# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1264499624 -3600
# Node ID da5e4b8317c77985fe5929a6c0e5cca9e1c4c1b8
# Parent  c96e007b512ffb7bf0530dd25bbe613a91afa0ba# Parent  0879d144aaa3961fda32ba2db8def0282e341223
merged

diff -r 0879d144aaa3 -r da5e4b8317c7 Quot/QuotBase.thy
--- a/Quot/QuotBase.thy	Tue Jan 26 09:54:43 2010 +0100
+++ b/Quot/QuotBase.thy	Tue Jan 26 10:53:44 2010 +0100
@@ -72,10 +72,9 @@
 where
   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
 
-lemma eq_comp_r: "(op = OO R OO op =) = R"
-  apply (rule ext)+
-  apply auto
-  done
+lemma eq_comp_r: 
+  shows "((op =) OOO R) = R"
+  by (auto simp add: expand_fun_eq)
 
 section {* Respects predicate *}
 
diff -r 0879d144aaa3 -r da5e4b8317c7 Quot/QuotList.thy
--- a/Quot/QuotList.thy	Tue Jan 26 09:54:43 2010 +0100
+++ b/Quot/QuotList.thy	Tue Jan 26 10:53:44 2010 +0100
@@ -1,4 +1,4 @@
-theory QuotList
+ theory QuotList
 imports QuotMain List
 begin
 
@@ -12,23 +12,60 @@
 
 declare [[map list = (map, list_rel)]]
 
+
+
+text {* should probably be in Sum_Type.thy *}
+lemma split_list_all: 
+  shows "(\<forall>x. P x) \<longleftrightarrow> P [] \<and> (\<forall>x xs. P (x#xs))"
+  apply(auto)
+  apply(case_tac x)
+  apply(simp_all)
+  done
+
+lemma map_id[id_simps]: 
+  shows "map id = id"
+  apply(simp add: expand_fun_eq)
+  apply(rule allI)
+  apply(induct_tac x)
+  apply(simp_all)
+  done
+
+
+lemma list_rel_reflp:
+  shows "equivp R \<Longrightarrow> list_rel R xs xs"
+  apply(induct xs)
+  apply(simp_all add: equivp_reflp)
+  done
+
+lemma list_rel_symp:
+  assumes a: "equivp R" 
+  shows "list_rel R xs ys \<Longrightarrow> list_rel R ys xs"
+  apply(induct xs ys rule: list_induct2')
+  apply(simp_all)
+  apply(rule equivp_symp[OF a])
+  apply(simp)
+  done
+
+lemma list_rel_transp:
+  assumes a: "equivp R" 
+  shows "list_rel R xs1 xs2 \<Longrightarrow> list_rel R xs2 xs3 \<Longrightarrow> list_rel R xs1 xs3"
+  apply(induct xs1 xs2 arbitrary: xs3 rule: list_induct2')
+  apply(simp_all)
+  apply(case_tac xs3)
+  apply(simp_all)
+  apply(rule equivp_transp[OF a])
+  apply(auto)
+  done
+
 lemma list_equivp[quot_equiv]:
   assumes a: "equivp R"
   shows "equivp (list_rel R)"
-  unfolding equivp_def
-  apply(rule allI)+
-  apply(induct_tac x y rule: list_induct2')
-  apply(simp_all add: expand_fun_eq)
-  apply(metis list_rel.simps(1) list_rel.simps(2))
-  apply(metis list_rel.simps(1) list_rel.simps(2))
-  apply(rule iffI)
-  apply(rule allI)
-  apply(case_tac x)
-  apply(simp_all)
-  using a
-  apply(unfold equivp_def)
-  apply(auto)[1]
-  apply(metis list_rel.simps(4))
+  apply(rule equivpI)
+  unfolding reflp_def symp_def transp_def
+  apply(subst split_list_all)
+  apply(simp add: equivp_reflp[OF a] list_rel_reflp[OF a])
+  apply(blast intro: list_rel_symp[OF a])
+  apply(blast intro: list_rel_transp[OF a])
   done
 
 lemma list_rel_rel:
@@ -44,11 +81,8 @@
   assumes q: "Quotient R Abs Rep"
   shows "Quotient (list_rel R) (map Abs) (map Rep)"
   unfolding Quotient_def
-  apply(rule conjI)
-  apply(rule allI)
-  apply(induct_tac a)
-  apply(simp)
-  apply(simp add: Quotient_abs_rep[OF q])
+  apply(subst split_list_all)
+  apply(simp add: Quotient_abs_rep[OF q] abs_o_rep[OF q] map_id)
   apply(rule conjI)
   apply(rule allI)
   apply(induct_tac a)
@@ -59,11 +93,6 @@
   apply(rule list_rel_rel[OF q])
   done
 
