%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL006-5 : TPTP v9.0.0. Released v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:00:34 AM UTC 2025
% Result : Unsatisfiable 0.14s 0.39s
% Output : Proof 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 25
% Syntax : Number of formulae : 82 ( 61 unt; 0 typ; 0 def)
% Number of atoms : 107 ( 104 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 42 ( 19 ~; 15 |; 0 &)
% ( 8 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 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 : 84 ( 79 !; 0 ?; 84 :)
% 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(k_type,type,
k: $i ).
tff(s_type,type,
s: $i ).
tff(1,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(2,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)],[1]) ).
tff(3,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(4,axiom,
! [Z: $i,Y: $i,X: $i] : ( apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',s_definition) ).
tff(5,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)],[4,3]) ).
tff(6,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)],[5]) ).
tff(7,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)],[6,2]) ).
tff(8,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,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(symmetry,[status(thm)],[9]) ).
tff(11,plain,
^ [Y: $i,X: $i] :
refl(( ( apply(apply(k,X),Y) = X )
<=> ( apply(apply(k,X),Y) = X ) )),
inference(bind,[status(th)],[]) ).
tff(12,plain,
( ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
<=> ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X ) ),
inference(quant_intro,[status(thm)],[11]) ).
tff(13,plain,
( ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
<=> ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(14,axiom,
! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',k_definition) ).
tff(15,plain,
! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X ),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X ),
inference(skolemize,[status(sab)],[15]) ).
tff(17,plain,
! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X ),
inference(modus_ponens,[status(thm)],[16,12]) ).
tff(18,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),
inference(unit_resolution,[status(thm)],[18,17]) ).
tff(20,plain,
apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)) = apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(symmetry,[status(thm)],[19]) ).
tff(21,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(22,plain,
apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),
inference(unit_resolution,[status(thm)],[21,17]) ).
tff(23,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,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(unit_resolution,[status(thm)],[23,7]) ).
tff(25,plain,
apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(transitivity,[status(thm)],[24,22,20,10]) ).
tff(26,plain,
apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)) = apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(transitivity,[status(thm)],[20,10]) ).
tff(27,plain,
apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(monotonicity,[status(thm)],[26,25]) ).
tff(28,plain,
apply(apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(symmetry,[status(thm)],[27]) ).
tff(29,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(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(30,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(unit_resolution,[status(thm)],[29,7]) ).
tff(31,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(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(32,plain,
apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),
inference(unit_resolution,[status(thm)],[31,7]) ).
tff(33,plain,
apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)) = apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt),
inference(symmetry,[status(thm)],[32]) ).
tff(34,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt) = apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(35,plain,
apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt) = apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),
inference(unit_resolution,[status(thm)],[34,17]) ).
tff(36,plain,
apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)) = apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt),
inference(symmetry,[status(thm)],[35]) ).
tff(37,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt),apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt)),
inference(monotonicity,[status(thm)],[36,33]) ).
tff(38,plain,
apply(apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt),apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt)) = apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(symmetry,[status(thm)],[37]) ).
tff(39,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(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt) = apply(apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt),apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(40,plain,
apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt) = apply(apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt),apply(apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k)),fixed_pt)),
inference(unit_resolution,[status(thm)],[39,7]) ).
tff(41,plain,
( ( strong_fixed_point = apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))) )
<=> ( strong_fixed_point = apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
strong_fixed_point = apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strong_fixed_point) ).
tff(43,plain,
strong_fixed_point = apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
apply(strong_fixed_point,fixed_pt) = apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(k,apply(apply(s,s),apply(s,k)))),apply(apply(s,apply(k,s)),k))),fixed_pt),
inference(monotonicity,[status(thm)],[43]) ).
tff(45,plain,
apply(strong_fixed_point,fixed_pt) = apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(transitivity,[status(thm)],[44,40,38,30,28]) ).
tff(46,plain,
apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(k,fixed_pt),apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))),
inference(monotonicity,[status(thm)],[24]) ).
tff(47,plain,
apply(apply(k,fixed_pt),apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))) = apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(symmetry,[status(thm)],[46]) ).
