wenzelm@35788: (* Title: HOL/Library/Quotient_List.thy kaliszyk@35222: Author: Cezary Kaliszyk and Christian Urban kaliszyk@35222: *) wenzelm@35788: wenzelm@35788: header {* Quotient infrastructure for the list type *} wenzelm@35788: kaliszyk@35222: theory Quotient_List kaliszyk@35222: imports Main Quotient_Syntax kaliszyk@35222: begin kaliszyk@35222: kaliszyk@37492: declare [[map list = (map, list_all2)]] kaliszyk@35222: kaliszyk@35222: lemma split_list_all: kaliszyk@35222: shows "(\x. P x) \ P [] \ (\x xs. P (x#xs))" kaliszyk@35222: apply(auto) kaliszyk@35222: apply(case_tac x) kaliszyk@35222: apply(simp_all) kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: lemma map_id[id_simps]: kaliszyk@35222: shows "map id = id" nipkow@39302: apply(simp add: fun_eq_iff) kaliszyk@35222: apply(rule allI) kaliszyk@35222: apply(induct_tac x) kaliszyk@35222: apply(simp_all) kaliszyk@35222: done kaliszyk@35222: kaliszyk@37492: lemma list_all2_reflp: kaliszyk@37492: shows "equivp R \ list_all2 R xs xs" kaliszyk@37492: by (induct xs, simp_all add: equivp_reflp) kaliszyk@35222: kaliszyk@37492: lemma list_all2_symp: kaliszyk@35222: assumes a: "equivp R" kaliszyk@37492: and b: "list_all2 R xs ys" kaliszyk@37492: shows "list_all2 R ys xs" kaliszyk@37492: using list_all2_lengthD[OF b] b kaliszyk@37492: apply(induct xs ys rule: list_induct2) kaliszyk@35222: apply(simp_all) kaliszyk@35222: apply(rule equivp_symp[OF a]) kaliszyk@35222: apply(simp) kaliszyk@35222: done kaliszyk@35222: kaliszyk@37492: thm list_induct3 kaliszyk@37492: kaliszyk@37492: lemma list_all2_transp: kaliszyk@35222: assumes a: "equivp R" kaliszyk@37492: and b: "list_all2 R xs1 xs2" kaliszyk@37492: and c: "list_all2 R xs2 xs3" kaliszyk@37492: shows "list_all2 R xs1 xs3" kaliszyk@37492: using list_all2_lengthD[OF b] list_all2_lengthD[OF c] b c kaliszyk@37492: apply(induct rule: list_induct3) kaliszyk@37492: apply(simp_all) kaliszyk@37492: apply(auto intro: equivp_transp[OF a]) kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: lemma list_equivp[quot_equiv]: kaliszyk@35222: assumes a: "equivp R" kaliszyk@37492: shows "equivp (list_all2 R)" kaliszyk@37492: apply (intro equivpI) kaliszyk@35222: unfolding reflp_def symp_def transp_def kaliszyk@37492: apply(simp add: list_all2_reflp[OF a]) kaliszyk@37492: apply(blast intro: list_all2_symp[OF a]) kaliszyk@37492: apply(blast intro: list_all2_transp[OF a]) kaliszyk@35222: done kaliszyk@35222: kaliszyk@37492: lemma list_all2_rel: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@37492: shows "list_all2 R r s = (list_all2 R r r \ list_all2 R s s \ (map Abs r = map Abs s))" kaliszyk@35222: apply(induct r s rule: list_induct2') kaliszyk@35222: apply(simp_all) kaliszyk@35222: using Quotient_rel[OF q] kaliszyk@35222: apply(metis) kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: lemma list_quotient[quot_thm]: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@37492: shows "Quotient (list_all2 R) (map Abs) (map Rep)" kaliszyk@35222: unfolding Quotient_def kaliszyk@35222: apply(subst split_list_all) kaliszyk@35222: apply(simp add: Quotient_abs_rep[OF q] abs_o_rep[OF q] map_id) kaliszyk@37492: apply(intro conjI allI) kaliszyk@35222: apply(induct_tac a) kaliszyk@37492: apply(simp_all add: Quotient_rep_reflp[OF q]) kaliszyk@37492: apply(rule list_all2_rel[OF q]) kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: lemma cons_prs_aux: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@35222: shows "(map Abs) ((Rep h) # (map Rep t)) = h # t" kaliszyk@35222: by (induct t) (simp_all add: Quotient_abs_rep[OF q]) kaliszyk@35222: kaliszyk@35222: lemma cons_prs[quot_preserve]: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@35222: shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)" nipkow@39302: by (simp only: fun_eq_iff fun_map_def cons_prs_aux[OF q]) kaliszyk@35222: (simp) kaliszyk@35222: kaliszyk@35222: lemma cons_rsp[quot_respect]: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@37492: shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)" kaliszyk@35222: by (auto) kaliszyk@35222: kaliszyk@35222: lemma nil_prs[quot_preserve]: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@35222: shows "map Abs [] = []" kaliszyk@35222: by simp kaliszyk@35222: kaliszyk@35222: lemma nil_rsp[quot_respect]: kaliszyk@35222: assumes q: "Quotient R Abs Rep" kaliszyk@37492: shows "list_all2 R [] []" kaliszyk@35222: by simp kaliszyk@35222: kaliszyk@35222: lemma map_prs_aux: kaliszyk@35222: assumes a: "Quotient R1 abs1 rep1" kaliszyk@35222: and b: "Quotient R2 abs2 rep2" kaliszyk@35222: shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l" kaliszyk@35222: by (induct l) kaliszyk@35222: (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b]) kaliszyk@35222: kaliszyk@35222: lemma map_prs[quot_preserve]: kaliszyk@35222: assumes a: "Quotient R1 abs1 rep1" kaliszyk@35222: and b: "Quotient R2 abs2 rep2" kaliszyk@35222: shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map" kaliszyk@36216: and "((abs1 ---> id) ---> map rep1 ---> id) map = map" nipkow@39302: by (simp_all only: fun_eq_iff fun_map_def map_prs_aux[OF a b]) kaliszyk@36216: (simp_all add: Quotient_abs_rep[OF a]) kaliszyk@35222: kaliszyk@35222: lemma map_rsp[quot_respect]: kaliszyk@35222: assumes q1: "Quotient R1 Abs1 Rep1" kaliszyk@35222: and q2: "Quotient R2 Abs2 Rep2" kaliszyk@37492: shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map" kaliszyk@37492: and "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map" kaliszyk@36216: apply simp_all kaliszyk@36216: apply(rule_tac [!] allI)+ kaliszyk@36216: apply(rule_tac [!] impI) kaliszyk@36216: apply(rule_tac [!] allI)+ kaliszyk@36216: apply (induct_tac [!] xa ya rule: list_induct2') kaliszyk@35222: apply simp_all kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: lemma foldr_prs_aux: kaliszyk@35222: assumes a: "Quotient R1 abs1 rep1" kaliszyk@35222: and b: "Quotient R2 abs2 rep2" kaliszyk@35222: shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e" kaliszyk@35222: by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b]) kaliszyk@35222: kaliszyk@35222: lemma foldr_prs[quot_preserve]: kaliszyk@35222: assumes a: "Quotient R1 abs1 rep1" kaliszyk@35222: and b: "Quotient R2 abs2 rep2" kaliszyk@35222: shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr" nipkow@39302: by (simp only: fun_eq_iff fun_map_def foldr_prs_aux[OF a b]) kaliszyk@35222: (simp) kaliszyk@35222: kaliszyk@35222: lemma foldl_prs_aux: kaliszyk@35222: assumes a: "Quotient R1 abs1 rep1" kaliszyk@35222: and b: "Quotient R2 abs2 rep2" kaliszyk@35222: shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l" kaliszyk@35222: by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b]) kaliszyk@35222: kaliszyk@35222: kaliszyk@35222: lemma foldl_prs[quot_preserve]: kaliszyk@35222: assumes a: "Quotient R1 abs1 rep1" kaliszyk@35222: and b: "Quotient R2 abs2 rep2" kaliszyk@35222: shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl" nipkow@39302: by (simp only: fun_eq_iff fun_map_def foldl_prs_aux[OF a b]) kaliszyk@35222: (simp) kaliszyk@35222: kaliszyk@37492: lemma list_all2_empty: kaliszyk@37492: shows "list_all2 R [] b \ length b = 0" kaliszyk@35222: by (induct b) (simp_all) kaliszyk@35222: kaliszyk@35222: (* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *) kaliszyk@35222: lemma foldl_rsp[quot_respect]: kaliszyk@35222: assumes q1: "Quotient R1 Abs1 Rep1" kaliszyk@35222: and q2: "Quotient R2 Abs2 Rep2" kaliszyk@37492: shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl" kaliszyk@35222: apply(auto) kaliszyk@37492: apply (subgoal_tac "R1 xa ya \ list_all2 R2 xb yb \ R1 (foldl x xa xb) (foldl y ya yb)") kaliszyk@35222: apply simp kaliszyk@35222: apply (rule_tac