--- a/QuotProd.thy	Mon Dec 07 14:00:36 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,80 +0,0 @@
-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