(*  Title:      HOL/Quotient_Examples/FSet.thy+
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    Author:     Cezary Kaliszyk, TU Munich+
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    Author:     Christian Urban, TU Munich+
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+
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    Type of finite sets.+
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*)+
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+
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theory FSet+
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imports Quotient_List+
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begin+
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+
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text {* Definiton of the list equivalence relation *}+
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+
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fun+
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  list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)+
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where+
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  "list_eq xs ys = (set xs = set ys)"+
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+
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lemma list_eq_equivp:+
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  shows "equivp list_eq"+
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  unfolding equivp_reflp_symp_transp+
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  unfolding reflp_def symp_def transp_def+
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  by auto+
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+
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text {* Fset type *}+
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+
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quotient_type+
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  'a fset = "'a list" / "list_eq"+
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  by (rule list_eq_equivp)+
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+
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text {* +
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  Definitions of membership, sublist, cardinality,+
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  intersection etc over lists+
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*}+
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+
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definition+
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  memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"+
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where+
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  "memb x xs \<equiv> x \<in> set xs"+
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+
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definition+
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  sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"+
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where+
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  "sub_list xs ys \<equiv> set xs \<subseteq> set ys"+
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+
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definition+
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  card_list :: "'a list \<Rightarrow> nat"+
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where+
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  "card_list xs = card (set xs)"+
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+
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primrec+
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  finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where+
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  "finter_raw [] ys = []"+
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| "finter_raw (x # xs) ys =+
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    (if x \<in> set ys then x # (finter_raw xs ys) else finter_raw xs ys)"+
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+
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+
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definition+
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  diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where+
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  "diff_list xs ys \<equiv> [x \<leftarrow> xs. x\<notin>set ys]"+
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+
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+
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definition+
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  rsp_fold+
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where+
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  "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"+
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+
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primrec+
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  ffold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"+
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where+
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  "ffold_raw f z [] = z"+
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| "ffold_raw f z (a # xs) =+
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     (if (rsp_fold f) then+
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       if a \<in> set xs then ffold_raw f z xs+
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       else f a (ffold_raw f z xs)+
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     else z)"+
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+
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+
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+
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lemma set_finter_raw[simp]:+
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  shows "set (finter_raw xs ys) = set xs \<inter> set ys"+
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  by (induct xs) (auto simp add: memb_def)+
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+
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lemma set_diff_list[simp]: +
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  shows "set (diff_list xs ys) = (set xs - set ys)"+
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  by (auto simp add: diff_list_def)+
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+
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+
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section {* Quotient composition lemmas *}+
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+
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lemma list_all2_refl1:+
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  shows "(list_all2 op \<approx>) r r"+
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  by (rule list_all2_refl) (metis equivp_def fset_equivp)+
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+
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lemma compose_list_refl:+
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  shows "(list_all2 op \<approx> OOO op \<approx>) r r"+
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proof+
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  have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])+
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  show "list_all2 op \<approx> r r" by (rule list_all2_refl1)+
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  with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..+
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qed+
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+
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lemma Quotient_fset_list:+
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  shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"+
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  by (fact list_quotient[OF Quotient_fset])+
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+
