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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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*)
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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 Main 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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by simp
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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 "(set_rel (set_rel A) ===> set_rel 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 "(set_rel (set_rel A) ===> set_rel 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 "(set_rel (set_rel A) ===> set_rel 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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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" "(\<And>a. a \<in> X \<Longrightarrow> z |\<subseteq>| a)"
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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" "(\<And>a. a \<in> X \<Longrightarrow> a |\<subseteq>| z)"
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then show "x |\<subseteq>| Sup X" 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, auto)+
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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 by simp
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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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lemma compose_rel_to_Domainp:
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assumes "left_unique R"
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assumes "(R ===> op=) P P'"
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shows "(R OO Lifting.invariant P' OO R\<inverse>\<inverse>) x y \<longleftrightarrow> Domainp R x \<and> P x \<and> x = y"
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using assms unfolding OO_def conversep_iff Domainp_iff left_unique_def fun_rel_def invariant_def
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by blast
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lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer
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by (subst compose_rel_to_Domainp [OF _ finite_transfer]) (auto intro: transfer_raw finite_subset
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simp add: fset.pcr_cr_eq[symmetric] Domainp_set fset.domain_eq)
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lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer by simp
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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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(* FIXME why is not invariant here unfolded ? *)
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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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unfolding invariant_def Set.bind_def by clarsimp metis
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lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer
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by (subst(asm) compose_rel_to_Domainp [OF _ finite_transfer])
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(auto intro: transfer_raw simp add: fset.pcr_cr_eq[symmetric] Domainp_set fset.domain_eq invariant_def)
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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]
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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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lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
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lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
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lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
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lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
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lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
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lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
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lemmas fimageI = imageI[Transfer.transferred]
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lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
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lemmas fimageE[elim!] = imageE[Transfer.transferred]
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lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
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lemmas fimage_funion = image_Un[Transfer.transferred]
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lemmas fimage_iff = image_iff[Transfer.transferred]
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lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
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lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
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lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
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lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
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lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