-lemma map_id[id_simps]: "map id = id"
-  apply (rule ext)
-  apply (rule_tac list="x" in list.induct)
-  apply (simp_all)
-  done
 
 lemma cons_prs_aux:
   assumes q: "Quotient R Abs Rep"
@@ -73,12 +102,13 @@
 lemma cons_prs[quot_preserve]:
   assumes q: "Quotient R Abs Rep"
   shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"
-  by (simp only: expand_fun_eq fun_map_def cons_prs_aux[OF q]) (simp)
+  by (simp only: expand_fun_eq fun_map_def cons_prs_aux[OF q])
+     (simp)
 
 lemma cons_rsp[quot_respect]:
   assumes q: "Quotient R Abs Rep"
-  shows "(R ===> list_rel R ===> list_rel R) op # op #"
-  by auto
+  shows "(R ===> list_rel R ===> list_rel R) (op #) (op #)"
+  by (auto)
 
 lemma nil_prs[quot_preserve]:
   assumes q: "Quotient R Abs Rep"
@@ -94,14 +124,16 @@
   assumes a: "Quotient R1 abs1 rep1"
   and     b: "Quotient R2 abs2 rep2"
   shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
-  by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
+  by (induct l) 
+     (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
 
 
 lemma map_prs[quot_preserve]:
   assumes a: "Quotient R1 abs1 rep1"
   and     b: "Quotient R2 abs2 rep2"
   shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map"
-  by (simp only: expand_fun_eq fun_map_def map_prs_aux[OF a b]) (simp)
+  by (simp only: expand_fun_eq fun_map_def map_prs_aux[OF a b])
+     (simp)
 
 
 lemma map_rsp[quot_respect]:
@@ -126,7 +158,8 @@
   assumes a: "Quotient R1 abs1 rep1"
   and     b: "Quotient R2 abs2 rep2"
   shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr"
-  by (simp only: expand_fun_eq fun_map_def foldr_prs_aux[OF a b]) (simp)
+  by (simp only: expand_fun_eq fun_map_def foldr_prs_aux[OF a b])
+     (simp)
 
 lemma foldl_prs_aux:
   assumes a: "Quotient R1 abs1 rep1"
@@ -134,18 +167,20 @@
   shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"
   by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
 
+
 lemma foldl_prs[quot_preserve]:
   assumes a: "Quotient R1 abs1 rep1"
   and     b: "Quotient R2 abs2 rep2"
   shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl"
-  by (simp only: expand_fun_eq fun_map_def foldl_prs_aux[OF a b]) (simp)
+  by (simp only: expand_fun_eq fun_map_def foldl_prs_aux[OF a b])
+     (simp)
 
-lemma list_rel_empty:
-  "list_rel R [] b \<Longrightarrow> length b = 0"
+lemma list_rel_empty: 
+  shows "list_rel R [] b \<Longrightarrow> length b = 0"
   by (induct b) (simp_all)
 
-lemma list_rel_len:
-  "list_rel R a b \<Longrightarrow> length a = length b"
+lemma list_rel_len: 
+  shows "list_rel R a b \<Longrightarrow> length a = length b"
   apply (induct a arbitrary: b)
   apply (simp add: list_rel_empty)
   apply (case_tac b)
diff -r 0879d144aaa3 -r da5e4b8317c7 Quot/QuotOption.thy
--- a/Quot/QuotOption.thy	Tue Jan 26 09:54:43 2010 +0100
+++ b/Quot/QuotOption.thy	Tue Jan 26 10:53:44 2010 +0100
@@ -1,7 +1,12 @@
+(*  Title:      QuotOption.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
 theory QuotOption
 imports QuotMain
 begin
 
