(* Title: LK/LK0.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
There may be printing problems if a seqent is in expanded normal form
(eta-expanded, beta-contracted)
*)
header {* Classical First-Order Sequent Calculus *}
theory LK0
imports Sequents
begin
global
classes "term"
defaultsort "term"
consts
Trueprop :: "two_seqi"
True :: o
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)
"<->" :: "[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)
parse_translation {* [("@Trueprop", two_seq_tr "Trueprop")] *}
print_translation {* [("Trueprop", two_seq_tr' "@Trueprop")] *}
translations
"x ~= y" == "~ (x = y)"
syntax (xsymbols)
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 "\<longrightarrow>" 25)
"op <->" :: "[o, o] => o" (infixr "\<longleftrightarrow>" 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)
"_not_equal" :: "['a, 'a] => o" (infixl "\<noteq>" 50)
syntax (HTML output)
Not :: "o => o" ("\<not> _" [40] 40)
"op &" :: "[o, o] => o" (infixr "\<and>" 35)
"op |" :: "[o, o] => o" (infixr "\<or>" 30)
"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)
"_not_equal" :: "['a, 'a] => o" (infixl "\<noteq>" 50)
local
axioms
(*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
ML {* use_legacy_bindings (the_context ()) *}
end