--- a/src/HOL/Library/Quotient_List.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_List.thy Tue Aug 13 15:59:22 2013 +0200
@@ -1,5 +1,5 @@
(* Title: HOL/Library/Quotient_List.thy
- Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
+ Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the list type *}
@@ -8,13 +8,13 @@
imports Main Quotient_Set Quotient_Product Quotient_Option
begin
-subsection {* Relator for list type *}
+subsection {* Rules for the Quotient package *}
lemma map_id [id_simps]:
"map id = id"
by (fact List.map.id)
-lemma list_all2_eq [id_simps, relator_eq]:
+lemma list_all2_eq [id_simps]:
"list_all2 (op =) = (op =)"
proof (rule ext)+
fix xs ys
@@ -22,66 +22,6 @@
by (induct xs ys rule: list_induct2') simp_all
qed
-lemma list_all2_mono[relator_mono]:
- assumes "A \<le> B"
- shows "(list_all2 A) \<le> (list_all2 B)"
-using assms by (auto intro: list_all2_mono)
-
-lemma list_all2_OO[relator_distr]: "list_all2 A OO list_all2 B = list_all2 (A OO B)"
-proof (intro ext iffI)
- fix xs ys
- assume "list_all2 (A OO B) xs ys"
- thus "(list_all2 A OO list_all2 B) xs ys"
- unfolding OO_def
- by (induct, simp, simp add: list_all2_Cons1 list_all2_Cons2, fast)
-next
- fix xs ys
- assume "(list_all2 A OO list_all2 B) xs ys"
- then obtain zs where "list_all2 A xs zs" and "list_all2 B zs ys" ..
- thus "list_all2 (A OO B) xs ys"
- by (induct arbitrary: ys, simp, clarsimp simp add: list_all2_Cons1, fast)
-qed
-
-lemma Domainp_list[relator_domain]:
- assumes "Domainp A = P"
- shows "Domainp (list_all2 A) = (list_all P)"
-proof -
- {
- fix x
- have *: "\<And>x. (\<exists>y. A x y) = P x" using assms unfolding Domainp_iff by blast
- have "(\<exists>y. (list_all2 A x y)) = list_all P x"
- by (induction x) (simp_all add: * list_all2_Cons1)
- }
- then show ?thesis
- unfolding Domainp_iff[abs_def]
- by (auto iff: fun_eq_iff)
-qed
-
-lemma reflp_list_all2[reflexivity_rule]:
- assumes "reflp R"
- shows "reflp (list_all2 R)"
-proof (rule reflpI)
- from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
- fix xs
- show "list_all2 R xs xs"
- by (induct xs) (simp_all add: *)
-qed
-
-lemma left_total_list_all2[reflexivity_rule]:
- "left_total R \<Longrightarrow> left_total (list_all2 R)"
- unfolding left_total_def
- apply safe
- apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
-done
-
-lemma left_unique_list_all2 [reflexivity_rule]:
- "left_unique R \<Longrightarrow> left_unique (list_all2 R)"
- unfolding left_unique_def
- apply (subst (2) all_comm, subst (1) all_comm)
- apply (rule allI, rename_tac zs, induct_tac zs)
- apply (auto simp add: list_all2_Cons2)
- done
-
lemma list_symp:
assumes "symp R"
shows "symp (list_all2 R)"
@@ -108,272 +48,6 @@
"equivp R \<Longrightarrow> equivp (list_all2 R)"
by (blast intro: equivpI reflp_list_all2 list_symp list_transp elim: equivpE)
-lemma right_total_list_all2 [transfer_rule]:
- "right_total R \<Longrightarrow> right_total (list_all2 R)"
- unfolding right_total_def
- by (rule allI, induct_tac y, simp, simp add: list_all2_Cons2)
-
-lemma right_unique_list_all2 [transfer_rule]:
- "right_unique R \<Longrightarrow> right_unique (list_all2 R)"
- unfolding right_unique_def
- apply (rule allI, rename_tac xs, induct_tac xs)
- apply (auto simp add: list_all2_Cons1)
- done
-
-lemma bi_total_list_all2 [transfer_rule]:
- "bi_total A \<Longrightarrow> bi_total (list_all2 A)"
- unfolding bi_total_def
- apply safe
- apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
- apply (rename_tac ys, induct_tac ys, simp, simp add: list_all2_Cons2)
- done
-
-lemma bi_unique_list_all2 [transfer_rule]:
- "bi_unique A \<Longrightarrow> bi_unique (list_all2 A)"
- unfolding bi_unique_def
- apply (rule conjI)
- apply (rule allI, rename_tac xs, induct_tac xs)
- apply (simp, force simp add: list_all2_Cons1)
- apply (subst (2) all_comm, subst (1) all_comm)
- apply (rule allI, rename_tac xs, induct_tac xs)
- apply (simp, force simp add: list_all2_Cons2)
- done
-
-subsection {* Transfer rules for transfer package *}
-
-lemma Nil_transfer [transfer_rule]: "(list_all2 A) [] []"
- by simp
-
-lemma Cons_transfer [transfer_rule]:
- "(A ===> list_all2 A ===> list_all2 A) Cons Cons"
- unfolding fun_rel_def by simp
-
-lemma list_case_transfer [transfer_rule]:
- "(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B)
- list_case list_case"
- unfolding fun_rel_def by (simp split: list.split)
-
-lemma list_rec_transfer [transfer_rule]:
- "(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B)
- list_rec list_rec"
- unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all)
-
-lemma tl_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 A) tl tl"
- unfolding tl_def by transfer_prover
-
-lemma butlast_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 A) butlast butlast"
- by (rule fun_relI, erule list_all2_induct, auto)
-
-lemma set_transfer [transfer_rule]:
- "(list_all2 A ===> set_rel A) set set"
- unfolding set_def by transfer_prover
-
-lemma map_transfer [transfer_rule]:
- "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"
- unfolding List.map_def by transfer_prover
-
-lemma append_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 A ===> list_all2 A) append append"
- unfolding List.append_def by transfer_prover
-
-lemma rev_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 A) rev rev"
- unfolding List.rev_def by transfer_prover
-
-lemma filter_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> list_all2 A) filter filter"
- unfolding List.filter_def by transfer_prover
-
-lemma fold_transfer [transfer_rule]:
- "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) fold fold"
- unfolding List.fold_def by transfer_prover
-
-lemma foldr_transfer [transfer_rule]:
- "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr"
- unfolding List.foldr_def by transfer_prover
-
-lemma foldl_transfer [transfer_rule]:
- "((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl"
- unfolding List.foldl_def by transfer_prover
-
-lemma concat_transfer [transfer_rule]:
- "(list_all2 (list_all2 A) ===> list_all2 A) concat concat"
- unfolding List.concat_def by transfer_prover
-
-lemma drop_transfer [transfer_rule]:
- "(op = ===> list_all2 A ===> list_all2 A) drop drop"
- unfolding List.drop_def by transfer_prover
-
-lemma take_transfer [transfer_rule]:
- "(op = ===> list_all2 A ===> list_all2 A) take take"
- unfolding List.take_def by transfer_prover
-
-lemma list_update_transfer [transfer_rule]:
- "(list_all2 A ===> op = ===> A ===> list_all2 A) list_update list_update"
- unfolding list_update_def by transfer_prover
-
-lemma takeWhile_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> list_all2 A) takeWhile takeWhile"
- unfolding takeWhile_def by transfer_prover
-
-lemma dropWhile_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> list_all2 A) dropWhile dropWhile"
- unfolding dropWhile_def by transfer_prover
-
-lemma zip_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 B ===> list_all2 (prod_rel A B)) zip zip"
- unfolding zip_def by transfer_prover
-
-lemma insert_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "(A ===> list_all2 A ===> list_all2 A) List.insert List.insert"
- unfolding List.insert_def [abs_def] by transfer_prover
-
-lemma find_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> option_rel A) List.find List.find"
- unfolding List.find_def by transfer_prover
-
-lemma remove1_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "(A ===> list_all2 A ===> list_all2 A) remove1 remove1"
- unfolding remove1_def by transfer_prover
-
-lemma removeAll_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "(A ===> list_all2 A ===> list_all2 A) removeAll removeAll"
- unfolding removeAll_def by transfer_prover
-
-lemma distinct_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> op =) distinct distinct"
- unfolding distinct_def by transfer_prover
-
-lemma remdups_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A) remdups remdups"
- unfolding remdups_def by transfer_prover
-
-lemma replicate_transfer [transfer_rule]:
- "(op = ===> A ===> list_all2 A) replicate replicate"
- unfolding replicate_def by transfer_prover
-
-lemma length_transfer [transfer_rule]:
- "(list_all2 A ===> op =) length length"
- unfolding list_size_overloaded_def by transfer_prover
-
-lemma rotate1_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 A) rotate1 rotate1"
- unfolding rotate1_def by transfer_prover
-
-lemma funpow_transfer [transfer_rule]: (* FIXME: move to Transfer.thy *)
- "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"
- unfolding funpow_def by transfer_prover
-
-lemma rotate_transfer [transfer_rule]:
- "(op = ===> list_all2 A ===> list_all2 A) rotate rotate"
- unfolding rotate_def [abs_def] by transfer_prover
-
-lemma list_all2_transfer [transfer_rule]:
- "((A ===> B ===> op =) ===> list_all2 A ===> list_all2 B ===> op =)
- list_all2 list_all2"
- apply (subst (4) list_all2_def [abs_def])
- apply (subst (3) list_all2_def [abs_def])
- apply transfer_prover
