%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL006-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 0.13s 0.39s
% Output : Proof 0.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 29
% Syntax : Number of formulae : 92 ( 61 unt; 0 typ; 0 def)
% Number of atoms : 151 ( 119 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 102 ( 47 ~; 42 |; 0 &)
% ( 13 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of FOOLs : 4 ( 4 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 : 4 ( 4 usr; 3 con; 0-2 aty)
% Number of variables : 101 ( 94 !; 0 ?; 101 :)
% Comments :
%------------------------------------------------------------------------------
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(fixed_pt_type,type,
fixed_pt: $i ).
tff(k_type,type,
k: $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(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/sandbox/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/sandbox/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,
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(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)],[40,38,30,28]) ).
tff(42,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(43,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)],[42]) ).
tff(44,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(45,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)],[44,17]) ).
tff(46,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)],[45]) ).
tff(47,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)],[46,43]) ).
tff(48,plain,
apply(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)) = 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)],[47,41]) ).
tff(49,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(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(symmetry,[status(thm)],[48]) ).
tff(50,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(51,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)],[50,7]) ).
tff(52,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,s),fixed_pt) = s ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(53,plain,
apply(apply(k,s),fixed_pt) = s,
inference(unit_resolution,[status(thm)],[52,17]) ).
tff(54,plain,
s = apply(apply(k,s),fixed_pt),
inference(symmetry,[status(thm)],[53]) ).
tff(55,plain,
apply(s,apply(k,fixed_pt)) = apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),
inference(monotonicity,[status(thm)],[54]) ).
tff(56,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)],[55]) ).
tff(57,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)],[56]) ).
tff(58,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)],[57]) ).
tff(59,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(60,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)],[59,7]) ).
tff(61,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(62,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)],[61,7]) ).
tff(63,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)],[62]) ).
tff(64,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(65,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)],[64,17]) ).
tff(66,plain,
apply(apply(s,s),apply(s,k)) = apply(apply(k,apply(apply(s,s),apply(s,k))),fixed_pt),
inference(symmetry,[status(thm)],[65]) ).
tff(67,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)],[66,63]) ).
tff(68,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(69,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)],[68,7]) ).
tff(70,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)],[69]) ).
tff(71,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)],[70,67,20,10]) ).
tff(72,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)],[71,26]) ).
tff(73,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)],[72]) ).
tff(74,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(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(transitivity,[status(thm)],[40,38,30,73,60,58,51,49]) ).
tff(75,plain,
( ~ 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))))
<=> ~ 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(76,axiom,
~ 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/sandbox/benchmark/theBenchmark.p',prove_strong_fixed_point) ).
tff(77,plain,
~ 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)],[76,75]) ).
tff(78,plain,
^ [Strong_fixed_point: $i] :
refl(( ( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
<=> ( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(79,plain,
( ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
<=> ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ) ),
inference(quant_intro,[status(thm)],[78]) ).
tff(80,plain,
( ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
<=> ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,plain,
^ [Strong_fixed_point: $i] :
rewrite(( ( ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) )
| fixed_point(Strong_fixed_point) )
<=> ( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(82,plain,
( ! [Strong_fixed_point: $i] :
( ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) )
| fixed_point(Strong_fixed_point) )
<=> ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ) ),
inference(quant_intro,[status(thm)],[81]) ).
tff(83,axiom,
! [Strong_fixed_point: $i] :
( ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) )
| fixed_point(Strong_fixed_point) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',strong_fixed_point) ).
tff(84,plain,
! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ),
inference(modus_ponens,[status(thm)],[83,82]) ).
tff(85,plain,
! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ),
inference(modus_ponens,[status(thm)],[84,80]) ).
tff(86,plain,
! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ),
inference(skolemize,[status(sab)],[85]) ).
tff(87,plain,
! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) ),
inference(modus_ponens,[status(thm)],[86,79]) ).
tff(88,plain,
( ( ~ ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
| 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))))
| ( 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(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)) ) )
<=> ( ~ ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
| 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))))
| ( 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(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(rewrite,[status(thm)],[]) ).
tff(89,plain,
( ~ ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
| 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))))
| ( 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(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(quant_inst,[status(thm)],[]) ).
tff(90,plain,
( ~ ! [Strong_fixed_point: $i] :
( fixed_point(Strong_fixed_point)
| ( apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ) )
| 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))))
| ( 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(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(modus_ponens,[status(thm)],[89,88]) ).
tff(91,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(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(unit_resolution,[status(thm)],[90,87,77]) ).
tff(92,plain,
$false,
inference(unit_resolution,[status(thm)],[91,74]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.14 % Problem : COL006-2 : TPTP v9.0.0. Released v1.0.0.
% 0.13/0.14 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n015.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Jun 19 12:02:40 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.13/0.39 % SZS status Unsatisfiable
% 0.13/0.39 % SZS output start Proof
% See solution above
% 0.23/0.44 % E exiting
%------------------------------------------------------------------------------