%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : COL003-5 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.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 01:01:35 PM UTC 2026
% Result : Unsatisfiable 25.32s 8.88s
% Output : Proof 25.32s
% Verified :
% 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(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(t8,axiom,
sFlat4 = apply(b,b),
introduced(definition) ).
cnf(t36,plain,
apply(b,b) = sFlat4,
inference(orient,[status(thm)],[t8]) ).
cnf(t37,plain,
apply(b,apply(X1,X2)) = apply(apply(sFlat4,X1),X2),
inference(cp,[status(thm)],[t13,t36]) ).
cnf(t77,plain,
apply(apply(sFlat4,X1),X2) = apply(b,apply(X1,X2)),
inference(orient,[status(thm)],[t37]) ).
cnf(t83,plain,
apply(apply(w,apply(sFlat4,X1)),X2) = apply(apply(b,apply(X1,X2)),X2),
inference(cp,[status(thm)],[t14,t77]) ).
cnf(t260,plain,
apply(apply(b,apply(X1,X2)),X2) = apply(apply(w,apply(sFlat4,X1)),X2),
inference(orient,[status(thm)],[t83]) ).
cnf(t4,axiom,
sFlat0 = apply(w,w),
introduced(definition) ).
cnf(t39,plain,
apply(w,w) = sFlat0,
inference(orient,[status(thm)],[t4]) ).
cnf(t277,plain,
apply(apply(w,apply(sFlat4,w)),w) = apply(apply(b,sFlat0),w),
inference(cp,[status(thm)],[t260,t39]) ).
cnf(t5,axiom,
sFlat1 = apply(b,apply(w,w)),
introduced(definition) ).
cnf(t32,plain,
apply(b,apply(w,w)) = sFlat1,
inference(orient,[status(thm)],[t5]) ).
cnf(t1096,plain,
apply(b,sFlat0) = sFlat1,
inference(step,[status(thm)],[t32,t39]) ).
cnf(t42,plain,
apply(b,sFlat0) = sFlat1,
inference(rw,[status(thm)],[t1096]) ).
cnf(t53,plain,
apply(b,sFlat0) = sFlat1,
inference(orient,[status(thm)],[t42]) ).
cnf(t1106,plain,
apply(apply(w,apply(sFlat4,w)),w) = apply(sFlat1,w),
inference(step,[status(thm)],[t277,t53]) ).
cnf(t6,axiom,
sFlat2 = apply(apply(b,apply(w,w)),w),
introduced(definition) ).
cnf(t23,plain,
apply(apply(b,apply(w,w)),w) = sFlat2,
inference(orient,[status(thm)],[t6]) ).
cnf(t1094,plain,
apply(sFlat1,w) = sFlat2,
inference(step,[status(thm)],[t23,t32]) ).
cnf(t35,plain,
apply(sFlat1,w) = sFlat2,
inference(rw,[status(thm)],[t1094]) ).
cnf(t50,plain,
apply(sFlat1,w) = sFlat2,
inference(orient,[status(thm)],[t35]) ).
cnf(t1107,plain,
apply(apply(w,apply(sFlat4,w)),w) = sFlat2,
inference(step,[status(thm)],[t1106,t50]) ).
cnf(t301,plain,
apply(apply(w,apply(sFlat4,w)),w) = sFlat2,
inference(orient,[status(thm)],[t1107]) ).
cnf(t7,axiom,
sFlat3 = apply(b,apply(apply(b,apply(w,w)),w)),
introduced(definition) ).
cnf(t1093,plain,
sFlat3 = apply(b,sFlat2),
inference(step,[status(thm)],[t7,t23]) ).
cnf(t29,plain,
apply(b,sFlat2) = sFlat3,
inference(orient,[status(thm)],[t1093]) ).
