Theory Multiset

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theory Multiset
imports Lemmas

(*  Title:      HOL/IOA/NTP/Multiset.thy
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)"


lemma count_empty:
"count {|} x = 0"

by (simp add: Multiset.count_def Multiset.countm_empty_def)

lemma count_addm_simp:
"count (addm M x) y = (if y=x then Suc(count M y) else count M y)"

by (simp add: Multiset.count_def Multiset.countm_nonempty_def)

lemma count_leq_addm: "count M y <= count (addm M x) y"
by (simp add: count_addm_simp)

lemma count_delm_simp:
"count (delm M x) y = (if y=x then count M y - 1 else count M y)"

apply (unfold Multiset.count_def)
apply (rule_tac M = "M" in Multiset.induction)
apply (simp (no_asm_simp) add: Multiset.delm_empty_def Multiset.countm_empty_def)
apply (simp add: Multiset.delm_nonempty_def Multiset.countm_nonempty_def)
apply safe
apply simp
done

lemma countm_props: "!!M. (!x. P(x) --> Q(x)) ==> (countm M P <= countm M Q)"
apply (rule_tac M = "M" in Multiset.induction)
apply (simp (no_asm) add: Multiset.countm_empty_def)
apply (simp (no_asm) add: Multiset.countm_nonempty_def)
apply auto
done

lemma countm_spurious_delm: "!!P. ~P(obj) ==> countm M P = countm (delm M obj) P"
apply (rule_tac M = "M" in Multiset.induction)
apply (simp (no_asm) add: Multiset.delm_empty_def Multiset.countm_empty_def)
apply (simp (no_asm_simp) add: Multiset.countm_nonempty_def Multiset.delm_nonempty_def)
done


lemma pos_count_imp_pos_countm [rule_format (no_asm)]: "!!P. P(x) ==> 0<count M x --> countm M P > 0"
apply (rule_tac M = "M" in Multiset.induction)
apply (simp (no_asm) add: Multiset.delm_empty_def Multiset.count_def Multiset.countm_empty_def)
apply (simp add: Multiset.count_def Multiset.delm_nonempty_def Multiset.countm_nonempty_def)
done

lemma countm_done_delm:
"!!P. P(x) ==> 0<count M x --> countm (delm M x) P = countm M P - 1"

apply (rule_tac M = "M" in Multiset.induction)
apply (simp (no_asm) add: Multiset.delm_empty_def Multiset.countm_empty_def)
apply (simp (no_asm_simp) add: count_addm_simp Multiset.delm_nonempty_def Multiset.countm_nonempty_def pos_count_imp_pos_countm)
apply auto
done


declare count_addm_simp [simp] count_delm_simp [simp]
Multiset.countm_empty_def [simp] Multiset.delm_empty_def [simp] count_empty [simp]


end