src/HOL/Library/Mapping.thy

-(* Author: Florian Haftmann, TU Muenchen *)
+(*  Title:      HOL/Library/Mapping.thy
+    Author:     Florian Haftmann and Ondrej Kuncar
+*)
 
 header {* An abstract view on maps for code generation. *}
 
 theory Mapping
-imports Main
+imports Main "~~/src/HOL/Library/Quotient_Option"
 begin
 
 subsection {* Type definition and primitive operations *}
 
 typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
-  morphisms lookup Mapping ..
-
-lemma lookup_Mapping [simp]:
-  "lookup (Mapping f) = f"
-  by (rule Mapping_inverse) rule
-
-lemma Mapping_lookup [simp]:
-  "Mapping (lookup m) = m"
-  by (fact lookup_inverse)
-
-lemma Mapping_inject [simp]:
-  "Mapping f = Mapping g \<longleftrightarrow> f = g"
-  by (simp add: Mapping_inject)
-
-lemma mapping_eq_iff:
-  "m = n \<longleftrightarrow> lookup m = lookup n"
-  by (simp add: lookup_inject)
-
-lemma mapping_eqI:
-  "lookup m = lookup n \<Longrightarrow> m = n"
-  by (simp add: mapping_eq_iff)
-
-definition empty :: "('a, 'b) mapping" where
-  "empty = Mapping (\<lambda>_. None)"
-
-definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
-  "update k v m = Mapping ((lookup m)(k \<mapsto> v))"
-
-definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
-  "delete k m = Mapping ((lookup m)(k := None))"
-
+  morphisms rep Mapping ..
+
+setup_lifting(no_code) type_definition_mapping
+
+lift_definition empty :: "('a, 'b) mapping" is "(\<lambda>_. None)" .
+
+lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option" is "\<lambda>m k. m k" .
+
+lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k v m. m(k \<mapsto> v)" .
+
+lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k m. m(k := None)" .
+
+lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" is dom .
+
+lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" is
+  "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" .
+
+lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" is
+  "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" .
+
+lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" is
+  "\<lambda>f g m. (Option.map g \<circ> m \<circ> f)" .
 
 subsection {* Functorial structure *}
 
-definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" where
-  "map f g m = Mapping (Option.map g \<circ> lookup m \<circ> f)"
-
-lemma lookup_map [simp]:
-  "lookup (map f g m) = Option.map g \<circ> lookup m \<circ> f"
-  by (simp add: map_def)
-
 enriched_type map: map
-  by (simp_all add: mapping_eq_iff fun_eq_iff Option.map.compositionality Option.map.id)
-
+  by (transfer, auto simp add: fun_eq_iff Option.map.compositionality Option.map.id)+
 
 subsection {* Derived operations *}
-
-definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" where
-  "keys m = dom (lookup m)"
 
 definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
   "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
 
 definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where

   "replace k v m = (if k \<in> keys m then update k v m else m)"
 
 definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
   "default k v m = (if k \<in> keys m then m else update k v m)"
 
-definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
-  "map_entry k f m = (case lookup m k of None \<Rightarrow> m
-    | Some v \<Rightarrow> update k (f v) m)" 
+lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
+  "\<lambda>k f m. (case m k of None \<Rightarrow> m
+    | Some v \<Rightarrow> m (k \<mapsto> (f v)))" .
+
+lemma map_entry_code [code]: "map_entry k f m = (case lookup m k of None \<Rightarrow> m
+    | Some v \<Rightarrow> update k (f v) m)" by transfer rule
 
 definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
   "map_default k v f m = map_entry k f (default k v m)" 
 
-definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" where
-  "tabulate ks f = Mapping (map_of (List.map (\<lambda>k. (k, f k)) ks))"
-
-definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" where
-  "bulkload xs = Mapping (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
-
-
 subsection {* Properties *}
 
