(*  Title:      Quotient_Product.thy+
−
    Author:     Cezary Kaliszyk and Christian Urban+
−
*)+
−
theory Quotient_Product+
−
imports Quotient Quotient_Syntax+
−
begin+
−
+
−
section {* Quotient infrastructure for the product type. *}+
−
+
−
fun+
−
  prod_rel+
−
where+
−
  "prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"+
−
+
−
declare [[map * = (prod_fun, prod_rel)]]+
−
+
−
+
−
lemma prod_equivp[quot_equiv]:+
−
  assumes a: "equivp R1"+
−
  assumes b: "equivp R2"+
−
  shows "equivp (prod_rel R1 R2)"+
−
  apply(rule equivpI)+
−
  unfolding reflp_def symp_def transp_def+
−
  apply(simp_all add: split_paired_all)+
−
  apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])+
−
  apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])+
−
  apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])+
−
  done+
−
+
−
lemma prod_quotient[quot_thm]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  assumes q2: "Quotient R2 Abs2 Rep2"+
−
  shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"+
−
  unfolding Quotient_def+
−
  apply(simp add: split_paired_all)+
−
  apply(simp add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])+
−
  apply(simp add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])+
−
  using q1 q2+
−
  unfolding Quotient_def+
−
  apply(blast)+
−
  done+
−
+
−
lemma Pair_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  assumes q2: "Quotient R2 Abs2 Rep2"+
−
  shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"+
−
  by simp+
−
+
−
lemma Pair_prs[quot_preserve]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  assumes q2: "Quotient R2 Abs2 Rep2"+
−
  shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"+
−
  apply(simp add: expand_fun_eq)+
−
  apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])+
−
  done+
−
+
−
lemma fst_rsp[quot_respect]:+
−
  assumes "Quotient R1 Abs1 Rep1"+
−
  assumes "Quotient R2 Abs2 Rep2"+
−
  shows "(prod_rel R1 R2 ===> R1) fst fst"+
−
  by simp+
−
+
−
lemma fst_prs[quot_preserve]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  assumes q2: "Quotient R2 Abs2 Rep2"+
−
  shows "(prod_fun Rep1 Rep2 ---> Abs1) fst = fst"+
−
  apply(simp add: expand_fun_eq)+
−
  apply(simp add: Quotient_abs_rep[OF q1])+
−
  done+
−
+
−
lemma snd_rsp[quot_respect]:+
−
  assumes "Quotient R1 Abs1 Rep1"+
−
  assumes "Quotient R2 Abs2 Rep2"+
−
  shows "(prod_rel R1 R2 ===> R2) snd snd"+
−
  by simp+
−
+
−
lemma snd_prs[quot_preserve]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  assumes q2: "Quotient R2 Abs2 Rep2"+
−
  shows "(prod_fun Rep1 Rep2 ---> Abs2) snd = snd"+
−
  apply(simp add: expand_fun_eq)+
−
  apply(simp add: Quotient_abs_rep[OF q2])+
−
  done+
−
+
−
lemma split_rsp[quot_respect]:+
−
  shows "((R1 ===> R2 ===> (op =)) ===> (prod_rel R1 R2) ===> (op =)) split split"+
−
  by auto+
−
+
−
lemma split_prs[quot_preserve]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "(((Abs1 ---> Abs2 ---> id) ---> prod_fun Rep1 Rep2 ---> id) split) = split"+
−
  by (simp add: expand_fun_eq Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])+
−
+
−
lemma prod_fun_id[id_simps]:+
−
  shows "prod_fun id id = id"+
−
  by (simp add: prod_fun_def)+
−
+
−
lemma prod_rel_eq[id_simps]:+
−
  shows "prod_rel (op =) (op =) = (op =)"+
−
  by (simp add: expand_fun_eq)+
−
+
−
+
−
end+
−