# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1264466566 -3600
# Node ID c96e007b512ffb7bf0530dd25bbe613a91afa0ba
# Parent  0b15b83ded4a392b6a5a3c1fada6cc5ca82e4068
cleaning of QuotProd; a little cleaning of QuotList

diff -r 0b15b83ded4a -r c96e007b512f Quot/QuotList.thy
--- a/Quot/QuotList.thy	Tue Jan 26 01:00:35 2010 +0100
+++ b/Quot/QuotList.thy	Tue Jan 26 01:42:46 2010 +0100
@@ -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]: "map id \<equiv> id"
+  apply(rule eq_reflection)
+  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)
@@ -59,142 +93,139 @@
   apply(rule list_rel_rel[OF q])
   done
 
-lemma map_id[id_simps]: "map id \<equiv> id"
-  apply (rule eq_reflection)
-  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"
   shows "(map Abs) ((Rep h) # (map Rep t)) = h # t"
-by (induct t) (simp_all add: Quotient_abs_rep[OF q])
+  by (induct t) (simp_all add: Quotient_abs_rep[OF q])
 
 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"
   shows "map Abs [] \<equiv> []"
-by (simp)
+  by (simp)
 
 lemma nil_rsp[quot_respect]:
   assumes q: "Quotient R Abs Rep"
   shows "list_rel R [] []"
-by simp
+  by simp
 
 lemma map_prs_aux:
   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]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   and     q2: "Quotient R2 Abs2 Rep2"
   shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"
-apply(simp)
-apply(rule allI)+
-apply(rule impI)
-apply(rule allI)+
-apply (induct_tac xa ya rule: list_induct2')
-apply simp_all
-done
+  apply(simp)
+  apply(rule allI)+
+  apply(rule impI)
+  apply(rule allI)+
+  apply (induct_tac xa ya rule: list_induct2')
+  apply simp_all
+  done
 
 lemma foldr_prs_aux:
   assumes a: "Quotient R1 abs1 rep1"
   and     b: "Quotient R2 abs2 rep2"
   shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"
-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 foldr_prs[quot_preserve]:
   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"
   and     b: "Quotient R2 abs2 rep2"
   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])
+  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"
-by (induct b) (simp_all)
+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"
-apply (induct a arbitrary: b)
-apply (simp add: list_rel_empty)
-apply (case_tac b)
-apply simp_all
-done
+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)
+  apply simp_all
+  done
 
 (* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
 lemma foldl_rsp[quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   and     q2: "Quotient R2 Abs2 Rep2"
   shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"
-apply(auto)
-apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
-apply simp
-apply (rule_tac x="xa" in spec)
-apply (rule_tac x="ya" in spec)
-apply (rule_tac xs="xb" and ys="yb" in list_induct2)
-apply (rule list_rel_len)
-apply (simp_all)
-done
+  apply(auto)
+  apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
+  apply simp
+  apply (rule_tac x="xa" in spec)
+  apply (rule_tac x="ya" in spec)
+  apply (rule_tac xs="xb" and ys="yb" in list_induct2)
+  apply (rule list_rel_len)
+  apply (simp_all)
+  done
 
 lemma foldr_rsp[quot_respect]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   and     q2: "Quotient R2 Abs2 Rep2"
   shows "((R1 ===> R2 ===> R2) ===> list_rel R1 ===> R2 ===> R2) foldr foldr"
-apply auto
-apply(subgoal_tac "R2 xb yb \<longrightarrow> list_rel R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")
-apply simp
-apply (rule_tac xs="xa" and ys="ya" in list_induct2)
-apply (rule list_rel_len)
-apply (simp_all)
-done
+  apply auto
+  apply(subgoal_tac "R2 xb yb \<longrightarrow> list_rel R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")
+  apply simp
+  apply (rule_tac xs="xa" and ys="ya" in list_induct2)
+  apply (rule list_rel_len)
+  apply (simp_all)
+  done
 
 lemma list_rel_eq[id_simps]:
   shows "list_rel (op =) \<equiv> (op =)"
-apply(rule eq_reflection)
-unfolding expand_fun_eq
-apply(rule allI)+
-apply(induct_tac x xa rule: list_induct2')
-apply(simp_all)
-done
+  apply(rule eq_reflection)
+  unfolding expand_fun_eq
+  apply(rule allI)+
+  apply(induct_tac x xa rule: list_induct2')
+  apply(simp_all)
+  done
 
 lemma list_rel_refl:
   assumes a: "\<And>x y. R x y = (R x = R y)"
   shows "list_rel R x x"
-by (induct x) (auto simp add: a)
+  by (induct x) (auto simp add: a)
 
 end
diff -r 0b15b83ded4a -r c96e007b512f Quot/QuotProd.thy
--- a/Quot/QuotProd.thy	Tue Jan 26 01:00:35 2010 +0100
+++ b/Quot/QuotProd.thy	Tue Jan 26 01:42:46 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)]]
 
@@ -14,37 +19,40 @@
   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
-done
+  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]:
   assumes q1: "Quotient R1 Abs1 Rep1"
   assumes q2: "Quotient R2 Abs2 Rep2"
   shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"
-unfolding Quotient_def
-using q1 q2
-apply (simp add: Quotient_abs_rep Quotient_abs_rep Quotient_rel_rep Quotient_rel_rep)
-using Quotient_rel[OF q1] Quotient_rel[OF q2] 
-by blast
+  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
+  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
 
-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])
-done
+  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]:
   assumes "Quotient R1 Abs1 Rep1"
@@ -56,9 +64,9 @@
   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])
-done
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q1])
+  done
 
 lemma snd_rsp[quot_respect]:
   assumes "Quotient R1 Abs1 Rep1"
@@ -70,20 +78,18 @@
   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])
-done
+  apply(simp add: expand_fun_eq)
+  apply(simp add: Quotient_abs_rep[OF q2])
+  done
 
 lemma prod_fun_id[id_simps]: 
   shows "prod_fun id id \<equiv> id"
-  by (rule eq_reflection) 
-     (simp add: prod_fun_def)
+  by (rule eq_reflection) (simp add: prod_fun_def)
 
 lemma prod_rel_eq[id_simps]: 
   shows "prod_rel (op =) (op =) \<equiv> (op =)"
   apply (rule eq_reflection)
-  apply (rule ext)+
-  apply auto
+  apply (simp add: expand_fun_eq)
   done
 
 end