%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : KLE004+1 : TPTP v9.0.0. Released v4.0.0.
% 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 05:12:49 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : KLE004+1 : TPTP v9.0.0. Released v4.0.0.
% 0.12/0.12 % 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 18:49:38 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(complement_type, type, (
% 0.12/0.39 complement: ( $i * $i ) > $o)).
% 0.12/0.39 tff(zero_type, type, (
% 0.12/0.39 zero: $i)).
% 0.12/0.39 tff(addition_type, type, (
% 0.12/0.39 addition: ( $i * $i ) > $i)).
% 0.12/0.39 tff(one_type, type, (
% 0.12/0.39 one: $i)).
% 0.12/0.39 tff(multiplication_type, type, (
% 0.12/0.39 multiplication: ( $i * $i ) > $i)).
% 0.12/0.39 tff(test_type, type, (
% 0.12/0.39 test: $i > $o)).
% 0.12/0.39 tff(c_type, type, (
% 0.12/0.39 c: $i > $i)).
% 0.12/0.39 tff(tptp_fun_X1_0_type, type, (
% 0.12/0.39 tptp_fun_X1_0: $i > $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[A: $i] : refl((addition(A, zero) = A) <=> (addition(A, zero) = A))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![A: $i] : (addition(A, zero) = A) <=> ![A: $i] : (addition(A, zero) = A)),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![A: $i] : (addition(A, zero) = A) <=> ![A: $i] : (addition(A, zero) = A)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(4,axiom,(![A: $i] : (addition(A, zero) = A)), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+0.ax','additive_identity')).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![A: $i] : (addition(A, zero) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.12/0.39 tff(6,plain,(
% 0.12/0.39 ![A: $i] : (addition(A, zero) = A)),
% 0.12/0.39 inference(skolemize,[status(sab)],[5])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![A: $i] : (addition(A, zero) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 ((~![A: $i] : (addition(A, zero) = A)) | (addition(one, zero) = one)),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (addition(one, zero) = one),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (complement(addition(one, zero), zero) <=> complement(one, zero)),
% 0.12/0.39 inference(monotonicity,[status(thm)],[9])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 (complement(one, zero) <=> complement(addition(one, zero), zero)),
% 0.12/0.39 inference(symmetry,[status(thm)],[10])).
% 0.12/0.39 tff(12,assumption,(~(multiplication(one, zero) = zero)), introduced(assumption)).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (^[A: $i] : refl((multiplication(one, A) = A) <=> (multiplication(one, A) = A))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (![A: $i] : (multiplication(one, A) = A) <=> ![A: $i] : (multiplication(one, A) = A)),
% 0.12/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![A: $i] : (multiplication(one, A) = A) <=> ![A: $i] : (multiplication(one, A) = A)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(16,axiom,(![A: $i] : (multiplication(one, A) = A)), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+0.ax','multiplicative_left_identity')).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (![A: $i] : (multiplication(one, A) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.12/0.39 tff(18,plain,(
% 0.12/0.39 ![A: $i] : (multiplication(one, A) = A)),
% 0.12/0.39 inference(skolemize,[status(sab)],[17])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (![A: $i] : (multiplication(one, A) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 ((~![A: $i] : (multiplication(one, A) = A)) | (multiplication(one, zero) = zero)),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[20, 19, 12])).
% 0.12/0.39 tff(22,plain,(multiplication(one, zero) = zero), inference(lemma,lemma(discharge,[]))).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (^[A: $i] : refl((multiplication(A, one) = A) <=> (multiplication(A, one) = A))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (![A: $i] : (multiplication(A, one) = A) <=> ![A: $i] : (multiplication(A, one) = A)),
% 0.12/0.39 inference(quant_intro,[status(thm)],[23])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (![A: $i] : (multiplication(A, one) = A) <=> ![A: $i] : (multiplication(A, one) = A)),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(26,axiom,(![A: $i] : (multiplication(A, one) = A)), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+0.ax','multiplicative_right_identity')).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (![A: $i] : (multiplication(A, one) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.12/0.39 tff(28,plain,(
% 0.12/0.39 ![A: $i] : (multiplication(A, one) = A)),
% 0.12/0.39 inference(skolemize,[status(sab)],[27])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (![A: $i] : (multiplication(A, one) = A)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 ((~![A: $i] : (multiplication(A, one) = A)) | (multiplication(zero, one) = zero)),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (multiplication(zero, one) = zero),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 (^[A: $i, B: $i] : refl((addition(A, B) = addition(B, A)) <=> (addition(A, B) = addition(B, A)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(33,plain,
% 0.12/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[32])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(35,axiom,(![A: $i, B: $i] : (addition(A, B) = addition(B, A))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+0.ax','additive_commutativity')).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.39 tff(37,plain,(
% 0.12/0.39 ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.12/0.39 inference(skolemize,[status(sab)],[36])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(zero, one) = addition(one, zero))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 (addition(zero, one) = addition(one, zero)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[39, 38])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 ((addition(zero, one) = one) <=> (addition(one, zero) = one)),
% 0.12/0.39 inference(monotonicity,[status(thm)],[40])).
