%------------------------------------------------------------------------------
% File : FindProof---0.1
% Problem : COL003-6 : 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 01:01:36 PM UTC 2026
% Result : Unsatisfiable 28.76s 4.14s
% Output : Proof 28.76s
% Verified :
% Comments :
%------------------------------------------------------------------------------
cnf(t11,axiom,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,apply(apply(b,apply(w,w)),w)),b)),b)),
introduced(definition) ).
cnf(t4,axiom,
sFlat0 = apply(w,w),
introduced(definition) ).
cnf(t32,plain,
apply(w,w) = sFlat0,
inference(orient,[status(thm)],[t4]) ).
cnf(t27642,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,apply(apply(b,sFlat0),w)),b)),b)),
inference(step,[status(thm)],[t11,t32]) ).
cnf(t5,axiom,
sFlat1 = apply(b,apply(w,w)),
introduced(definition) ).
cnf(t27640,plain,
sFlat1 = apply(b,sFlat0),
inference(step,[status(thm)],[t5,t32]) ).
cnf(t42,plain,
apply(b,sFlat0) = sFlat1,
inference(orient,[status(thm)],[t27640]) ).
cnf(t27643,plain,
sFlat7 = fixed_point(apply(apply(b,apply(apply(b,apply(sFlat1,w)),b)),b)),
inference(step,[status(thm)],[t27642,t42]) ).
cnf(t53,plain,
fixed_point(apply(apply(b,apply(apply(b,apply(sFlat1,w)),b)),b)) = sFlat7,
inference(orient,[status(thm)],[t27643]) ).
cnf(t6,axiom,
sFlat2 = apply(apply(b,apply(w,w)),w),
introduced(definition) ).
cnf(t28,plain,
apply(apply(b,apply(w,w)),w) = sFlat2,
inference(orient,[status(thm)],[t6]) ).
cnf(t27636,plain,
apply(apply(b,sFlat0),w) = sFlat2,
inference(step,[status(thm)],[t28,t32]) ).
cnf(t34,plain,
apply(apply(b,sFlat0),w) = sFlat2,
inference(rw,[status(thm)],[t27636]) ).
cnf(t27644,plain,
apply(sFlat1,w) = sFlat2,
inference(step,[status(thm)],[t34,t42]) ).
cnf(t61,plain,
apply(sFlat1,w) = sFlat2,
inference(orient,[status(thm)],[t27644]) ).
cnf(t27645,plain,
fixed_point(apply(apply(b,apply(apply(b,sFlat2),b)),b)) = sFlat7,
inference(step,[status(thm)],[t53,t61]) ).
cnf(t8,axiom,
sFlat4 = apply(apply(b,apply(apply(b,apply(w,w)),w)),b),
introduced(definition) ).
cnf(t24,plain,
apply(apply(b,apply(apply(b,apply(w,w)),w)),b) = sFlat4,
inference(orient,[status(thm)],[t8]) ).
cnf(t27635,plain,
apply(apply(b,sFlat2),b) = sFlat4,
inference(step,[status(thm)],[t24,t28]) ).
cnf(t31,plain,
apply(apply(b,sFlat2),b) = sFlat4,
inference(rw,[status(thm)],[t27635]) ).
cnf(t58,plain,
apply(apply(b,sFlat2),b) = sFlat4,
inference(orient,[status(thm)],[t31]) ).
cnf(t27646,plain,
fixed_point(apply(apply(b,sFlat4),b)) = sFlat7,
inference(step,[status(thm)],[t27645,t58]) ).
cnf(t10,axiom,
sFlat6 = apply(apply(b,apply(apply(b,apply(apply(b,apply(w,w)),w)),b)),b),
introduced(definition) ).
cnf(t21,plain,
apply(apply(b,apply(apply(b,apply(apply(b,apply(w,w)),w)),b)),b) = sFlat6,
inference(orient,[status(thm)],[t10]) ).
cnf(t27634,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)],[t27634]) ).
cnf(t55,plain,
apply(apply(b,sFlat4),b) = sFlat6,
inference(orient,[status(thm)],[t27]) ).
cnf(t27647,plain,
fixed_point(sFlat6) = sFlat7,
inference(step,[status(thm)],[t27646,t55]) ).
cnf(t62,plain,
fixed_point(sFlat6) = sFlat7,
inference(rw,[status(thm)],[t27647]) ).
cnf(t63,plain,
fixed_point(sFlat6) = sFlat7,
inference(orient,[status(thm)],[t62]) ).
