src/HOL/Library/FSet.thy
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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 :: (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]
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lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
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lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
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lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
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lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
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lemmas pfsubsetD = psubsetD[Transfer.transferred]
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lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
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lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
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lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
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lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
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lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
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lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
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lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
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lemmas fsubset_finsert = subset_insert[Transfer.transferred]
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lemmas funion_upper1 = Un_upper1[Transfer.transferred]
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parents:
diff changeset
   309
lemmas funion_upper2 = Un_upper2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   310
lemmas funion_least = Un_least[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   311
lemmas finter_lower1 = Int_lower1[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   312
lemmas finter_lower2 = Int_lower2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   313
lemmas finter_greatest = Int_greatest[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   314
lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   315
lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   316
lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   317
lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   318
lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   319
lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   320
lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   321
lemmas finsert_absorb = insert_absorb[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   322
lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   323
lemmas finsert_commute = insert_commute[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   324
lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   325
lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   326
lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   327
lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   328
lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   329
lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   330
lemmas fimage_constant = image_constant[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   331
lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   332
lemmas fimage_fimage = image_image[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   333
lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   334
lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   335
lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   336
lemmas fimage_cong = image_cong[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   337
lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   338
lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   339
lemmas finter_absorb = Int_absorb[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   340
lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   341
lemmas finter_commute = Int_commute[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   342
lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   343
lemmas finter_assoc = Int_assoc[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   344
lemmas finter_ac = Int_ac[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   345
lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   346
lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   347
lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   348
lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   349
lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   350
lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   351
lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   352
lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   353
lemmas funion_absorb = Un_absorb[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   354
lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   355
lemmas funion_commute = Un_commute[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   356
lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   357
lemmas funion_assoc = Un_assoc[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   358
lemmas funion_ac = Un_ac[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   359
lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   360
lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   361
lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   362
lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   363
lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   364
lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   365
lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   366
lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   367
lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   368
lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   369
lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   370
lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   371
lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   372
lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   373
lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   374
lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   375
lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   376
lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   377
lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   378
lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   379
lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   380
lemmas fBall_funion = ball_Un[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   381
lemmas fBex_funion = bex_Un[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   382
lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   383
lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   384
lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   385
lemmas fminus_triv = Diff_triv[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   386
lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   387
lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   388
lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   389
lemmas fminus_finsert = Diff_insert[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   390
lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   391
lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   392
lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   393
lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   394
lemmas finsert_fminus = insert_Diff[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   395
lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   396
lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   397
lemmas fminus_partition = Diff_partition[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   398
lemmas double_fminus = double_diff[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   399
lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   400
lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   401
lemmas fminus_funion = Diff_Un[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   402
lemmas fminus_finter = Diff_Int[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   403
lemmas funion_fminus = Un_Diff[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   404
lemmas finter_fminus = Int_Diff[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   405
lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   406
lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   407
lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   408
lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   409
lemmas fPow_finsert = Pow_insert[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   410
lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   411
lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   412
lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   413
lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   414
lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   415
lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   416
lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   417
lemmas fimage_mono = image_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   418
lemmas fPow_mono = Pow_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   419
lemmas finsert_mono = insert_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   420
lemmas funion_mono = Un_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   421
lemmas finter_mono = Int_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   422
lemmas fminus_mono = Diff_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   423
lemmas fin_mono = in_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   424
lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   425
lemmas fLeast_mono = Least_mono[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   426
lemmas fbind_fbind = bind_bind[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   427
lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   428
lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   429
lemmas fbind_const = bind_const[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   430
lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   431
lemmas fequalityI = equalityI[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   432
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   433
subsection {* Additional lemmas*}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   434
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   435
subsubsection {* fsingleton *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   436
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   437
lemmas fsingletonE = fsingletonD [elim_format]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   438
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   439
subsubsection {* femepty *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   440
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   441
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   442
by transfer auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   443
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   444
(* FIXME, transferred doesn't work here *)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   445
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   446
  by simp
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   447
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   448
subsubsection {* fset *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   449
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   450
lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   451
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   452
lemma finite_fset [simp]: 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   453
  shows "finite (fset S)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   454
  by transfer simp
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   455
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   456
lemmas fset_cong = fset_inject
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   457
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   458
lemma filter_fset [simp]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   459
  shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   460
  by transfer auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   461
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   462
lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   463
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   464
lemmas inter_fset[simp] = inf_fset.rep_eq
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   465
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   466
lemmas union_fset[simp] = sup_fset.rep_eq
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   467
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   468
lemmas minus_fset[simp] = minus_fset.rep_eq
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   469
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   470
subsubsection {* filter_fset *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   471
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   472
lemma subset_ffilter: 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   473
  "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   474
  by transfer auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   475
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   476
lemma eq_ffilter: 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   477
  "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   478
  by transfer auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   479
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   480
lemma pfsubset_ffilter:
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   481
  "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow> 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   482
    ffilter P A |\<subset>| ffilter Q A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   483
  unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   484
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   485
subsubsection {* finsert *}
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   486
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   487
(* FIXME, transferred doesn't work here *)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   488
lemma set_finsert:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   489
  assumes "x |\<in>| A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   490
  obtains B where "A = finsert x B" and "x |\<notin>| B"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   491
using assms by transfer (metis Set.set_insert finite_insert)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   492
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   493
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   494
  by (rule_tac x = "A |-| {|a|}" in exI, blast)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   495
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   496
subsubsection {* fimage *}
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   497
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   498
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   499
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   500
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   501
subsubsection {* bounded quantification *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   502
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   503
lemma bex_simps [simp, no_atp]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   504
  "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   505
  "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   506
  "\<And>P. fBex {||} P = False" 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   507
  "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   508
  "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   509
  "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   510
by auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   511
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   512
lemma ball_simps [simp, no_atp]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   513
  "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   514
  "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   515
  "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   516
  "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   517
  "\<And>P. fBall {||} P = True"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   518
  "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   519
  "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   520
  "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   521
by auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   522
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   523
lemma atomize_fBall:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   524
    "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   525
apply (simp only: atomize_all atomize_imp)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   526
apply (rule equal_intr_rule)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   527
by (transfer, simp)+
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   528
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   529
end
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   530
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   531
subsubsection {* fcard *}
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   532
53964
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   533
(* FIXME: improve transferred to handle bounded meta quantification *)