% 0.12/0.39 tff(42,plain,
% 0.12/0.39 ((addition(one, zero) = one) <=> (addition(zero, one) = one)),
% 0.12/0.39 inference(symmetry,[status(thm)],[41])).
% 0.12/0.39 tff(43,plain,
% 0.12/0.39 (addition(zero, one) = one),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[9, 42])).
% 0.12/0.39 tff(44,plain,
% 0.12/0.39 ((~((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one)))) | (~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(45,plain,
% 0.12/0.39 ((~((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one)))) | (~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[44, 43])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 (~((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[45, 31, 22])).
% 0.12/0.39 tff(47,plain,
% 0.12/0.39 (^[X0: $i, X1: $i] : refl((~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(48,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))) <=> ![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[47])).
% 0.12/0.39 tff(49,plain,
% 0.12/0.39 (^[X0: $i, X1: $i] : trans(trans(monotonicity(rewrite(((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))))), ((complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> (complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))))))), rewrite((complement(X1, X0) <=> (~((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))), ((complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))))), rewrite((~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))), ((complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(50,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> ![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[49])).
% 0.12/0.39 tff(51,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))) <=> ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(52,plain,
% 0.12/0.39 (^[X0: $i, X1: $i] : rewrite((complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one))) <=> (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(53,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one))) <=> ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[52])).
% 0.12/0.39 tff(54,axiom,(![X0: $i, X1: $i] : (complement(X1, X0) <=> (((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero)) & (addition(X0, X1) = one)))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+1.ax','test_2')).
% 0.12/0.39 tff(55,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.12/0.39 tff(56,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[55, 51])).
% 0.12/0.39 tff(57,plain,(
% 0.12/0.39 ![X0: $i, X1: $i] : (complement(X1, X0) <=> ((multiplication(X0, X1) = zero) & (multiplication(X1, X0) = zero) & (addition(X0, X1) = one)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[56])).
% 0.12/0.39 tff(58,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[57, 50])).
% 0.12/0.39 tff(59,plain,
% 0.12/0.39 (![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[58, 48])).
% 0.12/0.39 tff(60,plain,
% 0.12/0.39 ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one))) <=> complement(one, zero)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(61,plain,
% 0.12/0.39 (~(((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one))) <=> complement(one, zero))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[60, 59])).
% 0.12/0.39 tff(62,plain,
% 0.12/0.39 ((((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one))) <=> complement(one, zero)) | ((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one))) | complement(one, zero)),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(63,plain,
% 0.12/0.39 (((~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero)) | (~(addition(zero, one) = one))) | complement(one, zero)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.12/0.39 tff(64,plain,
% 0.12/0.39 (complement(one, zero)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[63, 46])).
% 0.12/0.39 tff(65,plain,
% 0.12/0.39 (complement(addition(one, zero), zero)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[64, 11])).
% 0.12/0.39 tff(66,plain,
% 0.12/0.39 ((~((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero)))) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(67,plain,
% 0.12/0.39 ((~((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero)))) | (~(multiplication(zero, one) = zero)) | (~(multiplication(one, zero) = zero))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[66, 9])).
% 0.12/0.39 tff(68,plain,
% 0.12/0.39 (~((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[67, 31, 22])).
% 0.12/0.39 tff(69,plain,
% 0.12/0.39 (((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one)))) <=> ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(70,plain,
% 0.12/0.39 ((~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one))) <=> (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(71,plain,
% 0.12/0.39 ((((~(multiplication(one, zero) = zero)) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one))) <=> complement(zero, one)) <=> (((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(72,plain,
% 0.12/0.39 ((~(((~(multiplication(one, zero) = zero)) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one))) <=> complement(zero, one))) <=> (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one)))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[71])).
% 0.12/0.39 tff(73,plain,
% 0.12/0.39 ((~(((~(multiplication(one, zero) = zero)) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one))) <=> complement(zero, one))) <=> (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[72, 70])).
