%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : COL003-2 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 25 01:01:32 PM UTC 2026
% Result : Unsatisfiable 30.73s 4.53s
% Output : Proof 30.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 32
% Number of leaves : 10
% Syntax : Number of formulae : 89 ( 85 unt; 0 def)
% Number of atoms : 93 ( 84 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 17 ( 13 ~; 4 |; 0 &)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 4 ( 1 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 9 con; 0-4 aty)
% Number of variables : 89 ( 2 sgn 12 !; 0 ?)
% Comments :
%------------------------------------------------------------------------------
cnf(f1,axiom,
apply(apply(w,X),Y) = apply(apply(X,Y),Y),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',w_definition) ).
fof(f1_nnf,plain,
! [X,Y] : apply(apply(w,X),Y) = apply(apply(X,Y),Y),
inference(nnf_transformation,[status(thm)],[f1]) ).
fof(f1_sk,plain,
! [X,Y] : apply(apply(w,X),Y) = apply(apply(X,Y),Y),
inference(skolemisation,[status(esa)],[f1_nnf]) ).
cnf(c1,plain,
apply(apply(w,X0),X1) = apply(apply(X0,X1),X1),
inference(cnf_transformation,[status(esa)],[f1_sk]) ).
cnf(t1,plain,
apply(apply(w,X1),X2) = apply(apply(X1,X2),X2),
inference(equality_encoding,[status(esa)],[c1]) ).
cnf(t14,plain,
apply(apply(X1,X2),X2) = apply(apply(w,X1),X2),
inference(orient,[status(thm)],[t1]) ).
cnf(f2,axiom,
( fixed_point(Strong_fixed_point)
| apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',strong_fixed_point) ).
fof(f2_nnf,plain,
! [Strong_fixed_point] :
( fixed_point(Strong_fixed_point)
| apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ),
inference(nnf_transformation,[status(thm)],[f2]) ).
fof(f2_sk,plain,
! [Strong_fixed_point] :
( fixed_point(Strong_fixed_point)
| apply(Strong_fixed_point,fixed_pt) != apply(fixed_pt,apply(Strong_fixed_point,fixed_pt)) ),
inference(skolemisation,[status(esa)],[f2_nnf]) ).
cnf(c2,plain,
( fixed_point(X0)
| apply(X0,fixed_pt) != apply(fixed_pt,apply(X0,fixed_pt)) ),
inference(cnf_transformation,[status(esa)],[f2_sk]) ).
cnf(t3,plain,
ifeq(apply(X1,fixed_pt),apply(fixed_pt,apply(X1,fixed_pt)),fixed_point(X1),true) = true,
inference(equality_encoding,[status(esa)],[c2]) ).
cnf(t43,plain,
ifeq(apply(X1,fixed_pt),apply(fixed_pt,apply(X1,fixed_pt)),fixed_point(X1),true) = true,
inference(orient,[status(thm)],[t3]) ).
cnf(f0,axiom,
apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_definition) ).
fof(f0_nnf,plain,
! [X,Y,Z] : apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)),
inference(nnf_transformation,[status(thm)],[f0]) ).
fof(f0_sk,plain,
! [X,Y,Z] : apply(apply(apply(b,X),Y),Z) = apply(X,apply(Y,Z)),
inference(skolemisation,[status(esa)],[f0_nnf]) ).
cnf(c0,plain,
apply(apply(apply(b,X0),X1),X2) = apply(X0,apply(X1,X2)),
inference(cnf_transformation,[status(esa)],[f0_sk]) ).
cnf(t2,plain,
apply(apply(apply(b,X1),X2),X3) = apply(X1,apply(X2,X3)),
inference(equality_encoding,[status(esa)],[c0]) ).
cnf(t13,plain,
apply(apply(apply(b,X1),X2),X3) = apply(X1,apply(X2,X3)),
inference(orient,[status(thm)],[t2]) ).
cnf(t15,plain,
apply(apply(w,apply(b,X1)),X2) = apply(X1,apply(X2,X2)),
inference(cp,[status(thm)],[t14,t13]) ).
cnf(t137,plain,
apply(apply(w,apply(b,X1)),X2) = apply(X1,apply(X2,X2)),
inference(orient,[status(thm)],[t15]) ).
cnf(t154,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(w,apply(b,apply(b,X1))),X2),X3),
inference(cp,[status(thm)],[t13,t137]) ).
cnf(t7,axiom,
sFlat3 = apply(b,b),
introduced(definition) ).
cnf(t32,plain,
apply(b,b) = sFlat3,
inference(orient,[status(thm)],[t7]) ).
cnf(t33,plain,
apply(b,apply(X1,X2)) = apply(apply(sFlat3,X1),X2),
inference(cp,[status(thm)],[t13,t32]) ).
cnf(t74,plain,
apply(apply(sFlat3,X1),X2) = apply(b,apply(X1,X2)),
inference(orient,[status(thm)],[t33]) ).