x="xa" in spec) kaliszyk@35222: apply (rule_tac x="ya" in spec) kaliszyk@35222: apply (rule_tac xs="xb" and ys="yb" in list_induct2) kaliszyk@37492: apply (rule list_all2_lengthD) kaliszyk@35222: apply (simp_all) kaliszyk@35222: done kaliszyk@35222: kaliszyk@35222: lemma foldr_rsp[quot_respect]: kaliszyk@35222: assumes q1: "Quotient R1 Abs1 Rep1" kaliszyk@35222: and q2: "Quotient R2 Abs2 Rep2" kaliszyk@37492: shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr" kaliszyk@35222: apply auto kaliszyk@37492: apply(subgoal_tac "R2 xb yb \ list_all2 R1 xa ya \ R2 (foldr x xa xb) (foldr y ya yb)") kaliszyk@35222: apply simp kaliszyk@35222: apply (rule_tac xs="xa" and ys="ya" in list_induct2) kaliszyk@37492: apply (rule list_all2_lengthD) kaliszyk@35222: apply (simp_all) kaliszyk@35222: done kaliszyk@35222: kaliszyk@37492: lemma list_all2_rsp: kaliszyk@36154: assumes r: "\x y. R x y \ (\a b. R a b \ S x a = T y b)" kaliszyk@37492: and l1: "list_all2 R x y" kaliszyk@37492: and l2: "list_all2 R a b" kaliszyk@37492: shows "list_all2 S x a = list_all2 T y b" kaliszyk@36154: proof - kaliszyk@37492: have a: "length y = length x" by (rule list_all2_lengthD[OF l1, symmetric]) kaliszyk@37492: have c: "length a = length b" by (rule list_all2_lengthD[OF l2]) kaliszyk@36154: show ?thesis proof (cases "length x = length a") kaliszyk@36154: case True kaliszyk@36154: have b: "length x = length a" by fact kaliszyk@36154: show ?thesis using a b c r l1 l2 proof (induct rule: list_induct4) kaliszyk@36154: case Nil kaliszyk@36154: show ?case using assms by simp kaliszyk@36154: next kaliszyk@36154: case (Cons h t) kaliszyk@36154: then show ?case by auto kaliszyk@36154: qed kaliszyk@36154: next kaliszyk@36154: case False kaliszyk@36154: have d: "length x \ length a" by fact kaliszyk@37492: then have e: "\list_all2 S x a" using list_all2_lengthD by auto kaliszyk@36154: have "length y \ length b" using d a c by simp kaliszyk@37492: then have "\list_all2 T y b" using list_all2_lengthD by auto kaliszyk@36154: then show ?thesis using e by simp kaliszyk@36154: qed kaliszyk@36154: qed kaliszyk@36154: kaliszyk@36154: lemma[quot_respect]: kaliszyk@37492: "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2" kaliszyk@37492: by (simp add: list_all2_rsp) kaliszyk@36154: kaliszyk@36154: lemma[quot_preserve]: kaliszyk@36154: assumes a: "Quotient R abs1 rep1" kaliszyk@37492: shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2" nipkow@39302: apply (simp add: fun_eq_iff) kaliszyk@36154: apply clarify kaliszyk@36154: apply (induct_tac xa xb rule: list_induct2') kaliszyk@36154: apply (simp_all add: Quotient_abs_rep[OF a]) kaliszyk@36154: done kaliszyk@36154: kaliszyk@36154: lemma[quot_preserve]: kaliszyk@36154: assumes a: "Quotient R abs1 rep1" kaliszyk@37492: shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)" kaliszyk@36154: by (induct l m rule: list_induct2') (simp_all add: Quotient_rel_rep[OF a]) kaliszyk@36154: kaliszyk@37492: lemma list_all2_eq[id_simps]: kaliszyk@37492: shows "(list_all2 (op =)) = (op =)" nipkow@39302: unfolding fun_eq_iff kaliszyk@35222: apply(rule allI)+ kaliszyk@35222: apply(induct_tac x xa rule: list_induct2') kaliszyk@35222: apply(simp_all) kaliszyk@35222: done kaliszyk@35222: kaliszyk@37492: lemma list_all2_find_element: kaliszyk@36276: assumes a: "x \ set a" kaliszyk@37492: and b: "list_all2 R a b" kaliszyk@36276: shows "\y. (y \ set b \ R x y)" kaliszyk@36276: proof - kaliszyk@37492: have "length a = length b" using b by (rule list_all2_lengthD) kaliszyk@36276: then show ?thesis using a b by (induct a b rule: list_induct2) auto kaliszyk@36276: qed kaliszyk@36276: kaliszyk@37492: lemma list_all2_refl: kaliszyk@35222: assumes a: "\x y. R x y = (R x = R y)" kaliszyk@37492: shows "list_all2 R x x" kaliszyk@35222: by (induct x) (auto simp add: a) kaliszyk@35222: kaliszyk@35222: end