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(t46,plain,
ifeq(apply(X1,fixed_pt),apply(fixed_pt,apply(X1,fixed_pt)),fixed_point(X1),true) = true,
inference(orient,[status(thm)],[t3]) ).
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(t33,plain,
apply(apply(w,X1),X1) = apply(sFlat0,X1),
inference(cp,[status(thm)],[t14,t32]) ).
cnf(t79,plain,
apply(apply(w,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t33]) ).
cnf(t81,plain,
true = ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(apply(w,fixed_pt),fixed_pt)),fixed_point(apply(w,fixed_pt)),true),
inference(cp,[status(thm)],[t46,t79]) ).
cnf(t27756,plain,
true = ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(w,fixed_pt)),true),
inference(step,[status(thm)],[t81,t79]) ).
cnf(t6360,plain,
ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(w,fixed_pt)),true) = true,
inference(orient,[status(thm)],[t27756]) ).
cnf(t80,plain,
apply(sFlat0,X1) = apply(apply(X1,X1),X1),
inference(cp,[status(thm)],[t79,t14]) ).
cnf(t90,plain,
apply(apply(X1,X1),X1) = apply(sFlat0,X1),
inference(orient,[status(thm)],[t80]) ).
cnf(t94,plain,
true = ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(apply(fixed_pt,fixed_pt),fixed_pt)),fixed_point(apply(fixed_pt,fixed_pt)),true),
inference(cp,[status(thm)],[t46,t90]) ).
cnf(t27757,plain,
true = ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(fixed_pt,fixed_pt)),true),
inference(step,[status(thm)],[t94,t90]) ).
cnf(t6361,plain,
ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(fixed_pt,fixed_pt)),true) = true,
inference(orient,[status(thm)],[t27757]) ).
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(t220,plain,
apply(apply(w,apply(b,X1)),X2) = apply(X1,apply(X2,X2)),
inference(orient,[status(thm)],[t15]) ).
cnf(t229,plain,
apply(apply(w,apply(w,apply(b,X1))),X2) = apply(apply(X1,apply(X2,X2)),X2),
inference(cp,[status(thm)],[t14,t220]) ).
cnf(t25280,plain,
apply(apply(w,apply(w,apply(b,X1))),X2) = apply(apply(X1,apply(X2,X2)),X2),
inference(orient,[status(thm)],[t229]) ).
cnf(t25367,plain,
apply(apply(w,apply(w,apply(b,apply(b,X1)))),X2) = apply(X1,apply(apply(X2,X2),X2)),
inference(cp,[status(thm)],[t25280,t13]) ).
cnf(t28002,plain,
apply(apply(w,apply(w,apply(b,apply(b,X1)))),X2) = apply(X1,apply(sFlat0,X2)),
inference(step,[status(thm)],[t25367,t90]) ).
cnf(t27227,plain,
apply(apply(w,apply(w,apply(b,apply(b,X1)))),X2) = apply(X1,apply(sFlat0,X2)),
inference(orient,[status(thm)],[t28002]) ).
cnf(t27230,plain,
apply(X1,apply(sFlat0,apply(w,apply(b,apply(b,X1))))) = apply(sFlat0,apply(w,apply(b,apply(b,X1)))),
inference(cp,[status(thm)],[t27227,t79]) ).
cnf(t43,plain,
apply(sFlat0,apply(X1,X2)) = apply(apply(sFlat1,X1),X2),
inference(cp,[status(thm)],[t13,t42]) ).
cnf(t97,plain,
apply(apply(sFlat1,X1),X2) = apply(sFlat0,apply(X1,X2)),
inference(orient,[status(thm)],[t43]) ).