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lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"+
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  unfolding list_eq.simps+
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  by (simp only: set_map)+
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+
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lemma quotient_compose_list[quot_thm]:+
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  shows  "Quotient ((list_all2 op \<approx>) OOO (op \<approx>))+
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    (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"+
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  unfolding Quotient_def comp_def+
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proof (intro conjI allI)+
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  fix a r s+
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  show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"+
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    by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)+
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  have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"+
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    by (rule list_all2_refl1)+
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  have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"+
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    by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)+
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  show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"+
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    by (rule, rule list_all2_refl1) (rule c)+
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  show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and>+
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        (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"+
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  proof (intro iffI conjI)+
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    show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)+
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    show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)+
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  next+
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    assume a: "(list_all2 op \<approx> OOO op \<approx>) r s"+
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    then have b: "map abs_fset r \<approx> map abs_fset s"+
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    proof (elim pred_compE)+
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      fix b ba+
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      assume c: "list_all2 op \<approx> r b"+
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      assume d: "b \<approx> ba"+
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      assume e: "list_all2 op \<approx> ba s"+
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      have f: "map abs_fset r = map abs_fset b"+
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        using Quotient_rel[OF Quotient_fset_list] c by blast+
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      have "map abs_fset ba = map abs_fset s"+
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        using Quotient_rel[OF Quotient_fset_list] e by blast+
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      then have g: "map abs_fset s = map abs_fset ba" by simp+
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      then show "map abs_fset r \<approx> map abs_fset s" using d f map_rel_cong by simp+
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    qed+
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    then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"+
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      using Quotient_rel[OF Quotient_fset] by blast+
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  next+
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    assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s+
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      \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"+
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    then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp+
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    have d: "map abs_fset r \<approx> map abs_fset s"+
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      by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)+
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    have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"+
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      by (rule map_rel_cong[OF d])+
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    have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s"+
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      by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl1[of s]])+
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    have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s"+
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      by (rule pred_compI) (rule b, rule y)+
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    have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))"+
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      by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl1[of r]])+
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    then show "(list_all2 op \<approx> OOO op \<approx>) r s"+
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      using a c pred_compI by simp+
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  qed+
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qed+
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+
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+
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text {* Respectfulness *}+
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+
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lemma append_rsp[quot_respect]:+
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  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) append append"+
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  by simp+
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+
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lemma sub_list_rsp[quot_respect]:+
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  shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list"+
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  by (auto simp add: sub_list_def)+
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+
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lemma memb_rsp[quot_respect]:+
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  shows "(op = ===> op \<approx> ===> op =) memb memb"+
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  by (auto simp add: memb_def)+
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+
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lemma nil_rsp[quot_respect]:+
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  shows "(op \<approx>) Nil Nil"+
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  by simp+
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+
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lemma cons_rsp[quot_respect]:+
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  shows "(op = ===> op \<approx> ===> op \<approx>) Cons Cons"+
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  by simp+
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+
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lemma map_rsp[quot_respect]:+
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  shows "(op = ===> op \<approx> ===> op \<approx>) map map"+
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  by auto+
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+
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lemma set_rsp[quot_respect]:+
