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1 (* Title: LK/LK0 |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1993 University of Cambridge |
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5 |
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6 Classical First-Order Sequent Calculus |
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7 |
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8 There may be printing problems if a seqent is in expanded normal form |
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9 (eta-expanded, beta-contracted) |
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10 *) |
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11 |
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12 LK0 = Sequents + |
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13 |
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14 global |
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15 |
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16 classes |
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17 term < logic |
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18 |
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19 default |
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20 term |
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21 |
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22 consts |
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23 |
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24 Trueprop :: "two_seqi" |
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25 "@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5) |
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26 |
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27 (*Constant to allow definitions of SEQUENCES of formulas*) |
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28 "@Side" :: "seq=>(seq'=>seq')" ("<<(_)>>") |
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29 |
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30 True,False :: o |
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31 "=" :: ['a,'a] => o (infixl 50) |
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32 Not :: o => o ("~ _" [40] 40) |
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33 "&" :: [o,o] => o (infixr 35) |
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34 "|" :: [o,o] => o (infixr 30) |
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35 "-->","<->" :: [o,o] => o (infixr 25) |
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36 The :: ('a => o) => 'a (binder "THE " 10) |
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37 All :: ('a => o) => o (binder "ALL " 10) |
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38 Ex :: ('a => o) => o (binder "EX " 10) |
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39 |
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40 syntax |
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41 "~=" :: ['a, 'a] => o (infixl 50) |
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42 |
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43 translations |
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44 "x ~= y" == "~ (x = y)" |
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45 |
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46 syntax (symbols) |
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47 Not :: o => o ("\\<not> _" [40] 40) |
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48 "op &" :: [o, o] => o (infixr "\\<and>" 35) |
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49 "op |" :: [o, o] => o (infixr "\\<or>" 30) |
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50 "op -->" :: [o, o] => o (infixr "\\<midarrow>\\<rightarrow>" 25) |
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51 "op <->" :: [o, o] => o (infixr "\\<leftarrow>\\<rightarrow>" 25) |
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52 "ALL " :: [idts, o] => o ("(3\\<forall>_./ _)" [0, 10] 10) |
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53 "EX " :: [idts, o] => o ("(3\\<exists>_./ _)" [0, 10] 10) |
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54 "EX! " :: [idts, o] => o ("(3\\<exists>!_./ _)" [0, 10] 10) |
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55 "op ~=" :: ['a, 'a] => o (infixl "\\<noteq>" 50) |
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56 |
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57 syntax (xsymbols) |
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58 "op -->" :: [o, o] => o (infixr "\\<longrightarrow>" 25) |
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59 "op <->" :: [o, o] => o (infixr "\\<longleftrightarrow>" 25) |
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60 |
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61 syntax (HTML output) |
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62 Not :: o => o ("\\<not> _" [40] 40) |
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63 |
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64 |
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65 local |
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66 |
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67 rules |
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68 |
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69 (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *) |
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70 |
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71 contRS "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F" |
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72 contLS "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E" |
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73 |
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74 thinRS "$H |- $E, $F ==> $H |- $E, $S, $F" |
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75 thinLS "$H, $G |- $E ==> $H, $S, $G |- $E" |
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76 |
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77 exchRS "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F" |
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78 exchLS "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E" |
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79 |
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80 cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E" |
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81 |
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82 (*Propositional rules*) |
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83 |
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84 basic "$H, P, $G |- $E, P, $F" |
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85 |
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86 conjR "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" |
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87 conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" |
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88 |
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89 disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" |
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90 disjL "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" |
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91 |
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92 impR "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" |
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93 impL "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" |
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94 |
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95 notR "$H, P |- $E, $F ==> $H |- $E, ~P, $F" |
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96 notL "$H, $G |- $E, P ==> $H, ~P, $G |- $E" |
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97 |
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98 FalseL "$H, False, $G |- $E" |
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99 |
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100 True_def "True == False-->False" |
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101 iff_def "P<->Q == (P-->Q) & (Q-->P)" |
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102 |
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103 (*Quantifiers*) |
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104 |
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105 allR "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F" |
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106 allL "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E" |
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107 |
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108 exR "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F" |
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109 exL "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E" |
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110 |
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111 (*Equality*) |
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112 |
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113 refl "$H |- $E, a=a, $F" |
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114 subst "$H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)" |
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115 |
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116 (* Reflection *) |
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117 |
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118 eq_reflection "|- x=y ==> (x==y)" |
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119 iff_reflection "|- P<->Q ==> (P==Q)" |
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120 |
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121 (*Descriptions*) |
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122 |
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123 The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==> |
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124 $H |- $E, P(THE x. P(x)), $F" |
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125 |
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126 constdefs |
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127 If :: [o, 'a, 'a] => 'a ("(if (_)/ then (_)/ else (_))" 10) |
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128 "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)" |
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129 |
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130 |
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131 setup |
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132 Simplifier.setup |
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133 |
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134 setup |
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135 prover_setup |
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136 |
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137 end |
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138 |
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139 ML |
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140 |
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141 val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")]; |
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142 val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")]; |