--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/AList.thy Tue Jan 10 10:17:07 2012 +0100
@@ -0,0 +1,178 @@
+(* Title: HOL/Library/AList.thy
+ Author: Lukas Bulwahn, TU Muenchen *)
+
+header {* Abstract type of association lists with unique keys *}
+
+theory AList
+imports AList_Impl
+begin
+
+text {* This was based on some existing fragments in the AFP-Collection framework. *}
+
+subsection {* Type @{text "('key, 'value) alist" } *}
+
+typedef (open) ('key, 'value) alist = "{xs :: ('key \<times> 'value) list. distinct (map fst xs)}"
+morphisms impl_of Alist
+by(rule exI[where x="[]"]) simp
+
+lemma alist_ext: "impl_of xs = impl_of ys \<Longrightarrow> xs = ys"
+by(simp add: impl_of_inject)
+
+lemma alist_eq_iff: "xs = ys \<longleftrightarrow> impl_of xs = impl_of ys"
+by(simp add: impl_of_inject)
+
+lemma impl_of_distinct [simp, intro]: "distinct (map fst (impl_of xs))"
+using impl_of[of xs] by simp
+
+lemma Alist_impl_of [code abstype]: "Alist (impl_of xs) = xs"
+by(rule impl_of_inverse)
+
+subsection {* Primitive operations *}
+
+definition lookup :: "('key, 'value) alist \<Rightarrow> 'key \<Rightarrow> 'value option"
+where [code]: "lookup xs = map_of (impl_of xs)"
+
+definition empty :: "('key, 'value) alist"
+where [code del]: "empty = Alist []"
+
+definition update :: "'key \<Rightarrow> 'value \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
+where [code del]: "update k v xs = Alist (AList_Impl.update k v (impl_of xs))"
+
+(* FIXME: we use an unoptimised delete operation. *)
+definition delete :: "'key \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
+where [code del]: "delete k xs = Alist (AList_Impl.delete k (impl_of xs))"
+
+definition map_entry :: "'key \<Rightarrow> ('value \<Rightarrow> 'value) \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
+where [code del]: "map_entry k f xs = Alist (AList_Impl.map_entry k f (impl_of xs))"
+
+definition filter :: "('key \<times> 'value \<Rightarrow> bool) \<Rightarrow> ('key, 'value) alist \<Rightarrow> ('key, 'value) alist"
+where [code del]: "filter P xs = Alist (List.filter P (impl_of xs))"
+
+definition map_default :: "'key => 'value => ('value => 'value) => ('key, 'value) alist => ('key, 'value) alist"
+where
+ "map_default k v f xs = Alist (AList_Impl.map_default k v f (impl_of xs))"
+
+lemma impl_of_empty [code abstract]: "impl_of empty = []"
+by (simp add: empty_def Alist_inverse)
+
+lemma impl_of_update [code abstract]: "impl_of (update k v xs) = AList_Impl.update k v (impl_of xs)"
+by (simp add: update_def Alist_inverse distinct_update)
+
+lemma impl_of_delete [code abstract]:
+ "impl_of (delete k al) = AList_Impl.delete k (impl_of al)"
+unfolding delete_def by (simp add: Alist_inverse distinct_delete)
+
+lemma impl_of_map_entry [code abstract]:
+ "impl_of (map_entry k f xs) = AList_Impl.map_entry k f (impl_of xs)"
+unfolding map_entry_def by (simp add: Alist_inverse distinct_map_entry)
+
+lemma distinct_map_fst_filter:
+ "distinct (map fst xs) ==> distinct (map fst (List.filter P xs))"
+by (induct xs) auto
+
+lemma impl_of_filter [code abstract]:
+ "impl_of (filter P xs) = List.filter P (impl_of xs)"
+unfolding filter_def by (simp add: Alist_inverse distinct_map_fst_filter)
+
+lemma impl_of_map_default [code abstract]:
+ "impl_of (map_default k v f xs) = AList_Impl.map_default k v f (impl_of xs)"
+by (auto simp add: map_default_def Alist_inverse distinct_map_default)
+
+subsection {* Abstract operation properties *}
+
+(* FIXME: to be completed *)
+
+lemma lookup_empty [simp]: "lookup empty k = None"
+by(simp add: empty_def lookup_def Alist_inverse)
+
+lemma lookup_delete [simp]: "lookup (delete k al) = (lookup al)(k := None)"
+by (simp add: lookup_def delete_def Alist_inverse distinct_delete delete_conv')
+
+subsection {* Further operations *}
+
+subsubsection {* Equality *}
+
+instantiation alist :: (equal, equal) equal begin
+
+definition "HOL.equal (xs :: ('a, 'b) alist) ys == impl_of xs = impl_of ys"
+
+instance
+proof
+qed (simp add: equal_alist_def impl_of_inject)
+
+end
+
+subsubsection {* Size *}
+
+instantiation alist :: (type, type) size begin
+
+definition "size (al :: ('a, 'b) alist) = length (impl_of al)"
+
+instance ..
