%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : KLE143+2 : TPTP v9.0.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 05:13:05 AM UTC 2025
% Result : Theorem 0.69s 0.85s
% Output : Proof 0.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE143+2 : TPTP v9.0.0. Released v4.0.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n010.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 19:04:56 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.69/0.85 % SZS status Theorem
% 0.69/0.85 % SZS output start Proof
% 0.69/0.85 tff(leq_type, type, (
% 0.69/0.85 leq: ( $i * $i ) > $o)).
% 0.69/0.85 tff(strong_iteration_type, type, (
% 0.69/0.85 strong_iteration: $i > $i)).
% 0.69/0.85 tff(tptp_fun_X0_0_type, type, (
% 0.69/0.85 tptp_fun_X0_0: $i)).
% 0.69/0.85 tff(multiplication_type, type, (
% 0.69/0.85 multiplication: ( $i * $i ) > $i)).
% 0.69/0.85 tff(star_type, type, (
% 0.69/0.85 star: $i > $i)).
% 0.69/0.85 tff(addition_type, type, (
% 0.69/0.85 addition: ( $i * $i ) > $i)).
% 0.69/0.85 tff(one_type, type, (
% 0.69/0.85 one: $i)).
% 0.69/0.85 tff(zero_type, type, (
% 0.69/0.85 zero: $i)).
% 0.69/0.85 tff(1,plain,
% 0.69/0.85 (^[A: $i] : refl((strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)) <=> (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)))),
% 0.69/0.85 inference(bind,[status(th)],[])).
% 0.69/0.85 tff(2,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)) <=> ![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.69/0.85 inference(quant_intro,[status(thm)],[1])).
% 0.69/0.85 tff(3,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one)) <=> ![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.69/0.85 inference(rewrite,[status(thm)],[])).
% 0.69/0.85 tff(4,axiom,(![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','infty_unfold1')).
% 0.69/0.85 tff(5,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.69/0.85 tff(6,plain,(
% 0.69/0.85 ![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.69/0.85 inference(skolemize,[status(sab)],[5])).
% 0.69/0.85 tff(7,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.69/0.85 tff(8,plain,
% 0.69/0.85 ((~![A: $i] : (strong_iteration(A) = addition(multiplication(A, strong_iteration(A)), one))) | (strong_iteration(X0!0) = addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.85 inference(quant_inst,[status(thm)],[])).
% 0.69/0.85 tff(9,plain,
% 0.69/0.85 (strong_iteration(X0!0) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.85 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.69/0.85 tff(10,plain,
% 0.69/0.85 (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = strong_iteration(X0!0)),
% 0.69/0.85 inference(symmetry,[status(thm)],[9])).
% 0.69/0.85 tff(11,plain,
% 0.69/0.85 (^[A: $i] : refl((strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero))) <=> (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero))))),
% 0.69/0.85 inference(bind,[status(th)],[])).
% 0.69/0.85 tff(12,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero))) <=> ![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))),
% 0.69/0.85 inference(quant_intro,[status(thm)],[11])).
% 0.69/0.85 tff(13,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero))) <=> ![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))),
% 0.69/0.85 inference(rewrite,[status(thm)],[])).
% 0.69/0.85 tff(14,axiom,(![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','isolation')).
% 0.69/0.85 tff(15,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.69/0.85 tff(16,plain,(
% 0.69/0.85 ![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))),
% 0.69/0.85 inference(skolemize,[status(sab)],[15])).
% 0.69/0.85 tff(17,plain,
% 0.69/0.85 (![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.69/0.85 tff(18,plain,
% 0.69/0.85 ((~![A: $i] : (strong_iteration(A) = addition(star(A), multiplication(strong_iteration(A), zero)))) | (strong_iteration(X0!0) = addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)))),
% 0.69/0.85 inference(quant_inst,[status(thm)],[])).
% 0.69/0.85 tff(19,plain,
% 0.69/0.85 (strong_iteration(X0!0) = addition(star(X0!0), multiplication(strong_iteration(X0!0), zero))),
% 0.69/0.85 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.69/0.85 tff(20,plain,
% 0.69/0.85 (addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)) = strong_iteration(X0!0)),
% 0.69/0.85 inference(symmetry,[status(thm)],[19])).
% 0.69/0.85 tff(21,plain,
% 0.69/0.85 (multiplication(addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(strong_iteration(X0!0), strong_iteration(X0!0))),
% 0.69/0.85 inference(monotonicity,[status(thm)],[20, 10])).
% 0.69/0.85 tff(22,plain,
% 0.69/0.85 (^[A: $i, B: $i, C: $i] : refl((multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C))) <=> (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C))))),
% 0.69/0.85 inference(bind,[status(th)],[])).
% 0.69/0.85 tff(23,plain,
% 0.69/0.85 (![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C))) <=> ![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))),
% 0.69/0.85 inference(quant_intro,[status(thm)],[22])).
% 0.69/0.85 tff(24,plain,
% 0.69/0.85 (![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C))) <=> ![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))),
% 0.69/0.85 inference(rewrite,[status(thm)],[])).
% 0.69/0.85 tff(25,axiom,(![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','distributivity2')).
% 0.69/0.85 tff(26,plain,
% 0.69/0.85 (![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.69/0.85 tff(27,plain,(
% 0.69/0.85 ![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))),
% 0.69/0.85 inference(skolemize,[status(sab)],[26])).
% 0.69/0.85 tff(28,plain,
% 0.69/0.85 (![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.69/0.85 tff(29,plain,
% 0.69/0.85 ((~![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))) | (multiplication(addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(star(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))))),
% 0.69/0.85 inference(quant_inst,[status(thm)],[])).
% 0.69/0.85 tff(30,plain,
% 0.69/0.85 (multiplication(addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(star(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.85 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.69/0.85 tff(31,plain,
% 0.69/0.85 (addition(multiplication(star(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = multiplication(addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.85 inference(symmetry,[status(thm)],[30])).
% 0.69/0.85 tff(32,plain,
% 0.69/0.85 (multiplication(star(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(star(X0!0), strong_iteration(X0!0))),
% 0.69/0.85 inference(monotonicity,[status(thm)],[10])).
% 0.69/0.85 tff(33,plain,
% 0.69/0.85 (multiplication(star(X0!0), strong_iteration(X0!0)) = multiplication(star(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.85 inference(symmetry,[status(thm)],[32])).