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  "(op \<approx> ===> op =) set set"+
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  by auto+
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+
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lemma list_equiv_rsp[quot_respect]:+
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  shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"+
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  by auto+
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+
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lemma finter_raw_rsp[quot_respect]:+
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  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"+
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  by simp+
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+
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lemma removeAll_rsp[quot_respect]:+
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  shows "(op = ===> op \<approx> ===> op \<approx>) removeAll removeAll"+
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  by simp+
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+
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lemma diff_list_rsp[quot_respect]:+
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  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) diff_list diff_list"+
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  by simp+
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+
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lemma card_list_rsp[quot_respect]:+
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  shows "(op \<approx> ===> op =) card_list card_list"+
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  by (simp add: card_list_def)+
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+
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+
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+
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lemma memb_commute_ffold_raw:+
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  "rsp_fold f \<Longrightarrow> h \<in> set b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (removeAll h b))"+
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  apply (induct b)+
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  apply (auto simp add: rsp_fold_def)+
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  done+
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+
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lemma ffold_raw_rsp_pre:+
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  "set a = set b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"+
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  apply (induct a arbitrary: b)+
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  apply (simp)+
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  apply (simp (no_asm_use))+
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  apply (rule conjI)+
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  apply (rule_tac [!] impI)+
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  apply (rule_tac [!] conjI)+
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  apply (rule_tac [!] impI)+
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  apply (metis insert_absorb)+
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  apply (metis List.insert_def List.set.simps(2) List.set_insert ffold_raw.simps(2))+
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  apply (metis Diff_insert_absorb insertI1 memb_commute_ffold_raw set_removeAll)+
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  apply(drule_tac x="removeAll a1 b" in meta_spec)+
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  apply(auto)+
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  apply(drule meta_mp)+
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  apply(blast)+
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  by (metis List.set.simps(2) emptyE ffold_raw.simps(2) in_listsp_conv_set listsp.simps mem_def)+
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+
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lemma ffold_raw_rsp[quot_respect]:+
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  shows "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"+
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  unfolding fun_rel_def+
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  by(auto intro: ffold_raw_rsp_pre)+
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+
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lemma concat_rsp_pre:+
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  assumes a: "list_all2 op \<approx> x x'"+
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  and     b: "x' \<approx> y'"+
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  and     c: "list_all2 op \<approx> y' y"+
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  and     d: "\<exists>x\<in>set x. xa \<in> set x"+
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  shows "\<exists>x\<in>set y. xa \<in> set x"+
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proof -+
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  obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto+
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  have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])+
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  then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto+
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  have "ya \<in> set y'" using b h by simp+
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  then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)+
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  then show ?thesis using f i by auto+
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qed+
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+
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lemma concat_rsp[quot_respect]:+
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  shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"+
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proof (rule fun_relI, elim pred_compE)+
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  fix a b ba bb+
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  assume a: "list_all2 op \<approx> a ba"+
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  assume b: "ba \<approx> bb"+
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  assume c: "list_all2 op \<approx> bb b"+
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  have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" +
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  proof+
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    fix x+
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    show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" +
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    proof+
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      assume d: "\<exists>xa\<in>set a. x \<in> set xa"+
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      show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])+
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    next+
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      assume e: "\<exists>xa\<in>set b. x \<in> set xa"+
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      have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])+
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      have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])+
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      have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])+
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      show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])+
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    qed+
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  qed+
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  then show "concat a \<approx> concat b" by auto+
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qed+
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+
