restoring notion of primitive vs. derived operations in terms of generated code;
established _paramatric suffix for parametricity rules
--- a/src/HOL/Library/Mapping.thy Wed Apr 09 14:08:18 2014 +0200
+++ b/src/HOL/Library/Mapping.thy Wed Apr 09 14:08:25 2014 +0200
@@ -10,25 +10,29 @@
subsection {* Parametricity transfer rules *}
+lemma map_of_foldr: -- {* FIXME move *}
+ "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"
+ using map_add_map_of_foldr [of Map.empty] by auto
+
context
begin
interpretation lifting_syntax .
-lemma empty_transfer:
+lemma empty_parametric:
"(A ===> rel_option B) Map.empty Map.empty"
by transfer_prover
-lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
+lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
by transfer_prover
-lemma update_transfer:
+lemma update_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
(\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
by transfer_prover
-lemma delete_transfer:
+lemma delete_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
(\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
@@ -38,23 +42,31 @@
"(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
by (auto simp add: is_none_def rel_fun_def rel_option_iff split: option.split)
-lemma dom_transfer:
+lemma dom_parametric:
assumes [transfer_rule]: "bi_total A"
shows "((A ===> rel_option B) ===> rel_set A) dom dom"
unfolding dom_def [abs_def] is_none_def [symmetric] by transfer_prover
-lemma map_of_transfer [transfer_rule]:
+lemma map_of_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique R1"
shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
unfolding map_of_def by transfer_prover
-lemma tabulate_transfer:
+lemma map_entry_parametric [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique A"
+ shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
+ (\<lambda>k f m. (case m k of None \<Rightarrow> m
+ | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
+ | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
+ by transfer_prover
+
+lemma tabulate_parametric:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
(\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
by transfer_prover
-lemma bulkload_transfer:
+lemma bulkload_parametric:
"(list_all2 A ===> HOL.eq ===> rel_option A)
(\<lambda>xs k. if k < length xs then Some (xs ! k) else None) (\<lambda>xs k. if k < length xs then Some (xs ! k) else None)"
proof
@@ -72,20 +84,13 @@
done
qed
-lemma map_transfer:
+lemma map_parametric:
"((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
(\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
by transfer_prover
-lemma map_entry_transfer:
- assumes [transfer_rule]: "bi_unique A"
- shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B)
- (\<lambda>k f m. (case m k of None \<Rightarrow> m
- | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
- | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
- by transfer_prover
+end
-end
subsection {* Type definition and primitive operations *}
@@ -96,28 +101,28 @@
setup_lifting (no_code) type_definition_mapping
lift_definition empty :: "('a, 'b) mapping"
- is Map.empty parametric empty_transfer .
+ is Map.empty parametric empty_parametric .
lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option"
- is "\<lambda>m k. m k" parametric lookup_transfer .
+ is "\<lambda>m k. m k" parametric lookup_parametric .
lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
- is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_transfer .
+ is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric .
lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
- is "\<lambda>k m. m(k := None)" parametric delete_transfer .
+ is "\<lambda>k m. m(k := None)" parametric delete_parametric .
lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
- is dom parametric dom_transfer .
+ is dom parametric dom_parametric .
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))" parametric tabulate_transfer .
+ is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_parametric .
lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping"
- is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .
+ is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric .
lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
- is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_transfer .
+ is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
subsection {* Functorial structure *}
@@ -148,11 +153,14 @@
where
"default k v m = (if k \<in> keys m then m else update k v m)"
+text {* Manual derivation of transfer rule is non-trivial *}
+
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)))" parametric map_entry_transfer .
+ | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_parametric .
-lemma map_entry_code [code]: "map_entry k f m = (case lookup m k of None \<Rightarrow> m
+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
@@ -160,12 +168,9 @@
where
"map_default k v f m = map_entry k f (default k v m)"
-lift_definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
- is map_of parametric map_of_transfer .
-
-lemma of_alist_code [code]:
+definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
+where
"of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
- by transfer (simp add: map_add_map_of_foldr [symmetric])
instantiation mapping :: (type, type) equal
begin
@@ -189,6 +194,11 @@
shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
by (unfold equal) transfer_prover
+lemma of_alist_transfer [transfer_rule]:
+ assumes [transfer_rule]: "bi_unique R1"
+ shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist"
+ unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover
+
end
@@ -380,12 +390,8 @@
"tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
proof transfer
fix f :: "'a \<Rightarrow> 'b" and xs
- from map_add_map_of_foldr
- have "Map.empty ++ map_of (List.map (\<lambda>k. (k, f k)) xs) =
- foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) (List.map (\<lambda>k. (k, f k)) xs) Map.empty"
- .
- then have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
- by (simp add: foldr_map comp_def)
+ have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
+ by (simp add: foldr_map comp_def map_of_foldr)
also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs"
by (rule foldr_fold) (simp add: fun_eq_iff)
ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"