src/HOL/Library/FSet.thy
author haftmann
Fri Nov 01 18:51:14 2013 +0100 (2013-11-01)
changeset 54230 b1d955791529
parent 54014 21dac9a60f0c
child 54258 adfc759263ab
permissions -rw-r--r--
more simplification rules on unary and binary minus
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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]
kuncar@53964
   282
lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
kuncar@53953
   283
lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
kuncar@53953
   284
lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
kuncar@53953
   285
lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
kuncar@53953
   286
lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
kuncar@53953
   287
lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
kuncar@53953
   288
lemmas fimageI = imageI[Transfer.transferred]
kuncar@53953
   289
lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
kuncar@53953
   290
lemmas fimageE[elim!] = imageE[Transfer.transferred]
kuncar@53953
   291
lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
kuncar@53953
   292
lemmas fimage_funion = image_Un[Transfer.transferred]
kuncar@53953
   293
lemmas fimage_iff = image_iff[Transfer.transferred]
kuncar@53964
   294
lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
kuncar@53964
   295
lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
kuncar@53953
   296
lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
kuncar@53953
   297
lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
kuncar@53953
   298
lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
kuncar@53964
   299
lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
kuncar@53964
   300
lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
kuncar@53964
   301
lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
kuncar@53964
   302
lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
kuncar@53964
   303
lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
kuncar@53964
   304
lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
kuncar@53964
   305
lemmas pfsubsetD = psubsetD[Transfer.transferred]
kuncar@53964
   306
lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
kuncar@53964
   307
lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
kuncar@53964
   308
lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
kuncar@53953
   309
lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
kuncar@53953
   310
lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
kuncar@53964
   311
lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
kuncar@53964
   312
lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
kuncar@53964
   313
lemmas fsubset_finsert = subset_insert[Transfer.transferred]
kuncar@53953
   314
lemmas funion_upper1 = Un_upper1[Transfer.transferred]
kuncar@53953
   315
lemmas funion_upper2 = Un_upper2[Transfer.transferred]
kuncar@53953
   316
lemmas funion_least = Un_least[Transfer.transferred]
kuncar@53953
   317
lemmas finter_lower1 = Int_lower1[Transfer.transferred]
kuncar@53953
   318
lemmas finter_lower2 = Int_lower2[Transfer.transferred]
kuncar@53953
   319
lemmas finter_greatest = Int_greatest[Transfer.transferred]
kuncar@53964
   320
lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
kuncar@53964
   321
lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
kuncar@53964
   322
lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
kuncar@53964
   323
lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
kuncar@53953
   324
lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
kuncar@53953
   325
lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
kuncar@53953
   326
lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
kuncar@53953
   327
lemmas finsert_absorb = insert_absorb[Transfer.transferred]
kuncar@53953
   328
lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
kuncar@53953
   329
lemmas finsert_commute = insert_commute[Transfer.transferred]
kuncar@53964
   330
lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
kuncar@53953
   331
lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
kuncar@53953
   332
lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
kuncar@53953
   333
lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
kuncar@53953
   334
lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
kuncar@53953
   335
lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
kuncar@53953
   336
lemmas fimage_constant = image_constant[Transfer.transferred]
kuncar@53953
   337
lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
kuncar@53953
   338
lemmas fimage_fimage = image_image[Transfer.transferred]
kuncar@53953
   339
lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
kuncar@53953
   340
lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
kuncar@53953
   341
lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
kuncar@53953
   342
lemmas fimage_cong = image_cong[Transfer.transferred]
kuncar@53964
   343
lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
kuncar@53964
   344
lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
kuncar@53953
   345
lemmas finter_absorb = Int_absorb[Transfer.transferred]
kuncar@53953
   346
lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
kuncar@53953
   347
lemmas finter_commute = Int_commute[Transfer.transferred]
kuncar@53953
   348
lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
kuncar@53953
   349
lemmas finter_assoc = Int_assoc[Transfer.transferred]
kuncar@53953
   350
lemmas finter_ac = Int_ac[Transfer.transferred]
