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1 (**** FOL examples ****) |
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2 |
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3 Pretty.setmargin 72; (*existing macros just allow this margin*) |
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4 print_depth 0; |
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5 |
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6 (*** Intuitionistic examples ***) |
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7 |
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8 context IFOL.thy; |
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9 |
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10 (*Quantifier example from Logic&Computation*) |
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11 Goal "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"; |
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12 by (resolve_tac [impI] 1); |
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13 by (resolve_tac [allI] 1); |
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14 by (resolve_tac [exI] 1); |
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15 by (eresolve_tac [exE] 1); |
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16 choplev 2; |
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17 by (eresolve_tac [exE] 1); |
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18 by (resolve_tac [exI] 1); |
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19 by (eresolve_tac [allE] 1); |
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20 by (assume_tac 1); |
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21 |
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22 |
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23 (*Example of Dyckhoff's method*) |
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24 Goalw [not_def] "~ ~ ((P-->Q) | (Q-->P))"; |
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25 by (resolve_tac [impI] 1); |
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26 by (eresolve_tac [disj_impE] 1); |
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27 by (eresolve_tac [imp_impE] 1); |
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28 by (eresolve_tac [imp_impE] 1); |
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29 by (REPEAT (eresolve_tac [FalseE] 2)); |
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30 by (assume_tac 1); |
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31 |
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32 |
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33 |
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34 |
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35 |
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36 (*** Classical examples ***) |
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37 |
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38 context FOL.thy; |
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39 |
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40 Goal "EX y. ALL x. P(y)-->P(x)"; |
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41 by (resolve_tac [exCI] 1); |
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42 by (resolve_tac [allI] 1); |
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43 by (resolve_tac [impI] 1); |
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44 by (eresolve_tac [allE] 1); |
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45 prth (allI RSN (2,swap)); |
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46 by (eresolve_tac [it] 1); |
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47 by (resolve_tac [impI] 1); |
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48 by (eresolve_tac [notE] 1); |
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49 by (assume_tac 1); |
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50 Goal "EX y. ALL x. P(y)-->P(x)"; |
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51 by (Blast_tac 1); |
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52 |
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53 |
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54 |
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55 - Goal "EX y. ALL x. P(y)-->P(x)"; |
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56 Level 0 |
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57 EX y. ALL x. P(y) --> P(x) |
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58 1. EX y. ALL x. P(y) --> P(x) |
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59 - by (resolve_tac [exCI] 1); |
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60 Level 1 |
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61 EX y. ALL x. P(y) --> P(x) |
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62 1. ALL y. ~(ALL x. P(y) --> P(x)) ==> ALL x. P(?a) --> P(x) |
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63 - by (resolve_tac [allI] 1); |
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64 Level 2 |
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65 EX y. ALL x. P(y) --> P(x) |
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66 1. !!x. ALL y. ~(ALL x. P(y) --> P(x)) ==> P(?a) --> P(x) |
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67 - by (resolve_tac [impI] 1); |
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68 Level 3 |
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69 EX y. ALL x. P(y) --> P(x) |
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70 1. !!x. [| ALL y. ~(ALL x. P(y) --> P(x)); P(?a) |] ==> P(x) |
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71 - by (eresolve_tac [allE] 1); |
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72 Level 4 |
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73 EX y. ALL x. P(y) --> P(x) |
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74 1. !!x. [| P(?a); ~(ALL xa. P(?y3(x)) --> P(xa)) |] ==> P(x) |
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75 - prth (allI RSN (2,swap)); |
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76 [| ~(ALL x. ?P1(x)); !!x. ~?Q ==> ?P1(x) |] ==> ?Q |
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77 - by (eresolve_tac [it] 1); |
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78 Level 5 |
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79 EX y. ALL x. P(y) --> P(x) |
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80 1. !!x xa. [| P(?a); ~P(x) |] ==> P(?y3(x)) --> P(xa) |
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81 - by (resolve_tac [impI] 1); |
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82 Level 6 |
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83 EX y. ALL x. P(y) --> P(x) |
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84 1. !!x xa. [| P(?a); ~P(x); P(?y3(x)) |] ==> P(xa) |
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85 - by (eresolve_tac [notE] 1); |
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86 Level 7 |
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87 EX y. ALL x. P(y) --> P(x) |
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88 1. !!x xa. [| P(?a); P(?y3(x)) |] ==> P(x) |
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89 - by (assume_tac 1); |
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90 Level 8 |
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91 EX y. ALL x. P(y) --> P(x) |
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92 No subgoals! |
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93 - Goal "EX y. ALL x. P(y)-->P(x)"; |
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94 Level 0 |
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95 EX y. ALL x. P(y) --> P(x) |
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96 1. EX y. ALL x. P(y) --> P(x) |
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97 - by (best_tac FOL_dup_cs 1); |
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98 Level 1 |
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99 EX y. ALL x. P(y) --> P(x) |
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100 No subgoals! |
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101 |
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102 |
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103 (**** finally, the example FOL/ex/if.ML ****) |
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104 |
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105 > val prems = goalw if_thy [if_def] |
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106 # "[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)"; |
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107 Level 0 |
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108 if(P,Q,R) |
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109 1. P & Q | ~P & R |
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110 > by (Classical.fast_tac (FOL_cs addIs prems) 1); |
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111 Level 1 |
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112 if(P,Q,R) |
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113 No subgoals! |
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114 > val ifI = result(); |
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115 |
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116 |
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117 > val major::prems = goalw if_thy [if_def] |
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118 # "[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S"; |
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119 Level 0 |
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120 S |
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121 1. S |
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122 > by (cut_facts_tac [major] 1); |
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123 Level 1 |
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124 S |
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125 1. P & Q | ~P & R ==> S |
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126 > by (Classical.fast_tac (FOL_cs addIs prems) 1); |
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127 Level 2 |
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128 S |
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129 No subgoals! |
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130 > val ifE = result(); |
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131 |
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132 > goal if_thy "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))"; |
