(* 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 = Sequents +
consts
Trueprop :: "two_seqi"
"@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
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)
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
val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];