kuncar@53953
   351
lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
kuncar@53953
   352
lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
kuncar@53953
   353
lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
kuncar@53953
   354
lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
kuncar@53953
   355
lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
kuncar@53953
   356
lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
kuncar@53953
   357
lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
kuncar@53964
   358
lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
kuncar@53953
   359
lemmas funion_absorb = Un_absorb[Transfer.transferred]
kuncar@53953
   360
lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
kuncar@53953
   361
lemmas funion_commute = Un_commute[Transfer.transferred]
kuncar@53953
   362
lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
kuncar@53953
   363
lemmas funion_assoc = Un_assoc[Transfer.transferred]
kuncar@53953
   364
lemmas funion_ac = Un_ac[Transfer.transferred]
kuncar@53953
   365
lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
kuncar@53953
   366
lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
kuncar@53953
   367
lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
kuncar@53953
   368
lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
kuncar@53953
   369
lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
kuncar@53953
   370
lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
kuncar@53953
   371
lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
kuncar@53953
   372
lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
kuncar@53953
   373
lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
kuncar@53953
   374
lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
kuncar@53953
   375
lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
kuncar@53953
   376
lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
kuncar@53953
   377
lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
kuncar@53953
   378
lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
kuncar@53953
   379
lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
kuncar@53964
   380
lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
kuncar@53953
   381
lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
kuncar@53964
   382
lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
kuncar@53953
   383
lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
kuncar@53953
   384
lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
kuncar@53953
   385
lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
kuncar@53953
   386
lemmas fBall_funion = ball_Un[Transfer.transferred]
kuncar@53953
   387
lemmas fBex_funion = bex_Un[Transfer.transferred]
kuncar@53953
   388
lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
kuncar@53953
   389
lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
kuncar@53953
   390
lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
kuncar@53953
   391
lemmas fminus_triv = Diff_triv[Transfer.transferred]
kuncar@53953
   392
lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
kuncar@53953
   393
lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
kuncar@53953
   394
lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
kuncar@53953
   395
lemmas fminus_finsert = Diff_insert[Transfer.transferred]
kuncar@53953
   396
lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
kuncar@53953
   397
lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
kuncar@53953
   398
lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
kuncar@53953
   399
lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
kuncar@53953
   400
lemmas finsert_fminus = insert_Diff[Transfer.transferred]
kuncar@53953
   401
lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
kuncar@53953
   402
lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
kuncar@53953
   403
lemmas fminus_partition = Diff_partition[Transfer.transferred]
kuncar@53953
   404
lemmas double_fminus = double_diff[Transfer.transferred]
kuncar@53953
   405
lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
kuncar@53953
   406
lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
kuncar@53953
   407
lemmas fminus_funion = Diff_Un[Transfer.transferred]
kuncar@53953
   408
lemmas fminus_finter = Diff_Int[Transfer.transferred]
kuncar@53953
   409
lemmas funion_fminus = Un_Diff[Transfer.transferred]
kuncar@53953
   410
lemmas finter_fminus = Int_Diff[Transfer.transferred]
kuncar@53953
   411
lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
kuncar@53953
   412
lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
kuncar@53953
   413
lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
kuncar@53953
   414
lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
kuncar@53953
   415
lemmas fPow_finsert = Pow_insert[Transfer.transferred]
kuncar@53964
   416
lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
kuncar@53953
   417
lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
kuncar@53964
   418
lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
kuncar@53964
   419
lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
kuncar@53964
   420
lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
kuncar@53953
   421
lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
kuncar@53953
   422
lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
kuncar@53953
   423
lemmas fimage_mono = image_mono[Transfer.transferred]
kuncar@53953
   424
