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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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header {* Type of finite sets defined as a subtype of sets *}
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theory FSet
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imports Conditionally_Complete_Lattices
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begin
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subsection {* Definition of the type *}
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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 {* Basic operations and type class instantiations *}
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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 default (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 default (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 by simp
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instance by default (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 by (default, 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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subsection {* Other operations *}
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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 {* Transferred lemmas from Set.thy *}
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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]
|
kuncar@55090
|
246 |
lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]
|
kuncar@55090
|
247 |
lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]
|
kuncar@55090
|
248 |
lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]
|
kuncar@55090
|
249 |
lemmas fPowI = PowI[Transfer.transferred]
|
kuncar@55090
|
250 |
lemmas fPowD = PowD[Transfer.transferred]
|
kuncar@55090
|
251 |
lemmas fPow_bottom = Pow_bottom[Transfer.transferred]
|
kuncar@55090
|
252 |
lemmas fPow_top = Pow_top[Transfer.transferred]
|
kuncar@55090
|
253 |
lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]
|
kuncar@55090
|
254 |
lemmas finter_iff[simp] = Int_iff[Transfer.transferred]
|
kuncar@55090
|
255 |
lemmas finterI[intro!] = IntI[Transfer.transferred]
|
kuncar@55090
|
256 |
lemmas finterD1 = IntD1[Transfer.transferred]
|
kuncar@55090
|
257 |
lemmas finterD2 = IntD2[Transfer.transferred]
|
kuncar@55090
|
258 |
lemmas finterE[elim!] = IntE[Transfer.transferred]
|
kuncar@55090
|
259 |
lemmas funion_iff[simp] = Un_iff[Transfer.transferred]
|
kuncar@55090
|
260 |
lemmas funionI1[elim?] = UnI1[Transfer.transferred]
|
kuncar@55090
|
261 |
lemmas funionI2[elim?] = UnI2[Transfer.transferred]
|
kuncar@55090
|
262 |
lemmas funionCI[intro!] = UnCI[Transfer.transferred]
|
kuncar@55090
|
263 |
lemmas funionE[elim!] = UnE[Transfer.transferred]
|
kuncar@55090
|
264 |
lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]
|
kuncar@55090
|
265 |
lemmas fminusI[intro!] = DiffI[Transfer.transferred]
|
kuncar@55090
|
266 |
lemmas fminusD1 = DiffD1[Transfer.transferred]
|
kuncar@55090
|
267 |
lemmas fminusD2 = DiffD2[Transfer.transferred]
|
kuncar@55090
|
268 |
lemmas fminusE[elim!] = DiffE[Transfer.transferred]
|
kuncar@55090
|
269 |
lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]
|
kuncar@55090
|
270 |
lemmas finsertI1 = insertI1[Transfer.transferred]
|
kuncar@55090
|
271 |
lemmas finsertI2 = insertI2[Transfer.transferred]
|
kuncar@55090
|
272 |
lemmas finsertE[elim!] = insertE[Transfer.transferred]
|
kuncar@55090
|
273 |
lemmas finsertCI[intro!] = insertCI[Transfer.transferred]
|
kuncar@55101
|
274 |
lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]
|
kuncar@55090
|
275 |
lemmas finsert_ident = insert_ident[Transfer.transferred]
|
kuncar@55090
|
276 |
lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
|
kuncar@55090
|
277 |
lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
|
kuncar@55090
|
278 |
lemmas fsingleton_iff = singleton_iff[Transfer.transferred]
|
kuncar@55090
|
279 |
lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]
|
kuncar@55090
|
280 |
lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
|
kuncar@55090
|
281 |
lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
|
kuncar@55101
|
282 |
lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
|
kuncar@55090
|
283 |
lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
|
kuncar@55090
|
284 |
lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
|
kuncar@55090
|
285 |
lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
|
kuncar@55090
|
286 |
lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
|
kuncar@55090
|
287 |
lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
|
kuncar@55090
|
288 |
lemmas fimageI = imageI[Transfer.transferred]
|
kuncar@55090
|
289 |
lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
|
kuncar@55090
|
290 |
lemmas fimageE[elim!] = imageE[Transfer.transferred]
|
kuncar@55090
|
291 |
lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
|
kuncar@55090
|
292 |
lemmas fimage_funion = image_Un[Transfer.transferred]
|
kuncar@55090
|
293 |
lemmas fimage_iff = image_iff[Transfer.transferred]
|
kuncar@55101
|
294 |
lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
|
kuncar@55101
|
295 |
lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
|
kuncar@55090
|
296 |
lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
|
kuncar@55090
|
297 |
lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
|
kuncar@55090
|
298 |
lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
|
kuncar@55101
|
299 |
lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
|
kuncar@55101
|
300 |
lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
|
kuncar@55101
|
301 |
lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
|
kuncar@55101
|
302 |
lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
|
kuncar@55101
|
303 |
lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
|
kuncar@55101
|
304 |
lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
|
kuncar@55101
|
305 |
lemmas pfsubsetD = psubsetD[Transfer.transferred]
|
kuncar@55101
|
306 |
lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
|
kuncar@55101
|
307 |
lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
|
kuncar@55101
|
308 |
lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
|
kuncar@55090
|
309 |
lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
|
kuncar@55090
|
310 |
lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
|
kuncar@55101
|
311 |
lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