ac0e4ca891f9 tuned names
kuncar
parents: 53963
diff changeset
   534
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   535
lemma fcard_fempty:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   536
  "fcard {||} = 0"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   537
  by transfer (rule card_empty)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   538
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   539
lemma fcard_finsert_disjoint:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   540
  "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   541
  by transfer (rule card_insert_disjoint)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   542
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   543
lemma fcard_finsert_if:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   544
  "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   545
  by transfer (rule card_insert_if)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   546
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   547
lemma card_0_eq [simp, no_atp]:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   548
  "fcard A = 0 \<longleftrightarrow> A = {||}"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   549
  by transfer (rule card_0_eq)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   550
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   551
lemma fcard_Suc_fminus1:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   552
  "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   553
  by transfer (rule card_Suc_Diff1)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   554
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   555
lemma fcard_fminus_fsingleton:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   556
  "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   557
  by transfer (rule card_Diff_singleton)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   558
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   559
lemma fcard_fminus_fsingleton_if:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   560
  "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   561
  by transfer (rule card_Diff_singleton_if)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   562
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   563
lemma fcard_fminus_finsert[simp]:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   564
  assumes "a |\<in>| A" and "a |\<notin>| B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   565
  shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   566
using assms by transfer (rule card_Diff_insert)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   567
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   568
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   569
by transfer (rule card_insert)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   570
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   571
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   572
by transfer (rule card_insert_le)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   573
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   574
lemma fcard_mono:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   575
  "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   576
by transfer (rule card_mono)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   577
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   578
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   579
by transfer (rule card_seteq)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   580
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   581
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   582
by transfer (rule psubset_card_mono)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   583
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   584
lemma fcard_funion_finter: 
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   585
  "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   586
by transfer (rule card_Un_Int)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   587
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   588
lemma fcard_funion_disjoint:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   589
  "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   590
by transfer (rule card_Un_disjoint)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   591
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   592
lemma fcard_funion_fsubset:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   593
  "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   594
by transfer (rule card_Diff_subset)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   595
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   596
lemma diff_fcard_le_fcard_fminus:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   597
  "fcard A - fcard B \<le> fcard(A |-| B)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   598
by transfer (rule diff_card_le_card_Diff)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   599
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   600
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   601
by transfer (rule card_Diff1_less)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   602
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   603
lemma fcard_fminus2_less:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   604
  "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   605
by transfer (rule card_Diff2_less)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   606
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   607
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   608
by transfer (rule card_Diff1_le)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   609
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   610
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   611
by transfer (rule card_psubset)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   612
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   613
subsubsection {* ffold *}
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   614
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   615
(* FIXME: improve transferred to handle bounded meta quantification *)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   616
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   617
context comp_fun_commute
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   618
begin
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   619
  lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   620
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   621
  lemma ffold_finsert [simp]:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   622
    assumes "x |\<notin>| A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   623
    shows "ffold f z (finsert x A) = f x (ffold f z A)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   624
    using assms by (transfer fixing: f) (rule fold_insert)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   625
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   626
  lemma ffold_fun_left_comm:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   627
    "f x (ffold f z A) = ffold f (f x z) A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   628
    by (transfer fixing: f) (rule fold_fun_left_comm)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   629
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   630
  lemma ffold_finsert2:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   631
    "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A)  = ffold f (f x z) A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   632
    by (transfer fixing: f) (rule fold_insert2)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   633
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   634
  lemma ffold_rec:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   635
    assumes "x |\<in>| A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   636
    shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   637
    using assms by (transfer fixing: f) (rule fold_rec)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   638
  
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   639
  lemma ffold_finsert_fremove:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   640
    "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   641
     by (transfer fixing: f) (rule fold_insert_remove)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   642
end
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   643
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   644
lemma ffold_fimage:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   645
  assumes "inj_on g (fset A)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   646
  shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   647
using assms by transfer' (rule fold_image)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   648
