%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL002-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 : 300s
% DateTime : Sat Jun 21 05:00:30 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.51s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COL002-2 : TPTP v9.0.0. Released v1.0.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n004.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
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 11:59:53 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.51 % SZS status Unsatisfiable
% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 tff(fixed_point_type, type, (
% 0.20/0.51 fixed_point: $i > $o)).
% 0.20/0.51 tff(apply_type, type, (
% 0.20/0.51 apply: ( $i * $i ) > $i)).
% 0.20/0.51 tff(i_type, type, (
% 0.20/0.51 i: $i)).
% 0.20/0.51 tff(fixed_pt_type, type, (
% 0.20/0.51 fixed_pt: $i)).
% 0.20/0.51 tff(b_type, type, (
% 0.20/0.51 b: $i)).
% 0.20/0.51 tff(s_type, type, (
% 0.20/0.51 s: $i)).
% 0.20/0.51 tff(1,plain,
% 0.20/0.51 (^[Z: $i, Y: $i, X: $i] : refl((apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z))) <=> (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(2,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.51 tff(3,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(4,axiom,(![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','b_definition')).
% 0.20/0.51 tff(5,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.51 tff(6,plain,(
% 0.20/0.51 ![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))),
% 0.20/0.51 inference(skolemize,[status(sab)],[5])).
% 0.20/0.51 tff(7,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.51 tff(8,plain,
% 0.20/0.51 ((~![Z: $i, Y: $i, X: $i] : (apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)))) | (apply(apply(apply(b, fixed_pt), apply(apply(s, apply(b, fixed_pt)), i)), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(9,plain,
% 0.20/0.51 (apply(apply(apply(b, fixed_pt), apply(apply(s, apply(b, fixed_pt)), i)), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)))),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.51 tff(10,plain,
% 0.20/0.51 (^[X: $i] : refl((apply(i, X) = X) <=> (apply(i, X) = X))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(11,plain,
% 0.20/0.51 (![X: $i] : (apply(i, X) = X) <=> ![X: $i] : (apply(i, X) = X)),
% 0.20/0.51 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.51 tff(12,plain,
% 0.20/0.51 (![X: $i] : (apply(i, X) = X) <=> ![X: $i] : (apply(i, X) = X)),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(13,axiom,(![X: $i] : (apply(i, X) = X)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','i_definition')).
% 0.20/0.51 tff(14,plain,
% 0.20/0.51 (![X: $i] : (apply(i, X) = X)),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.20/0.51 tff(15,plain,(
% 0.20/0.51 ![X: $i] : (apply(i, X) = X)),
% 0.20/0.51 inference(skolemize,[status(sab)],[14])).
% 0.20/0.51 tff(16,plain,
% 0.20/0.51 (![X: $i] : (apply(i, X) = X)),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.20/0.51 tff(17,plain,
% 0.20/0.51 ((~![X: $i] : (apply(i, X) = X)) | (apply(i, apply(apply(s, apply(b, fixed_pt)), i)) = apply(apply(s, apply(b, fixed_pt)), i))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(18,plain,
% 0.20/0.51 (apply(i, apply(apply(s, apply(b, fixed_pt)), i)) = apply(apply(s, apply(b, fixed_pt)), i)),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.51 tff(19,plain,
% 0.20/0.51 (apply(apply(apply(b, fixed_pt), apply(apply(s, apply(b, fixed_pt)), i)), apply(i, apply(apply(s, apply(b, fixed_pt)), i))) = apply(apply(apply(b, fixed_pt), apply(apply(s, apply(b, fixed_pt)), i)), apply(apply(s, apply(b, fixed_pt)), i))),
% 0.20/0.51 inference(monotonicity,[status(thm)],[18])).
% 0.20/0.51 tff(20,plain,
% 0.20/0.51 (^[Z: $i, Y: $i, X: $i] : refl((apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z))) <=> (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(21,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[20])).
% 0.20/0.51 tff(22,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(23,axiom,(![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','s_definition')).
% 0.20/0.51 tff(24,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.20/0.51 tff(25,plain,(
% 0.20/0.51 ![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))),
% 0.20/0.51 inference(skolemize,[status(sab)],[24])).