tff(48,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,fixed_pt),apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))) = fixed_pt ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(49,plain,
apply(apply(k,fixed_pt),apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))) = fixed_pt,
inference(unit_resolution,[status(thm)],[48,17]) ).
tff(50,plain,
fixed_pt = apply(apply(k,fixed_pt),apply(apply(k,apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))),
inference(symmetry,[status(thm)],[49]) ).
tff(51,plain,
fixed_pt = apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(transitivity,[status(thm)],[50,47]) ).
tff(52,plain,
apply(fixed_pt,apply(strong_fixed_point,fixed_pt)) = apply(apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))),
inference(monotonicity,[status(thm)],[51,45]) ).
tff(53,plain,
apply(apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))) = apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
inference(symmetry,[status(thm)],[52]) ).
tff(54,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(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(55,plain,
apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(k,fixed_pt),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),apply(apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))))),
inference(unit_resolution,[status(thm)],[54,7]) ).
tff(56,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,s),fixed_pt) = s ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(57,plain,
apply(apply(k,s),fixed_pt) = s,
inference(unit_resolution,[status(thm)],[56,17]) ).
tff(58,plain,
s = apply(apply(k,s),fixed_pt),
inference(symmetry,[status(thm)],[57]) ).
tff(59,plain,
apply(s,apply(k,fixed_pt)) = apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),
inference(monotonicity,[status(thm)],[58]) ).
tff(60,plain,
apply(apply(s,apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(monotonicity,[status(thm)],[59]) ).
tff(61,plain,
apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(monotonicity,[status(thm)],[60]) ).
tff(62,plain,
apply(apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(symmetry,[status(thm)],[61]) ).
tff(63,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(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(64,plain,
apply(apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(unit_resolution,[status(thm)],[63,7]) ).
tff(65,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(k,s)),k),fixed_pt) = apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(66,plain,
apply(apply(apply(s,apply(k,s)),k),fixed_pt) = apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),
inference(unit_resolution,[status(thm)],[65,7]) ).
tff(67,plain,
apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)) = apply(apply(apply(s,apply(k,s)),k),fixed_pt),
inference(symmetry,[status(thm)],[66]) ).
tff(68,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt) = apply(apply(s,s),apply(s,k)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(69,plain,
apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt) = apply(apply(s,s),apply(s,k)),
inference(unit_resolution,[status(thm)],[68,17]) ).
tff(70,plain,
apply(apply(s,s),apply(s,k)) = apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),
inference(symmetry,[status(thm)],[69]) ).
tff(71,plain,
apply(apply(apply(s,s),apply(s,k)),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))) = apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)),
inference(monotonicity,[status(thm)],[70,67]) ).
tff(72,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,s),apply(s,k)),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))) = apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(73,plain,
apply(apply(apply(s,s),apply(s,k)),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))) = apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))),
inference(unit_resolution,[status(thm)],[72,7]) ).
tff(74,plain,
apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))) = apply(apply(apply(s,s),apply(s,k)),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),
inference(symmetry,[status(thm)],[73]) ).
tff(75,plain,
apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))) = apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(transitivity,[status(thm)],[74,71,20,10]) ).
tff(76,plain,
apply(apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))),
inference(monotonicity,[status(thm)],[75,26]) ).
tff(77,plain,
apply(apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt)))) = apply(apply(apply(s,apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt))),apply(apply(s,k),apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)))),apply(apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),apply(apply(apply(s,apply(k,s)),k),fixed_pt))),
inference(symmetry,[status(thm)],[76]) ).
tff(78,plain,
apply(strong_fixed_point,fixed_pt) = apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
inference(transitivity,[status(thm)],[44,40,38,30,77,64,62,55,53]) ).
tff(79,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(80,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(81,plain,
apply(strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(strong_fixed_point,fixed_pt)),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
$false,
inference(unit_resolution,[status(thm)],[81,78]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : COL006-5 : TPTP v9.0.0. Released v2.1.0.
% 0.12/0.13 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n016.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu Jun 19 12:02:32 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.14/0.39 % SZS status Unsatisfiable
% 0.14/0.39 % SZS output start Proof
% See solution above
% 0.22/0.43 % E exiting
%------------------------------------------------------------------------------