar@53963
   573
  by transfer (rule card_Suc_Diff1)
kuncar@53963
   574
kuncar@53963
   575
lemma fcard_fminus_fsingleton:
kuncar@53963
   576
  "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
kuncar@53963
   577
  by transfer (rule card_Diff_singleton)
kuncar@53963
   578
kuncar@53963
   579
lemma fcard_fminus_fsingleton_if:
kuncar@53963
   580
  "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
kuncar@53963
   581
  by transfer (rule card_Diff_singleton_if)
kuncar@53963
   582
kuncar@53963
   583
lemma fcard_fminus_finsert[simp]:
kuncar@53963
   584
  assumes "a |\<in>| A" and "a |\<notin>| B"
kuncar@53963
   585
  shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
kuncar@53963
   586
using assms by transfer (rule card_Diff_insert)
kuncar@53963
   587
kuncar@53963
   588
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
kuncar@53963
   589
by transfer (rule card_insert)
kuncar@53963
   590
kuncar@53963
   591
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
kuncar@53963
   592
by transfer (rule card_insert_le)
kuncar@53963
   593
kuncar@53963
   594
lemma fcard_mono:
kuncar@53963
   595
  "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
kuncar@53963
   596
by transfer (rule card_mono)
kuncar@53963
   597
kuncar@53963
   598
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
kuncar@53963
   599
by transfer (rule card_seteq)
kuncar@53963
   600
kuncar@53963
   601
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
kuncar@53963
   602
by transfer (rule psubset_card_mono)
kuncar@53963
   603
kuncar@53963
   604
lemma fcard_funion_finter: 
kuncar@53963
   605
  "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
kuncar@53963
   606
by transfer (rule card_Un_Int)
kuncar@53963
   607
kuncar@53963
   608
lemma fcard_funion_disjoint:
kuncar@53963
   609
  "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
kuncar@53963
   610
by transfer (rule card_Un_disjoint)
kuncar@53963
   611
kuncar@53963
   612
lemma fcard_funion_fsubset:
kuncar@53963
   613
  "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
kuncar@53963
   614
by transfer (rule card_Diff_subset)
kuncar@53963
   615
kuncar@53963
   616
lemma diff_fcard_le_fcard_fminus:
kuncar@53963
   617
  "fcard A - fcard B \<le> fcard(A |-| B)"
kuncar@53963
   618
by transfer (rule diff_card_le_card_Diff)
kuncar@53963
   619
kuncar@53963
   620
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
kuncar@53963
   621
by transfer (rule card_Diff1_less)
kuncar@53963
   622
kuncar@53963
   623
lemma fcard_fminus2_less:
kuncar@53963
   624
  "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
kuncar@53963
   625
by transfer (rule card_Diff2_less)
kuncar@53963
   626
kuncar@53963
   627
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
kuncar@53963
   628
by transfer (rule card_Diff1_le)
kuncar@53963
   629
kuncar@53963
   630
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
kuncar@53963
   631
by transfer (rule card_psubset)
kuncar@53963
   632
blanchet@55129
   633
wenzelm@60500
   634
subsubsection \<open>@{text ffold}\<close>
kuncar@53963
   635
kuncar@53963
   636
(* FIXME: improve transferred to handle bounded meta quantification *)
kuncar@53963
   637
kuncar@53963
   638
context comp_fun_commute
kuncar@53963
   639
begin
kuncar@53963
   640
  lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
kuncar@53963
   641
kuncar@53963
   642
  lemma ffold_finsert [simp]:
kuncar@53963
   643
    assumes "x |\<notin>| A"
kuncar@53963
   644
    shows "ffold f z (finsert x A) = f x (ffold f z A)"
kuncar@53963
   645
    using assms by (transfer fixing: f) (rule fold_insert)
kuncar@53963
   646
kuncar@53963
   647
  lemma ffold_fun_left_comm:
kuncar@53963
   648
    "f x (ffold f z A) = ffold f (f x z) A"
kuncar@53963
   649
    by (transfer fixing: f) (rule fold_fun_left_comm)
kuncar@53963
   650
kuncar@53963
   651
  lemma ffold_finsert2:
blanchet@56646
   652
    "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A) = ffold f (f x z) A"
