%------------------------------------------------------------------------------
% File : Metis---2.4
% Problem : COL004-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : metis --show proof --show saturation %s
% Computer : n020.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 : 600s
% DateTime : Fri Jul 15 00:36:12 EDT 2022
% Result : Unsatisfiable 0.12s 0.34s
% Output : CNFRefutation 0.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 20
% Syntax : Number of clauses : 69 ( 35 unt; 0 nHn; 66 RR)
% Number of literals : 121 ( 120 equ; 54 neg)
% Maximal clause size : 3 ( 1 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 3 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 4 con; 0-2 aty)
% Number of variables : 16 ( 1 sgn)
% Comments :
%------------------------------------------------------------------------------
cnf(s_definition,axiom,
apply(apply(apply(s,X),Y),Z) = apply(apply(X,Z),apply(Y,Z)) ).
cnf(k_definition,axiom,
apply(apply(k,X),Y) = X ).
cnf(prove_u_combinator,negated_conjecture,
apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) != apply(y,apply(apply(x,x),y)) ).
cnf(refute_0_0,plain,
apply(apply(k,y),apply(k,y)) = y,
inference(subst,[],[k_definition:[bind(X,$fot(y)),bind(Y,$fot(apply(k,y)))]]) ).
cnf(refute_0_1,plain,
apply(apply(apply(s,k),k),y) = apply(apply(k,y),apply(k,y)),
inference(subst,[],[s_definition:[bind(X,$fot(k)),bind(Y,$fot(k)),bind(Z,$fot(y))]]) ).
cnf(refute_0_2,plain,
X0 = X0,
introduced(tautology,[refl,[$fot(X0)]]) ).
cnf(refute_0_3,plain,
( X0 != X0
| X0 != Y0
| Y0 = X0 ),
introduced(tautology,[equality,[$cnf( $equal(X0,X0) ),[0],$fot(Y0)]]) ).
cnf(refute_0_4,plain,
( X0 != Y0
| Y0 = X0 ),
inference(resolve,[$cnf( $equal(X0,X0) )],[refute_0_2,refute_0_3]) ).
cnf(refute_0_5,plain,
( Y0 != X0
| Y0 != Z0
| X0 = Z0 ),
introduced(tautology,[equality,[$cnf( $equal(Y0,Z0) ),[0],$fot(X0)]]) ).
cnf(refute_0_6,plain,
( X0 != Y0
| Y0 != Z0
| X0 = Z0 ),
inference(resolve,[$cnf( $equal(Y0,X0) )],[refute_0_4,refute_0_5]) ).
cnf(refute_0_7,plain,
( apply(apply(apply(s,k),k),y) != apply(apply(k,y),apply(k,y))
| apply(apply(k,y),apply(k,y)) != y
| apply(apply(apply(s,k),k),y) = y ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(s,k),k),y))),bind(Y0,$fot(apply(apply(k,y),apply(k,y)))),bind(Z0,$fot(y))]]) ).
cnf(refute_0_8,plain,
( apply(apply(k,y),apply(k,y)) != y
| apply(apply(apply(s,k),k),y) = y ),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,k),k),y),apply(apply(k,y),apply(k,y))) )],[refute_0_1,refute_0_7]) ).
cnf(refute_0_9,plain,
apply(apply(apply(s,k),k),y) = y,
inference(resolve,[$cnf( $equal(apply(apply(k,y),apply(k,y)),y) )],[refute_0_0,refute_0_8]) ).
cnf(refute_0_10,plain,
apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) = apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)),
introduced(tautology,[refl,[$fot(apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)))]]) ).
cnf(refute_0_11,plain,
( apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) != apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y))
| apply(apply(apply(s,k),k),y) != y
| apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) = apply(y,apply(apply(x,x),y)) ),
introduced(tautology,[equality,[$cnf( $equal(apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)),apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y))) ),[1,0],$fot(y)]]) ).
cnf(refute_0_12,plain,
( apply(apply(apply(s,k),k),y) != y
| apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) = apply(y,apply(apply(x,x),y)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)),apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y))) )],[refute_0_10,refute_0_11]) ).
cnf(refute_0_13,plain,
apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) = apply(y,apply(apply(x,x),y)),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,k),k),y),y) )],[refute_0_9,refute_0_12]) ).
cnf(refute_0_14,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) = apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)),
inference(subst,[],[s_definition:[bind(X,$fot(apply(apply(s,k),k))),bind(Y,$fot(apply(x,x))),bind(Z,$fot(y))]]) ).
