"isabelle make test" makes all major examples....they work up to supp theorems (excluding)
(* Title: HOL/Quotient_Examples/FSet.thy Author: Cezary Kaliszyk, TU Munich Author: Christian Urban, TU MunichA reasoning infrastructure for the type of finite sets.*)theory FSetimports Quotient_List Quotient_Productbegintext {* Definiton of List relation and the quotient type *}fun list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)where "list_eq xs ys = (\<forall>x. x \<in> set xs \<longleftrightarrow> x \<in> set ys)"lemma list_eq_equivp: shows "equivp list_eq" unfolding equivp_reflp_symp_transp unfolding reflp_def symp_def transp_def by autoquotient_type 'a fset = "'a list" / "list_eq" by (rule list_eq_equivp)text {* Raw definitions of membership, sublist, cardinality, intersection*}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> (\<forall>x. x \<in> set xs \<longrightarrow> x \<in> set ys)"fun fcard_raw :: "'a list \<Rightarrow> nat"where fcard_raw_nil: "fcard_raw [] = 0"| fcard_raw_cons: "fcard_raw (x # xs) = (if memb x xs then fcard_raw xs else Suc (fcard_raw xs))"primrec finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"where "finter_raw [] l = []"| "finter_raw (h # t) l = (if memb h l then h # (finter_raw t l) else finter_raw t l)"primrec delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"where "delete_raw [] x = []"| "delete_raw (a # xs) x = (if (a = x) then delete_raw xs x else a # (delete_raw xs x))"primrec fminus_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"where "fminus_raw l [] = l"| "fminus_raw l (h # t) = fminus_raw (delete_raw l h) t"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 memb a xs then ffold_raw f z xs else f a (ffold_raw f z xs) else z)"text {* Composition Quotient *}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 set_in_eq: "(\<forall>e. ((e \<in> xs) \<longleftrightarrow> (e \<in> ys))) \<equiv> xs = ys" by (rule eq_reflection) autolemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba" unfolding list_eq.simps by (simp only: set_map set_in_eq)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 {* Respectfullness *}lemma append_rsp[quot_respect]: shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @" apply(simp del: list_eq.simps) by auto lemma append_rsp_unfolded: "\<lbrakk>x \<approx> y; v \<approx> w\<rbrakk> \<Longrightarrow> x @ v \<approx> y @ w" by autolemma [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 "[] \<approx> []" by simplemma cons_rsp[quot_respect]: shows "(op = ===> op \<approx> ===> op \<approx>) op # op #" 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 not_memb_nil: shows "\<not> memb x []" by (simp add: memb_def)lemma memb_cons_iff: shows "memb x (y # xs) = (x = y \<or> memb x xs)" by (induct xs) (auto simp add: memb_def)lemma memb_finter_raw: "memb x (finter_raw xs ys) \<longleftrightarrow> memb x xs \<and> memb x ys" by (induct xs) (auto simp add: not_memb_nil memb_cons_iff)lemma [quot_respect]: "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw" by (simp add: memb_def[symmetric] memb_finter_raw)lemma memb_delete_raw: "memb x (delete_raw xs y) = (memb x xs \<and> x \<noteq> y)" by (induct xs arbitrary: x y) (auto simp add: memb_def)lemma [quot_respect]: "(op \<approx> ===> op = ===> op \<approx>) delete_raw delete_raw" by (simp add: memb_def[symmetric] memb_delete_raw)lemma fminus_raw_memb: "memb x (fminus_raw xs ys) = (memb x xs \<and> \<not> memb x ys)" by (induct ys arbitrary: xs) (simp_all add: not_memb_nil memb_delete_raw memb_cons_iff)lemma [quot_respect]: "(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw" by (simp add: memb_def[symmetric] fminus_raw_memb)lemma