- done
-
-lemma sublist_transfer [transfer_rule]:
- "(list_all2 A ===> set_rel (op =) ===> list_all2 A) sublist sublist"
- unfolding sublist_def [abs_def] by transfer_prover
-
-lemma partition_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> prod_rel (list_all2 A) (list_all2 A))
- partition partition"
- unfolding partition_def by transfer_prover
-
-lemma lists_transfer [transfer_rule]:
- "(set_rel A ===> set_rel (list_all2 A)) lists lists"
- apply (rule fun_relI, rule set_relI)
- apply (erule lists.induct, simp)
- apply (simp only: set_rel_def list_all2_Cons1, metis lists.Cons)
- apply (erule lists.induct, simp)
- apply (simp only: set_rel_def list_all2_Cons2, metis lists.Cons)
- done
-
-lemma set_Cons_transfer [transfer_rule]:
- "(set_rel A ===> set_rel (list_all2 A) ===> set_rel (list_all2 A))
- set_Cons set_Cons"
- unfolding fun_rel_def set_rel_def set_Cons_def
- apply safe
- apply (simp add: list_all2_Cons1, fast)
- apply (simp add: list_all2_Cons2, fast)
- done
-
-lemma listset_transfer [transfer_rule]:
- "(list_all2 (set_rel A) ===> set_rel (list_all2 A)) listset listset"
- unfolding listset_def by transfer_prover
-
-lemma null_transfer [transfer_rule]:
- "(list_all2 A ===> op =) List.null List.null"
- unfolding fun_rel_def List.null_def by auto
-
-lemma list_all_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> op =) list_all list_all"
- unfolding list_all_iff [abs_def] by transfer_prover
-
-lemma list_ex_transfer [transfer_rule]:
- "((A ===> op =) ===> list_all2 A ===> op =) list_ex list_ex"
- unfolding list_ex_iff [abs_def] by transfer_prover
-
-lemma splice_transfer [transfer_rule]:
- "(list_all2 A ===> list_all2 A ===> list_all2 A) splice splice"
- apply (rule fun_relI, erule list_all2_induct, simp add: fun_rel_def, simp)
- apply (rule fun_relI)
- apply (erule_tac xs=x in list_all2_induct, simp, simp add: fun_rel_def)
- done
-
-lemma listsum_transfer[transfer_rule]:
- assumes [transfer_rule]: "A 0 0"
- assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
- shows "(list_all2 A ===> A) listsum listsum"
- unfolding listsum_def[abs_def]
- by transfer_prover
-
-subsection {* Setup for lifting package *}
-
-lemma Quotient_list[quot_map]:
- assumes "Quotient R Abs Rep T"
- shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)"
-proof (unfold Quotient_alt_def, intro conjI allI impI)
- from assms have 1: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
- unfolding Quotient_alt_def by simp
- fix xs ys assume "list_all2 T xs ys" thus "map Abs xs = ys"
- by (induct, simp, simp add: 1)
-next
- from assms have 2: "\<And>x. T (Rep x) x"
- unfolding Quotient_alt_def by simp
- fix xs show "list_all2 T (map Rep xs) xs"
- by (induct xs, simp, simp add: 2)
-next
- from assms have 3: "\<And>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y"
- unfolding Quotient_alt_def by simp
- fix xs ys show "list_all2 R xs ys \<longleftrightarrow> list_all2 T xs (map Abs xs) \<and>
- list_all2 T ys (map Abs ys) \<and> map Abs xs = map Abs ys"
- by (induct xs ys rule: list_induct2', simp_all, metis 3)
-qed
-
-lemma list_invariant_commute [invariant_commute]:
- "list_all2 (Lifting.invariant P) = Lifting.invariant (list_all P)"
- apply (simp add: fun_eq_iff list_all2_def list_all_iff Lifting.invariant_def Ball_def)
- apply (intro allI)
- apply (induct_tac rule: list_induct2')
- apply simp_all
- apply metis
-done
-
-subsection {* Rules for quotient package *}
-
lemma list_quotient3 [quot_thm]:
assumes "Quotient3 R Abs Rep"
shows "Quotient3 (list_all2 R) (map Abs) (map Rep)"
--- a/src/HOL/Library/Quotient_Option.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_Option.thy Tue Aug 13 15:59:22 2013 +0200
@@ -1,5 +1,5 @@
(* Title: HOL/Library/Quotient_Option.thy
- Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
+ Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the option type *}
@@ -8,31 +8,7 @@
imports Main Quotient_Syntax
begin
-subsection {* Relator for option type *}
-
-fun
- option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
-where
- "option_rel R None None = True"
-| "option_rel R (Some x) None = False"
-| "option_rel R None (Some x) = False"
-| "option_rel R (Some x) (Some y) = R x y"
-
-lemma option_rel_unfold:
- "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
- | (Some x, Some y) \<Rightarrow> R x y
- | _ \<Rightarrow> False)"
- by (cases x) (cases y, simp_all)+
-
-fun option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
-where
- "option_pred R None = True"
-| "option_pred R (Some x) = R x"
-
-lemma option_pred_unfold:
- "option_pred P x = (case x of None \<Rightarrow> True
- | Some x \<Rightarrow> P x)"
-by (cases x) simp_all
+subsection {* Rules for the Quotient package *}
lemma option_rel_map1:
"option_rel R (Option.map f x) y \<longleftrightarrow> option_rel (\<lambda>x. R (f x)) x y"
@@ -46,37 +22,10 @@
"Option.map id = id"
by (simp add: id_def Option.map.identity fun_eq_iff)
-lemma option_rel_eq [id_simps, relator_eq]:
+lemma option_rel_eq [id_simps]:
"option_rel (op =) = (op =)"
by (simp add: option_rel_unfold fun_eq_iff split: option.split)
-lemma option_rel_mono[relator_mono]:
- assumes "A \<le> B"
- shows "(option_rel A) \<le> (option_rel B)"
-using assms by (auto simp: option_rel_unfold split: option.splits)
-
-lemma option_rel_OO[relator_distr]:
- "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
-by (rule ext)+ (auto simp: option_rel_unfold OO_def split: option.split)
-
-lemma Domainp_option[relator_domain]:
- assumes "Domainp A = P"
- shows "Domainp (option_rel A) = (option_pred P)"
-using assms unfolding Domainp_iff[abs_def] option_rel_unfold[abs_def] option_pred_unfold[abs_def]
-by (auto iff: fun_eq_iff split: option.split)
-
-lemma reflp_option_rel[reflexivity_rule]:
- "reflp R \<Longrightarrow> reflp (option_rel R)"
- unfolding reflp_def split_option_all by simp
-
-lemma left_total_option_rel[reflexivity_rule]:
- "left_total R \<Longrightarrow> left_total (option_rel R)"
- unfolding left_total_def split_option_all split_option_ex by simp
-
-lemma left_unique_option_rel [reflexivity_rule]:
- "left_unique R \<Longrightarrow> left_unique (option_rel R)"
- unfolding left_unique_def split_option_all by simp
-
lemma option_symp:
"symp R \<Longrightarrow> symp (option_rel R)"
unfolding symp_def split_option_all option_rel.simps by fast
@@ -89,65 +38,6 @@
"equivp R \<Longrightarrow> equivp (option_rel R)"
by (blast intro: equivpI reflp_option_rel option_symp option_transp elim: equivpE)
-lemma right_total_option_rel [transfer_rule]:
- "right_total R \<Longrightarrow> right_total (option_rel R)"
- unfolding right_total_def split_option_all split_option_ex by simp
-
-lemma right_unique_option_rel [transfer_rule]:
- "right_unique R \<Longrightarrow> right_unique (option_rel R)"
- unfolding right_unique_def split_option_all by simp
-
-lemma bi_total_option_rel [transfer_rule]:
- "bi_total R \<Longrightarrow> bi_total (option_rel R)"
- unfolding bi_total_def split_option_all split_option_ex by simp
-
-lemma bi_unique_option_rel [transfer_rule]:
- "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
- unfolding bi_unique_def split_option_all by simp
-
-subsection {* Transfer rules for transfer package *}
-
-lemma None_transfer [transfer_rule]: "(option_rel A) None None"
- by simp
-
-lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
- unfolding fun_rel_def by simp
-
-lemma option_case_transfer [transfer_rule]:
- "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
- unfolding fun_rel_def split_option_all by simp
-
-lemma option_map_transfer [transfer_rule]:
- "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
- unfolding Option.map_def by transfer_prover
-
-lemma option_bind_transfer [transfer_rule]:
- "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
- Option.bind Option.bind"
- unfolding fun_rel_def split_option_all by simp
-
-subsection {* Setup for lifting package *}
-
-lemma Quotient_option[quot_map]:
- assumes "Quotient R Abs Rep T"
- shows "Quotient (option_rel R) (Option.map Abs)
- (Option.map Rep) (option_rel T)"
- using assms unfolding Quotient_alt_def option_rel_unfold
- by (simp split: option.split)
-
-lemma option_invariant_commute [invariant_commute]:
- "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
- apply (simp add: fun_eq_iff Lifting.invariant_def)
- apply (intro allI)
- apply (case_tac x rule: option.exhaust)
- apply (case_tac xa rule: option.exhaust)
- apply auto[2]
- apply (case_tac xa rule: option.exhaust)
- apply auto
-done
-
-subsection {* Rules for quotient package *}
-
lemma option_quotient [quot_thm]:
assumes "Quotient3 R Abs Rep"
shows "Quotient3 (option_rel R) (Option.map Abs) (Option.map Rep)"
--- a/src/HOL/Library/Quotient_Product.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_Product.thy Tue Aug 13 15:59:22 2013 +0200
@@ -1,5 +1,5 @@
(* Title: HOL/Library/Quotient_Product.thy
- Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
+ Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the product type *}
