(* 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 default term consts Trueprop :: "two_seqi" 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 "@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5) "_not_equal" :: ['a, 'a] => o (infixl "~=" 50) translations "x ~= y" == "~ (x = y)" syntax (xsymbols) Not :: o => o ("\\ _" [40] 40) "op &" :: [o, o] => o (infixr "\\" 35) "op |" :: [o, o] => o (infixr "\\" 30) "op -->" :: [o, o] => o (infixr "\\" 25) "op <->" :: [o, o] => o (infixr "\\" 25) "ALL " :: [idts, o] => o ("(3\\_./ _)" [0, 10] 10) "EX " :: [idts, o] => o ("(3\\_./ _)" [0, 10] 10) "EX! " :: [idts, o] => o ("(3\\!_./ _)" [0, 10] 10) "_not_equal" :: ['a, 'a] => o (infixl "\\" 50) syntax (HTML output) Not :: o => o ("\\ _" [40] 40) "op &" :: [o, o] => o (infixr "\\" 35) "op |" :: [o, o] => o (infixr "\\" 30) "ALL " :: [idts, o] => o ("(3\\_./ _)" [0, 10] 10) "EX " :: [idts, o] => o ("(3\\_./ _)" [0, 10] 10) "EX! " :: [idts, o] => o ("(3\\!_./ _)" [0, 10] 10) "_not_equal" :: ['a, 'a] => o (infixl "\\" 50) 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 prover_setup end ML val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")]; val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];