fcard_raw_gt_0: assumes a: "x \<in> set xs" shows "0 < fcard_raw xs" using a by (induct xs) (auto simp add: memb_def)lemma fcard_raw_delete_one: shows "fcard_raw ([x \<leftarrow> xs. x \<noteq> y]) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)" by (induct xs) (auto dest: fcard_raw_gt_0 simp add: memb_def)lemma fcard_raw_rsp_aux: assumes a: "xs \<approx> ys" shows "fcard_raw xs = fcard_raw ys" using a proof (induct xs arbitrary: ys) case Nil show ?case using Nil.prems by simp next case (Cons a xs) have a: "a # xs \<approx> ys" by fact have b: "\<And>ys. xs \<approx> ys \<Longrightarrow> fcard_raw xs = fcard_raw ys" by fact show ?case proof (cases "a \<in> set xs") assume c: "a \<in> set xs" have "\<forall>x. (x \<in> set xs) = (x \<in> set ys)" proof (intro allI iffI) fix x assume "x \<in> set xs" then show "x \<in> set ys" using a by auto next fix x assume d: "x \<in> set ys" have e: "(x \<in> set (a # xs)) = (x \<in> set ys)" using a by simp show "x \<in> set xs" using c d e unfolding list_eq.simps by simp blast qed then show ?thesis using b c by (simp add: memb_def) next assume c: "a \<notin> set xs" have d: "xs \<approx> [x\<leftarrow>ys . x \<noteq> a] \<Longrightarrow> fcard_raw xs = fcard_raw [x\<leftarrow>ys . x \<noteq> a]" using b by simp have "Suc (fcard_raw xs) = fcard_raw ys" proof (cases "a \<in> set ys") assume e: "a \<in> set ys" have f: "\<forall>x. (x \<in> set xs) = (x \<in> set ys \<and> x \<noteq> a)" using a c by (auto simp add: fcard_raw_delete_one) have "fcard_raw ys = Suc (fcard_raw ys - 1)" by (rule Suc_pred'[OF fcard_raw_gt_0]) (rule e) then show ?thesis using d e f by (simp_all add: fcard_raw_delete_one memb_def) next case False then show ?thesis using a c d by auto qed then show ?thesis using a c d by (simp add: memb_def) qedqedlemma fcard_raw_rsp[quot_respect]: shows "(op \<approx> ===> op =) fcard_raw fcard_raw" by (simp add: fcard_raw_rsp_aux)lemma memb_absorb: shows "memb x xs \<Longrightarrow> x # xs \<approx> xs" by (induct xs) (auto simp add: memb_def)lemma none_memb_nil: "(\<forall>x. \<not> memb x xs) = (xs \<approx> [])" by (simp add: memb_def)lemma not_memb_delete_raw_ident: shows "\<not> memb x xs \<Longrightarrow> delete_raw xs x = xs" by (induct xs) (auto simp add: memb_def)lemma memb_commute_ffold_raw: "rsp_fold f \<Longrightarrow> memb h b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (delete_raw b h))" apply (induct b) apply (simp_all add: not_memb_nil) apply (auto) apply (simp_all add: memb_delete_raw not_memb_delete_raw_ident rsp_fold_def memb_cons_iff) donelemma ffold_raw_rsp_pre: "\<forall>e. memb e a = memb e b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b" apply (induct a arbitrary: b) apply (simp add: memb_absorb memb_def none_memb_nil) apply (simp) apply (rule conjI) apply (rule_tac [!] impI) apply (rule_tac [!] conjI) apply (rule_tac [!] impI) apply (subgoal_tac "\<forall>e. memb e a2 = memb e b") apply (simp) apply (simp add: memb_cons_iff memb_def) apply (auto)[1] apply (drule_tac x="e" in spec) apply (blast) apply (case_tac b) apply (simp_all) apply (subgoal_tac "ffold_raw f z b = f a1 (ffold_raw f z (delete_raw b a1))") apply (simp only:) apply (rule_tac f="f a1" in arg_cong) apply (subgoal_tac "\<forall>e. memb e a2 = memb e (delete_raw b a1)") apply (simp) apply (simp add: memb_delete_raw) apply (auto simp add: memb_cons_iff)[1] apply (erule memb_commute_ffold_raw) apply (drule_tac x="a1" in spec) apply (simp add: memb_cons_iff) apply (simp add: memb_cons_iff) apply (case_tac b) apply (simp_all) donelemma [quot_respect]: "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw" by (simp add: