14 HOL system proofs may be found in |
14 HOL system proofs may be found in |
15 /usr/groups/theory/hvg-aftp/contrib/rule-induction/cl.ml |
15 /usr/groups/theory/hvg-aftp/contrib/rule-induction/cl.ml |
16 *) |
16 *) |
17 |
17 |
18 open Comb; |
18 open Comb; |
19 |
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20 val [_,_,comb_Ap] = comb.intrs; |
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21 val Ap_E = comb.mk_cases comb.con_defs "p#q : comb"; |
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22 |
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23 |
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24 (*** Results about Contraction ***) |
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25 |
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26 (*For type checking: replaces a-1->b by a,b:comb *) |
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27 val contract_combE2 = contract.dom_subset RS subsetD RS SigmaE2; |
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28 val contract_combD1 = contract.dom_subset RS subsetD RS SigmaD1; |
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29 val contract_combD2 = contract.dom_subset RS subsetD RS SigmaD2; |
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30 |
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31 goal Comb.thy "field(contract) = comb"; |
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32 by (blast_tac (!claset addIs [contract.K] addSEs [contract_combE2]) 1); |
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33 qed "field_contract_eq"; |
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34 |
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35 bind_thm ("reduction_refl", |
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36 (field_contract_eq RS equalityD2 RS subsetD RS rtrancl_refl)); |
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37 |
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38 bind_thm ("rtrancl_into_rtrancl2", |
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39 (r_into_rtrancl RS (trans_rtrancl RS transD))); |
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40 |
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41 val reduction_rls = [reduction_refl, contract.K, contract.S, |
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42 contract.K RS rtrancl_into_rtrancl2, |
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43 contract.S RS rtrancl_into_rtrancl2, |
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44 contract.Ap1 RS rtrancl_into_rtrancl2, |
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45 contract.Ap2 RS rtrancl_into_rtrancl2]; |
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46 |
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47 (*Example only: not used*) |
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48 goalw Comb.thy [I_def] "!!p. p:comb ==> I#p ---> p"; |
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49 by (REPEAT (ares_tac (comb.intrs @ reduction_rls) 1)); |
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50 qed "reduce_I"; |
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51 |
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52 goalw Comb.thy [I_def] "I: comb"; |
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53 by (REPEAT (ares_tac comb.intrs 1)); |
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54 qed "comb_I"; |
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55 |
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56 (** Non-contraction results **) |
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57 |
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58 (*Derive a case for each combinator constructor*) |
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59 val K_contractE = contract.mk_cases comb.con_defs "K -1-> r"; |
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60 val S_contractE = contract.mk_cases comb.con_defs "S -1-> r"; |
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61 val Ap_contractE = contract.mk_cases comb.con_defs "p#q -1-> r"; |
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62 |
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63 AddSIs comb.intrs; |
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64 AddIs contract.intrs; |
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65 AddSEs [contract_combD1,contract_combD2]; (*type checking*) |
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66 AddSEs [K_contractE, S_contractE, Ap_contractE]; |
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67 AddSEs comb.free_SEs; |
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68 |
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69 goalw Comb.thy [I_def] "!!r. I -1-> r ==> P"; |
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70 by (Blast_tac 1); |
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71 qed "I_contract_E"; |
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72 |
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73 goal Comb.thy "!!p r. K#p -1-> r ==> (EX q. r = K#q & p -1-> q)"; |
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74 by (Blast_tac 1); |
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75 qed "K1_contractD"; |
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76 |
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77 goal Comb.thy "!!p r. [| p ---> q; r: comb |] ==> p#r ---> q#r"; |
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78 by (forward_tac [rtrancl_type RS subsetD RS SigmaD1] 1); |
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79 by (dtac (field_contract_eq RS equalityD1 RS subsetD) 1); |
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80 by (etac rtrancl_induct 1); |