% 0.69/0.85 tff(34,plain,
% 0.69/0.85 (addition(multiplication(star(X0!0), strong_iteration(X0!0)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(multiplication(star(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.85 inference(monotonicity,[status(thm)],[33])).
% 0.69/0.85 tff(35,plain,
% 0.69/0.85 (^[A: $i, B: $i] : refl((addition(A, B) = addition(B, A)) <=> (addition(A, B) = addition(B, A)))),
% 0.69/0.85 inference(bind,[status(th)],[])).
% 0.69/0.85 tff(36,plain,
% 0.69/0.85 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.69/0.85 inference(quant_intro,[status(thm)],[35])).
% 0.69/0.85 tff(37,plain,
% 0.69/0.85 (![A: $i, B: $i] : (addition(A, B) = addition(B, A)) <=> ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.69/0.85 inference(rewrite,[status(thm)],[])).
% 0.69/0.85 tff(38,axiom,(![A: $i, B: $i] : (addition(A, B) = addition(B, A))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','additive_commutativity')).
% 0.69/0.85 tff(39,plain,
% 0.69/0.85 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.69/0.85 tff(40,plain,(
% 0.69/0.85 ![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.69/0.85 inference(skolemize,[status(sab)],[39])).
% 0.69/0.85 tff(41,plain,
% 0.69/0.85 (![A: $i, B: $i] : (addition(A, B) = addition(B, A))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.69/0.85 tff(42,plain,
% 0.69/0.85 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(multiplication(star(X0!0), strong_iteration(X0!0)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(star(X0!0), strong_iteration(X0!0))))),
% 0.69/0.85 inference(quant_inst,[status(thm)],[])).
% 0.69/0.85 tff(43,plain,
% 0.69/0.85 (addition(multiplication(star(X0!0), strong_iteration(X0!0)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(star(X0!0), strong_iteration(X0!0)))),
% 0.69/0.85 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.69/0.85 tff(44,plain,
% 0.69/0.85 (addition(multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(star(X0!0), strong_iteration(X0!0))) = addition(multiplication(star(X0!0), strong_iteration(X0!0)), multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.85 inference(symmetry,[status(thm)],[43])).
% 0.69/0.85 tff(45,plain,
% 0.69/0.85 (^[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))))),
% 0.69/0.85 inference(bind,[status(th)],[])).
% 0.69/0.85 tff(46,plain,
% 0.69/0.85 (![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)))),
% 0.69/0.85 inference(quant_intro,[status(thm)],[45])).
% 0.69/0.85 tff(47,plain,
% 0.69/0.85 (![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)))),
% 0.69/0.85 inference(rewrite,[status(thm)],[])).
% 0.69/0.85 tff(48,axiom,(![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','distributivity1')).
% 0.69/0.85 tff(49,plain,
% 0.69/0.85 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.69/0.85 tff(50,plain,(
% 0.69/0.85 ![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.69/0.85 inference(skolemize,[status(sab)],[49])).
% 0.69/0.85 tff(51,plain,
% 0.69/0.85 (![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.69/0.85 tff(52,plain,
% 0.69/0.85 ((~![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))) | (multiplication(X0!0, addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.85 inference(quant_inst,[status(thm)],[])).
% 0.69/0.85 tff(53,plain,
% 0.69/0.85 (multiplication(X0!0, addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))),
% 0.69/0.85 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.69/0.85 tff(54,plain,
% 0.69/0.85 (multiplication(X0!0, strong_iteration(X0!0)) = multiplication(X0!0, addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.85 inference(monotonicity,[status(thm)],[9])).
% 0.69/0.85 tff(55,plain,
% 0.69/0.85 (multiplication(X0!0, strong_iteration(X0!0)) = addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))),
% 0.69/0.85 inference(transitivity,[status(thm)],[54, 53])).
% 0.69/0.85 tff(56,plain,
% 0.69/0.85 (multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero) = multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero)),
% 0.69/0.85 inference(monotonicity,[status(thm)],[55])).
% 0.69/0.85 tff(57,plain,
% 0.69/0.85 (multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero) = multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero)),
% 0.69/0.85 inference(symmetry,[status(thm)],[56])).
% 0.69/0.85 tff(58,plain,
% 0.69/0.85 ((~![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))) | (multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero) = addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)))),
% 0.69/0.85 inference(quant_inst,[status(thm)],[])).
% 0.69/0.85 tff(59,plain,
% 0.69/0.85 (multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero) = addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero))),
% 0.69/0.85 inference(unit_resolution,[status(thm)],[58, 28])).
% 0.69/0.85 tff(60,plain,
% 0.69/0.85 (addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)) = multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero)),
% 0.69/0.85 inference(symmetry,[status(thm)],[59])).
% 0.69/0.85 tff(61,plain,
% 0.69/0.85 (^[A: $i] : refl((addition(A, zero) = A) <=> (addition(A, zero) = A))),
% 0.69/0.85 inference(bind,[status(th)],[])).
% 0.69/0.85 tff(62,plain,
% 0.69/0.85 (![A: $i] : (addition(A, zero) = A) <=> ![A: $i] : (addition(A, zero) = A)),
% 0.69/0.85 inference(quant_intro,[status(thm)],[61])).
% 0.69/0.85 tff(63,plain,
% 0.69/0.85 (![A: $i] : (addition(A, zero) = A) <=> ![A: $i] : (addition(A, zero) = A)),
% 0.69/0.85 inference(rewrite,[status(thm)],[])).
% 0.69/0.85 tff(64,axiom,(![A: $i] : (addition(A, zero) = A)), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','additive_identity')).
% 0.69/0.85 tff(65,plain,
% 0.69/0.85 (![A: $i] : (addition(A, zero) = A)),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.69/0.85 tff(66,plain,(
% 0.69/0.85 ![A: $i] : (addition(A, zero) = A)),
% 0.69/0.85 inference(skolemize,[status(sab)],[65])).
% 0.69/0.85 tff(67,plain,
% 0.69/0.85 (![A: $i] : (addition(A, zero) = A)),
% 0.69/0.85 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.69/0.86 tff(68,plain,
% 0.69/0.86 ((~![A: $i] : (addition(A, zero) = A)) | (addition(addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)), zero) = addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(69,plain,
% 0.69/0.86 (addition(addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)), zero) = addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[68, 67])).