memb_def[symmetric] 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 simpqedlemma concat_rsp_unfolded: "\<lbrakk>list_all2 op \<approx> a ba; ba \<approx> bb; list_all2 op \<approx> bb b\<rbrakk> \<Longrightarrow> concat a \<approx> concat b"proof - 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 simpqedlemma [quot_respect]: "((op =) ===> op \<approx> ===> op \<approx>) filter filter" by autotext {* Distributive lattice with bot *}lemma sub_list_not_eq: "(sub_list x y \<and> \<not> list_eq x y) = (sub_list x y \<and> \<not> sub_list y x)" by (auto simp add: sub_list_def)lemma sub_list_refl: "sub_list x x" by (simp add: sub_list_def)lemma sub_list_trans: "sub_list x y \<Longrightarrow> sub_list y z \<Longrightarrow> sub_list x z" by (simp add: sub_list_def)lemma sub_list_empty: "sub_list [] x" by (simp add: sub_list_def)lemma sub_list_append_left: "sub_list x (x @ y)" by (simp add: sub_list_def)lemma sub_list_append_right: "sub_list y (x @ y)" by (simp add: sub_list_def)lemma sub_list_inter_left: shows "sub_list (finter_raw x y) x" by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)lemma sub_list_inter_right: shows "sub_list (finter_raw x y) y" by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)lemma sub_list_list_eq: "sub_list x y \<Longrightarrow> sub_list y x \<Longrightarrow> list_eq x y" unfolding sub_list_def list_eq.simps by blastlemma sub_list_append: "sub_list y x \<Longrightarrow> sub_list z x \<Longrightarrow> sub_list (y @ z) x" unfolding sub_list_def by autolemma sub_list_inter: "sub_list x y \<Longrightarrow> sub_list x z \<Longrightarrow> sub_list x (finter_raw y z)" by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)lemma append_inter_distrib: "x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)" apply (induct x) apply (simp_all add: memb_def) apply (simp add: memb_def[symmetric] memb_finter_raw) apply (auto simp add: memb_def) doneinstantiation 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: "(xs :: 'a fset) < ys \<equiv> xs \<le> ys \<and> xs \<noteq> ys"abbreviation f_subset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)where "xs |\<subset>| ys \<equiv> xs < ys"quotient_definition "sup \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"is "(op @) \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"abbreviation funion (infixl "|\<union>|" 65)where "xs |\<union>| ys \<equiv> sup (xs :: 'a fset) ys"quotient_definition "inf \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"is "finter_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"abbreviation finter (infixl "|\<inter>|" 65)where "xs |\<inter>| ys \<equiv> inf (xs :: 'a fset) ys"quotient_definition "minus \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"is "fminus_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"instanceproof fix x y z :: "'a fset" show "(x |\<subset>| y) = (x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x)" unfolding less_fset by (lifting sub_list_not_eq) show "x |\<subseteq>| x" by (lifting sub_list_refl) show "{||} |\<subseteq>| x" by (lifting sub_list_empty) show "x |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_left) show "y |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_right) show "x |\<inter>| y |\<subseteq>| x" by (lifting sub_list_inter_left) show "x |\<inter>| y |\<subseteq>| y" by (lifting sub_list_inter_right) show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" by (lifting append_inter_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 (lifting sub_list_trans)next fix x y :: "'a fset" assume a: "x |\<subseteq>| y" assume b: "y |\<subseteq>| x" show "x = y" using a b by (lifting