@@ -8,137 +8,22 @@
imports Main Quotient_Syntax
begin
-subsection {* Relator for product type *}
-
-definition
- prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
-where
- "prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
-
-definition prod_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
-where "prod_pred R1 R2 = (\<lambda>(a, b). R1 a \<and> R2 b)"
-
-lemma prod_rel_apply [simp]:
- "prod_rel R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
- by (simp add: prod_rel_def)
-
-lemma prod_pred_apply [simp]:
- "prod_pred P1 P2 (a, b) \<longleftrightarrow> P1 a \<and> P2 b"
- by (simp add: prod_pred_def)
+subsection {* Rules for the Quotient package *}
lemma map_pair_id [id_simps]:
shows "map_pair id id = id"
by (simp add: fun_eq_iff)
-lemma prod_rel_eq [id_simps, relator_eq]:
+lemma prod_rel_eq [id_simps]:
shows "prod_rel (op =) (op =) = (op =)"
by (simp add: fun_eq_iff)
-lemma prod_rel_mono[relator_mono]:
- assumes "A \<le> C"
- assumes "B \<le> D"
- shows "(prod_rel A B) \<le> (prod_rel C D)"
-using assms by (auto simp: prod_rel_def)
-
-lemma prod_rel_OO[relator_distr]:
- "(prod_rel A B) OO (prod_rel C D) = prod_rel (A OO C) (B OO D)"
-by (rule ext)+ (auto simp: prod_rel_def OO_def)
-
-lemma Domainp_prod[relator_domain]:
- assumes "Domainp T1 = P1"
- assumes "Domainp T2 = P2"
- shows "Domainp (prod_rel T1 T2) = (prod_pred P1 P2)"
-using assms unfolding prod_rel_def prod_pred_def by blast
-
-lemma reflp_prod_rel [reflexivity_rule]:
- assumes "reflp R1"
- assumes "reflp R2"
- shows "reflp (prod_rel R1 R2)"
-using assms by (auto intro!: reflpI elim: reflpE)
-
-lemma left_total_prod_rel [reflexivity_rule]:
- assumes "left_total R1"
- assumes "left_total R2"
- shows "left_total (prod_rel R1 R2)"
- using assms unfolding left_total_def prod_rel_def by auto
-
-lemma left_unique_prod_rel [reflexivity_rule]:
- assumes "left_unique R1" and "left_unique R2"
- shows "left_unique (prod_rel R1 R2)"
- using assms unfolding left_unique_def prod_rel_def by auto
-
lemma prod_equivp [quot_equiv]:
assumes "equivp R1"
assumes "equivp R2"
shows "equivp (prod_rel R1 R2)"
using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE)
-lemma right_total_prod_rel [transfer_rule]:
- assumes "right_total R1" and "right_total R2"
- shows "right_total (prod_rel R1 R2)"
- using assms unfolding right_total_def prod_rel_def by auto
-
-lemma right_unique_prod_rel [transfer_rule]:
- assumes "right_unique R1" and "right_unique R2"
- shows "right_unique (prod_rel R1 R2)"
- using assms unfolding right_unique_def prod_rel_def by auto
-
-lemma bi_total_prod_rel [transfer_rule]:
- assumes "bi_total R1" and "bi_total R2"
- shows "bi_total (prod_rel R1 R2)"
- using assms unfolding bi_total_def prod_rel_def by auto
-
-lemma bi_unique_prod_rel [transfer_rule]:
- assumes "bi_unique R1" and "bi_unique R2"
- shows "bi_unique (prod_rel R1 R2)"
- using assms unfolding bi_unique_def prod_rel_def by auto
-
-subsection {* Transfer rules for transfer package *}
-
-lemma Pair_transfer [transfer_rule]: "(A ===> B ===> prod_rel A B) Pair Pair"
- unfolding fun_rel_def prod_rel_def by simp
-
-lemma fst_transfer [transfer_rule]: "(prod_rel A B ===> A) fst fst"
- unfolding fun_rel_def prod_rel_def by simp
-
-lemma snd_transfer [transfer_rule]: "(prod_rel A B ===> B) snd snd"
- unfolding fun_rel_def prod_rel_def by simp
-
-lemma prod_case_transfer [transfer_rule]:
- "((A ===> B ===> C) ===> prod_rel A B ===> C) prod_case prod_case"
- unfolding fun_rel_def prod_rel_def by simp
-
-lemma curry_transfer [transfer_rule]:
- "((prod_rel A B ===> C) ===> A ===> B ===> C) curry curry"
- unfolding curry_def by transfer_prover
-
-lemma map_pair_transfer [transfer_rule]:
- "((A ===> C) ===> (B ===> D) ===> prod_rel A B ===> prod_rel C D)
- map_pair map_pair"
- unfolding map_pair_def [abs_def] by transfer_prover
-
-lemma prod_rel_transfer [transfer_rule]:
- "((A ===> B ===> op =) ===> (C ===> D ===> op =) ===>
- prod_rel A C ===> prod_rel B D ===> op =) prod_rel prod_rel"
- unfolding fun_rel_def by auto
-
-subsection {* Setup for lifting package *}
-
-lemma Quotient_prod[quot_map]:
- assumes "Quotient R1 Abs1 Rep1 T1"
- assumes "Quotient R2 Abs2 Rep2 T2"
- shows "Quotient (prod_rel R1 R2) (map_pair Abs1 Abs2)
- (map_pair Rep1 Rep2) (prod_rel T1 T2)"
- using assms unfolding Quotient_alt_def by auto
-
-lemma prod_invariant_commute [invariant_commute]:
- "prod_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (prod_pred P1 P2)"
- apply (simp add: fun_eq_iff prod_rel_def prod_pred_def Lifting.invariant_def)
- apply blast
-done
-
-subsection {* Rules for quotient package *}
-
lemma prod_quotient [quot_thm]:
assumes "Quotient3 R1 Abs1 Rep1"
assumes "Quotient3 R2 Abs2 Rep2"
--- a/src/HOL/Library/Quotient_Set.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_Set.thy Tue Aug 13 15:59:22 2013 +0200
@@ -8,273 +8,7 @@
imports Main Quotient_Syntax
begin
-subsection {* Relator for set type *}
-
-definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
- where "set_rel R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
-
-lemma set_relI:
- assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
- assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
- shows "set_rel R A B"
- using assms unfolding set_rel_def by simp
-
-lemma set_rel_conversep: "set_rel (conversep R) = conversep (set_rel R)"
- unfolding set_rel_def by auto
-
-lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)"
- unfolding set_rel_def fun_eq_iff by auto
-
-lemma set_rel_mono[relator_mono]:
- assumes "A \<le> B"
- shows "set_rel A \<le> set_rel B"
-using assms unfolding set_rel_def by blast
-
-lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (R OO S)"
- apply (rule sym)
- apply (intro ext, rename_tac X Z)
- apply (rule iffI)
- apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
- apply (simp add: set_rel_def, fast)
- apply (simp add: set_rel_def, fast)
- apply (simp add: set_rel_def, fast)
- done
-
-lemma Domainp_set[relator_domain]:
- assumes "Domainp T = R"
- shows "Domainp (set_rel T) = (\<lambda>A. Ball A R)"
-using assms unfolding set_rel_def Domainp_iff[abs_def]
-apply (intro ext)
-apply (rule iffI)
-apply blast
-apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
-done
-
-lemma reflp_set_rel[reflexivity_rule]: "reflp R \<Longrightarrow> reflp (set_rel R)"
- unfolding reflp_def set_rel_def by fast
-
-lemma left_total_set_rel[reflexivity_rule]:
- "left_total A \<Longrightarrow> left_total (set_rel A)"
- unfolding left_total_def set_rel_def
- apply safe
- apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
-done
-
-lemma left_unique_set_rel[reflexivity_rule]:
- "left_unique A \<Longrightarrow> left_unique (set_rel A)"
- unfolding left_unique_def set_rel_def
- by fast
-
-lemma symp_set_rel: "symp R \<Longrightarrow> symp (set_rel R)"
- unfolding symp_def set_rel_def by fast
-
-lemma transp_set_rel: "transp R \<Longrightarrow> transp (set_rel R)"
- unfolding transp_def set_rel_def by fast
-
-lemma equivp_set_rel: "equivp R \<Longrightarrow> equivp (set_rel R)"
- by (blast intro: equivpI reflp_set_rel symp_set_rel transp_set_rel
- elim: equivpE)
-
-lemma right_total_set_rel [transfer_rule]:
- "right_total A \<Longrightarrow> right_total (set_rel A)"
- unfolding right_total_def set_rel_def
- by (rule allI, rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast)
-
-lemma right_unique_set_rel [transfer_rule]:
- "right_unique A \<Longrightarrow> right_unique (set_rel A)"
- unfolding right_unique_def set_rel_def by fast
-
-lemma bi_total_set_rel [transfer_rule]:
- "bi_total A \<Longrightarrow> bi_total (set_rel A)"
- unfolding bi_total_def set_rel_def
- apply safe
- apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
- apply (rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast)
- done
-
-lemma bi_unique_set_rel [transfer_rule]:
- "bi_unique A \<Longrightarrow> bi_unique (set_rel A)"
- unfolding bi_unique_def set_rel_def by fast
-
-subsection {* Transfer rules for transfer package *}
-
-subsubsection {* Unconditional transfer rules *}
-
-lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}"
- unfolding set_rel_def by simp
-
-lemma insert_transfer [transfer_rule]:
- "(A ===> set_rel A ===> set_rel A) insert insert"
- unfolding fun_rel_def set_rel_def by auto
-
-lemma union_transfer [transfer_rule]:
- "(set_rel A ===> set_rel A ===> set_rel A) union union"
- unfolding fun_rel_def set_rel_def by auto
-
-lemma Union_transfer [transfer_rule]:
- "(set_rel (set_rel A) ===> set_rel A) Union Union"
- unfolding fun_rel_def set_rel_def by simp fast
-
-lemma image_transfer [transfer_rule]:
- "((A ===> B) ===> set_rel A ===> set_rel B) image image"
- unfolding fun_rel_def set_rel_def by simp fast
-
-lemma UNION_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION"
- unfolding SUP_def [abs_def] by transfer_prover
-
-lemma Ball_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> op =) ===> op =) Ball Ball"