kuncar@53963
   653
    by (transfer fixing: f) (rule fold_insert2)
kuncar@53963
   654
kuncar@53963
   655
  lemma ffold_rec:
kuncar@53963
   656
    assumes "x |\<in>| A"
kuncar@53963
   657
    shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
kuncar@53963
   658
    using assms by (transfer fixing: f) (rule fold_rec)
kuncar@53963
   659
  
kuncar@53963
   660
  lemma ffold_finsert_fremove:
kuncar@53963
   661
    "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
kuncar@53963
   662
     by (transfer fixing: f) (rule fold_insert_remove)
kuncar@53963
   663
end
kuncar@53963
   664
kuncar@53963
   665
lemma ffold_fimage:
kuncar@53963
   666
  assumes "inj_on g (fset A)"
kuncar@53963
   667
  shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
kuncar@53963
   668
using assms by transfer' (rule fold_image)
kuncar@53963
   669
kuncar@53963
   670
lemma ffold_cong:
kuncar@53963
   671
  assumes "comp_fun_commute f" "comp_fun_commute g"
kuncar@53963
   672
  "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
kuncar@53963
   673
    and "s = t" and "A = B"
kuncar@53963
   674
  shows "ffold f s A = ffold g t B"
kuncar@53963
   675
using assms by transfer (metis Finite_Set.fold_cong)
kuncar@53963
   676
kuncar@53963
   677
context comp_fun_idem
kuncar@53963
   678
begin
kuncar@53963
   679
kuncar@53963
   680
  lemma ffold_finsert_idem:
blanchet@56646
   681
    "ffold f z (finsert x A) = f x (ffold f z A)"
kuncar@53963
   682
    by (transfer fixing: f) (rule fold_insert_idem)
kuncar@53963
   683
  
kuncar@53963
   684
  declare ffold_finsert [simp del] ffold_finsert_idem [simp]
kuncar@53963
   685
  
kuncar@53963
   686
  lemma ffold_finsert_idem2:
kuncar@53963
   687
    "ffold f z (finsert x A) = ffold f (f x z) A"
kuncar@53963
   688
    by (transfer fixing: f) (rule fold_insert_idem2)
kuncar@53963
   689
kuncar@53963
   690
end
kuncar@53963
   691
blanchet@55129
   692
wenzelm@60500
   693
subsection \<open>Choice in fsets\<close>
kuncar@53953
   694
kuncar@53953
   695
lemma fset_choice: 
kuncar@53953
   696
  assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
kuncar@53953
   697
  shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
kuncar@53953
   698
  using assms by transfer metis
kuncar@53953
   699
blanchet@55129
   700
wenzelm@60500
   701
subsection \<open>Induction and Cases rules for fsets\<close>
kuncar@53953
   702
kuncar@53953
   703
lemma fset_exhaust [case_names empty insert, cases type: fset]:
kuncar@53953
   704
  assumes fempty_case: "S = {||} \<Longrightarrow> P" 
kuncar@53953
   705
  and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
kuncar@53953
   706
  shows "P"
kuncar@53953
   707
  using assms by transfer blast
kuncar@53953
   708
kuncar@53953
   709
lemma fset_induct [case_names empty insert]:
kuncar@53953
   710
  assumes fempty_case: "P {||}"
kuncar@53953
   711
  and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
kuncar@53953
   712
  shows "P S"
kuncar@53953
   713
proof -
kuncar@53953
   714
  (* FIXME transfer and right_total vs. bi_total *)
kuncar@53953
   715
  note Domainp_forall_transfer[transfer_rule]
kuncar@53953
   716
  show ?thesis
kuncar@53953
   717
  using assms by transfer (auto intro: finite_induct)
kuncar@53953
   718
qed
kuncar@53953
   719
kuncar@53953
   720
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
kuncar@53953
   721
  assumes empty_fset_case: "P {||}"
kuncar@53953
   722
  and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
kuncar@53953
   723
  shows "P S"
kuncar@53953
   724
proof -
kuncar@53953
   725
  (* FIXME transfer and right_total vs. bi_total *)
kuncar@53953
   726
  note Domainp_forall_transfer[transfer_rule]
kuncar@53953
   727
  show ?thesis
kuncar@53953
   728
  using assms by transfer (auto intro: finite_induct)
kuncar@53953
   729
qed
kuncar@53953
   730
kuncar@53953
   731
lemma fset_card_induct:
kuncar@53953
   732
  assumes empty_fset_case: "P {||}"
kuncar@53953