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81 by (blast_tac (!claset addIs reduction_rls) 1); |
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82 by (etac (trans_rtrancl RS transD) 1); |
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83 by (fast_tac (!claset addIs reduction_rls) 1); |
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84 qed "Ap_reduce1"; |
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85 |
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86 goal Comb.thy "!!p r. [| p ---> q; r: comb |] ==> r#p ---> r#q"; |
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87 by (forward_tac [rtrancl_type RS subsetD RS SigmaD1] 1); |
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88 by (dtac (field_contract_eq RS equalityD1 RS subsetD) 1); |
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89 by (etac rtrancl_induct 1); |
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90 by (blast_tac (!claset addIs reduction_rls) 1); |
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91 by (etac (trans_rtrancl RS transD) 1); |
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92 by (fast_tac (!claset addIs reduction_rls) 1); |
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93 qed "Ap_reduce2"; |
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94 |
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95 (** Counterexample to the diamond property for -1-> **) |
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96 |
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97 goal Comb.thy "K#I#(I#I) -1-> I"; |
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98 by (REPEAT (ares_tac [contract.K, comb_I, comb_Ap] 1)); |
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99 qed "KIII_contract1"; |
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100 |
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101 goalw Comb.thy [I_def] "K#I#(I#I) -1-> K#I#((K#I)#(K#I))"; |
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102 by (DEPTH_SOLVE (resolve_tac (comb.intrs @ contract.intrs) 1)); |
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103 qed "KIII_contract2"; |
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104 |
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105 goal Comb.thy "K#I#((K#I)#(K#I)) -1-> I"; |
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106 by (REPEAT (ares_tac (comb.intrs @ [contract.K, comb_I]) 1)); |
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107 qed "KIII_contract3"; |
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108 |
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109 goalw Comb.thy [diamond_def] "~ diamond(contract)"; |
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110 by (blast_tac (!claset addIs [KIII_contract1,KIII_contract2,KIII_contract3] |
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111 addSEs [I_contract_E]) 1); |
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112 qed "not_diamond_contract"; |
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113 |
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114 |
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115 |
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116 (*** Results about Parallel Contraction ***) |
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117 |
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118 (*For type checking: replaces a=1=>b by a,b:comb *) |
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119 val parcontract_combE2 = parcontract.dom_subset RS subsetD RS SigmaE2; |
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120 val parcontract_combD1 = parcontract.dom_subset RS subsetD RS SigmaD1; |
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121 val parcontract_combD2 = parcontract.dom_subset RS subsetD RS SigmaD2; |
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122 |
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123 goal Comb.thy "field(parcontract) = comb"; |
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124 by (blast_tac (!claset addIs [parcontract.K] |
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125 addSEs [parcontract_combE2]) 1); |
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126 qed "field_parcontract_eq"; |
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127 |
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128 (*Derive a case for each combinator constructor*) |
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129 val K_parcontractE = parcontract.mk_cases comb.con_defs "K =1=> r"; |
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130 val S_parcontractE = parcontract.mk_cases comb.con_defs "S =1=> r"; |
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131 val Ap_parcontractE = parcontract.mk_cases comb.con_defs "p#q =1=> r"; |
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132 |
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133 AddSIs comb.intrs; |
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134 AddIs parcontract.intrs; |
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135 AddSEs [Ap_E, K_parcontractE, S_parcontractE, Ap_parcontractE]; |
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136 AddSEs [parcontract_combD1, parcontract_combD2]; (*type checking*) |
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137 AddSEs comb.free_SEs; |
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138 |
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139 (*** Basic properties of parallel contraction ***) |
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140 |
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141 goal Comb.thy "!!p r. K#p =1=> r ==> (EX p'. r = K#p' & p =1=> p')"; |
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142 by (Blast_tac 1); |
