%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : COL074-3 : TPTP v9.0.0. Released v1.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:00:44 AM UTC 2025
% Result : Unsatisfiable 4.08s 4.26s
% Output : Proof 4.08s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : COL074-3 : TPTP v9.0.0. Released v1.2.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 12:11:39 EDT 2025
% 0.12/0.34 % CPUTime :
% 4.08/4.26 % SZS status Unsatisfiable
% 4.08/4.26 % SZS output start Proof
% 4.08/4.26 tff(apply_type, type, (
% 4.08/4.26 apply: ( $i * $i ) > $i)).
% 4.08/4.26 tff(pair_type, type, (
% 4.08/4.26 pair: ( $i * $i ) > $i)).
% 4.08/4.26 tff(projection2_type, type, (
% 4.08/4.26 projection2: $i)).
% 4.08/4.26 tff(k_type, type, (
% 4.08/4.26 k: $i)).
% 4.08/4.26 tff(s_type, type, (
% 4.08/4.26 s: $i)).
% 4.08/4.26 tff(eq_type, type, (
% 4.08/4.26 eq: $i)).
% 4.08/4.26 tff(k_projection2_type, type, (
% 4.08/4.26 k_projection2: $i)).
% 4.08/4.26 tff(projection1_type, type, (
% 4.08/4.26 projection1: $i)).
% 4.08/4.26 tff(1,plain,
% 4.08/4.26 (^[Y: $i, X: $i] : refl((apply(apply(k, X), Y) = X) <=> (apply(apply(k, X), Y) = X))),
% 4.08/4.26 inference(bind,[status(th)],[])).
% 4.08/4.26 tff(2,plain,
% 4.08/4.26 (![Y: $i, X: $i] : (apply(apply(k, X), Y) = X) <=> ![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)),
% 4.08/4.26 inference(quant_intro,[status(thm)],[1])).
% 4.08/4.26 tff(3,plain,
% 4.08/4.26 (![Y: $i, X: $i] : (apply(apply(k, X), Y) = X) <=> ![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(4,axiom,(![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','k_definition')).
% 4.08/4.26 tff(5,plain,
% 4.08/4.26 (![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[4, 3])).
% 4.08/4.26 tff(6,plain,(
% 4.08/4.26 ![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)),
% 4.08/4.26 inference(skolemize,[status(sab)],[5])).
% 4.08/4.26 tff(7,plain,
% 4.08/4.26 (![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[6, 2])).
% 4.08/4.26 tff(8,plain,
% 4.08/4.26 ((~![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)) | (apply(apply(k, projection2), pair(apply(k, s), apply(k, projection2))) = projection2)),
% 4.08/4.26 inference(quant_inst,[status(thm)],[])).
% 4.08/4.26 tff(9,plain,
% 4.08/4.26 (apply(apply(k, projection2), pair(apply(k, s), apply(k, projection2))) = projection2),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[8, 7])).
% 4.08/4.26 tff(10,plain,
% 4.08/4.26 ((apply(k, projection2) = k_projection2) <=> (apply(k, projection2) = k_projection2)),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(11,axiom,(apply(k, projection2) = k_projection2), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','k_projection2')).
% 4.08/4.26 tff(12,plain,
% 4.08/4.26 (apply(k, projection2) = k_projection2),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[11, 10])).
% 4.08/4.26 tff(13,plain,
% 4.08/4.26 (k_projection2 = apply(k, projection2)),
% 4.08/4.26 inference(symmetry,[status(thm)],[12])).
% 4.08/4.26 tff(14,plain,
% 4.08/4.26 (apply(k_projection2, pair(apply(k, s), apply(k, projection2))) = apply(apply(k, projection2), pair(apply(k, s), apply(k, projection2)))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[13])).
% 4.08/4.26 tff(15,assumption,(~(apply(k, s) = k_projection2)), introduced(assumption)).
% 4.08/4.26 tff(16,plain,
% 4.08/4.26 (^[Y: $i, X: $i] : refl(((X = Y) | (apply(eq, pair(X, Y)) = projection2)) <=> ((X = Y) | (apply(eq, pair(X, Y)) = projection2)))),
% 4.08/4.26 inference(bind,[status(th)],[])).
% 4.08/4.26 tff(17,plain,
% 4.08/4.26 (![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2)) <=> ![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))),
% 4.08/4.26 inference(quant_intro,[status(thm)],[16])).
% 4.08/4.26 tff(18,plain,
% 4.08/4.26 (![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2)) <=> ![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(19,axiom,(![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','extensionality1')).
% 4.08/4.26 tff(20,plain,
% 4.08/4.26 (![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[19, 18])).
% 4.08/4.26 tff(21,plain,(
% 4.08/4.26 ![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))),
% 4.08/4.26 inference(skolemize,[status(sab)],[20])).
% 4.08/4.26 tff(22,plain,
% 4.08/4.26 (![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[21, 17])).