|
kuncar@55101
|
312 |
lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
|
kuncar@55101
|
313 |
lemmas fsubset_finsert = subset_insert[Transfer.transferred]
|
kuncar@55090
|
314 |
lemmas funion_upper1 = Un_upper1[Transfer.transferred]
|
kuncar@55090
|
315 |
lemmas funion_upper2 = Un_upper2[Transfer.transferred]
|
kuncar@55090
|
316 |
lemmas funion_least = Un_least[Transfer.transferred]
|
kuncar@55090
|
317 |
lemmas finter_lower1 = Int_lower1[Transfer.transferred]
|
kuncar@55090
|
318 |
lemmas finter_lower2 = Int_lower2[Transfer.transferred]
|
kuncar@55090
|
319 |
lemmas finter_greatest = Int_greatest[Transfer.transferred]
|
kuncar@55101
|
320 |
lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
|
kuncar@55101
|
321 |
lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
|
kuncar@55101
|
322 |
lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
|
kuncar@55101
|
323 |
lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
|
kuncar@55090
|
324 |
lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
|
kuncar@55090
|
325 |
lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
|
kuncar@55090
|
326 |
lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
|
kuncar@55090
|
327 |
lemmas finsert_absorb = insert_absorb[Transfer.transferred]
|
kuncar@55090
|
328 |
lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
|
kuncar@55090
|
329 |
lemmas finsert_commute = insert_commute[Transfer.transferred]
|
kuncar@55101
|
330 |
lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
|
kuncar@55090
|
331 |
lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
|
kuncar@55090
|
332 |
lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
|
kuncar@55090
|
333 |
lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
|
kuncar@55090
|
334 |
lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
|
kuncar@55090
|
335 |
lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
|
kuncar@55090
|
336 |
lemmas fimage_constant = image_constant[Transfer.transferred]
|
kuncar@55090
|
337 |
lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
|
kuncar@55090
|
338 |
lemmas fimage_fimage = image_image[Transfer.transferred]
|
kuncar@55090
|
339 |
lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
|
kuncar@55090
|
340 |
lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
|
kuncar@55090
|
341 |
lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
|
kuncar@55090
|
342 |
lemmas fimage_cong = image_cong[Transfer.transferred]
|
kuncar@55101
|
343 |
lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
|
kuncar@55101
|
344 |
lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
|
kuncar@55090
|
345 |
lemmas finter_absorb = Int_absorb[Transfer.transferred]
|
kuncar@55090
|
346 |
lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
|
kuncar@55090
|
347 |
lemmas finter_commute = Int_commute[Transfer.transferred]
|
kuncar@55090
|
348 |
lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
|
kuncar@55090
|
349 |
lemmas finter_assoc = Int_assoc[Transfer.transferred]
|
kuncar@55090
|
350 |
lemmas finter_ac = Int_ac[Transfer.transferred]
|
kuncar@55090
|
351 |
lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
|
kuncar@55090
|
352 |
lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
|
kuncar@55090
|
353 |
lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
|
kuncar@55090
|
354 |
lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
|
kuncar@55090
|
355 |
lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
|
kuncar@55090
|
356 |
lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
|
kuncar@55090
|
357 |
lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
|
kuncar@55101
|
358 |
lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
|
kuncar@55090
|
359 |
lemmas funion_absorb = Un_absorb[Transfer.transferred]
|
kuncar@55090
|
360 |
lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
|
kuncar@55090
|
361 |
lemmas funion_commute = Un_commute[Transfer.transferred]
|
kuncar@55090
|
362 |
lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
|
kuncar@55090
|
363 |
lemmas funion_assoc = Un_assoc[Transfer.transferred]
|
kuncar@55090
|
364 |
lemmas funion_ac = Un_ac[Transfer.transferred]
|
kuncar@55090
|
365 |
lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
|
kuncar@55090
|
366 |
lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
|
kuncar@55090
|
367 |
lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
|
kuncar@55090
|
368 |
lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
|
kuncar@55090
|
369 |
lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
|
kuncar@55090
|
370 |
lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
|
kuncar@55090
|
371 |
lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
|
kuncar@55090
|
372 |
lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
|
kuncar@55090
|
373 |
lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
|
kuncar@55090
|
374 |
lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
|
kuncar@55090
|
375 |
lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
|
kuncar@55090
|
376 |
lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
|
kuncar@55090
|
377 |
lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
|
kuncar@55090
|
378 |
lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
|
kuncar@55090
|
379 |
lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
|
kuncar@55101
|
380 |
lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
|
kuncar@55090
|
381 |
lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
|
kuncar@55101
|
382 |
lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
|
kuncar@55090
|
383 |
lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
|
kuncar@55090
|
384 |
lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
|
kuncar@55090
|
385 |
lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
|
kuncar@55090
|
386 |
lemmas fBall_funion = ball_Un[Transfer.transferred]
|
kuncar@55090
|
387 |
lemmas fBex_funion = bex_Un[Transfer.transferred]
|
kuncar@55090
|
388 |
lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
|
kuncar@55090
|
389 |
lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
|
kuncar@55090
|
390 |
lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
|
kuncar@55090
|
391 |
lemmas fminus_triv = Diff_triv[Transfer.transferred]
|
kuncar@55090
|
392 |
lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
|
kuncar@55090
|
393 |
lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
|
kuncar@55090
|
394 |
lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
