--- 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