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143 qed "K1_parcontractD"; |
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144 |
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145 goal Comb.thy "!!p r. S#p =1=> r ==> (EX p'. r = S#p' & p =1=> p')"; |
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146 by (Blast_tac 1); |
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147 qed "S1_parcontractD"; |
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148 |
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149 goal Comb.thy |
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150 "!!p q r. S#p#q =1=> r ==> (EX p' q'. r = S#p'#q' & p =1=> p' & q =1=> q')"; |
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151 by (blast_tac (!claset addSDs [S1_parcontractD]) 1); |
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152 qed "S2_parcontractD"; |
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153 |
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154 (*Church-Rosser property for parallel contraction*) |
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155 goalw Comb.thy [diamond_def] "diamond(parcontract)"; |
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156 by (rtac (impI RS allI RS allI) 1); |
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157 by (etac parcontract.induct 1); |
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158 by (ALLGOALS |
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159 (fast_tac (!claset addSDs [K1_parcontractD,S2_parcontractD]))); |
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160 qed "diamond_parcontract"; |
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161 |
19 |
162 (*** Transitive closure preserves the Church-Rosser property ***) |
20 (*** Transitive closure preserves the Church-Rosser property ***) |
163 |
21 |
164 goalw Comb.thy [diamond_def] |
22 goalw Comb.thy [diamond_def] |
165 "!!x y r. [| diamond(r); <x,y>:r^+ |] ==> \ |
23 "!!x y r. [| diamond(r); <x,y>:r^+ |] ==> \ |
172 |
30 |
173 val [major] = goal Comb.thy "diamond(r) ==> diamond(r^+)"; |
31 val [major] = goal Comb.thy "diamond(r) ==> diamond(r^+)"; |
174 by (rewtac diamond_def); (*unfold only in goal, not in premise!*) |
32 by (rewtac diamond_def); (*unfold only in goal, not in premise!*) |
175 by (rtac (impI RS allI RS allI) 1); |
33 by (rtac (impI RS allI RS allI) 1); |
176 by (etac trancl_induct 1); |
34 by (etac trancl_induct 1); |
177 by (ALLGOALS |
35 by (ALLGOALS (*Seems to be a brittle, undirected search*) |
178 (slow_best_tac (!claset addIs [r_into_trancl, trans_trancl RS transD] |
36 (slow_best_tac (!claset addIs [r_into_trancl, trans_trancl RS transD] |
179 addEs [major RS diamond_strip_lemmaE]))); |
37 addEs [major RS diamond_strip_lemmaE]))); |
180 qed "diamond_trancl"; |
38 qed "diamond_trancl"; |
181 |
39 |
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40 |
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41 val [_,_,comb_Ap] = comb.intrs; |
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42 val Ap_E = comb.mk_cases comb.con_defs "p#q : comb"; |
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43 |
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44 AddSIs comb.intrs; |
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45 AddSEs comb.free_SEs; |
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46 |
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47 |
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48 (*** Results about Contraction ***) |
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49 |
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50 (*For type checking: replaces a-1->b by a,b:comb *) |
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51 val contract_combE2 = contract.dom_subset RS subsetD RS SigmaE2; |
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52 val contract_combD1 = contract.dom_subset RS subsetD RS SigmaD1; |
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53 val contract_combD2 = contract.dom_subset RS subsetD RS SigmaD2; |
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54 |
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55 goal Comb.thy "field(contract) = comb"; |
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56 by (blast_tac (!claset addIs [contract.K] addSEs [contract_combE2]) 1); |
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57 qed "field_contract_eq"; |
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58 |
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59 bind_thm ("reduction_refl", |
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60 (field_contract_eq RS equalityD2 RS subsetD RS rtrancl_refl)); |
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61 |
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62 bind_thm ("rtrancl_into_rtrancl2", |
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63 (r_into_rtrancl RS (trans_rtrancl RS transD))); |
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64 |
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65 AddSIs [reduction_refl, contract.K, contract.S]; |
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66 |
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67 val reduction_rls = [contract.K RS rtrancl_into_rtrancl2, |
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68 contract.S RS rtrancl_into_rtrancl2, |
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69 contract.Ap1 RS rtrancl_into_rtrancl2, |
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70 contract.Ap2 RS rtrancl_into_rtrancl2]; |
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71 |
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72 (*Example only: not used*) |