% 0.69/0.86 tff(70,plain,
% 0.69/0.86 (^[A: $i] : refl((multiplication(one, A) = A) <=> (multiplication(one, A) = A))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(71,plain,
% 0.69/0.86 (![A: $i] : (multiplication(one, A) = A) <=> ![A: $i] : (multiplication(one, A) = A)),
% 0.69/0.86 inference(quant_intro,[status(thm)],[70])).
% 0.69/0.86 tff(72,plain,
% 0.69/0.86 (![A: $i] : (multiplication(one, A) = A) <=> ![A: $i] : (multiplication(one, A) = A)),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(73,axiom,(![A: $i] : (multiplication(one, A) = A)), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','multiplicative_left_identity')).
% 0.69/0.86 tff(74,plain,
% 0.69/0.86 (![A: $i] : (multiplication(one, A) = A)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.69/0.86 tff(75,plain,(
% 0.69/0.86 ![A: $i] : (multiplication(one, A) = A)),
% 0.69/0.86 inference(skolemize,[status(sab)],[74])).
% 0.69/0.86 tff(76,plain,
% 0.69/0.86 (![A: $i] : (multiplication(one, A) = A)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.69/0.86 tff(77,plain,
% 0.69/0.86 ((~![A: $i] : (multiplication(one, A) = A)) | (multiplication(one, zero) = zero)),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(78,plain,
% 0.69/0.86 (multiplication(one, zero) = zero),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.69/0.86 tff(79,plain,
% 0.69/0.86 (zero = multiplication(one, zero)),
% 0.69/0.86 inference(symmetry,[status(thm)],[78])).
% 0.69/0.86 tff(80,plain,
% 0.69/0.86 (addition(addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)), zero) = addition(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), multiplication(one, zero))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[60, 79])).
% 0.69/0.86 tff(81,plain,
% 0.69/0.86 (addition(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), multiplication(one, zero)) = addition(addition(multiplication(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), zero), multiplication(multiplication(X0!0, one), zero)), zero)),
% 0.69/0.86 inference(symmetry,[status(thm)],[80])).
% 0.69/0.86 tff(82,plain,
% 0.69/0.86 (addition(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), multiplication(one, zero)) = addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[57])).
% 0.69/0.86 tff(83,plain,
% 0.69/0.86 (addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)) = addition(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), multiplication(one, zero))),
% 0.69/0.86 inference(symmetry,[status(thm)],[82])).
% 0.69/0.86 tff(84,plain,
% 0.69/0.86 ((~![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))) | (multiplication(addition(multiplication(X0!0, strong_iteration(X0!0)), one), zero) = addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(85,plain,
% 0.69/0.86 (multiplication(addition(multiplication(X0!0, strong_iteration(X0!0)), one), zero) = addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[84, 28])).
% 0.69/0.86 tff(86,plain,
% 0.69/0.86 (multiplication(strong_iteration(X0!0), zero) = multiplication(addition(multiplication(X0!0, strong_iteration(X0!0)), one), zero)),
% 0.69/0.86 inference(monotonicity,[status(thm)],[9])).
% 0.69/0.86 tff(87,plain,
% 0.69/0.86 (multiplication(strong_iteration(X0!0), zero) = multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero)),
% 0.69/0.86 inference(transitivity,[status(thm)],[86, 85, 83, 81, 69, 60, 57])).
% 0.69/0.86 tff(88,plain,
% 0.69/0.86 (multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[87])).
% 0.69/0.86 tff(89,plain,
% 0.69/0.86 (multiplication(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(symmetry,[status(thm)],[88])).
% 0.69/0.86 tff(90,plain,
% 0.69/0.86 (multiplication(X0!0, addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(X0!0, strong_iteration(X0!0))),
% 0.69/0.86 inference(symmetry,[status(thm)],[54])).
% 0.69/0.86 tff(91,plain,
% 0.69/0.86 (addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)) = multiplication(X0!0, addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(symmetry,[status(thm)],[53])).
% 0.69/0.86 tff(92,plain,
% 0.69/0.86 (addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)) = multiplication(X0!0, strong_iteration(X0!0))),
% 0.69/0.86 inference(transitivity,[status(thm)],[91, 90])).
% 0.69/0.86 tff(93,plain,
% 0.69/0.86 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = addition(one, multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(94,plain,
% 0.69/0.86 (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = addition(one, multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[93, 41])).
% 0.69/0.86 tff(95,plain,
% 0.69/0.86 (addition(one, multiplication(X0!0, strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.86 inference(symmetry,[status(thm)],[94])).
% 0.69/0.86 tff(96,plain,
% 0.69/0.86 (addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(one, multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[92])).
% 0.69/0.86 tff(97,plain,
% 0.69/0.86 (addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = strong_iteration(X0!0)),
% 0.69/0.86 inference(transitivity,[status(thm)],[96, 95, 10])).
% 0.69/0.86 tff(98,plain,
% 0.69/0.86 (addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[97, 92])).
% 0.69/0.86 tff(99,plain,
% 0.69/0.86 (^[C: $i, B: $i, A: $i] : refl((addition(A, addition(B, C)) = addition(addition(A, B), C)) <=> (addition(A, addition(B, C)) = addition(addition(A, B), C)))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(100,plain,
% 0.69/0.86 (![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C)) <=> ![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))),
% 0.69/0.86 inference(quant_intro,[status(thm)],[99])).
% 0.69/0.86 tff(101,plain,
% 0.69/0.86 (![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C)) <=> ![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(102,axiom,(![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','additive_associativity')).
% 0.69/0.86 tff(103,plain,
% 0.69/0.86 (![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.69/0.86 tff(104,plain,(
% 0.69/0.86 ![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))),
% 0.69/0.86 inference(skolemize,[status(sab)],[103])).
% 0.69/0.86 tff(105,plain,
% 0.69/0.86 (![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[104, 100])).
% 0.69/0.86 tff(106,plain,
% 0.69/0.86 ((~![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))) | (addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))) = addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(107,plain,
% 0.69/0.86 (addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))) = addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[106, 105])).
% 0.69/0.86 tff(108,plain,
% 0.69/0.86 (^[A: $i] : refl((addition(A, A) = A) <=> (addition(A, A) = A))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(109,plain,
% 0.69/0.86 (![A: $i] : (addition(A, A) = A) <=> ![A: $i] : (addition(A, A) = A)),
% 0.69/0.86 inference(quant_intro,[status(thm)],[108])).
% 0.69/0.86 tff(110,plain,
% 0.69/0.86 (![A: $i] : (addition(A, A) = A) <=> ![A: $i] : (addition(A, A) = A)),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(111,axiom,(![A: $i] : (addition(A, A) = A)), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','idempotence')).