cnf(t100,plain,
apply(sFlat0,apply(w,X1)) = apply(sFlat2,X1),
inference(cp,[status(thm)],[t97,t61]) ).
cnf(t107,plain,
apply(sFlat0,apply(w,X1)) = apply(sFlat2,X1),
inference(orient,[status(thm)],[t100]) ).
cnf(t28004,plain,
apply(X1,apply(sFlat2,apply(b,apply(b,X1)))) = apply(sFlat0,apply(w,apply(b,apply(b,X1)))),
inference(step,[status(thm)],[t27230,t107]) ).
cnf(t7,axiom,
sFlat3 = apply(b,apply(apply(b,apply(w,w)),w)),
introduced(definition) ).
cnf(t27638,plain,
sFlat3 = apply(b,apply(apply(b,sFlat0),w)),
inference(step,[status(thm)],[t7,t32]) ).
cnf(t38,plain,
apply(b,apply(apply(b,sFlat0),w)) = sFlat3,
inference(orient,[status(thm)],[t27638]) ).
cnf(t27641,plain,
apply(b,apply(sFlat1,w)) = sFlat3,
inference(step,[status(thm)],[t38,t42]) ).
cnf(t45,plain,
apply(b,apply(sFlat1,w)) = sFlat3,
inference(rw,[status(thm)],[t27641]) ).
cnf(t27648,plain,
apply(b,sFlat2) = sFlat3,
inference(step,[status(thm)],[t45,t61]) ).
cnf(t68,plain,
apply(b,sFlat2) = sFlat3,
inference(orient,[status(thm)],[t27648]) ).
cnf(t69,plain,
apply(sFlat2,apply(X1,X2)) = apply(apply(sFlat3,X1),X2),
inference(cp,[status(thm)],[t13,t68]) ).
cnf(t111,plain,
apply(apply(sFlat3,X1),X2) = apply(sFlat2,apply(X1,X2)),
inference(orient,[status(thm)],[t69]) ).
cnf(t27649,plain,
apply(sFlat3,b) = sFlat4,
inference(step,[status(thm)],[t58,t68]) ).
cnf(t71,plain,
apply(sFlat3,b) = sFlat4,
inference(rw,[status(thm)],[t27649]) ).
cnf(t72,plain,
apply(sFlat3,b) = sFlat4,
inference(orient,[status(thm)],[t71]) ).
cnf(t114,plain,
apply(sFlat2,apply(b,X1)) = apply(sFlat4,X1),
inference(cp,[status(thm)],[t111,t72]) ).
cnf(t121,plain,
apply(sFlat2,apply(b,X1)) = apply(sFlat4,X1),
inference(orient,[status(thm)],[t114]) ).
cnf(t28005,plain,
apply(X1,apply(sFlat4,apply(b,X1))) = apply(sFlat0,apply(w,apply(b,apply(b,X1)))),
inference(step,[status(thm)],[t28004,t121]) ).
cnf(t9,axiom,
sFlat5 = apply(b,apply(apply(b,apply(apply(b,apply(w,w)),w)),b)),
introduced(definition) ).
cnf(t27637,plain,
sFlat5 = apply(b,apply(apply(b,apply(apply(b,sFlat0),w)),b)),
inference(step,[status(thm)],[t9,t32]) ).
cnf(t35,plain,
apply(b,apply(apply(b,apply(apply(b,sFlat0),w)),b)) = sFlat5,
inference(orient,[status(thm)],[t27637]) ).
cnf(t27639,plain,
apply(b,apply(sFlat3,b)) = sFlat5,
inference(step,[status(thm)],[t35,t38]) ).
cnf(t41,plain,
apply(b,apply(sFlat3,b)) = sFlat5,
inference(rw,[status(thm)],[t27639]) ).
cnf(t65,plain,
apply(b,apply(sFlat3,b)) = sFlat5,
inference(orient,[status(thm)],[t41]) ).
cnf(t27650,plain,
apply(b,sFlat4) = sFlat5,
inference(step,[status(thm)],[t65,t72]) ).