+section {* Quotient infrastructure for option type *}
+
 fun
   option_rel
 where
@@ -10,28 +15,25 @@
 | "option_rel R None (Some x) = False"
 | "option_rel R (Some x) (Some y) = R x y"
 
-fun
-  option_map
-where
-  "option_map f None = None"
-| "option_map f (Some x) = Some (f x)"
+declare [[map option = (Option.map, option_rel)]]
 
-declare [[map option = (option_map, option_rel)]]
-
+text {* should probably be in Option.thy *}
+lemma split_option_all: 
+  shows "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>a. P (Some a))"
+apply(auto)
+apply(case_tac x)
+apply(simp_all)
+done
 
 lemma option_quotient[quot_thm]:
   assumes q: "Quotient R Abs Rep"
-  shows "Quotient (option_rel R) (option_map Abs) (option_map Rep)"
-  apply(unfold Quotient_def)
-  apply(auto)
-  apply(case_tac a, simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
-  apply(case_tac a, simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
-  apply(case_tac [!] r)
-  apply(case_tac [!] s)
-  apply(simp_all add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q] )
+  shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
+  unfolding Quotient_def
+  apply(simp add: split_option_all)
+  apply(simp add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
   using q
   unfolding Quotient_def
-  apply(blast)+
+  apply(blast)
   done
   
 lemma option_equivp[quot_equiv]:
@@ -39,16 +41,10 @@
   shows "equivp (option_rel R)"
   apply(rule equivpI)
   unfolding reflp_def symp_def transp_def
-  apply(auto)
-  apply(case_tac [!] x)
-  apply(simp_all add: equivp_reflp[OF a])
-  apply(case_tac [!] y)
-  apply(simp_all add: equivp_symp[OF a])
-  apply(case_tac [!] z)
-  apply(simp_all)
-  apply(clarify)
-  apply(rule equivp_transp[OF a])
-  apply(assumption)+
+  apply(simp_all add: split_option_all)
+  apply(blast intro: equivp_reflp[OF a])
+  apply(blast intro: equivp_symp[OF a])
+  apply(blast intro: equivp_transp[OF a])
   done
 
 lemma option_None_rsp[quot_respect]:
@@ -63,29 +59,22 @@
 
 lemma option_None_prs[quot_preserve]:
   assumes q: "Quotient R Abs Rep"
-  shows "option_map Abs None = None"
+  shows "Option.map Abs None = None"
   by simp
 
 lemma option_Some_prs[quot_respect]:
   assumes q: "Quotient R Abs Rep"
-  shows "(Rep ---> option_map Abs) Some = Some"
+  shows "(Rep ---> Option.map Abs) Some = Some"
   apply(simp add: expand_fun_eq)
   apply(simp add: Quotient_abs_rep[OF q])
   done
 
-lemma option_map_id[id_simps]:
-  shows "option_map id = id"
-  apply (auto simp add: expand_fun_eq)
-  apply (case_tac x)
-  apply (auto)
-  done
+lemma option_map_id[id_simps]: 
+  shows "Option.map id = id"
+  by (simp add: expand_fun_eq split_option_all)
 
-lemma option_rel_eq[id_simps]:
+lemma option_rel_eq[id_simps]: 
   shows "option_rel (op =) = (op =)"
-  apply(auto simp add: expand_fun_eq)
-  apply(case_tac x)
-  apply(case_tac [!] xa)
-  apply(auto)
-  done
+  by (simp add: expand_fun_eq split_option_all)
 
 end
diff -r 0879d144aaa3 -r da5e4b8317c7 Quot/QuotProd.thy
--- a/Quot/QuotProd.thy	Tue Jan 26 09:54:43 2010 +0100
+++ b/Quot/QuotProd.thy	Tue Jan 26 10:53:44 2010 +0100
@@ -1,11 +1,16 @@
+(*  Title:      QuotProd.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
 theory QuotProd
 imports QuotMain
 begin
 
+section {* Quotient infrastructure for product type *}
+
 fun
   prod_rel
 where
-  "prod_rel R1 R2 = (\<lambda>(a,b) (c,d). R1 a c \<and> R2 b d)"
+  "prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
 
 declare [[map * = (prod_fun, prod_rel)]]
 