-lemma keys_is_none_lookup [code_unfold]:
+lemma keys_is_none_rep [code_unfold]:
   "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
-  by (auto simp add: keys_def is_none_def)
-
-lemma lookup_empty [simp]:
-  "lookup empty = Map.empty"
-  by (simp add: empty_def)
-
-lemma lookup_update [simp]:
-  "lookup (update k v m) = (lookup m) (k \<mapsto> v)"
-  by (simp add: update_def)
-
-lemma lookup_delete [simp]:
-  "lookup (delete k m) = (lookup m) (k := None)"
-  by (simp add: delete_def)
-
-lemma lookup_map_entry [simp]:
-  "lookup (map_entry k f m) = (lookup m) (k := Option.map f (lookup m k))"
-  by (cases "lookup m k") (simp_all add: map_entry_def fun_eq_iff)
-
-lemma lookup_tabulate [simp]:
-  "lookup (tabulate ks f) = (Some o f) |` set ks"
-  by (induct ks) (auto simp add: tabulate_def restrict_map_def fun_eq_iff)
-
-lemma lookup_bulkload [simp]:
-  "lookup (bulkload xs) = (\<lambda>k. if k < length xs then Some (xs ! k) else None)"
-  by (simp add: bulkload_def)
+  by transfer (auto simp add: is_none_def)
+
+lemma tabulate_alt_def:
+  "map_of (List.map (\<lambda>k. (k, f k)) ks) = (Some o f) |` set ks"
+  by (induct ks) (auto simp add: tabulate_def restrict_map_def)
 
 lemma update_update:
   "update k v (update k w m) = update k v m"
   "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
-  by (rule mapping_eqI, simp add: fun_upd_twist)+
+  by (transfer, simp add: fun_upd_twist)+
 
 lemma update_delete [simp]:
   "update k v (delete k m) = update k v m"
-  by (rule mapping_eqI) simp
+  by transfer simp
 
 lemma delete_update:
   "delete k (update k v m) = delete k m"
   "k \<noteq> l \<Longrightarrow> delete k (update l v m) = update l v (delete k m)"
-  by (rule mapping_eqI, simp add: fun_upd_twist)+
+  by (transfer, simp add: fun_upd_twist)+
 
 lemma delete_empty [simp]:
   "delete k empty = empty"
-  by (rule mapping_eqI) simp
+  by transfer simp
 
 lemma replace_update:
   "k \<notin> keys m \<Longrightarrow> replace k v m = m"
   "k \<in> keys m \<Longrightarrow> replace k v m = update k v m"
-  by (rule mapping_eqI) (auto simp add: replace_def fun_upd_twist)+
+  by (transfer, auto simp add: replace_def fun_upd_twist)+
 
 lemma size_empty [simp]:
   "size empty = 0"
-  by (simp add: size_def keys_def)
+  unfolding size_def by transfer simp
 
 lemma size_update:
   "finite (keys m) \<Longrightarrow> size (update k v m) =
     (if k \<in> keys m then size m else Suc (size m))"
-  by (auto simp add: size_def insert_dom keys_def)
+  unfolding size_def by transfer (auto simp add: insert_dom)
 
 lemma size_delete:
   "size (delete k m) = (if k \<in> keys m then size m - 1 else size m)"
-  by (simp add: size_def keys_def)
+  unfolding size_def by transfer simp
 
 lemma size_tabulate [simp]:
   "size (tabulate ks f) = length (remdups ks)"
-  by (simp add: size_def distinct_card [of "remdups ks", symmetric] comp_def keys_def)
+  unfolding size_def by transfer (auto simp add: tabulate_alt_def card_set comp_def)
 
 lemma bulkload_tabulate:
   "bulkload xs = tabulate [0..<length xs] (nth xs)"
-  by (rule mapping_eqI) (simp add: fun_eq_iff)
-
-lemma is_empty_empty: (*FIXME*)
-  "is_empty m \<longleftrightarrow> m = Mapping Map.empty"
-  by (cases m) (simp add: is_empty_def keys_def)
-
-lemma is_empty_empty' [simp]:
+  by transfer (auto simp add: tabulate_alt_def)
+
+lemma is_empty_empty [simp]:
   "is_empty empty"
-  by (simp add: is_empty_empty empty_def) 
+  unfolding is_empty_def by transfer simp 
 
