1.1 --- a/src/HOL/Library/Mapping.thy Thu Apr 10 17:48:54 2014 +0200
1.2 +++ b/src/HOL/Library/Mapping.thy Wed Apr 09 14:08:18 2014 +0200
1.3 @@ -12,61 +12,70 @@
1.4
1.5 context
1.6 begin
1.7 +
1.8 interpretation lifting_syntax .
1.9
1.10 -lemma empty_transfer: "(A ===> rel_option B) Map.empty Map.empty" by transfer_prover
1.11 +lemma empty_transfer:
1.12 + "(A ===> rel_option B) Map.empty Map.empty"
1.13 + by transfer_prover
1.14
1.15 -lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)" by transfer_prover
1.16 +lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
1.17 + by transfer_prover
1.18
1.19 lemma update_transfer:
1.20 assumes [transfer_rule]: "bi_unique A"
1.21 - shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
1.22 - (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
1.23 -by transfer_prover
1.24 + shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
1.25 + (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
1.26 + by transfer_prover
1.27
1.28 lemma delete_transfer:
1.29 assumes [transfer_rule]: "bi_unique A"
1.30 shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B)
1.31 - (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
1.32 -by transfer_prover
1.33 + (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
1.34 + by transfer_prover
1.35
1.36 -definition equal_None :: "'a option \<Rightarrow> bool" where "equal_None x \<equiv> x = None"
1.37 -
1.38 -lemma [transfer_rule]: "(rel_option A ===> op=) equal_None equal_None"
1.39 -unfolding rel_fun_def rel_option_iff equal_None_def by (auto split: option.split)
1.40 +lemma is_none_parametric [transfer_rule]:
1.41 + "(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
1.42 + by (auto simp add: is_none_def rel_fun_def rel_option_iff split: option.split)
1.43
1.44 lemma dom_transfer:
1.45 assumes [transfer_rule]: "bi_total A"
1.46 shows "((A ===> rel_option B) ===> rel_set A) dom dom"
1.47 -unfolding dom_def[abs_def] equal_None_def[symmetric]
1.48 -by transfer_prover
1.49 + unfolding dom_def [abs_def] is_none_def [symmetric] by transfer_prover
1.50
1.51 lemma map_of_transfer [transfer_rule]:
1.52 assumes [transfer_rule]: "bi_unique R1"
1.53 shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
1.54 -unfolding map_of_def by transfer_prover
1.55 + unfolding map_of_def by transfer_prover
1.56
1.57 lemma tabulate_transfer:
1.58 assumes [transfer_rule]: "bi_unique A"
1.59 shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B)
1.60 - (\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks)))"
1.61 -by transfer_prover
1.62 + (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
1.63 + by transfer_prover
1.64
1.65 lemma bulkload_transfer:
1.66 - "(list_all2 A ===> op= ===> rel_option A)
1.67 + "(list_all2 A ===> HOL.eq ===> rel_option A)
1.68 (\<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)"
1.69 -unfolding rel_fun_def
1.70 -apply clarsimp
1.71 -apply (erule list_all2_induct)
1.72 - apply simp
1.73 -apply (case_tac xa)
1.74 - apply simp
1.75 -by (auto dest: list_all2_lengthD list_all2_nthD)
1.76 +proof
1.77 + fix xs ys
1.78 + assume "list_all2 A xs ys"
1.79 + then show "(HOL.eq ===> rel_option A)
1.80 + (\<lambda>k. if k < length xs then Some (xs ! k) else None)
1.81 + (\<lambda>k. if k < length ys then Some (ys ! k) else None)"
1.82 + apply induct
1.83 + apply auto
1.84 + unfolding rel_fun_def
1.85 + apply clarsimp
1.86 + apply (case_tac xa)
1.87 + apply (auto dest: list_all2_lengthD list_all2_nthD)
1.88 + done
1.89 +qed
1.90
1.91 lemma map_transfer:
1.92 "((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D)
1.93 - (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
1.94 -by transfer_prover
1.95 + (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
1.96 + by transfer_prover
1.97
1.98 lemma map_entry_transfer:
1.99 assumes [transfer_rule]: "bi_unique A"
1.100 @@ -74,38 +83,41 @@
1.101 (\<lambda>k f m. (case m k of None \<Rightarrow> m
1.102 | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
1.103 | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
1.104 -by transfer_prover
1.105 + by transfer_prover
1.106
1.107 end
1.108
1.109 subsection {* Type definition and primitive operations *}
1.110
1.111 typedef ('a, 'b) mapping = "UNIV :: ('a \<rightharpoonup> 'b) set"
1.112 - morphisms rep Mapping ..