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lemma [quot_respect]:+
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  shows "(op = ===> op \<approx> ===> op \<approx>) filter filter"+
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  by auto+
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+
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+
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instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"+
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begin+
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+
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quotient_definition+
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  "bot :: 'a fset" is "[] :: 'a list"+
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+
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abbreviation+
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  fempty  ("{||}")+
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where+
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  "{||} \<equiv> bot :: 'a fset"+
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+
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quotient_definition+
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  "less_eq_fset \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"+
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is+
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  "sub_list \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> bool)"+
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+
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abbreviation+
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  f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)+
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where+
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  "xs |\<subseteq>| ys \<equiv> xs \<le> ys"+
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+
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definition+
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  less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"+
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where  +
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  "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"+
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+
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abbreviation+
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  fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)+
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where+
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  "xs |\<subset>| ys \<equiv> xs < ys"+
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+
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quotient_definition+
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  "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+
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is+
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  "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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+
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abbreviation+
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  funion (infixl "|\<union>|" 65)+
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where+
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  "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"+
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+
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quotient_definition+
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  "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+
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is+
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  "finter_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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+
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abbreviation+
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  finter (infixl "|\<inter>|" 65)+
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where+
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  "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"+
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+
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quotient_definition+
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  "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+
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is+
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  "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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+
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lemma append_finter_raw_distrib:+
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  "x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)"+
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  apply (induct x)+
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  apply (auto)+
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  done+
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+
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instance+
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proof+
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  fix x y z :: "'a fset"+
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  show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"+
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    unfolding less_fset_def +
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    by (descending) (auto simp add: sub_list_def)+
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  show "x |\<subseteq>| x"  by (descending) (simp add: sub_list_def)+
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  show "{||} |\<subseteq>| x" by (descending) (simp add: sub_list_def)+
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  show "x |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)+
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  show "y |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)+
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  show "x |\<inter>| y |\<subseteq>| x"+
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    by (descending) (simp add: sub_list_def memb_def[symmetric])+
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  show "x |\<inter>| y |\<subseteq>| y" +
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    by (descending) (simp add: sub_list_def memb_def[symmetric])+
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  show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" +
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    by (descending) (rule append_finter_raw_distrib)+
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next+
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  fix x y z :: "'a fset"+
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  assume a: "x |\<subseteq>| y"+
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  assume b: "y |\<subseteq>| z"+
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  show "x |\<subseteq>| z" using a b +
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    by (descending) (simp add: sub_list_def)+
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next+
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  fix x y :: "'a fset"+
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  assume a: "x |\<subseteq>| y"+
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  assume b: "y |\<subseteq>| x"+
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  show "x = y" using a b +
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    by (descending) (unfold sub_list_def list_eq.simps, blast)+
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next+
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  fix x y z :: "'a fset"+
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  assume a: "y |\<subseteq>| x"+
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  assume b: "z |\<subseteq>| x"+
