(* Title: HOL/Quotient_Examples/FSet.thy Author: Cezary Kaliszyk, TU Munich Author: Christian Urban, TU Munich Type of finite sets.*)theory FSetimports Quotient_Listbegintext {* Definiton of the list equivalence relation *}fun list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)where "list_eq xs ys = (set xs = set ys)"lemma list_eq_equivp: shows "equivp list_eq" unfolding equivp_reflp_symp_transp unfolding reflp_def symp_def transp_def by autotext {* Fset type *}quotient_type 'a fset = "'a list" / "list_eq" by (rule list_eq_equivp)text {* Definitions of membership, sublist, cardinality, intersection etc over lists*}definition memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"where "memb x xs \<equiv> x \<in> set xs"definition sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"where "sub_list xs ys \<equiv> set xs \<subseteq> set ys"definition fcard_raw :: "'a list \<Rightarrow> nat"where "fcard_raw xs = card (set xs)"primrec finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"where "finter_raw [] ys = []"| "finter_raw (x # xs) ys = (if x \<in> set ys then x # (finter_raw xs ys) else finter_raw xs ys)"definition fminus_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"where "fminus_raw xs ys \<equiv> [x \<leftarrow> xs. x\<notin>set ys]"definition rsp_foldwhere "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"primrec ffold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"where "ffold_raw f z [] = z"| "ffold_raw f z (a # xs) = (if (rsp_fold f) then if a \<in> set xs then ffold_raw f z xs else f a (ffold_raw f z xs) else z)"lemma set_finter_raw[simp]: shows "set (finter_raw xs ys) = set xs \<inter> set ys" by (induct xs) (auto simp add: memb_def)lemma set_fminus_raw[simp]: shows "set (fminus_raw xs ys) = (set xs - set ys)" by (auto simp add: fminus_raw_def)section {* Quotient composition lemmas *}lemma list_all2_refl1: shows "(list_all2 op \<approx>) r r" by (rule list_all2_refl) (metis equivp_def fset_equivp)lemma compose_list_refl: shows "(list_all2 op \<approx> OOO op \<approx>) r r"proof have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp]) show "list_all2 op \<approx> r r" by (rule list_all2_refl1) with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..qedlemma Quotient_fset_list: shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)" by (fact list_quotient[OF Quotient_fset])lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba" unfolding list_eq.simps by (simp only: set_map)lemma quotient_compose_list[quot_thm]: shows "Quotient ((list_all2 op \<approx>) OOO (op \<approx>)) (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)" unfolding Quotient_def comp_defproof (intro conjI allI) fix a r s show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a" by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id) have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))" by (rule list_all2_refl1) have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))" by (rule, rule equivp_reflp[OF fset_equivp]) (rule b) show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))" by (rule, rule list_all2_refl1) (rule c) show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))" proof (intro iffI conjI) show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl) show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl) next assume a: "(list_all2 op \<approx> OOO op \<approx>) r s" then have b: "map abs_fset r \<approx> map abs_fset s" proof (elim pred_compE) fix b ba assume c: "list_all2 op \<approx> r b" assume d: "b \<approx> ba" assume e: "list_all2 op \<approx> ba s" have f: "map abs_fset r = map abs_fset b" using Quotient_rel[OF Quotient_fset_list] c by blast have "map abs_fset ba = map abs_fset s" using Quotient_rel[OF Quotient_fset_list] e by blast then have g: "map abs_fset s = map abs_fset ba" by simp then show "map abs_fset r \<approx> map abs_fset s" using d f map_rel_cong by simp qed then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)" using Quotient_rel[OF Quotient_fset] by blast next assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)" then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp have d: "map abs_fset r \<approx> map abs_fset s" by (subst Quotient_rel[OF Quotient_fset]) (simp add: a) have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)" by (rule map_rel_cong[OF d]) have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s" by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl1[of s]]) have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s" by (rule pred_compI) (rule b, rule y) have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))" by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl1[of r]]) then show "(list_all2 op \<approx> OOO op \<approx>) r s" using a c pred_compI by simp qedqedtext {* Respectfulness *}lemma append_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op \<approx>) append append" by simplemma sub_list_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list" by (auto simp add: sub_list_def)lemma memb_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op =) memb memb" by (auto simp add: memb_def)lemma nil_rsp[quot_respect]: shows "(op \<approx>) Nil Nil" by simplemma