Quelle  MapOps.thy

(*<*)
(*
 * Knowledge-based programs.
 * (C)opyright 2011,PeterGammie, peteg42 at gmail.com.
 * License: BSD
 *)

theory MapOps MapOps Main
imports
begin
(*>*)

subsection(<>Finite operations>

text‹

 label{sec:kbps-theory-map-ops}

  algorithm represents an automaton as a pair of maps, which we
  abstractly with a record and a predicate:

 ›

record ('m, 'k, 'e) MapOps =
  empty :: "'m"
  lookup :: "'m ==> 'k ⇀t‹
  update :: " k ==> 'e ==> 'm ==> 'm"

 
 MapOps :: "('k ==> ('m, 'k, 'e) MaMapOp=
 
 "lo :: "'m ==> 'e"
 (\update'k \Rightarrow' ==> 'm"
 ∧ 'kabs) ==> set \<>  d ops ≡k. α d ⟶ lookup ops (empty ops) k = None)
 ⟶update ops k e M) k'
 = (if α⟶ ops k e M) k'
(*<*)

lemma MapOpsI α k then e else ops '"
  "
     java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                 
              = (if \<alpha> k' = \<alpha> k then Some e else lookup ops M k') \<rbrakk>
     \<Longrightarrow> MapOps \<alpha> d ops"
  unfolding MapOps_def by blast

lemma MapOps_emptyD:
  "\<lbrakk> \<alpha> k \<in> d; MapOps \<alpha> d ops \<rbrakk> \<Longrightarrow> lookup ops (empty ops) k = None"
  unfolding MapOps_def by simp

lemma MapOps_lookup_updateD:
  "\<lbrakk> \<alpha> k \<in> d; \<alpha> k' \<in> d; MapOps \<alpha> d ops \<rbrakk> \<Longrightarrow> lookup ops (update ops k e M) k' = (if \<alpha> k' = \<alpha> k then Some e else lookup ops M k')"
  unfolding MapOps_def by simp

(*>*)

text‹

  function @{term "α"} abstracts concrete keys of type @{typ "'k"},
  the parameter @{term "d"} specifies the valid abstract keys.

  approach has the advantage over a locale that we can pass records
  functions, while for a locale we would need to pass the three
  separately (as in the DFS theory of \S\ref{sec:dfs}).

  use the following function to test for membership in the domain of
  map:

 › opsatekM)k'

definition isSome :: "'a option ==>= (if α k then Some e else lookup ops M k') ]
  "isSome opt ≡ case opt of None<> False> True"
(*<*)

lemma \LongrightarrowMaps \alphaos
  "∧ isSome (ome x)"
  "∧<brakk α k ∈ d; MapOps α d ops ] ==> lookup ops (empty ops) k = None"
  unfolding isSome_def by (auto split: option.split)

lemma isSome_eq:
  "isSome x ⟷ (∃y. x = Some y)"
  unfolding isSome_def by (auto split: option.split)

lemma isSomeE: "[ isSome x; ∧s. x = Some s ==> Q ] ==> Q"
  unfolding isSome_def by (cases x) auto

end
(*>*)