% 0.69/0.86 tff(112,plain,
% 0.69/0.86 (![A: $i] : (addition(A, A) = A)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.69/0.86 tff(113,plain,(
% 0.69/0.86 ![A: $i] : (addition(A, A) = A)),
% 0.69/0.86 inference(skolemize,[status(sab)],[112])).
% 0.69/0.86 tff(114,plain,
% 0.69/0.86 (![A: $i] : (addition(A, A) = A)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[113, 109])).
% 0.69/0.86 tff(115,plain,
% 0.69/0.86 ((~![A: $i] : (addition(A, A) = A)) | (addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(116,plain,
% 0.69/0.86 (addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[115, 114])).
% 0.69/0.86 tff(117,plain,
% 0.69/0.86 (addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)) = addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.86 inference(symmetry,[status(thm)],[116])).
% 0.69/0.86 tff(118,plain,
% 0.69/0.86 (multiplication(X0!0, strong_iteration(X0!0)) = addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.86 inference(transitivity,[status(thm)],[54, 53, 117])).
% 0.69/0.86 tff(119,plain,
% 0.69/0.86 (addition(one, multiplication(X0!0, strong_iteration(X0!0))) = addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[118])).
% 0.69/0.86 tff(120,plain,
% 0.69/0.86 (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.86 inference(transitivity,[status(thm)],[94, 119, 107, 98])).
% 0.69/0.86 tff(121,plain,
% 0.69/0.86 (multiplication(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[56, 120])).
% 0.69/0.86 tff(122,plain,
% 0.69/0.86 (multiplication(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = multiplication(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(symmetry,[status(thm)],[121])).
% 0.69/0.86 tff(123,plain,
% 0.69/0.86 (^[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)))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(124,plain,
% 0.69/0.86 (![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))),
% 0.69/0.86 inference(quant_intro,[status(thm)],[123])).
% 0.69/0.86 tff(125,plain,
% 0.69/0.86 (![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))),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(126,axiom,(![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','multiplicative_associativity')).
% 0.69/0.86 tff(127,plain,
% 0.69/0.86 (![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.69/0.86 tff(128,plain,(
% 0.69/0.86 ![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.69/0.86 inference(skolemize,[status(sab)],[127])).
% 0.69/0.86 tff(129,plain,
% 0.69/0.86 (![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[128, 124])).
% 0.69/0.86 tff(130,plain,
% 0.69/0.86 ((~![A: $i, B: $i, C: $i] : (multiplication(A, multiplication(B, C)) = multiplication(multiplication(A, B), C))) | (multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), multiplication(zero, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = multiplication(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(131,plain,
% 0.69/0.86 (multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), multiplication(zero, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = multiplication(multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), zero), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[130, 129])).
% 0.69/0.86 tff(132,plain,
% 0.69/0.86 (^[A: $i] : refl((multiplication(zero, A) = zero) <=> (multiplication(zero, A) = zero))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(133,plain,
% 0.69/0.86 (![A: $i] : (multiplication(zero, A) = zero) <=> ![A: $i] : (multiplication(zero, A) = zero)),
% 0.69/0.86 inference(quant_intro,[status(thm)],[132])).
% 0.69/0.86 tff(134,plain,
% 0.69/0.86 (![A: $i] : (multiplication(zero, A) = zero) <=> ![A: $i] : (multiplication(zero, A) = zero)),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(135,axiom,(![A: $i] : (multiplication(zero, A) = zero)), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','left_annihilation')).
% 0.69/0.86 tff(136,plain,
% 0.69/0.86 (![A: $i] : (multiplication(zero, A) = zero)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.69/0.86 tff(137,plain,(
% 0.69/0.86 ![A: $i] : (multiplication(zero, A) = zero)),
% 0.69/0.86 inference(skolemize,[status(sab)],[136])).
% 0.69/0.86 tff(138,plain,
% 0.69/0.86 (![A: $i] : (multiplication(zero, A) = zero)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[137, 133])).
% 0.69/0.86 tff(139,plain,
% 0.69/0.86 ((~![A: $i] : (multiplication(zero, A) = zero)) | (multiplication(zero, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = zero)),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(140,plain,
% 0.69/0.86 (multiplication(zero, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = zero),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[139, 138])).
% 0.69/0.86 tff(141,plain,
% 0.69/0.86 (multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), multiplication(zero, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero)),
% 0.69/0.86 inference(monotonicity,[status(thm)],[92, 140])).
% 0.69/0.86 tff(142,plain,
% 0.69/0.86 (multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero) = multiplication(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), multiplication(zero, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))))),
% 0.69/0.86 inference(symmetry,[status(thm)],[141])).
% 0.69/0.86 tff(143,plain,
% 0.69/0.86 (addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)) = multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(transitivity,[status(thm)],[83, 81, 69, 60, 57, 142, 131, 122, 89])).
% 0.69/0.86 tff(144,plain,
% 0.69/0.86 (addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), multiplication(star(X0!0), strong_iteration(X0!0))) = addition(multiplication(multiplication(strong_iteration(X0!0), zero), addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(star(X0!0), strong_iteration(X0!0)))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[143])).
% 0.69/0.86 tff(145,plain,
% 0.69/0.86 ((~![A: $i] : (addition(A, A) = A)) | (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(146,plain,
% 0.69/0.86 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[145, 114])).
% 0.69/0.86 tff(147,plain,
% 0.69/0.86 (addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.86 inference(transitivity,[status(thm)],[96, 95])).
% 0.69/0.86 tff(148,plain,
% 0.69/0.86 (addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[147])).
% 0.69/0.86 tff(149,plain,
% 0.69/0.86 ((~![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))) | (addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(150,plain,
% 0.69/0.86 (addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[149, 105])).
% 0.69/0.86 tff(151,plain,
% 0.69/0.86 (addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[92, 10])).
% 0.69/0.86 tff(152,plain,
% 0.69/0.86 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(symmetry,[status(thm)],[151])).
% 0.69/0.86 tff(153,plain,
% 0.69/0.86 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(154,plain,
% 0.69/0.86 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[153, 41])).
% 0.69/0.86 tff(155,plain,
% 0.69/0.86 (addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.86 inference(symmetry,[status(thm)],[154])).
% 0.69/0.86 tff(156,plain,
% 0.69/0.86 (addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))) = addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(transitivity,[status(thm)],[155, 152])).
