1 (* Title: LK/lk.thy |
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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 |
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9 LK = Pure + |
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10 |
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11 classes term < logic |
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12 |
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13 default term |
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14 |
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15 types |
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16 o sequence seqobj seqcont sequ sobj |
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17 |
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18 arities |
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19 o :: logic |
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20 |
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21 consts |
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22 True,False :: "o" |
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23 "=" :: "['a,'a] => o" (infixl 50) |
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24 "Not" :: "o => o" ("~ _" [40] 40) |
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25 "&" :: "[o,o] => o" (infixr 35) |
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26 "|" :: "[o,o] => o" (infixr 30) |
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27 "-->","<->" :: "[o,o] => o" (infixr 25) |
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28 The :: "('a => o) => 'a" (binder "THE " 10) |
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29 All :: "('a => o) => o" (binder "ALL " 10) |
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30 Ex :: "('a => o) => o" (binder "EX " 10) |
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31 |
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32 (*Representation of sequents*) |
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33 Trueprop :: "[sobj=>sobj,sobj=>sobj] => prop" |
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34 Seqof :: "o => sobj=>sobj" |
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35 "@Trueprop" :: "[sequence,sequence] => prop" ("((_)/ |- (_))" [6,6] 5) |
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36 "@MtSeq" :: "sequence" ("" [] 1000) |
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37 "@NmtSeq" :: "[seqobj,seqcont] => sequence" ("__" [] 1000) |
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38 "@MtSeqCont" :: "seqcont" ("" [] 1000) |
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39 "@SeqCont" :: "[seqobj,seqcont] => seqcont" (",/ __" [] 1000) |
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40 "" :: "o => seqobj" ("_" [] 1000) |
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41 "@SeqId" :: "id => seqobj" ("$_" [] 1000) |
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42 "@SeqVar" :: "var => seqobj" ("$_") |
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43 |
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44 rules |
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45 (*Structural rules*) |
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46 |
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47 basic "$H, P, $G |- $E, P, $F" |
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48 |
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49 thinR "$H |- $E, $F ==> $H |- $E, P, $F" |
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50 thinL "$H, $G |- $E ==> $H, P, $G |- $E" |
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51 |
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52 cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E" |
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53 |
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54 (*Propositional rules*) |
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55 |
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56 conjR "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" |
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57 conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" |
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58 |
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59 disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" |
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60 disjL "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" |
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61 |
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62 impR "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" |
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63 impL "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" |
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64 |
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65 notR "$H, P |- $E, $F ==> $H |- $E, ~P, $F" |
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66 notL "$H, $G |- $E, P ==> $H, ~P, $G |- $E" |
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67 |
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68 FalseL "$H, False, $G |- $E" |
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69 |
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70 True_def "True == False-->False" |
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71 iff_def "P<->Q == (P-->Q) & (Q-->P)" |
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72 |
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73 (*Quantifiers*) |
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74 |
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75 allR "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x.P(x), $F" |
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76 allL "$H, P(x), $G, ALL x.P(x) |- $E ==> $H, ALL x.P(x), $G |- $E" |
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77 |
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78 exR "$H |- $E, P(x), $F, EX x.P(x) ==> $H |- $E, EX x.P(x), $F" |
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79 exL "(!!x.$H, P(x), $G |- $E) ==> $H, EX x.P(x), $G |- $E" |
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80 |
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81 (*Equality*) |
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82 |
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83 refl "$H |- $E, a=a, $F" |
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84 sym "$H |- $E, a=b, $F ==> $H |- $E, b=a, $F" |
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85 trans "[| $H|- $E, a=b, $F; $H|- $E, b=c, $F |] ==> $H|- $E, a=c, $F" |
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86 |
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87 |
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88 (*Descriptions*) |
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89 |
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90 The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==> \ |
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91 \ $H |- $E, P(THE x.P(x)), $F" |
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92 end |
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93 |
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94 ML |
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95 |
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96 (*Abstract over "sobj" -- representation of a sequence of formulae *) |
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97 fun abs_sobj t = Abs("sobj", Type("sobj",[]), t); |
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98 |
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99 (*Representation of empty sequence*) |
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100 val Sempty = abs_sobj (Bound 0); |
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101 |
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102 fun seq_obj_tr(Const("@SeqId",_)$id) = id | |
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103 seq_obj_tr(Const("@SeqVar",_)$id) = id | |
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104 seq_obj_tr(fm) = Const("Seqof",dummyT)$fm; |
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105 |
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106 fun seq_tr(_$obj$seq) = seq_obj_tr(obj)$seq_tr(seq) | |
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107 seq_tr(_) = Bound 0; |
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108 |
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109 fun seq_tr1(Const("@MtSeq",_)) = Sempty | |
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110 seq_tr1(seq) = abs_sobj(seq_tr seq); |
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111 |
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112 fun true_tr[s1,s2] = Const("Trueprop",dummyT)$seq_tr1 s1$seq_tr1 s2; |
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113 |
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114 fun seq_obj_tr'(Const("Seqof",_)$fm) = fm | |
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115 seq_obj_tr'(id) = Const("@SeqId",dummyT)$id; |
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116 |
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117 fun seq_tr'(obj$sq,C) = |
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118 let val sq' = case sq of |
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119 Bound 0 => Const("@MtSeqCont",dummyT) | |
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120 _ => seq_tr'(sq,Const("@SeqCont",dummyT)) |
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121 in C $ seq_obj_tr' obj $ sq' end; |
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122 |
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123 fun seq_tr1'(Bound 0) = Const("@MtSeq",dummyT) | |
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124 seq_tr1' s = seq_tr'(s,Const("@NmtSeq",dummyT)); |
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125 |
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126 fun true_tr'[Abs(_,_,s1),Abs(_,_,s2)] = |
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127 Const("@Trueprop",dummyT)$seq_tr1' s1$seq_tr1' s2; |
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128 |
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129 val parse_translation = [("@Trueprop",true_tr)]; |
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130 val print_translation = [("Trueprop",true_tr')]; |
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