--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Quotient_Sum.thy	Thu Feb 11 10:06:02 2010 +0100
@@ -0,0 +1,96 @@
+(*  Title:      Quotient_Sum.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
+theory Quotient_Sum
+imports Quotient
+begin
+
+section {* Quotient infrastructure for the sum type. *}
+
+fun
+  sum_rel
+where
+  "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
+| "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
+| "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
+| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
+
+fun
+  sum_map
+where
+  "sum_map f1 f2 (Inl a) = Inl (f1 a)"
+| "sum_map f1 f2 (Inr a) = Inr (f2 a)"
+
+declare [[map "+" = (sum_map, sum_rel)]]
+
+
+text {* should probably be in Sum_Type.thy *}
+lemma split_sum_all:
+  shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
+  apply(auto)
+  apply(case_tac x)
+  apply(simp_all)
+  done
+
+lemma sum_equivp[quot_equiv]:
+  assumes a: "equivp R1"
+  assumes b: "equivp R2"
+  shows "equivp (sum_rel R1 R2)"
+  apply(rule equivpI)
+  unfolding reflp_def symp_def transp_def
+  apply(simp_all add: split_sum_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 sum_quotient[quot_thm]:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
+  unfolding Quotient_def
+  apply(simp add: split_sum_all)
+  apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
+  apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
+  using q1 q2
+  unfolding Quotient_def
+  apply(blast)+
+  done
+
+lemma sum_Inl_rsp[quot_respect]:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "(R1 ===> sum_rel R1 R2) Inl Inl"
+  by simp
+
+lemma sum_Inr_rsp[quot_respect]:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "(R2 ===> sum_rel R1 R2) Inr Inr"
+  by simp
+
+lemma sum_Inl_prs[quot_preserve]:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q1])
+  done
+
+lemma sum_Inr_prs[quot_preserve]:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q2])
+  done
+
+lemma sum_map_id[id_simps]:
+  shows "sum_map id id = id"
+  by (simp add: expand_fun_eq split_sum_all)
+
+lemma sum_rel_eq[id_simps]:
+  shows "sum_rel (op =) (op =) = (op =)"
+  by (simp add: expand_fun_eq split_sum_all)
+
+end