(* Title: HOL/Library/Quotient_List.thy
Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
*)
header {* Quotient infrastructure for the list type *}
theory Quotient_List
imports Main Quotient_Set
begin
subsection {* Relator for list type *}
lemma map_id [id_simps]:
"map id = id"
by (fact List.map.id)
lemma list_all2_eq [id_simps, relator_eq]:
"list_all2 (op =) = (op =)"
proof (rule ext)+
fix xs ys
show "list_all2 (op =) xs ys \<longleftrightarrow> xs = ys"
by (induct xs ys rule: list_induct2') simp_all
qed
lemma list_all2_OO: "list_all2 (A OO B) = list_all2 A OO list_all2 B"
proof (intro ext iffI)
fix xs ys
assume "list_all2 (A OO B) xs ys"
thus "(list_all2 A OO list_all2 B) xs ys"
unfolding OO_def
by (induct, simp, simp add: list_all2_Cons1 list_all2_Cons2, fast)
next
fix xs ys
assume "(list_all2 A OO list_all2 B) xs ys"
then obtain zs where "list_all2 A xs zs" and "list_all2 B zs ys" ..
thus "list_all2 (A OO B) xs ys"
by (induct arbitrary: ys, simp, clarsimp simp add: list_all2_Cons1, fast)
qed
lemma list_reflp:
assumes "reflp R"
shows "reflp (list_all2 R)"
proof (rule reflpI)
from assms have *: "\<And>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 *: "\<And>xs ys. R xs ys \<Longrightarrow> 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 *: "\<And>xs ys zs. R xs ys \<Longrightarrow> R ys zs \<Longrightarrow> 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 \<Longrightarrow> equivp (list_all2 R)"
by (blast intro: equivpI list_reflp list_symp list_transp elim: equivpE)
lemma right_total_list_all2 [transfer_rule]:
"right_total R \<Longrightarrow> right_total (list_all2 R)"
unfolding right_total_def
by (rule allI, induct_tac y, simp, simp add: list_all2_Cons2)
lemma right_unique_list_all2 [transfer_rule]:
"right_unique R \<Longrightarrow> right_unique (list_all2 R)"
unfolding right_unique_def
apply (rule allI, rename_tac xs, induct_tac xs)
apply (auto simp add: list_all2_Cons1)
done
lemma bi_total_list_all2 [transfer_rule]:
"bi_total A \<Longrightarrow> bi_total (list_all2 A)"
unfolding bi_total_def
apply safe
apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
apply (rename_tac ys, induct_tac ys, simp, simp add: list_all2_Cons2)
done
lemma bi_unique_list_all2 [transfer_rule]:
"bi_unique A \<Longrightarrow> bi_unique (list_all2 A)"
unfolding bi_unique_def
apply (rule conjI)
apply (rule allI, rename_tac xs, induct_tac xs)
apply (simp, force simp add: list_all2_Cons1)
apply (subst (2) all_comm, subst (1) all_comm)
apply (rule allI, rename_tac xs, induct_tac xs)
apply (simp, force simp add: list_all2_Cons2)
done
subsection {* Transfer rules for transfer package *}
lemma Nil_transfer [transfer_rule]: "(list_all2 A) [] []"
by simp
lemma Cons_transfer [transfer_rule]:
"(A ===> list_all2 A ===> list_all2 A) Cons Cons"
unfolding fun_rel_def by simp
lemma list_case_transfer [transfer_rule]:
"(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B)
list_case list_case"
unfolding fun_rel_def by (simp split: list.split)
lemma list_rec_transfer [transfer_rule]:
"(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B)
list_rec list_rec"
unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all)
lemma map_transfer [transfer_rule]:
"((A ===> B) ===> list_all2 A ===> list_all2 B) map map"
unfolding List.map_def by transfer_prover
lemma append_transfer [transfer_rule]:
"(list_all2 A ===> list_all2 A ===> list_all2 A) append append"
unfolding List.append_def by transfer_prover
lemma filter_transfer [transfer_rule]:
"((A ===> op =) ===> list_all2 A ===> list_all2 A) filter filter"
unfolding List.filter_def by transfer_prover
lemma foldr_transfer [transfer_rule]:
"((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr"
unfolding List.foldr_def by transfer_prover
lemma foldl_transfer [transfer_rule]:
"((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl"
unfolding List.foldl_def by transfer_prover