+
+end
+
+subsection {* Quickcheck generators *}
+
+notation fcomp (infixl "\<circ>>" 60)
+notation scomp (infixl "\<circ>\<rightarrow>" 60)
+
+definition (in term_syntax)
+ valterm_empty :: "('key :: typerep, 'value :: typerep) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)"
+where
+ "valterm_empty = Code_Evaluation.valtermify empty"
+
+definition (in term_syntax)
+ valterm_update :: "'key :: typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
+ 'value :: typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
+ ('key, 'value) alist \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
+ ('key, 'value) alist \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
+ [code_unfold]: "valterm_update k v a = Code_Evaluation.valtermify update {\<cdot>} k {\<cdot>} v {\<cdot>}a"
+
+fun (in term_syntax) random_aux_alist
+where
+ "random_aux_alist i j = (if i = 0 then Pair valterm_empty else Quickcheck.collapse (Random.select_weight [(i, Quickcheck.random j \<circ>\<rightarrow> (%k. Quickcheck.random j \<circ>\<rightarrow> (%v. random_aux_alist (i - 1) j \<circ>\<rightarrow> (%a. Pair (valterm_update k v a))))), (1, Pair valterm_empty)]))"
+
+instantiation alist :: (random, random) random
+begin
+
+definition random_alist
+where
+ "random_alist i = random_aux_alist i i"
+
+instance ..
+
+end
+
+no_notation fcomp (infixl "\<circ>>" 60)
+no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
+
+instantiation alist :: (exhaustive, exhaustive) exhaustive
+begin
+
+fun exhaustive_alist :: "(('a, 'b) alist => (bool * term list) option) => code_numeral => (bool * term list) option"
+where
+ "exhaustive_alist f i = (if i = 0 then None else case f empty of Some ts => Some ts | None =>
+ exhaustive_alist (%a. Quickcheck_Exhaustive.exhaustive (%k. Quickcheck_Exhaustive.exhaustive (%v. f (update k v a)) (i - 1)) (i - 1)) (i - 1))"
+
+instance ..
+
+end
+
+instantiation alist :: (full_exhaustive, full_exhaustive) full_exhaustive
+begin
+
+fun full_exhaustive_alist :: "(('a, 'b) alist * (unit => term) => (bool * term list) option) => code_numeral => (bool * term list) option"
+where
+ "full_exhaustive_alist f i = (if i = 0 then None else case f valterm_empty of Some ts => Some ts | None =>
+ full_exhaustive_alist (%a. Quickcheck_Exhaustive.full_exhaustive (%k. Quickcheck_Exhaustive.full_exhaustive (%v. f (valterm_update k v a)) (i - 1)) (i - 1)) (i - 1))"
+
+instance ..
+
+end
+
+hide_const valterm_empty valterm_update random_aux_alist
+
+hide_const (open) impl_of lookup empty update delete map_entry filter map_default
+
+end
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