% 0.69/0.86 tff(157,plain,
% 0.69/0.86 (addition(one, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[156])).
% 0.69/0.86 tff(158,plain,
% 0.69/0.86 (addition(one, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = strong_iteration(X0!0)),
% 0.69/0.86 inference(transitivity,[status(thm)],[157, 150, 148, 146, 10])).
% 0.69/0.86 tff(159,plain,
% 0.69/0.86 (multiplication(star(X0!0), addition(one, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = multiplication(star(X0!0), strong_iteration(X0!0))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[158])).
% 0.69/0.86 tff(160,plain,
% 0.69/0.86 ((~![A: $i, B: $i, C: $i] : (multiplication(A, addition(B, C)) = addition(multiplication(A, B), multiplication(A, C)))) | (multiplication(star(X0!0), addition(one, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = addition(multiplication(star(X0!0), one), multiplication(star(X0!0), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(161,plain,
% 0.69/0.86 (multiplication(star(X0!0), addition(one, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = addition(multiplication(star(X0!0), one), multiplication(star(X0!0), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[160, 51])).
% 0.69/0.86 tff(162,plain,
% 0.69/0.86 (addition(multiplication(star(X0!0), one), multiplication(star(X0!0), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))) = multiplication(star(X0!0), addition(one, addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))))),
% 0.69/0.86 inference(symmetry,[status(thm)],[161])).
% 0.69/0.86 tff(163,plain,
% 0.69/0.86 (addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))) = addition(one, multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.86 inference(symmetry,[status(thm)],[119])).
% 0.69/0.86 tff(164,plain,
% 0.69/0.86 (addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))))),
% 0.69/0.86 inference(symmetry,[status(thm)],[107])).
% 0.69/0.86 tff(165,plain,
% 0.69/0.86 (addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))) = addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.86 inference(symmetry,[status(thm)],[98])).
% 0.69/0.86 tff(166,plain,
% 0.69/0.86 (addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))) = strong_iteration(X0!0)),
% 0.69/0.86 inference(transitivity,[status(thm)],[165, 164, 163, 95, 10])).
% 0.69/0.86 tff(167,plain,
% 0.69/0.86 (multiplication(star(X0!0), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = multiplication(star(X0!0), strong_iteration(X0!0))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[166])).
% 0.69/0.86 tff(168,plain,
% 0.69/0.86 (multiplication(star(X0!0), strong_iteration(X0!0)) = multiplication(star(X0!0), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.86 inference(symmetry,[status(thm)],[167])).
% 0.69/0.86 tff(169,plain,
% 0.69/0.86 (^[A: $i] : refl((multiplication(A, one) = A) <=> (multiplication(A, one) = A))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(170,plain,
% 0.69/0.86 (![A: $i] : (multiplication(A, one) = A) <=> ![A: $i] : (multiplication(A, one) = A)),
% 0.69/0.86 inference(quant_intro,[status(thm)],[169])).
% 0.69/0.86 tff(171,plain,
% 0.69/0.86 (![A: $i] : (multiplication(A, one) = A) <=> ![A: $i] : (multiplication(A, one) = A)),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(172,axiom,(![A: $i] : (multiplication(A, one) = A)), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','multiplicative_right_identity')).
% 0.69/0.86 tff(173,plain,
% 0.69/0.86 (![A: $i] : (multiplication(A, one) = A)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.69/0.86 tff(174,plain,(
% 0.69/0.86 ![A: $i] : (multiplication(A, one) = A)),
% 0.69/0.86 inference(skolemize,[status(sab)],[173])).
% 0.69/0.86 tff(175,plain,
% 0.69/0.86 (![A: $i] : (multiplication(A, one) = A)),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[174, 170])).
% 0.69/0.86 tff(176,plain,
% 0.69/0.86 ((~![A: $i] : (multiplication(A, one) = A)) | (multiplication(star(X0!0), one) = star(X0!0))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(177,plain,
% 0.69/0.86 (multiplication(star(X0!0), one) = star(X0!0)),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[176, 175])).
% 0.69/0.86 tff(178,plain,
% 0.69/0.86 (star(X0!0) = multiplication(star(X0!0), one)),
% 0.69/0.86 inference(symmetry,[status(thm)],[177])).
% 0.69/0.86 tff(179,plain,
% 0.69/0.86 (addition(star(X0!0), multiplication(star(X0!0), strong_iteration(X0!0))) = addition(multiplication(star(X0!0), one), multiplication(star(X0!0), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[178, 168])).
% 0.69/0.86 tff(180,plain,
% 0.69/0.86 (addition(star(X0!0), multiplication(star(X0!0), strong_iteration(X0!0))) = multiplication(star(X0!0), strong_iteration(X0!0))),
% 0.69/0.86 inference(transitivity,[status(thm)],[179, 162, 159])).
% 0.69/0.86 tff(181,plain,
% 0.69/0.86 (addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), addition(star(X0!0), multiplication(star(X0!0), strong_iteration(X0!0)))) = addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), multiplication(star(X0!0), strong_iteration(X0!0)))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[180])).
% 0.69/0.86 tff(182,plain,
% 0.69/0.86 ((~![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))) | (addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), addition(star(X0!0), multiplication(star(X0!0), strong_iteration(X0!0)))) = addition(addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0)), multiplication(star(X0!0), strong_iteration(X0!0))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(183,plain,
% 0.69/0.86 (addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), addition(star(X0!0), multiplication(star(X0!0), strong_iteration(X0!0)))) = addition(addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0)), multiplication(star(X0!0), strong_iteration(X0!0)))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[182, 105])).
% 0.69/0.86 tff(184,plain,
% 0.69/0.86 (addition(addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0)), multiplication(star(X0!0), strong_iteration(X0!0))) = addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), addition(star(X0!0), multiplication(star(X0!0), strong_iteration(X0!0))))),
% 0.69/0.86 inference(symmetry,[status(thm)],[183])).
% 0.69/0.86 tff(185,plain,
% 0.69/0.86 (multiplication(addition(multiplication(X0!0, strong_iteration(X0!0)), one), zero) = multiplication(strong_iteration(X0!0), zero)),
% 0.69/0.86 inference(symmetry,[status(thm)],[86])).
% 0.69/0.86 tff(186,plain,
% 0.69/0.86 (addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)) = multiplication(addition(multiplication(X0!0, strong_iteration(X0!0)), one), zero)),
% 0.69/0.86 inference(symmetry,[status(thm)],[85])).