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  show "y |\<union>| z |\<subseteq>| x" using a b +
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    by (descending) (simp add: sub_list_def)+
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next+
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  fix x y z :: "'a fset"+
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  assume a: "x |\<subseteq>| y"+
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  assume b: "x |\<subseteq>| z"+
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  show "x |\<subseteq>| y |\<inter>| z" using a b +
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    by (descending) (simp add: sub_list_def memb_def[symmetric])+
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qed+
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+
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end+
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+
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+
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section {* Definitions for fsets *}+
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+
−
+
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quotient_definition+
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  "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+
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  is "Cons"+
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+
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syntax+
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  "@Finset"     :: "args => 'a fset"  ("{|(_)|}")+
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+
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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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+
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quotient_definition+
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  fin (infix "|\<in>|" 50)+
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where+
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  "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"+
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+
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abbreviation+
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  fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)+
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where+
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  "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"+
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+
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section {* Other constants on the Quotient Type *}+
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+
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quotient_definition+
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  "fcard :: 'a fset \<Rightarrow> nat"+
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  is card_list+
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+
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quotient_definition+
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  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"+
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  is map+
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+
−
quotient_definition+
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  "fdelete :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+
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  is removeAll+
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+
−
quotient_definition+
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  "fset :: 'a fset \<Rightarrow> 'a set"+
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  is "set"+
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+
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quotient_definition+
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  "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"+
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  is ffold_raw+
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+
−
quotient_definition+
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  "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"+
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  is concat+
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+
−
quotient_definition+
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  "ffilter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+
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  is filter+
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+
−
+
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subsection {* Compositional Respectfulness and Preservation *}+
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+
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lemma [quot_respect]: "(list_all2 op \<approx> OOO op \<approx>) [] []"+
−
  by (fact compose_list_refl)+
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+
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lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"+
−
  by simp+
−
+
−
lemma [quot_respect]:+
−
  shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"+
−
  apply auto+
−
  apply (rule_tac b="x # b" in pred_compI)+
−
  apply auto+
−
  apply (rule_tac b="x # ba" in pred_compI)+
−
  apply auto+
−
  done+
−
+
−
lemma [quot_preserve]:+
−
  "(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert"+
−
  by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]+
−
      abs_o_rep[OF Quotient_fset] map_id finsert_def)+
−
+
−
lemma [quot_preserve]:+
−
  "((map rep_fset \<circ> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op @ = funion"+
−
  by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]+
−
      abs_o_rep[OF Quotient_fset] map_id sup_fset_def)+
−
+
−
lemma list_all2_app_l:+
−
  assumes a: "reflp R"+
−
  and b: "list_all2 R l r"+
−
  shows "list_all2 R (z @ l) (z @ r)"+
−
  by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])+
−
+
−
lemma append_rsp2_pre0:+
−
  assumes a:"list_all2 op \<approx> x x'"+
−
  shows "list_all2 op \<approx> (x @ z) (x' @ z)"+
−
  using a apply (induct x x' rule: list_induct2')+
−
  by simp_all (rule list_all2_refl1)+
−
+
−
lemma append_rsp2_pre1:+
−
  assumes a:"list_all2 op \<approx> x x'"+
−
  shows "list_all2 op \<approx> (z @ x) (z @ x')"+
−
  using a apply (induct x x' arbitrary: z rule: list_induct2')+
−
  apply (rule list_all2_refl1)+
−
  apply (simp_all del: list_eq.simps)+
−
  apply (rule list_all2_app_l)+
−
  apply (simp_all add: reflp_def)+
−
  done+
−
+
−
lemma append_rsp2_pre:+
−
  assumes a:"list_all2 op \<approx> x x'"+
−
  and     b: "list_all2 op \<approx> z z'"+
−
  shows "list_all2 op \<approx> (x @ z) (x' @ z')"+
−
  apply (rule list_all2_transp[OF fset_equivp])+
−
  apply (rule append_rsp2_pre0)+
−
  apply (rule a)+
−
  using b apply (induct z z' rule: list_induct2')+
−
  apply (simp_all only: append_Nil2)+
−
  apply (rule list_all2_refl1)+
−
  apply simp_all+
−
  apply (rule append_rsp2_pre1)+
−
  apply simp+
−
  done+
−
+
−
lemma [quot_respect]:+
−
  "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op @ op @"+
−
proof (intro fun_relI, elim pred_compE)+
−
  fix x y z w x' z' y' w' :: "'a list list"+
−
  assume a:"list_all2 op \<approx> x x'"+
−
  and b:    "x' \<approx> y'"+
−
  and c:    "list_all2 op \<approx> y' y"+
−
  assume aa: "list_all2 op \<approx> z z'"+
−
  and bb:   "z' \<approx> w'"+
−
  and cc:   "list_all2 op \<approx> w' w"+
−
  have a': "list_all2 op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto+
−
  have b': "x' @ z' \<approx> y' @ w'" using b bb by simp+
−
  have c': "list_all2 op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto+
−
  have d': "(op \<approx> OO list_all2 op \<approx>) (x' @ z') (y @ w)"+
−
    by (rule pred_compI) (rule b', rule c')+
−
  show "(list_all2 op \<approx> OOO op \<approx>) (x @ z) (y @ w)"+
−
    by (rule pred_compI) (rule a', rule d')+
−
qed+
−
+
−
+
−
+
−
section {* Cases *}+
−
+
−
+
−
lemma fset_raw_strong_cases:+
−
  obtains "xs = []"+
