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(* Title: LK/lk.thy


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ID: $Id$


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Author: Lawrence C Paulson, Cambridge University Computer Laboratory


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Copyright 1993 University of Cambridge


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Classical FirstOrder Sequent Calculus


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*)


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LK = Pure +


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classes term < logic


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default term


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types o 0


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sequence,seqobj,seqcont,sequ,sobj 0


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arities o :: logic


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consts


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True,False :: "o"


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"=" :: "['a,'a] => o" (infixl 50)


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"Not" :: "o => o" ("~ _" [40] 40)


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"&" :: "[o,o] => o" (infixr 35)


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"" :: "[o,o] => o" (infixr 30)


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">","<>" :: "[o,o] => o" (infixr 25)


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The :: "('a => o) => 'a" (binder "THE " 10)


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All :: "('a => o) => o" (binder "ALL " 10)


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Ex :: "('a => o) => o" (binder "EX " 10)


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(*Representation of sequents*)


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Trueprop :: "[sobj=>sobj,sobj=>sobj] => prop"


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Seqof :: "o => sobj=>sobj"


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"@Trueprop" :: "[sequence,sequence] => prop" ("((_)/  (_))" [6,6] 5)


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"@MtSeq" :: "sequence" ("" [] 1000)


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"@NmtSeq" :: "[seqobj,seqcont] => sequence" ("__" [] 1000)


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"@MtSeqCont" :: "seqcont" ("" [] 1000)


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"@SeqCont" :: "[seqobj,seqcont] => seqcont" (",/ __" [] 1000)


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"" :: "o => seqobj" ("_" [] 1000)


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"@SeqId" :: "id => seqobj" ("$_" [] 1000)


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"@SeqVar" :: "var => seqobj" ("$_")


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rules


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(*Structural rules*)


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basic "$H, P, $G  $E, P, $F"


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thinR "$H  $E, $F ==> $H  $E, P, $F"


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thinL "$H, $G  $E ==> $H, P, $G  $E"


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cut "[ $H  $E, P; $H, P  $E ] ==> $H  $E"


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(*Propositional rules*)


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conjR "[ $H $E, P, $F; $H $E, Q, $F ] ==> $H $E, P&Q, $F"


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conjL "$H, P, Q, $G  $E ==> $H, P & Q, $G  $E"


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disjR "$H  $E, P, Q, $F ==> $H  $E, PQ, $F"


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disjL "[ $H, P, $G  $E; $H, Q, $G  $E ] ==> $H, PQ, $G  $E"


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impR "$H, P  $E, Q, $F ==> $H  $E, P>Q, $F"


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impL "[ $H,$G  $E,P; $H, Q, $G  $E ] ==> $H, P>Q, $G  $E"


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notR "$H, P  $E, $F ==> $H  $E, ~P, $F"


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notL "$H, $G  $E, P ==> $H, ~P, $G  $E"


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FalseL "$H, False, $G  $E"


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True_def "True == False>False"


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iff_def "P<>Q == (P>Q) & (Q>P)"


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(*Quantifiers*)


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allR "(!!x.$H  $E, P(x), $F) ==> $H  $E, ALL x.P(x), $F"


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allL "$H, P(x), $G, ALL x.P(x)  $E ==> $H, ALL x.P(x), $G  $E"


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exR "$H  $E, P(x), $F, EX x.P(x) ==> $H  $E, EX x.P(x), $F"


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exL "(!!x.$H, P(x), $G  $E) ==> $H, EX x.P(x), $G  $E"


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(*Equality*)


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refl "$H  $E, a=a, $F"


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sym "$H  $E, a=b, $F ==> $H  $E, b=a, $F"


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trans "[ $H $E, a=b, $F; $H $E, b=c, $F ] ==> $H $E, a=c, $F"


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(*Descriptions*)


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The "[ $H  $E, P(a), $F; !!x.$H, P(x)  $E, x=a, $F ] ==> \


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\ $H  $E, P(THE x.P(x)), $F"


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end


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ML


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(*Abstract over "sobj"  representation of a sequence of formulae *)


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fun abs_sobj t = Abs("sobj", Type("sobj",[]), t);


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(*Representation of empty sequence*)


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val Sempty = abs_sobj (Bound 0);


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fun seq_obj_tr(Const("@SeqId",_)$id) = id 


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seq_obj_tr(Const("@SeqVar",_)$id) = id 


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seq_obj_tr(fm) = Const("Seqof",dummyT)$fm;


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fun seq_tr(_$obj$seq) = seq_obj_tr(obj)$seq_tr(seq) 


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seq_tr(_) = Bound 0;


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fun seq_tr1(Const("@MtSeq",_)) = Sempty 


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seq_tr1(seq) = abs_sobj(seq_tr seq);


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fun true_tr[s1,s2] = Const("Trueprop",dummyT)$seq_tr1 s1$seq_tr1 s2;


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fun seq_obj_tr'(Const("Seqof",_)$fm) = fm 


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seq_obj_tr'(id) = Const("@SeqId",dummyT)$id;


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fun seq_tr'(obj$sq,C) =


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let val sq' = case sq of


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Bound 0 => Const("@MtSeqCont",dummyT) 


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_ => seq_tr'(sq,Const("@SeqCont",dummyT))


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in C $ seq_obj_tr' obj $ sq' end;


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fun seq_tr1'(Bound 0) = Const("@MtSeq",dummyT) 


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seq_tr1' s = seq_tr'(s,Const("@NmtSeq",dummyT));


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fun true_tr'[Abs(_,_,s1),Abs(_,_,s2)] =


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Const("@Trueprop",dummyT)$seq_tr1' s1$seq_tr1' s2;


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val parse_translation = [("@Trueprop",true_tr)];


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val print_translation = [("Trueprop",true_tr')];