- unfolding set_rel_def fun_rel_def by fast
-
-lemma Bex_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex"
- unfolding set_rel_def fun_rel_def by fast
-
-lemma Pow_transfer [transfer_rule]:
- "(set_rel A ===> set_rel (set_rel A)) Pow Pow"
- apply (rule fun_relI, rename_tac X Y, rule set_relI)
- apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
- apply (simp add: set_rel_def, fast)
- apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
- apply (simp add: set_rel_def, fast)
- done
-
-lemma set_rel_transfer [transfer_rule]:
- "((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =)
- set_rel set_rel"
- unfolding fun_rel_def set_rel_def by fast
-
-
-subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
-
-lemma member_transfer [transfer_rule]:
- assumes "bi_unique A"
- shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)"
- using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
-
-lemma right_total_Collect_transfer[transfer_rule]:
- assumes "right_total A"
- shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
- using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast
-
-lemma Collect_transfer [transfer_rule]:
- assumes "bi_total A"
- shows "((A ===> op =) ===> set_rel A) Collect Collect"
- using assms unfolding fun_rel_def set_rel_def bi_total_def by fast
-
-lemma inter_transfer [transfer_rule]:
- assumes "bi_unique A"
- shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter"
- using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
-
-lemma Diff_transfer [transfer_rule]:
- assumes "bi_unique A"
- shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)"
- using assms unfolding fun_rel_def set_rel_def bi_unique_def
- unfolding Ball_def Bex_def Diff_eq
- by (safe, simp, metis, simp, metis)
-
-lemma subset_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)"
- unfolding subset_eq [abs_def] by transfer_prover
-
-lemma right_total_UNIV_transfer[transfer_rule]:
- assumes "right_total A"
- shows "(set_rel A) (Collect (Domainp A)) UNIV"
- using assms unfolding right_total_def set_rel_def Domainp_iff by blast
-
-lemma UNIV_transfer [transfer_rule]:
- assumes "bi_total A"
- shows "(set_rel A) UNIV UNIV"
- using assms unfolding set_rel_def bi_total_def by simp
-
-lemma right_total_Compl_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
- shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
- unfolding Compl_eq [abs_def]
- by (subst Collect_conj_eq[symmetric]) transfer_prover
-
-lemma Compl_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
- shows "(set_rel A ===> set_rel A) uminus uminus"
- unfolding Compl_eq [abs_def] by transfer_prover
-
-lemma right_total_Inter_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
- shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
- unfolding Inter_eq[abs_def]
- by (subst Collect_conj_eq[symmetric]) transfer_prover
-
-lemma Inter_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
- shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter"
- unfolding Inter_eq [abs_def] by transfer_prover
-
-lemma filter_transfer [transfer_rule]:
- assumes [transfer_rule]: "bi_unique A"
- shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter"
- unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast
-
-lemma bi_unique_set_rel_lemma:
- assumes "bi_unique R" and "set_rel R X Y"
- obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
-proof
- let ?f = "\<lambda>x. THE y. R x y"
- from assms show f: "\<forall>x\<in>X. R x (?f x)"
- apply (clarsimp simp add: set_rel_def)
- apply (drule (1) bspec, clarify)
- apply (rule theI2, assumption)
- apply (simp add: bi_unique_def)
- apply assumption
- done
- from assms show "Y = image ?f X"
- apply safe
- apply (clarsimp simp add: set_rel_def)
- apply (drule (1) bspec, clarify)
- apply (rule image_eqI)
- apply (rule the_equality [symmetric], assumption)
- apply (simp add: bi_unique_def)
- apply assumption
- apply (clarsimp simp add: set_rel_def)
- apply (frule (1) bspec, clarify)
- apply (rule theI2, assumption)
- apply (clarsimp simp add: bi_unique_def)
- apply (simp add: bi_unique_def, metis)
- done
- show "inj_on ?f X"
- apply (rule inj_onI)
- apply (drule f [rule_format])
- apply (drule f [rule_format])
- apply (simp add: assms(1) [unfolded bi_unique_def])
- done
-qed
-
-lemma finite_transfer [transfer_rule]:
- "bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite"
- by (rule fun_relI, erule (1) bi_unique_set_rel_lemma,
- auto dest: finite_imageD)
-
-lemma card_transfer [transfer_rule]:
- "bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card"
- by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, simp add: card_image)
-
-subsection {* Setup for lifting package *}
-
-lemma Quotient_set[quot_map]:
- assumes "Quotient R Abs Rep T"
- shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)"
- using assms unfolding Quotient_alt_def4
- apply (simp add: set_rel_OO[symmetric] set_rel_conversep)
- apply (simp add: set_rel_def, fast)
- done
-
-lemma set_invariant_commute [invariant_commute]:
- "set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
- unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast
-
-subsection {* Contravariant set map (vimage) and set relator *}
+subsection {* Contravariant set map (vimage) and set relator, rules for the Quotient package *}
definition "vset_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
--- a/src/HOL/Library/Quotient_Sum.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_Sum.thy Tue Aug 13 15:59:22 2013 +0200
@@ -1,5 +1,5 @@
(* Title: HOL/Library/Quotient_Sum.thy
- Author: Cezary Kaliszyk, Christian Urban and Brian Huffman
+ Author: Cezary Kaliszyk and Christian Urban
*)
header {* Quotient infrastructure for the sum type *}
@@ -8,31 +8,7 @@
imports Main Quotient_Syntax
begin
-subsection {* Relator for sum type *}
-
-fun
- sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
-where
- "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
-| "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
-| "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
-| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
-
-lemma sum_rel_unfold:
- "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
- | (Inr x, Inr y) \<Rightarrow> R2 x y
- | _ \<Rightarrow> False)"
- by (cases x) (cases y, simp_all)+
-
-fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool"
-where
- "sum_pred P1 P2 (Inl a) = P1 a"
-| "sum_pred P1 P2 (Inr a) = P2 a"
-
-lemma sum_pred_unfold:
- "sum_pred P1 P2 x = (case x of Inl x \<Rightarrow> P1 x
- | Inr x \<Rightarrow> P2 x)"
-by (cases x) simp_all
+subsection {* Rules for the Quotient package *}
lemma sum_rel_map1:
"sum_rel R1 R2 (sum_map f1 f2 x) y \<longleftrightarrow> sum_rel (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
@@ -46,39 +22,10 @@
"sum_map id id = id"
by (simp add: id_def sum_map.identity fun_eq_iff)
-lemma sum_rel_eq [id_simps, relator_eq]:
+lemma sum_rel_eq [id_simps]:
"sum_rel (op =) (op =) = (op =)"
by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
-lemma sum_rel_mono[relator_mono]:
- assumes "A \<le> C"
- assumes "B \<le> D"
- shows "(sum_rel A B) \<le> (sum_rel C D)"
-using assms by (auto simp: sum_rel_unfold split: sum.splits)
-
-lemma sum_rel_OO[relator_distr]:
- "(sum_rel A B) OO (sum_rel C D) = sum_rel (A OO C) (B OO D)"
-by (rule ext)+ (auto simp add: sum_rel_unfold OO_def split_sum_ex split: sum.split)
-
-lemma Domainp_sum[relator_domain]:
- assumes "Domainp R1 = P1"
- assumes "Domainp R2 = P2"
- shows "Domainp (sum_rel R1 R2) = (sum_pred P1 P2)"
-using assms
-by (auto simp add: Domainp_iff split_sum_ex sum_pred_unfold iff: fun_eq_iff split: sum.split)
-
-lemma reflp_sum_rel[reflexivity_rule]:
- "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
- unfolding reflp_def split_sum_all sum_rel.simps by fast
-
-lemma left_total_sum_rel[reflexivity_rule]:
- "left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"
- using assms unfolding left_total_def split_sum_all split_sum_ex by simp
-
-lemma left_unique_sum_rel [reflexivity_rule]:
- "left_unique R1 \<Longrightarrow> left_unique R2 \<Longrightarrow> left_unique (sum_rel R1 R2)"
- using assms unfolding left_unique_def split_sum_all by simp
-
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
@@ -91,50 +38,6 @@
"equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
by (blast intro: equivpI reflp_sum_rel sum_symp sum_transp elim: equivpE)
-lemma right_total_sum_rel [transfer_rule]:
- "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"
- unfolding right_total_def split_sum_all split_sum_ex by simp
-
-lemma right_unique_sum_rel [transfer_rule]:
- "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"
- unfolding right_unique_def split_sum_all by simp
-
-lemma bi_total_sum_rel [transfer_rule]:
- "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"
- using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
-
-lemma bi_unique_sum_rel [transfer_rule]:
- "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"