|
kuncar@55090
|
395 |
lemmas fminus_finsert = Diff_insert[Transfer.transferred]
|
kuncar@55090
|
396 |
lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
|
kuncar@55090
|
397 |
lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
|
kuncar@55090
|
398 |
lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
|
kuncar@55090
|
399 |
lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
|
kuncar@55090
|
400 |
lemmas finsert_fminus = insert_Diff[Transfer.transferred]
|
kuncar@55090
|
401 |
lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
|
kuncar@55090
|
402 |
lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
|
kuncar@55090
|
403 |
lemmas fminus_partition = Diff_partition[Transfer.transferred]
|
kuncar@55090
|
404 |
lemmas double_fminus = double_diff[Transfer.transferred]
|
kuncar@55090
|
405 |
lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
|
kuncar@55090
|
406 |
lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
|
kuncar@55090
|
407 |
lemmas fminus_funion = Diff_Un[Transfer.transferred]
|
kuncar@55090
|
408 |
lemmas fminus_finter = Diff_Int[Transfer.transferred]
|
kuncar@55090
|
409 |
lemmas funion_fminus = Un_Diff[Transfer.transferred]
|
kuncar@55090
|
410 |
lemmas finter_fminus = Int_Diff[Transfer.transferred]
|
kuncar@55090
|
411 |
lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
|
kuncar@55090
|
412 |
lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
|
kuncar@55090
|
413 |
lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
|
kuncar@55090
|
414 |
lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
|
kuncar@55090
|
415 |
lemmas fPow_finsert = Pow_insert[Transfer.transferred]
|
kuncar@55101
|
416 |
lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
|
kuncar@55090
|
417 |
lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
|
kuncar@55101
|
418 |
lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
|
kuncar@55101
|
419 |
lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
|
kuncar@55101
|
420 |
lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
|
kuncar@55090
|
421 |
lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
|
kuncar@55090
|
422 |
lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
|
kuncar@55090
|
423 |
lemmas fimage_mono = image_mono[Transfer.transferred]
|
kuncar@55090
|
424 |
lemmas fPow_mono = Pow_mono[Transfer.transferred]
|
kuncar@55090
|
425 |
lemmas finsert_mono = insert_mono[Transfer.transferred]
|
kuncar@55090
|
426 |
lemmas funion_mono = Un_mono[Transfer.transferred]
|
kuncar@55090
|
427 |
lemmas finter_mono = Int_mono[Transfer.transferred]
|
kuncar@55090
|
428 |
lemmas fminus_mono = Diff_mono[Transfer.transferred]
|
kuncar@55090
|
429 |
lemmas fin_mono = in_mono[Transfer.transferred]
|
kuncar@55090
|
430 |
lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
|
kuncar@55090
|
431 |
lemmas fLeast_mono = Least_mono[Transfer.transferred]
|
kuncar@55090
|
432 |
lemmas fbind_fbind = bind_bind[Transfer.transferred]
|
kuncar@55090
|
433 |
lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
|
kuncar@55090
|
434 |
lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
|
kuncar@55090
|
435 |
lemmas fbind_const = bind_const[Transfer.transferred]
|
kuncar@55090
|
436 |
lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
|
kuncar@55090
|
437 |
lemmas fequalityI = equalityI[Transfer.transferred]
|
kuncar@55090
|
438 |
|
blanchet@56471
|
439 |
|
kuncar@55090
|
440 |
subsection {* Additional lemmas*}
|
kuncar@55090
|
441 |
|
wenzelm@55106
|
442 |
subsubsection {* @{text fsingleton} *}
|
kuncar@55090
|
443 |
|
kuncar@55090
|
444 |
lemmas fsingletonE = fsingletonD [elim_format]
|
kuncar@55090
|
445 |
|
blanchet@56471
|
446 |
|
wenzelm@55106
|
447 |
subsubsection {* @{text femepty} *}
|
kuncar@55090
|
448 |
|
kuncar@55090
|
449 |
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
|
kuncar@55090
|
450 |
by transfer auto
|
kuncar@55090
|
451 |
|
kuncar@55090
|
452 |
(* FIXME, transferred doesn't work here *)
|
kuncar@55090
|
453 |
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
|
kuncar@55090
|
454 |
by simp
|
kuncar@55090
|
455 |
|
blanchet@56471
|
456 |
|
wenzelm@55106
|
457 |
subsubsection {* @{text fset} *}
|
kuncar@55090
|
458 |
|
kuncar@55100
|
459 |
lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
|
kuncar@55090
|
460 |
|
kuncar@55090
|
461 |
lemma finite_fset [simp]:
|
kuncar@55090
|
462 |
shows "finite (fset S)"
|
kuncar@55090
|
463 |
by transfer simp
|
kuncar@55090
|
464 |
|
kuncar@55100
|
465 |
lemmas fset_cong = fset_inject
|
kuncar@55090
|
466 |
|
kuncar@55090
|
467 |
lemma filter_fset [simp]:
|
kuncar@55090
|
468 |
shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
|
kuncar@55090
|
469 |
by transfer auto
|
kuncar@55090
|
470 |
|
kuncar@55100
|
471 |
lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
|
kuncar@55090
|
472 |
|
kuncar@55100
|
473 |
lemmas inter_fset[simp] = inf_fset.rep_eq
|
kuncar@55090
|
474 |
|
kuncar@55100
|
475 |
lemmas union_fset[simp] = sup_fset.rep_eq
|
kuncar@55100
|
476 |
|
kuncar@55100
|
477 |
lemmas minus_fset[simp] = minus_fset.rep_eq
|
kuncar@55090
|
478 |
|
blanchet@56471
|
479 |
|
wenzelm@55106
|
480 |
subsubsection {* @{text filter_fset} *}
|
kuncar@55090
|
481 |
|
kuncar@55090
|
482 |
lemma subset_ffilter:
|
kuncar@55090
|
483 |
"ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
|
kuncar@55090
|
484 |
by transfer auto
|
kuncar@55090
|
485 |
|
kuncar@55090
|
486 |
lemma eq_ffilter:
|
kuncar@55090
|
487 |
"(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
|
kuncar@55090
|
488 |
by transfer auto
|
kuncar@55090
|
489 |
|
kuncar@55101
|
490 |
lemma pfsubset_ffilter:
|
kuncar@55090
|
491 |
"(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow>
|
kuncar@55090
|
492 |
ffilter P A |\<subset>| ffilter Q A"
|
kuncar@55090
|
493 |
unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
|
kuncar@55090
|
494 |
|
blanchet@56471
|
495 |
|
wenzelm@55106
|
496 |
subsubsection {* @{text finsert} *}
|
kuncar@55090
|
497 |
|
kuncar@55090
|
498 |
(* FIXME, transferred doesn't work here *)
|
kuncar@55090
|
499 |
lemma set_finsert:
|
kuncar@55090
|
500 |
assumes "x |\<in>| A"
|
kuncar@55090
|
501 |
obtains B where "A = finsert x B" and "x |\<notin>| B"
|
kuncar@55090
|
502 |
using assms by transfer (metis Set.set_insert finite_insert)
|
kuncar@55090
|
503 |
|
kuncar@55090
|
504 |
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
|
kuncar@55090
|
505 |
by (rule_tac x = "A |-| {|a|}" in exI, blast)
|
kuncar@55090
|
506 |
