(* Title: HOL/Library/Quotient3_Product.thy
Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the product type *}
theory Quotient_Product
imports Main Quotient_Syntax
begin
definition
prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
where
"prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
lemma prod_rel_apply [simp]:
"prod_rel R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
by (simp add: prod_rel_def)
lemma map_pair_id [id_simps]:
shows "map_pair id id = id"
by (simp add: fun_eq_iff)
lemma prod_rel_eq [id_simps]:
shows "prod_rel (op =) (op =) = (op =)"
by (simp add: fun_eq_iff)
lemma prod_equivp [quot_equiv]:
assumes "equivp R1"
assumes "equivp R2"
shows "equivp (prod_rel R1 R2)"
using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE)
lemma prod_quotient [quot_thm]:
assumes "Quotient3 R1 Abs1 Rep1"
assumes "Quotient3 R2 Abs2 Rep2"
shows "Quotient3 (prod_rel R1 R2) (map_pair Abs1 Abs2) (map_pair Rep1 Rep2)"
apply (rule Quotient3I)
apply (simp add: map_pair.compositionality comp_def map_pair.identity
Quotient3_abs_rep [OF assms(1)] Quotient3_abs_rep [OF assms(2)])
apply (simp add: split_paired_all Quotient3_rel_rep [OF assms(1)] Quotient3_rel_rep [OF assms(2)])
using Quotient3_rel [OF assms(1)] Quotient3_rel [OF assms(2)]
apply (auto simp add: split_paired_all)
done
declare [[mapQ3 prod = (prod_rel, prod_quotient)]]
lemma Pair_rsp [quot_respect]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
by (auto simp add: prod_rel_def)
lemma Pair_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> Rep2 ---> (map_pair Abs1 Abs2)) Pair = Pair"
apply(simp add: fun_eq_iff)
apply(simp add: Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
done
lemma fst_rsp [quot_respect]:
assumes "Quotient3 R1 Abs1 Rep1"
assumes "Quotient3 R2 Abs2 Rep2"
shows "(prod_rel R1 R2 ===> R1) fst fst"
by auto
lemma fst_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(map_pair Rep1 Rep2 ---> Abs1) fst = fst"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1])
lemma snd_rsp [quot_respect]:
assumes "Quotient3 R1 Abs1 Rep1"
assumes "Quotient3 R2 Abs2 Rep2"
shows "(prod_rel R1 R2 ===> R2) snd snd"
by auto
lemma snd_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(map_pair Rep1 Rep2 ---> Abs2) snd = snd"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q2])
lemma split_rsp [quot_respect]:
shows "((R1 ===> R2 ===> (op =)) ===> (prod_rel R1 R2) ===> (op =)) split split"
unfolding prod_rel_def fun_rel_def
by auto
lemma split_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and q2: "Quotient3 R2 Abs2 Rep2"
shows "(((Abs1 ---> Abs2 ---> id) ---> map_pair Rep1 Rep2 ---> id) split) = split"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
lemma [quot_respect]:
shows "((R2 ===> R2 ===> op =) ===> (R1 ===> R1 ===> op =) ===>
prod_rel R2 R1 ===> prod_rel R2 R1 ===> op =) prod_rel prod_rel"
by (auto simp add: fun_rel_def)
lemma [quot_preserve]:
assumes q1: "Quotient3 R1 abs1 rep1"
and q2: "Quotient3 R2 abs2 rep2"
shows "((abs1 ---> abs1 ---> id) ---> (abs2 ---> abs2 ---> id) --->
map_pair rep1 rep2 ---> map_pair rep1 rep2 ---> id) prod_rel = prod_rel"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
lemma [quot_preserve]:
shows"(prod_rel ((rep1 ---> rep1 ---> id) R1) ((rep2 ---> rep2 ---> id) R2)
(l1, l2) (r1, r2)) = (R1 (rep1 l1) (rep1 r1) \<and> R2 (rep2 l2) (rep2 r2))"
by simp
declare Pair_eq[quot_preserve]
end