--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/LK/LK.thy Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,124 @@
+(* Title: LK/lk.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+Classical First-Order Sequent Calculus
+*)
+
+LK = Pure +
+classes term < logic
+default term
+types o 0
+ sequence,seqobj,seqcont,sequ,sobj 0
+arities o :: logic
+consts
+ True,False :: "o"
+ "=" :: "['a,'a] => o" (infixl 50)
+ "Not" :: "o => o" ("~ _" [40] 40)
+ "&" :: "[o,o] => o" (infixr 35)
+ "|" :: "[o,o] => o" (infixr 30)
+ "-->","<->" :: "[o,o] => o" (infixr 25)
+ The :: "('a => o) => 'a" (binder "THE " 10)
+ All :: "('a => o) => o" (binder "ALL " 10)
+ Ex :: "('a => o) => o" (binder "EX " 10)
+
+ (*Representation of sequents*)
+ Trueprop :: "[sobj=>sobj,sobj=>sobj] => prop"
+ Seqof :: "o => sobj=>sobj"
+ "@Trueprop" :: "[sequence,sequence] => prop" ("((_)/ |- (_))" [6,6] 5)
+ "@MtSeq" :: "sequence" ("" [] 1000)
+ "@NmtSeq" :: "[seqobj,seqcont] => sequence" ("__" [] 1000)
+ "@MtSeqCont" :: "seqcont" ("" [] 1000)
+ "@SeqCont" :: "[seqobj,seqcont] => seqcont" (",/ __" [] 1000)
+ "" :: "o => seqobj" ("_" [] 1000)
+ "@SeqId" :: "id => seqobj" ("$_" [] 1000)
+ "@SeqVar" :: "var => seqobj" ("$_")
+
+rules
+ (*Structural rules*)
+
+ basic "$H, P, $G |- $E, P, $F"
+
+ thinR "$H |- $E, $F ==> $H |- $E, P, $F"
+ thinL "$H, $G |- $E ==> $H, P, $G |- $E"
+
+ cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E"
+
+ (*Propositional rules*)
+
+ conjR "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F"
+ conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E"
+
+ disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F"
+ disjL "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E"
+
+ impR "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F"
+ impL "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E"
+
+ notR "$H, P |- $E, $F ==> $H |- $E, ~P, $F"
+ notL "$H, $G |- $E, P ==> $H, ~P, $G |- $E"
+
+ FalseL "$H, False, $G |- $E"
+
+ True_def "True == False-->False"
+ iff_def "P<->Q == (P-->Q) & (Q-->P)"
+
+ (*Quantifiers*)
+
+ allR "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x.P(x), $F"
+ allL "$H, P(x), $G, ALL x.P(x) |- $E ==> $H, ALL x.P(x), $G |- $E"
+
+ exR "$H |- $E, P(x), $F, EX x.P(x) ==> $H |- $E, EX x.P(x), $F"
+ exL "(!!x.$H, P(x), $G |- $E) ==> $H, EX x.P(x), $G |- $E"
+
+ (*Equality*)
+
+ refl "$H |- $E, a=a, $F"
+ sym "$H |- $E, a=b, $F ==> $H |- $E, b=a, $F"
+ trans "[| $H|- $E, a=b, $F; $H|- $E, b=c, $F |] ==> $H|- $E, a=c, $F"
+
+
+ (*Descriptions*)
+
+ The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==> \
+\ $H |- $E, P(THE x.P(x)), $F"
+end
+
+ML
+
+(*Abstract over "sobj" -- representation of a sequence of formulae *)
+fun abs_sobj t = Abs("sobj", Type("sobj",[]), t);
+
+(*Representation of empty sequence*)
+val Sempty = abs_sobj (Bound 0);
+
+fun seq_obj_tr(Const("@SeqId",_)$id) = id |
+ seq_obj_tr(Const("@SeqVar",_)$id) = id |
+ seq_obj_tr(fm) = Const("Seqof",dummyT)$fm;
+
+fun seq_tr(_$obj$seq) = seq_obj_tr(obj)$seq_tr(seq) |
+ seq_tr(_) = Bound 0;
+
+fun seq_tr1(Const("@MtSeq",_)) = Sempty |
+ seq_tr1(seq) = abs_sobj(seq_tr seq);
+
+fun true_tr[s1,s2] = Const("Trueprop",dummyT)$seq_tr1 s1$seq_tr1 s2;
+
+fun seq_obj_tr'(Const("Seqof",_)$fm) = fm |
+ seq_obj_tr'(id) = Const("@SeqId",dummyT)$id;
+
+fun seq_tr'(obj$sq,C) =
+ let val sq' = case sq of
+ Bound 0 => Const("@MtSeqCont",dummyT) |
+ _ => seq_tr'(sq,Const("@SeqCont",dummyT))
+ in C $ seq_obj_tr' obj $ sq' end;
+
+fun seq_tr1'(Bound 0) = Const("@MtSeq",dummyT) |
+ seq_tr1' s = seq_tr'(s,Const("@NmtSeq",dummyT));
+
+fun true_tr'[Abs(_,_,s1),Abs(_,_,s2)] =
+ Const("@Trueprop",dummyT)$seq_tr1' s1$seq_tr1' s2;
+
+val parse_translation = [("@Trueprop",true_tr)];
+val print_translation = [("Trueprop",true_tr')];