% 0.12/0.39 tff(74,plain,
% 0.12/0.39 (((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(one, zero) = zero)) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one))) <=> complement(zero, one)))) <=> ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[73])).
% 0.12/0.40 tff(75,plain,
% 0.12/0.40 (((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(one, zero) = zero)) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one))) <=> complement(zero, one)))) <=> ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[74, 69])).
% 0.12/0.40 tff(76,plain,
% 0.12/0.40 ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(one, zero) = zero)) | (~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one))) <=> complement(zero, one)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(77,plain,
% 0.12/0.40 ((~![X0: $i, X1: $i] : (~(((~(multiplication(X0, X1) = zero)) | (~(multiplication(X1, X0) = zero)) | (~(addition(X0, X1) = one))) <=> complement(X1, X0)))) | (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.12/0.40 tff(78,plain,
% 0.12/0.40 (~(((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[77, 59])).
% 0.12/0.40 tff(79,plain,
% 0.12/0.40 ((((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) <=> complement(zero, one)) | ((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) | complement(zero, one)),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(80,plain,
% 0.12/0.40 (((~(multiplication(zero, one) = zero)) | (~(addition(one, zero) = one)) | (~(multiplication(one, zero) = zero))) | complement(zero, one)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.12/0.40 tff(81,plain,
% 0.12/0.40 (complement(zero, one)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[80, 68])).
% 0.12/0.40 tff(82,plain,
% 0.12/0.40 ((~(c(zero) = one)) <=> (~(c(zero) = one))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(83,axiom,(~(c(zero) = one)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','goals')).
% 0.12/0.40 tff(84,plain,
% 0.12/0.40 (~(c(zero) = one)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.12/0.40 tff(85,plain,
% 0.12/0.40 ((~((c(zero) = one) <=> complement(zero, one))) | (c(zero) = one) | (~complement(zero, one))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(86,plain,
% 0.12/0.40 ((~((c(zero) = one) <=> complement(zero, one))) | (~complement(zero, one))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.12/0.40 tff(87,plain,
% 0.12/0.40 (~((c(zero) = one) <=> complement(zero, one))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[86, 81])).
% 0.12/0.40 tff(88,plain,
% 0.12/0.40 (^[X0: $i, X1: $i] : refl(((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(89,plain,
% 0.12/0.40 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[88])).
% 0.12/0.40 tff(90,plain,
% 0.12/0.40 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(91,plain,
% 0.12/0.40 (^[X0: $i, X1: $i] : rewrite((test(X0) => ((c(X0) = X1) <=> complement(X0, X1))) <=> ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(92,plain,
% 0.12/0.40 (![X0: $i, X1: $i] : (test(X0) => ((c(X0) = X1) <=> complement(X0, X1))) <=> ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[91])).
% 0.12/0.40 tff(93,axiom,(![X0: $i, X1: $i] : (test(X0) => ((c(X0) = X1) <=> complement(X0, X1)))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+1.ax','test_3')).
% 0.12/0.40 tff(94,plain,
% 0.12/0.40 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.12/0.40 tff(95,plain,
% 0.12/0.40 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[94, 90])).
% 0.12/0.40 tff(96,plain,(
% 0.12/0.40 ![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(skolemize,[status(sab)],[95])).
% 0.12/0.40 tff(97,plain,
% 0.12/0.40 (![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[96, 89])).
% 0.12/0.40 tff(98,plain,
% 0.12/0.40 (((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(zero)) | ((c(zero) = one) <=> complement(zero, one)))) <=> ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(zero)) | ((c(zero) = one) <=> complement(zero, one)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(99,plain,
% 0.12/0.40 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | ((~test(zero)) | ((c(zero) = one) <=> complement(zero, one)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(100,plain,
% 0.12/0.40 ((~![X0: $i, X1: $i] : ((~test(X0)) | ((c(X0) = X1) <=> complement(X0, X1)))) | (~test(zero)) | ((c(zero) = one) <=> complement(zero, one))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.12/0.40 tff(101,plain,
% 0.12/0.40 ((~test(zero)) | ((c(zero) = one) <=> complement(zero, one))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[100, 97])).
% 0.12/0.40 tff(102,plain,
% 0.12/0.40 (~test(zero)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[101, 87])).
% 0.12/0.40 tff(103,plain,
% 0.12/0.40 (^[X0: $i] : refl((~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))) <=> (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(104,plain,
% 0.12/0.40 (![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))) <=> ![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[103])).
% 0.12/0.40 tff(105,plain,
% 0.12/0.40 (^[X0: $i] : rewrite((~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))) <=> (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(106,plain,
% 0.12/0.40 (![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))) <=> ![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[105])).