−
    | x ys where "\<not> memb x ys" and "xs \<approx> x # ys"+
−
proof (induct xs arbitrary: x ys)+
−
  case Nil+
−
  then show thesis by simp+
−
next+
−
  case (Cons a xs)+
−
  have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis" by fact+
−
  have b: "\<And>x' ys'. \<lbrakk>\<not> memb x' ys'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact+
−
  have c: "xs = [] \<Longrightarrow> thesis" using b unfolding memb_def+
−
    by (metis in_set_conv_nth less_zeroE list.size(3) list_eq.simps member_set)+
−
  have "\<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"+
−
  proof -+
−
    fix x :: 'a+
−
    fix ys :: "'a list"+
−
    assume d:"\<not> memb x ys"+
−
    assume e:"xs \<approx> x # ys"+
−
    show thesis+
−
    proof (cases "x = a")+
−
      assume h: "x = a"+
−
      then have f: "\<not> memb a ys" using d by simp+
−
      have g: "a # xs \<approx> a # ys" using e h by auto+
−
      show thesis using b f g by simp+
−
    next+
−
      assume h: "x \<noteq> a"+
−
      then have f: "\<not> memb x (a # ys)" using d unfolding memb_def by auto+
−
      have g: "a # xs \<approx> x # (a # ys)" using e h by auto+
−
      show thesis using b f g by simp+
−
    qed+
−
  qed+
−
  then show thesis using a c by blast+
−
qed+
−
+
−
+
−
text {* alternate formulation with a different decomposition principle+
−
  and a proof of equivalence *}+
−
+
−
inductive+
−
  list_eq2+
−
where+
−
  "list_eq2 (a # b # xs) (b # a # xs)"+
−
| "list_eq2 [] []"+
−
| "list_eq2 xs ys \<Longrightarrow> list_eq2 ys xs"+
−
| "list_eq2 (a # a # xs) (a # xs)"+
−
| "list_eq2 xs ys \<Longrightarrow> list_eq2 (a # xs) (a # ys)"+
−
| "\<lbrakk>list_eq2 xs1 xs2; list_eq2 xs2 xs3\<rbrakk> \<Longrightarrow> list_eq2 xs1 xs3"+
−
+
−
lemma list_eq2_refl:+
−
  shows "list_eq2 xs xs"+
−
  by (induct xs) (auto intro: list_eq2.intros)+
−
+
−
lemma cons_delete_list_eq2:+
−
  shows "list_eq2 (a # (removeAll a A)) (if memb a A then A else a # A)"+
−
  apply (induct A)+
−
  apply (simp add: memb_def list_eq2_refl)+
−
  apply (case_tac "memb a (aa # A)")+
−
  apply (simp_all only: memb_def)+
−
  apply (case_tac [!] "a = aa")+
−
  apply (simp_all)+
−
  apply (case_tac "memb a A")+
−
  apply (auto simp add: memb_def)[2]+
−
  apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))+
−
  apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))+
−
  apply (auto simp add: list_eq2_refl memb_def)+
−
  done+
−
+
−
lemma memb_delete_list_eq2:+
−
  assumes a: "memb e r"+
−
  shows "list_eq2 (e # removeAll e r) r"+
−
  using a cons_delete_list_eq2[of e r]+
−
  by simp+
−
+
−
lemma list_eq2_equiv:+
−
  "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"+
−
proof+
−
  show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto+
−
next+
−
  {+
−
    fix n+
−
    assume a: "card_list l = n" and b: "l \<approx> r"+
−
    have "list_eq2 l r"+
−
      using a b+
−
    proof (induct n arbitrary: l r)+
−
      case 0+
−
      have "card_list l = 0" by fact+
−
      then have "\<forall>x. \<not> memb x l" unfolding card_list_def memb_def by auto+
−
      then have z: "l = []" unfolding memb_def by auto+
−
      then have "r = []" using `l \<approx> r` by simp+
−
      then show ?case using z list_eq2_refl by simp+
−
    next+
−
      case (Suc m)+
−
      have b: "l \<approx> r" by fact+
−
      have d: "card_list l = Suc m" by fact+
−
      then have "\<exists>a. memb a l" +
−
	apply(simp add: card_list_def memb_def)+
−
	apply(drule card_eq_SucD)+
−
	apply(blast)+
−
	done+
−
      then obtain a where e: "memb a l" by auto+
−
      then have e': "memb a r" using list_eq.simps[simplified memb_def[symmetric], of l r] b +
−
	unfolding memb_def by auto+
−
      have f: "card_list (removeAll a l) = m" using e d by (simp add: card_list_def memb_def)+
−
      have g: "removeAll a l \<approx> removeAll a r" using removeAll_rsp b by simp+
−
      have "list_eq2 (removeAll a l) (removeAll a r)" by (rule Suc.hyps[OF f g])+
−
      then have h: "list_eq2 (a # removeAll a l) (a # removeAll a r)" by (rule list_eq2.intros(5))+
−
      have i: "list_eq2 l (a # removeAll a l)"+
−
        by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]])+
−
      have "list_eq2 l (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])+
−
      then show ?case using list_eq2.intros(6)[OF _ memb_delete_list_eq2[OF e']] by simp+
−
    qed+
−
    }+
−
  then show "l \<approx> r \<Longrightarrow> list_eq2 l r" by blast+
−
qed+
−
+
−
+
−
section {* Lifted theorems *}+
−
+
−
+
−
subsection {* fin *}+
−
+
−
lemma not_fin_fnil: +
−
  shows "x |\<notin>| {||}"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fin_set: +
−
  shows "x |\<in>| S \<longleftrightarrow> x \<in> fset S"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fnotin_set: +
−
  shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fset_eq_iff:+
−
  shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"+
−
  by (descending) (auto simp add: memb_def)+
−
+
−
lemma none_fin_fempty:+
−
  shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"+
−
  by (descending) (simp add: memb_def)+
−
+
−
+
−
subsection {* finsert *}+
−
+
−
lemma fin_finsert_iff[simp]:+
−
  shows "x |\<in>| finsert y S \<longleftrightarrow> x = y \<or> x |\<in>| S"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma+
−
  shows finsertI1: "x |\<in>| finsert x S"+
−
  and   finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S"+
−
  by (descending, simp add: memb_def)++
−
+
−
lemma finsert_absorb[simp]:+
−
  shows "x |\<in>| S \<Longrightarrow> finsert x S = S"+
−
  by (descending) (auto simp add: memb_def)+
−
+
−
lemma fempty_not_finsert[simp]:+
−
  shows "{||} \<noteq> finsert x S"+
−
  and   "finsert x S \<noteq> {||}"+
−
  by (descending, simp)++
−
+
−
lemma finsert_left_comm:+
−
  shows "finsert x (finsert y S) = finsert y (finsert x S)"+
−
  by (descending) (auto)+
−
+
−
lemma finsert_left_idem:+
−
  shows "finsert x (finsert x S) = finsert x S"+
−
  by (descending) (auto)+
−
+
−
lemma fsingleton_eq[simp]:+
−
  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"+
−
  by (descending) (auto)+
−
+
−
+
−
(* FIXME: is this in any case a useful lemma *)+
−
lemma fin_mdef:+
−
  shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> F = finsert x (F - {|x|})"+
−
  by (descending) (auto simp add: memb_def diff_list_def)+
−
+
−
+
−
subsection {* fset *}+
−
+
−
lemma fset_simps[simp]:+
−
  "fset {||} = ({} :: 'a set)"+
−
  "fset (finsert (x :: 'a) S) = insert x (fset S)"+
−
  by (lifting set.simps)+
−
+
−
lemma finite_fset [simp]: +
−
  shows "finite (fset S)"+
−
  by (descending) (simp)+
−
+
−
lemma fset_cong:+
−
  shows "fset S = fset T \<longleftrightarrow> S = T"+
−
  by (descending) (simp)+
−
+
−
lemma ffilter_set [simp]: +
−
  shows "fset (ffilter P xs) = P \<inter> fset xs"+
−
  by (descending) (auto simp add: mem_def)+
−
+
−
lemma fdelete_set [simp]: +
−
  shows "fset (fdelete x xs) = fset xs - {x}"+
−
  by (descending) (simp)+
−
+
−
lemma finter_set [simp]: +
−
  shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"+
−
  by (lifting set_finter_raw)+
−
+
−
lemma funion_set [simp]: +
−
  shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"+
−
  by (lifting set_append)+
−
+
−
lemma fminus_set [simp]: +
−
  shows "fset (xs - ys) = fset xs - fset ys"+
−
  by (lifting set_diff_list)+
−
+
−
+
−
subsection {* funion *}+
−
+
−
lemmas [simp] =+
−
  sup_bot_left[where 'a="'a fset", standard]+
−
  sup_bot_right[where 'a="'a fset", standard]+
−
+
−
lemma funion_finsert[simp]:+
−
  shows "finsert x S |\<union>| T = finsert x (S |\<union>| T)"+
−
  by (lifting append.simps(2))+
−
+
−
lemma singleton_funion_left:+
−
  shows "{|a|} |\<union>| S = finsert a S"+
−
  by simp+
−
+
−
lemma singleton_funion_right:+
−
  shows "S |\<union>| {|a|} = finsert a S"+
−
  by (subst sup.commute) simp+
−
+
−
lemma fin_funion:+
−
  shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"+
−
  by (descending) (simp add: memb_def)+
−
+
−
+
−
subsection {* fminus *}+
−
+
−
lemma fminus_fin: +
−
  shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fminus_red: +
−
  shows "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))"+
−
  by (descending) (auto simp add: memb_def)+
−
+
−
lemma fminus_red_fin[simp]: +
