Theory Multiset

Up to index of Isabelle/HOLCF/IOA/NTP

theory Multiset
imports Lemmas
uses [Multiset.ML]
begin
(*  Title:      HOL/IOA/NTP/Multiset.thy
    ID:         $Id: Multiset.thy,v 1.7 2005/09/03 14:50:24 wenzelm Exp $
    Author:     Tobias Nipkow & Konrad Slind
*)

header {* Axiomatic multisets *}

theory Multiset
imports Lemmas
begin

typedecl
  'a multiset

consts

  "{|}"  :: "'a multiset"                        ("{|}")
  addm   :: "['a multiset, 'a] => 'a multiset"
  delm   :: "['a multiset, 'a] => 'a multiset"
  countm :: "['a multiset, 'a => bool] => nat"
  count  :: "['a multiset, 'a] => nat"

axioms

delm_empty_def:
  "delm {|} x = {|}"

delm_nonempty_def:
  "delm (addm M x) y == (if x=y then M else addm (delm M y) x)"

countm_empty_def:
   "countm {|} P == 0"

countm_nonempty_def:
   "countm (addm M x) P == countm M P + (if P x then Suc 0 else 0)"

count_def:
   "count M x == countm M (%y. y = x)"

"induction":
   "[| P({|}); !!M x. P(M) ==> P(addm M x) |] ==> P(M)"

ML {* use_text Context.ml_output true
  ("structure Multiset = struct " ^ legacy_bindings (the_context ()) ^ " end") *}
ML {* open Multiset *}

end

theorem count_empty: 

  count {|} x = 0

theorem count_addm_simp: 

  count (addm M x) y = (if y = x then Suc (count M y) else count M y)

theorem count_leq_addm: 

  count M y ≤ count (addm M x) y

theorem count_delm_simp: 

  count (delm M x) y = (if y = x then count M y - 1 else count M y)

theorem countm_props: 

  ∀x. P x --> Q x ==> countm M P ≤ countm M Q

theorem countm_spurious_delm: 

  ¬ P obj ==> countm M P = countm (delm M obj) P

theorem pos_count_imp_pos_countm: 

  [| P x; 0 < count M x |] ==> 0 < countm M P

theorem countm_done_delm: 

  P x ==> 0 < count M x --> countm (delm M x) P = countm M P - 1