1.1 --- a/src/HOL/Library/Mapping.thy	Wed Apr 09 14:08:18 2014 +0200
     1.2 +++ b/src/HOL/Library/Mapping.thy	Wed Apr 09 14:08:25 2014 +0200
     1.3 @@ -10,25 +10,29 @@
     1.4  
     1.5  subsection {* Parametricity transfer rules *}
     1.6  
     1.7 +lemma map_of_foldr: -- {* FIXME move *}
     1.8 +  "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"
     1.9 +  using map_add_map_of_foldr [of Map.empty] by auto
    1.10 +
    1.11  context
    1.12  begin
    1.13  
    1.14  interpretation lifting_syntax .
    1.15  
    1.16 -lemma empty_transfer:
    1.17 +lemma empty_parametric:
    1.18    "(A ===> rel_option B) Map.empty Map.empty"
    1.19    by transfer_prover
    1.20  
    1.21 -lemma lookup_transfer: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
    1.22 +lemma lookup_parametric: "((A ===> B) ===> A ===> B) (\<lambda>m k. m k) (\<lambda>m k. m k)"
    1.23    by transfer_prover
    1.24  
    1.25 -lemma update_transfer:
    1.26 +lemma update_parametric:
    1.27    assumes [transfer_rule]: "bi_unique A"
    1.28    shows "(A ===> B ===> (A ===> rel_option B) ===> A ===> rel_option B)
    1.29      (\<lambda>k v m. m(k \<mapsto> v)) (\<lambda>k v m. m(k \<mapsto> v))"
    1.30    by transfer_prover
    1.31  
    1.32 -lemma delete_transfer:
    1.33 +lemma delete_parametric:
    1.34    assumes [transfer_rule]: "bi_unique A"
    1.35    shows "(A ===> (A ===> rel_option B) ===> A ===> rel_option B) 
    1.36      (\<lambda>k m. m(k := None)) (\<lambda>k m. m(k := None))"
    1.37 @@ -38,23 +42,31 @@
    1.38    "(rel_option A ===> HOL.eq) Option.is_none Option.is_none"
    1.39    by (auto simp add: is_none_def rel_fun_def rel_option_iff split: option.split)
    1.40  
    1.41 -lemma dom_transfer:
    1.42 +lemma dom_parametric:
    1.43    assumes [transfer_rule]: "bi_total A"
    1.44    shows "((A ===> rel_option B) ===> rel_set A) dom dom" 
    1.45    unfolding dom_def [abs_def] is_none_def [symmetric] by transfer_prover
    1.46  
    1.47 -lemma map_of_transfer [transfer_rule]:
    1.48 +lemma map_of_parametric [transfer_rule]:
    1.49    assumes [transfer_rule]: "bi_unique R1"
    1.50    shows "(list_all2 (rel_prod R1 R2) ===> R1 ===> rel_option R2) map_of map_of"
    1.51    unfolding map_of_def by transfer_prover
    1.52  
    1.53 -lemma tabulate_transfer: 
    1.54 +lemma map_entry_parametric [transfer_rule]:
    1.55 +  assumes [transfer_rule]: "bi_unique A"
    1.56 +  shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B) 
    1.57 +    (\<lambda>k f m. (case m k of None \<Rightarrow> m
    1.58 +      | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
    1.59 +      | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
    1.60 +  by transfer_prover
    1.61 +
    1.62 +lemma tabulate_parametric: 
    1.63    assumes [transfer_rule]: "bi_unique A"
    1.64    shows "(list_all2 A ===> (A ===> B) ===> A ===> rel_option B) 
    1.65      (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks))) (\<lambda>ks f. (map_of (map (\<lambda>k. (k, f k)) ks)))"
    1.66    by transfer_prover
    1.67  
    1.68 -lemma bulkload_transfer: 
    1.69 +lemma bulkload_parametric: 
    1.70    "(list_all2 A ===> HOL.eq ===> rel_option A) 
    1.71      (\<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)"
    1.72  proof
    1.73 @@ -72,20 +84,13 @@
    1.74      done
    1.75  qed
    1.76  
    1.77 -lemma map_transfer: 
    1.78 +lemma map_parametric: 
    1.79    "((A ===> B) ===> (C ===> D) ===> (B ===> rel_option C) ===> A ===> rel_option D) 
    1.80       (\<lambda>f g m. (map_option g \<circ> m \<circ> f)) (\<lambda>f g m. (map_option g \<circ> m \<circ> f))"
    1.81    by transfer_prover
    1.82  
    1.83 -lemma map_entry_transfer:
    1.84 -  assumes [transfer_rule]: "bi_unique A"
    1.85 -  shows "(A ===> (B ===> B) ===> (A ===> rel_option B) ===> A ===> rel_option B) 
    1.86 -    (\<lambda>k f m. (case m k of None \<Rightarrow> m
    1.87 -      | Some v \<Rightarrow> m (k \<mapsto> (f v)))) (\<lambda>k f m. (case m k of None \<Rightarrow> m
    1.88 -      | Some v \<Rightarrow> m (k \<mapsto> (f v))))"
    1.89 -  by transfer_prover
    1.90 +end
    1.91  
    1.92 -end
    1.93  
    1.94  subsection {* Type definition and primitive operations *}
    1.95  
    1.96 @@ -96,28 +101,28 @@
    1.97  setup_lifting (no_code) type_definition_mapping
    1.98  
    1.99  lift_definition empty :: "('a, 'b) mapping"
   1.100 -  is Map.empty parametric empty_transfer .