−
  shows "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys"+
−
  by (simp add: fminus_red)+
−
+
−
lemma fminus_red_fnotin[simp]: +
−
  shows "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)"+
−
  by (simp add: fminus_red)+
−
+
−
lemma fin_fminus_fnotin: +
−
  shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"+
−
  unfolding fin_set fminus_set+
−
  by blast+
−
+
−
lemma fin_fnotin_fminus: +
−
  shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"+
−
  unfolding fin_set fminus_set+
−
  by blast+
−
+
−
+
−
section {* fdelete *}+
−
+
−
lemma fin_fdelete:+
−
  shows "x |\<in>| fdelete y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fnotin_fdelete:+
−
  shows "x |\<notin>| fdelete x S"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fnotin_fdelete_ident:+
−
  shows "x |\<notin>| S \<Longrightarrow> fdelete x S = S"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fset_fdelete_cases:+
−
  shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete x S))"+
−
  by (descending) (auto simp add: memb_def insert_absorb)+
−
  +
−
+
−
section {* finter *}+
−
+
−
lemma finter_empty_l:+
−
  shows "{||} |\<inter>| S = {||}"+
−
  by simp+
−
+
−
lemma finter_empty_r:+
−
  shows "S |\<inter>| {||} = {||}"+
−
  by simp+
−
+
−
lemma finter_finsert:+
−
  shows "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fin_finter:+
−
  shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"+
−
  by (descending) (simp add: memb_def)+
−
+
−
+
−
subsection {* fsubset *}+
−
+
−
lemma fsubseteq_set: +
−
  shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"+
−
  by (descending) (simp add: sub_list_def)+
−
+
−
lemma fsubset_set: +
−
  shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"+
−
  unfolding less_fset_def +
−
  by (descending) (auto simp add: sub_list_def)+
−
+
−
lemma fsubseteq_finsert:+
−
  shows "(finsert x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"+
−
  by (descending) (simp add: sub_list_def memb_def)+
−
+
−
lemma fsubset_fin: +
−
  shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"+
−
  by (descending) (auto simp add: sub_list_def memb_def)+
−
+
−
(* FIXME: no type ord *)+
−
(*+
−
lemma fsubset_finsert:+
−
  shows "(finsert x xs) |\<subset>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subset>| ys"+
−
  by (descending) (simp add: sub_list_def memb_def)+
−
*)+
−
+
−
lemma fsubseteq_fempty:+
−
  shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"+
−
  by (descending) (simp add: sub_list_def)+
−
+
−
(* also problem with ord *)+
−
lemma not_fsubset_fnil: +
−
  shows "\<not> xs |\<subset>| {||}"+
−
  by (metis fset_simps(1) fsubset_set not_psubset_empty)+
−
+
−
+
−
section {* fmap *}+
−
+
−
lemma fmap_simps [simp]:+
−
   shows "fmap f {||} = {||}"+
−
  and   "fmap f (finsert x S) = finsert (f x) (fmap f S)"+
−
  by (descending, simp)++
−
+
−
lemma fmap_set_image [simp]:+
−
  shows "fset (fmap f S) = f ` (fset S)"+
−
  by (descending) (simp)+
−
+
−
lemma inj_fmap_eq_iff:+
−
  shows "inj f \<Longrightarrow> fmap f S = fmap f T \<longleftrightarrow> S = T"+
−
  by (descending) (metis inj_vimage_image_eq list_eq.simps set_map)+
−
+
−
lemma fmap_funion: +
−
  shows "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T"+
−
  by (descending) (simp)+
−
+
−
+
−
subsection {* fcard *}+
−
+
−
lemma fcard_set: +
−
  shows "fcard xs = card (fset xs)"+
−
  by (lifting card_list_def)+
−
+
−
lemma fcard_finsert_if [simp]:+
−
  shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"+
−
  by (descending) (auto simp add: card_list_def memb_def insert_absorb)+
−
+
−
lemma fcard_0[simp]:+
−
  shows "fcard S = 0 \<longleftrightarrow> S = {||}"+
−
  by (descending) (simp add: card_list_def)+
−
+
−
lemma fcard_fempty[simp]:+
−
  shows "fcard {||} = 0"+
−
  by (simp add: fcard_0)+
−
+
−
lemma fcard_1:+
−
  shows "fcard S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"+
−
  by (descending) (auto simp add: card_list_def card_Suc_eq)+
−
+
−
lemma fcard_gt_0:+
−
  shows "x \<in> fset S \<Longrightarrow> 0 < fcard S"+
−
  by (descending) (auto simp add: card_list_def card_gt_0_iff)+
−
  +
−
lemma fcard_not_fin:+
−
  shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))"+
−
  by (descending) (auto simp add: memb_def card_list_def insert_absorb)+
−
+
−
lemma fcard_suc: +
−
  shows "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"+
−
  apply(descending)+
−
  apply(auto simp add: card_list_def memb_def)+
−
  apply(drule card_eq_SucD)+
−
  apply(auto)+
−
  apply(rule_tac x="b" in exI)+
−
  apply(rule_tac x="removeAll b S" in exI)+
−
  apply(auto)+
−
  done+
−
+
−
lemma fcard_delete:+
−
  shows "fcard (fdelete y S) = (if y |\<in>| S then fcard S - 1 else fcard S)"+
−
  by (descending) (simp add: card_list_def memb_def)+
−
+
−
lemma fcard_suc_memb: +
−
  shows "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"+
−
  apply(descending)+
−
  apply(simp add: card_list_def memb_def)+
−
  apply(drule card_eq_SucD)+
−
  apply(auto)+
−
  done+
−
+
−
lemma fin_fcard_not_0: +
−
  shows "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"+
−
  by (descending) (auto simp add: card_list_def memb_def)+
−
+
−
lemma fcard_mono: +
−
  shows "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys"+
−
  unfolding fcard_set fsubseteq_set+
−
  by (simp add: card_mono[OF finite_fset])+
−
+
−
lemma fcard_fsubset_eq: +
−
  shows "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys"+
−
  unfolding fcard_set fsubseteq_set+
−
  by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)+
−
+
−
lemma psubset_fcard_mono: +
−
  shows "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys"+
−
  unfolding fcard_set fsubset_set+
−
  by (rule psubset_card_mono[OF finite_fset])+
−
+
−
lemma fcard_funion_finter: +
−
  shows "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)"+
−
  unfolding fcard_set funion_set finter_set+
−
  by (rule card_Un_Int[OF finite_fset finite_fset])+
−
+
−
lemma fcard_funion_disjoint: +
−
  shows "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys"+
−
  unfolding fcard_set funion_set +
−
  apply (rule card_Un_disjoint[OF finite_fset finite_fset])+
−
  by (metis finter_set fset_simps(1))+
−
+
−
lemma fcard_delete1_less: +
−
  shows "x |\<in>| xs \<Longrightarrow> fcard (fdelete x xs) < fcard xs"+
−
  unfolding fcard_set fin_set fdelete_set +
−
  by (rule card_Diff1_less[OF finite_fset])+
−
+
−
lemma fcard_delete2_less: +
−
  shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete y (fdelete x xs)) < fcard xs"+
−
  unfolding fcard_set fdelete_set fin_set+
−
  by (rule card_Diff2_less[OF finite_fset])+
−
+
−
lemma fcard_delete1_le: +
−
  shows "fcard (fdelete x xs) \<le> fcard xs"+
−
  unfolding fdelete_set fcard_set+
−
  by (rule card_Diff1_le[OF finite_fset])+
−
+
−
lemma fcard_psubset: +
−
  shows "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs"+
−
  unfolding fcard_set fsubseteq_set fsubset_set+
−
  by (rule card_psubset[OF finite_fset])+
−
+
−
lemma fcard_fmap_le: +
−
  shows "fcard (fmap f xs) \<le> fcard xs"+
−
  unfolding fcard_set fmap_set_image+
−
  by (rule card_image_le[OF finite_fset])+
−
+
−
lemma fcard_fminus_finsert[simp]:+
−
  assumes "a |\<in>| A" and "a |\<notin>| B"+
−
  shows "fcard (A - finsert a B) = fcard (A - B) - 1"+
−
  using assms +
−
  unfolding fin_set fcard_set fminus_set+
−
  by (simp add: card_Diff_insert[OF finite_fset])+
−
+
−
lemma fcard_fminus_fsubset:+
−
  assumes "B |\<subseteq>| A"+
−
  shows "fcard (A - B) = fcard A - fcard B"+
−
  using assms +
−
  unfolding fsubseteq_set fcard_set fminus_set+
−
  by (rule card_Diff_subset[OF finite_fset])+
−
+
−
lemma fcard_fminus_subset_finter:+
−
  shows "fcard (A - B) = fcard A - fcard (A |\<inter>| B)"+
−
  unfolding finter_set fcard_set fminus_set+
−
  by (rule card_Diff_subset_Int) (simp)+
−
+
−
+
−
section {* fconcat *}+
−
+
−
lemma fconcat_fempty:+
−
  shows "fconcat {||} = {||}"+
−
  by (lifting concat.simps(1))+
−
+
−
lemma fconcat_finsert:+
−
  shows "fconcat (finsert x S) = x |\<union>| fconcat S"+
−
  by (lifting concat.simps(2))+
−
+
−
lemma fconcat_finter:+
−
  shows "fconcat (xs |\<union>| ys) = fconcat xs |\<union>| fconcat ys"+
−
  by (lifting concat_append)+
−
+
−
+
−
section {* ffilter *}+
−
+