|
blanchet@56471
|
507 |
|
wenzelm@55106
|
508 |
subsubsection {* @{text fimage} *}
|
kuncar@55090
|
509 |
|
kuncar@55090
|
510 |
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
|
kuncar@55090
|
511 |
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
|
kuncar@55090
|
512 |
|
blanchet@56471
|
513 |
|
kuncar@55090
|
514 |
subsubsection {* bounded quantification *}
|
kuncar@55090
|
515 |
|
kuncar@55090
|
516 |
lemma bex_simps [simp, no_atp]:
|
kuncar@55090
|
517 |
"\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)"
|
kuncar@55090
|
518 |
"\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
|
kuncar@55090
|
519 |
"\<And>P. fBex {||} P = False"
|
kuncar@55090
|
520 |
"\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
|
kuncar@55090
|
521 |
"\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
|
kuncar@55090
|
522 |
"\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
|
kuncar@55090
|
523 |
by auto
|
kuncar@55090
|
524 |
|
kuncar@55090
|
525 |
lemma ball_simps [simp, no_atp]:
|
kuncar@55090
|
526 |
"\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
|
kuncar@55090
|
527 |
"\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
|
kuncar@55090
|
528 |
"\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
|
kuncar@55090
|
529 |
"\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
|
kuncar@55090
|
530 |
"\<And>P. fBall {||} P = True"
|
kuncar@55090
|
531 |
"\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
|
kuncar@55090
|
532 |
"\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
|
kuncar@55090
|
533 |
"\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
|
kuncar@55090
|
534 |
by auto
|
kuncar@55090
|
535 |
|
kuncar@55090
|
536 |
lemma atomize_fBall:
|
kuncar@55090
|
537 |
"(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
|
kuncar@55090
|
538 |
apply (simp only: atomize_all atomize_imp)
|
kuncar@55090
|
539 |
apply (rule equal_intr_rule)
|
kuncar@55090
|
540 |
by (transfer, simp)+
|
kuncar@55090
|
541 |
|
kuncar@55100
|
542 |
end
|
kuncar@55100
|
543 |
|
blanchet@56471
|
544 |
|
wenzelm@55106
|
545 |
subsubsection {* @{text fcard} *}
|
kuncar@55100
|
546 |
|
kuncar@55101
|
547 |
(* FIXME: improve transferred to handle bounded meta quantification *)
|
kuncar@55101
|
548 |
|
kuncar@55100
|
549 |
lemma fcard_fempty:
|
kuncar@55100
|
550 |
"fcard {||} = 0"
|
kuncar@55100
|
551 |
by transfer (rule card_empty)
|
kuncar@55100
|
552 |
|
kuncar@55100
|
553 |
lemma fcard_finsert_disjoint:
|
kuncar@55100
|
554 |
"x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
|
kuncar@55100
|
555 |
by transfer (rule card_insert_disjoint)
|
kuncar@55100
|
556 |
|
kuncar@55100
|
557 |
lemma fcard_finsert_if:
|
kuncar@55100
|
558 |
"fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
|
kuncar@55100
|
559 |
by transfer (rule card_insert_if)
|
kuncar@55100
|
560 |
|
kuncar@55100
|
561 |
lemma card_0_eq [simp, no_atp]:
|
kuncar@55100
|
562 |
"fcard A = 0 \<longleftrightarrow> A = {||}"
|
kuncar@55100
|
563 |
by transfer (rule card_0_eq)
|
kuncar@55100
|
564 |
|
kuncar@55100
|
565 |
lemma fcard_Suc_fminus1:
|
kuncar@55100
|
566 |
"x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
|
kuncar@55100
|
567 |
by transfer (rule card_Suc_Diff1)
|
kuncar@55100
|
568 |
|
kuncar@55100
|
569 |
lemma fcard_fminus_fsingleton:
|
kuncar@55100
|
570 |
"x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
|
kuncar@55100
|
571 |
by transfer (rule card_Diff_singleton)
|
kuncar@55100
|
572 |
|
kuncar@55100
|
573 |
lemma fcard_fminus_fsingleton_if:
|
kuncar@55100
|
574 |
"fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
|
kuncar@55100
|
575 |
by transfer (rule card_Diff_singleton_if)
|
kuncar@55100
|
576 |
|
kuncar@55100
|
577 |
lemma fcard_fminus_finsert[simp]:
|
kuncar@55100
|
578 |
assumes "a |\<in>| A" and "a |\<notin>| B"
|
kuncar@55100
|
579 |
shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
|
kuncar@55100
|
580 |
using assms by transfer (rule card_Diff_insert)
|
kuncar@55100
|
581 |
|
kuncar@55100
|
582 |
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
|
kuncar@55100
|
583 |
by transfer (rule card_insert)
|
kuncar@55100
|
584 |
|
kuncar@55100
|
585 |
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
|
kuncar@55100
|
586 |
by transfer (rule card_insert_le)
|
kuncar@55100
|
587 |
|
kuncar@55100
|
588 |
lemma fcard_mono:
|
kuncar@55100
|
589 |
"A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
|
kuncar@55100
|
590 |
by transfer (rule card_mono)
|
kuncar@55100
|
591 |
|
kuncar@55100
|
592 |
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
|
kuncar@55100
|
593 |
by transfer (rule card_seteq)
|
kuncar@55100
|
594 |
|
kuncar@55100
|
595 |
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
|
kuncar@55100
|
596 |
by transfer (rule psubset_card_mono)
|
kuncar@55100
|
597 |
|
kuncar@55100
|
598 |
lemma fcard_funion_finter:
|
kuncar@55100
|
599 |
"fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
|
kuncar@55100
|
600 |
by transfer (rule card_Un_Int)
|
kuncar@55100
|
601 |
|
kuncar@55100
|
602 |
lemma fcard_funion_disjoint:
|
kuncar@55100
|
603 |
"A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
|
kuncar@55100
|
604 |
by transfer (rule card_Un_disjoint)
|
kuncar@55100
|
605 |
|
kuncar@55100
|
606 |
lemma fcard_funion_fsubset:
|
kuncar@55100
|
607 |
"B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
|
kuncar@55100
|
608 |
by transfer (rule card_Diff_subset)
|
kuncar@55100
|
609 |
|
kuncar@55100
|
610 |
lemma diff_fcard_le_fcard_fminus:
|
kuncar@55100
|
611 |
"fcard A - fcard B \<le> fcard(A |-| B)"
|
kuncar@55100
|
612 |
by transfer (rule diff_card_le_card_Diff)
|
kuncar@55100
|
613 |
|
kuncar@55100
|
614 |
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
|
kuncar@55100
|
615 |
by transfer (rule card_Diff1_less)
|
kuncar@55100
|
616 |
|
kuncar@55100
|
617 |
lemma fcard_fminus2_less:
|
kuncar@55100
|
618 |
"x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
|
kuncar@55100
|
619 |
by transfer (rule card_Diff2_less)
|
kuncar@55100
|
620 |
|
kuncar@55100
|
621 |
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
|
kuncar@55100
|
622 |
by transfer (rule card_Diff1_le)
|
kuncar@55100
|
623 |
|
kuncar@55100
|
624 |
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
|
kuncar@55100
|
625 |
by transfer (rule card_psubset)
|
kuncar@55100
|
626 |
|
blanchet@56471
|
627 |
|
wenzelm@55106
|
628 |
subsubsection {* @{text ffold} *}
|
kuncar@55100
|
629 |
|
kuncar@55100
|
630 |
(* FIXME: improve transferred to handle bounded meta quantification *)