@@ -14,12 +19,12 @@
   assumes a: "equivp R1"
   assumes b: "equivp R2"
   shows "equivp (prod_rel R1 R2)"
-  unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
-  apply (auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])
-  apply (simp only: equivp_symp[OF a])
-  apply (simp only: equivp_symp[OF b])
-  using equivp_transp[OF a] apply blast
-  using equivp_transp[OF b] apply blast
+  apply(rule equivpI)
+  unfolding reflp_def symp_def transp_def
+  apply(simp_all add: split_paired_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 prod_quotient[quot_thm]:
@@ -27,23 +32,26 @@
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"
   unfolding Quotient_def
+  apply(simp add: split_paired_all)
+  apply(simp add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
+  apply(simp add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
   using q1 q2
-  apply (simp add: Quotient_abs_rep Quotient_rel_rep)
-  using Quotient_rel[OF q1] Quotient_rel[OF q2]
-  by blast
+  unfolding Quotient_def
+  apply(blast)
+  done
 
-lemma pair_rsp[quot_respect]:
+lemma Pair_rsp[quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
-  by simp
+by simp
 
-lemma pair_prs[quot_preserve]:
+lemma Pair_prs[quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"
-  apply (simp add: expand_fun_eq)
-  apply (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
   done
 
 lemma fst_rsp[quot_respect]:
@@ -56,8 +64,8 @@
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(prod_fun Rep1 Rep2 ---> Abs1) fst = fst"
-  apply (simp add: expand_fun_eq)
-  apply (simp add: Quotient_abs_rep[OF q1])
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q1])
   done
 
 lemma snd_rsp[quot_respect]:
@@ -65,33 +73,32 @@
   assumes "Quotient R2 Abs2 Rep2"
   shows "(prod_rel R1 R2 ===> R2) snd snd"
   by simp
-
+  
 lemma snd_prs[quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "(prod_fun Rep1 Rep2 ---> Abs2) snd = snd"
-  apply (simp add: expand_fun_eq)
-  apply (simp add: Quotient_abs_rep[OF q2])
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q2])
   done
 
 lemma split_rsp[quot_respect]:
-  "((R1 ===> R2 ===> op =) ===> (prod_rel R1 R2) ===> op =) split split"
+  shows "((R1 ===> R2 ===> (op =)) ===> (prod_rel R1 R2) ===> (op =)) split split"
   by auto
-
+  
 lemma split_prs[quot_preserve]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   and     q2: "Quotient R2 Abs2 Rep2"
   shows "(((Abs1 ---> Abs2 ---> id) ---> prod_fun Rep1 Rep2 ---> id) split) = split"
   by (simp add: expand_fun_eq Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
 
-lemma prod_fun_id[id_simps]:
+lemma prod_fun_id[id_simps]: 
   shows "prod_fun id id = id"
   by (simp add: prod_fun_def)
 
-lemma prod_rel_eq[id_simps]:
-  shows "(prod_rel (op =) (op =)) = (op =)"
-  apply (rule ext)+
-  apply auto
-  done
+lemma prod_rel_eq[id_simps]: 
+  shows "prod_rel (op =) (op =) = (op =)"
+  by (simp add: expand_fun_eq)
+ 
 
 end
diff -r 0879d144aaa3 -r da5e4b8317c7 Quot/QuotSum.thy
--- a/Quot/QuotSum.thy	Tue Jan 26 09:54:43 2010 +0100
+++ b/Quot/QuotSum.thy	Tue Jan 26 10:53:44 2010 +0100
@@ -1,3 +1,6 @@
+(*  Title:      QuotSum.thy
+    Author:     Cezary Kaliszyk and Christian Urban
+*)
 theory QuotSum
 imports QuotMain
 begin
@@ -19,79 +22,73 @@
 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_map_id[id_simps]:
-  shows "sum_map id id = id"
-  apply (rule ext)
-  apply (case_tac x)
-  apply (auto)
+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_rel_eq[id_simps]:
-  "(sum_rel op = op =) = op ="
-  apply (rule ext)+
-  apply (case_tac x)
-  apply auto
-  apply (case_tac xa)
-  apply auto
-  apply (case_tac xa)
-  apply auto
+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 = 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