%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : COL003-4 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n004.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 01:01:34 PM UTC 2026
% Result : Unsatisfiable 8.37s 6.71s
% Output : Proof 8.37s
% Verified :
% Comments :
%------------------------------------------------------------------------------
cnf(t11,axiom,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),b))),b)),
introduced(definition) ).
cnf(t4,axiom,
sFlat0 = apply(w,w),
introduced(definition) ).
cnf(t43,plain,
apply(w,w) = sFlat0,
inference(orient,[status(thm)],[t4]) ).
cnf(t11406,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,sFlat0),apply(apply(b,w),b))),b)),
inference(step,[status(thm)],[t11,t43]) ).
cnf(t6,axiom,
sFlat2 = apply(b,w),
introduced(definition) ).
cnf(t39,plain,
apply(b,w) = sFlat2,
inference(orient,[status(thm)],[t6]) ).
cnf(t11407,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,sFlat0),apply(sFlat2,b))),b)),
inference(step,[status(thm)],[t11406,t39]) ).
cnf(t53,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat0),apply(sFlat2,b))),b)) = sFlat7,
inference(orient,[status(thm)],[t11407]) ).
cnf(t7,axiom,
sFlat3 = apply(apply(b,w),b),
introduced(definition) ).
cnf(t28,plain,
apply(apply(b,w),b) = sFlat3,
inference(orient,[status(thm)],[t7]) ).
cnf(t11404,plain,
apply(sFlat2,b) = sFlat3,
inference(step,[status(thm)],[t28,t39]) ).
cnf(t42,plain,
apply(sFlat2,b) = sFlat3,
inference(rw,[status(thm)],[t11404]) ).
cnf(t55,plain,
apply(sFlat2,b) = sFlat3,
inference(orient,[status(thm)],[t42]) ).
cnf(t11408,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat0),sFlat3)),b)) = sFlat7,
inference(step,[status(thm)],[t53,t55]) ).
cnf(t56,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat0),sFlat3)),b)) = sFlat7,
inference(rw,[status(thm)],[t11408]) ).
cnf(t5,axiom,
sFlat1 = apply(b,apply(w,w)),
introduced(definition) ).
cnf(t35,plain,
apply(b,apply(w,w)) = sFlat1,
inference(orient,[status(thm)],[t5]) ).
cnf(t11405,plain,
apply(b,sFlat0) = sFlat1,
inference(step,[status(thm)],[t35,t43]) ).
cnf(t45,plain,
apply(b,sFlat0) = sFlat1,
inference(rw,[status(thm)],[t11405]) ).
cnf(t57,plain,
apply(b,sFlat0) = sFlat1,
inference(orient,[status(thm)],[t45]) ).
cnf(t11440,plain,
fixed_point(apply(apply(b,apply(sFlat1,sFlat3)),b)) = sFlat7,
inference(step,[status(thm)],[t56,t57]) ).
cnf(t8,axiom,
sFlat4 = apply(apply(b,apply(w,w)),apply(apply(b,w),b)),
introduced(definition) ).
cnf(t24,plain,
apply(apply(b,apply(w,w)),apply(apply(b,w),b)) = sFlat4,
inference(orient,[status(thm)],[t8]) ).
cnf(t11401,plain,
apply(apply(b,apply(w,w)),sFlat3) = sFlat4,
inference(step,[status(thm)],[t24,t28]) ).
cnf(t31,plain,
apply(apply(b,apply(w,w)),sFlat3) = sFlat4,
inference(rw,[status(thm)],[t11401]) ).
cnf(t11409,plain,
apply(apply(b,sFlat0),sFlat3) = sFlat4,
inference(step,[status(thm)],[t31,t43]) ).
cnf(t11410,plain,
apply(sFlat1,sFlat3) = sFlat4,
inference(step,[status(thm)],[t11409,t57]) ).
cnf(t134,plain,
apply(sFlat1,sFlat3) = sFlat4,
inference(orient,[status(thm)],[t11410]) ).
cnf(t11441,plain,
fixed_point(apply(apply(b,sFlat4),b)) = sFlat7,
inference(step,[status(thm)],[t11440,t134]) ).
cnf(t9,axiom,
sFlat5 = apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),b))),
introduced(definition) ).
cnf(t11402,plain,
sFlat5 = apply(b,apply(apply(b,apply(w,w)),sFlat3)),
inference(step,[status(thm)],[t9,t28]) ).
cnf(t32,plain,
apply(b,apply(apply(b,apply(w,w)),sFlat3)) = sFlat5,
inference(orient,[status(thm)],[t11402]) ).
