(* Title: LK/LK0
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)
*)
LK0 = Sequents +
global
classes
term < logic
default
term
consts
Trueprop :: "two_seqi"
"@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
(*Constant to allow definitions of SEQUENCES of formulas*)
"@Side" :: "seq=>(seq'=>seq')" ("<<(_)>>")
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)
syntax
"~=" :: ['a, 'a] => o (infixl 50)
translations
"x ~= y" == "~ (x = y)"
syntax (symbols)
Not :: o => o ("\\<not> _" [40] 40)
"op &" :: [o, o] => o (infixr "\\<and>" 35)
"op |" :: [o, o] => o (infixr "\\<or>" 30)
"op -->" :: [o, o] => o (infixr "\\<midarrow>\\<rightarrow>" 25)
"op <->" :: [o, o] => o (infixr "\\<leftarrow>\\<rightarrow>" 25)
"ALL " :: [idts, o] => o ("(3\\<forall>_./ _)" [0, 10] 10)
"EX " :: [idts, o] => o ("(3\\<exists>_./ _)" [0, 10] 10)
"EX! " :: [idts, o] => o ("(3\\<exists>!_./ _)" [0, 10] 10)
"op ~=" :: ['a, 'a] => o (infixl "\\<noteq>" 50)
syntax (xsymbols)
"op -->" :: [o, o] => o (infixr "\\<longrightarrow>" 25)
"op <->" :: [o, o] => o (infixr "\\<longleftrightarrow>" 25)
syntax (HTML output)
Not :: o => o ("\\<not> _" [40] 40)
local
rules
(*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)
contRS "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F"
contLS "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E"
thinRS "$H |- $E, $F ==> $H |- $E, $S, $F"
thinLS "$H, $G |- $E ==> $H, $S, $G |- $E"
exchRS "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F"
exchLS "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E"
cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E"
(*Propositional rules*)
basic "$H, P, $G |- $E, P, $F"
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, $F"
subst "$H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)"
(* Reflection *)
eq_reflection "|- x=y ==> (x==y)"
iff_reflection "|- P<->Q ==> (P==Q)"
(*Descriptions*)
The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==>
$H |- $E, P(THE x. P(x)), $F"
constdefs
If :: [o, 'a, 'a] => 'a ("(if (_)/ then (_)/ else (_))" 10)
"If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"
setup
Simplifier.setup
setup
prover_setup
end
ML
val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];