   733
  and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
kuncar@53953
   734
  shows "P S"
kuncar@53953
   735
proof (induct S)
kuncar@53953
   736
  case empty
kuncar@53953
   737
  show "P {||}" by (rule empty_fset_case)
kuncar@53953
   738
next
kuncar@53953
   739
  case (insert x S)
kuncar@53953
   740
  have h: "P S" by fact
kuncar@53953
   741
  have "x |\<notin>| S" by fact
kuncar@53953
   742
  then have "Suc (fcard S) = fcard (finsert x S)" 
kuncar@53953
   743
    by transfer auto
kuncar@53953
   744
  then show "P (finsert x S)" 
kuncar@53953
   745
    using h card_fset_Suc_case by simp
kuncar@53953
   746
qed
kuncar@53953
   747
kuncar@53953
   748
lemma fset_strong_cases:
kuncar@53953
   749
  obtains "xs = {||}"
kuncar@53953
   750
    | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
kuncar@53953
   751
by transfer blast
kuncar@53953
   752
kuncar@53953
   753
lemma fset_induct2:
kuncar@53953
   754
  "P {||} {||} \<Longrightarrow>
kuncar@53953
   755
  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
kuncar@53953
   756
  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
kuncar@53953
   757
  (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow>
kuncar@53953
   758
  P xsa ysa"
kuncar@53953
   759
  apply (induct xsa arbitrary: ysa)
kuncar@53953
   760
  apply (induct_tac x rule: fset_induct_stronger)
kuncar@53953
   761
  apply simp_all
kuncar@53953
   762
  apply (induct_tac xa rule: fset_induct_stronger)
kuncar@53953
   763
  apply simp_all
kuncar@53953
   764
  done
kuncar@53953
   765
blanchet@55129
   766
wenzelm@60500
   767
subsection \<open>Setup for Lifting/Transfer\<close>
kuncar@53953
   768
wenzelm@60500
   769
subsubsection \<open>Relator and predicator properties\<close>
kuncar@53953
   770
blanchet@55938
   771
lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set
blanchet@55938
   772
parametric rel_set_transfer .
kuncar@53953
   773
blanchet@55933
   774
lemma rel_fset_alt_def: "rel_fset R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y) 
kuncar@53953
   775
  \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
kuncar@53953
   776
apply (rule ext)+
kuncar@53953
   777
apply transfer'
blanchet@55938
   778
apply (subst rel_set_def[unfolded fun_eq_iff]) 
kuncar@53953
   779
by blast
kuncar@53953
   780
blanchet@55938
   781
lemma finite_rel_set:
kuncar@53953
   782
  assumes fin: "finite X" "finite Z"
blanchet@55938
   783
  assumes R_S: "rel_set (R OO S) X Z"
blanchet@55938
   784
  shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"
kuncar@53953
   785
proof -
kuncar@53953
   786
  obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
kuncar@53953
   787
  apply atomize_elim
kuncar@53953
   788
  apply (subst bchoice_iff[symmetric])
blanchet@55938
   789
  using R_S[unfolded rel_set_def OO_def] by blast
kuncar@53953
   790
  
blanchet@56646
   791
  obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))"
kuncar@53953
   792
  apply atomize_elim
kuncar@53953
   793
  apply (subst bchoice_iff[symmetric])
blanchet@55938
   794
  using R_S[unfolded rel_set_def OO_def] by blast
kuncar@53953
   795
  
kuncar@53953
   796
  let ?Y = "f ` X \<union> g ` Z"
kuncar@53953
   797
  have "finite ?Y" by (simp add: fin)
blanchet@55938
   798
  moreover have "rel_set R X ?Y"
blanchet@55938
   799
    unfolding rel_set_def
kuncar@53953
   800
    using f g by clarsimp blast
blanchet@55938
   801
  moreover have "rel_set S ?Y Z"
blanchet@55938
   802
    unfolding rel_set_def
kuncar@53953
   803
    using f g by clarsimp blast
kuncar@53953
   804
  ultimately show ?thesis by metis
kuncar@53953
   805
qed
kuncar@53953
   806
wenzelm@60500
   807
subsubsection \<open>Transfer rules for the Transfer package\<close>
kuncar@53953
   808
wenzelm@60500
   809
text \<open>Unconditional transfer rules\<close>
kuncar@53953
   810
kuncar@53963
   811
context
kuncar@53963
   812
begin
kuncar@53963
   813
kuncar@53963
   814
interpretation lifting_syntax .
