src/HOL/Library/Quotient_Sum.thy

--- a/src/HOL/Library/Quotient_Sum.thy	Thu Mar 06 15:14:09 2014 +0100
+++ b/src/HOL/Library/Quotient_Sum.thy	Thu Mar 06 15:25:21 2014 +0100
@@ -10,61 +10,61 @@
 
 subsection {* Rules for the Quotient package *}
 
-lemma sum_rel_map1:
-  "sum_rel R1 R2 (map_sum f1 f2 x) y \<longleftrightarrow> sum_rel (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
-  by (simp add: sum_rel_def split: sum.split)
+lemma rel_sum_map1:
+  "rel_sum R1 R2 (map_sum f1 f2 x) y \<longleftrightarrow> rel_sum (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
+  by (simp add: rel_sum_def split: sum.split)
 
-lemma sum_rel_map2:
-  "sum_rel R1 R2 x (map_sum 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_def split: sum.split)
+lemma rel_sum_map2:
+  "rel_sum R1 R2 x (map_sum f1 f2 y) \<longleftrightarrow> rel_sum (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
+  by (simp add: rel_sum_def split: sum.split)
 
 lemma map_sum_id [id_simps]:
   "map_sum id id = id"
   by (simp add: id_def map_sum.identity fun_eq_iff)
 
-lemma sum_rel_eq [id_simps]:
-  "sum_rel (op =) (op =) = (op =)"
-  by (simp add: sum_rel_def fun_eq_iff split: sum.split)
+lemma rel_sum_eq [id_simps]:
+  "rel_sum (op =) (op =) = (op =)"
+  by (simp add: rel_sum_def fun_eq_iff split: sum.split)
 
-lemma reflp_sum_rel:
-  "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
-  unfolding reflp_def split_sum_all sum_rel_simps by fast
+lemma reflp_rel_sum:
+  "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (rel_sum R1 R2)"
+  unfolding reflp_def split_sum_all rel_sum_simps by fast
 
 lemma sum_symp:
-  "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
-  unfolding symp_def split_sum_all sum_rel_simps by fast
+  "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (rel_sum R1 R2)"
+  unfolding symp_def split_sum_all rel_sum_simps by fast
 
 lemma sum_transp:
-  "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
-  unfolding transp_def split_sum_all sum_rel_simps by fast
+  "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (rel_sum R1 R2)"
+  unfolding transp_def split_sum_all rel_sum_simps by fast
 
 lemma sum_equivp [quot_equiv]:
-  "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
-  by (blast intro: equivpI reflp_sum_rel sum_symp sum_transp elim: equivpE)
+  "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (rel_sum R1 R2)"
+  by (blast intro: equivpI reflp_rel_sum sum_symp sum_transp elim: equivpE)
 
 lemma sum_quotient [quot_thm]:
   assumes q1: "Quotient3 R1 Abs1 Rep1"
   assumes q2: "Quotient3 R2 Abs2 Rep2"
-  shows "Quotient3 (sum_rel R1 R2) (map_sum Abs1 Abs2) (map_sum Rep1 Rep2)"
+  shows "Quotient3 (rel_sum R1 R2) (map_sum Abs1 Abs2) (map_sum Rep1 Rep2)"
   apply (rule Quotient3I)
-  apply (simp_all add: map_sum.compositionality comp_def map_sum.identity sum_rel_eq sum_rel_map1 sum_rel_map2
+  apply (simp_all add: map_sum.compositionality comp_def map_sum.identity rel_sum_eq rel_sum_map1 rel_sum_map2
     Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
   using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
-  apply (simp add: sum_rel_def comp_def split: sum.split)
+  apply (simp add: rel_sum_def comp_def split: sum.split)
   done
 
-declare [[mapQ3 sum = (sum_rel, sum_quotient)]]
+declare [[mapQ3 sum = (rel_sum, sum_quotient)]]
 
 lemma sum_Inl_rsp [quot_respect]:
   assumes q1: "Quotient3 R1 Abs1 Rep1"
   assumes q2: "Quotient3 R2 Abs2 Rep2"
-  shows "(R1 ===> sum_rel R1 R2) Inl Inl"
+  shows "(R1 ===> rel_sum R1 R2) Inl Inl"
   by auto
 
 lemma sum_Inr_rsp [quot_respect]:
   assumes q1: "Quotient3 R1 Abs1 Rep1"
   assumes q2: "Quotient3 R2 Abs2 Rep2"
-  shows "(R2 ===> sum_rel R1 R2) Inr Inr"
+  shows "(R2 ===> rel_sum R1 R2) Inr Inr"
   by auto
 
 lemma sum_Inl_prs [quot_preserve]: