src/HOL/Library/Quotient_Product.thy
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(*  Title:      HOL/Library/Quotient_Product.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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header {* Quotient infrastructure for the product type *}
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theory Quotient_Product
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imports Main Quotient_Syntax
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begin
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subsection {* Rules for the Quotient package *}
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lemma map_prod_id [id_simps]:
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  shows "map_prod id id = id"
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  by (simp add: fun_eq_iff)
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lemma rel_prod_eq [id_simps]:
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  shows "rel_prod (op =) (op =) = (op =)"
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  by (simp add: fun_eq_iff)
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lemma prod_equivp [quot_equiv]:
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  assumes "equivp R1"
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  assumes "equivp R2"
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  shows "equivp (rel_prod R1 R2)"
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  using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE)
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lemma prod_quotient [quot_thm]:
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  assumes "Quotient3 R1 Abs1 Rep1"
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  assumes "Quotient3 R2 Abs2 Rep2"
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  shows "Quotient3 (rel_prod R1 R2) (map_prod Abs1 Abs2) (map_prod Rep1 Rep2)"
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  apply (rule Quotient3I)
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  apply (simp add: map_prod.compositionality comp_def map_prod.identity
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     Quotient3_abs_rep [OF assms(1)] Quotient3_abs_rep [OF assms(2)])
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  apply (simp add: split_paired_all Quotient3_rel_rep [OF assms(1)] Quotient3_rel_rep [OF assms(2)])
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  using Quotient3_rel [OF assms(1)] Quotient3_rel [OF assms(2)]
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  apply (auto simp add: split_paired_all)
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  done
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declare [[mapQ3 prod = (rel_prod, prod_quotient)]]
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lemma Pair_rsp [quot_respect]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(R1 ===> R2 ===> rel_prod R1 R2) Pair Pair"
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  by (rule Pair_transfer)
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lemma Pair_prs [quot_preserve]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(Rep1 ---> Rep2 ---> (map_prod Abs1 Abs2)) Pair = Pair"
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  apply(simp add: fun_eq_iff)
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  apply(simp add: Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
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  done
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lemma fst_rsp [quot_respect]:
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  assumes "Quotient3 R1 Abs1 Rep1"
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  assumes "Quotient3 R2 Abs2 Rep2"
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  shows "(rel_prod R1 R2 ===> R1) fst fst"
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  by auto
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lemma fst_prs [quot_preserve]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(map_prod Rep1 Rep2 ---> Abs1) fst = fst"
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  by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1])
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lemma snd_rsp [quot_respect]:
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  assumes "Quotient3 R1 Abs1 Rep1"
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  assumes "Quotient3 R2 Abs2 Rep2"
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  shows "(rel_prod R1 R2 ===> R2) snd snd"
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  by auto
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lemma snd_prs [quot_preserve]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(map_prod Rep1 Rep2 ---> Abs2) snd = snd"
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  by (simp add: fun_eq_iff Quotient3_abs_rep[OF q2])
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lemma split_rsp [quot_respect]:
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  shows "((R1 ===> R2 ===> (op =)) ===> (rel_prod R1 R2) ===> (op =)) split split"
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  by (rule case_prod_transfer)
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lemma split_prs [quot_preserve]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(((Abs1 ---> Abs2 ---> id) ---> map_prod Rep1 Rep2 ---> id) split) = split"
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  by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
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lemma [quot_respect]:
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  shows "((R2 ===> R2 ===> op =) ===> (R1 ===> R1 ===> op =) ===>
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  rel_prod R2 R1 ===> rel_prod R2 R1 ===> op =) rel_prod rel_prod"
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  by (rule rel_prod_transfer)
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lemma [quot_preserve]:
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  assumes q1: "Quotient3 R1 abs1 rep1"
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  and     q2: "Quotient3 R2 abs2 rep2"
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  shows "((abs1 ---> abs1 ---> id) ---> (abs2 ---> abs2 ---> id) --->
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  map_prod rep1 rep2 ---> map_prod rep1 rep2 ---> id) rel_prod = rel_prod"
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  by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
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lemma [quot_preserve]:
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  shows"(rel_prod ((rep1 ---> rep1 ---> id) R1) ((rep2 ---> rep2 ---> id) R2)
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  (l1, l2) (r1, r2)) = (R1 (rep1 l1) (rep1 r1) \<and> R2 (rep2 l2) (rep2 r2))"
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  by simp
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declare Pair_eq[quot_preserve]
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end