kuncar@53963
   815
kuncar@53953
   816
lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
kuncar@53953
   817
kuncar@53953
   818
lemma finsert_transfer [transfer_rule]:
blanchet@55933
   819
  "(A ===> rel_fset A ===> rel_fset A) finsert finsert"
blanchet@55945
   820
  unfolding rel_fun_def rel_fset_alt_def by blast
kuncar@53953
   821
kuncar@53953
   822
lemma funion_transfer [transfer_rule]:
blanchet@55933
   823
  "(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion"
blanchet@55945
   824
  unfolding rel_fun_def rel_fset_alt_def by blast
kuncar@53953
   825
kuncar@53953
   826
lemma ffUnion_transfer [transfer_rule]:
blanchet@55933
   827
  "(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion"
blanchet@55945
   828
  unfolding rel_fun_def rel_fset_alt_def by transfer (simp, fast)
kuncar@53953
   829
kuncar@53953
   830
lemma fimage_transfer [transfer_rule]:
blanchet@55933
   831
  "((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage"
blanchet@55945
   832
  unfolding rel_fun_def rel_fset_alt_def by simp blast
kuncar@53953
   833
kuncar@53953
   834
lemma fBall_transfer [transfer_rule]:
blanchet@55933
   835
  "(rel_fset A ===> (A ===> op =) ===> op =) fBall fBall"
blanchet@55945
   836
  unfolding rel_fset_alt_def rel_fun_def by blast
kuncar@53953
   837
kuncar@53953
   838
lemma fBex_transfer [transfer_rule]:
blanchet@55933
   839
  "(rel_fset A ===> (A ===> op =) ===> op =) fBex fBex"
blanchet@55945
   840
  unfolding rel_fset_alt_def rel_fun_def by blast
kuncar@53953
   841
kuncar@53953
   842
(* FIXME transfer doesn't work here *)
kuncar@53953
   843
lemma fPow_transfer [transfer_rule]:
blanchet@55933
   844
  "(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow"
blanchet@55945
   845
  unfolding rel_fun_def
blanchet@55945
   846
  using Pow_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred]
kuncar@53953
   847
  by blast
kuncar@53953
   848
blanchet@55933
   849
lemma rel_fset_transfer [transfer_rule]:
blanchet@55933
   850
  "((A ===> B ===> op =) ===> rel_fset A ===> rel_fset B ===> op =)
blanchet@55933
   851
    rel_fset rel_fset"
blanchet@55945
   852
  unfolding rel_fun_def
blanchet@55945
   853
  using rel_set_transfer[unfolded rel_fun_def,rule_format, Transfer.transferred, where A = A and B = B]
kuncar@53953
   854
  by simp
kuncar@53953
   855
kuncar@53953
   856
lemma bind_transfer [transfer_rule]:
blanchet@55933
   857
  "(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind"
blanchet@55945
   858
  using assms unfolding rel_fun_def
blanchet@55945
   859
  using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   860
wenzelm@60500
   861
text \<open>Rules requiring bi-unique, bi-total or right-total relations\<close>
kuncar@53953
   862
kuncar@53953
   863
lemma fmember_transfer [transfer_rule]:
kuncar@53953
   864
  assumes "bi_unique A"
blanchet@55933
   865
  shows "(A ===> rel_fset A ===> op =) (op |\<in>|) (op |\<in>|)"
blanchet@55945
   866
  using assms unfolding rel_fun_def rel_fset_alt_def bi_unique_def by metis
kuncar@53953
   867
kuncar@53953
   868
lemma finter_transfer [transfer_rule]:
kuncar@53953
   869
  assumes "bi_unique A"
blanchet@55933
   870
  shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter"
blanchet@55945
   871
  using assms unfolding rel_fun_def
blanchet@55945
