%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL002-4 : TPTP v9.0.0. Bugfixed v3.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:00:31 AM UTC 2025
% Result : Unsatisfiable 0.21s 0.37s
% Output : Proof 0.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 18
% Syntax : Number of formulae : 44 ( 31 unt; 0 typ; 0 def)
% Number of atoms : 65 ( 60 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 25 ( 8 ~; 4 |; 0 &)
% ( 13 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 4 ( 4 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 4 con; 0-2 aty)
% Number of variables : 80 ( 72 !; 0 ?; 80 :)
% Comments :
%------------------------------------------------------------------------------
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(weak_fixed_point_type,type,
weak_fixed_point: $i > $i ).
tff(fixed_pt_type,type,
fixed_pt: $i ).
tff(i_type,type,
i: $i ).
tff(b_type,type,
b: $i ).
tff(s_type,type,
s: $i ).
tff(1,plain,
^ [X: $i] :
refl(( ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) )
<=> ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) )
<=> ! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) )
<=> ! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',weak_fixed_point) ).
tff(5,plain,
! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( weak_fixed_point(X) = apply(apply(apply(s,apply(b,X)),i),apply(apply(s,apply(b,X)),i)) )
| ( weak_fixed_point(fixed_pt) = apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
weak_fixed_point(fixed_pt) = apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
apply(fixed_pt,weak_fixed_point(fixed_pt)) = apply(fixed_pt,apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i))),
inference(monotonicity,[status(thm)],[9]) ).
tff(11,plain,
apply(fixed_pt,apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i))) = apply(fixed_pt,weak_fixed_point(fixed_pt)),
inference(symmetry,[status(thm)],[10]) ).
tff(12,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
<=> ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(13,plain,
( ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ) ),
inference(quant_intro,[status(thm)],[12]) ).
tff(14,plain,
( ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,axiom,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_definition) ).
tff(16,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(skolemize,[status(sab)],[16]) ).
tff(18,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[17,13]) ).
tff(19,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
| ( apply(apply(apply(b,fixed_pt),apply(apply(s,apply(b,fixed_pt)),i)),apply(apply(s,apply(b,fixed_pt)),i)) = apply(fixed_pt,apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(20,plain,
apply(apply(apply(b,fixed_pt),apply(apply(s,apply(b,fixed_pt)),i)),apply(apply(s,apply(b,fixed_pt)),i)) = apply(fixed_pt,apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i))),
inference(unit_resolution,[status(thm)],[19,18]) ).
tff(21,plain,
^ [X: $i] :
refl(( ( apply(i,X) = X )
<=> ( apply(i,X) = X ) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [X: $i] : ( apply(i,X) = X )
<=> ! [X: $i] : ( apply(i,X) = X ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,plain,
( ! [X: $i] : ( apply(i,X) = X )
<=> ! [X: $i] : ( apply(i,X) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,axiom,
! [X: $i] : ( apply(i,X) = X ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',i_definition) ).
tff(25,plain,
! [X: $i] : ( apply(i,X) = X ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [X: $i] : ( apply(i,X) = X ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [X: $i] : ( apply(i,X) = X ),
inference(modus_ponens,[status(thm)],[26,22]) ).
tff(28,plain,
( ~ ! [X: $i] : ( apply(i,X) = X )
| ( apply(i,apply(apply(s,apply(b,fixed_pt)),i)) = apply(apply(s,apply(b,fixed_pt)),i) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(29,plain,
apply(i,apply(apply(s,apply(b,fixed_pt)),i)) = apply(apply(s,apply(b,fixed_pt)),i),
inference(unit_resolution,[status(thm)],[28,27]) ).
tff(30,plain,
apply(apply(apply(b,fixed_pt),apply(apply(s,apply(b,fixed_pt)),i)),apply(i,apply(apply(s,apply(b,fixed_pt)),i))) = apply(apply(apply(b,fixed_pt),apply(apply(s,apply(b,fixed_pt)),i)),apply(apply(s,apply(b,fixed_pt)),i)),
inference(monotonicity,[status(thm)],[29]) ).
tff(31,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) )
<=> ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(32,plain,
( ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ) ),
inference(quant_intro,[status(thm)],[31]) ).
tff(33,plain,
( ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,axiom,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',s_definition) ).
tff(35,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ),
inference(skolemize,[status(sab)],[35]) ).
tff(37,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) )
| ( apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i)) = apply(apply(apply(b,fixed_pt),apply(apply(s,apply(b,fixed_pt)),i)),apply(i,apply(apply(s,apply(b,fixed_pt)),i))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
apply(apply(apply(s,apply(b,fixed_pt)),i),apply(apply(s,apply(b,fixed_pt)),i)) = apply(apply(apply(b,fixed_pt),apply(apply(s,apply(b,fixed_pt)),i)),apply(i,apply(apply(s,apply(b,fixed_pt)),i))),
inference(unit_resolution,[status(thm)],[38,37]) ).
tff(40,plain,
weak_fixed_point(fixed_pt) = apply(fixed_pt,weak_fixed_point(fixed_pt)),
inference(transitivity,[status(thm)],[9,39,30,20,11]) ).
tff(41,plain,
( ( weak_fixed_point(fixed_pt) != apply(fixed_pt,weak_fixed_point(fixed_pt)) )
<=> ( weak_fixed_point(fixed_pt) != apply(fixed_pt,weak_fixed_point(fixed_pt)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
weak_fixed_point(fixed_pt) != apply(fixed_pt,weak_fixed_point(fixed_pt)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_weak_fixed_point) ).
tff(43,plain,
weak_fixed_point(fixed_pt) != apply(fixed_pt,weak_fixed_point(fixed_pt)),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
$false,
inference(unit_resolution,[status(thm)],[43,40]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : COL002-4 : TPTP v9.0.0. Bugfixed v3.1.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 12:02:31 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.21/0.37 % SZS status Unsatisfiable
% 0.21/0.37 % SZS output start Proof
% See solution above
% 0.21/0.38 % E exiting
%------------------------------------------------------------------------------