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lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
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lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
|
|
301 |
lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
|
|
302 |
lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
|
|
303 |
lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
|
|
304 |
lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
|
|
305 |
lemmas pfsubsetD = psubsetD[Transfer.transferred]
|
|
306 |
lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
|
|
307 |
lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
|
|
308 |
lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
|
53953
|
309 |
lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
|
|
310 |
lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
|
53964
|
311 |
lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
|
|
312 |
lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
|
|
313 |
lemmas fsubset_finsert = subset_insert[Transfer.transferred]
|
53953
|
314 |
lemmas funion_upper1 = Un_upper1[Transfer.transferred]
|
|
315 |
lemmas funion_upper2 = Un_upper2[Transfer.transferred]
|
|
316 |
lemmas funion_least = Un_least[Transfer.transferred]
|
|
317 |
lemmas finter_lower1 = Int_lower1[Transfer.transferred]
|
|
318 |
lemmas finter_lower2 = Int_lower2[Transfer.transferred]
|
|
319 |
lemmas finter_greatest = Int_greatest[Transfer.transferred]
|
53964
|
320 |
lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
|
|
321 |
lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
|
|
322 |
lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
|
|
323 |
lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
|
53953
|
324 |
lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
|
|
325 |
lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
|
|
326 |
lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
|
|
327 |
lemmas finsert_absorb = insert_absorb[Transfer.transferred]
|
|
328 |
lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
|
|
329 |
lemmas finsert_commute = insert_commute[Transfer.transferred]
|
53964
|
330 |
lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
|
53953
|
331 |
lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
|
|
332 |
lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
|
|
333 |
lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
|
|
334 |
lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
|
|
335 |
lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
|
|
336 |
lemmas fimage_constant = image_constant[Transfer.transferred]
|
|
337 |
lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
|
|
338 |
lemmas fimage_fimage = image_image[Transfer.transferred]
|
|
339 |
lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
|
|
340 |
lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
|
|
341 |
lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
|
|
342 |
lemmas fimage_cong = image_cong[Transfer.transferred]
|
53964
|
343 |
lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
|
|
344 |
lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
|
53953
|
345 |
lemmas finter_absorb = Int_absorb[Transfer.transferred]
|
|
346 |
lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
|
|
347 |
lemmas finter_commute = Int_commute[Transfer.transferred]
|
|
348 |
lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
|
|
349 |
lemmas finter_assoc = Int_assoc[Transfer.transferred]
|
|
350 |
lemmas finter_ac = Int_ac[Transfer.transferred]
|
|
351 |
lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
|
|
352 |
lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
|
|
353 |
lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
|
|
354 |
lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
|
|
355 |
lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
|
|
356 |
lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
|
|
357 |
lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
|
53964
|
358 |
lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
|
53953
|
359 |
lemmas funion_absorb = Un_absorb[Transfer.transferred]
|
|
360 |
lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
|
|
361 |
lemmas funion_commute = Un_commute[Transfer.transferred]
|
|
362 |
lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
|
|
363 |
lemmas funion_assoc = Un_assoc[Transfer.transferred]
|
|
364 |
lemmas funion_ac = Un_ac[Transfer.transferred]
|
|
365 |
lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
|
|
366 |
lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
|
|
367 |
lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
|
|
368 |
lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
|
|
369 |
lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
|
|
370 |
lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
|
|
371 |
lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
|
|
372 |
lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
|
|
373 |
lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
|
|
374 |
lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
|
|
375 |
lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
|
|
376 |
lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
|
|
377 |
lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
|
|
378 |
lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