lemmas fPow_mono = Pow_mono[Transfer.transferred]
kuncar@53953
   425
lemmas finsert_mono = insert_mono[Transfer.transferred]
kuncar@53953
   426
lemmas funion_mono = Un_mono[Transfer.transferred]
kuncar@53953
   427
lemmas finter_mono = Int_mono[Transfer.transferred]
kuncar@53953
   428
lemmas fminus_mono = Diff_mono[Transfer.transferred]
kuncar@53953
   429
lemmas fin_mono = in_mono[Transfer.transferred]
kuncar@53953
   430
lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
kuncar@53953
   431
lemmas fLeast_mono = Least_mono[Transfer.transferred]
kuncar@53953
   432
lemmas fbind_fbind = bind_bind[Transfer.transferred]
kuncar@53953
   433
lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
kuncar@53953
   434
lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
kuncar@53953
   435
lemmas fbind_const = bind_const[Transfer.transferred]
kuncar@53953
   436
lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
kuncar@53953
   437
lemmas fequalityI = equalityI[Transfer.transferred]
kuncar@53953
   438
kuncar@53953
   439
subsection {* Additional lemmas*}
kuncar@53953
   440
wenzelm@53969
   441
subsubsection {* @{text fsingleton} *}
kuncar@53953
   442
kuncar@53953
   443
lemmas fsingletonE = fsingletonD [elim_format]
kuncar@53953
   444
wenzelm@53969
   445
subsubsection {* @{text femepty} *}
kuncar@53953
   446
kuncar@53953
   447
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
kuncar@53953
   448
by transfer auto
kuncar@53953
   449
kuncar@53953
   450
(* FIXME, transferred doesn't work here *)
kuncar@53953
   451
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
kuncar@53953
   452
  by simp
kuncar@53953
   453
wenzelm@53969
   454
subsubsection {* @{text fset} *}
kuncar@53953
   455
kuncar@53963
   456
lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
kuncar@53953
   457
kuncar@53953
   458
lemma finite_fset [simp]: 
kuncar@53953
   459
  shows "finite (fset S)"
kuncar@53953
   460
  by transfer simp
kuncar@53953
   461
kuncar@53963
   462
lemmas fset_cong = fset_inject
kuncar@53953
   463
kuncar@53953
   464
lemma filter_fset [simp]:
kuncar@53953
   465
  shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
kuncar@53953
   466
  by transfer auto
kuncar@53953
   467
kuncar@53963
   468
lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
kuncar@53963
   469
kuncar@53963
   470
lemmas inter_fset[simp] = inf_fset.rep_eq
kuncar@53953
   471
kuncar@53963
   472
lemmas union_fset[simp] = sup_fset.rep_eq
kuncar@53953
   473
kuncar@53963
   474
lemmas minus_fset[simp] = minus_fset.rep_eq
kuncar@53953
   475
wenzelm@53969
   476
subsubsection {* @{text filter_fset} *}
kuncar@53953
   477
kuncar@53953
   478
lemma subset_ffilter: 
kuncar@53953
   479
  "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
kuncar@53953
   480
  by transfer auto
kuncar@53953
   481
kuncar@53953
   482
lemma eq_ffilter: 
kuncar@53953
   483
  "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
kuncar@53953
   484
  by transfer auto
kuncar@53953
   485
kuncar@53964
   486
lemma pfsubset_ffilter:
kuncar@53953
   487
  "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow> 
kuncar@53953
   488
    ffilter P A |\<subset>| ffilter Q A"
kuncar@53953
   489
  unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
kuncar@53953
   490
wenzelm@53969
   491
subsubsection {* @{text finsert} *}
kuncar@53953
   492
kuncar@53953
   493
(* FIXME, transferred doesn't work here *)
kuncar@53953
   494
lemma set_finsert:
kuncar@53953
   495
  assumes "x |\<in>| A"
kuncar@53953
   496
  obtains B where "A = finsert x B" and "x |\<notin>| B"
kuncar@53953
   497
using assms by transfer (metis Set.set_insert finite_insert)
kuncar@53953
   498
kuncar@53953
   499
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
kuncar@53953
   500
  by (rule_tac x = "A |-| {|a|}" in exI, blast)
kuncar@53953
   501
wenzelm@53969
   502
subsubsection {* @{text fimage} *}
kuncar@53953
   503
kuncar@53953
   504
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
kuncar@53953
   505
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
kuncar@53953
   506
kuncar@53953
   507
subsubsection {* bounded quantification *}
kuncar@53953
   508
kuncar@53953
   509
lemma bex_simps [simp, no_atp]:
kuncar@53953
   510
  "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" 
kuncar@53953
   511
  "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
kuncar@53953
   512
  "\<And>P. fBex {||} P = False" 
kuncar@53953
   513
  "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
kuncar@53953
   514
  "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
kuncar@53953
   515
  "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
kuncar@53953
   516
by auto
kuncar@53953
   517
kuncar@53953
   518
lemma ball_simps [simp, no_atp]:
kuncar@53953
   519
  "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
kuncar@53953
   520
  "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
kuncar@53953
   521
  "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
kuncar@53953
   522
  "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
kuncar@53953
   523
  "\<And>P. fBall {||} P = True"
kuncar@53953
   524
  "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
kuncar@53953
   525
  "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
kuncar@53953
   526
  "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
kuncar@53953
   527
by auto
kuncar@53953
   528
kuncar@53953
   529
lemma atomize_fBall:
kuncar@53953
   530
    "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
