--- 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 *}
--- 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)
--- 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
--- 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
--- 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