|
|
379 |
lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
|
53964
|
380 |
lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
|
53953
|
381 |
lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
|
53964
|
382 |
lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
|
53953
|
383 |
lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
|
|
384 |
lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
|
|
385 |
lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
|
|
386 |
lemmas fBall_funion = ball_Un[Transfer.transferred]
|
|
387 |
lemmas fBex_funion = bex_Un[Transfer.transferred]
|
|
388 |
lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
|
|
389 |
lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
|
|
390 |
lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
|
|
391 |
lemmas fminus_triv = Diff_triv[Transfer.transferred]
|
|
392 |
lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
|
|
393 |
lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
|
|
394 |
lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
|
|
395 |
lemmas fminus_finsert = Diff_insert[Transfer.transferred]
|
|
396 |
lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
|
|
397 |
lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
|
|
398 |
lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
|
|
399 |
lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
|
|
400 |
lemmas finsert_fminus = insert_Diff[Transfer.transferred]
|
|
401 |
lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
|
|
402 |
lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
|
|
403 |
lemmas fminus_partition = Diff_partition[Transfer.transferred]
|
|
404 |
lemmas double_fminus = double_diff[Transfer.transferred]
|
|
405 |
lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
|
|
406 |
lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
|
|
407 |
lemmas fminus_funion = Diff_Un[Transfer.transferred]
|
|
408 |
lemmas fminus_finter = Diff_Int[Transfer.transferred]
|
|
409 |
lemmas funion_fminus = Un_Diff[Transfer.transferred]
|
|
410 |
lemmas finter_fminus = Int_Diff[Transfer.transferred]
|
|
411 |
lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
|
|
412 |
lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
|
|
413 |
lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
|
|
414 |
lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
|
|
415 |
lemmas fPow_finsert = Pow_insert[Transfer.transferred]
|
53964
|
416 |
lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
|
53953
|
417 |
lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
|
53964
|
418 |
lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
|
|
419 |
lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
|
|
420 |
lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
|
53953
|
421 |
lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
|
|
422 |
lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
|
|
423 |
lemmas fimage_mono = image_mono[Transfer.transferred]
|
|
424 |
lemmas fPow_mono = Pow_mono[Transfer.transferred]
|
|
425 |
lemmas finsert_mono = insert_mono[Transfer.transferred]
|
|
426 |
lemmas funion_mono = Un_mono[Transfer.transferred]
|
|
427 |
lemmas finter_mono = Int_mono[Transfer.transferred]
|
|
428 |
lemmas fminus_mono = Diff_mono[Transfer.transferred]
|
|
429 |
lemmas fin_mono = in_mono[Transfer.transferred]
|
|
430 |
lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
|
|
431 |
lemmas fLeast_mono = Least_mono[Transfer.transferred]
|
|
432 |
lemmas fbind_fbind = bind_bind[Transfer.transferred]
|
|
433 |
lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
|
|
434 |
lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
|
|
435 |
lemmas fbind_const = bind_const[Transfer.transferred]
|
|
436 |
lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
|
|
437 |
lemmas fequalityI = equalityI[Transfer.transferred]
|
|
438 |
|
|
439 |
subsection {* Additional lemmas*}
|
|
440 |
|
53969
|
441 |
subsubsection {* @{text fsingleton} *}
|
53953
|
442 |
|
|
443 |
lemmas fsingletonE = fsingletonD [elim_format]
|
|
444 |
|
53969
|
445 |
subsubsection {* @{text femepty} *}
|
53953
|
446 |
|
|
447 |
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
|
|
448 |
by transfer auto
|
|
449 |
|
|
450 |
(* FIXME, transferred doesn't work here *)
|
|
451 |
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
|
|
452 |
by simp
|
|
453 |
|
53969
|
454 |
subsubsection {* @{text fset} *}
|
53953
|
455 |
|
53963
|
456 |
lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
|
53953
|
457 |
|
|
458 |
lemma finite_fset [simp]:
|
|
459 |
shows "finite (fset S)"
|
|
460 |
by transfer simp
|
|
461 |
|
53963
|
462 |
lemmas fset_cong = fset_inject
|
53953
|
463 |
|
|
464 |
lemma filter_fset [simp]:
|
|
465 |
shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
|
|
466 |
by transfer auto
|
|
467 |
|
53963
|
468 |
lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
|
|
469 |
|
|
470 |
lemmas inter_fset[simp] = inf_fset.rep_eq
|
53953
|
471 |
|
53963
|
472 |
lemmas union_fset[simp] = sup_fset.rep_eq
|
53953
|
473 |
|
53963
|
474 |
lemmas minus_fset[simp] = minus_fset.rep_eq
|
53953
|
475 |
|
53969
|
476 |
subsubsection {* @{text filter_fset} *}
|
53953
|
477 |
|
|
478 |
lemma subset_ffilter:
|
|
479 |
"ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
|
|
480 |
by transfer auto
|
|
481 |
|
|
482 |
lemma eq_ffilter:
|
|
483 |