cnf(refute_0_15,plain,
( apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) != apply(y,apply(apply(x,x),y))
| apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) != apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y))
| apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) = apply(y,apply(apply(x,x),y)) ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y))),bind(Y0,$fot(apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)))),bind(Z0,$fot(apply(y,apply(apply(x,x),y))))]]) ).
cnf(refute_0_16,plain,
( apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)) != apply(y,apply(apply(x,x),y))
| apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) = apply(y,apply(apply(x,x),y)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y),apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y))) )],[refute_0_14,refute_0_15]) ).
cnf(refute_0_17,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) = apply(y,apply(apply(x,x),y)),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,k),k),y),apply(apply(x,x),y)),apply(y,apply(apply(x,x),y))) )],[refute_0_13,refute_0_16]) ).
cnf(refute_0_18,plain,
apply(apply(k,x),apply(k,x)) = x,
inference(subst,[],[k_definition:[bind(X,$fot(x)),bind(Y,$fot(apply(k,x)))]]) ).
cnf(refute_0_19,plain,
apply(apply(apply(s,k),k),x) = apply(apply(k,x),apply(k,x)),
inference(subst,[],[s_definition:[bind(X,$fot(k)),bind(Y,$fot(k)),bind(Z,$fot(x))]]) ).
cnf(refute_0_20,plain,
( apply(apply(apply(s,k),k),x) != apply(apply(k,x),apply(k,x))
| apply(apply(k,x),apply(k,x)) != x
| apply(apply(apply(s,k),k),x) = x ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(s,k),k),x))),bind(Y0,$fot(apply(apply(k,x),apply(k,x)))),bind(Z0,$fot(x))]]) ).
cnf(refute_0_21,plain,
( apply(apply(k,x),apply(k,x)) != x
| apply(apply(apply(s,k),k),x) = x ),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,k),k),x),apply(apply(k,x),apply(k,x))) )],[refute_0_19,refute_0_20]) ).
cnf(refute_0_22,plain,
apply(apply(apply(s,k),k),x) = x,
inference(resolve,[$cnf( $equal(apply(apply(k,x),apply(k,x)),x) )],[refute_0_18,refute_0_21]) ).
cnf(refute_0_23,plain,
apply(x,apply(apply(apply(s,k),k),x)) = apply(x,apply(apply(apply(s,k),k),x)),
introduced(tautology,[refl,[$fot(apply(x,apply(apply(apply(s,k),k),x)))]]) ).
cnf(refute_0_24,plain,
( apply(apply(apply(s,k),k),x) != x
| apply(x,apply(apply(apply(s,k),k),x)) != apply(x,apply(apply(apply(s,k),k),x))
| apply(x,apply(apply(apply(s,k),k),x)) = apply(x,x) ),
introduced(tautology,[equality,[$cnf( $equal(apply(x,apply(apply(apply(s,k),k),x)),apply(x,apply(apply(apply(s,k),k),x))) ),[1,1],$fot(x)]]) ).
cnf(refute_0_25,plain,
( apply(apply(apply(s,k),k),x) != x
| apply(x,apply(apply(apply(s,k),k),x)) = apply(x,x) ),
inference(resolve,[$cnf( $equal(apply(x,apply(apply(apply(s,k),k),x)),apply(x,apply(apply(apply(s,k),k),x))) )],[refute_0_23,refute_0_24]) ).
cnf(refute_0_26,plain,
apply(x,apply(apply(apply(s,k),k),x)) = apply(x,x),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,k),k),x),x) )],[refute_0_22,refute_0_25]) ).
cnf(refute_0_27,plain,
apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)),
introduced(tautology,[refl,[$fot(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)))]]) ).
cnf(refute_0_28,plain,
( apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) != apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x))
| apply(apply(apply(s,k),k),x) != x
| apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(x,apply(apply(apply(s,k),k),x)) ),
introduced(tautology,[equality,[$cnf( $equal(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)),apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x))) ),[1,0],$fot(x)]]) ).
cnf(refute_0_29,plain,
( apply(apply(apply(s,k),k),x) != x
| apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(x,apply(apply(apply(s,k),k),x)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)),apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x))) )],[refute_0_27,refute_0_28]) ).
cnf(refute_0_30,plain,
apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(x,apply(apply(apply(s,k),k),x)),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,k),k),x),x) )],[refute_0_22,refute_0_29]) ).
cnf(refute_0_31,plain,
( apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) != apply(x,apply(apply(apply(s,k),k),x))
| apply(x,apply(apply(apply(s,k),k),x)) != apply(x,x)
| apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(x,x) ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)))),bind(Y0,$fot(apply(x,apply(apply(apply(s,k),k),x)))),bind(Z0,$fot(apply(x,x)))]]) ).