cnf(t34,plain,
apply(apply(w,b),b) = apply(sFlat3,b),
inference(cp,[status(thm)],[t14,t32]) ).
cnf(t4,axiom,
sFlat0 = apply(w,w),
introduced(definition) ).
cnf(t39,plain,
apply(w,w) = sFlat0,
inference(orient,[status(thm)],[t4]) ).
cnf(t40,plain,
apply(apply(w,X1),X1) = apply(sFlat0,X1),
inference(cp,[status(thm)],[t14,t39]) ).
cnf(t56,plain,
apply(apply(w,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t40]) ).
cnf(t1272,plain,
apply(sFlat0,b) = apply(sFlat3,b),
inference(step,[status(thm)],[t34,t56]) ).
cnf(t8,axiom,
sFlat4 = apply(apply(b,b),b),
introduced(definition) ).
cnf(t1256,plain,
sFlat4 = apply(apply(w,b),b),
inference(step,[status(thm)],[t8,t14]) ).
cnf(t1257,plain,
sFlat4 = apply(apply(w,w),b),
inference(step,[status(thm)],[t1256,t14]) ).
cnf(t23,plain,
apply(apply(w,w),b) = sFlat4,
inference(orient,[status(thm)],[t1257]) ).
cnf(t1263,plain,
apply(sFlat0,b) = sFlat4,
inference(step,[status(thm)],[t23,t39]) ).
cnf(t41,plain,
apply(sFlat0,b) = sFlat4,
inference(rw,[status(thm)],[t1263]) ).
cnf(t51,plain,
apply(sFlat0,b) = sFlat4,
inference(orient,[status(thm)],[t41]) ).
cnf(t1273,plain,
sFlat4 = apply(sFlat3,b),
inference(step,[status(thm)],[t1272,t51]) ).
cnf(t64,plain,
apply(sFlat3,b) = sFlat4,
inference(orient,[status(thm)],[t1273]) ).
cnf(t76,plain,
apply(b,apply(b,X1)) = apply(sFlat4,X1),
inference(cp,[status(thm)],[t74,t64]) ).
cnf(t87,plain,
apply(b,apply(b,X1)) = apply(sFlat4,X1),
inference(orient,[status(thm)],[t76]) ).
cnf(t1281,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(w,apply(sFlat4,X1)),X2),X3),
inference(step,[status(thm)],[t154,t87]) ).
cnf(t9,axiom,
sFlat5 = apply(apply(b,w),apply(apply(b,b),b)),
introduced(definition) ).
cnf(t1259,plain,
sFlat5 = apply(apply(b,w),apply(apply(w,b),b)),
inference(step,[status(thm)],[t9,t14]) ).
cnf(t1260,plain,
sFlat5 = apply(apply(b,w),apply(apply(w,w),b)),
inference(step,[status(thm)],[t1259,t14]) ).
cnf(t1261,plain,
sFlat5 = apply(apply(b,w),sFlat4),
inference(step,[status(thm)],[t1260,t23]) ).
cnf(t26,plain,
apply(apply(b,w),sFlat4) = sFlat5,
inference(orient,[status(thm)],[t1261]) ).
cnf(t27,plain,
apply(w,apply(sFlat4,X1)) = apply(sFlat5,X1),
inference(cp,[status(thm)],[t13,t26]) ).
cnf(t61,plain,
apply(w,apply(sFlat4,X1)) = apply(sFlat5,X1),
inference(orient,[status(thm)],[t27]) ).
cnf(t1282,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(sFlat5,X1),X2),X3),
inference(step,[status(thm)],[t1281,t61]) ).
cnf(t287,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(sFlat5,X1),X2),X3),
inference(orient,[status(thm)],[t1282]) ).
cnf(t298,plain,
apply(X1,apply(apply(X2,X2),X2)) = apply(apply(w,apply(sFlat5,X1)),X2),
inference(cp,[status(thm)],[t287,t14]) ).
cnf(t57,plain,
apply(sFlat0,X1) = apply(apply(X1,X1),X1),
inference(cp,[status(thm)],[t56,t14]) ).
cnf(t67,plain,
apply(apply(X1,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t57]) ).
cnf(t1302,plain,
apply(X1,apply(sFlat0,X2)) = apply(apply(w,apply(sFlat5,X1)),X2),
inference(step,[status(thm)],[t298,t67]) ).
cnf(t697,plain,
apply(apply(w,apply(sFlat5,X1)),X2) = apply(X1,apply(sFlat0,X2)),
inference(orient,[status(thm)],[t1302]) ).
cnf(t699,plain,
apply(X1,apply(sFlat0,apply(sFlat5,X1))) = apply(sFlat0,apply(sFlat5,X1)),
inference(cp,[status(thm)],[t697,t56]) ).
cnf(t868,plain,
apply(X1,apply(sFlat0,apply(sFlat5,X1))) = apply(sFlat0,apply(sFlat5,X1)),
inference(orient,[status(thm)],[t699]) ).