"(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
|
|
484 |
by transfer auto
|
|
485 |
|
53964
|
486 |
lemma pfsubset_ffilter:
|
53953
|
487 |
"(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow>
|
|
488 |
ffilter P A |\<subset>| ffilter Q A"
|
|
489 |
unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
|
|
490 |
|
53969
|
491 |
subsubsection {* @{text finsert} *}
|
53953
|
492 |
|
|
493 |
(* FIXME, transferred doesn't work here *)
|
|
494 |
lemma set_finsert:
|
|
495 |
assumes "x |\<in>| A"
|
|
496 |
obtains B where "A = finsert x B" and "x |\<notin>| B"
|
|
497 |
using assms by transfer (metis Set.set_insert finite_insert)
|
|
498 |
|
|
499 |
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
|
|
500 |
by (rule_tac x = "A |-| {|a|}" in exI, blast)
|
|
501 |
|
53969
|
502 |
subsubsection {* @{text fimage} *}
|
53953
|
503 |
|
|
504 |
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
|
|
505 |
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
|
|
506 |
|
|
507 |
subsubsection {* bounded quantification *}
|
|
508 |
|
|
509 |
lemma bex_simps [simp, no_atp]:
|
|
510 |
"\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)"
|
|
511 |
"\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
|
|
512 |
"\<And>P. fBex {||} P = False"
|
|
513 |
"\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
|
|
514 |
"\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
|
|
515 |
"\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
|
|
516 |
by auto
|
|
517 |
|
|
518 |
lemma ball_simps [simp, no_atp]:
|
|
519 |
"\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
|
|
520 |
"\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
|
|
521 |
"\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
|
|
522 |
"\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
|
|
523 |
"\<And>P. fBall {||} P = True"
|
|
524 |
"\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
|
|
525 |
"\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
|
|
526 |
"\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
|
|
527 |
by auto
|
|
528 |
|
|
529 |
lemma atomize_fBall:
|
|
530 |
"(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
|
|
531 |
apply (simp only: atomize_all atomize_imp)
|
|
532 |
apply (rule equal_intr_rule)
|
|
533 |
by (transfer, simp)+
|
|
534 |
|
53963
|
535 |
end
|
|
536 |
|
53969
|
537 |
subsubsection {* @{text fcard} *}
|
53963
|
538 |
|
53964
|
539 |
(* FIXME: improve transferred to handle bounded meta quantification *)
|
|
540 |
|
53963
|
541 |
lemma fcard_fempty:
|
|
542 |
"fcard {||} = 0"
|
|
543 |
by transfer (rule card_empty)
|
|
544 |
|
|
545 |
lemma fcard_finsert_disjoint:
|
|
546 |
"x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
|
|
547 |
by transfer (rule card_insert_disjoint)
|
|
548 |
|
|
549 |
lemma fcard_finsert_if:
|
|
550 |
"fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
|
|
551 |
by transfer (rule card_insert_if)
|
|
552 |
|
|
553 |
lemma card_0_eq [simp, no_atp]:
|
|
554 |
"fcard A = 0 \<longleftrightarrow> A = {||}"
|
|
555 |
by transfer (rule card_0_eq)
|
|
556 |
|
|
557 |
lemma fcard_Suc_fminus1:
|
|
558 |
"x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
|
|
559 |
by transfer (rule card_Suc_Diff1)
|
|
560 |
|
|
561 |
lemma fcard_fminus_fsingleton:
|
|
562 |
"x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
|
|
563 |
by transfer (rule card_Diff_singleton)
|
|
564 |
|
|
565 |
lemma fcard_fminus_fsingleton_if:
|
|
566 |
"fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
|
|
567 |
by transfer (rule card_Diff_singleton_if)
|
|
568 |
|
|
569 |
lemma fcard_fminus_finsert[simp]:
|
|
570 |
assumes "a |\<in>| A" and "a |\<notin>| B"
|
|
571 |
shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
|
|
572 |
using assms by transfer (rule card_Diff_insert)
|
|
573 |
|
|
574 |
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
|
|
575 |
by transfer (rule card_insert)
|
|
576 |
|
|
577 |
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
|
|
578 |
by transfer (rule card_insert_le)
|
|
579 |
|
|
580 |
lemma fcard_mono:
|
|
581 |
"A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
|
|
582 |
by transfer (rule card_mono)
|
|
583 |
|
|
584 |
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
|
|
585 |
by transfer (rule card_seteq)
|
|
586 |
|
|
587 |
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
|
|
588 |
by transfer (rule psubset_card_mono)
|
|
589 |
|
|
590 |
lemma fcard_funion_finter:
|
|
591 |
"fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
|
|
592 |
by transfer (rule card_Un_Int)
|
|
593 |
|
|
594 |
lemma fcard_funion_disjoint:
|
|
595 |
"A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
|
|
596 |
by transfer (rule card_Un_disjoint)
|
|
597 |
|
|
598 |
lemma fcard_funion_fsubset:
|
|
599 |
"B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
|
|
600 |
by transfer (rule card_Diff_subset)
|
|
601 |
|
|
602 |
lemma diff_fcard_le_fcard_fminus:
|
|
603 |
"fcard A - fcard B \<le> fcard(A |-| B)"
|
|
604 |
by transfer (rule diff_card_le_card_Diff)
|
|
605 |
|
|
606 |
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
|
|
607 |
by transfer (rule card_Diff1_less)
|
|
608 |
|
|
609 |
lemma fcard_fminus2_less:
|
|
610 |
"x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
|
|
611 |
by transfer (rule card_Diff2_less)
|
|
612 |
|
|
613 |
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
|
|
614 |
