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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 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 LK = Sequents + |
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13 |
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14 |
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15 consts |
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16 |
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17 Trueprop :: "two_seqi" |
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18 "@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5) |
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19 |
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20 |
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21 True,False :: o |
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22 "=" :: ['a,'a] => o (infixl 50) |
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23 Not :: o => o ("~ _" [40] 40) |
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24 "&" :: [o,o] => o (infixr 35) |
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25 "|" :: [o,o] => o (infixr 30) |
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26 "-->","<->" :: [o,o] => o (infixr 25) |
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27 The :: ('a => o) => 'a (binder "THE " 10) |
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28 All :: ('a => o) => o (binder "ALL " 10) |
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29 Ex :: ('a => o) => o (binder "EX " 10) |
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30 |
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31 rules |
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32 (*Structural rules*) |
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33 |
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34 basic "$H, P, $G |- $E, P, $F" |
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35 |
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36 thinR "$H |- $E, $F ==> $H |- $E, P, $F" |
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37 thinL "$H, $G |- $E ==> $H, P, $G |- $E" |
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38 |
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39 cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E" |
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40 |
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41 (*Propositional rules*) |
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42 |
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43 conjR "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" |
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44 conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" |
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45 |
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46 disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" |
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47 disjL "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" |
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48 |
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49 impR "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" |
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50 impL "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" |
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51 |
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52 notR "$H, P |- $E, $F ==> $H |- $E, ~P, $F" |
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53 notL "$H, $G |- $E, P ==> $H, ~P, $G |- $E" |
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54 |
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55 FalseL "$H, False, $G |- $E" |
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56 |
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57 True_def "True == False-->False" |
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58 iff_def "P<->Q == (P-->Q) & (Q-->P)" |
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59 |
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60 (*Quantifiers*) |
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61 |
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62 allR "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x.P(x), $F" |
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63 allL "$H, P(x), $G, ALL x.P(x) |- $E ==> $H, ALL x.P(x), $G |- $E" |
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64 |
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65 exR "$H |- $E, P(x), $F, EX x.P(x) ==> $H |- $E, EX x.P(x), $F" |
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66 exL "(!!x.$H, P(x), $G |- $E) ==> $H, EX x.P(x), $G |- $E" |
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67 |
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68 (*Equality*) |
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69 |
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70 refl "$H |- $E, a=a, $F" |
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71 sym "$H |- $E, a=b, $F ==> $H |- $E, b=a, $F" |
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72 trans "[| $H|- $E, a=b, $F; $H|- $E, b=c, $F |] ==> $H|- $E, a=c, $F" |
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73 |
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74 |
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75 (*Descriptions*) |
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76 |
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77 The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==> |
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78 $H |- $E, P(THE x.P(x)), $F" |
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79 end |
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80 |
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81 ML |
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82 |
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83 val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")]; |
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84 val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")]; |