cnf(refute_0_32,plain,
( apply(x,apply(apply(apply(s,k),k),x)) != apply(x,x)
| apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(x,x) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)),apply(x,apply(apply(apply(s,k),k),x))) )],[refute_0_30,refute_0_31]) ).
cnf(refute_0_33,plain,
apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) = apply(x,x),
inference(resolve,[$cnf( $equal(apply(x,apply(apply(apply(s,k),k),x)),apply(x,x)) )],[refute_0_26,refute_0_32]) ).
cnf(refute_0_34,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) = apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)),
inference(subst,[],[s_definition:[bind(X,$fot(apply(apply(s,k),k))),bind(Y,$fot(apply(apply(s,k),k))),bind(Z,$fot(x))]]) ).
cnf(refute_0_35,plain,
( apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) != apply(x,x)
| apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) != apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x))
| apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) = apply(x,x) ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))),bind(Y0,$fot(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)))),bind(Z0,$fot(apply(x,x)))]]) ).
cnf(refute_0_36,plain,
( apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)) != apply(x,x)
| apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) = apply(x,x) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x),apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x))) )],[refute_0_34,refute_0_35]) ).
cnf(refute_0_37,plain,
apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) = apply(x,x),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,k),k),x),apply(apply(apply(s,k),k),x)),apply(x,x)) )],[refute_0_33,refute_0_36]) ).
cnf(refute_0_38,plain,
apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),
introduced(tautology,[refl,[$fot(apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)))]]) ).
cnf(refute_0_39,plain,
( apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) != apply(x,x)
| apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))
| apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)) ),
introduced(tautology,[equality,[$cnf( $equal(apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))) ),[1,1],$fot(apply(x,x))]]) ).
cnf(refute_0_40,plain,
( apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x) != apply(x,x)
| apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)) ),
inference(resolve,[$cnf( $equal(apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))) )],[refute_0_38,refute_0_39]) ).
cnf(refute_0_41,plain,
apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x),apply(x,x)) )],[refute_0_37,refute_0_40]) ).
cnf(refute_0_42,plain,
apply(apply(k,apply(s,apply(apply(s,k),k))),x) = apply(s,apply(apply(s,k),k)),
inference(subst,[],[k_definition:[bind(X,$fot(apply(s,apply(apply(s,k),k)))),bind(Y,$fot(x))]]) ).
cnf(refute_0_43,plain,
apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),
introduced(tautology,[refl,[$fot(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)))]]) ).
cnf(refute_0_44,plain,
( apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))
| apply(apply(k,apply(s,apply(apply(s,k),k))),x) != apply(s,apply(apply(s,k),k))
| apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) ),
introduced(tautology,[equality,[$cnf( $equal(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))) ),[1,0],$fot(apply(s,apply(apply(s,k),k)))]]) ).
cnf(refute_0_45,plain,
( apply(apply(k,apply(s,apply(apply(s,k),k))),x) != apply(s,apply(apply(s,k),k))
| apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))) )],[refute_0_43,refute_0_44]) ).
cnf(refute_0_46,plain,
apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),
inference(resolve,[$cnf( $equal(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(s,apply(apply(s,k),k))) )],[refute_0_42,refute_0_45]) ).
cnf(refute_0_47,plain,
( apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))
| apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(s,apply(apply(s,k),k)),apply(x,x))
| apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)) ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)))),bind(Y0,$fot(apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)))),bind(Z0,$fot(apply(apply(s,apply(apply(s,k),k)),apply(x,x))))]]) ).
cnf(refute_0_48,plain,
( apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(s,apply(apply(s,k),k)),apply(x,x))
| apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))) )],[refute_0_46,refute_0_47]) ).
cnf(refute_0_49,plain,
apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)),
inference(resolve,[$cnf( $equal(apply(apply(s,apply(apply(s,k),k)),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(s,apply(apply(s,k),k)),apply(x,x))) )],[refute_0_41,refute_0_48]) ).
cnf(refute_0_50,plain,
apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) = apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),
inference(subst,[],[s_definition:[bind(X,$fot(apply(k,apply(s,apply(apply(s,k),k))))),bind(Y,$fot(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)))),bind(Z,$fot(x))]]) ).
cnf(refute_0_51,plain,
( apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(s,apply(apply(s,k),k)),apply(x,x))
| apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) != apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))
| apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)) ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x))),bind(Y0,$fot(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)))),bind(Z0,$fot(apply(apply(s,apply(apply(s,k),k)),apply(x,x))))]]) ).