by transfer (rule card_Diff1_le)
|
|
615 |
|
|
616 |
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
|
|
617 |
by transfer (rule card_psubset)
|
|
618 |
|
53969
|
619 |
subsubsection {* @{text ffold} *}
|
53963
|
620 |
|
|
621 |
(* FIXME: improve transferred to handle bounded meta quantification *)
|
|
622 |
|
|
623 |
context comp_fun_commute
|
|
624 |
begin
|
|
625 |
lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
|
|
626 |
|
|
627 |
lemma ffold_finsert [simp]:
|
|
628 |
assumes "x |\<notin>| A"
|
|
629 |
shows "ffold f z (finsert x A) = f x (ffold f z A)"
|
|
630 |
using assms by (transfer fixing: f) (rule fold_insert)
|
|
631 |
|
|
632 |
lemma ffold_fun_left_comm:
|
|
633 |
"f x (ffold f z A) = ffold f (f x z) A"
|
|
634 |
by (transfer fixing: f) (rule fold_fun_left_comm)
|
|
635 |
|
|
636 |
lemma ffold_finsert2:
|
|
637 |
"x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A) = ffold f (f x z) A"
|
|
638 |
by (transfer fixing: f) (rule fold_insert2)
|
|
639 |
|
|
640 |
lemma ffold_rec:
|
|
641 |
assumes "x |\<in>| A"
|
|
642 |
shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
|
|
643 |
using assms by (transfer fixing: f) (rule fold_rec)
|
|
644 |
|
|
645 |
lemma ffold_finsert_fremove:
|
|
646 |
"ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
|
|
647 |
by (transfer fixing: f) (rule fold_insert_remove)
|
|
648 |
end
|
|
649 |
|
|
650 |
lemma ffold_fimage:
|
|
651 |
assumes "inj_on g (fset A)"
|
|
652 |
shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
|
|
653 |
using assms by transfer' (rule fold_image)
|
|
654 |
|
|
655 |
lemma ffold_cong:
|
|
656 |
assumes "comp_fun_commute f" "comp_fun_commute g"
|
|
657 |
"\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
|
|
658 |
and "s = t" and "A = B"
|
|
659 |
shows "ffold f s A = ffold g t B"
|
|
660 |
using assms by transfer (metis Finite_Set.fold_cong)
|
|
661 |
|
|
662 |
context comp_fun_idem
|
|
663 |
begin
|
|
664 |
|
|
665 |
lemma ffold_finsert_idem:
|
|
666 |
"ffold f z (finsert x A) = f x (ffold f z A)"
|
|
667 |
by (transfer fixing: f) (rule fold_insert_idem)
|
|
668 |
|
|
669 |
declare ffold_finsert [simp del] ffold_finsert_idem [simp]
|
|
670 |
|
|
671 |
lemma ffold_finsert_idem2:
|
|
672 |
"ffold f z (finsert x A) = ffold f (f x z) A"
|
|
673 |
by (transfer fixing: f) (rule fold_insert_idem2)
|
|
674 |
|
|
675 |
end
|
|
676 |
|
53953
|
677 |
subsection {* Choice in fsets *}
|
|
678 |
|
|
679 |
lemma fset_choice:
|
|
680 |
assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
|
|
681 |
shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
|
|
682 |
using assms by transfer metis
|
|
683 |
|
|
684 |
subsection {* Induction and Cases rules for fsets *}
|
|
685 |
|
|
686 |
lemma fset_exhaust [case_names empty insert, cases type: fset]:
|
|
687 |
assumes fempty_case: "S = {||} \<Longrightarrow> P"
|
|
688 |
and finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
|
|
689 |
shows "P"
|
|
690 |
using assms by transfer blast
|
|
691 |
|
|
692 |
lemma fset_induct [case_names empty insert]:
|
|
693 |
assumes fempty_case: "P {||}"
|
|
694 |
and finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
|
|
695 |
shows "P S"
|
|
696 |
proof -
|
|
697 |
(* FIXME transfer and right_total vs. bi_total *)
|
|
698 |
note Domainp_forall_transfer[transfer_rule]
|
|
699 |
show ?thesis
|
|
700 |
using assms by transfer (auto intro: finite_induct)
|
|
701 |
qed
|
|
702 |
|
|
703 |
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
|
|
704 |
assumes empty_fset_case: "P {||}"
|
|
705 |
and insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
|
|
706 |
shows "P S"
|
|
707 |
proof -
|
|
708 |
(* FIXME transfer and right_total vs. bi_total *)
|
|
709 |
note Domainp_forall_transfer[transfer_rule]
|
|
710 |
show ?thesis
|
|
711 |
using assms by transfer (auto intro: finite_induct)
|
|
712 |
qed
|
|
713 |
|
|
714 |
lemma fset_card_induct:
|
|
715 |
assumes empty_fset_case: "P {||}"
|
|
716 |
and card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
|
|
717 |
shows "P S"
|
|
718 |
proof (induct S)
|
|
719 |
case empty
|
|
720 |
show "P {||}" by (rule empty_fset_case)
|
|
721 |
next
|
|
722 |
case (insert x S)
|
|
723 |
have h: "P S" by fact
|
|
724 |
have "x |\<notin>| S" by fact
|
|
725 |
then have "Suc (fcard S) = fcard (finsert x S)"
|
|
726 |
by transfer auto
|
|
727 |
then show "P (finsert x S)"
|
|
728 |
using h card_fset_Suc_case by simp
|
|
729 |
qed
|
|
730 |
|
|
731 |
lemma fset_strong_cases:
|
|
732 |
obtains "xs = {||}"
|
|
733 |
| ys x where "x |\<notin>| ys" and "xs = finsert x ys"
|
|
734 |
by transfer blast
|
|
735 |
|
|
736 |
lemma fset_induct2:
|
|
737 |
"P {||} {||} \<Longrightarrow>
|
|
738 |
(\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
|
|
739 |
(\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
|
|
740 |
(\<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>
|
|
741 |
P xsa ysa"
|
|
742 |
apply (induct xsa arbitrary: ysa)
|
|
743 |
apply (induct_tac x rule: fset_induct_stronger)
|
|
744 |
apply simp_all
|
|
745 |
apply (induct_tac xa rule: fset_induct_stronger)
|
|
746 |
apply simp_all
|
|
747 |
done
|
|
748 |
|
|
749 |
subsection {* Setup for Lifting/Transfer *}
|
|
750 |
|
|
751 |
subsubsection {* Relator and predicator properties *}
|
|
752 |
|
|
753 |
lift_definition fset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is set_rel
|
|
754 |
parametric set_rel_transfer ..
