(* Title: HOL/Library/Quotient_List.thy Author: Cezary Kaliszyk and Christian Urban *) section ‹Quotient infrastructure for the list type› theory Quotient_List imports Quotient_Set Quotient_Product Quotient_Option begin subsection ‹Rules for the Quotient package› lemma map_id [id_simps]: "map id = id" by (fact List.map.id) lemma list_all2_eq [id_simps]: "list_all2 (=) = (=)" proof (rule ext)+ fix xs ys show "list_all2 (=) xs ys ⟷ xs = ys" by (induct xs ys rule: list_induct2') simp_all qed lemma reflp_list_all2: assumes "reflp R" shows "reflp (list_all2 R)" proof (rule reflpI) from assms have *: "⋀xs. R xs xs" by (rule reflpE) fix xs show "list_all2 R xs xs" by (induct xs) (simp_all add: *) qed lemma list_symp: assumes "symp R" shows "symp (list_all2 R)" proof (rule sympI) from assms have *: "⋀xs ys. R xs ys ⟹ R ys xs" by (rule sympE) fix xs ys assume "list_all2 R xs ys" then show "list_all2 R ys xs" by (induct xs ys rule: list_induct2') (simp_all add: *) qed lemma list_transp: assumes "transp R" shows "transp (list_all2 R)" proof (rule transpI) from assms have *: "⋀xs ys zs. R xs ys ⟹ R ys zs ⟹ R xs zs" by (rule transpE) fix xs ys zs assume "list_all2 R xs ys" and "list_all2 R ys zs" then show "list_all2 R xs zs" by (induct arbitrary: zs) (auto simp: list_all2_Cons1 intro: *) qed lemma list_equivp [quot_equiv]: "equivp R ⟹ equivp (list_all2 R)" by (blast intro: equivpI reflp_list_all2 list_symp list_transp elim: equivpE) lemma list_quotient3 [quot_thm]: assumes "Quotient3 R Abs Rep" shows "Quotient3 (list_all2 R) (map Abs) (map Rep)" proof (rule Quotient3I) from assms have "⋀x. Abs (Rep x) = x" by (rule Quotient3_abs_rep) then show "⋀xs. map Abs (map Rep xs) = xs" by (simp add: comp_def) next from assms have "⋀x y. R (Rep x) (Rep y) ⟷ x = y" by (rule Quotient3_rel_rep) then show "⋀xs. list_all2 R (map Rep xs) (map Rep xs)" by (simp add: list_all2_map1 list_all2_map2 list_all2_eq) next fix xs ys from assms have "⋀x y. R x x ∧ R y y ∧ Abs x = Abs y ⟷ R x y" by (rule Quotient3_rel) then show "list_all2 R xs ys ⟷ list_all2 R xs xs ∧ list_all2 R ys ys ∧ map Abs xs = map Abs ys" by (induct xs ys rule: list_induct2') auto qed declare [[mapQ3 list = (list_all2, list_quotient3)]] lemma cons_prs [quot_preserve]: assumes q: "Quotient3 R Abs Rep" shows "(Rep ---> (map Rep) ---> (map Abs)) (#) = (#)" by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q]) lemma cons_rsp [quot_respect]: assumes q: "Quotient3 R Abs Rep" shows "(R ===> list_all2 R ===> list_all2 R) (#) (#)" by auto lemma nil_prs [quot_preserve]: assumes q: "Quotient3 R Abs Rep" shows "map Abs [] = []" by simp lemma nil_rsp [quot_respect]: assumes q: "Quotient3 R Abs Rep" shows "list_all2 R [] []" by simp lemma map_prs_aux: assumes a: "Quotient3 R1 abs1 rep1" and b: "Quotient3 R2 abs2 rep2" shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l" by (induct l) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b]) lemma map_prs [quot_preserve]: assumes a: "Quotient3 R1 abs1 rep1" and b: "Quotient3 R2 abs2 rep2" shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map" and "((abs1 ---> id) ---> map rep1 ---> id) map = map" by (simp_all only: fun_eq_iff map_prs_aux[OF a b] comp_def) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b]) lemma map_rsp [quot_respect]: assumes q1: "Quotient3 R1 Abs1 Rep1" and q2: "Quotient3 R2 Abs2 Rep2" shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map" and "((R1 ===> (=)) ===> (list_all2 R1) ===> (=)) map map" unfolding list_all2_eq [symmetric] by (rule list.map_transfer)+ lemma foldr_prs_aux: assumes a: "Quotient3 R1 abs1 rep1" and b: "Quotient3 R2 abs2 rep2" shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e" by (induct l) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b]) lemma foldr_prs [quot_preserve]: assumes a: "Quotient3 R1 abs1 rep1" and b: "Quotient3 R2 abs2 rep2" shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr" apply (simp add: fun_eq_iff) by (simp only: fun_eq_iff foldr_prs_aux[OF a b]) (simp) lemma foldl_prs_aux: assumes a: "Quotient3 R1 abs1 rep1" and b: "Quotient3 R2 abs2 rep2" shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l" by (induct l arbitrary:e) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b]) lemma foldl_prs [quot_preserve]: assumes a: "Quotient3 R1 abs1 rep1" and b: "Quotient3 R2 abs2 rep2" shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl" by (simp add: fun_eq_iff foldl_prs_aux [OF a b]) lemma foldl_rsp[quot_respect]: assumes q1: "Quotient3 R1 Abs1 Rep1" and q2: "Quotient3 R2 Abs2 Rep2" shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl" by (rule foldl_transfer) lemma foldr_rsp[quot_respect]: assumes q1: "Quotient3 R1 Abs1 Rep1" and q2: "Quotient3 R2 Abs2 Rep2" shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr" by (rule foldr_transfer) lemma list_all2_rsp: assumes r: "∀x y. R x y ⟶ (∀a b. R a b ⟶ S x a = T y b)" and l1: "list_all2 R x y" and l2: "list_all2 R a b" shows "list_all2 S x a = list_all2 T y b" using l1 l2 by (induct arbitrary: a b rule: list_all2_induct, auto simp: list_all2_Cons1 list_all2_Cons2 r) lemma [quot_respect]: "((R ===> R ===> (=)) ===> list_all2 R ===> list_all2 R ===> (=)) list_all2 list_all2" by (rule list.rel_transfer) lemma [quot_preserve]: assumes a: "Quotient3 R abs1 rep1" shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2" apply (simp add: fun_eq_iff) apply clarify apply (induct_tac xa xb rule: list_induct2') apply (simp_all add: Quotient3_abs_rep[OF a]) done lemma [quot_preserve]: assumes a: "Quotient3 R abs1 rep1" shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)" by (induct l m rule: list_induct2') (simp_all add: Quotient3_rel_rep[OF a]) lemma list_all2_find_element: assumes a: "x ∈ set a" and b: "list_all2 R a b" shows "∃y. (y ∈ set b ∧ R x y)" using b a by induct auto lemma list_all2_refl: assumes a: "⋀x y. R x y = (R x = R y)" shows "list_all2 R x x" by (induct x) (auto simp add: a) end