 lemma is_empty_update [simp]:
   "\<not> is_empty (update k v m)"
-  by (simp add: is_empty_empty update_def)
+  unfolding is_empty_def by transfer simp
 
 lemma is_empty_delete:
   "is_empty (delete k m) \<longleftrightarrow> is_empty m \<or> keys m = {k}"
-  by (auto simp add: delete_def is_empty_def keys_def simp del: dom_eq_empty_conv)
+  unfolding is_empty_def by transfer (auto simp del: dom_eq_empty_conv)
 
 lemma is_empty_replace [simp]:
   "is_empty (replace k v m) \<longleftrightarrow> is_empty m"
-  by (auto simp add: replace_def) (simp add: is_empty_def)
+  unfolding is_empty_def replace_def by transfer auto
 
 lemma is_empty_default [simp]:
   "\<not> is_empty (default k v m)"
-  by (auto simp add: default_def) (simp add: is_empty_def)
+  unfolding is_empty_def default_def by transfer auto
 
 lemma is_empty_map_entry [simp]:
   "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
-  by (cases "lookup m k")
-    (auto simp add: map_entry_def, simp add: is_empty_empty)
+  unfolding is_empty_def 
+  apply transfer by (case_tac "m k") auto
 
 lemma is_empty_map_default [simp]:
   "\<not> is_empty (map_default k v f m)"
   by (simp add: map_default_def)
 
 lemma keys_empty [simp]:
   "keys empty = {}"
-  by (simp add: keys_def)
+  by transfer simp
 
 lemma keys_update [simp]:
   "keys (update k v m) = insert k (keys m)"
-  by (simp add: keys_def)
+  by transfer simp
 
 lemma keys_delete [simp]:
   "keys (delete k m) = keys m - {k}"
-  by (simp add: keys_def)
+  by transfer simp
 
 lemma keys_replace [simp]:
   "keys (replace k v m) = keys m"
-  by (auto simp add: keys_def replace_def)
+  unfolding replace_def by transfer (simp add: insert_absorb)
 
 lemma keys_default [simp]:
   "keys (default k v m) = insert k (keys m)"
-  by (auto simp add: keys_def default_def)
+  unfolding default_def by transfer (simp add: insert_absorb)
 
 lemma keys_map_entry [simp]:
   "keys (map_entry k f m) = keys m"
-  by (auto simp add: keys_def)
+  apply transfer by (case_tac "m k") auto
 
 lemma keys_map_default [simp]:
   "keys (map_default k v f m) = insert k (keys m)"
   by (simp add: map_default_def)
 
 lemma keys_tabulate [simp]:
   "keys (tabulate ks f) = set ks"
-  by (simp add: tabulate_def keys_def map_of_map_restrict o_def)
+  by transfer (simp add: map_of_map_restrict o_def)
 
 lemma keys_bulkload [simp]:
   "keys (bulkload xs) = {0..<length xs}"
   by (simp add: keys_tabulate bulkload_tabulate)
 

 code_datatype empty update
 
 instantiation mapping :: (type, type) equal
 begin
 
-definition [code del]:
-  "HOL.equal m n \<longleftrightarrow> lookup m = lookup n"
+lift_definition equal_mapping :: "('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping \<Rightarrow> bool" is "op=" .
 
 instance proof
-qed (simp add: equal_mapping_def mapping_eq_iff)
+qed(transfer, rule)
 
 end
 
 
-hide_const (open) empty is_empty lookup update delete ordered_keys keys size
+hide_const (open) empty is_empty rep lookup update delete ordered_keys keys size
   replace default map_entry map_default tabulate bulkload map
 
 end