cnf(t10,axiom,
sFlat6 = apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(b,b),b))),
introduced(definition) ).
cnf(t1254,plain,
sFlat6 = apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(w,b),b))),
inference(step,[status(thm)],[t10,t14]) ).
cnf(t1255,plain,
sFlat6 = apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(w,w),b))),
inference(step,[status(thm)],[t1254,t14]) ).
cnf(t20,plain,
apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(w,w),b))) = sFlat6,
inference(orient,[status(thm)],[t1255]) ).
cnf(t21,plain,
apply(apply(w,w),apply(apply(apply(b,w),apply(apply(w,w),b)),X1)) = apply(sFlat6,X1),
inference(cp,[status(thm)],[t13,t20]) ).
cnf(t1321,plain,
apply(sFlat0,apply(apply(apply(b,w),apply(apply(w,w),b)),X1)) = apply(sFlat6,X1),
inference(step,[status(thm)],[t21,t39]) ).
cnf(t1322,plain,
apply(sFlat0,apply(w,apply(apply(apply(w,w),b),X1))) = apply(sFlat6,X1),
inference(step,[status(thm)],[t1321,t13]) ).
cnf(t1323,plain,
apply(sFlat0,apply(w,apply(apply(sFlat0,b),X1))) = apply(sFlat6,X1),
inference(step,[status(thm)],[t1322,t39]) ).
cnf(t1324,plain,
apply(sFlat0,apply(w,apply(sFlat4,X1))) = apply(sFlat6,X1),
inference(step,[status(thm)],[t1323,t51]) ).
cnf(t1325,plain,
apply(sFlat0,apply(sFlat5,X1)) = apply(sFlat6,X1),
inference(step,[status(thm)],[t1324,t61]) ).
cnf(t1188,plain,
apply(sFlat0,apply(sFlat5,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t1325]) ).
cnf(t1326,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat0,apply(sFlat5,X1)),
inference(step,[status(thm)],[t868,t1188]) ).
cnf(t1327,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(step,[status(thm)],[t1326,t1188]) ).
cnf(t1196,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(rw,[status(thm)],[t1327]) ).
cnf(t1197,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t1196]) ).
cnf(t1238,plain,
true = ifeq(apply(sFlat6,fixed_pt),apply(sFlat6,fixed_pt),fixed_point(sFlat6),true),
inference(cp,[status(thm)],[t43,t1197]) ).
cnf(t0,plain,
ifeq(X1,X1,X2,X3) = X2,
introduced(definition) ).
cnf(t12,plain,
ifeq(X1,X1,X2,X3) = X2,
inference(orient,[status(thm)],[t0]) ).
cnf(t1328,plain,
true = fixed_point(sFlat6),
inference(step,[status(thm)],[t1238,t12]) ).
cnf(t1253,plain,
fixed_point(sFlat6) = true,
inference(orient,[status(thm)],[t1328]) ).
cnf(f3,negated_conjecture,
~ fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(b,b),b)))),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_strong_fixed_point) ).
fof(f3_nnf,plain,
~ fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(b,b),b)))),
inference(nnf_transformation,[status(thm)],[f3]) ).
fof(f3_sk,plain,
~ fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(b,b),b)))),
inference(skolemisation,[status(esa)],[f3_nnf]) ).
cnf(c3,plain,
~ fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(b,b),b)))),
inference(cnf_transformation,[status(esa)],[f3_sk]) ).
cnf(goal_0,negated_conjecture,
fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(b,b),b)))) != true,
inference(equality_encoding,[status(esa)],[c3]) ).
cnf(g0_0,plain,
fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(w,b),b)))) != true,
inference(rw,[status(thm)],[goal_0,t14]) ).
cnf(g0_1,plain,
fixed_point(apply(apply(b,apply(w,w)),apply(apply(b,w),apply(apply(w,w),b)))) != true,
inference(rw,[status(thm)],[g0_0,t14]) ).
cnf(g0_2,plain,
fixed_point(sFlat6) != true,
inference(rw,[status(thm)],[g0_1]) ).
cnf(g0_3,plain,
fixed_point(sFlat6) != fixed_point(sFlat6),
inference(rw,[status(thm)],[g0_2,t1253]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_3]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL003-2 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.08/0.54 % Computer : n020.cluster.edu
% 0.08/0.54 % Model : x86_64 x86_64
% 0.08/0.54 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.54 % Memory : 8046.5625MB
% 0.08/0.54 % OS : Linux 6.8.0-71-generic
% 0.08/0.54 % CPULimit : 300
% 0.08/0.54 % WCLimit : 300
% 0.08/0.54 % DateTime : Fri Sep 25 07:09:18 UTC 2026
% 0.08/0.54 % CPUTime :
% 0.08/0.54 Running run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 30.73/4.53 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p
% 30.73/4.53 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------