TPTP Problem File: SEU695^2.p
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% File : SEU695^2 : TPTP v9.0.0. Released v3.7.0.
% Domain : Set Theory
% Problem : Functions - Extensionality and Beta Reduction
% Version : Especial > Reduced > Especial.
% English : (! A:i.! B:i.! f:i>i.(! x:i.in x A -> in (f x) B) ->
% (! x:i.in x A -> ap A B (lam A B (^ y:i.f y)) x = f x))
% Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source : [Bro08]
% Names : ZFC197l [Bro08]
% : ZFC200l [Bro08]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.45 v7.5.0, 0.29 v7.4.0, 0.22 v7.2.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.43 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 0.67 v3.7.0
% Syntax : Number of formulae : 14 ( 4 unt; 9 typ; 4 def)
% Number of atoms : 26 ( 7 equ; 0 cnn)
% Maximal formula atoms : 7 ( 5 avg)
% Number of connectives : 77 ( 0 ~; 0 |; 0 &; 62 @)
% ( 0 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% Number of symbols : 10 ( 9 usr; 3 con; 0-4 aty)
% Number of variables : 28 ( 9 ^; 19 !; 0 ?; 28 :)
% SPC : TH0_THM_EQU_NAR
% Comments : http://mathgate.info/detsetitem.php?id=248
% : http://mathgate.info/detsetitem.php?id=250
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thf(in_type,type,
in: $i > $i > $o ).
thf(kpair_type,type,
kpair: $i > $i > $i ).
thf(dpsetconstr_type,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(dpsetconstrI_type,type,
dpsetconstrI: $o ).
thf(dpsetconstrI,definition,
( dpsetconstrI
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xphi @ Xx @ Xy )
=> ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
thf(func_type,type,
func: $i > $i > $i > $o ).
thf(ap_type,type,
ap: $i > $i > $i > $i > $i ).
thf(lam_type,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(lam,definition,
( lam
= ( ^ [A: $i,B: $i,Xf: $i > $i] :
( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] :
( ( Xf @ Xx )
= Xy ) ) ) ) ).
thf(lamp_type,type,
lamp: $o ).
thf(lamp,definition,
( lamp
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( func @ A @ B
@ ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
thf(funcGraphProp2_type,type,
funcGraphProp2: $o ).
thf(funcGraphProp2,definition,
( funcGraphProp2
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
=> ( ( ap @ A @ B @ Xf @ Xx )
= Xy ) ) ) ) ) ) ) ).
thf(beta1,conjecture,
( dpsetconstrI
=> ( lamp
=> ( funcGraphProp2
=> ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B
@ ( lam @ A @ B
@ ^ [Xy: $i] : ( Xf @ Xy ) )
@ Xx )
= ( Xf @ Xx ) ) ) ) ) ) ) ).
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