cons_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) Cons Cons" by simplemma map_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) map map" by autolemma set_rsp[quot_respect]: "(op \<approx> ===> op =) set set" by autolemma list_equiv_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>" by autolemma finter_raw_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw" by simplemma removeAll_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) removeAll removeAll" by simplemma fminus_raw_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw" by simplemma fcard_raw_rsp[quot_respect]: shows "(op \<approx> ===> op =) fcard_raw fcard_raw" by (simp add: fcard_raw_def)lemma memb_commute_ffold_raw: "rsp_fold f \<Longrightarrow> h \<in> set b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (removeAll h b))" apply (induct b) apply (auto simp add: rsp_fold_def) donelemma ffold_raw_rsp_pre: "set a = set b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b" apply (induct a arbitrary: b) apply (simp) apply (simp (no_asm_use)) apply (rule conjI) apply (rule_tac [!] impI) apply (rule_tac [!] conjI) apply (rule_tac [!] impI) apply (metis insert_absorb) apply (metis List.insert_def List.set.simps(2) List.set_insert ffold_raw.simps(2)) apply (metis Diff_insert_absorb insertI1 memb_commute_ffold_raw set_removeAll) apply(drule_tac x="removeAll a1 b" in meta_spec) apply(auto) apply(drule meta_mp) apply(blast) by (metis List.set.simps(2) emptyE ffold_raw.simps(2) in_listsp_conv_set listsp.simps mem_def)lemma ffold_raw_rsp[quot_respect]: shows "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw" unfolding fun_rel_def by(auto intro: ffold_raw_rsp_pre)lemma concat_rsp_pre: assumes a: "list_all2 op \<approx> x x'" and b: "x' \<approx> y'" and c: "list_all2 op \<approx> y' y" and d: "\<exists>x\<in>set x. xa \<in> set x" shows "\<exists>x\<in>set y. xa \<in> set x"proof - obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a]) then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto have "ya \<in> set y'" using b h by simp then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element) then show ?thesis using f i by autoqedlemma concat_rsp[quot_respect]: shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"proof (rule fun_relI, elim pred_compE) fix a b ba bb assume a: "list_all2 op \<approx> a ba" assume b: "ba \<approx> bb" assume c: "list_all2 op \<approx> bb b" have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof fix x show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof assume d: "\<exists>xa\<in>set a. x \<in> set xa" show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d]) next assume e: "\<exists>xa\<in>set b. x \<in> set xa" have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a]) have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b]) have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c]) show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e]) qed qed then show "concat a \<approx> concat b" by autoqedlemma [quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) filter filter" by autoinstantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"beginquotient_definition "bot :: 'a fset" is "[] :: 'a list"abbreviation fempty ("{||}")where "{||} \<equiv> bot :: 'a fset"quotient_definition "less_eq_fset \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"is "sub_list \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> bool)"abbreviation f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)where "xs |\<subset>| ys \<equiv> xs < ys"quotient_definition "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"abbreviation funion (infixl "|\<union>|" 65)where "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"quotient_definition "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "finter_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"abbreviation finter (infixl "|\<inter>|" 65)where "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"quotient_definition "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "fminus_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"lemma append_finter_raw_distrib: "x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)" apply (induct x) apply (auto) doneinstanceproof fix x y z :: "'a fset" show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x" unfolding less_fset_def by (descending) (auto simp add: sub_list_def) show "x |\<subseteq>| x" by (descending) (simp add: sub_list_def) show "{||} |\<subseteq>| x" by (descending) (simp add: sub_list_def) show "x |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def) show "y |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def) show "x |\<inter>| y |\<subseteq>| x" by (descending) (simp add: sub_list_def memb_def[symmetric]) show "x |\<inter>| y |\<subseteq>| y" by (descending) (simp add: sub_list_def memb_def[symmetric]) show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" by (descending) (rule append_finter_raw_distrib)next fix x y z :: "'a fset" assume a: "x |\<subseteq>| y" assume b: "y |\<subseteq>| z" show "x |\<subseteq>| z" using a b by (descending) (simp add: sub_list_def)next fix x y :: "'a fset" assume a: "x |\<subseteq>| y" assume b: "y |\<subseteq>| x" show "x = y" using a b by (descending) (unfold sub_list_def list_eq.simps, blast)next fix x y z :: "'a fset" assume a: "y |\<subseteq>| x" assume b: "z |\<subseteq>| x" show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp add: sub_list_def)next fix x y z :: "'a fset" assume a: "x |\<subseteq>| y" assume b: "x |\<subseteq>| z" show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (simp add: sub_list_def memb_def[symmetric])qedendsection {* Definitions for fsets *}quotient_definition "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is "Cons"syntax "@Finset" :: "args => 'a fset" ("{|(_)|}")translations "{|x, xs|}" == "CONST finsert x {|xs|}" "{|x|}" == "CONST finsert x {||}"quotient_definition fin (infix "|\<in>|" 50)where "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"abbreviation fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)where "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"section {* Other constants on the Quotient Type *}quotient_definition "fcard :: 'a fset \<Rightarrow> nat" is fcard_rawquotient_definition "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" is mapquotient_definition "fdelete :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is removeAllquotient_definition "fset :: 'a fset \<Rightarrow> 'a set" is "set"quotient_definition "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is ffold_rawquotient_definition "fconcat :: ('a fset) fset \<Rightarrow> 'a fset" is concatquotient_definition "ffilter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is filtersubsection {* Compositional Respectfulness and Preservation *}lemma [quot_respect]: "(list_all2 op \<approx> OOO op \<approx>) [] []" by (fact compose_list_refl)lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []" by simplemma [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 donelemma [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) donelemma 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 donelemma [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')qedsection {* 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 simpnext 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 blastqedtext {* alternate formulation with a different decomposition principle and a proof of equivalence *}inductive list_eq2where "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) donelemma 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 simplemma 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) autonext { fix n assume a: "fcard_raw l = n" and b: "l \<approx> r" have "list_eq2 l r" using a b proof (induct n arbitrary: l r) case 0 have "fcard_raw l = 0" by fact then have "\<forall>x. \<not> memb x l" unfolding fcard_raw_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: "fcard_raw l = Suc m" by fact then have "\<exists>a. memb a l" apply(simp add: fcard_raw_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: "fcard_raw (removeAll a l) = m" using e d by (simp add: fcard_raw_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 blastqedsection {* 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 fminus_raw_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_fminus_raw)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 simplemma singleton_funion_right: shows "S |\<union>| {|a|} = finsert a S" by (subst sup.commute) simplemma 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 blastlemma fin_fnotin_fminus: shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S" unfolding fin_set fminus_set by blastsection {* 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 simplemma finter_empty_r: shows "S |\<inter>| {||} = {||}" by simplemma 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 fcard_raw_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: fcard_raw_def memb_def insert_absorb)lemma fcard_0[simp]: shows "fcard S = 0 \<longleftrightarrow> S = {||}" by (descending) (simp add: fcard_raw_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: fcard_raw_def card_Suc_eq)lemma fcard_gt_0: shows "x \<in> fset S \<Longrightarrow> 0 < fcard S" by (descending) (auto simp add: fcard_raw_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 fcard_raw_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: fcard_raw_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) donelemma fcard_delete: shows "fcard (fdelete y S) = (if y |\<in>| S then fcard S - 1 else fcard S)" by (descending) (simp add: fcard_raw_def memb_def)lemma fcard_suc_memb: shows "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A" apply(descending) apply(simp add: fcard_raw_def memb_def) apply(drule card_eq_SucD) apply(auto) donelemma fin_fcard_not_0: shows "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0" by (descending) (auto simp add: fcard_raw_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)qedlemma 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 donelemma 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 simpqed(* 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" ..qedlemma 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_defproof (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 qedqedML {*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