% 0.69/0.86 tff(187,plain,
% 0.69/0.86 (addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)) = multiplication(strong_iteration(X0!0), zero)),
% 0.69/0.86 inference(transitivity,[status(thm)],[186, 185])).
% 0.69/0.86 tff(188,plain,
% 0.69/0.86 (addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0)) = addition(multiplication(strong_iteration(X0!0), zero), star(X0!0))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[187])).
% 0.69/0.86 tff(189,plain,
% 0.69/0.86 (addition(multiplication(strong_iteration(X0!0), zero), star(X0!0)) = addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0))),
% 0.69/0.86 inference(symmetry,[status(thm)],[188])).
% 0.69/0.86 tff(190,plain,
% 0.69/0.86 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)) = addition(multiplication(strong_iteration(X0!0), zero), star(X0!0)))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(191,plain,
% 0.69/0.86 (addition(star(X0!0), multiplication(strong_iteration(X0!0), zero)) = addition(multiplication(strong_iteration(X0!0), zero), star(X0!0))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[190, 41])).
% 0.69/0.86 tff(192,plain,
% 0.69/0.86 (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0))),
% 0.69/0.86 inference(transitivity,[status(thm)],[10, 19, 191, 189])).
% 0.69/0.86 tff(193,plain,
% 0.69/0.86 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(star(X0!0), strong_iteration(X0!0))) = addition(addition(addition(multiplication(multiplication(X0!0, strong_iteration(X0!0)), zero), multiplication(one, zero)), star(X0!0)), multiplication(star(X0!0), strong_iteration(X0!0)))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[192])).
% 0.69/0.86 tff(194,plain,
% 0.69/0.86 (^[A: $i] : refl((addition(one, multiplication(A, star(A))) = star(A)) <=> (addition(one, multiplication(A, star(A))) = star(A)))),
% 0.69/0.86 inference(bind,[status(th)],[])).
% 0.69/0.86 tff(195,plain,
% 0.69/0.86 (![A: $i] : (addition(one, multiplication(A, star(A))) = star(A)) <=> ![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))),
% 0.69/0.86 inference(quant_intro,[status(thm)],[194])).
% 0.69/0.86 tff(196,plain,
% 0.69/0.86 (![A: $i] : (addition(one, multiplication(A, star(A))) = star(A)) <=> ![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))),
% 0.69/0.86 inference(rewrite,[status(thm)],[])).
% 0.69/0.86 tff(197,axiom,(![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','star_unfold1')).
% 0.69/0.86 tff(198,plain,
% 0.69/0.86 (![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[197, 196])).
% 0.69/0.86 tff(199,plain,(
% 0.69/0.86 ![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))),
% 0.69/0.86 inference(skolemize,[status(sab)],[198])).
% 0.69/0.86 tff(200,plain,
% 0.69/0.86 (![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))),
% 0.69/0.86 inference(modus_ponens,[status(thm)],[199, 195])).
% 0.69/0.86 tff(201,plain,
% 0.69/0.86 ((~![A: $i] : (addition(one, multiplication(A, star(A))) = star(A))) | (addition(one, multiplication(X0!0, star(X0!0))) = star(X0!0))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(202,plain,
% 0.69/0.86 (addition(one, multiplication(X0!0, star(X0!0))) = star(X0!0)),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[201, 200])).
% 0.69/0.86 tff(203,plain,
% 0.69/0.86 (star(X0!0) = addition(one, multiplication(X0!0, star(X0!0)))),
% 0.69/0.86 inference(symmetry,[status(thm)],[202])).
% 0.69/0.86 tff(204,plain,
% 0.69/0.86 (multiplication(star(X0!0), strong_iteration(X0!0)) = multiplication(addition(one, multiplication(X0!0, star(X0!0))), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.86 inference(monotonicity,[status(thm)],[203, 9])).
% 0.69/0.86 tff(205,plain,
% 0.69/0.86 (multiplication(addition(one, multiplication(X0!0, star(X0!0))), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = multiplication(star(X0!0), strong_iteration(X0!0))),
% 0.69/0.86 inference(symmetry,[status(thm)],[204])).
% 0.69/0.86 tff(206,plain,
% 0.69/0.86 ((~![A: $i, B: $i, C: $i] : (multiplication(addition(A, B), C) = addition(multiplication(A, C), multiplication(B, C)))) | (multiplication(addition(one, multiplication(X0!0, star(X0!0))), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))))),
% 0.69/0.86 inference(quant_inst,[status(thm)],[])).
% 0.69/0.86 tff(207,plain,
% 0.69/0.86 (multiplication(addition(one, multiplication(X0!0, star(X0!0))), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.86 inference(unit_resolution,[status(thm)],[206, 28])).
% 0.69/0.86 tff(208,plain,
% 0.69/0.86 (addition(multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = multiplication(addition(one, multiplication(X0!0, star(X0!0))), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(symmetry,[status(thm)],[207])).
% 0.69/0.87 tff(209,plain,
% 0.69/0.87 ((~![A: $i] : (multiplication(one, A) = A)) | (multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(210,plain,
% 0.69/0.87 (multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[209, 76])).
% 0.69/0.87 tff(211,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(symmetry,[status(thm)],[210])).
% 0.69/0.87 tff(212,plain,
% 0.69/0.87 (addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))) = multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(transitivity,[status(thm)],[165, 164, 163, 95, 211])).
% 0.69/0.87 tff(213,plain,
% 0.69/0.87 (addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[212])).
% 0.69/0.87 tff(214,plain,
% 0.69/0.87 (addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = multiplication(star(X0!0), strong_iteration(X0!0))),
% 0.69/0.87 inference(transitivity,[status(thm)],[213, 208, 205])).
% 0.69/0.87 tff(215,plain,
% 0.69/0.87 (addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.87 inference(transitivity,[status(thm)],[165, 164, 163, 95])).
% 0.69/0.87 tff(216,plain,
% 0.69/0.87 (addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(star(X0!0), strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[215, 214])).
% 0.69/0.87 tff(217,plain,
% 0.69/0.87 ((~![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))) | (addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))) = addition(addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(218,plain,
% 0.69/0.87 (addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))) = addition(addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[217, 105])).
% 0.69/0.87 tff(219,plain,
% 0.69/0.87 (addition(addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))))),
% 0.69/0.87 inference(symmetry,[status(thm)],[218])).
% 0.69/0.87 tff(220,plain,
% 0.69/0.87 (addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))) = addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[166, 155])).
