src/HOL/Library/Quotient_Sum.thy

--- a/src/HOL/Library/Quotient_Sum.thy	Tue Nov 30 15:58:09 2010 +0100
+++ b/src/HOL/Library/Quotient_Sum.thy	Tue Nov 30 15:58:09 2010 +0100
@@ -18,53 +18,68 @@
 
 declare [[map sum = (sum_map, sum_rel)]]
 
+lemma sum_rel_unfold:
+  "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
+    | (Inr x, Inr y) \<Rightarrow> R2 x y
+    | _ \<Rightarrow> False)"
+  by (cases x) (cases y, simp_all)+
 
-text {* should probably be in @{theory Sum_Type} *}
-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_rel_map1:
+  "sum_rel R1 R2 (sum_map f1 f2 x) y \<longleftrightarrow> sum_rel (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
+  by (simp add: sum_rel_unfold split: sum.split)
+
+lemma sum_rel_map2:
+  "sum_rel R1 R2 x (sum_map f1 f2 y) \<longleftrightarrow> sum_rel (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
+  by (simp add: sum_rel_unfold split: sum.split)
+
+lemma sum_map_id [id_simps]:
+  "sum_map id id = id"
+  by (simp add: id_def sum_map.identity fun_eq_iff)
 
-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_rel_eq [id_simps]:
+  "sum_rel (op =) (op =) = (op =)"
+  by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
+
+lemma sum_reflp:
+  "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
+  by (auto simp add: sum_rel_unfold split: sum.splits intro!: reflpI elim: reflpE)
 
-lemma sum_quotient[quot_thm]:
+lemma sum_symp:
+  "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
+  by (auto simp add: sum_rel_unfold split: sum.splits intro!: sympI elim: sympE)
+
+lemma sum_transp:
+  "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
+  by (auto simp add: sum_rel_unfold split: sum.splits intro!: transpI elim: transpE)
+
+lemma sum_equivp [quot_equiv]:
+  "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
+  by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)
+  
+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)+
+  apply (rule QuotientI)
+  apply (simp_all add: sum_map.compositionality sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
+    Quotient_abs_rep [OF q1] Quotient_rel_rep [OF q1] Quotient_abs_rep [OF q2] Quotient_rel_rep [OF q2])
+  using Quotient_rel [OF q1] Quotient_rel [OF q2]
+  apply (simp add: sum_rel_unfold split: sum.split)
   done
 
-lemma sum_Inl_rsp[quot_respect]:
+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 auto
 
-lemma sum_Inr_rsp[quot_respect]:
+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 auto
 
-lemma sum_Inl_prs[quot_preserve]:
+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"
@@ -72,7 +87,7 @@
   apply(simp add: Quotient_abs_rep[OF q1])
   done
 
-lemma sum_Inr_prs[quot_preserve]:
+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"
@@ -80,12 +95,4 @@
   apply(simp add: Quotient_abs_rep[OF q2])
   done
 
-lemma sum_map_id[id_simps]:
-  shows "sum_map id id = id"
-  by (simp add: fun_eq_iff split_sum_all)
-
-lemma sum_rel_eq[id_simps]:
-  shows "sum_rel (op =) (op =) = (op =)"
-  by (simp add: fun_eq_iff split_sum_all)
-
 end