%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL003-19 : TPTP v9.0.0. Released v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 05:00:32 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 19
% Syntax : Number of formulae : 56 ( 41 unt; 0 typ; 0 def)
% Number of atoms : 75 ( 72 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 30 ( 13 ~; 9 |; 0 &)
% ( 8 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of FOOLs : 2 ( 2 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 : 5 ( 5 usr; 4 con; 0-2 aty)
% Number of variables : 69 ( 64 !; 0 ?; 69 :)
% Comments :
%------------------------------------------------------------------------------
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(fixed_pt_type,type,
fixed_pt: $i ).
tff(strong_fixed_point_type,type,
strong_fixed_point: $i ).
tff(b_type,type,
b: $i ).
tff(w_type,type,
w: $i ).
tff(1,plain,
( ( strong_fixed_point = apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b) )
<=> ( strong_fixed_point = apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
strong_fixed_point = apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strong_fixed_point) ).
tff(3,plain,
strong_fixed_point = apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
apply(strong_fixed_point,fixed_pt) = apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt),
inference(monotonicity,[status(thm)],[3]) ).
tff(5,plain,
apply(fixed_pt,apply(strong_fixed_point,fixed_pt)) = apply(fixed_pt,apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt)),
inference(monotonicity,[status(thm)],[4]) ).
tff(6,plain,
apply(fixed_pt,apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt)) = apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
inference(symmetry,[status(thm)],[5]) ).
tff(7,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(8,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)],[7]) ).
tff(9,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(10,axiom,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_definition) ).
tff(11,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[10,9]) ).
tff(12,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[12,8]) ).
tff(14,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
| ( apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt) = apply(apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b)),apply(b,fixed_pt)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(15,plain,
apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt) = apply(apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b)),apply(b,fixed_pt)),
inference(unit_resolution,[status(thm)],[14,13]) ).
tff(16,plain,
apply(apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b)),apply(b,fixed_pt)) = apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt),
inference(symmetry,[status(thm)],[15]) ).
tff(17,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
| ( apply(apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b)),apply(b,fixed_pt)) = apply(apply(w,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
apply(apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b)),apply(b,fixed_pt)) = apply(apply(w,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),
inference(unit_resolution,[status(thm)],[17,13]) ).
tff(19,plain,
apply(apply(w,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) = apply(apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b)),apply(b,fixed_pt)),
inference(symmetry,[status(thm)],[18]) ).
tff(20,plain,
^ [Y: $i,X: $i] :
refl(( ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) )
<=> ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(21,plain,
( ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) )
<=> ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ) ),
inference(quant_intro,[status(thm)],[20]) ).
tff(22,plain,
( ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) )
<=> ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,axiom,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',w_definition) ).
tff(24,plain,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
inference(skolemize,[status(sab)],[24]) ).
tff(26,plain,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) )
| ( apply(apply(w,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) = apply(apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
apply(apply(w,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) = apply(apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),
inference(unit_resolution,[status(thm)],[27,26]) ).
tff(29,plain,
apply(apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) = apply(apply(w,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),
inference(symmetry,[status(thm)],[28]) ).
tff(30,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
| ( apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt)) = apply(apply(b,w),apply(b,apply(b,fixed_pt))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(31,plain,
apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt)) = apply(apply(b,w),apply(b,apply(b,fixed_pt))),
inference(unit_resolution,[status(thm)],[30,13]) ).
tff(32,plain,
apply(apply(b,w),apply(b,apply(b,fixed_pt))) = apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt)),
inference(symmetry,[status(thm)],[31]) ).
tff(33,plain,
apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),
inference(monotonicity,[status(thm)],[32]) ).
tff(34,plain,
apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) = apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(symmetry,[status(thm)],[33]) ).
tff(35,plain,
apply(apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))) = apply(apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(monotonicity,[status(thm)],[34,31]) ).
tff(36,plain,
apply(apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(w,apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),
inference(symmetry,[status(thm)],[35]) ).
tff(37,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) )
| ( apply(apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,plain,
apply(apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(unit_resolution,[status(thm)],[37,26]) ).
tff(39,plain,
apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(w,apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(symmetry,[status(thm)],[38]) ).
tff(40,plain,
apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt),
inference(transitivity,[status(thm)],[39,36,29,19,16]) ).
tff(41,plain,
apply(fixed_pt,apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))) = apply(fixed_pt,apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b),fixed_pt)),
inference(monotonicity,[status(thm)],[40]) ).
tff(42,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(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(fixed_pt,apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(43,plain,
apply(apply(apply(b,fixed_pt),apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(fixed_pt,apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[42,13]) ).
tff(44,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
| ( apply(apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(b,fixed_pt),apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(45,plain,
apply(apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(b,fixed_pt),apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[44,13]) ).
tff(46,plain,
apply(apply(apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(apply(b,fixed_pt),apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(monotonicity,[status(thm)],[45]) ).
tff(47,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) )
| ( apply(apply(w,apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(48,plain,
apply(apply(w,apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(unit_resolution,[status(thm)],[47,26]) ).
tff(49,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) )
| ( apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(w,apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(w,apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),
inference(unit_resolution,[status(thm)],[49,13]) ).
tff(51,plain,
apply(apply(apply(apply(b,w),apply(b,apply(b,fixed_pt))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))) = apply(apply(w,apply(apply(b,apply(b,fixed_pt)),apply(apply(b,w),apply(b,apply(b,fixed_pt))))),apply(apply(b,w),apply(b,apply(b,fixed_pt)))),
inference(monotonicity,[status(thm)],[50]) ).
tff(52,plain,
apply(strong_fixed_point,fixed_pt) = apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
inference(transitivity,[status(thm)],[4,15,18,28,35,38,51,48,46,43,41,6]) ).
tff(53,plain,
( ( apply(strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(strong_fixed_point,fixed_pt)) )
<=> ( apply(strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(strong_fixed_point,fixed_pt)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(54,axiom,
apply(strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_strong_fixed_point) ).
tff(55,plain,
apply(strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
$false,
inference(unit_resolution,[status(thm)],[55,52]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.12 % Problem : COL003-19 : TPTP v9.0.0. Released v2.1.0.
% 0.08/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n008.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 12:01:48 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.43 % SZS status Unsatisfiable
% 0.19/0.43 % SZS output start Proof
% See solution above
% 0.19/0.45 % E exiting
%------------------------------------------------------------------------------