src/LK/LK.thy

New unified treatment of sequent calculi by Sara Kalvala
combines the old LK and Modal with the new ILL (Int. Linear Logic)

(*  Title:      LK/lk.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

Classical First-Order Sequent Calculus

There may be printing problems if a seqent is in expanded normal form
	(eta-expanded, beta-contracted)
*)

LK = Pure +

classes term < logic

default term

types
  o sequence seqobj seqcont sobj

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

syntax
  "@Trueprop"  :: [sequence,sequence] => prop     ("((_)/ |- (_))" [6,6] 5)
  NullSeq      :: sequence                        ("" [] 1000)
  NonNullSeq   :: [seqobj,seqcont] => sequence    ("__" [] 1000)
  NullSeqCont  :: 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("NullSeq",_)) = 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("NullSeqCont",dummyT)
  |         _ => seq_tr'(sq,Const("SeqCont",dummyT))
      in C $ seq_obj_tr' obj $ sq' end;

fun seq_tr1' (Bound 0) = Const("NullSeq",dummyT)
  | seq_tr1' s = seq_tr'(s,Const("NonNullSeq",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')];