kuncar@53953
   531
apply (simp only: atomize_all atomize_imp)
kuncar@53953
   532
apply (rule equal_intr_rule)
kuncar@53953
   533
by (transfer, simp)+
kuncar@53953
   534
kuncar@53963
   535
end
kuncar@53963
   536
wenzelm@53969
   537
subsubsection {* @{text fcard} *}
kuncar@53963
   538
kuncar@53964
   539
(* FIXME: improve transferred to handle bounded meta quantification *)
kuncar@53964
   540
kuncar@53963
   541
lemma fcard_fempty:
kuncar@53963
   542
  "fcard {||} = 0"
kuncar@53963
   543
  by transfer (rule card_empty)
kuncar@53963
   544
kuncar@53963
   545
lemma fcard_finsert_disjoint:
kuncar@53963
   546
  "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
kuncar@53963
   547
  by transfer (rule card_insert_disjoint)
kuncar@53963
   548
kuncar@53963
   549
lemma fcard_finsert_if:
kuncar@53963
   550
  "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
kuncar@53963
   551
  by transfer (rule card_insert_if)
kuncar@53963
   552
kuncar@53963
   553
lemma card_0_eq [simp, no_atp]:
kuncar@53963
   554
  "fcard A = 0 \<longleftrightarrow> A = {||}"
kuncar@53963
   555
  by transfer (rule card_0_eq)
kuncar@53963
   556
kuncar@53963
   557
lemma fcard_Suc_fminus1:
kuncar@53963
   558
  "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
kuncar@53963
   559
  by transfer (rule card_Suc_Diff1)
kuncar@53963
   560
kuncar@53963
   561
lemma fcard_fminus_fsingleton:
kuncar@53963
   562
  "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
kuncar@53963
   563
  by transfer (rule card_Diff_singleton)
kuncar@53963
   564
kuncar@53963
   565
lemma fcard_fminus_fsingleton_if:
kuncar@53963
   566
  "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
kuncar@53963
   567
  by transfer (rule card_Diff_singleton_if)
kuncar@53963
   568
kuncar@53963
   569
lemma fcard_fminus_finsert[simp]:
kuncar@53963
   570
  assumes "a |\<in>| A" and "a |\<notin>| B"
kuncar@53963
   571
  shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
kuncar@53963
   572
using assms by transfer (rule card_Diff_insert)
kuncar@53963
   573
kuncar@53963
   574
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
kuncar@53963
   575
by transfer (rule card_insert)
kuncar@53963
   576
kuncar@53963
   577
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
kuncar@53963
   578
by transfer (rule card_insert_le)
kuncar@53963
   579
kuncar@53963
   580
lemma fcard_mono:
kuncar@53963
   581
  "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
kuncar@53963
   582
by transfer (rule card_mono)
kuncar@53963
   583
kuncar@53963
   584
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
kuncar@53963
   585
by transfer (rule card_seteq)
kuncar@53963
   586
kuncar@53963
   587
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
kuncar@53963
   588
by transfer (rule psubset_card_mono)
kuncar@53963
   589
kuncar@53963
   590
lemma fcard_funion_finter: 
kuncar@53963
   591
  "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
kuncar@53963
   592
by transfer (rule card_Un_Int)
kuncar@53963
   593
kuncar@53963
   594
lemma fcard_funion_disjoint:
kuncar@53963
   595
  "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
kuncar@53963
   596
by transfer (rule card_Un_disjoint)
kuncar@53963
   597
kuncar@53963
   598
lemma fcard_funion_fsubset:
kuncar@53963
   599
  "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
kuncar@53963
   600
by transfer (rule card_Diff_subset)
kuncar@53963
   601
kuncar@53963
   602
lemma diff_fcard_le_fcard_fminus:
kuncar@53963
   603
  "fcard A - fcard B \<le> fcard(A |-| B)"
kuncar@53963
   604
by transfer (rule diff_card_le_card_Diff)
kuncar@53963
   605
kuncar@53963
   606
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
kuncar@53963
   607
by transfer (rule card_Diff1_less)
kuncar@53963
   608
kuncar@53963
   609
lemma fcard_fminus2_less:
kuncar@53963
   610
  "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
kuncar@53963
   611
by transfer (rule card_Diff2_less)
kuncar@53963
   612
kuncar@53963
   613
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
kuncar@53963
   614
by transfer (rule card_Diff1_le)
kuncar@53963
   615
kuncar@53963
   616
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
kuncar@53963
   617
by transfer (rule card_psubset)
kuncar@53963
   618
wenzelm@53969
   619
subsubsection {* @{text ffold} *}
kuncar@53963
   620
kuncar@53963
   621
(* FIXME: improve transferred to handle bounded meta quantification *)
kuncar@53963
   622
kuncar@53963
   623
context comp_fun_commute
kuncar@53963
   624
begin
kuncar@53963
   625
  lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
kuncar@53963
   626
kuncar@53963
   627
  lemma ffold_finsert [simp]:
kuncar@53963
   628
    assumes "x |\<notin>| A"
kuncar@53963
   629
    shows "ffold f z (finsert x A) = f x (ffold f z A)"
kuncar@53963
   630
    using assms by (transfer fixing: f) (rule fold_insert)
kuncar@53963
   631
kuncar@53963
   632
  lemma ffold_fun_left_comm:
kuncar@53963
   633
    "f x (ffold f z A) = ffold f (f x z) A"
kuncar@53963
   634
    by (transfer fixing: f) (rule fold_fun_left_comm)
kuncar@53963
   635
kuncar@53963
   636
  lemma ffold_finsert2:
kuncar@53963
   637
    "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A)  = ffold f (f x z) A"
kuncar@53963
   638
    by (transfer fixing: f) (rule fold_insert2)
kuncar@53963
   639
kuncar@53963
   640
  lemma ffold_rec:
kuncar@53963
   641
    assumes "x |\<in>| A"
kuncar@53963
   642
    shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
kuncar@53963