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(t57,plain,
apply(sFlat0,X1) = apply(apply(X1,X1),X1),
inference(cp,[status(thm)],[t56,t14]) ).
cnf(t64,plain,
apply(apply(X1,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t57]) ).
cnf(t9,axiom,
sFlat5 = apply(apply(b,b),b),
introduced(definition) ).
cnf(t1091,plain,
sFlat5 = apply(apply(w,b),b),
inference(step,[status(thm)],[t9,t14]) ).
cnf(t1092,plain,
sFlat5 = apply(apply(w,w),b),
inference(step,[status(thm)],[t1091,t14]) ).
cnf(t27,plain,
apply(apply(w,w),b) = sFlat5,
inference(orient,[status(thm)],[t1092]) ).
cnf(t1095,plain,
apply(sFlat0,b) = sFlat5,
inference(step,[status(thm)],[t27,t39]) ).
cnf(t41,plain,
apply(sFlat0,b) = sFlat5,
inference(rw,[status(thm)],[t1095]) ).
cnf(t51,plain,
apply(sFlat0,b) = sFlat5,
inference(orient,[status(thm)],[t41]) ).
cnf(t10,axiom,
sFlat6 = apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(b,b),b)),
introduced(definition) ).
cnf(t1088,plain,
sFlat6 = apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(w,b),b)),
inference(step,[status(thm)],[t10,t14]) ).
cnf(t1089,plain,
sFlat6 = apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(w,w),b)),
inference(step,[status(thm)],[t1088,t14]) ).
cnf(t20,plain,
apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(w,w),b)) = sFlat6,
inference(orient,[status(thm)],[t1089]) ).
cnf(t1090,plain,
apply(apply(b,sFlat2),apply(apply(w,w),b)) = sFlat6,
inference(step,[status(thm)],[t20,t23]) ).
cnf(t26,plain,
apply(apply(b,sFlat2),apply(apply(w,w),b)) = sFlat6,
inference(rw,[status(thm)],[t1090]) ).
cnf(t1109,plain,
apply(sFlat3,apply(apply(w,w),b)) = sFlat6,
inference(step,[status(thm)],[t26,t29]) ).
cnf(t1110,plain,
apply(sFlat3,apply(sFlat0,b)) = sFlat6,
inference(step,[status(thm)],[t1109,t39]) ).
cnf(t1111,plain,
apply(sFlat3,sFlat5) = sFlat6,
inference(step,[status(thm)],[t1110,t51]) ).
cnf(t329,plain,
apply(sFlat3,sFlat5) = sFlat6,
inference(orient,[status(thm)],[t1111]) ).
cnf(t11,axiom,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(b,b),b))),
introduced(definition) ).
cnf(t1097,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),apply(apply(b,b),b))),
inference(step,[status(thm)],[t11,t39]) ).
cnf(t1098,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),apply(apply(w,b),b))),
inference(step,[status(thm)],[t1097,t14]) ).
cnf(t1099,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),apply(apply(w,w),b))),
inference(step,[status(thm)],[t1098,t14]) ).
cnf(t1100,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),apply(sFlat0,b))),
inference(step,[status(thm)],[t1099,t39]) ).
cnf(t48,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),apply(sFlat0,b))) = sFlat7,
inference(orient,[status(thm)],[t1100]) ).
cnf(t1101,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),sFlat5)) = sFlat7,
inference(step,[status(thm)],[t48,t51]) ).
cnf(t52,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat0),w)),sFlat5)) = sFlat7,
inference(rw,[status(thm)],[t1101]) ).
cnf(t1114,plain,
fixed_point(apply(apply(b,apply(sFlat1,w)),sFlat5)) = sFlat7,
inference(step,[status(thm)],[t52,t53]) ).
cnf(t1115,plain,
fixed_point(apply(apply(b,sFlat2),sFlat5)) = sFlat7,
inference(step,[status(thm)],[t1114,t50]) ).
cnf(t1116,plain,
fixed_point(apply(sFlat3,sFlat5)) = sFlat7,
inference(step,[status(thm)],[t1115,t29]) ).
cnf(t1117,plain,
fixed_point(sFlat6) = sFlat7,
inference(step,[status(thm)],[t1116,t329]) ).
cnf(t569,plain,
fixed_point(sFlat6) = sFlat7,
inference(orient,[status(thm)],[t1117]) ).
