--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/QuotSum.thy	Thu Dec 10 14:35:06 2009 +0100
@@ -0,0 +1,84 @@
+theory QuotSum
+imports QuotMain
+begin
+
+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)]]
+
+lemma sum_equivp[quot_equiv]:
+  assumes a: "equivp R1"
+  assumes b: "equivp R2"
+  shows "equivp (sum_rel R1 R2)"
+unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
+apply(auto)
+apply(case_tac x)
+apply(auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])
+apply(case_tac x)
+apply(case_tac y)
+prefer 3
+apply(case_tac y)
+apply(auto simp add: equivp_symp[OF a] equivp_symp[OF b])
+apply(case_tac x)
+apply(case_tac y)
+apply(case_tac z)
+prefer 3
+apply(case_tac z)
+prefer 5
+apply(case_tac y)
+apply(case_tac z)
+prefer 3
+apply(case_tac z)
+apply(auto)
+apply(metis equivp_transp[OF b])
+apply(metis equivp_transp[OF a])
+done
+
+lemma sum_fun_fun:
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  assumes q2: "Quotient R2 Abs2 Rep2"
+  shows  "sum_rel R1 R2 r s =
+          (sum_rel R1 R2 r r \<and> sum_rel R1 R2 s s \<and> sum_map Abs1 Abs2 r = sum_map Abs1 Abs2 s)"
+  using q1 q2
+  apply(case_tac r)
+  apply(case_tac s)
+  apply(simp_all)
+  prefer 2
+  apply(case_tac s)
+  apply(auto)
+  unfolding Quotient_def 
+  apply metis+
+  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(rule conjI)
+  apply(rule allI)
+  apply(case_tac a)
+  apply(simp add: Quotient_abs_rep[OF q1])
+  apply(simp add: Quotient_abs_rep[OF q2])
+  apply(rule conjI)
+  apply(rule allI)
+  apply(case_tac a)
+  apply(simp add: Quotient_rel_rep[OF q1])
+  apply(simp add: Quotient_rel_rep[OF q2])
+  apply(rule allI)+
+  apply(rule sum_fun_fun[OF q1 q2])
+  done
+
+end
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