cnf(refute_0_52,plain,
( apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)) != apply(apply(s,apply(apply(s,k),k)),apply(x,x))
| apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x))) )],[refute_0_50,refute_0_51]) ).
cnf(refute_0_53,plain,
apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) = apply(apply(s,apply(apply(s,k),k)),apply(x,x)),
inference(resolve,[$cnf( $equal(apply(apply(apply(k,apply(s,apply(apply(s,k),k))),x),apply(apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k)),x)),apply(apply(s,apply(apply(s,k),k)),apply(x,x))) )],[refute_0_49,refute_0_52]) ).
cnf(refute_0_54,plain,
apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),
introduced(tautology,[refl,[$fot(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y))]]) ).
cnf(refute_0_55,plain,
( apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) != apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y)
| apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) != apply(apply(s,apply(apply(s,k),k)),apply(x,x))
| apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) ),
introduced(tautology,[equality,[$cnf( $equal(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y)) ),[1,0],$fot(apply(apply(s,apply(apply(s,k),k)),apply(x,x)))]]) ).
cnf(refute_0_56,plain,
( apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x) != apply(apply(s,apply(apply(s,k),k)),apply(x,x))
| apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y)) )],[refute_0_54,refute_0_55]) ).
cnf(refute_0_57,plain,
apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),apply(apply(s,apply(apply(s,k),k)),apply(x,x))) )],[refute_0_53,refute_0_56]) ).
cnf(refute_0_58,plain,
( apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) != apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y)
| apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) != apply(y,apply(apply(x,x),y))
| apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(y,apply(apply(x,x),y)) ),
inference(subst,[],[refute_0_6:[bind(X0,$fot(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y))),bind(Y0,$fot(apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y))),bind(Z0,$fot(apply(y,apply(apply(x,x),y))))]]) ).
cnf(refute_0_59,plain,
( apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y) != apply(y,apply(apply(x,x),y))
| apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(y,apply(apply(x,x),y)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y)) )],[refute_0_57,refute_0_58]) ).
cnf(refute_0_60,plain,
apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(y,apply(apply(x,x),y)),
inference(resolve,[$cnf( $equal(apply(apply(apply(s,apply(apply(s,k),k)),apply(x,x)),y),apply(y,apply(apply(x,x),y))) )],[refute_0_17,refute_0_59]) ).
cnf(refute_0_61,plain,
( apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) != apply(y,apply(apply(x,x),y))
| apply(y,apply(apply(x,x),y)) != apply(y,apply(apply(x,x),y))
| apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(y,apply(apply(x,x),y)) ),
introduced(tautology,[equality,[$cnf( ~ $equal(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),apply(y,apply(apply(x,x),y))) ),[0],$fot(apply(y,apply(apply(x,x),y)))]]) ).
cnf(refute_0_62,plain,
( apply(y,apply(apply(x,x),y)) != apply(y,apply(apply(x,x),y))
| apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y) = apply(y,apply(apply(x,x),y)) ),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),apply(y,apply(apply(x,x),y))) )],[refute_0_60,refute_0_61]) ).
cnf(refute_0_63,plain,
apply(y,apply(apply(x,x),y)) != apply(y,apply(apply(x,x),y)),
inference(resolve,[$cnf( $equal(apply(apply(apply(apply(s,apply(k,apply(s,apply(apply(s,k),k)))),apply(apply(s,apply(apply(s,k),k)),apply(apply(s,k),k))),x),y),apply(y,apply(apply(x,x),y))) )],[refute_0_62,prove_u_combinator]) ).
cnf(refute_0_64,plain,
apply(y,apply(apply(x,x),y)) = apply(y,apply(apply(x,x),y)),
introduced(tautology,[refl,[$fot(apply(y,apply(apply(x,x),y)))]]) ).
cnf(refute_0_65,plain,
$false,
inference(resolve,[$cnf( $equal(apply(y,apply(apply(x,x),y)),apply(y,apply(apply(x,x),y))) )],[refute_0_64,refute_0_63]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : COL004-3 : TPTP v8.1.0. Released v1.0.0.
% 0.07/0.12 % Command : metis --show proof --show saturation %s
% 0.12/0.33 % Computer : n020.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 600
% 0.12/0.34 % DateTime : Tue May 31 12:52:08 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 0.12/0.34 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.12/0.34
% 0.12/0.34 % SZS output start CNFRefutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 0.12/0.37
%------------------------------------------------------------------------------