   643
    using assms by (transfer fixing: f) (rule fold_rec)
kuncar@53963
   644
  
kuncar@53963
   645
  lemma ffold_finsert_fremove:
kuncar@53963
   646
    "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
kuncar@53963
   647
     by (transfer fixing: f) (rule fold_insert_remove)
kuncar@53963
   648
end
kuncar@53963
   649
kuncar@53963
   650
lemma ffold_fimage:
kuncar@53963
   651
  assumes "inj_on g (fset A)"
kuncar@53963
   652
  shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
kuncar@53963
   653
using assms by transfer' (rule fold_image)
kuncar@53963
   654
kuncar@53963
   655
lemma ffold_cong:
kuncar@53963
   656
  assumes "comp_fun_commute f" "comp_fun_commute g"
kuncar@53963
   657
  "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
kuncar@53963
   658
    and "s = t" and "A = B"
kuncar@53963
   659
  shows "ffold f s A = ffold g t B"
kuncar@53963
   660
using assms by transfer (metis Finite_Set.fold_cong)
kuncar@53963
   661
kuncar@53963
   662
context comp_fun_idem
kuncar@53963
   663
begin
kuncar@53963
   664
kuncar@53963
   665
  lemma ffold_finsert_idem:
kuncar@53963
   666
    "ffold f z (finsert x A)  = f x (ffold f z A)"
kuncar@53963
   667
    by (transfer fixing: f) (rule fold_insert_idem)
kuncar@53963
   668
  
kuncar@53963
   669
  declare ffold_finsert [simp del] ffold_finsert_idem [simp]
kuncar@53963
   670
  
kuncar@53963
   671
  lemma ffold_finsert_idem2:
kuncar@53963
   672
    "ffold f z (finsert x A) = ffold f (f x z) A"
kuncar@53963
   673
    by (transfer fixing: f) (rule fold_insert_idem2)
kuncar@53963
   674
kuncar@53963
   675
end
kuncar@53963
   676
kuncar@53953
   677
subsection {* Choice in fsets *}
kuncar@53953
   678
kuncar@53953
   679
lemma fset_choice: 
kuncar@53953
   680
  assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
kuncar@53953
   681
  shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
kuncar@53953
   682
  using assms by transfer metis
kuncar@53953
   683
kuncar@53953
   684
subsection {* Induction and Cases rules for fsets *}
kuncar@53953
   685
kuncar@53953
   686
lemma fset_exhaust [case_names empty insert, cases type: fset]:
kuncar@53953
   687
  assumes fempty_case: "S = {||} \<Longrightarrow> P" 
kuncar@53953
   688
  and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
kuncar@53953
   689
  shows "P"
kuncar@53953
   690
  using assms by transfer blast
kuncar@53953
   691
kuncar@53953
   692
lemma fset_induct [case_names empty insert]:
kuncar@53953
   693
  assumes fempty_case: "P {||}"
kuncar@53953
   694
  and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
kuncar@53953
   695
  shows "P S"
kuncar@53953
   696
proof -
kuncar@53953
   697
  (* FIXME transfer and right_total vs. bi_total *)
kuncar@53953
   698
  note Domainp_forall_transfer[transfer_rule]
kuncar@53953
   699
  show ?thesis
kuncar@53953
   700
  using assms by transfer (auto intro: finite_induct)
kuncar@53953
   701
qed
kuncar@53953
   702
kuncar@53953
   703
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
kuncar@53953
   704
  assumes empty_fset_case: "P {||}"
kuncar@53953
   705
  and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
kuncar@53953
   706
  shows "P S"
kuncar@53953
   707
proof -
kuncar@53953
   708
  (* FIXME transfer and right_total vs. bi_total *)
kuncar@53953
   709
  note Domainp_forall_transfer[transfer_rule]
kuncar@53953
   710
  show ?thesis
kuncar@53953
   711
  using assms by transfer (auto intro: finite_induct)
kuncar@53953
   712
qed
kuncar@53953
   713
kuncar@53953
   714
lemma fset_card_induct:
kuncar@53953
   715
  assumes empty_fset_case: "P {||}"
kuncar@53953
   716
  and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
kuncar@53953
   717
  shows "P S"
kuncar@53953
   718
proof (induct S)
kuncar@53953
   719
  case empty
kuncar@53953
   720
  show "P {||}" by (rule empty_fset_case)
kuncar@53953
   721
next
kuncar@53953
   722
  case (insert x S)
kuncar@53953
   723
  have h: "P S" by fact
kuncar@53953
   724
  have "x |\<notin>| S" by fact
kuncar@53953
   725
  then have "Suc (fcard S) = fcard (finsert x S)" 
kuncar@53953
   726
    by transfer auto
kuncar@53953
   727
  then show "P (finsert x S)" 
kuncar@53953
   728
    using h card_fset_Suc_case by simp
kuncar@53953
   729
qed
kuncar@53953
   730
kuncar@53953
   731
lemma fset_strong_cases:
kuncar@53953
   732
  obtains "xs = {||}"
kuncar@53953
   733
    | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
kuncar@53953
   734
by transfer blast
kuncar@53953
   735
kuncar@53953
   736
lemma fset_induct2:
kuncar@53953
   737
  "P {||} {||} \<Longrightarrow>
kuncar@53953
   738
  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
kuncar@53953
   739
  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
kuncar@53953
   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>
kuncar@53953
   741
  P xsa ysa"
kuncar@53953
   742
  apply (induct xsa arbitrary: ysa)
kuncar@53953
   743
  apply (induct_tac x rule: fset_induct_stronger)
kuncar@53953
   744
  apply simp_all
kuncar@53953
   745
  apply (induct_tac xa rule: fset_induct_stronger)
kuncar@53953
   746
  apply simp_all
kuncar@53953
   747
  done
kuncar@53953
   748
kuncar@53953
   749
subsection {* Setup for Lifting/Transfer *}
kuncar@53953
   750
kuncar@53953
   751
subsubsection {* Relator and predicator properties *}
kuncar@53953
   752
kuncar@53953
   753
lift_definition fset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is set_rel
kuncar@53953
   754
parametric set_rel_transfer ..