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73 goalw Comb.thy [I_def] "!!p. p:comb ==> I#p ---> p"; |
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74 by (blast_tac (!claset addIs reduction_rls) 1); |
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75 qed "reduce_I"; |
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76 |
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77 goalw Comb.thy [I_def] "I: comb"; |
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78 by (Blast_tac 1); |
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79 qed "comb_I"; |
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80 |
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81 (** Non-contraction results **) |
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82 |
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83 (*Derive a case for each combinator constructor*) |
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84 val K_contractE = contract.mk_cases comb.con_defs "K -1-> r"; |
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85 val S_contractE = contract.mk_cases comb.con_defs "S -1-> r"; |
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86 val Ap_contractE = contract.mk_cases comb.con_defs "p#q -1-> r"; |
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87 |
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88 AddIs [contract.Ap1, contract.Ap2]; |
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89 AddSEs [K_contractE, S_contractE, Ap_contractE]; |
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90 |
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91 goalw Comb.thy [I_def] "!!r. I -1-> r ==> P"; |
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92 by (Blast_tac 1); |
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93 qed "I_contract_E"; |
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94 |
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95 goal Comb.thy "!!p r. K#p -1-> r ==> (EX q. r = K#q & p -1-> q)"; |
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96 by (Blast_tac 1); |
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97 qed "K1_contractD"; |
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98 |
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99 goal Comb.thy "!!p r. [| p ---> q; r: comb |] ==> p#r ---> q#r"; |
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100 by (forward_tac [rtrancl_type RS subsetD RS SigmaD1] 1); |
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101 by (dtac (field_contract_eq RS equalityD1 RS subsetD) 1); |
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102 by (etac rtrancl_induct 1); |
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103 by (blast_tac (!claset addIs reduction_rls) 1); |
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104 by (etac (trans_rtrancl RS transD) 1); |
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105 by (blast_tac (!claset addIs (contract_combD2::reduction_rls)) 1); |
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106 qed "Ap_reduce1"; |
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107 |
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108 goal Comb.thy "!!p r. [| p ---> q; r: comb |] ==> r#p ---> r#q"; |
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109 by (forward_tac [rtrancl_type RS subsetD RS SigmaD1] 1); |
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110 by (dtac (field_contract_eq RS equalityD1 RS subsetD) 1); |
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111 by (etac rtrancl_induct 1); |
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112 by (blast_tac (!claset addIs reduction_rls) 1); |
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113 by (etac (trans_rtrancl RS transD) 1); |
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114 by (blast_tac (!claset addIs (contract_combD2::reduction_rls)) 1); |
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115 qed "Ap_reduce2"; |
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116 |
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117 (** Counterexample to the diamond property for -1-> **) |
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118 |
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119 goal Comb.thy "K#I#(I#I) -1-> I"; |
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120 by (REPEAT (ares_tac [contract.K, comb_I, comb_Ap] 1)); |
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121 qed "KIII_contract1"; |
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122 |
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123 goalw Comb.thy [I_def] "K#I#(I#I) -1-> K#I#((K#I)#(K#I))"; |
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124 by (DEPTH_SOLVE (resolve_tac (comb.intrs @ contract.intrs) 1)); |
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125 qed "KIII_contract2"; |
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126 |
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127 goal Comb.thy "K#I#((K#I)#(K#I)) -1-> I"; |
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128 by (REPEAT (ares_tac (comb.intrs @ [contract.K, comb_I]) 1)); |
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129 qed "KIII_contract3"; |
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130 |
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131 goalw Comb.thy [diamond_def] "~ diamond(contract)"; |
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132 by (blast_tac (!claset addIs [KIII_contract1,KIII_contract2,KIII_contract3] |
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133 addSEs [I_contract_E]) 1); |
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134 qed "not_diamond_contract"; |
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135 |