   872
  using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   873
kuncar@53963
   874
lemma fminus_transfer [transfer_rule]:
kuncar@53953
   875
  assumes "bi_unique A"
blanchet@55933
   876
  shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (op |-|) (op |-|)"
blanchet@55945
   877
  using assms unfolding rel_fun_def
blanchet@55945
   878
  using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   879
kuncar@53953
   880
lemma fsubset_transfer [transfer_rule]:
kuncar@53953
   881
  assumes "bi_unique A"
blanchet@55933
   882
  shows "(rel_fset A ===> rel_fset A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
blanchet@55945
   883
  using assms unfolding rel_fun_def
blanchet@55945
   884
  using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   885
kuncar@53953
   886
lemma fSup_transfer [transfer_rule]:
blanchet@55938
   887
  "bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"
blanchet@55945
   888
  using assms unfolding rel_fun_def
kuncar@53953
   889
  apply clarify
kuncar@53953
   890
  apply transfer'
blanchet@55945
   891
  using Sup_fset_transfer[unfolded rel_fun_def] by blast
kuncar@53953
   892
kuncar@53953
   893
(* FIXME: add right_total_fInf_transfer *)
kuncar@53953
   894
kuncar@53953
   895
lemma fInf_transfer [transfer_rule]:
kuncar@53953
   896
  assumes "bi_unique A" and "bi_total A"
blanchet@55938
   897
  shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"
blanchet@55945
   898
  using assms unfolding rel_fun_def
kuncar@53953
   899
  apply clarify
kuncar@53953
   900
  apply transfer'
blanchet@55945
   901
  using Inf_fset_transfer[unfolded rel_fun_def] by blast
kuncar@53953
   902
kuncar@53953
   903
lemma ffilter_transfer [transfer_rule]:
kuncar@53953
   904
  assumes "bi_unique A"
blanchet@55933
   905
  shows "((A ===> op=) ===> rel_fset A ===> rel_fset A) ffilter ffilter"
blanchet@55945
   906
  using assms unfolding rel_fun_def
blanchet@55945
   907
  using Lifting_Set.filter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   908
kuncar@53953
   909
lemma card_transfer [transfer_rule]:
blanchet@55933
   910
  "bi_unique A \<Longrightarrow> (rel_fset A ===> op =) fcard fcard"
blanchet@55945
   911
  using assms unfolding rel_fun_def
blanchet@55945
   912
  using card_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   913
kuncar@53953
   914
end
kuncar@53953
   915
kuncar@53953
   916
lifting_update fset.lifting
kuncar@53953
   917
lifting_forget fset.lifting
kuncar@53953
   918
blanchet@55129
   919
wenzelm@60500
   920
subsection \<open>BNF setup\<close>
blanchet@55129
   921
blanchet@55129
   922
context
blanchet@55129
   923
includes fset.lifting
blanchet@55129
   924
begin
blanchet@55129
   925
blanchet@55933
   926
lemma rel_fset_alt:
blanchet@55933
   927
  "rel_fset R a b \<longleftrightarrow> (\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>t \<in> fset b. \<exists>u \<in> fset a. R u t)"
blanchet@55938
   928
by transfer (simp add: rel_set_def)
blanchet@55129
   929
blanchet@55129
   930
lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
blanchet@55129
   931
apply (rule f_the_inv_into_f[unfolded inj_on_def])
blanchet@55129
   932
apply (simp add: fset_inject)
blanchet@55129
   933
apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
blanchet@55129
   934
.
