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1 goalw Multiset.thy [Multiset.count_def, Multiset.countm_empty_def] |
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2 "count({|},x) = 0"; |
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3 by (rtac refl 1); |
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4 val count_empty = result(); |
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5 |
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6 goal Multiset.thy |
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7 "count(addm(M,x),y) = if(y=x,Suc(count(M,y)),count(M,y))"; |
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8 by (asm_simp_tac (arith_ss addsimps |
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9 [Multiset.count_def,Multiset.countm_nonempty_def] |
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10 setloop (split_tac [expand_if])) 1); |
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11 val count_addm_simp = result(); |
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12 |
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13 goal Multiset.thy "count(M,y) <= count(addm(M,x),y)"; |
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14 by (simp_tac (arith_ss addsimps [count_addm_simp] |
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15 setloop (split_tac [expand_if])) 1); |
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16 by (rtac impI 1); |
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17 by (rtac (le_refl RS (leq_suc RS mp)) 1); |
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18 val count_leq_addm = result(); |
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19 |
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20 goalw Multiset.thy [Multiset.count_def] |
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21 "count(delm(M,x),y) = if(y=x,pred(count(M,y)),count(M,y))"; |
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22 by (res_inst_tac [("M","M")] Multiset.induction 1); |
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23 by (asm_simp_tac (arith_ss |
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24 addsimps [Multiset.delm_empty_def,Multiset.countm_empty_def] |
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25 setloop (split_tac [expand_if])) 1); |
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26 by (asm_full_simp_tac (arith_ss |
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27 addsimps [Multiset.delm_nonempty_def, |
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28 Multiset.countm_nonempty_def] |
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29 setloop (split_tac [expand_if])) 1); |
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30 by (safe_tac HOL_cs); |
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31 by (asm_full_simp_tac HOL_ss 1); |
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32 val count_delm_simp = result(); |
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33 |
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34 goal Multiset.thy "!!M. (!x. P(x) --> Q(x)) ==> (countm(M,P) <= countm(M,Q))"; |
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35 by (res_inst_tac [("M","M")] Multiset.induction 1); |
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36 by (simp_tac (arith_ss addsimps [Multiset.countm_empty_def]) 1); |
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37 by (simp_tac (arith_ss addsimps[Multiset.countm_nonempty_def]) 1); |
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38 by (etac (less_eq_add_cong RS mp RS mp) 1); |
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39 by (asm_full_simp_tac (arith_ss addsimps [le_eq_less_or_eq] |
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40 setloop (split_tac [expand_if])) 1); |
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41 val countm_props = result(); |
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42 |
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43 goal Multiset.thy "!!P. ~P(obj) ==> countm(M,P) = countm(delm(M,obj),P)"; |
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44 by (res_inst_tac [("M","M")] Multiset.induction 1); |
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45 by (simp_tac (arith_ss addsimps [Multiset.delm_empty_def, |
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46 Multiset.countm_empty_def]) 1); |
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47 by (asm_simp_tac (arith_ss addsimps[Multiset.countm_nonempty_def, |
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48 Multiset.delm_nonempty_def] |
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49 setloop (split_tac [expand_if])) 1); |
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50 val countm_spurious_delm = result(); |
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51 |
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52 |
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53 goal Multiset.thy "!!P. P(x) ==> 0<count(M,x) --> 0<countm(M,P)"; |
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54 by (res_inst_tac [("M","M")] Multiset.induction 1); |
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55 by (simp_tac (arith_ss addsimps |
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56 [Multiset.delm_empty_def,Multiset.count_def, |
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57 Multiset.countm_empty_def]) 1); |
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58 by (asm_simp_tac (arith_ss addsimps |
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59 [Multiset.count_def,Multiset.delm_nonempty_def, |
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60 Multiset.countm_nonempty_def] |
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61 setloop (split_tac [expand_if])) 1); |
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62 val pos_count_imp_pos_countm = standard(result() RS mp); |
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63 |
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64 goal Multiset.thy |
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65 "!!P. P(x) ==> 0<count(M,x) --> countm(delm(M,x),P) = pred(countm(M,P))"; |
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66 by (res_inst_tac [("M","M")] Multiset.induction 1); |
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67 by (simp_tac (arith_ss addsimps |
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68 [Multiset.delm_empty_def, |
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69 Multiset.countm_empty_def]) 1); |
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70 by (asm_simp_tac (arith_ss addsimps |
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71 [eq_sym_conv,count_addm_simp,Multiset.delm_nonempty_def, |
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72 Multiset.countm_nonempty_def,pos_count_imp_pos_countm, |
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73 suc_pred_id] |
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74 setloop (split_tac [expand_if])) 1); |
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75 val countm_done_delm = result(); |