--- a/Quot/QuotSum.thy	Tue Jan 26 00:47:40 2010 +0100
+++ b/Quot/QuotSum.thy	Tue Jan 26 01:00:35 2010 +0100
@@ -1,3 +1,6 @@
+(*  Title:      QuotSum.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
 theory QuotSum
 imports QuotMain
 begin
@@ -19,81 +22,77 @@
 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(auto)
-  apply(case_tac [!] x)
-  apply(simp_all add: equivp_reflp[OF a] equivp_reflp[OF b])
-  apply(case_tac [!] y)
-  apply(simp_all add: equivp_symp[OF a] equivp_symp[OF b])
-  apply(case_tac [!] z)
-  apply(simp_all)
-  apply(rule equivp_transp[OF a])
-  apply(assumption)+
-  apply(rule equivp_transp[OF b])
-  apply(assumption)+
+  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_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(auto)
-  apply(case_tac a)
-  apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1]
-                      Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
-  apply(case_tac a)
-  apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1]
-                      Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
-  apply(case_tac [!] r)
-  apply(case_tac [!] s)
-  apply(simp_all)
+  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_respect]:
+  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_respect]:
+  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 \<equiv> id"
   apply (rule eq_reflection)
-  apply (rule ext)
-  apply (case_tac x)
-  apply (auto)
+  apply (simp add: expand_fun_eq split_sum_all)
   done
 
 lemma sum_rel_eq[id_simps]: 
-  "sum_rel op = op = \<equiv> op ="
+  shows "sum_rel (op =) (op =) \<equiv> (op =)"
   apply (rule eq_reflection)
-  apply (rule ext)+
-  apply (case_tac x)
-  apply auto
-  apply (case_tac xa)
-  apply auto
-  apply (case_tac xa)
-  apply auto
+  apply(simp add: expand_fun_eq split_sum_all)
   done
 
 end