|
|
755 |
|
|
756 |
lemma fset_rel_alt_def: "fset_rel R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y)
|
|
757 |
\<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
|
|
758 |
apply (rule ext)+
|
|
759 |
apply transfer'
|
|
760 |
apply (subst set_rel_def[unfolded fun_eq_iff])
|
|
761 |
by blast
|
|
762 |
|
|
763 |
lemma fset_rel_conversep: "fset_rel (conversep R) = conversep (fset_rel R)"
|
|
764 |
unfolding fset_rel_alt_def by auto
|
|
765 |
|
|
766 |
lemmas fset_rel_eq [relator_eq] = set_rel_eq[Transfer.transferred]
|
|
767 |
|
|
768 |
lemma fset_rel_mono[relator_mono]: "A \<le> B \<Longrightarrow> fset_rel A \<le> fset_rel B"
|
|
769 |
unfolding fset_rel_alt_def by blast
|
|
770 |
|
|
771 |
lemma finite_set_rel:
|
|
772 |
assumes fin: "finite X" "finite Z"
|
|
773 |
assumes R_S: "set_rel (R OO S) X Z"
|
|
774 |
shows "\<exists>Y. finite Y \<and> set_rel R X Y \<and> set_rel S Y Z"
|
|
775 |
proof -
|
|
776 |
obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
|
|
777 |
apply atomize_elim
|
|
778 |
apply (subst bchoice_iff[symmetric])
|
|
779 |
using R_S[unfolded set_rel_def OO_def] by blast
|
|
780 |
|
|
781 |
obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))"
|
|
782 |
apply atomize_elim
|
|
783 |
apply (subst bchoice_iff[symmetric])
|
|
784 |
using R_S[unfolded set_rel_def OO_def] by blast
|
|
785 |
|
|
786 |
let ?Y = "f ` X \<union> g ` Z"
|
|
787 |
have "finite ?Y" by (simp add: fin)
|
|
788 |
moreover have "set_rel R X ?Y"
|
|
789 |
unfolding set_rel_def
|
|
790 |
using f g by clarsimp blast
|
|
791 |
moreover have "set_rel S ?Y Z"
|
|
792 |
unfolding set_rel_def
|
|
793 |
using f g by clarsimp blast
|
|
794 |
ultimately show ?thesis by metis
|
|
795 |
qed
|
|
796 |
|
|
797 |
lemma fset_rel_OO[relator_distr]: "fset_rel R OO fset_rel S = fset_rel (R OO S)"
|
|
798 |
apply (rule ext)+
|
|
799 |
by transfer (auto intro: finite_set_rel set_rel_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
|
|
800 |
|
|
801 |
lemma Domainp_fset[relator_domain]:
|
|
802 |
assumes "Domainp T = P"
|
|
803 |
shows "Domainp (fset_rel T) = (\<lambda>A. fBall A P)"
|
|
804 |
proof -
|
|
805 |
from assms obtain f where f: "\<forall>x\<in>Collect P. T x (f x)"
|
|
806 |
unfolding Domainp_iff[abs_def]
|
|
807 |
apply atomize_elim
|
|
808 |
by (subst bchoice_iff[symmetric]) auto
|
|
809 |
from assms f show ?thesis
|
|
810 |
unfolding fun_eq_iff fset_rel_alt_def Domainp_iff
|
|
811 |
apply clarify
|
|
812 |
apply (rule iffI)
|
|
813 |
apply blast
|
|
814 |
by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
|
|
815 |
qed
|
|
816 |
|
|
817 |
lemmas reflp_fset_rel[reflexivity_rule] = reflp_set_rel[Transfer.transferred]
|
|
818 |
|
|
819 |
lemma right_total_fset_rel[transfer_rule]: "right_total A \<Longrightarrow> right_total (fset_rel A)"
|
|
820 |
unfolding right_total_def
|
|
821 |
apply transfer
|
|
822 |
apply (subst(asm) choice_iff)
|
|
823 |
apply clarsimp
|
|
824 |
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
|
|
825 |
by (auto simp add: set_rel_def)
|
|
826 |
|
|
827 |
lemma left_total_fset_rel[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (fset_rel A)"
|
|
828 |
unfolding left_total_def