blanchet@55129
   935
blanchet@55933
   936
lemma rel_fset_aux:
blanchet@55129
   937
"(\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>u \<in> fset b. \<exists>t \<in> fset a. R t u) \<longleftrightarrow>
blanchet@57398
   938
 ((BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
blanchet@57398
   939
  BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
blanchet@55129
   940
proof
blanchet@55129
   941
  assume ?L
haftmann@61424
   942
  def R' \<equiv> "the_inv fset (Collect (case_prod R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?L'")
blanchet@55129
   943
  have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
blanchet@55129
   944
  hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
blanchet@55129
   945
  show ?R unfolding Grp_def relcompp.simps conversep.simps
blanchet@55414
   946
  proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
wenzelm@60500
   947
    from * show "a = fimage fst R'" using conjunct1[OF \<open>?L\<close>]
blanchet@55129
   948
      by (transfer, auto simp add: image_def Int_def split: prod.splits)
wenzelm@60500
   949
    from * show "b = fimage snd R'" using conjunct2[OF \<open>?L\<close>]
blanchet@55129
   950
      by (transfer, auto simp add: image_def Int_def split: prod.splits)
blanchet@55129
   951
  qed (auto simp add: *)
blanchet@55129
   952
next
blanchet@55129
   953
  assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
blanchet@55129
   954
  apply (simp add: subset_eq Ball_def)
blanchet@55129
   955
  apply (rule conjI)
blanchet@55129
   956
  apply (transfer, clarsimp, metis snd_conv)
blanchet@55129
   957
  by (transfer, clarsimp, metis fst_conv)
blanchet@55129
   958
qed
blanchet@55129
   959
blanchet@55129
   960
bnf "'a fset"
blanchet@55129
   961
  map: fimage
blanchet@55129
   962
  sets: fset 
blanchet@55129
   963
  bd: natLeq
blanchet@55129
   964
  wits: "{||}"
blanchet@55933
   965
  rel: rel_fset
blanchet@55129
   966
apply -
blanchet@55129
   967
          apply transfer' apply simp
blanchet@55129
   968
         apply transfer' apply force
blanchet@55129
   969
        apply transfer apply force
blanchet@55129
   970
       apply transfer' apply force
blanchet@55129
   971
      apply (rule natLeq_card_order)
blanchet@55129
   972
     apply (rule natLeq_cinfinite)
blanchet@55129
   973
    apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq)
blanchet@55933
   974
   apply (fastforce simp: rel_fset_alt)
blanchet@55933
   975
 apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt rel_fset_aux) 
blanchet@55129
   976
apply transfer apply simp
blanchet@55129
   977
done
blanchet@55129
   978
blanchet@55938
   979
lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
blanchet@55129
   980
  by transfer (rule refl)
blanchet@55129
   981
kuncar@53953
   982
end
blanchet@55129
   983
blanchet@55129
   984
lemmas [simp] = fset.map_comp fset.map_id fset.set_map
blanchet@55129
   985
blanchet@55129
   986
wenzelm@60500
   987
subsection \<open>Size setup\<close>
blanchet@56646
   988
blanchet@56646
   989
context includes fset.lifting begin
blanchet@56646
   990
lift_definition size_fset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a fset \<Rightarrow> nat" is "\<lambda>f. setsum (Suc \<circ> f)" .
blanchet@56646
   991
end
blanchet@56646
   992
blanchet@56646
   993
instantiation fset :: (type) size begin
blanchet@56646
   994
definition size_fset where
blanchet@56646
   995
  size_fset_overloaded_def: "size_fset = FSet.size_fset (\<lambda>_. 0)"
blanchet@56646
   996
instance ..
