%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL002-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 05:00:30 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.46s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 22
% Syntax : Number of formulae : 52 ( 29 unt; 0 typ; 0 def)
% Number of atoms : 107 ( 64 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 94 ( 45 ~; 31 |; 0 &)
% ( 18 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 4 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of FOOLs : 6 ( 6 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 4 con; 0-2 aty)
% Number of variables : 117 ( 105 !; 0 ?; 117 :)
% Comments :
%------------------------------------------------------------------------------
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(fixed_pt_type,type,
fixed_pt: $i ).
tff(b_type,type,
b: $i ).
tff(c_type,type,
c: $i ).
tff(s_type,type,
s: $i ).
tff(fixed_point_type,type,
fixed_point: $i > $o ).
tff(1,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(2,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)],[1]) ).
tff(3,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(4,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(5,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,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(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))),apply(s,apply(c,apply(b,fixed_pt)))) = apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
apply(apply(apply(b,fixed_pt),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))),apply(s,apply(c,apply(b,fixed_pt)))) = apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) )
<=> ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) )
<=> ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) )
<=> ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',c_definition) ).
tff(14,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(c,X),Y),Z) = apply(apply(X,Z),Y) )
| ( apply(apply(apply(c,apply(b,fixed_pt)),apply(s,apply(c,apply(b,fixed_pt)))),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))) = apply(apply(apply(b,fixed_pt),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))),apply(s,apply(c,apply(b,fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
apply(apply(apply(c,apply(b,fixed_pt)),apply(s,apply(c,apply(b,fixed_pt)))),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))) = apply(apply(apply(b,fixed_pt),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))),apply(s,apply(c,apply(b,fixed_pt)))),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,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(23,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)],[22,21]) ).
tff(24,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)],[23]) ).
tff(25,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)],[24,20]) ).
tff(26,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(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) = apply(apply(apply(c,apply(b,fixed_pt)),apply(s,apply(c,apply(b,fixed_pt)))),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(27,plain,
apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) = apply(apply(apply(c,apply(b,fixed_pt)),apply(s,apply(c,apply(b,fixed_pt)))),apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[26,25]) ).
tff(28,plain,
apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) = apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))),
inference(transitivity,[status(thm)],[27,18,9]) ).
tff(29,plain,
^ [X: $i] :
refl(( ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X)))))
<=> ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))) )),
inference(bind,[status(th)],[]) ).
tff(30,plain,
( ! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X)))))
<=> ! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))) ),
inference(quant_intro,[status(thm)],[29]) ).
tff(31,plain,
( ! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X)))))
<=> ! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,axiom,
! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_weak_fixed_point) ).
tff(33,plain,
! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))),
inference(skolemize,[status(sab)],[33]) ).
tff(35,plain,
! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X))))),
inference(modus_ponens,[status(thm)],[34,30]) ).
tff(36,plain,
( ~ ! [X: $i] : ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,X))),apply(s,apply(c,apply(b,X)))),apply(s,apply(c,apply(b,X)))))
| ~ fixed_point(apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(37,plain,
~ fixed_point(apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[36,35]) ).
tff(38,plain,
^ [Weak_sage: $i] :
refl(( ( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
<=> ( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
<=> ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
<=> ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
^ [Weak_sage: $i] :
rewrite(( ( ( Weak_sage != apply(fixed_pt,Weak_sage) )
| fixed_point(Weak_sage) )
<=> ( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ) )),
inference(bind,[status(th)],[]) ).
tff(42,plain,
( ! [Weak_sage: $i] :
( ( Weak_sage != apply(fixed_pt,Weak_sage) )
| fixed_point(Weak_sage) )
<=> ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,axiom,
! [Weak_sage: $i] :
( ( Weak_sage != apply(fixed_pt,Weak_sage) )
| fixed_point(Weak_sage) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',weak_fixed_point) ).
tff(44,plain,
! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) ),
inference(modus_ponens,[status(thm)],[46,39]) ).
tff(48,plain,
( ( ~ ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
| fixed_point(apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))))
| ( apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) != apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))) ) )
<=> ( ~ ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
| fixed_point(apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))))
| ( apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) != apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,plain,
( ~ ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
| fixed_point(apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))))
| ( apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) != apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [Weak_sage: $i] :
( fixed_point(Weak_sage)
| ( Weak_sage != apply(fixed_pt,Weak_sage) ) )
| fixed_point(apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))))
| ( apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) != apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))) ) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt)))) != apply(fixed_pt,apply(apply(apply(s,apply(c,apply(b,fixed_pt))),apply(s,apply(c,apply(b,fixed_pt)))),apply(s,apply(c,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[50,47,37]) ).
tff(52,plain,
$false,
inference(unit_resolution,[status(thm)],[51,28]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : COL002-3 : TPTP v9.0.0. Released v1.0.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n004.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 12:00:38 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.46 % SZS status Unsatisfiable
% 0.19/0.46 % SZS output start Proof
% See solution above
% 0.19/0.47 % E exiting
%------------------------------------------------------------------------------