|
|
829 |
apply transfer
|
|
830 |
apply (subst(asm) choice_iff)
|
|
831 |
apply clarsimp
|
|
832 |
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
|
|
833 |
by (auto simp add: set_rel_def)
|
|
834 |
|
|
835 |
lemmas right_unique_fset_rel[transfer_rule] = right_unique_set_rel[Transfer.transferred]
|
|
836 |
lemmas left_unique_fset_rel[reflexivity_rule] = left_unique_set_rel[Transfer.transferred]
|
|
837 |
|
|
838 |
thm right_unique_fset_rel left_unique_fset_rel
|
|
839 |
|
|
840 |
lemma bi_unique_fset_rel[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (fset_rel A)"
|
|
841 |
by (auto intro: right_unique_fset_rel left_unique_fset_rel iff: bi_unique_iff)
|
|
842 |
|
|
843 |
lemma bi_total_fset_rel[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (fset_rel A)"
|
|
844 |
by (auto intro: right_total_fset_rel left_total_fset_rel iff: bi_total_iff)
|
|
845 |
|
|
846 |
lemmas fset_invariant_commute [invariant_commute] = set_invariant_commute[Transfer.transferred]
|
|
847 |
|
|
848 |
subsubsection {* Quotient theorem for the Lifting package *}
|
|
849 |
|
|
850 |
lemma Quotient_fset_map[quot_map]:
|
|
851 |
assumes "Quotient R Abs Rep T"
|
|
852 |
shows "Quotient (fset_rel R) (fimage Abs) (fimage Rep) (fset_rel T)"
|
|
853 |
using assms unfolding Quotient_alt_def4
|
|
854 |
by (simp add: fset_rel_OO[symmetric] fset_rel_conversep) (simp add: fset_rel_alt_def, blast)
|
|
855 |
|
|
856 |
subsubsection {* Transfer rules for the Transfer package *}
|
|
857 |
|
|
858 |
text {* Unconditional transfer rules *}
|
|
859 |
|
53963
|
860 |
context
|
|
861 |
begin
|
|
862 |
|
|
863 |
interpretation lifting_syntax .
|
|
864 |
|
53953
|
865 |
lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
|
|
866 |
|
|
867 |
lemma finsert_transfer [transfer_rule]:
|
|
868 |
"(A ===> fset_rel A ===> fset_rel A) finsert finsert"
|
|
869 |
unfolding fun_rel_def fset_rel_alt_def by blast
|
|
870 |
|
|
871 |
lemma funion_transfer [transfer_rule]:
|
|
872 |
"(fset_rel A ===> fset_rel A ===> fset_rel A) funion funion"
|
|
873 |
unfolding fun_rel_def fset_rel_alt_def by blast
|
|
874 |
|
|
875 |
lemma ffUnion_transfer [transfer_rule]:
|
|
876 |
"(fset_rel (fset_rel A) ===> fset_rel A) ffUnion ffUnion"
|
|
877 |
unfolding fun_rel_def fset_rel_alt_def by transfer (simp, fast)
|
|
878 |
|
|
879 |
lemma fimage_transfer [transfer_rule]:
|
|
880 |
"((A ===> B) ===> fset_rel A ===> fset_rel B) fimage fimage"
|
|
881 |
unfolding fun_rel_def fset_rel_alt_def by simp blast
|
|
882 |
|
|
883 |
lemma fBall_transfer [transfer_rule]:
|
|
884 |
"(fset_rel A ===> (A ===> op =) ===> op =) fBall fBall"
|
|
885 |
unfolding fset_rel_alt_def fun_rel_def by blast
|
|
886 |
|
|
887 |
lemma fBex_transfer [transfer_rule]:
|
|
888 |
"(fset_rel A ===> (A ===> op =) ===> op =) fBex fBex"
|
|
889 |
unfolding fset_rel_alt_def fun_rel_def by blast
|
|
890 |
|
|
891 |
(* FIXME transfer doesn't work here *)
|
|
892 |
lemma fPow_transfer [transfer_rule]:
|
|
893 |
"(fset_rel A ===> fset_rel (fset_rel A)) fPow fPow"
|
|
894 |
unfolding fun_rel_def
|
|
895 |
using Pow_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred]
|
|
896 |
by blast
|
|
897 |