−
lemma subseteq_filter: +
−
  shows "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"+
−
  by  (descending) (auto simp add: memb_def sub_list_def)+
−
+
−
lemma eq_ffilter: +
−
  shows "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"+
−
  by (descending) (auto simp add: memb_def)+
−
+
−
lemma subset_ffilter:+
−
  shows "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow> ffilter P xs < ffilter Q xs"+
−
  unfolding less_fset_def by (auto simp add: subseteq_filter eq_ffilter)+
−
+
−
+
−
subsection {* ffold *}+
−
+
−
lemma ffold_nil: +
−
  shows "ffold f z {||} = z"+
−
  by (descending) (simp)+
−
+
−
lemma ffold_finsert: "ffold f z (finsert a A) =+
−
  (if rsp_fold f then if a |\<in>| A then ffold f z A else f a (ffold f z A) else z)"+
−
  by (descending) (simp add: memb_def)+
−
+
−
lemma fin_commute_ffold:+
−
  "\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete h b))"+
−
  by (descending) (simp add: memb_def memb_commute_ffold_raw)+
−
+
−
+
−
subsection {* Choice in fsets *}+
−
+
−
lemma fset_choice: +
−
  assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"+
−
  shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"+
−
  using a+
−
  apply(descending)+
−
  using finite_set_choice+
−
  by (auto simp add: memb_def Ball_def)+
−
+
−
+
−
(* FIXME: is that in any way useful *)  +
−
+
−
+
−
+
−
section {* Induction and Cases rules for fsets *}+
−
+
−
lemma fset_strong_cases:+
−
  obtains "xs = {||}"+
−
    | x ys where "x |\<notin>| ys" and "xs = finsert x ys"+
−
  by (lifting fset_raw_strong_cases)+
−
+
−
lemma fset_exhaust[case_names fempty finsert, cases type: fset]:+
−
  shows "\<lbrakk>S = {||} \<Longrightarrow> P; \<And>x S'. S = finsert x S' \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"+
−
  by (lifting list.exhaust)+
−
+
−
lemma fset_induct_weak[case_names fempty finsert]:+
−
  shows "\<lbrakk>P {||}; \<And>x S. P S \<Longrightarrow> P (finsert x S)\<rbrakk> \<Longrightarrow> P S"+
−
  by (lifting list.induct)+
−
+
−
lemma fset_induct[case_names fempty finsert, induct type: fset]:+
−
  assumes prem1: "P {||}"+
−
  and     prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"+
−
  shows "P S"+
−
proof(induct S rule: fset_induct_weak)+
−
  case fempty+
−
  show "P {||}" by (rule prem1)+
−
next+
−
  case (finsert x S)+
−
  have asm: "P S" by fact+
−
  show "P (finsert x S)"+
−
    by (cases "x |\<in>| S") (simp_all add: asm prem2)+
−
qed+
−
+
−
lemma fset_induct2:+
−
  "P {||} {||} \<Longrightarrow>+
−
  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>+
−
  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>+
−
  (\<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>+
−
  P xsa ysa"+
−
  apply (induct xsa arbitrary: ysa)+
−
  apply (induct_tac x rule: fset_induct)+
−
  apply simp_all+
−
  apply (induct_tac xa rule: fset_induct)+
−
  apply simp_all+
−
  done+
−
+
−
lemma fset_fcard_induct:+
−
  assumes a: "P {||}"+
−
  and     b: "\<And>xs ys. Suc (fcard xs) = (fcard ys) \<Longrightarrow> P xs \<Longrightarrow> P ys"+
−
  shows "P zs"+
−
proof (induct zs)+
−
  show "P {||}" by (rule a)+
−
next+
−
  fix x :: 'a and zs :: "'a fset"+
−
  assume h: "P zs"+
−
  assume "x |\<notin>| zs"+
−
  then have H1: "Suc (fcard zs) = fcard (finsert x zs)" using fcard_suc by auto+
−
  then show "P (finsert x zs)" using b h by simp+
−
qed+
−
+
−
+
−
(* We cannot write it as "assumes .. shows" since Isabelle changes+
−
   the quantifiers to schematic variables and reintroduces them in+
−
   a different order *)+
−
lemma fset_eq_cases:+
−
 "\<lbrakk>a1 = a2;+
−
   \<And>a b xs. \<lbrakk>a1 = finsert a (finsert b xs); a2 = finsert b (finsert a xs)\<rbrakk> \<Longrightarrow> P;+
−
   \<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;+
−
   \<And>a xs. \<lbrakk>a1 = finsert a (finsert a xs); a2 = finsert a xs\<rbrakk> \<Longrightarrow> P;+
−
   \<And>xs ys a. \<lbrakk>a1 = finsert a xs; a2 = finsert a ys; xs = ys\<rbrakk> \<Longrightarrow> P;+
−
   \<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>+
−
  \<Longrightarrow> P"+
−
  by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])+
−
+
−
lemma fset_eq_induct:+
−
  assumes "x1 = x2"+
−
  and "\<And>a b xs. P (finsert a (finsert b xs)) (finsert b (finsert a xs))"+
−
  and "P {||} {||}"+
−
  and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"+
−
  and "\<And>a xs. P (finsert a (finsert a xs)) (finsert a xs)"+
−
  and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (finsert a xs) (finsert a ys)"+
−
  and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"+
−
  shows "P x1 x2"+
−
  using assms+
−
  by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])+
−
+
−
+
−
+
−
section {* lemmas transferred from Finite_Set theory *}+
−
+
−
text {* finiteness for finite sets holds *}+
−
+
−
+
−
+
−
+
−
lemma list_all2_refl:+
−
  assumes q: "equivp R"+
−
  shows "(list_all2 R) r r"+
−
  by (rule list_all2_refl) (metis equivp_def q)+
−
+
−
lemma compose_list_refl2:+
−
  assumes q: "equivp R"+
−
  shows "(list_all2 R OOO op \<approx>) r r"+
−
proof+
−
  have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])+
−
  show "list_all2 R r r" by (rule list_all2_refl[OF q])+
−
  with * show "(op \<approx> OO list_all2 R) r r" ..+
−
qed+
−
+
−
lemma quotient_compose_list_g:+
−
  assumes q: "Quotient R Abs Rep"+
−
  and     e: "equivp R"+
−
  shows  "Quotient ((list_all2 R) OOO (op \<approx>))+
−
    (abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"+
−
  unfolding Quotient_def comp_def+
−
proof (intro conjI allI)+
−
  fix a r s+
−
  show "abs_fset (map Abs (map Rep (rep_fset a))) = a"+
−
    by (simp add: abs_o_rep[OF q] Quotient_abs_rep[OF Quotient_fset] map_id)+
−
  have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"+
−
    by (rule list_all2_refl[OF e])+
−
  have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"+
−
    by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)+
−
  show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"+
−
    by (rule, rule list_all2_refl[OF e]) (rule c)+
−
  show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>+
−
        (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"+
−
  proof (intro iffI conjI)+
−
    show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl2[OF e])+
−
    show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl2[OF e])+
−
  next+
−
    assume a: "(list_all2 R OOO op \<approx>) r s"+
−
    then have b: "map Abs r \<approx> map Abs s"+
−
    proof (elim pred_compE)+
−
      fix b ba+
−
      assume c: "list_all2 R r b"+
−
      assume d: "b \<approx> ba"+
−
      assume e: "list_all2 R ba s"+
−
      have f: "map Abs r = map Abs b"+
−
        using Quotient_rel[OF list_quotient[OF q]] c by blast+
−
      have "map Abs ba = map Abs s"+
−
        using Quotient_rel[OF list_quotient[OF q]] e by blast+
−
      then have g: "map Abs s = map Abs ba" by simp+
−
      then show "map Abs r \<approx> map Abs s" using d f map_rel_cong by simp+
−
    qed+
−
    then show "abs_fset (map Abs r) = abs_fset (map Abs s)"+
−
      using Quotient_rel[OF Quotient_fset] by blast+
−
  next+
−
    assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s+
−
      \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"+
−
    then have s: "(list_all2 R OOO op \<approx>) s s" by simp+
−
    have d: "map Abs r \<approx> map Abs s"+
−
      by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)+
−
    have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"+
−
      by (rule map_rel_cong[OF d])+
−
    have y: "list_all2 R (map Rep (map Abs s)) s"+
−
      by (fact rep_abs_rsp_left[OF list_quotient[OF q], OF list_all2_refl[OF e, of s]])+
−
    have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"+
−
      by (rule pred_compI) (rule b, rule y)+
−
    have z: "list_all2 R r (map Rep (map Abs r))"+
−
      by (fact rep_abs_rsp[OF list_quotient[OF q], OF list_all2_refl[OF e, of r]])+
−
    then show "(list_all2 R OOO op \<approx>) r s"+
−
      using a c pred_compI by simp+
−
  qed+
−
qed+
−
+
−
+
−
ML {*+
−
fun dest_fsetT (Type (@{type_name fset}, [T])) = T+
−
  | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);+
−
*}+
−
+
−
no_notation+
−
  list_eq (infix "\<approx>" 50)+
−
+
−
end+
−