%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : KLE151+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 05:13:06 AM UTC 2025
% Result : Theorem 0.22s 0.45s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : KLE151+1 : TPTP v9.0.0. Released v4.0.0.
% 0.13/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n004.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 19:04:23 EDT 2025
% 0.20/0.35 % CPUTime :
% 0.22/0.45 % SZS status Theorem
% 0.22/0.45 % SZS output start Proof
% 0.22/0.45 tff(addition_type, type, (
% 0.22/0.45 addition: ( $i * $i ) > $i)).
% 0.22/0.45 tff(tptp_fun_X0_1_type, type, (
% 0.22/0.45 tptp_fun_X0_1: $i)).
% 0.22/0.45 tff(multiplication_type, type, (
% 0.22/0.45 multiplication: ( $i * $i ) > $i)).
% 0.22/0.45 tff(strong_iteration_type, type, (
% 0.22/0.45 strong_iteration: $i > $i)).
% 0.22/0.45 tff(tptp_fun_X1_0_type, type, (
% 0.22/0.45 tptp_fun_X1_0: $i)).
% 0.22/0.45 tff(one_type, type, (
% 0.22/0.45 one: $i)).
% 0.22/0.45 tff(leq_type, type, (
% 0.22/0.45 leq: ( $i * $i ) > $o)).
% 0.22/0.45 tff(1,plain,
% 0.22/0.45 (^[A: $i] : refl((multiplication(A, one) = A) <=> (multiplication(A, one) = A))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(2,plain,
% 0.22/0.45 (![A: $i] : (multiplication(A, one) = A) <=> ![A: $i] : (multiplication(A, one) = A)),
% 0.22/0.45 inference(quant_intro,[status(thm)],[1])).
% 0.22/0.45 tff(3,plain,
% 0.22/0.45 (![A: $i] : (multiplication(A, one) = A) <=> ![A: $i] : (multiplication(A, one) = A)),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(4,axiom,(![A: $i] : (multiplication(A, one) = A)), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','multiplicative_right_identity')).
% 0.22/0.45 tff(5,plain,
% 0.22/0.45 (![A: $i] : (multiplication(A, one) = A)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.22/0.45 tff(6,plain,(
% 0.22/0.45 ![A: $i] : (multiplication(A, one) = A)),
% 0.22/0.45 inference(skolemize,[status(sab)],[5])).
% 0.22/0.45 tff(7,plain,
% 0.22/0.45 (![A: $i] : (multiplication(A, one) = A)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.22/0.45 tff(8,plain,
% 0.22/0.45 ((~![A: $i] : (multiplication(A, one) = A)) | (multiplication(X0!1, one) = X0!1)),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(9,plain,
% 0.22/0.45 (multiplication(X0!1, one) = X0!1),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.22/0.45 tff(10,plain,
% 0.22/0.45 (X0!1 = multiplication(X0!1, one)),
% 0.22/0.45 inference(symmetry,[status(thm)],[9])).
% 0.22/0.45 tff(11,plain,
% 0.22/0.45 (^[A: $i] : refl((strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)) <=> (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(12,plain,
% 0.22/0.45 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)) <=> ![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[11])).
% 0.22/0.45 tff(13,plain,
% 0.22/0.45 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)) <=> ![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(14,axiom,(![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','infty_unfold1')).
% 0.22/0.45 tff(15,plain,
% 0.22/0.45 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.22/0.45 tff(16,plain,(
% 0.22/0.45 ![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.22/0.45 inference(skolemize,[status(sab)],[15])).
% 0.22/0.45 tff(17,plain,
% 0.22/0.45 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.22/0.45 tff(18,plain,
% 0.22/0.45 ((~![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))) | (strong_iteration(multiplication(X1!0, X0!1)) = addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(19,plain,
% 0.22/0.45 (strong_iteration(multiplication(X1!0, X0!1)) = addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.22/0.45 tff(20,plain,
% 0.22/0.45 (multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))) = multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[19])).
% 0.22/0.45 tff(21,plain,
% 0.22/0.45 (multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))) = multiplication(X1!0, multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[20])).
% 0.22/0.45 tff(22,plain,
% 0.22/0.45 (multiplication(X1!0, multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))) = multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))))),
% 0.22/0.45 inference(symmetry,[status(thm)],[21])).
% 0.22/0.45 tff(23,plain,
% 0.22/0.45 (^[A: $i, B: $i, C: $i] : refl((multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C)) <=> (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C)))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(24,plain,
% 0.22/0.45 (![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C)) <=> ![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[23])).
% 0.22/0.45 tff(25,plain,
% 0.22/0.45 (![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C)) <=> ![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(26,axiom,(![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','multiplicative_associativity')).
% 0.22/0.45 tff(27,plain,
% 0.22/0.45 (![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.22/0.45 tff(28,plain,(
% 0.22/0.45 ![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.22/0.45 inference(skolemize,[status(sab)],[27])).
% 0.22/0.45 tff(29,plain,
% 0.22/0.45 (![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.22/0.45 tff(30,plain,
% 0.22/0.45 ((~![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))) | (multiplication(X1!0, multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))) = multiplication(multiplication(X1!0, X0!1), addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(31,plain,
% 0.22/0.45 (multiplication(X1!0, multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))) = multiplication(multiplication(X1!0, X0!1), addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.22/0.45 tff(32,plain,
% 0.22/0.45 (multiplication(multiplication(X1!0, X0!1), addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)) = multiplication(X1!0, multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)))),
% 0.22/0.45 inference(symmetry,[status(thm)],[31])).
% 0.22/0.45 tff(33,plain,
% 0.22/0.45 (multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))) = multiplication(multiplication(X1!0, X0!1), addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[19])).
% 0.22/0.45 tff(34,plain,
% 0.22/0.45 (multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))) = multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))))),
% 0.22/0.45 inference(transitivity,[status(thm)],[33, 32, 22])).
% 0.22/0.45 tff(35,plain,
% 0.22/0.45 (multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))) = multiplication(X0!1, multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[34])).
% 0.22/0.46 tff(36,plain,
% 0.22/0.46 (multiplication(X0!1, multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))))) = multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))))),
% 0.22/0.46 inference(symmetry,[status(thm)],[35])).
% 0.22/0.46 tff(37,plain,
% 0.22/0.46 ((~![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))) | (multiplication(X0!1, multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))))) = multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(38,plain,
% 0.22/0.46 (multiplication(X0!1, multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))))) = multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[37, 29])).
% 0.22/0.46 tff(39,plain,
% 0.22/0.46 (multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))) = multiplication(X0!1, multiplication(X1!0, multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))))),
% 0.22/0.46 inference(symmetry,[status(thm)],[38])).
% 0.22/0.46 tff(40,plain,
% 0.22/0.46 (multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))) = multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))))),
% 0.22/0.46 inference(transitivity,[status(thm)],[39, 36])).
% 0.22/0.46 tff(41,plain,
% 0.22/0.46 (addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1) = addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[40, 10])).
% 0.22/0.46 tff(42,plain,
% 0.22/0.46 (addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)),
% 0.22/0.46 inference(symmetry,[status(thm)],[41])).
% 0.22/0.46 tff(43,plain,
% 0.22/0.46 (^[A: $i] : refl((addition(A, A) = A) <=> (addition(A, A) = A))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(44,plain,
% 0.22/0.46 (![A: $i] : (addition(A, A) = A) <=> ![A: $i] : (addition(A, A) = A)),
% 0.22/0.46 inference(quant_intro,[status(thm)],[43])).
% 0.22/0.46 tff(45,plain,
% 0.22/0.46 (![A: $i] : (addition(A, A) = A) <=> ![A: $i] : (addition(A, A) = A)),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(46,axiom,(![A: $i] : (addition(A, A) = A)), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','idempotence')).
% 0.22/0.46 tff(47,plain,
% 0.22/0.46 (![A: $i] : (addition(A, A) = A)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.22/0.46 tff(48,plain,(
% 0.22/0.46 ![A: $i] : (addition(A, A) = A)),
% 0.22/0.46 inference(skolemize,[status(sab)],[47])).
% 0.22/0.46 tff(49,plain,
% 0.22/0.46 (![A: $i] : (addition(A, A) = A)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.22/0.46 tff(50,plain,
% 0.22/0.46 ((~![A: $i] : (addition(A, A) = A)) | (addition(addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)), addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one))) = addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(51,plain,
% 0.22/0.46 (addition(addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)), addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one))) = addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.22/0.46 tff(52,plain,
% 0.22/0.46 (^[A: $i, B: $i, C: $i] : refl((multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))) <=> (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(53,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))) <=> ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.22/0.46 inference(quant_intro,[status(thm)],[52])).
% 0.22/0.46 tff(54,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C))) <=> ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(55,axiom,(![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','distributivity1')).
% 0.22/0.46 tff(56,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.22/0.46 tff(57,plain,(
% 0.22/0.46 ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.22/0.46 inference(skolemize,[status(sab)],[56])).
% 0.22/0.46 tff(58,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.22/0.46 tff(59,plain,
% 0.22/0.46 ((~![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))) | (multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)) = addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(60,plain,
% 0.22/0.46 (multiplication(X0!1, addition(multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1))), one)) = addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.22/0.46 tff(61,plain,
% 0.22/0.46 (multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))) = addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one))),
% 0.22/0.46 inference(transitivity,[status(thm)],[20, 60])).
% 0.22/0.46 tff(62,plain,
% 0.22/0.46 (addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)), addition(multiplication(X0!1, multiplication(multiplication(X1!0, X0!1), strong_iteration(multiplication(X1!0, X0!1)))), multiplication(X0!1, one)))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[61, 41])).