% 0.69/0.87 tff(221,plain,
% 0.69/0.87 (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.87 inference(symmetry,[status(thm)],[220])).
% 0.69/0.87 tff(222,plain,
% 0.69/0.87 (^[A: $i, B: $i] : refl((leq(A, B) <=> (addition(A, B) = B)) <=> (leq(A, B) <=> (addition(A, B) = B)))),
% 0.69/0.87 inference(bind,[status(th)],[])).
% 0.69/0.87 tff(223,plain,
% 0.69/0.87 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B)) <=> ![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.69/0.87 inference(quant_intro,[status(thm)],[222])).
% 0.69/0.87 tff(224,plain,
% 0.69/0.87 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B)) <=> ![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(225,axiom,(![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','order')).
% 0.69/0.87 tff(226,plain,
% 0.69/0.87 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[225, 224])).
% 0.69/0.87 tff(227,plain,(
% 0.69/0.87 ![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.69/0.87 inference(skolemize,[status(sab)],[226])).
% 0.69/0.87 tff(228,plain,
% 0.69/0.87 (![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[227, 223])).
% 0.69/0.87 tff(229,plain,
% 0.69/0.87 ((~![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))) | (leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(230,plain,
% 0.69/0.87 (leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[229, 228])).
% 0.69/0.87 tff(231,plain,
% 0.69/0.87 ((~![C: $i, B: $i, A: $i] : (addition(A, addition(B, C)) = addition(addition(A, B), C))) | (addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(232,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[231, 105])).
% 0.69/0.87 tff(233,plain,
% 0.69/0.87 ((~![A: $i, B: $i] : (addition(A, B) = addition(B, A))) | (addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(234,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[233, 41])).
% 0.69/0.87 tff(235,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(X0!0, strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(symmetry,[status(thm)],[234])).
% 0.69/0.87 tff(236,plain,
% 0.69/0.87 (addition(one, multiplication(X0!0, strong_iteration(X0!0))) = addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.87 inference(symmetry,[status(thm)],[96])).
% 0.69/0.87 tff(237,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), one) = addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.87 inference(transitivity,[status(thm)],[94, 236])).
% 0.69/0.87 tff(238,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(X0!0, strong_iteration(X0!0))) = addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[237, 55])).
% 0.69/0.87 tff(239,plain,
% 0.69/0.87 (addition(addition(one, addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.87 inference(symmetry,[status(thm)],[238])).
% 0.69/0.87 tff(240,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))) = multiplication(X0!0, strong_iteration(X0!0))),
% 0.69/0.87 inference(transitivity,[status(thm)],[116, 91, 90])).
% 0.69/0.87 tff(241,plain,
% 0.69/0.87 (addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)))) = addition(one, multiplication(X0!0, strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[240])).
% 0.69/0.87 tff(242,plain,
% 0.69/0.87 (addition(one, multiplication(X0!0, strong_iteration(X0!0))) = addition(one, addition(addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one)), addition(multiplication(X0!0, multiplication(X0!0, strong_iteration(X0!0))), multiplication(X0!0, one))))),
% 0.69/0.87 inference(symmetry,[status(thm)],[241])).
% 0.69/0.87 tff(243,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[9])).
% 0.69/0.87 tff(244,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(multiplication(X0!0, strong_iteration(X0!0)), one)),
% 0.69/0.87 inference(transitivity,[status(thm)],[243, 234, 238, 164, 241, 95])).
% 0.69/0.87 tff(245,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[244])).
% 0.69/0.87 tff(246,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(symmetry,[status(thm)],[243])).
% 0.69/0.87 tff(247,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one) = addition(multiplication(X0!0, strong_iteration(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))),
% 0.69/0.87 inference(symmetry,[status(thm)],[232])).
% 0.69/0.87 tff(248,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(transitivity,[status(thm)],[247, 246])).
% 0.69/0.87 tff(249,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[248])).
% 0.69/0.87 tff(250,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)),
% 0.69/0.87 inference(transitivity,[status(thm)],[249, 245, 146, 94, 242, 107, 239, 235, 232])).
% 0.69/0.87 tff(251,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)),
% 0.69/0.87 inference(transitivity,[status(thm)],[243, 232])).
% 0.69/0.87 tff(252,plain,
% 0.69/0.87 (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one))),
% 0.69/0.87 inference(symmetry,[status(thm)],[249])).
% 0.69/0.87 tff(253,plain,
% 0.69/0.87 ((addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[252, 251])).
% 0.69/0.87 tff(254,plain,
% 0.69/0.87 ((~(addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) <=> (~(addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[253])).
% 0.69/0.87 tff(255,plain,
% 0.69/0.87 ((~![A: $i, B: $i] : (leq(A, B) <=> (addition(A, B) = B))) | (leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(256,plain,
% 0.69/0.87 (leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[255, 228])).
% 0.69/0.87 tff(257,plain,
% 0.69/0.87 (leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[10])).
% 0.69/0.87 tff(258,plain,
% 0.69/0.87 (leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(symmetry,[status(thm)],[257])).
% 0.69/0.87 tff(259,plain,
% 0.69/0.87 ((~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) <=> (~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[258])).
% 0.69/0.87 tff(260,assumption,(~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))), introduced(assumption)).
% 0.69/0.87 tff(261,plain,
% 0.69/0.87 (~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[260, 259])).
% 0.69/0.87 tff(262,plain,
% 0.69/0.87 ((~(leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))) | leq(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) | (~(addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))),
% 0.69/0.87 inference(tautology,[status(thm)],[])).
% 0.69/0.87 tff(263,plain,
% 0.69/0.87 (~(addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[262, 261, 256])).
% 0.69/0.87 tff(264,plain,
% 0.69/0.87 (~(addition(addition(multiplication(X0!0, strong_iteration(X0!0)), one), addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one)) = addition(addition(multiplication(X0!0, strong_iteration(X0!0)), multiplication(X0!0, strong_iteration(X0!0))), one))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[263, 254])).
% 0.69/0.87 tff(265,plain,
% 0.69/0.87 ($false),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[264, 250])).
% 0.69/0.87 tff(266,plain,(leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))), inference(lemma,lemma(discharge,[]))).
% 0.69/0.87 tff(267,plain,
% 0.69/0.87 ((~(leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))) | (~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) | (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(tautology,[status(thm)],[])).
% 0.69/0.87 tff(268,plain,
% 0.69/0.87 ((~(leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))))) | (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[267, 266])).
