TSTP Solution File: COL006-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : COL006-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n007.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 : Tue Sep 6 17:48:00 EDT 2022
% Result : Unsatisfiable 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 29
% Syntax : Number of formulae : 77 ( 46 unt; 5 typ; 0 def)
% Number of atoms : 126 ( 94 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 92 ( 42 ~; 37 |; 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 : 3 ( 2 >; 1 *; 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 : 88 ( 81 !; 0 ?; 88 :)
% 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,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),fixed_pt) = apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
apply(apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),fixed_pt) = apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)) = apply(apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(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(19,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)],[18,17]) ).
tff(20,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)],[19]) ).
tff(21,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),fixed_pt)),
inference(monotonicity,[status(thm)],[20,10]) ).
tff(22,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),fixed_pt)) = apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(symmetry,[status(thm)],[21]) ).
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,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(24,plain,
apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),fixed_pt)),
inference(unit_resolution,[status(thm)],[23,7]) ).
tff(25,plain,
apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt) = apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(transitivity,[status(thm)],[24,22]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = fixed_pt ) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = fixed_pt,
inference(unit_resolution,[status(thm)],[26,17]) ).
tff(28,plain,
fixed_pt = apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(symmetry,[status(thm)],[27]) ).
tff(29,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)) = apply(apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))),
inference(monotonicity,[status(thm)],[28,25]) ).
tff(30,plain,
apply(apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)),
inference(symmetry,[status(thm)],[29]) ).
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,fixed_pt)),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(32,plain,
apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(apply(k,fixed_pt),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))),
inference(unit_resolution,[status(thm)],[31,7]) ).
tff(33,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(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(34,plain,
apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))),
inference(unit_resolution,[status(thm)],[33,7]) ).
tff(35,plain,
apply(apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) = apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(symmetry,[status(thm)],[34]) ).
tff(36,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) = apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(37,plain,
apply(apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) = apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)),
inference(unit_resolution,[status(thm)],[36,17]) ).
tff(38,plain,
apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)) = apply(apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(k,apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))),
inference(symmetry,[status(thm)],[37]) ).
tff(39,plain,
apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)) = apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(transitivity,[status(thm)],[38,35]) ).
tff(40,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(41,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)],[40,7]) ).
tff(42,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)],[41]) ).
tff(43,plain,
( ~ ! [Y: $i,X: $i] : ( apply(apply(k,X),Y) = X )
| ( apply(apply(k,s),fixed_pt) = s ) ),
inference(quant_inst,[status(thm)],]) ).
tff(44,plain,
apply(apply(k,s),fixed_pt) = s,
inference(unit_resolution,[status(thm)],[43,17]) ).
tff(45,plain,
s = apply(apply(k,s),fixed_pt),
inference(symmetry,[status(thm)],[44]) ).
tff(46,plain,
apply(s,apply(k,fixed_pt)) = apply(apply(apply(k,s),fixed_pt),apply(k,fixed_pt)),
inference(monotonicity,[status(thm)],[45]) ).
tff(47,plain,
apply(s,apply(k,fixed_pt)) = apply(apply(apply(s,apply(k,s)),k),fixed_pt),
inference(transitivity,[status(thm)],[46,42]) ).
tff(48,plain,
apply(apply(s,apply(k,fixed_pt)),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))) = apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)),
inference(monotonicity,[status(thm)],[47,20]) ).
tff(49,plain,
apply(apply(s,apply(k,fixed_pt)),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))) = apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(transitivity,[status(thm)],[48,38,35]) ).
tff(50,plain,
apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))),
inference(monotonicity,[status(thm)],[49,39]) ).
tff(51,plain,
apply(apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) = apply(apply(apply(s,apply(k,fixed_pt)),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),
inference(symmetry,[status(thm)],[50]) ).
tff(52,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(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(53,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))) = apply(apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt))),apply(apply(apply(s,k),k),apply(apply(apply(apply(s,apply(k,s)),k),fixed_pt),apply(apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),fixed_pt)))),
inference(unit_resolution,[status(thm)],[52,7]) ).
tff(54,plain,
apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)),
inference(transitivity,[status(thm)],[24,22,53,51,32,30]) ).
tff(55,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))))) ),
inference(rewrite,[status(thm)],]) ).
tff(56,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))))),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_strong_fixed_point) ).
tff(57,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))))),
inference(modus_ponens,[status(thm)],[56,55]) ).
tff(58,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(59,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)],[58]) ).
tff(60,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(61,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(62,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)],[61]) ).
tff(63,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(64,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)],[63,62]) ).
tff(65,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)],[64,60]) ).
tff(66,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)],[65]) ).
tff(67,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)],[66,59]) ).
tff(68,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))))
| ( apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))))
| ( apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(69,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))))
| ( apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(70,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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))))
| ( apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)) ) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
apply(apply(apply(s,apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),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(apply(s,apply(k,s)),k)),apply(k,apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))))),fixed_pt)),
inference(unit_resolution,[status(thm)],[70,67,57]) ).
tff(72,plain,
$false,
inference(unit_resolution,[status(thm)],[71,54]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : COL006-3 : TPTP v8.1.0. Released v1.0.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.35 % Computer : n007.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 : Tue Aug 30 08:51:01 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.21/0.40 % SZS status Unsatisfiable
% 0.21/0.40 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------