% 0.22/0.46 tff(63,plain,
% 0.22/0.46 (addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)),
% 0.22/0.46 inference(transitivity,[status(thm)],[62, 51, 42])).
% 0.22/0.46 tff(64,plain,
% 0.22/0.46 (^[A: $i, B: $i] : refl((leq(A, B) <=> (addition(A, B) = B)) <=> (leq(A, B) <=> (addition(A, B) = B)))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(65,plain,
% 0.22/0.46 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B)) <=> ![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.22/0.46 inference(quant_intro,[status(thm)],[64])).
% 0.22/0.46 tff(66,plain,
% 0.22/0.46 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B)) <=> ![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(67,axiom,(![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','order')).
% 0.22/0.46 tff(68,plain,
% 0.22/0.46 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.22/0.46 tff(69,plain,(
% 0.22/0.46 ![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.22/0.46 inference(skolemize,[status(sab)],[68])).
% 0.22/0.46 tff(70,plain,
% 0.22/0.46 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.22/0.46 tff(71,plain,
% 0.22/0.46 ((~![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))) | (leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) <=> (addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(72,plain,
% 0.22/0.46 (leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) <=> (addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[71, 70])).
% 0.22/0.46 tff(73,plain,
% 0.22/0.46 ((~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))) <=> (~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(74,axiom,(~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','goals')).
% 0.22/0.46 tff(75,plain,
% 0.22/0.46 (~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.22/0.46 tff(76,plain,
% 0.22/0.46 (~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[75, 73])).
% 0.22/0.46 tff(77,plain,
% 0.22/0.46 (~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[76, 73])).
% 0.22/0.46 tff(78,plain,
% 0.22/0.46 (~![X0: $i, X1: $i] : leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.22/0.46 tff(79,plain,(
% 0.22/0.46 ~leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), multiplication(strong_iteration(multiplication(X0!1, X1!0)), X0!1))),
% 0.22/0.46 inference(skolemize,[status(sab)],[78])).
% 0.22/0.46 tff(80,plain,
% 0.22/0.46 (^[A: $i, B: $i, C: $i] : refl(((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B))) <=> ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B))))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(81,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B))) <=> ![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(quant_intro,[status(thm)],[80])).
% 0.22/0.46 tff(82,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B))) <=> ![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(83,plain,
% 0.22/0.46 (^[A: $i, B: $i, C: $i] : rewrite((leq(C, addition(multiplication(A, C), B)) => leq(C, multiplication(strong_iteration(A), B))) <=> ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B))))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(84,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : (leq(C, addition(multiplication(A, C), B)) => leq(C, multiplication(strong_iteration(A), B))) <=> ![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(quant_intro,[status(thm)],[83])).
% 0.22/0.46 tff(85,axiom,(![A: $i, B: $i, C: $i] : (leq(C, addition(multiplication(A, C), B)) => leq(C, multiplication(strong_iteration(A), B)))), file('/export/starexec/sandbox/benchmark/Axioms/KLE004+0.ax','infty_coinduction')).
% 0.22/0.46 tff(86,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.22/0.46 tff(87,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.22/0.46 tff(88,plain,(
% 0.22/0.46 ![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(skolemize,[status(sab)],[87])).
% 0.22/0.46 tff(89,plain,
% 0.22/0.46 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.22/0.46 tff(90,plain,
% 0.22/0.46 (((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | ((~leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))) | leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), multiplication(strong_iteration(multiplication(X0!1, X1!0)), X0!1)))) <=> ((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | (~leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))) | leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), multiplication(strong_iteration(multiplication(X0!1, X1!0)), X0!1)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(91,plain,
% 0.22/0.46 ((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | ((~leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))) | leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), multiplication(strong_iteration(multiplication(X0!1, X1!0)), X0!1)))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(92,plain,
% 0.22/0.46 ((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | (~leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))) | leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), multiplication(strong_iteration(multiplication(X0!1, X1!0)), X0!1))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.22/0.46 tff(93,plain,
% 0.22/0.46 (~leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[92, 89, 79])).
% 0.22/0.46 tff(94,plain,
% 0.22/0.46 ((~(leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) <=> (addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)))) | leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) | (~(addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)))),
% 0.22/0.46 inference(tautology,[status(thm)],[])).
% 0.22/0.46 tff(95,plain,
% 0.22/0.46 ((~(leq(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) <=> (addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)))) | (~(addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.22/0.46 tff(96,plain,
% 0.22/0.46 (~(addition(multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1))), addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1)) = addition(multiplication(multiplication(X0!1, X1!0), multiplication(X0!1, strong_iteration(multiplication(X1!0, X0!1)))), X0!1))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[95, 72])).
% 0.22/0.46 tff(97,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[96, 63])).
% 0.22/0.46 % SZS output end Proof
% 0.22/0.47 % E exiting
%------------------------------------------------------------------------------