kuncar@53953
   755
kuncar@53953
   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) 
kuncar@53953
   757
  \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
kuncar@53953
   758
apply (rule ext)+
kuncar@53953
   759
apply transfer'
kuncar@53953
   760
apply (subst set_rel_def[unfolded fun_eq_iff]) 
kuncar@53953
   761
by blast
kuncar@53953
   762
kuncar@53953
   763
lemma fset_rel_conversep: "fset_rel (conversep R) = conversep (fset_rel R)"
kuncar@53953
   764
  unfolding fset_rel_alt_def by auto
kuncar@53953
   765
kuncar@53953
   766
lemmas fset_rel_eq [relator_eq] = set_rel_eq[Transfer.transferred]
kuncar@53953
   767
kuncar@53953
   768
lemma fset_rel_mono[relator_mono]: "A \<le> B \<Longrightarrow> fset_rel A \<le> fset_rel B"
kuncar@53953
   769
unfolding fset_rel_alt_def by blast
kuncar@53953
   770
kuncar@53953
   771
lemma finite_set_rel:
kuncar@53953
   772
  assumes fin: "finite X" "finite Z"
kuncar@53953
   773
  assumes R_S: "set_rel (R OO S) X Z"
kuncar@53953
   774
  shows "\<exists>Y. finite Y \<and> set_rel R X Y \<and> set_rel S Y Z"
kuncar@53953
   775
proof -
kuncar@53953
   776
  obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
kuncar@53953
   777
  apply atomize_elim
kuncar@53953
   778
  apply (subst bchoice_iff[symmetric])
kuncar@53953
   779
  using R_S[unfolded set_rel_def OO_def] by blast
kuncar@53953
   780
  
kuncar@53953
   781
  obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"
kuncar@53953
   782
  apply atomize_elim
kuncar@53953
   783
  apply (subst bchoice_iff[symmetric])
kuncar@53953
   784
  using R_S[unfolded set_rel_def OO_def] by blast
kuncar@53953
   785
  
kuncar@53953
   786
  let ?Y = "f ` X \<union> g ` Z"
kuncar@53953
   787
  have "finite ?Y" by (simp add: fin)
kuncar@53953
   788
  moreover have "set_rel R X ?Y"
kuncar@53953
   789
    unfolding set_rel_def
kuncar@53953
   790
    using f g by clarsimp blast
kuncar@53953
   791
  moreover have "set_rel S ?Y Z"
kuncar@53953
   792
    unfolding set_rel_def
kuncar@53953
   793
    using f g by clarsimp blast
kuncar@53953
   794
  ultimately show ?thesis by metis
kuncar@53953
   795
qed
kuncar@53953
   796
kuncar@53953
   797
lemma fset_rel_OO[relator_distr]: "fset_rel R OO fset_rel S = fset_rel (R OO S)"
kuncar@53953
   798
apply (rule ext)+
kuncar@53953
   799
by transfer (auto intro: finite_set_rel set_rel_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
kuncar@53953
   800
kuncar@53953
   801
lemma Domainp_fset[relator_domain]:
kuncar@53953
   802
  assumes "Domainp T = P"
kuncar@53953
   803
  shows "Domainp (fset_rel T) = (\<lambda>A. fBall A P)"
kuncar@53953
   804
proof -
kuncar@53953
   805
  from assms obtain f where f: "\<forall>x\<in>Collect P. T x (f x)"
kuncar@53953
   806
    unfolding Domainp_iff[abs_def]
kuncar@53953
   807
    apply atomize_elim
kuncar@53953
   808
    by (subst bchoice_iff[symmetric]) auto
kuncar@53953
   809
  from assms f show ?thesis
kuncar@53953
   810
    unfolding fun_eq_iff fset_rel_alt_def Domainp_iff
kuncar@53953
   811
    apply clarify
kuncar@53953
   812
    apply (rule iffI)
kuncar@53953
   813
      apply blast
kuncar@53953
   814
    by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
kuncar@53953
   815
qed
kuncar@53953
   816
kuncar@53953
   817
lemmas reflp_fset_rel[reflexivity_rule] = reflp_set_rel[Transfer.transferred]
kuncar@53953
   818
kuncar@53953
   819
lemma right_total_fset_rel[transfer_rule]: "right_total A \<Longrightarrow> right_total (fset_rel A)"
kuncar@53953
   820
unfolding right_total_def 
kuncar@53953
   821
apply transfer
kuncar@53953
   822
apply (subst(asm) choice_iff)
kuncar@53953
   823
apply clarsimp
kuncar@53953
   824
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
kuncar@53953
   825
by (auto simp add: set_rel_def)
kuncar@53953
   826
kuncar@53953
   827
lemma left_total_fset_rel[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (fset_rel A)"
kuncar@53953
   828
unfolding left_total_def 
kuncar@53953
   829
apply transfer
kuncar@53953
   830
apply (subst(asm) choice_iff)
kuncar@53953
   831
apply clarsimp
kuncar@53953
   832
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
kuncar@53953
   833
by (auto simp add: set_rel_def)
kuncar@53953
   834
kuncar@53953
   835