% 0.12/0.40 tff(107,plain,
% 0.12/0.40 (![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0)))))) <=> ![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[106, 104])).
% 0.12/0.40 tff(108,plain,
% 0.12/0.40 (^[X0: $i] : trans(monotonicity(rewrite((test(X0) | ![X1: $i] : (~complement(X1, X0))) <=> (test(X0) | ![X1: $i] : (~complement(X1, X0)))), ((((~test(X0)) | complement(tptp_fun_X1_0(X0), X0)) & (test(X0) | ![X1: $i] : (~complement(X1, X0)))) <=> (((~test(X0)) | complement(tptp_fun_X1_0(X0), X0)) & (test(X0) | ![X1: $i] : (~complement(X1, X0)))))), rewrite((((~test(X0)) | complement(tptp_fun_X1_0(X0), X0)) & (test(X0) | ![X1: $i] : (~complement(X1, X0)))) <=> (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))), ((((~test(X0)) | complement(tptp_fun_X1_0(X0), X0)) & (test(X0) | ![X1: $i] : (~complement(X1, X0)))) <=> (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(109,plain,
% 0.12/0.40 (![X0: $i] : (((~test(X0)) | complement(tptp_fun_X1_0(X0), X0)) & (test(X0) | ![X1: $i] : (~complement(X1, X0)))) <=> ![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[108])).
% 0.12/0.40 tff(110,plain,
% 0.12/0.40 (![X0: $i] : (test(X0) <=> ?[X1: $i] : complement(X1, X0)) <=> ![X0: $i] : (test(X0) <=> ?[X1: $i] : complement(X1, X0))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(111,axiom,(![X0: $i] : (test(X0) <=> ?[X1: $i] : complement(X1, X0))), file('/export/starexec/sandbox/benchmark/Axioms/KLE001+1.ax','test_1')).
% 0.12/0.40 tff(112,plain,
% 0.12/0.40 (![X0: $i] : (test(X0) <=> ?[X1: $i] : complement(X1, X0))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.12/0.40 tff(113,plain,(
% 0.12/0.40 ![X0: $i] : (((~test(X0)) | complement(tptp_fun_X1_0(X0), X0)) & (test(X0) | ![X1: $i] : (~complement(X1, X0))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[112])).
% 0.12/0.40 tff(114,plain,
% 0.12/0.40 (![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[113, 109])).
% 0.12/0.40 tff(115,plain,
% 0.12/0.40 (![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[114, 107])).
% 0.12/0.40 tff(116,plain,
% 0.12/0.40 ((~![X0: $i] : (~((~((~test(X0)) | complement(tptp_fun_X1_0(X0), X0))) | (~(test(X0) | ![X1: $i] : (~complement(X1, X0))))))) | (~((~((~test(zero)) | complement(tptp_fun_X1_0(zero), zero))) | (~(test(zero) | ![X1: $i] : (~complement(X1, zero))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(117,plain,
% 0.12/0.40 (~((~((~test(zero)) | complement(tptp_fun_X1_0(zero), zero))) | (~(test(zero) | ![X1: $i] : (~complement(X1, zero)))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.12/0.40 tff(118,plain,
% 0.12/0.40 (((~((~test(zero)) | complement(tptp_fun_X1_0(zero), zero))) | (~(test(zero) | ![X1: $i] : (~complement(X1, zero))))) | (test(zero) | ![X1: $i] : (~complement(X1, zero)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(119,plain,
% 0.12/0.40 (test(zero) | ![X1: $i] : (~complement(X1, zero))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[118, 117])).
% 0.12/0.40 tff(120,plain,
% 0.12/0.40 ((~(test(zero) | ![X1: $i] : (~complement(X1, zero)))) | test(zero) | ![X1: $i] : (~complement(X1, zero))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(121,plain,
% 0.12/0.40 (test(zero) | ![X1: $i] : (~complement(X1, zero))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[120, 119])).
% 0.12/0.40 tff(122,plain,
% 0.12/0.40 (![X1: $i] : (~complement(X1, zero))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[121, 102])).
% 0.12/0.40 tff(123,plain,
% 0.12/0.40 ((~![X1: $i] : (~complement(X1, zero))) | (~complement(addition(one, zero), zero))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(124,plain,
% 0.12/0.40 ($false),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[123, 122, 65])).
% 0.12/0.40 % SZS output end Proof
% 0.12/0.41 % E exiting
%------------------------------------------------------------------------------