1.113 + morphisms rep Mapping
1.114 + ..
1.115
1.116 -setup_lifting(no_code) type_definition_mapping
1.117 +setup_lifting (no_code) type_definition_mapping
1.118
1.119 -lift_definition empty :: "('a, 'b) mapping" is Map.empty parametric empty_transfer .
1.120 +lift_definition empty :: "('a, 'b) mapping"
1.121 + is Map.empty parametric empty_transfer .
1.122
1.123 -lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option" is "\<lambda>m k. m k"
1.124 - parametric lookup_transfer .
1.125 +lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option"
1.126 + is "\<lambda>m k. m k" parametric lookup_transfer .
1.127
1.128 -lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k v m. m(k \<mapsto> v)"
1.129 - parametric update_transfer .
1.130 +lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
1.131 + is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_transfer .
1.132
1.133 -lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is "\<lambda>k m. m(k := None)"
1.134 - parametric delete_transfer .
1.135 +lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
1.136 + is "\<lambda>k m. m(k := None)" parametric delete_transfer .
1.137
1.138 -lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set" is dom parametric dom_transfer .
1.139 +lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
1.140 + is dom parametric dom_transfer .
1.141
1.142 -lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping" is
1.143 - "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_transfer .
1.144 +lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping"
1.145 + is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_transfer .
1.146
1.147 -lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping" is
1.148 - "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .
1.149 +lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping"
1.150 + is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .
1.151
1.152 -lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping" is
1.153 - "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_transfer .
1.154 +lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
1.155 + is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_transfer .
1.156
1.157
1.158 subsection {* Functorial structure *}
1.159 @@ -116,19 +128,24 @@
1.160
1.161 subsection {* Derived operations *}
1.162
1.163 -definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list" where
1.164 +definition ordered_keys :: "('a\<Colon>linorder, 'b) mapping \<Rightarrow> 'a list"
1.165 +where
1.166 "ordered_keys m = (if finite (keys m) then sorted_list_of_set (keys m) else [])"
1.167
1.168 -definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool" where
1.169 +definition is_empty :: "('a, 'b) mapping \<Rightarrow> bool"
1.170 +where
1.171 "is_empty m \<longleftrightarrow> keys m = {}"
1.172
1.173 -definition size :: "('a, 'b) mapping \<Rightarrow> nat" where
1.174 +definition size :: "('a, 'b) mapping \<Rightarrow> nat"
1.175 +where
1.176 "size m = (if finite (keys m) then card (keys m) else 0)"
1.177
1.178 -definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
1.179 +definition replace :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
1.180 +where
1.181 "replace k v m = (if k \<in> keys m then update k v m else m)"
1.182
1.183 -definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
1.184 +definition default :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
1.185 +where
1.186 "default k v m = (if k \<in> keys m then m else update k v m)"
1.187
1.188 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
1.189 @@ -139,15 +156,16 @@
1.190 | Some v \<Rightarrow> update k (f v) m)"
1.191 by transfer rule
1.192
1.193 -definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" where
1.194 +definition map_default :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
1.195 +where
1.196 "map_default k v f m = map_entry k f (default k v m)"
1.197
1.198 lift_definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
1.199 -is map_of parametric map_of_transfer .
1.200 + is map_of parametric map_of_transfer .
1.201
1.202 lemma of_alist_code [code]:
1.203 "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
1.204 -by transfer(simp add: map_add_map_of_foldr[symmetric])
1.205 + by transfer (simp add: map_add_map_of_foldr [symmetric])
1.206
1.207 instantiation mapping :: (type, type) equal
1.208 begin
1.209 @@ -162,35 +180,36 @@
1.210
1.211 context
1.212 begin
1.213 +
1.214 interpretation lifting_syntax .