% 4.08/4.26 tff(23,plain,
% 4.08/4.26 (((~![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))) | ((apply(k, s) = k_projection2) | (apply(eq, pair(apply(k, s), k_projection2)) = projection2))) <=> ((~![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))) | (apply(k, s) = k_projection2) | (apply(eq, pair(apply(k, s), k_projection2)) = projection2))),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(24,plain,
% 4.08/4.26 ((~![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))) | ((apply(k, s) = k_projection2) | (apply(eq, pair(apply(k, s), k_projection2)) = projection2))),
% 4.08/4.26 inference(quant_inst,[status(thm)],[])).
% 4.08/4.26 tff(25,plain,
% 4.08/4.26 ((~![Y: $i, X: $i] : ((X = Y) | (apply(eq, pair(X, Y)) = projection2))) | (apply(k, s) = k_projection2) | (apply(eq, pair(apply(k, s), k_projection2)) = projection2)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[24, 23])).
% 4.08/4.26 tff(26,plain,
% 4.08/4.26 ((apply(k, s) = k_projection2) | (apply(eq, pair(apply(k, s), k_projection2)) = projection2)),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[25, 22])).
% 4.08/4.26 tff(27,plain,
% 4.08/4.26 (apply(eq, pair(apply(k, s), k_projection2)) = projection2),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[26, 15])).
% 4.08/4.26 tff(28,plain,
% 4.08/4.26 (pair(apply(k, s), k_projection2) = pair(apply(k, s), apply(k, projection2))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[13])).
% 4.08/4.26 tff(29,plain,
% 4.08/4.26 (apply(eq, pair(apply(k, s), k_projection2)) = apply(eq, pair(apply(k, s), apply(k, projection2)))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[28])).
% 4.08/4.26 tff(30,plain,
% 4.08/4.26 (apply(eq, pair(apply(k, s), apply(k, projection2))) = apply(eq, pair(apply(k, s), k_projection2))),
% 4.08/4.26 inference(symmetry,[status(thm)],[29])).
% 4.08/4.26 tff(31,plain,
% 4.08/4.26 ((s = apply(eq, pair(apply(k, s), apply(k, projection2)))) <=> (s = apply(eq, pair(apply(k, s), apply(k, projection2))))),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(32,axiom,(s = apply(eq, pair(apply(k, s), apply(k, projection2)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','self_referential')).
% 4.08/4.26 tff(33,plain,
% 4.08/4.26 (s = apply(eq, pair(apply(k, s), apply(k, projection2)))),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[32, 31])).
% 4.08/4.26 tff(34,plain,
% 4.08/4.26 (s = projection2),
% 4.08/4.26 inference(transitivity,[status(thm)],[33, 30, 27])).
% 4.08/4.26 tff(35,plain,
% 4.08/4.26 (apply(k, s) = apply(k, projection2)),
% 4.08/4.26 inference(monotonicity,[status(thm)],[34])).
% 4.08/4.26 tff(36,plain,
% 4.08/4.26 (apply(k, s) = k_projection2),
% 4.08/4.26 inference(transitivity,[status(thm)],[35, 12])).
% 4.08/4.26 tff(37,plain,
% 4.08/4.26 ($false),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[15, 36])).
% 4.08/4.26 tff(38,plain,(apply(k, s) = k_projection2), inference(lemma,lemma(discharge,[]))).
% 4.08/4.26 tff(39,plain,
% 4.08/4.26 (apply(eq, pair(apply(k, s), apply(k, projection2))) = s),
% 4.08/4.26 inference(symmetry,[status(thm)],[33])).
% 4.08/4.26 tff(40,plain,
% 4.08/4.26 (apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))) = apply(k, s)),
% 4.08/4.26 inference(monotonicity,[status(thm)],[39])).
% 4.08/4.26 tff(41,plain,
% 4.08/4.26 (apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))) = k_projection2),
% 4.08/4.26 inference(transitivity,[status(thm)],[40, 38])).
% 4.08/4.26 tff(42,plain,
% 4.08/4.26 (apply(apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))), pair(apply(k, s), apply(k, projection2))) = apply(k_projection2, pair(apply(k, s), apply(k, projection2)))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[41])).
% 4.08/4.26 tff(43,assumption,(apply(apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))), pair(apply(k, s), apply(k, projection2))) = apply(eq, pair(apply(k, s), apply(k, projection2)))), introduced(assumption)).
% 4.08/4.26 tff(44,plain,
% 4.08/4.26 (apply(eq, pair(apply(k, s), apply(k, projection2))) = apply(apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))), pair(apply(k, s), apply(k, projection2)))),
% 4.08/4.26 inference(symmetry,[status(thm)],[43])).
% 4.08/4.26 tff(45,plain,
% 4.08/4.26 (apply(eq, pair(apply(k, s), k_projection2)) = projection2),
% 4.08/4.26 inference(transitivity,[status(thm)],[29, 44, 42, 14, 9])).
% 4.08/4.26 tff(46,plain,
% 4.08/4.26 (k_projection2 = apply(k, s)),
% 4.08/4.26 inference(symmetry,[status(thm)],[38])).