lemmas right_unique_fset_rel[transfer_rule] = right_unique_set_rel[Transfer.transferred]
kuncar@53953
   836
lemmas left_unique_fset_rel[reflexivity_rule] = left_unique_set_rel[Transfer.transferred]
kuncar@53953
   837
kuncar@53953
   838
thm right_unique_fset_rel left_unique_fset_rel
kuncar@53953
   839
kuncar@53953
   840
lemma bi_unique_fset_rel[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (fset_rel A)"
kuncar@53953
   841
by (auto intro: right_unique_fset_rel left_unique_fset_rel iff: bi_unique_iff)
kuncar@53953
   842
kuncar@53953
   843
lemma bi_total_fset_rel[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (fset_rel A)"
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   844
by (auto intro: right_total_fset_rel left_total_fset_rel iff: bi_total_iff)
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   845
kuncar@53953
   846
lemmas fset_invariant_commute [invariant_commute] = set_invariant_commute[Transfer.transferred]
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   847
kuncar@53953
   848
subsubsection {* Quotient theorem for the Lifting package *}
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   849
kuncar@53953
   850
lemma Quotient_fset_map[quot_map]:
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   851
  assumes "Quotient R Abs Rep T"
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   852
  shows "Quotient (fset_rel R) (fimage Abs) (fimage Rep) (fset_rel T)"
kuncar@53953
   853
  using assms unfolding Quotient_alt_def4
kuncar@53953
   854
  by (simp add: fset_rel_OO[symmetric] fset_rel_conversep) (simp add: fset_rel_alt_def, blast)
kuncar@53953
   855
kuncar@53953
   856
subsubsection {* Transfer rules for the Transfer package *}
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   857
kuncar@53953
   858
text {* Unconditional transfer rules *}
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   859
kuncar@53963
   860
context
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   861
begin
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   862
kuncar@53963
   863
interpretation lifting_syntax .
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   864
kuncar@53953
   865
lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
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   866
kuncar@53953
   867
lemma finsert_transfer [transfer_rule]:
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   868
  "(A ===> fset_rel A ===> fset_rel A) finsert finsert"
kuncar@53953
   869
  unfolding fun_rel_def fset_rel_alt_def by blast
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   870
kuncar@53953
   871
lemma funion_transfer [transfer_rule]:
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   872
  "(fset_rel A ===> fset_rel A ===> fset_rel A) funion funion"
kuncar@53953
   873
  unfolding fun_rel_def fset_rel_alt_def by blast
kuncar@53953
   874
kuncar@53953
   875
lemma ffUnion_transfer [transfer_rule]:
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   876
  "(fset_rel (fset_rel A) ===> fset_rel A) ffUnion ffUnion"
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   877
  unfolding fun_rel_def fset_rel_alt_def by transfer (simp, fast)
kuncar@53953
   878
kuncar@53953
   879
lemma fimage_transfer [transfer_rule]:
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   880
  "((A ===> B) ===> fset_rel A ===> fset_rel B) fimage fimage"
kuncar@53953
   881
  unfolding fun_rel_def fset_rel_alt_def by simp blast
kuncar@53953
   882
kuncar@53953
   883
lemma fBall_transfer [transfer_rule]:
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   884
  "(fset_rel A ===> (A ===> op =) ===> op =) fBall fBall"
kuncar@53953
   885
  unfolding fset_rel_alt_def fun_rel_def by blast
kuncar@53953
   886
kuncar@53953
   887
lemma fBex_transfer [transfer_rule]:
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   888
  "(fset_rel A ===> (A ===> op =) ===> op =) fBex fBex"
kuncar@53953
   889
  unfolding fset_rel_alt_def fun_rel_def by blast
kuncar@53953
   890
kuncar@53953
   891
(* FIXME transfer doesn't work here *)
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   892
lemma fPow_transfer [transfer_rule]:
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   893
  "(fset_rel A ===> fset_rel (fset_rel A)) fPow fPow"
kuncar@53953
   894
  unfolding fun_rel_def