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(t15,plain,
apply(apply(w,apply(b,X1)),X2) = apply(X1,apply(X2,X2)),
inference(cp,[status(thm)],[t14,t13]) ).
cnf(t136,plain,
apply(apply(w,apply(b,X1)),X2) = apply(X1,apply(X2,X2)),
inference(orient,[status(thm)],[t15]) ).
cnf(t153,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(w,apply(b,apply(b,X1))),X2),X3),
inference(cp,[status(thm)],[t13,t136]) ).
cnf(t38,plain,
apply(apply(w,b),b) = apply(sFlat4,b),
inference(cp,[status(thm)],[t14,t36]) ).
cnf(t1102,plain,
apply(sFlat0,b) = apply(sFlat4,b),
inference(step,[status(thm)],[t38,t56]) ).
cnf(t1103,plain,
sFlat5 = apply(sFlat4,b),
inference(step,[status(thm)],[t1102,t51]) ).
cnf(t63,plain,
apply(sFlat4,b) = sFlat5,
inference(orient,[status(thm)],[t1103]) ).
cnf(t79,plain,
apply(b,apply(b,X1)) = apply(sFlat5,X1),
inference(cp,[status(thm)],[t77,t63]) ).
cnf(t84,plain,
apply(b,apply(b,X1)) = apply(sFlat5,X1),
inference(orient,[status(thm)],[t79]) ).
cnf(t1121,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(w,apply(sFlat5,X1)),X2),X3),
inference(step,[status(thm)],[t153,t84]) ).
cnf(t675,plain,
apply(X1,apply(apply(X2,X2),X3)) = apply(apply(apply(w,apply(sFlat5,X1)),X2),X3),
inference(orient,[status(thm)],[t1121]) ).
cnf(t683,plain,
apply(X1,apply(apply(X2,X2),X2)) = apply(apply(w,apply(w,apply(sFlat5,X1))),X2),
inference(cp,[status(thm)],[t675,t14]) ).
cnf(t1124,plain,
apply(X1,apply(sFlat0,X2)) = apply(apply(w,apply(w,apply(sFlat5,X1))),X2),
inference(step,[status(thm)],[t683,t64]) ).
cnf(t993,plain,
apply(apply(w,apply(w,apply(sFlat5,X1))),X2) = apply(X1,apply(sFlat0,X2)),
inference(orient,[status(thm)],[t1124]) ).
cnf(t995,plain,
apply(X1,apply(sFlat0,apply(w,apply(sFlat5,X1)))) = apply(sFlat0,apply(w,apply(sFlat5,X1))),
inference(cp,[status(thm)],[t993,t56]) ).
cnf(t54,plain,
apply(sFlat0,apply(X1,X2)) = apply(apply(sFlat1,X1),X2),
inference(cp,[status(thm)],[t13,t53]) ).
cnf(t105,plain,
apply(apply(sFlat1,X1),X2) = apply(sFlat0,apply(X1,X2)),
inference(orient,[status(thm)],[t54]) ).
cnf(t107,plain,
apply(sFlat0,apply(w,X1)) = apply(sFlat2,X1),
inference(cp,[status(thm)],[t105,t50]) ).
cnf(t112,plain,
apply(sFlat0,apply(w,X1)) = apply(sFlat2,X1),
inference(orient,[status(thm)],[t107]) ).
cnf(t1125,plain,
apply(X1,apply(sFlat2,apply(sFlat5,X1))) = apply(sFlat0,apply(w,apply(sFlat5,X1))),
inference(step,[status(thm)],[t995,t112]) ).
cnf(t30,plain,
apply(sFlat2,apply(X1,X2)) = apply(apply(sFlat3,X1),X2),
inference(cp,[status(thm)],[t13,t29]) ).