1.215
1.216 lemma [transfer_rule]:
1.217 assumes [transfer_rule]: "bi_total A"
1.218 assumes [transfer_rule]: "bi_unique B"
1.219 - shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
1.220 -by (unfold equal) transfer_prover
1.221 + shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
1.222 + by (unfold equal) transfer_prover
1.223
1.224 end
1.225
1.226 +
1.227 subsection {* Properties *}
1.228
1.229 -lemma lookup_update: "lookup (update k v m) k = Some v"
1.230 +lemma lookup_update:
1.231 + "lookup (update k v m) k = Some v"
1.232 by transfer simp
1.233
1.234 -lemma lookup_update_neq: "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'"
1.235 +lemma lookup_update_neq:
1.236 + "k \<noteq> k' \<Longrightarrow> lookup (update k v m) k' = lookup m k'"
1.237 by transfer simp
1.238
1.239 -lemma lookup_empty: "lookup empty k = None"
1.240 +lemma lookup_empty:
1.241 + "lookup empty k = None"
1.242 by transfer simp
1.243
1.244 lemma keys_is_none_rep [code_unfold]:
1.245 "k \<in> keys m \<longleftrightarrow> \<not> (Option.is_none (lookup m k))"
1.246 by transfer (auto simp add: is_none_def)
1.247
1.248 -lemma tabulate_alt_def:
1.249 - "map_of (List.map (\<lambda>k. (k, f k)) ks) = (Some o f) |` set ks"
1.250 - by (induct ks) (auto simp add: tabulate_def restrict_map_def)
1.251 -
1.252 lemma update_update:
1.253 "update k v (update k w m) = update k v m"
1.254 "k \<noteq> l \<Longrightarrow> update k v (update l w m) = update l w (update k v m)"
1.255 @@ -229,11 +248,11 @@
1.256
1.257 lemma size_tabulate [simp]:
1.258 "size (tabulate ks f) = length (remdups ks)"
1.259 - unfolding size_def by transfer (auto simp add: tabulate_alt_def card_set comp_def)
1.260 + unfolding size_def by transfer (auto simp add: map_of_map_restrict card_set comp_def)
1.261
1.262 lemma bulkload_tabulate:
1.263 "bulkload xs = tabulate [0..<length xs] (nth xs)"
1.264 - by transfer (auto simp add: tabulate_alt_def)
1.265 + by transfer (auto simp add: map_of_map_restrict)
1.266
1.267 lemma is_empty_empty [simp]:
1.268 "is_empty empty"
1.269 @@ -257,8 +276,7 @@
1.270
1.271 lemma is_empty_map_entry [simp]:
1.272 "is_empty (map_entry k f m) \<longleftrightarrow> is_empty m"
1.273 - unfolding is_empty_def
1.274 - apply transfer by (case_tac "m k") auto
1.275 + unfolding is_empty_def by transfer (auto split: option.split)
1.276
1.277 lemma is_empty_map_default [simp]:
1.278 "\<not> is_empty (map_default k v f m)"
1.279 @@ -286,7 +304,7 @@
1.280
1.281 lemma keys_map_entry [simp]:
1.282 "keys (map_entry k f m) = keys m"
1.283 - apply transfer by (case_tac "m k") auto
1.284 + by transfer (auto split: option.split)
1.285
1.286 lemma keys_map_default [simp]:
1.287 "keys (map_default k v f m) = insert k (keys m)"
1.288 @@ -298,7 +316,7 @@
1.289
1.290 lemma keys_bulkload [simp]:
1.291 "keys (bulkload xs) = {0..<length xs}"
1.292 - by (simp add: keys_tabulate bulkload_tabulate)
1.293 + by (simp add: bulkload_tabulate)
1.294
1.295 lemma distinct_ordered_keys [simp]:
1.296 "distinct (ordered_keys m)"
1.297 @@ -358,6 +376,22 @@
1.298 "ordered_keys (bulkload ks) = [0..<length ks]"
1.299 by (simp add: ordered_keys_def)
1.300
1.301 +lemma tabulate_fold:
1.302 + "tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
1.303 +proof transfer
1.304 + fix f :: "'a \<Rightarrow> 'b" and xs
1.305 + from map_add_map_of_foldr
1.306 + have "Map.empty ++ map_of (List.map (\<lambda>k. (k, f k)) xs) =
1.307 + foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) (List.map (\<lambda>k. (k, f k)) xs) Map.empty"
1.308 + .
1.309 + then have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
1.310 + by (simp add: foldr_map comp_def)
1.311 + also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs"
1.312 + by (rule foldr_fold) (simp add: fun_eq_iff)
1.313 + ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
1.314 + by simp
1.315 +qed
1.316 +
1.317
1.318 subsection {* Code generator setup *}
1.319