lemma concat_transfer [transfer_rule]:
"(list_all2 (list_all2 A) ===> list_all2 A) concat concat"
unfolding List.concat_def by transfer_prover
lemma drop_transfer [transfer_rule]:
"(op = ===> list_all2 A ===> list_all2 A) drop drop"
unfolding List.drop_def by transfer_prover
lemma take_transfer [transfer_rule]:
"(op = ===> list_all2 A ===> list_all2 A) take take"
unfolding List.take_def by transfer_prover
lemma length_transfer [transfer_rule]:
"(list_all2 A ===> op =) length length"
unfolding list_size_overloaded_def by transfer_prover
lemma list_all_transfer [transfer_rule]:
"((A ===> op =) ===> list_all2 A ===> op =) list_all list_all"
unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all)
lemma list_all2_transfer [transfer_rule]:
"((A ===> B ===> op =) ===> list_all2 A ===> list_all2 B ===> op =)
list_all2 list_all2"
apply (rule fun_relI, rule fun_relI, erule list_all2_induct)
apply (rule fun_relI, erule list_all2_induct, simp, simp)
apply (rule fun_relI, erule list_all2_induct [of B])
apply (simp, simp add: fun_rel_def)
done
lemma set_transfer [transfer_rule]:
"(list_all2 A ===> set_rel A) set set"
unfolding set_def by transfer_prover
subsection {* Setup for lifting package *}
lemma Quotient_list[quot_map]:
assumes "Quotient R Abs Rep T"
shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)"
proof (unfold Quotient_alt_def, intro conjI allI impI)
from assms have 1: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
unfolding Quotient_alt_def by simp
fix xs ys assume "list_all2 T xs ys" thus "map Abs xs = ys"
by (induct, simp, simp add: 1)
next
from assms have 2: "\<And>x. T (Rep x) x"
unfolding Quotient_alt_def by simp
fix xs show "list_all2 T (map Rep xs) xs"
by (induct xs, simp, simp add: 2)
next
from assms have 3: "\<And>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y"
unfolding Quotient_alt_def by simp
fix xs ys show "list_all2 R xs ys \<longleftrightarrow> list_all2 T xs (map Abs xs) \<and>
list_all2 T ys (map Abs ys) \<and> map Abs xs = map Abs ys"
by (induct xs ys rule: list_induct2', simp_all, metis 3)
qed
lemma list_invariant_commute [invariant_commute]:
"list_all2 (Lifting.invariant P) = Lifting.invariant (list_all P)"
apply (simp add: fun_eq_iff list_all2_def list_all_iff Lifting.invariant_def Ball_def)
apply (intro allI)
apply (induct_tac rule: list_induct2')
apply simp_all
apply metis
done
subsection {* Rules for quotient package *}
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 "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
then show "\<And>xs. map Abs (map Rep xs) = xs" by (simp add: comp_def)
next
from assms have "\<And>x y. R (Rep x) (Rep y) \<longleftrightarrow> x = y" by (rule Quotient3_rel_rep)
then show "\<And>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 "\<And>x y. R x x \<and> R y y \<and> Abs x = Abs y \<longleftrightarrow> R x y" by (rule Quotient3_rel)
then show "list_all2 R xs ys \<longleftrightarrow> list_all2 R xs xs \<and> list_all2 R ys ys \<and> 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)) (op #) = (op #)"
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) (op #) (op #)"
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 ===> op =) ===> (list_all2 R1) ===> op =) map map"
unfolding list_all2_eq [symmetric] by (rule 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])
(* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
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: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> 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 ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
by (rule list_all2_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 \<in> set a"
and b: "list_all2 R a b"
shows "\<exists>y. (y \<in> set b \<and> R x y)"
using b a by induct auto
lemma list_all2_refl:
assumes a: "\<And>x y. R x y = (R x = R y)"
shows "list_all2 R x x"
by (induct x) (auto simp add: a)
end