- using assms unfolding bi_unique_def split_sum_all by simp
-
-subsection {* Transfer rules for transfer package *}
-
-lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl"
- unfolding fun_rel_def by simp
-
-lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"
- unfolding fun_rel_def by simp
-
-lemma sum_case_transfer [transfer_rule]:
- "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"
- unfolding fun_rel_def sum_rel_unfold by (simp split: sum.split)
-
-subsection {* Setup for lifting package *}
-
-lemma Quotient_sum[quot_map]:
- assumes "Quotient R1 Abs1 Rep1 T1"
- assumes "Quotient R2 Abs2 Rep2 T2"
- shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2)
- (sum_map Rep1 Rep2) (sum_rel T1 T2)"
- using assms unfolding Quotient_alt_def
- by (simp add: split_sum_all)
-
-lemma sum_invariant_commute [invariant_commute]:
- "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
- by (auto simp add: fun_eq_iff Lifting.invariant_def sum_rel_unfold sum_pred_unfold split: sum.split)
-
-subsection {* Rules for quotient package *}
-
lemma sum_quotient [quot_thm]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Lifting_Option.thy Tue Aug 13 15:59:22 2013 +0200
@@ -0,0 +1,125 @@
+(* Title: HOL/Lifting_Option.thy
+ Author: Brian Huffman and Ondrej Kuncar
+*)
+
+header {* Setup for Lifting/Transfer for the option type *}
+
+theory Lifting_Option
+imports Lifting FunDef
+begin
+
+subsection {* Relator and predicator properties *}
+
+fun
+ option_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> bool"
+where
+ "option_rel R None None = True"
+| "option_rel R (Some x) None = False"
+| "option_rel R None (Some x) = False"
+| "option_rel R (Some x) (Some y) = R x y"
+
+lemma option_rel_unfold:
+ "option_rel R x y = (case (x, y) of (None, None) \<Rightarrow> True
+ | (Some x, Some y) \<Rightarrow> R x y
+ | _ \<Rightarrow> False)"
+ by (cases x) (cases y, simp_all)+
+
+fun option_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a option \<Rightarrow> bool"
+where
+ "option_pred R None = True"
+| "option_pred R (Some x) = R x"
+
+lemma option_pred_unfold:
+ "option_pred P x = (case x of None \<Rightarrow> True
+ | Some x \<Rightarrow> P x)"
+by (cases x) simp_all
+
+lemma option_rel_eq [relator_eq]:
+ "option_rel (op =) = (op =)"
+ by (simp add: option_rel_unfold fun_eq_iff split: option.split)
+
+lemma option_rel_mono[relator_mono]:
+ assumes "A \<le> B"
+ shows "(option_rel A) \<le> (option_rel B)"
+using assms by (auto simp: option_rel_unfold split: option.splits)
+
+lemma option_rel_OO[relator_distr]:
+ "(option_rel A) OO (option_rel B) = option_rel (A OO B)"
+by (rule ext)+ (auto simp: option_rel_unfold OO_def split: option.split)
+
+lemma Domainp_option[relator_domain]:
+ assumes "Domainp A = P"
+ shows "Domainp (option_rel A) = (option_pred P)"
+using assms unfolding Domainp_iff[abs_def] option_rel_unfold[abs_def] option_pred_unfold[abs_def]
+by (auto iff: fun_eq_iff split: option.split)
+
+lemma reflp_option_rel[reflexivity_rule]:
+ "reflp R \<Longrightarrow> reflp (option_rel R)"
+ unfolding reflp_def split_option_all by simp
+
+lemma left_total_option_rel[reflexivity_rule]:
+ "left_total R \<Longrightarrow> left_total (option_rel R)"
+ unfolding left_total_def split_option_all split_option_ex by simp
+
+lemma left_unique_option_rel [reflexivity_rule]:
+ "left_unique R \<Longrightarrow> left_unique (option_rel R)"
+ unfolding left_unique_def split_option_all by simp
+
+lemma right_total_option_rel [transfer_rule]:
+ "right_total R \<Longrightarrow> right_total (option_rel R)"
+ unfolding right_total_def split_option_all split_option_ex by simp
+
+lemma right_unique_option_rel [transfer_rule]:
+ "right_unique R \<Longrightarrow> right_unique (option_rel R)"
+ unfolding right_unique_def split_option_all by simp
+
+lemma bi_total_option_rel [transfer_rule]:
+ "bi_total R \<Longrightarrow> bi_total (option_rel R)"
+ unfolding bi_total_def split_option_all split_option_ex by simp
+
+lemma bi_unique_option_rel [transfer_rule]:
+ "bi_unique R \<Longrightarrow> bi_unique (option_rel R)"
+ unfolding bi_unique_def split_option_all by simp
+
+lemma option_invariant_commute [invariant_commute]:
+ "option_rel (Lifting.invariant P) = Lifting.invariant (option_pred P)"
+ by (auto simp add: fun_eq_iff Lifting.invariant_def split_option_all)
+
+subsection {* Quotient theorem for the Lifting package *}
+
+lemma Quotient_option[quot_map]:
+ assumes "Quotient R Abs Rep T"
+ shows "Quotient (option_rel R) (Option.map Abs)
+ (Option.map Rep) (option_rel T)"
+ using assms unfolding Quotient_alt_def option_rel_unfold
+ by (simp split: option.split)
+
+subsection {* Transfer rules for the Transfer package *}
+
+context
+begin
+interpretation lifting_syntax .
+
+lemma None_transfer [transfer_rule]: "(option_rel A) None None"
+ by simp
+
+lemma Some_transfer [transfer_rule]: "(A ===> option_rel A) Some Some"
+ unfolding fun_rel_def by simp
+
+lemma option_case_transfer [transfer_rule]:
+ "(B ===> (A ===> B) ===> option_rel A ===> B) option_case option_case"
+ unfolding fun_rel_def split_option_all by simp
+
+lemma option_map_transfer [transfer_rule]:
+ "((A ===> B) ===> option_rel A ===> option_rel B) Option.map Option.map"
+ unfolding Option.map_def by transfer_prover
+
+lemma option_bind_transfer [transfer_rule]:
+ "(option_rel A ===> (A ===> option_rel B) ===> option_rel B)
+ Option.bind Option.bind"
+ unfolding fun_rel_def split_option_all by simp
+
+end
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Lifting_Product.thy Tue Aug 13 15:59:22 2013 +0200
@@ -0,0 +1,135 @@
+(* Title: HOL/Lifting_Product.thy
+ Author: Brian Huffman and Ondrej Kuncar
+*)
+
+header {* Setup for Lifting/Transfer for the product type *}
+
+theory Lifting_Product
+imports Lifting
+begin
+
+subsection {* Relator and predicator properties *}
+
+definition
+ prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
+where
+ "prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
+
+definition prod_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
+where "prod_pred R1 R2 = (\<lambda>(a, b). R1 a \<and> R2 b)"
+
+lemma prod_rel_apply [simp]:
+ "prod_rel R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
+ by (simp add: prod_rel_def)
+
+lemma prod_pred_apply [simp]:
+ "prod_pred P1 P2 (a, b) \<longleftrightarrow> P1 a \<and> P2 b"
+ by (simp add: prod_pred_def)
+
+lemma prod_rel_eq [relator_eq]:
+ shows "prod_rel (op =) (op =) = (op =)"
+ by (simp add: fun_eq_iff)
+
+lemma prod_rel_mono[relator_mono]:
+ assumes "A \<le> C"
+ assumes "B \<le> D"
+ shows "(prod_rel A B) \<le> (prod_rel C D)"
+using assms by (auto simp: prod_rel_def)
+
+lemma prod_rel_OO[relator_distr]:
+ "(prod_rel A B) OO (prod_rel C D) = prod_rel (A OO C) (B OO D)"
+by (rule ext)+ (auto simp: prod_rel_def OO_def)
+
+lemma Domainp_prod[relator_domain]:
+ assumes "Domainp T1 = P1"
+ assumes "Domainp T2 = P2"
+ shows "Domainp (prod_rel T1 T2) = (prod_pred P1 P2)"
+using assms unfolding prod_rel_def prod_pred_def by blast
+
+lemma reflp_prod_rel [reflexivity_rule]:
+ assumes "reflp R1"
+ assumes "reflp R2"
+ shows "reflp (prod_rel R1 R2)"
+using assms by (auto intro!: reflpI elim: reflpE)
+
+lemma left_total_prod_rel [reflexivity_rule]:
+ assumes "left_total R1"
+ assumes "left_total R2"
+ shows "left_total (prod_rel R1 R2)"
+ using assms unfolding left_total_def prod_rel_def by auto
+
+lemma left_unique_prod_rel [reflexivity_rule]:
+ assumes "left_unique R1" and "left_unique R2"
+ shows "left_unique (prod_rel R1 R2)"
+ using assms unfolding left_unique_def prod_rel_def by auto
+
+lemma right_total_prod_rel [transfer_rule]:
+ assumes "right_total R1" and "right_total R2"
+ shows "right_total (prod_rel R1 R2)"
+ using assms unfolding right_total_def prod_rel_def by auto
+
+lemma right_unique_prod_rel [transfer_rule]:
+ assumes "right_unique R1" and "right_unique R2"
+ shows "right_unique (prod_rel R1 R2)"
+ using assms unfolding right_unique_def prod_rel_def by auto
+
+lemma bi_total_prod_rel [transfer_rule]:
+ assumes "bi_total R1" and "bi_total R2"
+ shows "bi_total (prod_rel R1 R2)"
+ using assms unfolding bi_total_def prod_rel_def by auto
+
+lemma bi_unique_prod_rel [transfer_rule]:
+ assumes "bi_unique R1" and "bi_unique R2"
+ shows "bi_unique (prod_rel R1 R2)"
+ using assms unfolding bi_unique_def prod_rel_def by auto
+
+lemma prod_invariant_commute [invariant_commute]:
+ "prod_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (prod_pred P1 P2)"
+ by (simp add: fun_eq_iff prod_rel_def prod_pred_def Lifting.invariant_def) blast
+
+subsection {* Quotient theorem for the Lifting package *}
+
+lemma Quotient_prod[quot_map]:
+ assumes "Quotient R1 Abs1 Rep1 T1"
+ assumes "Quotient R2 Abs2 Rep2 T2"
+ shows "Quotient (prod_rel R1 R2) (map_pair Abs1 Abs2)
+ (map_pair Rep1 Rep2) (prod_rel T1 T2)"
+ using assms unfolding Quotient_alt_def by auto
+
+subsection {* Transfer rules for the Transfer package *}
+
+context
+begin
+interpretation lifting_syntax .