cnf(t71,plain,
apply(apply(sFlat3,X1),X2) = apply(sFlat2,apply(X1,X2)),
inference(orient,[status(thm)],[t30]) ).
cnf(t330,plain,
apply(sFlat2,apply(sFlat5,X1)) = apply(sFlat6,X1),
inference(cp,[status(thm)],[t71,t329]) ).
cnf(t333,plain,
apply(sFlat2,apply(sFlat5,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t330]) ).
cnf(t1126,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat0,apply(w,apply(sFlat5,X1))),
inference(step,[status(thm)],[t1125,t333]) ).
cnf(t1127,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat2,apply(sFlat5,X1)),
inference(step,[status(thm)],[t1126,t112]) ).
cnf(t1128,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(step,[status(thm)],[t1127,t333]) ).
cnf(t1025,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t1128]) ).
cnf(t1059,plain,
true = ifeq(apply(sFlat6,fixed_pt),apply(sFlat6,fixed_pt),fixed_point(sFlat6),true),
inference(cp,[status(thm)],[t43,t1025]) ).
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(t1129,plain,
true = fixed_point(sFlat6),
inference(step,[status(thm)],[t1059,t12]) ).
cnf(t1130,plain,
true = sFlat7,
inference(step,[status(thm)],[t1129,t569]) ).
cnf(t1086,plain,
true = sFlat7,
inference(orient,[status(thm)],[t1130]) ).
cnf(f3,negated_conjecture,
~ fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),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(apply(b,apply(w,w)),w)),apply(apply(b,b),b))),
inference(nnf_transformation,[status(thm)],[f3]) ).
fof(f3_sk,plain,
~ fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(b,b),b))),
inference(skolemisation,[status(esa)],[f3_nnf]) ).
cnf(c3,plain,
~ fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(b,b),b))),
inference(cnf_transformation,[status(esa)],[f3_sk]) ).
cnf(goal_0,negated_conjecture,
fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),w)),apply(apply(b,b),b))) != true,
inference(equality_encoding,[status(esa)],[c3]) ).
cnf(g0_0,plain,
fixed_point(apply(apply(b,apply(apply(w,apply(sFlat4,w)),w)),apply(apply(b,b),b))) != true,
inference(rw,[status(thm)],[goal_0,t260]) ).
cnf(g0_1,plain,
fixed_point(apply(apply(b,sFlat2),apply(apply(b,b),b))) != true,
inference(rw,[status(thm)],[g0_0,t301]) ).
cnf(g0_2,plain,
fixed_point(apply(sFlat3,apply(apply(b,b),b))) != true,
inference(rw,[status(thm)],[g0_1,t29]) ).
cnf(g0_3,plain,
fixed_point(apply(sFlat3,apply(sFlat0,b))) != true,
inference(rw,[status(thm)],[g0_2,t64]) ).
cnf(g0_4,plain,
fixed_point(apply(sFlat3,sFlat5)) != true,
inference(rw,[status(thm)],[g0_3,t51]) ).
cnf(g0_5,plain,
fixed_point(sFlat6) != true,
inference(rw,[status(thm)],[g0_4,t329]) ).
cnf(g0_6,plain,
sFlat7 != true,
inference(rw,[status(thm)],[g0_5,t569]) ).
cnf(g0_7,plain,
sFlat7 != sFlat7,
inference(rw,[status(thm)],[g0_6,t1086]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_7]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : COL003-5 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.03 % Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.09/5.56 % Computer : n005.cluster.edu
% 0.09/5.56 % Model : x86_64 x86_64
% 0.09/5.56 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/5.56 % Memory : 8046.5625MB
% 0.09/5.56 % OS : Linux 6.8.0-71-generic
% 0.09/5.56 % CPULimit : 300
% 0.09/5.56 % WCLimit : 300
% 0.09/5.56 % DateTime : Fri Sep 25 07:09:22 UTC 2026
% 0.09/5.56 % CPUTime :
% 0.09/5.56 Running run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 25.32/8.88 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p
% 25.32/8.88 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------