cnf(t11403,plain,
apply(b,apply(sFlat1,sFlat3)) = sFlat5,
inference(step,[status(thm)],[t32,t35]) ).
cnf(t38,plain,
apply(b,apply(sFlat1,sFlat3)) = sFlat5,
inference(rw,[status(thm)],[t11403]) ).
cnf(t63,plain,
apply(b,apply(sFlat1,sFlat3)) = sFlat5,
inference(orient,[status(thm)],[t38]) ).
cnf(t11413,plain,
apply(b,sFlat4) = sFlat5,
inference(step,[status(thm)],[t63,t134]) ).
cnf(t138,plain,
apply(b,sFlat4) = sFlat5,
inference(rw,[status(thm)],[t11413]) ).
cnf(t139,plain,
apply(b,sFlat4) = sFlat5,
inference(orient,[status(thm)],[t138]) ).
cnf(t11442,plain,
fixed_point(apply(sFlat5,b)) = sFlat7,
inference(step,[status(thm)],[t11441,t139]) ).
cnf(t10,axiom,
sFlat6 = apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),b))),b),
introduced(definition) ).
cnf(t21,plain,
apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),b))),b) = sFlat6,
inference(orient,[status(thm)],[t10]) ).
cnf(t11400,plain,
apply(apply(b,sFlat4),b) = sFlat6,
inference(step,[status(thm)],[t21,t24]) ).
cnf(t27,plain,
apply(apply(b,sFlat4),b) = sFlat6,
inference(rw,[status(thm)],[t11400]) ).
cnf(t60,plain,
apply(apply(b,sFlat4),b) = sFlat6,
inference(orient,[status(thm)],[t27]) ).
cnf(t11414,plain,
apply(sFlat5,b) = sFlat6,
inference(step,[status(thm)],[t60,t139]) ).
cnf(t144,plain,
apply(sFlat5,b) = sFlat6,
inference(rw,[status(thm)],[t11414]) ).
cnf(t145,plain,
apply(sFlat5,b) = sFlat6,
inference(orient,[status(thm)],[t144]) ).
cnf(t11443,plain,
fixed_point(sFlat6) = sFlat7,
inference(step,[status(thm)],[t11442,t145]) ).
cnf(t480,plain,
fixed_point(sFlat6) = sFlat7,
inference(orient,[status(thm)],[t11443]) ).
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(t15,plain,
apply(apply(w,apply(b,X1)),X2) = apply(X1,apply(X2,X2)),
inference(cp,[status(thm)],[t14,t13]) ).
cnf(t40,plain,
apply(w,apply(X1,X2)) = apply(apply(sFlat2,X1),X2),
inference(cp,[status(thm)],[t13,t39]) ).
cnf(t91,plain,
apply(apply(sFlat2,X1),X2) = apply(w,apply(X1,X2)),
inference(orient,[status(thm)],[t40]) ).
cnf(t94,plain,
apply(w,apply(b,X1)) = apply(sFlat3,X1),
inference(cp,[status(thm)],[t91,t55]) ).
cnf(t102,plain,
apply(w,apply(b,X1)) = apply(sFlat3,X1),
inference(orient,[status(thm)],[t94]) ).
cnf(t11420,plain,
apply(apply(sFlat3,X1),X2) = apply(X1,apply(X2,X2)),
inference(step,[status(thm)],[t15,t102]) ).
cnf(t203,plain,
apply(apply(sFlat3,X1),X2) = apply(X1,apply(X2,X2)),
inference(orient,[status(thm)],[t11420]) ).
cnf(t208,plain,
apply(apply(w,apply(sFlat3,X1)),X2) = apply(apply(X1,apply(X2,X2)),X2),
inference(cp,[status(thm)],[t14,t203]) ).
cnf(t10342,plain,
apply(apply(w,apply(sFlat3,X1)),X2) = apply(apply(X1,apply(X2,X2)),X2),
inference(orient,[status(thm)],[t208]) ).
cnf(t10395,plain,
apply(apply(w,apply(sFlat3,apply(b,X1))),X2) = apply(X1,apply(apply(X2,X2),X2)),
inference(cp,[status(thm)],[t10342,t13]) ).
cnf(t44,plain,
apply(apply(w,X1),X1) = apply(sFlat0,X1),
inference(cp,[status(thm)],[t14,t43]) ).
cnf(t68,plain,
apply(apply(w,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t44]) ).