+
+lemma Pair_transfer [transfer_rule]: "(A ===> B ===> prod_rel A B) Pair Pair"
+ unfolding fun_rel_def prod_rel_def by simp
+
+lemma fst_transfer [transfer_rule]: "(prod_rel A B ===> A) fst fst"
+ unfolding fun_rel_def prod_rel_def by simp
+
+lemma snd_transfer [transfer_rule]: "(prod_rel A B ===> B) snd snd"
+ unfolding fun_rel_def prod_rel_def by simp
+
+lemma prod_case_transfer [transfer_rule]:
+ "((A ===> B ===> C) ===> prod_rel A B ===> C) prod_case prod_case"
+ unfolding fun_rel_def prod_rel_def by simp
+
+lemma curry_transfer [transfer_rule]:
+ "((prod_rel A B ===> C) ===> A ===> B ===> C) curry curry"
+ unfolding curry_def by transfer_prover
+
+lemma map_pair_transfer [transfer_rule]:
+ "((A ===> C) ===> (B ===> D) ===> prod_rel A B ===> prod_rel C D)
+ map_pair map_pair"
+ unfolding map_pair_def [abs_def] by transfer_prover
+
+lemma prod_rel_transfer [transfer_rule]:
+ "((A ===> B ===> op =) ===> (C ===> D ===> op =) ===>
+ prod_rel A C ===> prod_rel B D ===> op =) prod_rel prod_rel"
+ unfolding fun_rel_def by auto
+
+end
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Lifting_Set.thy Tue Aug 13 15:59:22 2013 +0200
@@ -0,0 +1,273 @@
+(* Title: HOL/Lifting_Set.thy
+ Author: Brian Huffman and Ondrej Kuncar
+*)
+
+header {* Setup for Lifting/Transfer for the set type *}
+
+theory Lifting_Set
+imports Lifting
+begin
+
+subsection {* Relator and predicator properties *}
+
+definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
+ where "set_rel R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
+
+lemma set_relI:
+ assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
+ assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
+ shows "set_rel R A B"
+ using assms unfolding set_rel_def by simp
+
+lemma set_rel_conversep: "set_rel (conversep R) = conversep (set_rel R)"
+ unfolding set_rel_def by auto
+
+lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)"
+ unfolding set_rel_def fun_eq_iff by auto
+
+lemma set_rel_mono[relator_mono]:
+ assumes "A \<le> B"
+ shows "set_rel A \<le> set_rel B"
+using assms unfolding set_rel_def by blast
+
+lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (R OO S)"
+ apply (rule sym)
+ apply (intro ext, rename_tac X Z)
+ apply (rule iffI)
+ apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
+ apply (simp add: set_rel_def, fast)
+ apply (simp add: set_rel_def, fast)
+ apply (simp add: set_rel_def, fast)
+ done
+
+lemma Domainp_set[relator_domain]:
+ assumes "Domainp T = R"
+ shows "Domainp (set_rel T) = (\<lambda>A. Ball A R)"
+using assms unfolding set_rel_def Domainp_iff[abs_def]
+apply (intro ext)
+apply (rule iffI)
+apply blast
+apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
+done
+
+lemma reflp_set_rel[reflexivity_rule]: "reflp R \<Longrightarrow> reflp (set_rel R)"
+ unfolding reflp_def set_rel_def by fast
+
+lemma left_total_set_rel[reflexivity_rule]:
+ "left_total A \<Longrightarrow> left_total (set_rel A)"
+ unfolding left_total_def set_rel_def
+ apply safe
+ apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
+done
+
+lemma left_unique_set_rel[reflexivity_rule]:
+ "left_unique A \<Longrightarrow> left_unique (set_rel A)"
+ unfolding left_unique_def set_rel_def
+ by fast
+
+lemma right_total_set_rel [transfer_rule]:
+ "right_total A \<Longrightarrow> right_total (set_rel A)"
+ unfolding right_total_def set_rel_def
+ by (rule allI, rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast)
+
+lemma right_unique_set_rel [transfer_rule]:
+ "right_unique A \<Longrightarrow> right_unique (set_rel A)"
+ unfolding right_unique_def set_rel_def by fast
+
+lemma bi_total_set_rel [transfer_rule]:
+ "bi_total A \<Longrightarrow> bi_total (set_rel A)"
+ unfolding bi_total_def set_rel_def
+ apply safe
+ apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
+ apply (rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast)
+ done
+
+lemma bi_unique_set_rel [transfer_rule]:
+ "bi_unique A \<Longrightarrow> bi_unique (set_rel A)"
+ unfolding bi_unique_def set_rel_def by fast
+
+lemma set_invariant_commute [invariant_commute]:
+ "set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
+ unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast
+
+subsection {* Quotient theorem for the Lifting package *}
+
+lemma Quotient_set[quot_map]:
+ assumes "Quotient R Abs Rep T"
+ shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)"
+ using assms unfolding Quotient_alt_def4
+ apply (simp add: set_rel_OO[symmetric] set_rel_conversep)
+ apply (simp add: set_rel_def, fast)
+ done
+
+subsection {* Transfer rules for the Transfer package *}
+
+subsubsection {* Unconditional transfer rules *}
+
+context
+begin
+interpretation lifting_syntax .
+
+lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}"
+ unfolding set_rel_def by simp
+
+lemma insert_transfer [transfer_rule]:
+ "(A ===> set_rel A ===> set_rel A) insert insert"
+ unfolding fun_rel_def set_rel_def by auto
+
+lemma union_transfer [transfer_rule]:
+ "(set_rel A ===> set_rel A ===> set_rel A) union union"
+ unfolding fun_rel_def set_rel_def by auto
+
+lemma Union_transfer [transfer_rule]:
+ "(set_rel (set_rel A) ===> set_rel A) Union Union"
+ unfolding fun_rel_def set_rel_def by simp fast
+
+lemma image_transfer [transfer_rule]:
+ "((A ===> B) ===> set_rel A ===> set_rel B) image image"
+ unfolding fun_rel_def set_rel_def by simp fast
+
+lemma UNION_transfer [transfer_rule]:
+ "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION"
+ unfolding SUP_def [abs_def] by transfer_prover
+
+lemma Ball_transfer [transfer_rule]:
+ "(set_rel A ===> (A ===> op =) ===> op =) Ball Ball"
+ unfolding set_rel_def fun_rel_def by fast
+
+lemma Bex_transfer [transfer_rule]:
+ "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex"
+ unfolding set_rel_def fun_rel_def by fast
+
+lemma Pow_transfer [transfer_rule]:
+ "(set_rel A ===> set_rel (set_rel A)) Pow Pow"
+ apply (rule fun_relI, rename_tac X Y, rule set_relI)
+ apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
+ apply (simp add: set_rel_def, fast)
+ apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
+ apply (simp add: set_rel_def, fast)
+ done
+
+lemma set_rel_transfer [transfer_rule]:
+ "((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =)
+ set_rel set_rel"
+ unfolding fun_rel_def set_rel_def by fast
+
+
+subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
+
+lemma member_transfer [transfer_rule]:
+ assumes "bi_unique A"
+ shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)"
+ using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
+
+lemma right_total_Collect_transfer[transfer_rule]:
+ assumes "right_total A"
+ shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
+ using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast
+
+lemma Collect_transfer [transfer_rule]:
+ assumes "bi_total A"
+ shows "((A ===> op =) ===> set_rel A) Collect Collect"
+ using assms unfolding fun_rel_def set_rel_def bi_total_def by fast
+
+lemma inter_transfer [transfer_rule]:
+ assumes "bi_unique A"
+ shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter"
+ using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
+
+lemma Diff_transfer [transfer_rule]:
+ assumes "bi_unique A"
+ shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)"
+ using assms unfolding fun_rel_def set_rel_def bi_unique_def
+ unfolding Ball_def Bex_def Diff_eq
+ by (safe, simp, metis, simp, metis)
+
+lemma subset_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)"
+ unfolding subset_eq [abs_def] by transfer_prover
+
+lemma right_total_UNIV_transfer[transfer_rule]:
+ assumes "right_total A"
+ shows "(set_rel A) (Collect (Domainp A)) UNIV"
+ using assms unfolding right_total_def set_rel_def Domainp_iff by blast
+
+lemma UNIV_transfer [transfer_rule]:
+ assumes "bi_total A"
+ shows "(set_rel A) UNIV UNIV"
+ using assms unfolding set_rel_def bi_total_def by simp
+
+lemma right_total_Compl_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
+ shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
+ unfolding Compl_eq [abs_def]
+ by (subst Collect_conj_eq[symmetric]) transfer_prover
+
+lemma Compl_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
+ shows "(set_rel A ===> set_rel A) uminus uminus"
+ unfolding Compl_eq [abs_def] by transfer_prover
+
+lemma right_total_Inter_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
+ shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
+ unfolding Inter_eq[abs_def]
+ by (subst Collect_conj_eq[symmetric]) transfer_prover
+
+lemma Inter_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
+ shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter"
+ unfolding Inter_eq [abs_def] by transfer_prover
+
+lemma filter_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter"
+ unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast
+
+lemma bi_unique_set_rel_lemma:
+ assumes "bi_unique R" and "set_rel R X Y"
+ obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
+proof
+ let ?f = "\<lambda>x. THE y. R x y"
+ from assms show f: "\<forall>x\<in>X. R x (?f x)"
+ apply (clarsimp simp add: set_rel_def)
+ apply (drule (1) bspec, clarify)
+ apply (rule theI2, assumption)
+ apply (simp add: bi_unique_def)
+ apply assumption
+ done
+ from assms show "Y = image ?f X"
+ apply safe
+ apply (clarsimp simp add: set_rel_def)
+ apply (drule (1) bspec, clarify)
+ apply (rule image_eqI)
+ apply (rule the_equality [symmetric], assumption)
+ apply (simp add: bi_unique_def)
+ apply assumption
+ apply (clarsimp simp add: set_rel_def)
+ apply (frule (1) bspec, clarify)
+ apply (rule theI2, assumption)
+ apply (clarsimp simp add: bi_unique_def)
+ apply (simp add: bi_unique_def, metis)
+ done
+ show "inj_on ?f X"
