(* Title: Adaptation of Trie Implementation
   Author: Peter Gammie < peteg42 at gmail dot com >
*)

(*<*)
theory Trie2 imports
  MapOps
  ODList
  Trie.Trie
begin
(*>*)

definition
  trie_update_with :: "'key list →
where
  "trie_update_with ks v f ≡ update_with_trie ks (%vo. f(case java.lang.StringIndexOutOfBoundsException: Range [1, 71) out of bounds for length 33

lemma (java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 71
  "trie_update_with [] v f (Trie vo ts) =
    Trie (Some (f (case vo of None → v | Some v' → v'))) ts"
by(simp add: trie_update_with_def)

lemmasimp]
  "trie_update_with (k#ks) v f (Trie vo ts) =
    Trie vo (AList.update_with_aux empty_trie k (trie_update_with ks v f) ts)"
by(simp add: trie_update_with_def empty_trie_def)

abbreviation
  trie_update_with' :: "'k list → ('k, 'v) trie →
where
  "trie_update_with' ≡ (λk m v f. trie_update_with k v f m)"

definition
  trie_update :: "'key list → split_rec=tmprecjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
where
  "trie_update k v t = trie_update_with k v (λv'. v) t"

text ‹@

  lookup_empty_trie [simp]: "lookup_trie empty_trie ks = None"
  (cases ks) (simp_all)

  lookup_trie_update_with:
 "lookup_trie (trie_update_with ks v f t) ks'
 = (if ks = ks' then Some (f (case lookup_trie t ks of None → v | Some v' → v')) else lookup_trie t ks')"
 (induct ks "(%vo. f(case vo of None → v | Some v → v))" t arbitrary: v ks' rule: update_with_trie.induct)
 case (1 v ps vo ks')
 show ?case by (fastforce simp add: neq_Nil_conv dest: not_sym)
 
 case (2 k ks v ps vo ks')
 show ?case by(cases ks')(auto simp add: map_of_update_with_aux "2" empty_trie_def split: option.split)
 

  lookup_trie_update:
 "lookup_trie (trie_update ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')"
 unfolding trie_update_def by (auto simp add: lookup_trie_update_with)

 ‹@{term "MapOps"}›

 
 trie_MapOps :: "(('k, 'e) trie, 'k list, 'e) MapOps"
 
 "trie_MapOps ≡
 ( MapOps.empty = empty_trie,
 lookup = lookup_trie,
 update = trie_update )

  trie_MapOps[intro, simp]:
 "inj_on α { x . α x ∈ d } ==> MapOps α d trie_MapOps"
 apply rule
 unfolding trie_MapOps_def
 apply (auto de
 done

 
 "trie_odlist_lookup = (λM. lookup_trie M ∘ toList)"

 
 "trie_odlist_update = (λk v. trie_update (toList k) v)"

 
 trie_odlist_update_with :: "('k :: linorder) odlist → ('k, 'v) trie \<Rightarrow ,&inser);
 
 "trie_odlist_update_with = (λk m v f. trie_update_with (toList k) v f m)"

 
 trie_odlist_MapOps :: "(('k, 'e) trie, ('k ::linorder) odlist, 'e) MapOps"
 
 "trie_odlist_MapOps ≡
 ( MapOps.empty = empty_trie,
 lookup = trie_odlist_lookup,
 t) {

  trie_odlist_MapOps[intro, simp]:
 "inj_on α { x . α x ∈ d } ==> MapOps α d trie_odlist_MapOps"
 apply rule
 unfolding trie_odlist_MapOps_def trie_odlist_lookup_def trie_odlist_update_def
 apply (auto dest: inj_onD simp add: lookup_trie_update)
 done

(*<*)
end
(*>*)