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136 |
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137 |
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138 (*** Results about Parallel Contraction ***) |
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139 |
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140 (*For type checking: replaces a=1=>b by a,b:comb *) |
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141 val parcontract_combE2 = parcontract.dom_subset RS subsetD RS SigmaE2; |
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142 val parcontract_combD1 = parcontract.dom_subset RS subsetD RS SigmaD1; |
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143 val parcontract_combD2 = parcontract.dom_subset RS subsetD RS SigmaD2; |
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144 |
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145 goal Comb.thy "field(parcontract) = comb"; |
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146 by (blast_tac (!claset addIs [parcontract.K] |
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147 addSEs [parcontract_combE2]) 1); |
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148 qed "field_parcontract_eq"; |
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149 |
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150 (*Derive a case for each combinator constructor*) |
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151 val K_parcontractE = parcontract.mk_cases comb.con_defs "K =1=> r"; |
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152 val S_parcontractE = parcontract.mk_cases comb.con_defs "S =1=> r"; |
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153 val Ap_parcontractE = parcontract.mk_cases comb.con_defs "p#q =1=> r"; |
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154 |
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155 AddIs parcontract.intrs; |
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156 AddSEs [Ap_E, K_parcontractE, S_parcontractE, Ap_parcontractE]; |
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157 |
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158 (*** Basic properties of parallel contraction ***) |
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159 |
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160 goal Comb.thy "!!p r. K#p =1=> r ==> (EX p'. r = K#p' & p =1=> p')"; |
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161 by (Blast_tac 1); |
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162 qed "K1_parcontractD"; |
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163 |
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164 goal Comb.thy "!!p r. S#p =1=> r ==> (EX p'. r = S#p' & p =1=> p')"; |
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165 by (Blast_tac 1); |
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166 qed "S1_parcontractD"; |
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167 |
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168 goal Comb.thy |
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169 "!!p q r. S#p#q =1=> r ==> (EX p' q'. r = S#p'#q' & p =1=> p' & q =1=> q')"; |
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170 by (blast_tac (!claset addSDs [S1_parcontractD]) 1); |
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171 qed "S2_parcontractD"; |
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172 |
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173 (*Church-Rosser property for parallel contraction*) |
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174 goalw Comb.thy [diamond_def] "diamond(parcontract)"; |
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175 by (rtac (impI RS allI RS allI) 1); |
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176 by (etac parcontract.induct 1); |
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177 by (ALLGOALS |
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178 (blast_tac (!claset addSDs [K1_parcontractD, S2_parcontractD] |
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179 addIs [parcontract_combD2]))); |
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180 qed "diamond_parcontract"; |
182 |
181 |
183 (*** Equivalence of p--->q and p===>q ***) |
182 (*** Equivalence of p--->q and p===>q ***) |
184 |
183 |
185 goal Comb.thy "!!p q. p-1->q ==> p=1=>q"; |
184 goal Comb.thy "!!p q. p-1->q ==> p=1=>q"; |
186 by (etac contract.induct 1); |
185 by (etac contract.induct 1); |
200 goal Comb.thy "!!p q. p=1=>q ==> p--->q"; |
199 goal Comb.thy "!!p q. p=1=>q ==> p--->q"; |
201 by (etac parcontract.induct 1); |
200 by (etac parcontract.induct 1); |
202 by (blast_tac (!claset addIs reduction_rls) 1); |
201 by (blast_tac (!claset addIs reduction_rls) 1); |
203 by (blast_tac (!claset addIs reduction_rls) 1); |
202 by (blast_tac (!claset addIs reduction_rls) 1); |
204 by (blast_tac (!claset addIs reduction_rls) 1); |
203 by (blast_tac (!claset addIs reduction_rls) 1); |
205 by (rtac (trans_rtrancl RS transD) 1); |
204 by (blast_tac |
206 by (ALLGOALS |
205 (!claset addIs [trans_rtrancl RS transD, |
207 (blast_tac |
206 Ap_reduce1, Ap_reduce2, parcontract_combD1, |
208 (!claset addIs [Ap_reduce1, Ap_reduce2, parcontract_combD1, |
207 parcontract_combD2]) 1); |
209 parcontract_combD2]))); |
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210 qed "parcontract_imp_reduce"; |
208 qed "parcontract_imp_reduce"; |
211 |
209 |
212 goal Comb.thy "!!p q. p===>q ==> p--->q"; |
210 goal Comb.thy "!!p q. p===>q ==> p--->q"; |
213 by (forward_tac [trancl_type RS subsetD RS SigmaD1] 1); |
211 by (forward_tac [trancl_type RS subsetD RS SigmaD1] 1); |
214 by (dtac (field_parcontract_eq RS equalityD1 RS subsetD) 1); |
212 by (dtac (field_parcontract_eq RS equalityD1 RS subsetD) 1); |