|
kuncar@55100
|
631 |
|
kuncar@55100
|
632 |
context comp_fun_commute
|
kuncar@55100
|
633 |
begin
|
kuncar@55100
|
634 |
lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
|
kuncar@55100
|
635 |
|
kuncar@55100
|
636 |
lemma ffold_finsert [simp]:
|
kuncar@55100
|
637 |
assumes "x |\<notin>| A"
|
kuncar@55100
|
638 |
shows "ffold f z (finsert x A) = f x (ffold f z A)"
|
kuncar@55100
|
639 |
using assms by (transfer fixing: f) (rule fold_insert)
|
kuncar@55100
|
640 |
|
kuncar@55100
|
641 |
lemma ffold_fun_left_comm:
|
kuncar@55100
|
642 |
"f x (ffold f z A) = ffold f (f x z) A"
|
kuncar@55100
|
643 |
by (transfer fixing: f) (rule fold_fun_left_comm)
|
kuncar@55100
|
644 |
|
kuncar@55100
|
645 |
lemma ffold_finsert2:
|
kuncar@55100
|
646 |
"x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A) = ffold f (f x z) A"
|
kuncar@55100
|
647 |
by (transfer fixing: f) (rule fold_insert2)
|
kuncar@55100
|
648 |
|
kuncar@55100
|
649 |
lemma ffold_rec:
|
kuncar@55100
|
650 |
assumes "x |\<in>| A"
|
kuncar@55100
|
651 |
shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
|
kuncar@55100
|
652 |
using assms by (transfer fixing: f) (rule fold_rec)
|
kuncar@55100
|
653 |
|
kuncar@55100
|
654 |
lemma ffold_finsert_fremove:
|
kuncar@55100
|
655 |
"ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
|
kuncar@55100
|
656 |
by (transfer fixing: f) (rule fold_insert_remove)
|
kuncar@55100
|
657 |
end
|
kuncar@55100
|
658 |
|
kuncar@55100
|
659 |
lemma ffold_fimage:
|
kuncar@55100
|
660 |
assumes "inj_on g (fset A)"
|
kuncar@55100
|
661 |
shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
|
kuncar@55100
|
662 |
using assms by transfer' (rule fold_image)
|
kuncar@55100
|
663 |
|
kuncar@55100
|
664 |
lemma ffold_cong:
|
kuncar@55100
|
665 |
assumes "comp_fun_commute f" "comp_fun_commute g"
|
kuncar@55100
|
666 |
"\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
|
kuncar@55100
|
667 |
and "s = t" and "A = B"
|
kuncar@55100
|
668 |
shows "ffold f s A = ffold g t B"
|
kuncar@55100
|
669 |
using assms by transfer (metis Finite_Set.fold_cong)
|
kuncar@55100
|
670 |
|
kuncar@55100
|
671 |
context comp_fun_idem
|
kuncar@55100
|
672 |
begin
|
kuncar@55100
|
673 |
|
kuncar@55100
|
674 |
lemma ffold_finsert_idem:
|
kuncar@55100
|
675 |
"ffold f z (finsert x A) = f x (ffold f z A)"
|
kuncar@55100
|
676 |
by (transfer fixing: f) (rule fold_insert_idem)
|
kuncar@55100
|
677 |
|
kuncar@55100
|
678 |
declare ffold_finsert [simp del] ffold_finsert_idem [simp]
|
kuncar@55100
|
679 |
|
kuncar@55100
|
680 |
lemma ffold_finsert_idem2:
|
kuncar@55100
|
681 |
"ffold f z (finsert x A) = ffold f (f x z) A"
|
kuncar@55100
|
682 |
by (transfer fixing: f) (rule fold_insert_idem2)
|
kuncar@55100
|
683 |
|
kuncar@55100
|
684 |
end
|
kuncar@55100
|
685 |
|
blanchet@56471
|
686 |
|
kuncar@55090
|
687 |
subsection {* Choice in fsets *}
|
kuncar@55090
|
688 |
|
kuncar@55090
|
689 |
lemma fset_choice:
|
kuncar@55090
|
690 |
assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
|
kuncar@55090
|
691 |
shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
|
kuncar@55090
|
692 |
using assms by transfer metis
|
kuncar@55090
|
693 |
|
blanchet@56471
|
694 |
|
kuncar@55090
|
695 |
subsection {* Induction and Cases rules for fsets *}
|
kuncar@55090
|
696 |
|
kuncar@55090
|
697 |
lemma fset_exhaust [case_names empty insert, cases type: fset]:
|
kuncar@55090
|
698 |
assumes fempty_case: "S = {||} \<Longrightarrow> P"
|
kuncar@55090
|
699 |
and finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
|
kuncar@55090
|
700 |
shows "P"
|
kuncar@55090
|
701 |
using assms by transfer blast
|
kuncar@55090
|
702 |
|
kuncar@55090
|
703 |
lemma fset_induct [case_names empty insert]:
|
kuncar@55090
|
704 |
assumes fempty_case: "P {||}"
|
kuncar@55090
|
705 |
and finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
|
kuncar@55090
|
706 |
shows "P S"
|
kuncar@55090
|
707 |
proof -
|
kuncar@55090
|
708 |
(* FIXME transfer and right_total vs. bi_total *)
|
kuncar@55090
|
709 |
note Domainp_forall_transfer[transfer_rule]
|
kuncar@55090
|
710 |
show ?thesis
|
kuncar@55090
|
711 |
using assms by transfer (auto intro: finite_induct)
|
kuncar@55090
|
712 |
qed
|
kuncar@55090
|
713 |
|
kuncar@55090
|
714 |
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
|
kuncar@55090
|
715 |
assumes empty_fset_case: "P {||}"
|
kuncar@55090
|
716 |
and insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
|
kuncar@55090
|
717 |
shows "P S"
|
kuncar@55090
|
718 |
proof -
|
kuncar@55090
|
719 |
(* FIXME transfer and right_total vs. bi_total *)
|
kuncar@55090
|
720 |
note Domainp_forall_transfer[transfer_rule]
|
kuncar@55090
|
721 |
show ?thesis
|
kuncar@55090
|
722 |
using assms by transfer (auto intro: finite_induct)
|
kuncar@55090
|
723 |
qed
|
kuncar@55090
|
724 |
|
kuncar@55090
|
725 |
lemma fset_card_induct:
|
kuncar@55090
|
726 |
assumes empty_fset_case: "P {||}"
|
kuncar@55090
|
727 |
and card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
|
kuncar@55090
|
728 |
shows "P S"
|
kuncar@55090
|
729 |
proof (induct S)
|
kuncar@55090
|
730 |
case empty
|
kuncar@55090
|
731 |
show "P {||}" by (rule empty_fset_case)
|
kuncar@55090
|
732 |
next
|
kuncar@55090
|
733 |
case (insert x S)
|
kuncar@55090
|
734 |
have h: "P S" by fact
|
kuncar@55090
|
735 |
have "x |\<notin>| S" by fact
|
kuncar@55090
|
736 |
then have "Suc (fcard S) = fcard (finsert x S)"
|
kuncar@55090
|
737 |
by transfer auto
|
kuncar@55090
|
738 |
then show "P (finsert x S)"
|
kuncar@55090
|
739 |
using h card_fset_Suc_case by simp
|
kuncar@55090
|
740 |
qed
|
kuncar@55090
|
741 |
|
kuncar@55090
|
742 |
lemma fset_strong_cases:
|
kuncar@55090
|
743 |
obtains "xs = {||}"
|
kuncar@55090
|
744 |
| ys x where "x |\<notin>| ys" and "xs = finsert x ys"
|
kuncar@55090
|
745 |
by transfer blast
|
kuncar@55090
|
746 |
|
kuncar@55090
|
747 |
lemma fset_induct2:
|
kuncar@55090
|
748 |
"P {||} {||} \<Longrightarrow>
|
kuncar@55090
|
749 |
(\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
|
kuncar@55090
|
750 |
(\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
|
kuncar@55090
|
751 |
(\<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@55090
|
752 |
P xsa ysa"
|
kuncar@55090
|
753 |
apply (induct xsa arbitrary: ysa)
|
kuncar@55090
|
754 |
apply (induct_tac x rule: fset_induct_stronger)
|
kuncar@55090
|
755 |
apply simp_all
|
kuncar@55090
|
756 |
apply (induct_tac xa rule: fset_induct_stronger)
|
kuncar@55090
|
757 |
apply simp_all
|