|
|
898 |
lemma fset_rel_transfer [transfer_rule]:
|
|
899 |
"((A ===> B ===> op =) ===> fset_rel A ===> fset_rel B ===> op =)
|
|
900 |
fset_rel fset_rel"
|
|
901 |
unfolding fun_rel_def
|
|
902 |
using set_rel_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
|
|
903 |
by simp
|
|
904 |
|
|
905 |
lemma bind_transfer [transfer_rule]:
|
|
906 |
"(fset_rel A ===> (A ===> fset_rel B) ===> fset_rel B) fbind fbind"
|
|
907 |
using assms unfolding fun_rel_def
|
|
908 |
using bind_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
|
|
909 |
|
|
910 |
text {* Rules requiring bi-unique, bi-total or right-total relations *}
|
|
911 |
|
|
912 |
lemma fmember_transfer [transfer_rule]:
|
|
913 |
assumes "bi_unique A"
|
|
914 |
shows "(A ===> fset_rel A ===> op =) (op |\<in>|) (op |\<in>|)"
|
|
915 |
using assms unfolding fun_rel_def fset_rel_alt_def bi_unique_def by metis
|
|
916 |
|
|
917 |
lemma finter_transfer [transfer_rule]:
|
|
918 |
assumes "bi_unique A"
|
|
919 |
shows "(fset_rel A ===> fset_rel A ===> fset_rel A) finter finter"
|
|
920 |
using assms unfolding fun_rel_def
|
|
921 |
using inter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
|
|
922 |
|
53963
|
923 |
lemma fminus_transfer [transfer_rule]:
|
53953
|
924 |
assumes "bi_unique A"
|
|
925 |
shows "(fset_rel A ===> fset_rel A ===> fset_rel A) (op |-|) (op |-|)"
|
|
926 |
using assms unfolding fun_rel_def
|
|
927 |
using Diff_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
|
|
928 |
|
|
929 |
lemma fsubset_transfer [transfer_rule]:
|
|
930 |
assumes "bi_unique A"
|
|
931 |
shows "(fset_rel A ===> fset_rel A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
|
|
932 |
using assms unfolding fun_rel_def
|
|
933 |
using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
|
|
934 |
|
|
935 |
lemma fSup_transfer [transfer_rule]:
|
|
936 |
"bi_unique A \<Longrightarrow> (set_rel (fset_rel A) ===> fset_rel A) Sup Sup"
|
|
937 |
using assms unfolding fun_rel_def
|
|
938 |
apply clarify
|
|
939 |
apply transfer'
|
|
940 |
using Sup_fset_transfer[unfolded fun_rel_def] by blast
|
|
941 |
|
|
942 |
(* FIXME: add right_total_fInf_transfer *)
|
|
943 |
|
|
944 |
lemma fInf_transfer [transfer_rule]:
|
|
945 |
assumes "bi_unique A" and "bi_total A"
|
|
946 |
shows "(set_rel (fset_rel A) ===> fset_rel A) Inf Inf"
|
|
947 |
using assms unfolding fun_rel_def
|
|
948 |
apply clarify
|
|
949 |
apply transfer'
|
|
950 |
using Inf_fset_transfer[unfolded fun_rel_def] by blast
|
|
951 |
|
|
952 |
lemma ffilter_transfer [transfer_rule]:
|
|
953 |
assumes "bi_unique A"
|
|
954 |
shows "((A ===> op=) ===> fset_rel A ===> fset_rel A) ffilter ffilter"
|
|
955 |
using assms unfolding fun_rel_def
|
|
956 |
using Lifting_Set.filter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
|
|
957 |
|
|
958 |
lemma card_transfer [transfer_rule]:
|
|
959 |
"bi_unique A \<Longrightarrow> (fset_rel A ===> op =) fcard fcard"
|
|
960 |
using assms unfolding fun_rel_def
|
|
961 |
using card_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
|
|
962 |
|
|
963 |
end
|
|
964 |
|
|
965 |
lifting_update fset.lifting
|
|
966 |
lifting_forget fset.lifting
|
|
967 |
|
|
968 |
end
|