# Theory HOL-Library.Quotient_Product

```(*  Title:      HOL/Library/Quotient_Product.thy
Author:     Cezary Kaliszyk and Christian Urban
*)

section ‹Quotient infrastructure for the product type›

theory Quotient_Product
imports Quotient_Syntax
begin

subsection ‹Rules for the Quotient package›

lemma map_prod_id [id_simps]:
shows "map_prod id id = id"

lemma rel_prod_eq [id_simps]:
shows "rel_prod (=) (=) = (=)"

lemma prod_equivp [quot_equiv]:
assumes "equivp R1"
assumes "equivp R2"
shows "equivp (rel_prod 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 (rel_prod R1 R2) (map_prod Abs1 Abs2) (map_prod Rep1 Rep2)"
apply (rule Quotient3I)
apply (simp add: map_prod.compositionality comp_def map_prod.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)]
done

declare [[mapQ3 prod = (rel_prod, prod_quotient)]]

lemma Pair_rsp [quot_respect]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(R1 ===> R2 ===> rel_prod R1 R2) Pair Pair"
by (rule Pair_transfer)

lemma Pair_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> Rep2 ---> (map_prod Abs1 Abs2)) Pair = Pair"
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 "(rel_prod 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_prod 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 "(rel_prod 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_prod Rep1 Rep2 ---> Abs2) snd = snd"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q2])

lemma case_prod_rsp [quot_respect]:
shows "((R1 ===> R2 ===> (=)) ===> (rel_prod R1 R2) ===> (=)) case_prod case_prod"
by (rule case_prod_transfer)

lemma split_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 R2 Abs2 Rep2"
shows "(((Abs1 ---> Abs2 ---> id) ---> map_prod Rep1 Rep2 ---> id) case_prod) = case_prod"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])

lemma [quot_respect]:
shows "((R2 ===> R2 ===> (=)) ===> (R1 ===> R1 ===> (=)) ===>
rel_prod R2 R1 ===> rel_prod R2 R1 ===> (=)) rel_prod rel_prod"
by (rule prod.rel_transfer)

lemma [quot_preserve]:
assumes q1: "Quotient3 R1 abs1 rep1"
and     q2: "Quotient3 R2 abs2 rep2"
shows "((abs1 ---> abs1 ---> id) ---> (abs2 ---> abs2 ---> id) --->
map_prod rep1 rep2 ---> map_prod rep1 rep2 ---> id) rel_prod = rel_prod"
by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])

lemma [quot_preserve]:
shows"(rel_prod ((rep1 ---> rep1 ---> id) R1) ((rep2 ---> rep2 ---> id) R2)
(l1, l2) (r1, r2)) = (R1 (rep1 l1) (rep1 r1) ∧ R2 (rep2 l2) (rep2 r2))"
by simp

declare prod.inject[quot_preserve]

end
```