+ apply (rule inj_onI)
+ apply (drule f [rule_format])
+ apply (drule f [rule_format])
+ apply (simp add: assms(1) [unfolded bi_unique_def])
+ done
+qed
+
+lemma finite_transfer [transfer_rule]:
+ "bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite"
+ by (rule fun_relI, erule (1) bi_unique_set_rel_lemma,
+ auto dest: finite_imageD)
+
+lemma card_transfer [transfer_rule]:
+ "bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card"
+ by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, simp add: card_image)
+
+end
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Lifting_Sum.thy Tue Aug 13 15:59:22 2013 +0200
@@ -0,0 +1,119 @@
+(* Title: HOL/Lifting_Sum.thy
+ Author: Brian Huffman and Ondrej Kuncar
+*)
+
+header {* Setup for Lifting/Transfer for the sum type *}
+
+theory Lifting_Sum
+imports Lifting FunDef
+begin
+
+subsection {* Relator and predicator properties *}
+
+fun
+ sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
+where
+ "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
+| "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
+| "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
+| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
+
+lemma sum_rel_unfold:
+ "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
+ | (Inr x, Inr y) \<Rightarrow> R2 x y
+ | _ \<Rightarrow> False)"
+ by (cases x) (cases y, simp_all)+
+
+fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool"
+where
+ "sum_pred P1 P2 (Inl a) = P1 a"
+| "sum_pred P1 P2 (Inr a) = P2 a"
+
+lemma sum_pred_unfold:
+ "sum_pred P1 P2 x = (case x of Inl x \<Rightarrow> P1 x
+ | Inr x \<Rightarrow> P2 x)"
+by (cases x) simp_all
+
+lemma sum_rel_eq [relator_eq]:
+ "sum_rel (op =) (op =) = (op =)"
+ by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
+
+lemma sum_rel_mono[relator_mono]:
+ assumes "A \<le> C"
+ assumes "B \<le> D"
+ shows "(sum_rel A B) \<le> (sum_rel C D)"
+using assms by (auto simp: sum_rel_unfold split: sum.splits)
+
+lemma sum_rel_OO[relator_distr]:
+ "(sum_rel A B) OO (sum_rel C D) = sum_rel (A OO C) (B OO D)"
+by (rule ext)+ (auto simp add: sum_rel_unfold OO_def split_sum_ex split: sum.split)
+
+lemma Domainp_sum[relator_domain]:
+ assumes "Domainp R1 = P1"
+ assumes "Domainp R2 = P2"
+ shows "Domainp (sum_rel R1 R2) = (sum_pred P1 P2)"
+using assms
+by (auto simp add: Domainp_iff split_sum_ex sum_pred_unfold iff: fun_eq_iff split: sum.split)
+
+lemma reflp_sum_rel[reflexivity_rule]:
+ "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
+ unfolding reflp_def split_sum_all sum_rel.simps by fast
+
+lemma left_total_sum_rel[reflexivity_rule]:
+ "left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"
+ using assms unfolding left_total_def split_sum_all split_sum_ex by simp
+
+lemma left_unique_sum_rel [reflexivity_rule]:
+ "left_unique R1 \<Longrightarrow> left_unique R2 \<Longrightarrow> left_unique (sum_rel R1 R2)"
+ using assms unfolding left_unique_def split_sum_all by simp
+
+lemma right_total_sum_rel [transfer_rule]:
+ "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"
+ unfolding right_total_def split_sum_all split_sum_ex by simp
+
+lemma right_unique_sum_rel [transfer_rule]:
+ "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"
+ unfolding right_unique_def split_sum_all by simp
+
+lemma bi_total_sum_rel [transfer_rule]:
+ "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"
+ using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
+
+lemma bi_unique_sum_rel [transfer_rule]:
+ "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"
+ using assms unfolding bi_unique_def split_sum_all by simp
+
+lemma sum_invariant_commute [invariant_commute]:
+ "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
+ by (auto simp add: fun_eq_iff Lifting.invariant_def sum_rel_unfold sum_pred_unfold split: sum.split)
+
+subsection {* Quotient theorem for the Lifting package *}
+
+lemma Quotient_sum[quot_map]:
+ assumes "Quotient R1 Abs1 Rep1 T1"
+ assumes "Quotient R2 Abs2 Rep2 T2"
+ shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2)
+ (sum_map Rep1 Rep2) (sum_rel T1 T2)"
+ using assms unfolding Quotient_alt_def
+ by (simp add: split_sum_all)
+
+subsection {* Transfer rules for the Transfer package *}
+
+context
+begin
+interpretation lifting_syntax .
+
+lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl"
+ unfolding fun_rel_def by simp
+
+lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"
+ unfolding fun_rel_def by simp
+
+lemma sum_case_transfer [transfer_rule]:
+ "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"
+ unfolding fun_rel_def sum_rel_unfold by (simp split: sum.split)
+
+end
+
+end
+
--- a/src/HOL/List.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/List.thy Tue Aug 13 15:59:22 2013 +0200
@@ -5,7 +5,7 @@
header {* The datatype of finite lists *}
theory List
-imports Presburger Code_Numeral Quotient ATP
+imports Presburger Code_Numeral Quotient ATP Lifting_Set Lifting_Option Lifting_Product
begin
datatype 'a list =
@@ -6228,5 +6228,338 @@
"wf (set xs) = acyclic (set xs)"
by (simp add: wf_iff_acyclic_if_finite)
+subsection {* Setup for Lifting/Transfer *}
+
+subsubsection {* Relator and predicator properties *}
+
+lemma list_all2_eq'[relator_eq]:
+ "list_all2 (op =) = (op =)"
+by (rule ext)+ (simp add: list_all2_eq)
+
+lemma list_all2_mono'[relator_mono]:
+ assumes "A \<le> B"
+ shows "(list_all2 A) \<le> (list_all2 B)"
+using assms by (auto intro: list_all2_mono)
+
+lemma list_all2_OO[relator_distr]: "list_all2 A OO list_all2 B = list_all2 (A OO B)"
+proof (intro ext iffI)
+ fix xs ys
+ assume "list_all2 (A OO B) xs ys"
+ thus "(list_all2 A OO list_all2 B) xs ys"
+ unfolding OO_def
+ by (induct, simp, simp add: list_all2_Cons1 list_all2_Cons2, fast)
+next
+ fix xs ys
+ assume "(list_all2 A OO list_all2 B) xs ys"
+ then obtain zs where "list_all2 A xs zs" and "list_all2 B zs ys" ..
+ thus "list_all2 (A OO B) xs ys"
+ by (induct arbitrary: ys, simp, clarsimp simp add: list_all2_Cons1, fast)
+qed
+
+lemma Domainp_list[relator_domain]:
+ assumes "Domainp A = P"
+ shows "Domainp (list_all2 A) = (list_all P)"
+proof -
+ {
+ fix x
+ have *: "\<And>x. (\<exists>y. A x y) = P x" using assms unfolding Domainp_iff by blast
+ have "(\<exists>y. (list_all2 A x y)) = list_all P x"
+ by (induction x) (simp_all add: * list_all2_Cons1)
+ }
+ then show ?thesis
+ unfolding Domainp_iff[abs_def]
+ by (auto iff: fun_eq_iff)
+qed
+
+lemma reflp_list_all2[reflexivity_rule]:
+ assumes "reflp R"
+ shows "reflp (list_all2 R)"
+proof (rule reflpI)
+ from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
+ fix xs
+ show "list_all2 R xs xs"
+ by (induct xs) (simp_all add: *)
+qed
+
+lemma left_total_list_all2[reflexivity_rule]:
+ "left_total R \<Longrightarrow> left_total (list_all2 R)"
+ unfolding left_total_def
+ apply safe
+ apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
+done
+
+lemma left_unique_list_all2 [reflexivity_rule]:
+ "left_unique R \<Longrightarrow> left_unique (list_all2 R)"
+ unfolding left_unique_def
+ apply (subst (2) all_comm, subst (1) all_comm)
+ apply (rule allI, rename_tac zs, induct_tac zs)
+ apply (auto simp add: list_all2_Cons2)
+ done
+
+lemma right_total_list_all2 [transfer_rule]:
+ "right_total R \<Longrightarrow> right_total (list_all2 R)"
+ unfolding right_total_def
+ by (rule allI, induct_tac y, simp, simp add: list_all2_Cons2)
+
+lemma right_unique_list_all2 [transfer_rule]:
+ "right_unique R \<Longrightarrow> right_unique (list_all2 R)"
+ unfolding right_unique_def
+ apply (rule allI, rename_tac xs, induct_tac xs)
+ apply (auto simp add: list_all2_Cons1)
+ done
+
+lemma bi_total_list_all2 [transfer_rule]:
+ "bi_total A \<Longrightarrow> bi_total (list_all2 A)"
+ unfolding bi_total_def
+ apply safe
+ apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
+ apply (rename_tac ys, induct_tac ys, simp, simp add: list_all2_Cons2)
+ done
+
+lemma bi_unique_list_all2 [transfer_rule]:
+ "bi_unique A \<Longrightarrow> bi_unique (list_all2 A)"
+ unfolding bi_unique_def
+ apply (rule conjI)
+ apply (rule allI, rename_tac xs, induct_tac xs)
+ apply (simp, force simp add: list_all2_Cons1)
+ apply (subst (2) all_comm, subst (1) all_comm)
+ apply (rule allI, rename_tac xs, induct_tac xs)
+ apply (simp, force simp add: list_all2_Cons2)
+ done
+
+lemma list_invariant_commute [invariant_commute]:
+ "list_all2 (Lifting.invariant P) = Lifting.invariant (list_all P)"
+ apply (simp add: fun_eq_iff list_all2_def list_all_iff Lifting.invariant_def Ball_def)
+ apply (intro allI)
+ apply (induct_tac rule: list_induct2')
+ apply simp_all
+ apply fastforce
+done
+
+subsubsection {* Quotient theorem for the Lifting package *}
+
+lemma Quotient_list[quot_map]:
+ assumes "Quotient R Abs Rep T"
+ shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)"
+proof (unfold Quotient_alt_def, intro conjI allI impI)
+ from assms have 1: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
+ unfolding Quotient_alt_def by simp
+ fix xs ys assume "list_all2 T xs ys" thus "map Abs xs = ys"
+ by (induct, simp, simp add: 1)
+next
+ from assms have 2: "\<And>x. T (Rep x) x"
+ unfolding Quotient_alt_def by simp
+ fix xs show "list_all2 T (map Rep xs) xs"
+ by (induct xs, simp, simp add: 2)
+next
+ from assms have 3: "\<And>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y"
+ unfolding Quotient_alt_def by simp
+ fix xs ys show "list_all2 R xs ys \<longleftrightarrow> list_all2 T xs (map Abs xs) \<and>
+ list_all2 T ys (map Abs ys) \<and> map Abs xs = map Abs ys"
+ by (induct xs ys rule: list_induct2', simp_all, metis 3)
+qed
+
+subsubsection {* Transfer rules for the Transfer package *}
+
+context
+begin
+interpretation lifting_syntax .