blanchet@56646
   997
end
blanchet@56646
   998
blanchet@56646
   999
lemmas size_fset_simps[simp] =
blanchet@56646
  1000
  size_fset_def[THEN meta_eq_to_obj_eq, THEN fun_cong, THEN fun_cong,
blanchet@56646
  1001
    unfolded map_fun_def comp_def id_apply]
blanchet@56646
  1002
blanchet@56646
  1003
lemmas size_fset_overloaded_simps[simp] =
blanchet@56646
  1004
  size_fset_simps[of "\<lambda>_. 0", unfolded add_0_left add_0_right,
blanchet@56646
  1005
    folded size_fset_overloaded_def]
blanchet@56646
  1006
blanchet@56646
  1007
lemma fset_size_o_map: "inj f \<Longrightarrow> size_fset g \<circ> fimage f = size_fset (g \<circ> f)"
kuncar@60228
  1008
  apply (subst fun_eq_iff)
kuncar@60228
  1009
  including fset.lifting by transfer (auto intro: setsum.reindex_cong subset_inj_on)
kuncar@60228
  1010
  
wenzelm@60500
  1011
setup \<open>
blanchet@56651
  1012
BNF_LFP_Size.register_size_global @{type_name fset} @{const_name size_fset}
blanchet@56646
  1013
  @{thms size_fset_simps size_fset_overloaded_simps} @{thms fset_size_o_map}
wenzelm@60500
  1014
\<close>
blanchet@56646
  1015
kuncar@60228
  1016
lifting_update fset.lifting
kuncar@60228
  1017
lifting_forget fset.lifting
blanchet@56646
  1018
wenzelm@60500
  1019
subsection \<open>Advanced relator customization\<close>
blanchet@55129
  1020
blanchet@55129
  1021
(* Set vs. sum relators: *)
blanchet@55129
  1022
blanchet@55943
  1023
lemma rel_set_rel_sum[simp]: 
blanchet@55943
  1024
"rel_set (rel_sum \<chi> \<phi>) A1 A2 \<longleftrightarrow> 
blanchet@55938
  1025
 rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
blanchet@55129
  1026
(is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
blanchet@55129
  1027
proof safe
blanchet@55129
  1028
  assume L: "?L"
blanchet@55938
  1029
  show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
blanchet@55129
  1030
    fix l1 assume "Inl l1 \<in> A1"
blanchet@55943
  1031
    then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inl l1) a2"
blanchet@55938
  1032
    using L unfolding rel_set_def by auto
blanchet@55129
  1033
    then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
blanchet@55129
  1034
    thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
blanchet@55129
  1035
  next
blanchet@55129
  1036
    fix l2 assume "Inl l2 \<in> A2"
blanchet@55943
  1037
    then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inl l2)"
blanchet@55938
  1038
    using L unfolding rel_set_def by auto
blanchet@55129
  1039
    then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
blanchet@55129
  1040
    thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
blanchet@55129
  1041
  qed
blanchet@55938
  1042
  show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
blanchet@55129
  1043
    fix r1 assume "Inr r1 \<in> A1"
blanchet@55943
  1044
    then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inr r1) a2"
blanchet@55938
  1045
    using L unfolding rel_set_def by auto
blanchet@55129
  1046
    then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
blanchet@55129
  1047
    thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
blanchet@55129
  1048
  next
blanchet@55129
  1049
    fix r2 assume "Inr r2 \<in> A2"
blanchet@55943
  1050
    then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inr r2)"
blanchet@55938
  1051
    using L unfolding rel_set_def by auto
blanchet@55129
  1052
    then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
blanchet@55129
  1053
    thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
blanchet@55129
  1054
  qed
blanchet@55129
  1055
next
blanchet@55129
  1056
  assume Rl: "?Rl" and Rr: "?Rr"
blanchet@55938
  1057
  show ?L unfolding rel_set_def Bex_def vimage_eq proof safe
blanchet@55129
  1058
    fix a1 assume a1: "a1 \<in> A1"
blanchet@55943
  1059
    show "\<exists> a2. a2 \<in> A2 \<and> rel_sum \<chi> \<phi> a1 a2"
blanchet@55129
  1060
    proof(cases a1)
blanchet@55129
  1061
      case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
blanchet@55938
  1062
      using Rl a1 unfolding rel_set_def by blast
blanchet@55129
  1063
      thus ?thesis unfolding Inl by auto
blanchet@55129
  1064
    next
blanchet@55129
  1065
      case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
blanchet@55938
  1066
      using Rr a1 unfolding rel_set_def by blast
blanchet@55129
  1067
      thus ?thesis unfolding Inr by auto
blanchet@55129
  1068
    qed