sub_list_list_eq)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 (lifting sub_list_append)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 (lifting sub_list_inter)qedendsection {* Finsert and Membership *}quotient_definition "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "op #"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_raw"quotient_definition "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"is "map"quotient_definition "fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" is "delete_raw"quotient_definition "fset_to_set :: 'a fset \<Rightarrow> 'a set" is "set"quotient_definition "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is "ffold_raw"quotient_definition "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"is "concat"quotient_definition "ffilter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"is "filter"text {* Compositional Respectfullness 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]: "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op # op #" apply auto apply (simp add: set_in_eq) apply (rule_tac b="x # b" in pred_compI) apply auto apply (rule_tac b="x # ba" in pred_compI) apply auto donelemma insert_preserve2: shows "((rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op #) = (id ---> rep_fset ---> abs_fset) op #" by (simp add: expand_fun_eq abs_o_rep[OF Quotient_fset] map_id Quotient_abs_rep[OF Quotient_fset])lemma [quot_preserve]: "(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert" by (simp add: expand_fun_eq 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: expand_fun_eq 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')qedtext {* Raw theorems. Finsert, memb, singleron, sub_list *}lemma nil_not_cons: shows "\<not> ([] \<approx> x # xs)" and "\<not> (x # xs \<approx> [])" by autolemma no_memb_nil: "(\<forall>x. \<not> memb x xs) = (xs = [])" by (simp add: memb_def)lemma memb_consI1: shows "memb x (x # xs)" by (simp add: memb_def)lemma memb_consI2: shows "memb x xs \<Longrightarrow> memb x (y # xs)" by (simp add: memb_def)lemma singleton_list_eq: shows "[x] \<approx> [y] \<longleftrightarrow> x = y" by (simp add:) autolemma sub_list_cons: "sub_list (x # xs) ys = (memb x ys \<and> sub_list xs ys)" by (auto simp add: memb_def sub_list_def)lemma fminus_raw_red: "fminus_raw (x # xs) ys = (if memb x ys then fminus_raw xs ys else x # (fminus_raw xs ys))" by (induct ys arbitrary: xs x) (simp_all add: not_memb_nil memb_delete_raw memb_cons_iff)text {* Cardinality of finite sets *}lemma fcard_raw_0: shows "fcard_raw xs = 0 \<longleftrightarrow> xs \<approx> []" by (induct xs) (auto simp add: memb_def)lemma fcard_raw_not_memb: shows "\<not> memb x xs \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)" by autolemma fcard_raw_suc: assumes a: "fcard_raw xs = Suc n" shows "\<exists>x ys. \<not> (memb x ys) \<and> xs \<approx> (x # ys) \<and> fcard_raw ys = n" using a by (induct xs) (auto simp add: memb_def split_ifs)lemma singleton_fcard_1: shows "set xs = {x} \<Longrightarrow> fcard_raw xs = 1" by (induct xs) (auto simp add: memb_def subset_insert)lemma fcard_raw_1: shows "fcard_raw xs = 1 \<longleftrightarrow> (\<exists>x. xs \<approx> [x])" apply (auto dest!: fcard_raw_suc) apply (simp add: fcard_raw_0) apply (rule_tac x="x" in exI) apply simp apply (subgoal_tac "set xs = {x}") apply (drule singleton_fcard_1) apply auto donelemma fcard_raw_suc_memb: assumes a: "fcard_raw A = Suc n" shows "\<exists>a. memb a A" using a by (induct A) (auto simp add: memb_def)lemma memb_card_not_0: assumes a: "memb a A" shows "\<not>(fcard_raw A = 0)"proof - have "\<not>(\<forall>x. \<not> memb x A)" using a by auto