% 4.08/4.26 tff(47,plain,
% 4.08/4.26 (pair(k_projection2, k_projection2) = pair(apply(k, s), apply(k, projection2))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[46, 13])).
% 4.08/4.26 tff(48,plain,
% 4.08/4.26 (apply(eq, pair(k_projection2, k_projection2)) = apply(eq, pair(apply(k, s), apply(k, projection2)))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[47])).
% 4.08/4.26 tff(49,plain,
% 4.08/4.26 (^[X: $i] : refl((apply(eq, pair(X, X)) = projection1) <=> (apply(eq, pair(X, X)) = projection1))),
% 4.08/4.26 inference(bind,[status(th)],[])).
% 4.08/4.26 tff(50,plain,
% 4.08/4.26 (![X: $i] : (apply(eq, pair(X, X)) = projection1) <=> ![X: $i] : (apply(eq, pair(X, X)) = projection1)),
% 4.08/4.26 inference(quant_intro,[status(thm)],[49])).
% 4.08/4.26 tff(51,plain,
% 4.08/4.26 (![X: $i] : (apply(eq, pair(X, X)) = projection1) <=> ![X: $i] : (apply(eq, pair(X, X)) = projection1)),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(52,axiom,(![X: $i] : (apply(eq, pair(X, X)) = projection1)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','equality')).
% 4.08/4.26 tff(53,plain,
% 4.08/4.26 (![X: $i] : (apply(eq, pair(X, X)) = projection1)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[52, 51])).
% 4.08/4.26 tff(54,plain,(
% 4.08/4.26 ![X: $i] : (apply(eq, pair(X, X)) = projection1)),
% 4.08/4.26 inference(skolemize,[status(sab)],[53])).
% 4.08/4.26 tff(55,plain,
% 4.08/4.26 (![X: $i] : (apply(eq, pair(X, X)) = projection1)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[54, 50])).
% 4.08/4.26 tff(56,plain,
% 4.08/4.26 ((~![X: $i] : (apply(eq, pair(X, X)) = projection1)) | (apply(eq, pair(k_projection2, k_projection2)) = projection1)),
% 4.08/4.26 inference(quant_inst,[status(thm)],[])).
% 4.08/4.26 tff(57,plain,
% 4.08/4.26 (apply(eq, pair(k_projection2, k_projection2)) = projection1),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[56, 55])).
% 4.08/4.26 tff(58,plain,
% 4.08/4.26 (projection1 = apply(eq, pair(k_projection2, k_projection2))),
% 4.08/4.26 inference(symmetry,[status(thm)],[57])).
% 4.08/4.26 tff(59,plain,
% 4.08/4.26 (projection1 = apply(eq, pair(apply(k, s), k_projection2))),
% 4.08/4.26 inference(transitivity,[status(thm)],[58, 48, 30])).
% 4.08/4.26 tff(60,plain,
% 4.08/4.26 ((projection1 = projection2) <=> (apply(eq, pair(apply(k, s), k_projection2)) = projection2)),
% 4.08/4.26 inference(monotonicity,[status(thm)],[59])).
% 4.08/4.26 tff(61,plain,
% 4.08/4.26 ((~(projection1 = projection2)) <=> (~(apply(eq, pair(apply(k, s), k_projection2)) = projection2))),
% 4.08/4.26 inference(monotonicity,[status(thm)],[60])).
% 4.08/4.26 tff(62,plain,
% 4.08/4.26 ((~(projection1 = projection2)) <=> (~(projection1 = projection2))),
% 4.08/4.26 inference(rewrite,[status(thm)],[])).
% 4.08/4.26 tff(63,axiom,(~(projection1 = projection2)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','different_projections')).
% 4.08/4.26 tff(64,plain,
% 4.08/4.26 (~(projection1 = projection2)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[63, 62])).
% 4.08/4.26 tff(65,plain,
% 4.08/4.26 (~(apply(eq, pair(apply(k, s), k_projection2)) = projection2)),
% 4.08/4.26 inference(modus_ponens,[status(thm)],[64, 61])).
% 4.08/4.26 tff(66,plain,
% 4.08/4.26 ($false),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[65, 45])).
% 4.08/4.26 tff(67,plain,(~(apply(apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))), pair(apply(k, s), apply(k, projection2))) = apply(eq, pair(apply(k, s), apply(k, projection2))))), inference(lemma,lemma(discharge,[]))).
% 4.08/4.26 tff(68,plain,
% 4.08/4.26 ((~![Y: $i, X: $i] : (apply(apply(k, X), Y) = X)) | (apply(apply(k, apply(eq, pair(apply(k, s), apply(k, projection2)))), pair(apply(k, s), apply(k, projection2))) = apply(eq, pair(apply(k, s), apply(k, projection2))))),
% 4.08/4.26 inference(quant_inst,[status(thm)],[])).
% 4.08/4.26 tff(69,plain,
% 4.08/4.26 ($false),
% 4.08/4.26 inference(unit_resolution,[status(thm)],[68, 7, 67])).
% 4.08/4.26 % SZS output end Proof
% 4.08/4.27 % E exiting
%------------------------------------------------------------------------------