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1 |
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2 |
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3 (*** Deadlock examples from section 5.6 ***) |
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4 |
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5 (*Trivial, two-process case*) |
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6 goalw thy [constrains_def, stable_def] |
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7 "!!Acts. [| constrains Acts (A Int B) A; constrains Acts (B Int A) B |] \ |
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8 \ ==> stable Acts (A Int B)"; |
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9 by (Blast_tac 1); |
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10 result(); |
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11 |
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12 |
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13 goal thy "{i. i < Suc n} = insert n {i. i < n}"; |
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14 by (blast_tac (claset() addSEs [less_SucE] addIs [less_SucI]) 1); |
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15 qed "Collect_less_Suc_insert"; |
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16 |
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17 |
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18 goal thy "{i. i <= Suc n} = insert (Suc n) {i. i <= n}"; |
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19 by (blast_tac (claset() addSEs [le_SucE] addIs [le_SucI]) 1); |
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20 qed "Collect_le_Suc_insert"; |
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21 |
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22 |
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23 (*a simplification step*) |
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24 goal thy "(INT i:{i. i <= n}. A(Suc i) Int A i) = \ |
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25 \ (INT i:{i. i <= Suc n}. A i)"; |
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26 by (induct_tac "n" 1); |
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27 by (ALLGOALS (asm_simp_tac (simpset() addsimps [Collect_le_Suc_insert]))); |
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28 by (blast_tac (claset() addEs [le_SucE] addSEs [equalityE]) 1); |
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29 qed "Collect_le_Int_equals"; |
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30 |
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31 |
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32 (*Dual of the required property. Converse inclusion fails.*) |
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33 goal thy "(UN i:{i. i < n}. A i) Int Compl (A n) <= \ |
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34 \ (UN i:{i. i < n}. (A i) Int Compl (A(Suc i)))"; |
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35 by (induct_tac "n" 1); |
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36 by (Asm_simp_tac 1); |
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37 by (simp_tac (simpset() addsimps [Collect_less_Suc_insert]) 1); |
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38 by (Blast_tac 1); |
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39 qed "UN_Int_Compl_subset"; |
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40 |
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41 |
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42 (*Converse inclusion fails.*) |
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43 goal thy "(INT i:{i. i < n}. Compl(A i) Un A (Suc i)) <= \ |
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44 \ (INT i:{i. i < n}. Compl(A i)) Un A n"; |
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45 by (induct_tac "n" 1); |
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46 by (Asm_simp_tac 1); |
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47 by (asm_simp_tac (simpset() addsimps [Collect_less_Suc_insert]) 1); |
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48 by (Blast_tac 1); |
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49 qed "INT_Un_Compl_subset"; |
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50 |
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51 |
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52 (*Specialized rewriting*) |
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53 goal thy "A 0 Int (Compl (A n) Int \ |
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54 \ (INT i:{i. i < n}. Compl(A i) Un A (Suc i))) = {}"; |
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55 by (blast_tac (claset() addIs [gr0I] |
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56 addDs [impOfSubs INT_Un_Compl_subset]) 1); |
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57 val lemma = result(); |
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58 |
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59 (*Reverse direction makes it harder to invoke the ind hyp*) |
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60 goal thy "(INT i:{i. i <= n}. A i) = \ |
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61 \ A 0 Int (INT i:{i. i < n}. Compl(A i) Un A(Suc i))"; |
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62 by (induct_tac "n" 1); |
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63 by (Asm_simp_tac 1); |
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64 by (asm_simp_tac |
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65 (simpset() addsimps (Int_ac @ |
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66 [Int_Un_distrib, Int_Un_distrib2, lemma, |
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67 Collect_less_Suc_insert, Collect_le_Suc_insert])) 1); |
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68 qed "INT_le_equals_Int"; |
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69 |
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70 goal thy "(INT i:{i. i <= Suc n}. A i) = \ |
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71 \ A 0 Int (INT i:{i. i <= n}. Compl(A i) Un A(Suc i))"; |
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72 by (simp_tac (simpset() addsimps [le_eq_less_Suc RS sym, |
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73 INT_le_equals_Int]) 1); |
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74 qed "INT_le_Suc_equals_Int"; |
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75 |
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76 |
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77 (*The final deadlock example*) |
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78 val [zeroprem, allprem] = goalw thy [stable_def] |
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79 "[| constrains Acts (A 0 Int A (Suc n)) (A 0); \ |
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80 \ ALL i:{i. i <= n}. constrains Acts (A(Suc i) Int A i) \ |
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81 \ (Compl(A i) Un A(Suc i)) |] \ |
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82 \ ==> stable Acts (INT i:{i. i <= Suc n}. A i)"; |
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83 |
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84 by (rtac ([zeroprem, allprem RS ball_constrains_INT] MRS |
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85 constrains_Int RS constrains_weaken) 1); |
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86 by (simp_tac (simpset() addsimps [Collect_le_Int_equals, |
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87 Int_assoc, INT_absorb]) 1); |
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88 by (simp_tac (simpset() addsimps ([INT_le_Suc_equals_Int])) 1); |
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89 result(); |
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90 |