then have "\<not>A \<approx> []" using none_memb_nil[of A] by simp then show ?thesis using fcard_raw_0[of A] by simpqedtext {* fmap *}lemma map_append: "map f (xs @ ys) \<approx> (map f xs) @ (map f ys)" by simplemma memb_append: "memb x (xs @ ys) \<longleftrightarrow> memb x xs \<or> memb x ys" by (induct xs) (simp_all add: not_memb_nil memb_cons_iff)lemma cons_left_comm: "x # y # xs \<approx> y # x # xs" by autolemma cons_left_idem: "x # x # xs \<approx> x # xs" by autolemma 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" by (metis no_memb_nil singleton_list_eq b) 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 blastqedsection {* deletion *}lemma memb_delete_raw_ident: shows "\<not> memb x (delete_raw xs x)" by (induct xs) (auto simp add: memb_def)lemma fset_raw_delete_raw_cases: "xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # delete_raw xs x)" by (induct xs) (auto simp add: memb_def)lemma fdelete_raw_filter: "delete_raw xs y = [x \<leftarrow> xs. x \<noteq> y]" by (induct xs) simp_alllemma fcard_raw_delete: "fcard_raw (delete_raw xs y) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)" by (simp add: fdelete_raw_filter fcard_raw_delete_one)lemma finter_raw_empty: "finter_raw l [] = []" by (induct l) (simp_all add: not_memb_nil)lemma set_cong: shows "(x \<approx> y) = (set x = set y)" by autolemma inj_map_eq_iff: "inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)" by (simp add: expand_set_eq[symmetric] inj_image_eq_iff)text {* 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 # (delete_raw 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_cons_iff) 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 not_memb_delete_raw_ident) donelemma memb_delete_list_eq2: assumes a: "memb e r" shows "list_eq2 (e # delete_raw r e) r" using a cons_delete_list_eq2[of e r] by simplemma delete_raw_rsp: "xs \<approx> ys \<Longrightarrow> delete_raw xs x \<approx> delete_raw ys x" by (simp add: memb_def[symmetric] memb_delete_raw)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) 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" using memb_card_not_0[of _ l] by auto then have z: "l = []" using no_memb_nil 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" by (rule fcard_raw_suc_memb) 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 by auto have f: "fcard_raw (delete_raw l a) = m" using fcard_raw_delete[of l a] e d by simp have g: "delete_raw l a \<approx> delete_raw r a" using delete_raw_rsp[OF b] by simp have "list_eq2 (delete_raw l a) (delete_raw r a)" by (rule Suc.hyps[OF f g]) then have h: "list_eq2 (a # delete_raw l a) (a # delete_raw r a)" by (rule list_eq2.intros(5)) have i: "list_eq2 l (a # delete_raw l a)" by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]]) have "list_eq2 l (a # delete_raw r a)" 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 blastqedtext {* Set *}lemma sub_list_set: "sub_list xs ys = (set xs \<subseteq> set ys)" by (metis rev_append set_append set_cong set_rev sub_list_append sub_list_append_left sub_list_def sub_list_not_eq subset_Un_eq)lemma sub_list_neq_set: "(sub_list xs ys \<and> \<not> list_eq xs ys) = (set xs \<subset> set ys)" by (auto simp add: sub_list_set)lemma fcard_raw_set: "fcard_raw xs = card (set xs)" by (induct xs) (auto simp add: insert_absorb memb_def card_insert_disjoint)lemma memb_set: "memb x xs = (x \<in> set xs)" by (simp only: memb_def)lemma filter_set: "set (filter P xs) = P \<inter> (set xs)" by (induct xs, simp) (metis Int_insert_right_if0 Int_insert_right_if1 List.set.simps(2) filter.simps(2) mem_def)lemma delete_raw_set: "set (delete_raw xs x) = set xs - {x}" by (induct