blanchet@55129
  1069
  next
blanchet@55129
  1070
    fix a2 assume a2: "a2 \<in> A2"
blanchet@55943
  1071
    show "\<exists> a1. a1 \<in> A1 \<and> rel_sum \<chi> \<phi> a1 a2"
blanchet@55129
  1072
    proof(cases a2)
blanchet@55129
  1073
      case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
blanchet@55938
  1074
      using Rl a2 unfolding rel_set_def by blast
blanchet@55129
  1075
      thus ?thesis unfolding Inl by auto
blanchet@55129
  1076
    next
blanchet@55129
  1077
      case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
blanchet@55938
  1078
      using Rr a2 unfolding rel_set_def by blast
blanchet@55129
  1079
      thus ?thesis unfolding Inr by auto
blanchet@55129
  1080
    qed
blanchet@55129
  1081
  qed
blanchet@55129
  1082
qed
blanchet@55129
  1083
lars@60712
  1084
lars@60712
  1085
subsection \<open>Quickcheck setup\<close>
lars@60712
  1086
lars@60712
  1087
text \<open>Setup adapted from sets.\<close>
lars@60712
  1088
lars@60712
  1089
notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)
lars@60712
  1090
lars@60712
  1091
definition (in term_syntax) [code_unfold]:
lars@60712
  1092
"valterm_femptyset = Code_Evaluation.valtermify ({||} :: ('a :: typerep) fset)"
lars@60712
  1093
lars@60712
  1094
definition (in term_syntax) [code_unfold]:
lars@60712
  1095
"valtermify_finsert x s = Code_Evaluation.valtermify finsert {\<cdot>} (x :: ('a :: typerep * _)) {\<cdot>} s"
lars@60712
  1096
lars@60712
  1097
instantiation fset :: (exhaustive) exhaustive
lars@60712
  1098
begin
lars@60712
  1099
lars@60712
  1100
fun exhaustive_fset where
lars@60712
  1101
"exhaustive_fset f i = (if i = 0 then None else (f {||} orelse exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.exhaustive (\<lambda>x. if x |\<in>| A then None else f (finsert x A)) (i - 1)) (i - 1)))"
lars@60712
  1102
lars@60712
  1103
instance ..
lars@60712
  1104
blanchet@55129
  1105
end
lars@60712
  1106
lars@60712
  1107
instantiation fset :: (full_exhaustive) full_exhaustive
lars@60712
  1108
begin
lars@60712
  1109
lars@60712
  1110
fun full_exhaustive_fset where
lars@60712
  1111
"full_exhaustive_fset f i = (if i = 0 then None else (f valterm_femptyset orelse full_exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.full_exhaustive (\<lambda>x. if fst x |\<in>| fst A then None else f (valtermify_finsert x A)) (i - 1)) (i - 1)))"
lars@60712
  1112
lars@60712
  1113
instance ..
lars@60712
  1114
lars@60712
  1115
end
lars@60712
  1116
lars@60712
  1117
no_notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)
lars@60712
  1118
lars@60712
  1119
notation scomp (infixl "\<circ>\<rightarrow>" 60)
lars@60712
  1120
lars@60712
  1121
instantiation fset :: (random) random
lars@60712
  1122
begin
lars@60712
  1123
lars@60712
  1124
fun random_aux_fset :: "natural \<Rightarrow> natural \<Rightarrow> natural \<times> natural \<Rightarrow> ('a fset \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where
lars@60712
  1125
"random_aux_fset 0 j = Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset)])" |
lars@60712
  1126
"random_aux_fset (Code_Numeral.Suc i) j =
lars@60712
  1127
  Quickcheck_Random.collapse (Random.select_weight
lars@60712
  1128
    [(1, Pair valterm_femptyset),
lars@60712
  1129
     (Code_Numeral.Suc i,
lars@60712
  1130
      Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset i j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
lars@60712
  1131
lars@60712
  1132
lemma [code]:
lars@60712
  1133
  "random_aux_fset i j =
lars@60712
  1134
    Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset),
lars@60712
  1135
      (i, Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset (i - 1) j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
lars@60712
  1136
proof (induct i rule: natural.induct)
lars@60712
  1137
  case zero
lars@60712
  1138
  show ?case by (subst select_weight_drop_zero[symmetric]) (simp add: less_natural_def)
lars@60712
  1139
next
lars@60712
  1140
  case (Suc i)
lars@60712
  1141
  show ?case by (simp only: random_aux_fset.simps Suc_natural_minus_one)
lars@60712
  1142
qed
lars@60712
  1143
lars@60712
  1144
definition "random_fset i = random_aux_fset i i"
lars@60712
  1145
lars@60712
  1146
instance ..
lars@60712
  1147
lars@60712
  1148
end
lars@60712
  1149
lars@60712
  1150
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
lars@60712
  1151
lars@60712
  1152
end