--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/QuotProd.thy	Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,80 @@
+theory QuotProd
+imports QuotScript
+begin
+
+fun
+  prod_rel
+where
+  "prod_rel r1 r2 = (\<lambda>(a,b) (c,d). r1 a c \<and> r2 b d)"
+
+(* prod_fun is a good mapping function *)
+
+lemma prod_equivp:
+  assumes a: "equivp R1"
+  assumes b: "equivp R2"
+  shows "equivp (prod_rel R1 R2)"
+unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
+apply(auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])
+apply(simp only: equivp_symp[OF a])
+apply(simp only: equivp_symp[OF b])
+using equivp_transp[OF a] apply blast
+using equivp_transp[OF b] apply blast
+done
+
+lemma prod_quotient:
+  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: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q1] Quotient_rel_rep[OF q2])
+using Quotient_rel[OF q1] Quotient_rel[OF q2] by blast
+
+lemma pair_rsp:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
+by auto
+
+lemma pair_prs:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "(prod_fun Abs1 Abs2) (Rep1 l, Rep2 r) \<equiv> (l, r)"
+  by (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
+
+(* TODO: Is the quotient assumption q1 necessary? *)
+(* TODO: Aren't there hard to use later? *)
+lemma fst_rsp:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  assumes a: "(prod_rel R1 R2) p1 p2"
+  shows "R1 (fst p1) (fst p2)"
+  using a
+  apply(case_tac p1)
+  apply(case_tac p2)
+  apply(auto)
+  done
+
+lemma snd_rsp:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  assumes a: "(prod_rel R1 R2) p1 p2"
+  shows "R2 (snd p1) (snd p2)"
+  using a
+  apply(case_tac p1)
+  apply(case_tac p2)
+  apply(auto)
+  done
+
+lemma fst_prs:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "Abs1 (fst ((prod_fun Rep1 Rep2) p)) = fst p"
+by (case_tac p) (auto simp add: Quotient_abs_rep[OF q1])
+
+lemma snd_prs:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "Abs2 (snd ((prod_fun Rep1 Rep2) p)) = snd p"
+by (case_tac p) (auto simp add: Quotient_abs_rep[OF q2])
+
+end