xs) autolemma inter_raw_set: "set (finter_raw xs ys) = set xs \<inter> set ys" by (induct xs) (simp_all add: memb_def)lemma fminus_raw_set: "set (fminus_raw xs ys) = set xs - set ys" by (induct ys arbitrary: xs) (simp_all add: delete_raw_set, blast)text {* Raw theorems of ffilter *}lemma sub_list_filter: "sub_list (filter P xs) (filter Q xs) = (\<forall> x. memb x xs \<longrightarrow> P x \<longrightarrow> Q x)"unfolding sub_list_def memb_def by autolemma list_eq_filter: "list_eq (filter P xs) (filter Q xs) = (\<forall>x. memb x xs \<longrightarrow> P x = Q x)"unfolding memb_def by autotext {* Lifted theorems *}lemma not_fin_fnil: "x |\<notin>| {||}" by (lifting not_memb_nil)lemma fin_finsert_iff[simp]: "x |\<in>| finsert y S = (x = y \<or> x |\<in>| S)" by (lifting memb_cons_iff)lemma shows finsertI1: "x |\<in>| finsert x S" and finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S" by (lifting memb_consI1, lifting memb_consI2)lemma finsert_absorb[simp]: shows "x |\<in>| S \<Longrightarrow> finsert x S = S" by (lifting memb_absorb)lemma fempty_not_finsert[simp]: "{||} \<noteq> finsert x S" "finsert x S \<noteq> {||}" by (lifting nil_not_cons)lemma finsert_left_comm: "finsert x (finsert y S) = finsert y (finsert x S)" by (lifting cons_left_comm)lemma finsert_left_idem: "finsert x (finsert x S) = finsert x S" by (lifting cons_left_idem)lemma fsingleton_eq[simp]: shows "{|x|} = {|y|} \<longleftrightarrow> x = y" by (lifting singleton_list_eq)text {* fset_to_set *}lemma fset_to_set_simps[simp]: "fset_to_set {||} = ({} :: 'a set)" "fset_to_set (finsert (h :: 'a) t) = insert h (fset_to_set t)" by (lifting set.simps)lemma in_fset_to_set: "x \<in> fset_to_set S \<equiv> x |\<in>| S" by (lifting memb_def[symmetric])lemma none_fin_fempty: "(\<forall>x. x |\<notin>| S) = (S = {||})" by (lifting none_memb_nil)lemma fset_cong: "(S = T) = (fset_to_set S = fset_to_set T)" by (lifting set_cong)text {* fcard *}lemma fcard_fempty [simp]: shows "fcard {||} = 0" by (lifting fcard_raw_nil)lemma fcard_finsert_if [simp]: shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))" by (lifting fcard_raw_cons)lemma fcard_0: "(fcard S = 0) = (S = {||})" by (lifting fcard_raw_0)lemma fcard_1: shows "(fcard S = 1) = (\<exists>x. S = {|x|})" by (lifting fcard_raw_1)lemma fcard_gt_0: shows "x \<in> fset_to_set S \<Longrightarrow> 0 < fcard S" by (lifting fcard_raw_gt_0)lemma fcard_not_fin: shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))" by (lifting fcard_raw_not_memb)lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n" by (lifting fcard_raw_suc)lemma fcard_delete: "fcard (fdelete S y) = (if y |\<in>| S then fcard S - 1 else fcard S)" by (lifting fcard_raw_delete)lemma fcard_suc_memb: "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A" by (lifting fcard_raw_suc_memb)lemma fin_fcard_not_0: "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0" by (lifting memb_card_not_0)text {* 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_union_left: "{|a|} |\<union>| S = finsert a S" by simplemma singleton_union_right: "S |\<union>| {|a|} = finsert a S" by (subst sup.commute) simpsection {* Induction and Cases rules for finite sets *}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 simpqedtext {* fmap *}lemma fmap_simps[simp]: "fmap (f :: 'a \<Rightarrow> 'b) {||} = {||}" "fmap f (finsert x S) = finsert (f x) (fmap f S)" by (lifting map.simps)lemma fmap_set_image: "fset_to_set (fmap f S) = f ` (fset_to_set S)" by (induct S) simp_alllemma inj_fmap_eq_iff: "inj f \<Longrightarrow> (fmap f S = fmap f T) = (S = T)" by (lifting inj_map_eq_iff)lemma fmap_funion: "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T" by (lifting map_append)lemma fin_funion: "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T" by (lifting memb_append)text {* to_set *}lemma fin_set: "(x |\<in>| xs) = (x \<in> fset_to_set xs)" by (lifting memb_set)lemma fnotin_set: "(x |\<notin>| xs) = (x \<notin> fset_to_set xs)" by (simp add: fin_set)lemma fcard_set: "fcard xs = card (fset_to_set xs)" by (lifting fcard_raw_set)lemma fsubseteq_set: "(xs |\<subseteq>| ys) = (fset_to_set xs \<subseteq> fset_to_set ys)" by (lifting sub_list_set)lemma fsubset_set: "(xs |\<subset>| ys) = (fset_to_set xs \<subset> fset_to_set ys)" unfolding less_fset by (lifting sub_list_neq_set)lemma ffilter_set: "fset_to_set (ffilter P xs) = P \<inter> fset_to_set xs" by (lifting filter_set)lemma fdelete_set: "fset_to_set (fdelete xs x) = fset_to_set xs - {x}" by (lifting delete_raw_set)lemma inter_set: "fset_to_set (xs |\<inter>| ys) = fset_to_set xs \<inter> fset_to_set ys" by (lifting inter_raw_set)lemma union_set: "fset_to_set (xs |\<union>| ys) = fset_to_set xs \<union> fset_to_set ys" by (lifting set_append)lemma fminus_set: "fset_to_set (xs - ys) = fset_to_set xs - fset_to_set ys" by (lifting fminus_raw_set)lemmas fset_to_set_trans = fin_set fnotin_set fcard_set fsubseteq_set fsubset_set inter_set union_set ffilter_set fset_to_set_simps fset_cong fdelete_set fmap_set_image fminus_settext {* ffold *}lemma ffold_nil: "ffold f z {||} = z" by (lifting ffold_raw.simps(1)[where 'a="'b" and 'b="'a"])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 (lifting ffold_raw.simps(2)[where 'a="'b" and 'b="'a"])lemma fin_commute_ffold: "\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete b h))" by (lifting memb_commute_ffold_raw)text {* fdelete *}lemma fin_fdelete: shows "x |\<in>| fdelete S y \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y" by (lifting memb_delete_raw)lemma fin_fdelete_ident: shows "x |\<notin>| fdelete S x" by (lifting memb_delete_raw_ident)lemma not_memb_fdelete_ident: shows "x |\<notin>| S \<Longrightarrow> fdelete S x = S" by (lifting not_memb_delete_raw_ident)lemma fset_fdelete_cases: shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete S x))" by (lifting fset_raw_delete_raw_cases)text {* inter *}lemma finter_empty_l: "({||} |\<inter>| S) = {||}" by (lifting finter_raw.simps(1))lemma finter_empty_r: "(S |\<inter>| {||}) = {||}" by (lifting finter_raw_empty)lemma finter_finsert: "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)" by (lifting finter_raw.simps(2))lemma fin_finter: "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T" by (lifting memb_finter_raw)lemma fsubset_finsert: "(finsert x xs |\<subseteq>| ys) = (x |\<in>| ys \<and> xs |\<subseteq>| ys)" by (lifting sub_list_cons)lemma "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys" by (lifting sub_list_def[simplified memb_def[symmetric]])lemma fsubset_fin: "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"by (rule meta_eq_to_obj_eq) (lifting sub_list_def[simplified memb_def[symmetric]])lemma fminus_fin: "(x |\<in>| xs - ys) = (x |\<in>| xs \<and> x |\<notin>| ys)" by (lifting fminus_raw_memb)lemma fminus_red: "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))" by (lifting fminus_raw_red)lemma fminus_red_fin[simp]: "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys" by (simp add: fminus_red)lemma fminus_red_fnotin[simp]: "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)" by (simp add: fminus_red)lemma expand_fset_eq: "(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))" by (lifting list_eq.simps[simplified memb_def[symmetric]])(* 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]])text {* concat *}lemma fconcat_empty: shows "fconcat {||} = {||}" by (lifting concat.simps(1))lemma fconcat_insert: shows "fconcat (finsert x S) = x |\<union>| fconcat S" by (lifting concat.simps(2))text {* ffilter *}lemma subseteq_filter: "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"by (lifting sub_list_filter)lemma eq_ffilter: "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"by (lifting list_eq_filter)lemma subset_ffilter: "(\<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 by (auto simp add: subseteq_filter eq_ffilter)section {* lemmas transferred from Finite_Set theory *}text {* finiteness for finite sets holds *}lemma finite_fset: "finite (fset_to_set S)" by (induct S) autolemma fset_choice: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)" unfolding fset_to_set_trans by (rule finite_set_choice[simplified Ball_def, OF finite_fset])lemma fsubseteq_fnil: "xs |\<subseteq>| {||} = (xs = {||})" unfolding fset_to_set_trans by (rule subset_empty)lemma not_fsubset_fnil: "\<not> xs |\<subset>| {||}" unfolding fset_to_set_trans by (rule not_psubset_empty)lemma fcard_mono: "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys" unfolding fset_to_set_trans by (rule card_mono[OF finite_fset])lemma fcard_fseteq: "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys" unfolding fset_to_set_trans by (rule card_seteq[OF finite_fset])lemma psubset_fcard_mono: "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys" unfolding fset_to_set_trans by (rule psubset_card_mono[OF finite_fset])lemma fcard_funion_finter: "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)" unfolding fset_to_set_trans by (rule card_Un_Int[OF finite_fset finite_fset])lemma fcard_funion_disjoint: "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys" unfolding fset_to_set_trans by (rule card_Un_disjoint[OF finite_fset finite_fset])lemma fcard_delete1_less: "x |\<in>| xs \<Longrightarrow> fcard (fdelete xs x) < fcard xs" unfolding fset_to_set_trans by (rule card_Diff1_less[OF finite_fset])lemma fcard_delete2_less: "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete (fdelete xs x) y) < fcard xs" unfolding fset_to_set_trans by (rule card_Diff2_less[OF finite_fset])lemma fcard_delete1_le: "fcard (fdelete xs x) <= fcard xs" unfolding fset_to_set_trans by (rule card_Diff1_le[OF finite_fset])lemma fcard_psubset: "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs" unfolding fset_to_set_trans by (rule card_psubset[OF finite_fset])lemma fcard_fmap_le: "fcard (fmap f xs) \<le> fcard xs" unfolding fset_to_set_trans by (rule card_image_le[OF finite_fset])lemma fin_fminus_fnotin: "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S" unfolding fset_to_set_trans by blastlemma fin_fnotin_fminus: "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S" unfolding fset_to_set_trans by blastlemma fin_mdef: "x |\<in>| F = ((x |\<notin>| (F - {|x|})) & (F = finsert x (F - {|x|})))" unfolding fset_to_set_trans by blastlemma 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 fset_to_set_trans by (rule 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 fset_to_set_trans by (rule card_Diff_subset[OF finite_fset])lemma fcard_fminus_subset_finter: "fcard (A - B) = fcard A - fcard (A |\<inter>| B)" unfolding fset_to_set_trans by (rule card_Diff_subset_Int) (fold inter_set, rule finite_fset)lemma ball_reg_right_unfolded: "(\<forall>x. R x \<longrightarrow> P x \<longrightarrow> Q x) \<longrightarrow> (All P \<longrightarrow> Ball R Q)"apply ruleapply (rule ball_reg_right)apply autodonelemma 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 qedqedno_notation list_eq (infix "\<approx>" 50)end