(* Title: HOL/IOA/NTP/Multiset.ML
ID: $Id$
Author: Tobias Nipkow & Konrad Slind
Axiomatic multisets.
Should be done as a subtype and moved to a global place.
*)
Goalw [Multiset.count_def, Multiset.countm_empty_def]
"count {|} x = 0";
by (rtac refl 1);
qed "count_empty";
Goal
"count (addm M x) y = (if y=x then Suc(count M y) else count M y)";
by (asm_simp_tac (simpset() addsimps
[Multiset.count_def,Multiset.countm_nonempty_def]) 1);
qed "count_addm_simp";
Goal "count M y <= count (addm M x) y";
by (simp_tac (simpset() addsimps [count_addm_simp]) 1);
qed "count_leq_addm";
Goalw [Multiset.count_def]
"count (delm M x) y = (if y=x then count M y - 1 else count M y)";
by (res_inst_tac [("M","M")] Multiset.induction 1);
by (asm_simp_tac (simpset()
addsimps [Multiset.delm_empty_def,Multiset.countm_empty_def]) 1);
by (asm_full_simp_tac (simpset()
addsimps [Multiset.delm_nonempty_def,
Multiset.countm_nonempty_def]) 1);
by Safe_tac;
by (Asm_full_simp_tac 1);
qed "count_delm_simp";
Goal "!!M. (!x. P(x) --> Q(x)) ==> (countm M P <= countm M Q)";
by (res_inst_tac [("M","M")] Multiset.induction 1);
by (simp_tac (simpset() addsimps [Multiset.countm_empty_def]) 1);
by (simp_tac (simpset() addsimps[Multiset.countm_nonempty_def]) 1);
by Auto_tac;
qed "countm_props";
Goal "!!P. ~P(obj) ==> countm M P = countm (delm M obj) P";
by (res_inst_tac [("M","M")] Multiset.induction 1);
by (simp_tac (simpset() addsimps [Multiset.delm_empty_def,
Multiset.countm_empty_def]) 1);
by (asm_simp_tac (simpset() addsimps[Multiset.countm_nonempty_def,
Multiset.delm_nonempty_def]) 1);
qed "countm_spurious_delm";
Goal "!!P. P(x) ==> 0<count M x --> 0<countm M P";
by (res_inst_tac [("M","M")] Multiset.induction 1);
by (simp_tac (simpset() addsimps
[Multiset.delm_empty_def,Multiset.count_def,
Multiset.countm_empty_def]) 1);
by (asm_simp_tac (simpset() addsimps
[Multiset.count_def,Multiset.delm_nonempty_def,
Multiset.countm_nonempty_def]) 1);
qed_spec_mp "pos_count_imp_pos_countm";
Goal
"!!P. P(x) ==> 0<count M x --> countm (delm M x) P = countm M P - 1";
by (res_inst_tac [("M","M")] Multiset.induction 1);
by (simp_tac (simpset() addsimps
[Multiset.delm_empty_def,
Multiset.countm_empty_def]) 1);
by (asm_simp_tac (simpset() addsimps
[count_addm_simp,Multiset.delm_nonempty_def,
Multiset.countm_nonempty_def,pos_count_imp_pos_countm]) 1);
by (Auto_tac);
qed "countm_done_delm";
Addsimps [count_addm_simp, count_delm_simp,
Multiset.countm_empty_def, Multiset.delm_empty_def,
count_empty];