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   649
lemma ffold_cong:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   650
  assumes "comp_fun_commute f" "comp_fun_commute g"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   651
  "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   652
    and "s = t" and "A = B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   653
  shows "ffold f s A = ffold g t B"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   654
using assms by transfer (metis Finite_Set.fold_cong)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   655
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   656
context comp_fun_idem
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   657
begin
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   658
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   659
  lemma ffold_finsert_idem:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   660
    "ffold f z (finsert x A)  = f x (ffold f z A)"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   661
    by (transfer fixing: f) (rule fold_insert_idem)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   662
  
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   663
  declare ffold_finsert [simp del] ffold_finsert_idem [simp]
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   664
  
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   665
  lemma ffold_finsert_idem2:
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   666
    "ffold f z (finsert x A) = ffold f (f x z) A"
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   667
    by (transfer fixing: f) (rule fold_insert_idem2)
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   668
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   669
end
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   670
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   671
subsection {* Choice in fsets *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   672
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   673
lemma fset_choice: 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   674
  assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   675
  shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   676
  using assms by transfer metis
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   677
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   678
subsection {* Induction and Cases rules for fsets *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   679
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   680
lemma fset_exhaust [case_names empty insert, cases type: fset]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   681
  assumes fempty_case: "S = {||} \<Longrightarrow> P" 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   682
  and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   683
  shows "P"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   684
  using assms by transfer blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   685
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   686
lemma fset_induct [case_names empty insert]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   687
  assumes fempty_case: "P {||}"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   688
  and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   689
  shows "P S"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   690
proof -
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   691
  (* FIXME transfer and right_total vs. bi_total *)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   692
  note Domainp_forall_transfer[transfer_rule]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   693
  show ?thesis
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   694
  using assms by transfer (auto intro: finite_induct)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   695
qed
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   696
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   697
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   698
  assumes empty_fset_case: "P {||}"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   699
  and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   700
  shows "P S"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   701
proof -
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   702
  (* FIXME transfer and right_total vs. bi_total *)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   703
  note Domainp_forall_transfer[transfer_rule]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   704
  show ?thesis
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   705
  using assms by transfer (auto intro: finite_induct)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   706
qed
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   707
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   708
lemma fset_card_induct:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   709
  assumes empty_fset_case: "P {||}"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   710
  and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   711
  shows "P S"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   712
proof (induct S)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   713
  case empty
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   714
  show "P {||}" by (rule empty_fset_case)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   715
next
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   716
  case (insert x S)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   717
  have h: "P S" by fact
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   718
  have "x |\<notin>| S" by fact
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   719
  then have "Suc (fcard S) = fcard (finsert x S)" 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   720
    by transfer auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   721
  then show "P (finsert x S)" 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   722
    using h card_fset_Suc_case by simp
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   723
qed
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   724
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   725
lemma fset_strong_cases:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   726
  obtains "xs = {||}"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   727
    | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   728
by transfer blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   729
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   730
lemma fset_induct2:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   731
  "P {||} {||} \<Longrightarrow>
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   732
  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   733
  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   734
  (\<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>
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   735
  P xsa ysa"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   736
  apply (induct xsa arbitrary: ysa)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   737
  apply (induct_tac x rule: fset_induct_stronger)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   738
  apply simp_all
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   739
  apply (induct_tac xa rule: fset_induct_stronger)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   740
  apply simp_all
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   741
  done
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   742
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   743
subsection {* Setup for Lifting/Transfer *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   744
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   745
subsubsection {* Relator and predicator properties *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   746
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   747
lift_definition fset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is set_rel
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   748
parametric set_rel_transfer ..
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   749
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   750
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) 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   751
  \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   752
apply (rule ext)+
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   753
apply transfer'
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   754
apply (subst set_rel_def[unfolded fun_eq_iff]) 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   755
by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   756
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   757