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133 Level 0 |
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134 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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135 1. if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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136 > by (resolve_tac [iffI] 1); |
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137 Level 1 |
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138 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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139 1. if(P,if(Q,A,B),if(Q,C,D)) ==> if(Q,if(P,A,C),if(P,B,D)) |
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140 2. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D)) |
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141 > by (eresolve_tac [ifE] 1); |
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142 Level 2 |
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143 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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144 1. [| P; if(Q,A,B) |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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145 2. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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146 3. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D)) |
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147 > by (eresolve_tac [ifE] 1); |
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148 Level 3 |
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149 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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150 1. [| P; Q; A |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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151 2. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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152 3. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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153 4. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D)) |
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154 > by (resolve_tac [ifI] 1); |
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155 Level 4 |
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156 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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157 1. [| P; Q; A; Q |] ==> if(P,A,C) |
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158 2. [| P; Q; A; ~Q |] ==> if(P,B,D) |
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159 3. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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160 4. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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161 5. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D)) |
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162 > by (resolve_tac [ifI] 1); |
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163 Level 5 |
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164 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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165 1. [| P; Q; A; Q; P |] ==> A |
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166 2. [| P; Q; A; Q; ~P |] ==> C |
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167 3. [| P; Q; A; ~Q |] ==> if(P,B,D) |
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168 4. [| P; ~Q; B |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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169 5. [| ~P; if(Q,C,D) |] ==> if(Q,if(P,A,C),if(P,B,D)) |
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170 6. if(Q,if(P,A,C),if(P,B,D)) ==> if(P,if(Q,A,B),if(Q,C,D)) |
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171 |
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172 > choplev 0; |
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173 Level 0 |
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174 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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175 1. if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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176 > val if_cs = FOL_cs addSIs [ifI] addSEs[ifE]; |
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177 > by (Classical.fast_tac if_cs 1); |
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178 Level 1 |
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179 if(P,if(Q,A,B),if(Q,C,D)) <-> if(Q,if(P,A,C),if(P,B,D)) |
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180 No subgoals! |
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181 > val if_commute = result(); |
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182 |
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183 > goal if_thy "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))"; |
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184 Level 0 |
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185 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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186 1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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187 > by (Classical.fast_tac if_cs 1); |
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188 Level 1 |
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189 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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190 No subgoals! |
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191 > val nested_ifs = result(); |
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192 |
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193 |
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194 > choplev 0; |
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195 Level 0 |
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196 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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197 1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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198 > by (rewrite_goals_tac [if_def]); |
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199 Level 1 |
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200 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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201 1. (P & Q | ~P & R) & A | ~(P & Q | ~P & R) & B <-> |
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202 P & (Q & A | ~Q & B) | ~P & (R & A | ~R & B) |
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203 > by (Classical.fast_tac FOL_cs 1); |
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204 Level 2 |
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205 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B)) |
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206 No subgoals! |
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207 |
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208 |
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209 > goal if_thy "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))"; |
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210 Level 0 |
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211 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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212 1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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213 > by (REPEAT (Classical.step_tac if_cs 1)); |
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214 Level 1 |
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215 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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216 1. [| A; ~P; R; ~P; R |] ==> B |
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217 2. [| B; ~P; ~R; ~P; ~R |] ==> A |
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218 3. [| ~P; R; B; ~P; R |] ==> A |
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219 4. [| ~P; ~R; A; ~B; ~P |] ==> R |
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220 |
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221 > choplev 0; |
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222 Level 0 |
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223 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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224 1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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225 > by (rewrite_goals_tac [if_def]); |
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226 Level 1 |
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227 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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228 1. (P & Q | ~P & R) & A | ~(P & Q | ~P & R) & B <-> |
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229 P & (Q & A | ~Q & B) | ~P & (R & B | ~R & A) |
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230 > by (Classical.fast_tac FOL_cs 1); |
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231 by: tactic failed |
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232 Exception- ERROR raised |
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233 Exception failure raised |
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234 |
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235 > by (REPEAT (Classical.step_tac FOL_cs 1)); |
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236 Level 2 |
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237 if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,B,A)) |
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238 1. [| A; ~P; R; ~P; R; ~False |] ==> B |
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239 2. [| A; ~P; R; R; ~False; ~B; ~B |] ==> Q |
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240 3. [| B; ~P; ~R; ~P; ~A |] ==> R |
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241 4. [| B; ~P; ~R; ~Q; ~A |] ==> R |
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242 5. [| B; ~R; ~P; ~A; ~R; Q; ~False |] ==> A |
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243 6. [| ~P; R; B; ~P; R; ~False |] ==> A |
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244 7. [| ~P; ~R; A; ~B; ~R |] ==> P |
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245 8. [| ~P; ~R; A; ~B; ~R |] ==> Q |