cnf(t69,plain,
apply(sFlat0,X1) = apply(apply(X1,X1),X1),
inference(cp,[status(thm)],[t68,t14]) ).
cnf(t82,plain,
apply(apply(X1,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t69]) ).
cnf(t11769,plain,
apply(apply(w,apply(sFlat3,apply(b,X1))),X2) = apply(X1,apply(sFlat0,X2)),
inference(step,[status(thm)],[t10395,t82]) ).
cnf(t11133,plain,
apply(apply(w,apply(sFlat3,apply(b,X1))),X2) = apply(X1,apply(sFlat0,X2)),
inference(orient,[status(thm)],[t11769]) ).
cnf(t11135,plain,
apply(X1,apply(sFlat0,apply(sFlat3,apply(b,X1)))) = apply(sFlat0,apply(sFlat3,apply(b,X1))),
inference(cp,[status(thm)],[t11133,t68]) ).
cnf(t58,plain,
apply(sFlat0,apply(X1,X2)) = apply(apply(sFlat1,X1),X2),
inference(cp,[status(thm)],[t13,t57]) ).
cnf(t120,plain,
apply(apply(sFlat1,X1),X2) = apply(sFlat0,apply(X1,X2)),
inference(orient,[status(thm)],[t58]) ).
cnf(t135,plain,
apply(sFlat0,apply(sFlat3,X1)) = apply(sFlat4,X1),
inference(cp,[status(thm)],[t120,t134]) ).
cnf(t150,plain,
apply(sFlat0,apply(sFlat3,X1)) = apply(sFlat4,X1),
inference(orient,[status(thm)],[t135]) ).
cnf(t11770,plain,
apply(X1,apply(sFlat4,apply(b,X1))) = apply(sFlat0,apply(sFlat3,apply(b,X1))),
inference(step,[status(thm)],[t11135,t150]) ).
cnf(t61,plain,
apply(sFlat4,apply(b,X1)) = apply(sFlat6,X1),
inference(cp,[status(thm)],[t13,t60]) ).
cnf(t75,plain,
apply(sFlat4,apply(b,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t61]) ).
cnf(t11771,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat0,apply(sFlat3,apply(b,X1))),
inference(step,[status(thm)],[t11770,t75]) ).
cnf(t11772,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat4,apply(b,X1)),
inference(step,[status(thm)],[t11771,t150]) ).
cnf(t11773,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(step,[status(thm)],[t11772,t75]) ).
cnf(t11213,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t11773]) ).
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(f3,negated_conjecture,
~ fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),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)),apply(apply(b,w),b))),b)),
inference(nnf_transformation,[status(thm)],[f3]) ).
fof(f3_sk,plain,
~ fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),b))),b)),
inference(skolemisation,[status(esa)],[f3_nnf]) ).
cnf(c3,plain,
~ fixed_point(apply(apply(b,apply(apply(b,apply(w,w)),apply(apply(b,w),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)),apply(apply(b,w),b))),b)) != true,
inference(equality_encoding,[status(esa)],[c3]) ).
cnf(g0_0,plain,
fixed_point(sFlat6) != true,
inference(rw,[status(thm)],[goal_0]) ).
cnf(g0_1,plain,
sFlat7 != true,
inference(rw,[status(thm)],[g0_0,t480]) ).
cnf(g0_2,plain,
sFlat7 != ifeq(apply(sFlat6,fixed_pt),apply(fixed_pt,apply(sFlat6,fixed_pt)),sFlat7,true),
inference(rw,[status(thm)],[g0_1]) ).
cnf(g0_3,plain,
sFlat7 != ifeq(apply(sFlat6,fixed_pt),apply(sFlat6,fixed_pt),sFlat7,true),
inference(rw,[status(thm)],[g0_2,t11213]) ).
cnf(g0_4,plain,
sFlat7 != sFlat7,
inference(rw,[status(thm)],[g0_3,t12]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_4]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : COL003-4 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.09/5.36 % Computer : n004.cluster.edu
% 0.09/5.36 % Model : x86_64 x86_64
% 0.09/5.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/5.36 % Memory : 8046.5625MB
% 0.09/5.36 % OS : Linux 6.8.0-71-generic
% 0.09/5.36 % CPULimit : 300
% 0.09/5.36 % WCLimit : 300
% 0.09/5.36 % DateTime : Fri Sep 25 07:09:27 UTC 2026
% 0.09/5.37 % CPUTime :
% 0.09/5.37 Running run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 8.37/6.71 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p
% 8.37/6.71 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------