kuncar@55090
|
758 |
done
|
kuncar@55090
|
759 |
|
blanchet@56471
|
760 |
|
kuncar@55090
|
761 |
subsection {* Setup for Lifting/Transfer *}
|
kuncar@55090
|
762 |
|
kuncar@55090
|
763 |
subsubsection {* Relator and predicator properties *}
|
kuncar@55090
|
764 |
|
blanchet@57280
|
765 |
lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set
|
blanchet@57280
|
766 |
parametric rel_set_transfer .
|
kuncar@55090
|
767 |
|
blanchet@57275
|
768 |
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@55090
|
769 |
\<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
|
kuncar@55090
|
770 |
apply (rule ext)+
|
kuncar@55090
|
771 |
apply transfer'
|
blanchet@57280
|
772 |
apply (subst rel_set_def[unfolded fun_eq_iff])
|
kuncar@55090
|
773 |
by blast
|
kuncar@55090
|
774 |
|
blanchet@57275
|
775 |
lemma rel_fset_conversep: "rel_fset (conversep R) = conversep (rel_fset R)"
|
blanchet@57275
|
776 |
unfolding rel_fset_alt_def by auto
|
kuncar@55090
|
777 |
|
blanchet@57280
|
778 |
lemmas rel_fset_eq [relator_eq] = rel_set_eq[Transfer.transferred]
|
kuncar@55090
|
779 |
|
blanchet@57275
|
780 |
lemma rel_fset_mono[relator_mono]: "A \<le> B \<Longrightarrow> rel_fset A \<le> rel_fset B"
|
blanchet@57275
|
781 |
unfolding rel_fset_alt_def by blast
|
kuncar@55090
|
782 |
|
blanchet@57280
|
783 |
lemma finite_rel_set:
|
kuncar@55090
|
784 |
assumes fin: "finite X" "finite Z"
|
blanchet@57280
|
785 |
assumes R_S: "rel_set (R OO S) X Z"
|
blanchet@57280
|
786 |
shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"
|
kuncar@55090
|
787 |
proof -
|
kuncar@55090
|
788 |
obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
|
kuncar@55090
|
789 |
apply atomize_elim
|
kuncar@55090
|
790 |
apply (subst bchoice_iff[symmetric])
|
blanchet@57280
|
791 |
using R_S[unfolded rel_set_def OO_def] by blast
|
kuncar@55090
|
792 |
|
kuncar@55090
|
793 |
obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))"
|
kuncar@55090
|
794 |
apply atomize_elim
|
kuncar@55090
|
795 |
apply (subst bchoice_iff[symmetric])
|
blanchet@57280
|
796 |
using R_S[unfolded rel_set_def OO_def] by blast
|
kuncar@55090
|
797 |
|
kuncar@55090
|
798 |
let ?Y = "f ` X \<union> g ` Z"
|
kuncar@55090
|
799 |
have "finite ?Y" by (simp add: fin)
|
blanchet@57280
|
800 |
moreover have "rel_set R X ?Y"
|
blanchet@57280
|
801 |
unfolding rel_set_def
|
kuncar@55090
|
802 |
using f g by clarsimp blast
|
blanchet@57280
|
803 |
moreover have "rel_set S ?Y Z"
|
blanchet@57280
|
804 |
unfolding rel_set_def
|
kuncar@55090
|
805 |
using f g by clarsimp blast
|
kuncar@55090
|
806 |
ultimately show ?thesis by metis
|
kuncar@55090
|
807 |
qed
|
kuncar@55090
|
808 |
|
blanchet@57275
|
809 |
lemma rel_fset_OO[relator_distr]: "rel_fset R OO rel_fset S = rel_fset (R OO S)"
|
kuncar@55090
|
810 |
apply (rule ext)+
|
blanchet@57280
|
811 |
by transfer (auto intro: finite_rel_set rel_set_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
|
kuncar@55090
|
812 |
|
kuncar@57862
|
813 |
lemma Domainp_fset[relator_domain]: "Domainp (rel_fset T) = (\<lambda>A. fBall A (Domainp T))"
|
kuncar@55090
|
814 |
proof -
|
kuncar@57862
|
815 |
obtain f where f: "\<forall>x\<in>Collect (Domainp T). T x (f x)"
|
kuncar@55090
|
816 |
unfolding Domainp_iff[abs_def]
|
kuncar@55090
|
817 |
apply atomize_elim
|
kuncar@57862
|
818 |
by (subst bchoice_iff[symmetric]) (auto iff: bchoice_iff[symmetric])
|
kuncar@57862
|
819 |
from f show ?thesis
|
blanchet@57275
|
820 |
unfolding fun_eq_iff rel_fset_alt_def Domainp_iff
|
kuncar@55090
|
821 |
apply clarify
|
kuncar@55090
|
822 |
apply (rule iffI)
|
kuncar@55090
|
823 |
apply blast
|
kuncar@55090
|
824 |
by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
|
kuncar@55090
|
825 |
qed
|
kuncar@55090
|
826 |
|
blanchet@57275
|
827 |
lemma right_total_rel_fset[transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_fset A)"
|
kuncar@55090
|
828 |
unfolding right_total_def
|
kuncar@55090
|
829 |
apply transfer
|
kuncar@55090
|
830 |
apply (subst(asm) choice_iff)
|
kuncar@55090
|
831 |
apply clarsimp
|
kuncar@55090
|
832 |
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
|
blanchet@57280
|
833 |
by (auto simp add: rel_set_def)
|
kuncar@55090
|
834 |
|
kuncar@57860
|
835 |
lemma left_total_rel_fset[transfer_rule]: "left_total A \<Longrightarrow> left_total (rel_fset A)"
|
kuncar@55090
|
836 |
unfolding left_total_def
|
kuncar@55090
|
837 |
apply transfer
|
kuncar@55090
|
838 |
apply (subst(asm) choice_iff)
|
kuncar@55090
|
839 |
apply clarsimp
|
kuncar@55090
|
840 |
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
|
blanchet@57280
|
841 |
by (auto simp add: rel_set_def)
|
kuncar@55090
|
842 |
|
blanchet@57280
|
843 |
lemmas right_unique_rel_fset[transfer_rule] = right_unique_rel_set[Transfer.transferred]
|
kuncar@57860
|
844 |
lemmas left_unique_rel_fset[transfer_rule] = left_unique_rel_set[Transfer.transferred]
|
kuncar@55090
|
845 |
|
blanchet@57275
|
846 |
thm right_unique_rel_fset left_unique_rel_fset
|
kuncar@55090
|
847 |
|
blanchet@57275
|
848 |
lemma bi_unique_rel_fset[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_fset A)"
|
kuncar@57866
|
849 |
by (auto intro: right_unique_rel_fset left_unique_rel_fset iff: bi_unique_alt_def)
|
kuncar@55090
|
850 |
|
blanchet@57275
|
851 |
lemma bi_total_rel_fset[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_fset A)"
|
kuncar@57866
|
852 |
by (auto intro: right_total_rel_fset left_total_rel_fset iff: bi_total_alt_def)
|
kuncar@55090
|
853 |
|
kuncar@57861
|
854 |
lemmas fset_relator_eq_onp [relator_eq_onp] = set_relator_eq_onp[Transfer.transferred]
|
kuncar@55090
|
855 |
|
blanchet@56471
|
856 |
|
kuncar@55090
|
857 |
subsubsection {* Quotient theorem for the Lifting package *}
|
kuncar@55090
|
858 |
|
kuncar@55090
|
859 |
lemma Quotient_fset_map[quot_map]:
|
kuncar@55090
|
860 |
assumes "Quotient R Abs Rep T"
|
blanchet@57275
|
861 |
shows "Quotient (rel_fset R) (fimage Abs) (fimage Rep) (rel_fset T)"
|
kuncar@55090
|
862 |
using assms unfolding Quotient_alt_def4
|
blanchet@57275
|
863 |
by (simp add: rel_fset_OO[symmetric] rel_fset_conversep) (simp add: rel_fset_alt_def, blast)
|
kuncar@55090
|
864 |
|
blanchet@56471
|
865 |
|
kuncar@55090
|
866 |
subsubsection {* Transfer rules for the Transfer package *}
|
kuncar@55090
|
867 |
|
kuncar@55090
|
868 |
text {* Unconditional transfer rules *}
|
kuncar@55090
|
869 |
|
kuncar@55100
|
870 |
context
|
kuncar@55100
|
871 |
begin
|
kuncar@55100
|
872 |
|
kuncar@55100
|
873 |
interpretation lifting_syntax .
