src/Sequents/LK0.thy
 author wenzelm Sun, 22 May 2005 16:51:07 +0200 changeset 16019 0e1405402d53 parent 14854 61bdf2ae4dc5 child 17481 75166ebb619b permissions -rw-r--r--
```
(*  Title:      LK/LK0
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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               ("\\<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

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")];

```