cnf(t73,plain,
apply(b,sFlat4) = sFlat5,
inference(rw,[status(thm)],[t27650]) ).
cnf(t74,plain,
apply(b,sFlat4) = sFlat5,
inference(orient,[status(thm)],[t73]) ).
cnf(t75,plain,
apply(sFlat4,apply(X1,X2)) = apply(apply(sFlat5,X1),X2),
inference(cp,[status(thm)],[t13,t74]) ).
cnf(t132,plain,
apply(apply(sFlat5,X1),X2) = apply(sFlat4,apply(X1,X2)),
inference(orient,[status(thm)],[t75]) ).
cnf(t27651,plain,
apply(sFlat5,b) = sFlat6,
inference(step,[status(thm)],[t55,t74]) ).
cnf(t77,plain,
apply(sFlat5,b) = sFlat6,
inference(rw,[status(thm)],[t27651]) ).
cnf(t78,plain,
apply(sFlat5,b) = sFlat6,
inference(orient,[status(thm)],[t77]) ).
cnf(t135,plain,
apply(sFlat4,apply(b,X1)) = apply(sFlat6,X1),
inference(cp,[status(thm)],[t132,t78]) ).
cnf(t142,plain,
apply(sFlat4,apply(b,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t135]) ).
cnf(t28006,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat0,apply(w,apply(b,apply(b,X1)))),
inference(step,[status(thm)],[t28005,t142]) ).
cnf(t28007,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat2,apply(b,apply(b,X1))),
inference(step,[status(thm)],[t28006,t107]) ).
cnf(t28008,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat4,apply(b,X1)),
inference(step,[status(thm)],[t28007,t121]) ).
cnf(t28009,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(step,[status(thm)],[t28008,t142]) ).
cnf(t27392,plain,
apply(X1,apply(sFlat6,X1)) = apply(sFlat6,X1),
inference(orient,[status(thm)],[t28009]) ).
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(apply(b,apply(w,w)),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(apply(b,apply(w,w)),w)),b)),b)),
inference(nnf_transformation,[status(thm)],[f3]) ).
fof(f3_sk,plain,
~ fixed_point(apply(apply(b,apply(apply(b,apply(apply(b,apply(w,w)),w)),b)),b)),
inference(skolemisation,[status(esa)],[f3_nnf]) ).
cnf(c3,plain,
~ fixed_point(apply(apply(b,apply(apply(b,apply(apply(b,apply(w,w)),w)),b)),b)),
inference(cnf_transformation,[status(esa)],[f3_sk]) ).
cnf(goal_0,negated_conjecture,
fixed_point(apply(apply(b,apply(apply(b,apply(apply(b,apply(w,w)),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,t63]) ).
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(fixed_pt,apply(sFlat6,fixed_pt)),sFlat7,ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(w,fixed_pt)),true)),
inference(rw,[status(thm)],[g0_2,t6360]) ).
cnf(g0_4,plain,
sFlat7 != ifeq(apply(sFlat6,fixed_pt),apply(fixed_pt,apply(sFlat6,fixed_pt)),sFlat7,ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(w,fixed_pt)),ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(fixed_pt,fixed_pt)),true))),
inference(rw,[status(thm)],[g0_3,t6361]) ).
cnf(g0_5,plain,
sFlat7 != ifeq(apply(sFlat6,fixed_pt),apply(sFlat6,fixed_pt),sFlat7,ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(w,fixed_pt)),ifeq(apply(sFlat0,fixed_pt),apply(fixed_pt,apply(sFlat0,fixed_pt)),fixed_point(apply(fixed_pt,fixed_pt)),true))),
inference(rw,[status(thm)],[g0_4,t27392]) ).
cnf(g0_6,plain,
sFlat7 != sFlat7,
inference(rw,[status(thm)],[g0_5,t12]) ).
cnf(contradiction_0,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[g0_6]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : COL003-6 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.10/0.37 % Computer : n020.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Fri Sep 25 07:10:19 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_findproof /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 28.76/4.14 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p
% 28.76/4.14 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
%------------------------------------------------------------------------------