   1.101 +  is Map.empty parametric empty_parametric .
   1.102  
   1.103  lift_definition lookup :: "('a, 'b) mapping \<Rightarrow> 'a \<Rightarrow> 'b option"
   1.104 -  is "\<lambda>m k. m k" parametric lookup_transfer .
   1.105 +  is "\<lambda>m k. m k" parametric lookup_parametric .
   1.106  
   1.107  lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
   1.108 -  is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_transfer .
   1.109 +  is "\<lambda>k v m. m(k \<mapsto> v)" parametric update_parametric .
   1.110  
   1.111  lift_definition delete :: "'a \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping"
   1.112 -  is "\<lambda>k m. m(k := None)" parametric delete_transfer .
   1.113 +  is "\<lambda>k m. m(k := None)" parametric delete_parametric .
   1.114  
   1.115  lift_definition keys :: "('a, 'b) mapping \<Rightarrow> 'a set"
   1.116 -  is dom parametric dom_transfer .
   1.117 +  is dom parametric dom_parametric .
   1.118  
   1.119  lift_definition tabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping"
   1.120 -  is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_transfer .
   1.121 +  is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" parametric tabulate_parametric .
   1.122  
   1.123  lift_definition bulkload :: "'a list \<Rightarrow> (nat, 'a) mapping"
   1.124 -  is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_transfer .
   1.125 +  is "\<lambda>xs k. if k < length xs then Some (xs ! k) else None" parametric bulkload_parametric .
   1.126  
   1.127  lift_definition map :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('c, 'd) mapping"
   1.128 -  is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_transfer .
   1.129 +  is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
   1.130  
   1.131  
   1.132  subsection {* Functorial structure *}
   1.133 @@ -148,11 +153,14 @@
   1.134  where
   1.135    "default k v m = (if k \<in> keys m then m else update k v m)"
   1.136  
   1.137 +text {* Manual derivation of transfer rule is non-trivial *}
   1.138 +
   1.139  lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) mapping \<Rightarrow> ('a, 'b) mapping" is
   1.140    "\<lambda>k f m. (case m k of None \<Rightarrow> m
   1.141 -    | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_transfer .
   1.142 +    | Some v \<Rightarrow> m (k \<mapsto> (f v)))" parametric map_entry_parametric .
   1.143  
   1.144 -lemma map_entry_code [code]: "map_entry k f m = (case lookup m k of None \<Rightarrow> m
   1.145 +lemma map_entry_code [code]:
   1.146 +  "map_entry k f m = (case lookup m k of None \<Rightarrow> m
   1.147      | Some v \<Rightarrow> update k (f v) m)"
   1.148    by transfer rule
   1.149  
   1.150 @@ -160,12 +168,9 @@
   1.151  where
   1.152    "map_default k v f m = map_entry k f (default k v m)" 
   1.153  
   1.154 -lift_definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
   1.155 -  is map_of parametric map_of_transfer .
   1.156 -
   1.157 -lemma of_alist_code [code]:
   1.158 +definition of_alist :: "('k \<times> 'v) list \<Rightarrow> ('k, 'v) mapping"
   1.159 +where
   1.160    "of_alist xs = foldr (\<lambda>(k, v) m. update k v m) xs empty"
   1.161 -  by transfer (simp add: map_add_map_of_foldr [symmetric])
   1.162  
   1.163  instantiation mapping :: (type, type) equal
   1.164  begin
   1.165 @@ -189,6 +194,11 @@
   1.166    shows "(pcr_mapping A B ===> pcr_mapping A B ===> op=) HOL.eq HOL.equal"
   1.167    by (unfold equal) transfer_prover
   1.168  
   1.169 +lemma of_alist_transfer [transfer_rule]:
   1.170 +  assumes [transfer_rule]: "bi_unique R1"
   1.171 +  shows "(list_all2 (rel_prod R1 R2) ===> pcr_mapping R1 R2) map_of of_alist"
   1.172 +  unfolding of_alist_def [abs_def] map_of_foldr [abs_def] by transfer_prover
   1.173 +
   1.174  end
   1.175  
   1.176  
   1.177 @@ -380,12 +390,8 @@
   1.178    "tabulate xs f = fold (\<lambda>k m. update k (f k) m) xs empty"
   1.179  proof transfer
   1.180    fix f :: "'a \<Rightarrow> 'b" and xs
   1.181 -  from map_add_map_of_foldr
   1.182 -  have "Map.empty ++ map_of (List.map (\<lambda>k. (k, f k)) xs) =
   1.183 -    foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) (List.map (\<lambda>k. (k, f k)) xs) Map.empty"
   1.184 -    .
   1.185 -  then have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
   1.186 -    by (simp add: foldr_map comp_def)
   1.187 +  have "map_of (List.map (\<lambda>k. (k, f k)) xs) = foldr (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"
   1.188 +    by (simp add: foldr_map comp_def map_of_foldr)
   1.189    also have "foldr (\<lambda>k m. m(k \<mapsto> f k)) xs = fold (\<lambda>k m. m(k \<mapsto> f k)) xs"
   1.190      by (rule foldr_fold) (simp add: fun_eq_iff)
   1.191    ultimately show "map_of (List.map (\<lambda>k. (k, f k)) xs) = fold (\<lambda>k m. m(k \<mapsto> f k)) xs Map.empty"