|
kuncar@55100
|
874 |
|
kuncar@55090
|
875 |
lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
|
kuncar@55090
|
876 |
|
kuncar@55090
|
877 |
lemma finsert_transfer [transfer_rule]:
|
blanchet@57275
|
878 |
"(A ===> rel_fset A ===> rel_fset A) finsert finsert"
|
blanchet@57287
|
879 |
unfolding rel_fun_def rel_fset_alt_def by blast
|
kuncar@55090
|
880 |
|
kuncar@55090
|
881 |
lemma funion_transfer [transfer_rule]:
|
blanchet@57275
|
882 |
"(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion"
|
blanchet@57287
|
883 |
unfolding rel_fun_def rel_fset_alt_def by blast
|
kuncar@55090
|
884 |
|
kuncar@55090
|
885 |
lemma ffUnion_transfer [transfer_rule]:
|
blanchet@57275
|
886 |
"(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion"
|
blanchet@57287
|
887 |
unfolding rel_fun_def rel_fset_alt_def by transfer (simp, fast)
|
kuncar@55090
|
888 |
|
kuncar@55090
|
889 |
lemma fimage_transfer [transfer_rule]:
|
blanchet@57275
|
890 |
"((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage"
|
blanchet@57287
|
891 |
unfolding rel_fun_def rel_fset_alt_def by simp blast
|
kuncar@55090
|
892 |
|
kuncar@55090
|
893 |
lemma fBall_transfer [transfer_rule]:
|
blanchet@57275
|
894 |
"(rel_fset A ===> (A ===> op =) ===> op =) fBall fBall"
|
blanchet@57287
|
895 |
unfolding rel_fset_alt_def rel_fun_def by blast
|
kuncar@55090
|
896 |
|
kuncar@55090
|
897 |
lemma fBex_transfer [transfer_rule]:
|
blanchet@57275
|
898 |
"(rel_fset A ===> (A ===> op =) ===> op =) fBex fBex"
|
blanchet@57287
|
899 |
unfolding rel_fset_alt_def rel_fun_def by blast
|
kuncar@55090
|
900 |
|
kuncar@55090
|
901 |
(* FIXME transfer doesn't work here *)
|
kuncar@55090
|
902 |
lemma fPow_transfer [transfer_rule]:
|
blanchet@57275
|
903 |
"(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow"
|
blanchet@57287
|
904 |
unfolding rel_fun_def
|
blanchet@57287
|
905 |
using Pow_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred]
|
kuncar@55090
|
906 |
by blast
|
kuncar@55090
|
907 |
|
blanchet@57275
|
908 |
lemma rel_fset_transfer [transfer_rule]:
|
blanchet@57275
|
909 |
"((A ===> B ===> op =) ===> rel_fset A ===> rel_fset B ===> op =)
|
blanchet@57275
|
910 |
rel_fset rel_fset"
|
blanchet@57287
|
911 |
unfolding rel_fun_def
|
blanchet@57287
|
912 |
using rel_set_transfer[unfolded rel_fun_def,rule_format, Transfer.transferred, where A = A and B = B]
|
kuncar@55090
|
913 |
by simp
|
kuncar@55090
|
914 |
|
kuncar@55090
|
915 |
lemma bind_transfer [transfer_rule]:
|
blanchet@57275
|
916 |
"(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind"
|
blanchet@57287
|
917 |
using assms unfolding rel_fun_def
|
blanchet@57287
|
918 |
using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
|
kuncar@55090
|
919 |
|
kuncar@55090
|
920 |
text {* Rules requiring bi-unique, bi-total or right-total relations *}
|
kuncar@55090
|
921 |
|
kuncar@55090
|
922 |
lemma fmember_transfer [transfer_rule]:
|
kuncar@55090
|
923 |
assumes "bi_unique A"
|
blanchet@57275
|
924 |
shows "(A ===> rel_fset A ===> op =) (op |\<in>|) (op |\<in>|)"
|
blanchet@57287
|
925 |
using assms unfolding rel_fun_def rel_fset_alt_def bi_unique_def by metis
|
kuncar@55090
|
926 |
|
kuncar@55090
|
927 |
lemma finter_transfer [transfer_rule]:
|
kuncar@55090
|
928 |
assumes "bi_unique A"
|
blanchet@57275
|
929 |
shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter"
|
blanchet@57287
|
930 |
using assms unfolding rel_fun_def
|
blanchet@57287
|
931 |
using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
|
kuncar@55090
|
932 |
|
kuncar@55100
|
933 |
lemma fminus_transfer [transfer_rule]:
|
kuncar@55090
|
934 |
assumes "bi_unique A"
|
blanchet@57275
|
935 |
shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (op |-|) (op |-|)"
|
blanchet@57287
|
936 |
using assms unfolding rel_fun_def
|
blanchet@57287
|
937 |
using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
|
kuncar@55090
|
938 |
|
kuncar@55090
|
939 |
lemma fsubset_transfer [transfer_rule]:
|
kuncar@55090
|
940 |
assumes "bi_unique A"
|
blanchet@57275
|
941 |
shows "(rel_fset A ===> rel_fset A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
|
blanchet@57287
|
942 |
using assms unfolding rel_fun_def
|
blanchet@57287
|
943 |
using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
|
kuncar@55090
|
944 |
|
kuncar@55090
|
945 |
lemma fSup_transfer [transfer_rule]:
|
blanchet@57280
|
946 |
"bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"
|
blanchet@57287
|
947 |
using assms unfolding rel_fun_def
|
kuncar@55090
|
948 |
apply clarify
|
kuncar@55090
|
949 |
apply transfer'
|
blanchet@57287
|
950 |
using Sup_fset_transfer[unfolded rel_fun_def] by blast
|
kuncar@55090
|
951 |
|
kuncar@55090
|
952 |
(* FIXME: add right_total_fInf_transfer *)
|
kuncar@55090
|
953 |
|
kuncar@55090
|
954 |
lemma fInf_transfer [transfer_rule]:
|
kuncar@55090
|
955 |
assumes "bi_unique A" and "bi_total A"
|
blanchet@57280
|
956 |
shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"
|
blanchet@57287
|
957 |
using assms unfolding rel_fun_def
|
kuncar@55090
|
958 |
apply clarify
|
kuncar@55090
|
959 |
apply transfer'
|
blanchet@57287
|
960 |
using Inf_fset_transfer[unfolded rel_fun_def] by blast
|
kuncar@55090
|
961 |
|
kuncar@55090
|
962 |
lemma ffilter_transfer [transfer_rule]:
|
kuncar@55090
|
963 |
assumes "bi_unique A"
|
blanchet@57275
|
964 |
shows "((A ===> op=) ===> rel_fset A ===> rel_fset A) ffilter ffilter"
|
blanchet@57287
|
965 |
using assms unfolding rel_fun_def
|
blanchet@57287
|
966 |
using Lifting_Set.filter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
|
kuncar@55090
|
967 |
|
kuncar@55090
|
968 |
lemma card_transfer [transfer_rule]:
|
blanchet@57275
|
969 |
"bi_unique A \<Longrightarrow> (rel_fset A ===> op =) fcard fcard"
|
blanchet@57287
|
970 |
using assms unfolding rel_fun_def
|
blanchet@57287
|
971 |
using card_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
|
kuncar@55090
|
972 |
|
kuncar@55090
|
973 |
end
|
kuncar@55090
|
974 |
|
kuncar@55090
|
975 |
lifting_update fset.lifting
|
kuncar@55090
|
976 |
lifting_forget fset.lifting
|
kuncar@55090
|
977 |
|
blanchet@56471
|
978 |
|
blanchet@56471
|
979 |
subsection {* BNF setup *}
|
blanchet@56471
|
980 |
|
blanchet@56471
|
981 |
context
|
blanchet@56471
|
982 |
includes fset.lifting
|
blanchet@56471
|
983 |
begin
|
blanchet@56471
|
984 |
|
blanchet@57275
|
985 |
lemma rel_fset_alt:
|
blanchet@57275
|
986 |
"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@57280
|
987 |
by transfer (simp add: rel_set_def)
|
blanchet@56471
|
988 |
|
blanchet@56471
|
989 |
lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
|
blanchet@56471
|
990 |
apply (rule f_the_inv_into_f[unfolded inj_on_def])
|
blanchet@56471
|
991 |
apply (simp add: fset_inject)
|
blanchet@56471
|
992 |
apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
|
blanchet@56471
|
993 |
.