lemma fset_rel_conversep: "fset_rel (conversep R) = conversep (fset_rel R)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   758
  unfolding fset_rel_alt_def by auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   759
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   760
lemmas fset_rel_eq [relator_eq] = set_rel_eq[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   761
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   762
lemma fset_rel_mono[relator_mono]: "A \<le> B \<Longrightarrow> fset_rel A \<le> fset_rel B"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   763
unfolding fset_rel_alt_def by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   764
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   765
lemma finite_set_rel:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   766
  assumes fin: "finite X" "finite Z"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   767
  assumes R_S: "set_rel (R OO S) X Z"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   768
  shows "\<exists>Y. finite Y \<and> set_rel R X Y \<and> set_rel S Y Z"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   769
proof -
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   770
  obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   771
  apply atomize_elim
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   772
  apply (subst bchoice_iff[symmetric])
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   773
  using R_S[unfolded set_rel_def OO_def] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   774
  
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   775
  obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   776
  apply atomize_elim
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   777
  apply (subst bchoice_iff[symmetric])
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   778
  using R_S[unfolded set_rel_def OO_def] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   779
  
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   780
  let ?Y = "f ` X \<union> g ` Z"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   781
  have "finite ?Y" by (simp add: fin)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   782
  moreover have "set_rel R X ?Y"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   783
    unfolding set_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   784
    using f g by clarsimp blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   785
  moreover have "set_rel S ?Y Z"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   786
    unfolding set_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   787
    using f g by clarsimp blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   788
  ultimately show ?thesis by metis
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   789
qed
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   790
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   791
lemma fset_rel_OO[relator_distr]: "fset_rel R OO fset_rel S = fset_rel (R OO S)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   792
apply (rule ext)+
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   793
by transfer (auto intro: finite_set_rel set_rel_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   794
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   795
lemma Domainp_fset[relator_domain]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   796
  assumes "Domainp T = P"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   797
  shows "Domainp (fset_rel T) = (\<lambda>A. fBall A P)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   798
proof -
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   799
  from assms obtain f where f: "\<forall>x\<in>Collect P. T x (f x)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   800
    unfolding Domainp_iff[abs_def]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   801
    apply atomize_elim
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   802
    by (subst bchoice_iff[symmetric]) auto
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   803
  from assms f show ?thesis
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   804
    unfolding fun_eq_iff fset_rel_alt_def Domainp_iff
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   805
    apply clarify
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   806
    apply (rule iffI)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   807
      apply blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   808
    by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   809
qed
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   810
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   811
lemmas reflp_fset_rel[reflexivity_rule] = reflp_set_rel[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   812
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   813
lemma right_total_fset_rel[transfer_rule]: "right_total A \<Longrightarrow> right_total (fset_rel A)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   814
unfolding right_total_def 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   815
apply transfer
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   816
apply (subst(asm) choice_iff)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   817
apply clarsimp
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   818
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   819
by (auto simp add: set_rel_def)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   820
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   821
lemma left_total_fset_rel[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (fset_rel A)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   822
unfolding left_total_def 
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   823
apply transfer
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   824
apply (subst(asm) choice_iff)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   825
apply clarsimp
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   826
apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   827
by (auto simp add: set_rel_def)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   828
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   829
lemmas right_unique_fset_rel[transfer_rule] = right_unique_set_rel[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   830
lemmas left_unique_fset_rel[reflexivity_rule] = left_unique_set_rel[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   831
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   832
thm right_unique_fset_rel left_unique_fset_rel
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   833
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   834
lemma bi_unique_fset_rel[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (fset_rel A)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   835
by (auto intro: right_unique_fset_rel left_unique_fset_rel iff: bi_unique_iff)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   836
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   837
lemma bi_total_fset_rel[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (fset_rel A)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   838
by (auto intro: right_total_fset_rel left_total_fset_rel iff: bi_total_iff)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   839
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   840
lemmas fset_invariant_commute [invariant_commute] = set_invariant_commute[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   841
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   842
subsubsection {* Quotient theorem for the Lifting package *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   843
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   844
lemma Quotient_fset_map[quot_map]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   845
  assumes "Quotient R Abs Rep T"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   846
  shows "Quotient (fset_rel R) (fimage Abs) (fimage Rep) (fset_rel T)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   847
  using assms unfolding Quotient_alt_def4
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   848
  by (simp add: fset_rel_OO[symmetric] fset_rel_conversep) (simp add: fset_rel_alt_def, blast)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   849
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   850
subsubsection {* Transfer rules for the Transfer package *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   851
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   852
text {* Unconditional transfer rules *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   853
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   854
context
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   855
begin
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   856
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   857
interpretation lifting_syntax .
