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theory Multiset(* 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