+
+lemma Nil_transfer [transfer_rule]: "(list_all2 A) [] []"
+ by simp
+
+lemma Cons_transfer [transfer_rule]:
+ "(A ===> list_all2 A ===> list_all2 A) Cons Cons"
+ unfolding fun_rel_def by simp
+
+lemma list_case_transfer [transfer_rule]:
+ "(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B)
+ list_case list_case"
+ unfolding fun_rel_def by (simp split: list.split)
+
+lemma list_rec_transfer [transfer_rule]:
+ "(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B)
+ list_rec list_rec"
+ unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all)
+
+lemma tl_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 A) tl tl"
+ unfolding tl_def by transfer_prover
+
+lemma butlast_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 A) butlast butlast"
+ by (rule fun_relI, erule list_all2_induct, auto)
+
+lemma set_transfer [transfer_rule]:
+ "(list_all2 A ===> set_rel A) set set"
+ unfolding set_def by transfer_prover
+
+lemma map_transfer [transfer_rule]:
+ "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"
+ unfolding List.map_def by transfer_prover
+
+lemma append_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 A ===> list_all2 A) append append"
+ unfolding List.append_def by transfer_prover
+
+lemma rev_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 A) rev rev"
+ unfolding List.rev_def by transfer_prover
+
+lemma filter_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> list_all2 A) filter filter"
+ unfolding List.filter_def by transfer_prover
+
+lemma fold_transfer [transfer_rule]:
+ "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) fold fold"
+ unfolding List.fold_def by transfer_prover
+
+lemma foldr_transfer [transfer_rule]:
+ "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr"
+ unfolding List.foldr_def by transfer_prover
+
+lemma foldl_transfer [transfer_rule]:
+ "((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl"
+ unfolding List.foldl_def by transfer_prover
+
+lemma concat_transfer [transfer_rule]:
+ "(list_all2 (list_all2 A) ===> list_all2 A) concat concat"
+ unfolding List.concat_def by transfer_prover
+
+lemma drop_transfer [transfer_rule]:
+ "(op = ===> list_all2 A ===> list_all2 A) drop drop"
+ unfolding List.drop_def by transfer_prover
+
+lemma take_transfer [transfer_rule]:
+ "(op = ===> list_all2 A ===> list_all2 A) take take"
+ unfolding List.take_def by transfer_prover
+
+lemma list_update_transfer [transfer_rule]:
+ "(list_all2 A ===> op = ===> A ===> list_all2 A) list_update list_update"
+ unfolding list_update_def by transfer_prover
+
+lemma takeWhile_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> list_all2 A) takeWhile takeWhile"
+ unfolding takeWhile_def by transfer_prover
+
+lemma dropWhile_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> list_all2 A) dropWhile dropWhile"
+ unfolding dropWhile_def by transfer_prover
+
+lemma zip_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 B ===> list_all2 (prod_rel A B)) zip zip"
+ unfolding zip_def by transfer_prover
+
+lemma insert_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(A ===> list_all2 A ===> list_all2 A) List.insert List.insert"
+ unfolding List.insert_def [abs_def] by transfer_prover
+
+lemma find_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> option_rel A) List.find List.find"
+ unfolding List.find_def by transfer_prover
+
+lemma remove1_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(A ===> list_all2 A ===> list_all2 A) remove1 remove1"
+ unfolding remove1_def by transfer_prover
+
+lemma removeAll_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(A ===> list_all2 A ===> list_all2 A) removeAll removeAll"
+ unfolding removeAll_def by transfer_prover
+
+lemma distinct_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(list_all2 A ===> op =) distinct distinct"
+ unfolding distinct_def by transfer_prover
+
+lemma remdups_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(list_all2 A ===> list_all2 A) remdups remdups"
+ unfolding remdups_def by transfer_prover
+
+lemma replicate_transfer [transfer_rule]:
+ "(op = ===> A ===> list_all2 A) replicate replicate"
+ unfolding replicate_def by transfer_prover
+
+lemma length_transfer [transfer_rule]:
+ "(list_all2 A ===> op =) length length"
+ unfolding list_size_overloaded_def by transfer_prover
+
+lemma rotate1_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 A) rotate1 rotate1"
+ unfolding rotate1_def by transfer_prover
+
+lemma rotate_transfer [transfer_rule]:
+ "(op = ===> list_all2 A ===> list_all2 A) rotate rotate"
+ unfolding rotate_def [abs_def] by transfer_prover
+
+lemma list_all2_transfer [transfer_rule]:
+ "((A ===> B ===> op =) ===> list_all2 A ===> list_all2 B ===> op =)
+ list_all2 list_all2"
+ apply (subst (4) list_all2_def [abs_def])
+ apply (subst (3) list_all2_def [abs_def])
+ apply transfer_prover
+ done
+
+lemma sublist_transfer [transfer_rule]:
+ "(list_all2 A ===> set_rel (op =) ===> list_all2 A) sublist sublist"
+ unfolding sublist_def [abs_def] by transfer_prover
+
+lemma partition_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> prod_rel (list_all2 A) (list_all2 A))
+ partition partition"
+ unfolding partition_def by transfer_prover
+
+lemma lists_transfer [transfer_rule]:
+ "(set_rel A ===> set_rel (list_all2 A)) lists lists"
+ apply (rule fun_relI, rule set_relI)
+ apply (erule lists.induct, simp)
+ apply (simp only: set_rel_def list_all2_Cons1, metis lists.Cons)
+ apply (erule lists.induct, simp)
+ apply (simp only: set_rel_def list_all2_Cons2, metis lists.Cons)
+ done
+
+lemma set_Cons_transfer [transfer_rule]:
+ "(set_rel A ===> set_rel (list_all2 A) ===> set_rel (list_all2 A))
+ set_Cons set_Cons"
+ unfolding fun_rel_def set_rel_def set_Cons_def
+ apply safe
+ apply (simp add: list_all2_Cons1, fast)
+ apply (simp add: list_all2_Cons2, fast)
+ done
+
+lemma listset_transfer [transfer_rule]:
+ "(list_all2 (set_rel A) ===> set_rel (list_all2 A)) listset listset"
+ unfolding listset_def by transfer_prover
+
+lemma null_transfer [transfer_rule]:
+ "(list_all2 A ===> op =) List.null List.null"
+ unfolding fun_rel_def List.null_def by auto
+
+lemma list_all_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> op =) list_all list_all"
+ unfolding list_all_iff [abs_def] by transfer_prover
+
+lemma list_ex_transfer [transfer_rule]:
+ "((A ===> op =) ===> list_all2 A ===> op =) list_ex list_ex"
+ unfolding list_ex_iff [abs_def] by transfer_prover
+
+lemma splice_transfer [transfer_rule]:
+ "(list_all2 A ===> list_all2 A ===> list_all2 A) splice splice"
+ apply (rule fun_relI, erule list_all2_induct, simp add: fun_rel_def, simp)
+ apply (rule fun_relI)
+ apply (erule_tac xs=x in list_all2_induct, simp, simp add: fun_rel_def)
+ done
+
+lemma listsum_transfer[transfer_rule]:
+ assumes [transfer_rule]: "A 0 0"
+ assumes [transfer_rule]: "(A ===> A ===> A) op + op +"
+ shows "(list_all2 A ===> A) listsum listsum"
+ unfolding listsum_def[abs_def]
+ by transfer_prover
+
end
+end
--- a/src/HOL/Main.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Main.thy Tue Aug 13 15:59:22 2013 +0200
@@ -1,7 +1,7 @@
header {* Main HOL *}
theory Main
-imports Predicate_Compile Nitpick Extraction
+imports Predicate_Compile Nitpick Extraction Lifting_Sum
begin
text {*
--- a/src/HOL/Rat.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Rat.thy Tue Aug 13 15:59:22 2013 +0200
@@ -49,8 +49,8 @@
morphisms Rep_Rat Abs_Rat
by (rule part_equivp_ratrel)
-lemma Domainp_cr_rat [transfer_domain_rule]: "Domainp cr_rat = (\<lambda>x. snd x \<noteq> 0)"
-by (simp add: rat.domain)
+lemma Domainp_cr_rat [transfer_domain_rule]: "Domainp pcr_rat = (\<lambda>x. snd x \<noteq> 0)"
+by (simp add: rat.domain_eq)
subsubsection {* Representation and basic operations *}