IOA/example/Multiset.ML

added IOA to isabelle/HOL

goalw Multiset.thy [Multiset.count_def, Multiset.countm_empty_def]
   "count({|},x) = 0";
 by (rtac refl 1);
val count_empty = result();

goal Multiset.thy 
     "count(addm(M,x),y) = if(y=x,Suc(count(M,y)),count(M,y))";
  by (asm_simp_tac (arith_ss addsimps 
                    [Multiset.count_def,Multiset.countm_nonempty_def]
                    setloop (split_tac [expand_if])) 1);
val count_addm_simp = result();

goal Multiset.thy "count(M,y) <= count(addm(M,x),y)";
  by (simp_tac (arith_ss addsimps [count_addm_simp]
                         setloop (split_tac [expand_if])) 1);
  by (rtac impI 1);
  by (rtac (le_refl RS (leq_suc RS mp)) 1);
val count_leq_addm = result();

goalw Multiset.thy [Multiset.count_def] 
     "count(delm(M,x),y) = if(y=x,pred(count(M,y)),count(M,y))";
  by (res_inst_tac [("M","M")] Multiset.induction 1);
  by (asm_simp_tac (arith_ss 
                   addsimps [Multiset.delm_empty_def,Multiset.countm_empty_def]
                   setloop (split_tac [expand_if])) 1);
  by (asm_full_simp_tac (arith_ss 
                         addsimps [Multiset.delm_nonempty_def,
                                   Multiset.countm_nonempty_def]
                         setloop (split_tac [expand_if])) 1);
  by (safe_tac HOL_cs);
  by (asm_full_simp_tac HOL_ss 1);
val count_delm_simp = result();

goal Multiset.thy "!!M. (!x. P(x) --> Q(x)) ==> (countm(M,P) <= countm(M,Q))";
  by (res_inst_tac [("M","M")] Multiset.induction 1);
  by (simp_tac (arith_ss addsimps [Multiset.countm_empty_def]) 1);
  by (simp_tac (arith_ss addsimps[Multiset.countm_nonempty_def]) 1);
  by (etac (less_eq_add_cong RS mp RS mp) 1);
  by (asm_full_simp_tac (arith_ss addsimps [le_eq_less_or_eq]
                                  setloop (split_tac [expand_if])) 1);
val countm_props = result();

goal Multiset.thy "!!P. ~P(obj) ==> countm(M,P) = countm(delm(M,obj),P)";
  by (res_inst_tac [("M","M")] Multiset.induction 1);
  by (simp_tac (arith_ss addsimps [Multiset.delm_empty_def,
                                   Multiset.countm_empty_def]) 1);
  by (asm_simp_tac (arith_ss addsimps[Multiset.countm_nonempty_def,
                                      Multiset.delm_nonempty_def]
                             setloop (split_tac [expand_if])) 1);
val countm_spurious_delm = result();


goal Multiset.thy "!!P. P(x) ==> 0<count(M,x) --> 0<countm(M,P)";
  by (res_inst_tac [("M","M")] Multiset.induction 1);
  by (simp_tac (arith_ss addsimps 
                          [Multiset.delm_empty_def,Multiset.count_def,
                           Multiset.countm_empty_def]) 1);
  by (asm_simp_tac (arith_ss addsimps 
                       [Multiset.count_def,Multiset.delm_nonempty_def,
                        Multiset.countm_nonempty_def]
                    setloop (split_tac [expand_if])) 1);
val pos_count_imp_pos_countm = standard(result() RS mp);

goal Multiset.thy
   "!!P. P(x) ==> 0<count(M,x) --> countm(delm(M,x),P) = pred(countm(M,P))";
  by (res_inst_tac [("M","M")] Multiset.induction 1);
  by (simp_tac (arith_ss addsimps 
                          [Multiset.delm_empty_def,
                           Multiset.countm_empty_def]) 1);
  by (asm_simp_tac (arith_ss addsimps 
                      [eq_sym_conv,count_addm_simp,Multiset.delm_nonempty_def,
                       Multiset.countm_nonempty_def,pos_count_imp_pos_countm,
                       suc_pred_id]
                    setloop (split_tac [expand_if])) 1);
val countm_done_delm = result();