%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL003-10 : TPTP v9.0.0. Released v7.3.0.
% 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 05:00:31 AM UTC 2025
% Result : Unsatisfiable 2.71s 2.89s
% Output : Proof 2.71s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 26
% Syntax : Number of formulae : 82 ( 61 unt; 0 typ; 0 def)
% Number of atoms : 111 ( 106 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 41 ( 16 ~; 12 |; 0 &)
% ( 13 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 11 ( 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 : 7 ( 7 usr; 4 con; 0-4 aty)
% Number of variables : 112 ( 103 !; 0 ?; 112 :)
% Comments :
%------------------------------------------------------------------------------
tff(true_type,type,
true: $i ).
tff(fixed_point_type,type,
fixed_point: $i > $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(w_type,type,
w: $i ).
tff(ifeq_type,type,
ifeq: ( $i * $i * $i * $i ) > $i ).
tff(fixed_pt_type,type,
fixed_pt: $i ).
tff(1,plain,
^ [Strong_fixed_point: $i] :
refl(( ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true )
<=> ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true )
<=> ! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true )
<=> ! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',strong_fixed_point) ).
tff(5,plain,
! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Strong_fixed_point: $i] : ( ifeq(apply(Strong_fixed_point,fixed_pt),apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)),fixed_point(Strong_fixed_point),true) = true )
| ( ifeq(apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
ifeq(apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true) = true,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,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(11,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)],[10]) ).
tff(12,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(13,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(14,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)) ),
inference(skolemize,[status(sab)],[14]) ).
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)],[15,11]) ).
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,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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(quant_inst,[status(thm)],[]) ).
tff(18,plain,
apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt),
inference(symmetry,[status(thm)],[18]) ).
tff(20,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(21,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)],[20,16]) ).
tff(22,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)],[21]) ).
tff(23,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(24,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)],[23,16]) ).
tff(25,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)],[24]) ).
tff(26,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(27,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)],[26]) ).
tff(28,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(29,axiom,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',w_definition) ).
tff(30,plain,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
inference(skolemize,[status(sab)],[30]) ).
tff(32,plain,
! [Y: $i,X: $i] : ( apply(apply(w,X),Y) = apply(apply(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[31,27]) ).
tff(33,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(34,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)],[33,32]) ).
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,w),apply(apply(apply(b,apply(b,w)),b),apply(b,fixed_pt))),
inference(symmetry,[status(thm)],[34]) ).
tff(36,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(37,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)],[36,16]) ).
tff(38,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)],[37]) ).
tff(39,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)],[38]) ).
tff(40,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)],[39]) ).
tff(41,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)],[40,37]) ).
tff(42,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)],[41]) ).
tff(43,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(44,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)],[43,32]) ).
tff(45,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)],[44]) ).
tff(46,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)],[45,42,35,25,22]) ).
tff(47,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)],[46]) ).
tff(48,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(49,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)],[48,16]) ).
tff(50,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(51,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)],[50,16]) ).
tff(52,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)],[51]) ).
tff(53,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(54,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)],[53,32]) ).
tff(55,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(56,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)],[55,16]) ).
tff(57,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)],[56]) ).
tff(58,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,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt),
inference(transitivity,[status(thm)],[21,24,34,41,44,57,54,52,49,47,19]) ).
tff(59,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(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),
inference(monotonicity,[status(thm)],[58]) ).
tff(60,plain,
apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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(symmetry,[status(thm)],[59]) ).
tff(61,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,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(symmetry,[status(thm)],[57]) ).
tff(62,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(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(symmetry,[status(thm)],[54]) ).
tff(63,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(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(symmetry,[status(thm)],[52]) ).
tff(64,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(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(symmetry,[status(thm)],[49]) ).
tff(65,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(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(symmetry,[status(thm)],[47]) ).
tff(66,plain,
apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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)],[60,65,64,63,62,61,45,42,35,25,22]) ).
tff(67,plain,
ifeq(apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true) = ifeq(apply(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),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)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true),
inference(monotonicity,[status(thm)],[66,60]) ).
tff(68,plain,
^ [B: $i,A: $i,C: $i] :
refl(( ( ifeq(A,A,B,C) = B )
<=> ( ifeq(A,A,B,C) = B ) )),
inference(bind,[status(th)],[]) ).
tff(69,plain,
( ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
<=> ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ) ),
inference(quant_intro,[status(thm)],[68]) ).
tff(70,plain,
( ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
<=> ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ) ),
inference(rewrite,[status(thm)],[]) ).
tff(71,axiom,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ifeq_axiom) ).
tff(72,plain,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
inference(modus_ponens,[status(thm)],[71,70]) ).
tff(73,plain,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
inference(skolemize,[status(sab)],[72]) ).
tff(74,plain,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
inference(modus_ponens,[status(thm)],[73,69]) ).
tff(75,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true) = fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(76,plain,
ifeq(apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true) = fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),
inference(unit_resolution,[status(thm)],[75,74]) ).
tff(77,plain,
fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) = ifeq(apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),apply(fixed_pt,apply(apply(apply(b,fixed_pt),apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),fixed_pt)),fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)),true),
inference(symmetry,[status(thm)],[76]) ).
tff(78,plain,
fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) = true,
inference(transitivity,[status(thm)],[77,67,9]) ).
tff(79,plain,
( ( fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) != true )
<=> ( fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) != true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,axiom,
fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) != true,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_strong_fixed_point) ).
tff(81,plain,
fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,apply(b,w)),b))),b)) != true,
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
$false,
inference(unit_resolution,[status(thm)],[81,78]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : COL003-10 : TPTP v9.0.0. Released v7.3.0.
% 0.03/0.12 % 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 12:00:23 EDT 2025
% 0.11/0.33 % CPUTime :
% 2.71/2.89 % SZS status Unsatisfiable
% 2.71/2.89 % SZS output start Proof
% See solution above
% 2.71/2.91 % E exiting
%------------------------------------------------------------------------------