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   858
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   859
lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   860
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   861
lemma finsert_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   862
  "(A ===> fset_rel A ===> fset_rel A) finsert finsert"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   863
  unfolding fun_rel_def fset_rel_alt_def by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   864
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   865
lemma funion_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   866
  "(fset_rel A ===> fset_rel A ===> fset_rel A) funion funion"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   867
  unfolding fun_rel_def fset_rel_alt_def by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   868
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   869
lemma ffUnion_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   870
  "(fset_rel (fset_rel A) ===> fset_rel A) ffUnion ffUnion"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   871
  unfolding fun_rel_def fset_rel_alt_def by transfer (simp, fast)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   872
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   873
lemma fimage_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   874
  "((A ===> B) ===> fset_rel A ===> fset_rel B) fimage fimage"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   875
  unfolding fun_rel_def fset_rel_alt_def by simp blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   876
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   877
lemma fBall_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   878
  "(fset_rel A ===> (A ===> op =) ===> op =) fBall fBall"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   879
  unfolding fset_rel_alt_def fun_rel_def by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   880
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   881
lemma fBex_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   882
  "(fset_rel A ===> (A ===> op =) ===> op =) fBex fBex"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   883
  unfolding fset_rel_alt_def fun_rel_def by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   884
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   885
(* FIXME transfer doesn't work here *)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   886
lemma fPow_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   887
  "(fset_rel A ===> fset_rel (fset_rel A)) fPow fPow"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   888
  unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   889
  using Pow_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   890
  by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   891
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   892
lemma fset_rel_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   893
  "((A ===> B ===> op =) ===> fset_rel A ===> fset_rel B ===> op =)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   894
    fset_rel fset_rel"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   895
  unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   896
  using set_rel_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   897
  by simp
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   898
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   899
lemma bind_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   900
  "(fset_rel A ===> (A ===> fset_rel B) ===> fset_rel B) fbind fbind"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   901
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   902
  using bind_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   903
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   904
text {* Rules requiring bi-unique, bi-total or right-total relations *}
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   905
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   906
lemma fmember_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   907
  assumes "bi_unique A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   908
  shows "(A ===> fset_rel A ===> op =) (op |\<in>|) (op |\<in>|)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   909
  using assms unfolding fun_rel_def fset_rel_alt_def bi_unique_def by metis
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   910
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   911
lemma finter_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   912
  assumes "bi_unique A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   913
  shows "(fset_rel A ===> fset_rel A ===> fset_rel A) finter finter"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   914
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   915
  using inter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   916
53963
51e81874b6f6 fold and lemmas about cardinality
kuncar
parents: 53953
diff changeset
   917
lemma fminus_transfer [transfer_rule]:
53953
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   918
  assumes "bi_unique A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   919
  shows "(fset_rel A ===> fset_rel A ===> fset_rel A) (op |-|) (op |-|)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   920
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   921
  using Diff_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   922
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   923
lemma fsubset_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   924
  assumes "bi_unique A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   925
  shows "(fset_rel A ===> fset_rel A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   926
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   927
  using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   928
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   929
lemma fSup_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   930
  "bi_unique A \<Longrightarrow> (set_rel (fset_rel A) ===> fset_rel A) Sup Sup"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   931
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   932
  apply clarify
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   933
  apply transfer'
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   934
  using Sup_fset_transfer[unfolded fun_rel_def] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   935
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   936
(* FIXME: add right_total_fInf_transfer *)
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   937
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   938
lemma fInf_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   939
  assumes "bi_unique A" and "bi_total A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   940
  shows "(set_rel (fset_rel A) ===> fset_rel A) Inf Inf"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   941
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   942
  apply clarify
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   943
  apply transfer'
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   944
  using Inf_fset_transfer[unfolded fun_rel_def] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   945
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   946
lemma ffilter_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   947
  assumes "bi_unique A"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   948
  shows "((A ===> op=) ===> fset_rel A ===> fset_rel A) ffilter ffilter"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   949
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   950
  using Lifting_Set.filter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   951
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   952
lemma card_transfer [transfer_rule]:
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   953
  "bi_unique A \<Longrightarrow> (fset_rel A ===> op =) fcard fcard"
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   954
  using assms unfolding fun_rel_def
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   955
  using card_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   956
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   957
end
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   958
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   959
lifting_update fset.lifting
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   960
lifting_forget fset.lifting
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   961
2f103a894ebe new theory of finite sets as a subtype
kuncar
parents:
diff changeset
   962
end