% 0.20/0.51 tff(26,plain,
% 0.20/0.51 (![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.20/0.51 tff(27,plain,
% 0.20/0.51 ((~![Z: $i, Y: $i, X: $i] : (apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)))) | (apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(apply(apply(b, fixed_pt), apply(apply(s, apply(b, fixed_pt)), i)), apply(i, apply(apply(s, apply(b, fixed_pt)), i))))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(28,plain,
% 0.20/0.51 (apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(apply(apply(b, fixed_pt), apply(apply(s, apply(b, fixed_pt)), i)), apply(i, apply(apply(s, apply(b, fixed_pt)), i)))),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.20/0.51 tff(29,plain,
% 0.20/0.51 (apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)))),
% 0.20/0.51 inference(transitivity,[status(thm)],[28, 19, 9])).
% 0.20/0.51 tff(30,assumption,(~fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)))), introduced(assumption)).
% 0.20/0.51 tff(31,plain,
% 0.20/0.51 (^[Weak_sage: $i] : refl((fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage)))) <=> (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage)))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(32,plain,
% 0.20/0.51 (![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage)))) <=> ![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[31])).
% 0.20/0.51 tff(33,plain,
% 0.20/0.51 (![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage)))) <=> ![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(34,plain,
% 0.20/0.51 (^[Weak_sage: $i] : rewrite(((~(Weak_sage = apply(fixed_pt, Weak_sage))) | fixed_point(Weak_sage)) <=> (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage)))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(35,plain,
% 0.20/0.51 (![Weak_sage: $i] : ((~(Weak_sage = apply(fixed_pt, Weak_sage))) | fixed_point(Weak_sage)) <=> ![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[34])).
% 0.20/0.51 tff(36,axiom,(![Weak_sage: $i] : ((~(Weak_sage = apply(fixed_pt, Weak_sage))) | fixed_point(Weak_sage))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','weak_fixed_point')).
% 0.20/0.51 tff(37,plain,
% 0.20/0.51 (![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.20/0.51 tff(38,plain,
% 0.20/0.51 (![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.20/0.51 tff(39,plain,(
% 0.20/0.51 ![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(skolemize,[status(sab)],[38])).
% 0.20/0.51 tff(40,plain,
% 0.20/0.51 (![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.20/0.51 tff(41,plain,
% 0.20/0.51 (((~![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))) | (fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))) | (~(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))))))) <=> ((~![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))) | fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))) | (~(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(42,plain,
% 0.20/0.51 ((~![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))) | (fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))) | (~(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))))))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(43,plain,
% 0.20/0.51 ((~![Weak_sage: $i] : (fixed_point(Weak_sage) | (~(Weak_sage = apply(fixed_pt, Weak_sage))))) | fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))) | (~(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)))))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.20/0.51 tff(44,plain,
% 0.20/0.51 (~(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)) = apply(fixed_pt, apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))))),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[43, 40, 30])).
% 0.20/0.51 tff(45,plain,
% 0.20/0.51 ($false),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[44, 29])).
% 0.20/0.51 tff(46,plain,(fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i)))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.51 tff(47,plain,
% 0.20/0.51 (^[X: $i] : refl((~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i)))) <=> (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i)))))),
% 0.20/0.51 inference(bind,[status(th)],[])).
% 0.20/0.51 tff(48,plain,
% 0.20/0.51 (![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i)))) <=> ![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))),
% 0.20/0.51 inference(quant_intro,[status(thm)],[47])).
% 0.20/0.51 tff(49,plain,
% 0.20/0.51 (![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i)))) <=> ![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))),
% 0.20/0.51 inference(rewrite,[status(thm)],[])).
% 0.20/0.51 tff(50,axiom,(![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_weak_fixed_point')).
% 0.20/0.51 tff(51,plain,
% 0.20/0.51 (![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.20/0.51 tff(52,plain,(
% 0.20/0.51 ![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))),
% 0.20/0.51 inference(skolemize,[status(sab)],[51])).
% 0.20/0.51 tff(53,plain,
% 0.20/0.51 (![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))),
% 0.20/0.51 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.20/0.51 tff(54,plain,
% 0.20/0.51 ((~![X: $i] : (~fixed_point(apply(apply(apply(s, apply(b, X)), i), apply(apply(s, apply(b, X)), i))))) | (~fixed_point(apply(apply(apply(s, apply(b, fixed_pt)), i), apply(apply(s, apply(b, fixed_pt)), i))))),
% 0.20/0.51 inference(quant_inst,[status(thm)],[])).
% 0.20/0.51 tff(55,plain,
% 0.20/0.51 ($false),
% 0.20/0.51 inference(unit_resolution,[status(thm)],[54, 53, 46])).
% 0.20/0.51 % SZS output end Proof
% 0.20/0.52 % E exiting
%------------------------------------------------------------------------------