kuncar@53953
   895
  using Pow_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred]
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   896
  by blast
kuncar@53953
   897
kuncar@53953
   898
lemma fset_rel_transfer [transfer_rule]:
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   899
  "((A ===> B ===> op =) ===> fset_rel A ===> fset_rel B ===> op =)
kuncar@53953
   900
    fset_rel fset_rel"
kuncar@53953
   901
  unfolding fun_rel_def
kuncar@53953
   902
  using set_rel_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
kuncar@53953
   903
  by simp
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   904
kuncar@53953
   905
lemma bind_transfer [transfer_rule]:
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   906
  "(fset_rel A ===> (A ===> fset_rel B) ===> fset_rel B) fbind fbind"
kuncar@53953
   907
  using assms unfolding fun_rel_def
kuncar@53953
   908
  using bind_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   909
kuncar@53953
   910
text {* Rules requiring bi-unique, bi-total or right-total relations *}
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   911
kuncar@53953
   912
lemma fmember_transfer [transfer_rule]:
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   913
  assumes "bi_unique A"
kuncar@53953
   914
  shows "(A ===> fset_rel A ===> op =) (op |\<in>|) (op |\<in>|)"
kuncar@53953
   915
  using assms unfolding fun_rel_def fset_rel_alt_def bi_unique_def by metis
kuncar@53953
   916
kuncar@53953
   917
lemma finter_transfer [transfer_rule]:
kuncar@53953
   918
  assumes "bi_unique A"
kuncar@53953
   919
  shows "(fset_rel A ===> fset_rel A ===> fset_rel A) finter finter"
kuncar@53953
   920
  using assms unfolding fun_rel_def
kuncar@53953
   921
  using inter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   922
kuncar@53963
   923
lemma fminus_transfer [transfer_rule]:
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   924
  assumes "bi_unique A"
kuncar@53953
   925
  shows "(fset_rel A ===> fset_rel A ===> fset_rel A) (op |-|) (op |-|)"
kuncar@53953
   926
  using assms unfolding fun_rel_def
kuncar@53953
   927
  using Diff_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   928
kuncar@53953
   929
lemma fsubset_transfer [transfer_rule]:
kuncar@53953
   930
  assumes "bi_unique A"
kuncar@53953
   931
  shows "(fset_rel A ===> fset_rel A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
kuncar@53953
   932
  using assms unfolding fun_rel_def
kuncar@53953
   933
  using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   934
kuncar@53953
   935
lemma fSup_transfer [transfer_rule]:
kuncar@53953
   936
  "bi_unique A \<Longrightarrow> (set_rel (fset_rel A) ===> fset_rel A) Sup Sup"
kuncar@53953
   937
  using assms unfolding fun_rel_def
kuncar@53953
   938
  apply clarify
kuncar@53953
   939
  apply transfer'
kuncar@53953
   940
  using Sup_fset_transfer[unfolded fun_rel_def] by blast
kuncar@53953
   941
kuncar@53953
   942
(* FIXME: add right_total_fInf_transfer *)
kuncar@53953
   943
kuncar@53953
   944
lemma fInf_transfer [transfer_rule]:
kuncar@53953
   945
  assumes "bi_unique A" and "bi_total A"
kuncar@53953
   946
  shows "(set_rel (fset_rel A) ===> fset_rel A) Inf Inf"
kuncar@53953
   947
  using assms unfolding fun_rel_def
kuncar@53953
   948
  apply clarify
kuncar@53953
   949
  apply transfer'
kuncar@53953
   950
  using Inf_fset_transfer[unfolded fun_rel_def] by blast
kuncar@53953
   951
kuncar@53953
   952
lemma ffilter_transfer [transfer_rule]:
kuncar@53953
   953
  assumes "bi_unique A"
kuncar@53953
   954
  shows "((A ===> op=) ===> fset_rel A ===> fset_rel A) ffilter ffilter"
kuncar@53953
   955
  using assms unfolding fun_rel_def
kuncar@53953
   956
  using Lifting_Set.filter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   957
kuncar@53953
   958
lemma card_transfer [transfer_rule]:
kuncar@53953
   959
  "bi_unique A \<Longrightarrow> (fset_rel A ===> op =) fcard fcard"
kuncar@53953
   960
  using assms unfolding fun_rel_def
kuncar@53953
   961
  using card_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
kuncar@53953
   962
kuncar@53953
   963
end
kuncar@53953
   964
kuncar@53953
   965
lifting_update fset.lifting
kuncar@53953
   966
lifting_forget fset.lifting
kuncar@53953
   967
kuncar@53953
   968
end