|
blanchet@56471
|
994 |
|
blanchet@57275
|
995 |
lemma rel_fset_aux:
|
blanchet@56471
|
996 |
"(\<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@56471
|
997 |
((BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
|
blanchet@56471
|
998 |
BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
|
blanchet@56471
|
999 |
proof
|
blanchet@56471
|
1000 |
assume ?L
|
blanchet@56471
|
1001 |
def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?L'")
|
blanchet@56471
|
1002 |
have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
|
blanchet@56471
|
1003 |
hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
|
blanchet@56471
|
1004 |
show ?R unfolding Grp_def relcompp.simps conversep.simps
|
blanchet@56756
|
1005 |
proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
|
blanchet@56471
|
1006 |
from * show "a = fimage fst R'" using conjunct1[OF `?L`]
|
blanchet@56471
|
1007 |
by (transfer, auto simp add: image_def Int_def split: prod.splits)
|
blanchet@56471
|
1008 |
from * show "b = fimage snd R'" using conjunct2[OF `?L`]
|
blanchet@56471
|
1009 |
by (transfer, auto simp add: image_def Int_def split: prod.splits)
|
blanchet@56471
|
1010 |
qed (auto simp add: *)
|
blanchet@56471
|
1011 |
next
|
blanchet@56471
|
1012 |
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
|
blanchet@56471
|
1013 |
apply (simp add: subset_eq Ball_def)
|
blanchet@56471
|
1014 |
apply (rule conjI)
|
blanchet@56471
|
1015 |
apply (transfer, clarsimp, metis snd_conv)
|
blanchet@56471
|
1016 |
by (transfer, clarsimp, metis fst_conv)
|
blanchet@56471
|
1017 |
qed
|
blanchet@56471
|
1018 |
|
blanchet@56471
|
1019 |
bnf "'a fset"
|
blanchet@56471
|
1020 |
map: fimage
|
blanchet@56471
|
1021 |
sets: fset
|
blanchet@56471
|
1022 |
bd: natLeq
|
blanchet@56471
|
1023 |
wits: "{||}"
|
blanchet@57275
|
1024 |
rel: rel_fset
|
blanchet@56471
|
1025 |
apply -
|
blanchet@56471
|
1026 |
apply transfer' apply simp
|
blanchet@56471
|
1027 |
apply transfer' apply force
|
blanchet@56471
|
1028 |
apply transfer apply force
|
blanchet@56471
|
1029 |
apply transfer' apply force
|
blanchet@56471
|
1030 |
apply (rule natLeq_card_order)
|
blanchet@56471
|
1031 |
apply (rule natLeq_cinfinite)
|
blanchet@56471
|
1032 |
apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq)
|
blanchet@57275
|
1033 |
apply (fastforce simp: rel_fset_alt)
|
blanchet@57275
|
1034 |
apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt rel_fset_aux)
|
blanchet@56471
|
1035 |
apply transfer apply simp
|
blanchet@56471
|
1036 |
done
|
blanchet@56471
|
1037 |
|
blanchet@57280
|
1038 |
lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
|
blanchet@56471
|
1039 |
by transfer (rule refl)
|
blanchet@56471
|
1040 |
|
kuncar@55090
|
1041 |
end
|
blanchet@56471
|
1042 |
|
blanchet@56471
|
1043 |
lemmas [simp] = fset.map_comp fset.map_id fset.set_map
|
blanchet@56471
|
1044 |
|
blanchet@56471
|
1045 |
|
blanchet@56471
|
1046 |
subsection {* Advanced relator customization *}
|
blanchet@56471
|
1047 |
|
blanchet@56471
|
1048 |
(* Set vs. sum relators: *)
|
blanchet@56471
|
1049 |
|
blanchet@57285
|
1050 |
lemma rel_set_rel_sum[simp]:
|
blanchet@57285
|
1051 |
"rel_set (rel_sum \<chi> \<phi>) A1 A2 \<longleftrightarrow>
|
blanchet@57280
|
1052 |
rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
|
blanchet@56471
|
1053 |
(is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
|
blanchet@56471
|
1054 |
proof safe
|
blanchet@56471
|
1055 |
assume L: "?L"
|
blanchet@57280
|
1056 |
show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
|
blanchet@56471
|
1057 |
fix l1 assume "Inl l1 \<in> A1"
|
blanchet@57285
|
1058 |
then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inl l1) a2"
|
blanchet@57280
|
1059 |
using L unfolding rel_set_def by auto
|
blanchet@56471
|
1060 |
then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
|
blanchet@56471
|
1061 |
thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
|
blanchet@56471
|
1062 |
next
|
blanchet@56471
|
1063 |
fix l2 assume "Inl l2 \<in> A2"
|
blanchet@57285
|
1064 |
then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inl l2)"
|
blanchet@57280
|
1065 |
using L unfolding rel_set_def by auto
|
blanchet@56471
|
1066 |
then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
|
blanchet@56471
|
1067 |
thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
|
blanchet@56471
|
1068 |
qed
|
blanchet@57280
|
1069 |
show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
|
blanchet@56471
|
1070 |
fix r1 assume "Inr r1 \<in> A1"
|
blanchet@57285
|
1071 |
then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inr r1) a2"
|
blanchet@57280
|
1072 |
using L unfolding rel_set_def by auto
|
blanchet@56471
|
1073 |
then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
|
blanchet@56471
|
1074 |
thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
|
blanchet@56471
|
1075 |
next
|
blanchet@56471
|
1076 |
fix r2 assume "Inr r2 \<in> A2"
|
blanchet@57285
|
1077 |
then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inr r2)"
|
blanchet@57280
|
1078 |
using L unfolding rel_set_def by auto
|
blanchet@56471
|
1079 |
then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
|
blanchet@56471
|
1080 |
thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
|
blanchet@56471
|
1081 |
qed
|
blanchet@56471
|
1082 |
next
|
blanchet@56471
|
1083 |
assume Rl: "?Rl" and Rr: "?Rr"
|
blanchet@57280
|
1084 |
show ?L unfolding rel_set_def Bex_def vimage_eq proof safe
|
blanchet@56471
|
1085 |
fix a1 assume a1: "a1 \<in> A1"
|
blanchet@57285
|
1086 |
show "\<exists> a2. a2 \<in> A2 \<and> rel_sum \<chi> \<phi> a1 a2"
|
blanchet@56471
|
1087 |
proof(cases a1)
|
blanchet@56471
|
1088 |
case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
|
blanchet@57280
|
1089 |
using Rl a1 unfolding rel_set_def by blast
|
blanchet@56471
|
1090 |
thus ?thesis unfolding Inl by auto
|
blanchet@56471
|
1091 |
next
|
blanchet@56471
|
1092 |
case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
|
blanchet@57280
|
1093 |
using Rr a1 unfolding rel_set_def by blast
|
blanchet@56471
|
1094 |
thus ?thesis unfolding Inr by auto
|
blanchet@56471
|
1095 |
qed
|
blanchet@56471
|
1096 |
next
|
blanchet@56471
|
1097 |
fix a2 assume a2: "a2 \<in> A2"
|
blanchet@57285
|
1098 |
show "\<exists> a1. a1 \<in> A1 \<and> rel_sum \<chi> \<phi> a1 a2"
|
blanchet@56471
|
1099 |
proof(cases a2)
|
blanchet@56471
|
1100 |
case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
|
blanchet@57280
|
1101 |
using Rl a2 unfolding rel_set_def by blast
|
blanchet@56471
|
1102 |
thus ?thesis unfolding Inl by auto
|
blanchet@56471
|
1103 |
next
|
blanchet@56471
|
1104 |
case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
|
blanchet@57280
|
1105 |
using Rr a2 unfolding rel_set_def by blast
|
blanchet@56471
|
1106 |
thus ?thesis unfolding Inr by auto
|
blanchet@56471
|
1107 |
qed
|
blanchet@56471
|
1108 |
qed
|
blanchet@56471
|
1109 |
qed
|
blanchet@56471
|
1110 |
|
blanchet@56471
|
1111 |
end
|