% 0.69/0.87 tff(269,plain,
% 0.69/0.87 (addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[268, 230])).
% 0.69/0.87 tff(270,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = addition(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(symmetry,[status(thm)],[269])).
% 0.69/0.87 tff(271,plain,
% 0.69/0.87 (multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)) = addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))))),
% 0.69/0.87 inference(transitivity,[status(thm)],[210, 94, 119, 107, 98, 155, 270, 221])).
% 0.69/0.87 tff(272,plain,
% 0.69/0.87 (addition(multiplication(one, addition(multiplication(X0!0, strong_iteration(X0!0)), one)), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one))) = addition(addition(addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0))), addition(strong_iteration(X0!0), multiplication(X0!0, strong_iteration(X0!0)))), multiplication(multiplication(X0!0, star(X0!0)), addition(multiplication(X0!0, strong_iteration(X0!0)), one)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[271])).
% 0.69/0.87 tff(273,plain,
% 0.69/0.87 (multiplication(star(X0!0), strong_iteration(X0!0)) = multiplication(strong_iteration(X0!0), strong_iteration(X0!0))),
% 0.69/0.87 inference(transitivity,[status(thm)],[204, 207, 272, 219, 216, 193, 184, 181, 144, 44, 34, 31, 21])).
% 0.69/0.87 tff(274,plain,
% 0.69/0.87 (leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)) <=> leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[273])).
% 0.69/0.87 tff(275,plain,
% 0.69/0.87 (leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)) <=> leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(symmetry,[status(thm)],[274])).
% 0.69/0.87 tff(276,plain,
% 0.69/0.87 ((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) <=> (~leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[275])).
% 0.69/0.87 tff(277,plain,
% 0.69/0.87 (^[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))))),
% 0.69/0.87 inference(bind,[status(th)],[])).
% 0.69/0.87 tff(278,plain,
% 0.69/0.87 (![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)))),
% 0.69/0.87 inference(quant_intro,[status(thm)],[277])).
% 0.69/0.87 tff(279,plain,
% 0.69/0.87 (![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)))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(280,plain,
% 0.69/0.87 (^[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))))),
% 0.69/0.87 inference(bind,[status(th)],[])).
% 0.69/0.87 tff(281,plain,
% 0.69/0.87 (![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)))),
% 0.69/0.87 inference(quant_intro,[status(thm)],[280])).
% 0.69/0.87 tff(282,axiom,(![A: $i, B: $i, C: $i] : (leq(C, addition(multiplication(A, C), B)) => leq(C, multiplication(strong_iteration(A), B)))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','infty_coinduction')).
% 0.69/0.87 tff(283,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[282, 281])).
% 0.69/0.87 tff(284,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[283, 279])).
% 0.69/0.87 tff(285,plain,(
% 0.69/0.87 ![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.69/0.87 inference(skolemize,[status(sab)],[284])).
% 0.69/0.87 tff(286,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[285, 278])).
% 0.69/0.87 tff(287,plain,
% 0.69/0.87 (((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | ((~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) | leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))) <=> ((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | (~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) | leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(288,plain,
% 0.69/0.87 ((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | ((~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) | leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))),
% 0.69/0.87 inference(quant_inst,[status(thm)],[])).
% 0.69/0.87 tff(289,plain,
% 0.69/0.87 ((~![A: $i, B: $i, C: $i] : ((~leq(C, addition(multiplication(A, C), B))) | leq(C, multiplication(strong_iteration(A), B)))) | (~leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))) | leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0)))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[288, 287])).
% 0.69/0.87 tff(290,plain,
% 0.69/0.87 (leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0)))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[289, 286, 266])).
% 0.69/0.87 tff(291,plain,
% 0.69/0.87 ((~(~((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) | (~leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))))) <=> ((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) | (~leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0)))))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(292,plain,
% 0.69/0.87 ((leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)) & leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0)))) <=> (~((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) | (~leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(293,plain,
% 0.69/0.87 ((~(leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)) & leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))) <=> (~(~((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) | (~leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0)))))))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[292])).
% 0.69/0.87 tff(294,plain,
% 0.69/0.87 ((~(leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)) & leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))) <=> ((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) | (~leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0)))))),
% 0.69/0.87 inference(transitivity,[status(thm)],[293, 291])).
% 0.69/0.87 tff(295,plain,
% 0.69/0.87 ((~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))) <=> (~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0)))))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(296,axiom,(~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','goals')).
% 0.69/0.87 tff(297,plain,
% 0.69/0.87 (~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[296, 295])).
% 0.69/0.87 tff(298,plain,
% 0.69/0.87 (~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[297, 295])).
% 0.69/0.87 tff(299,plain,
% 0.69/0.87 (~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[298, 295])).
% 0.69/0.87 tff(300,plain,
% 0.69/0.87 (~![X0: $i] : (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[299, 295])).
% 0.69/0.87 tff(301,plain,(
% 0.69/0.87 ~(leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)) & leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))),
% 0.69/0.87 inference(skolemize,[status(sab)],[300])).
% 0.69/0.87 tff(302,plain,
% 0.69/0.87 ((~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))) | (~leq(strong_iteration(X0!0), multiplication(strong_iteration(X0!0), strong_iteration(X0!0))))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[301, 294])).
% 0.69/0.87 tff(303,plain,
% 0.69/0.87 (~leq(multiplication(strong_iteration(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(unit_resolution,[status(thm)],[302, 290])).
% 0.69/0.87 tff(304,plain,
% 0.69/0.87 (~leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[303, 276])).
% 0.69/0.87 tff(305,plain,
% 0.69/0.87 (strong_iteration(X0!0) = addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(transitivity,[status(thm)],[9, 94, 119, 107, 98, 155])).
% 0.69/0.87 tff(306,plain,
% 0.69/0.87 (addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)) = strong_iteration(X0!0)),
% 0.69/0.87 inference(transitivity,[status(thm)],[154, 165, 164, 163, 95, 10])).
% 0.69/0.87 tff(307,plain,
% 0.69/0.87 (leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0)) <=> leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(monotonicity,[status(thm)],[306, 305])).
% 0.69/0.87 tff(308,plain,
% 0.69/0.87 (leq(strong_iteration(X0!0), addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0))) <=> leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(symmetry,[status(thm)],[307])).
% 0.69/0.87 tff(309,plain,
% 0.69/0.87 (leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[266, 308])).
% 0.69/0.87 tff(310,plain,
% 0.69/0.87 (^[A: $i, B: $i, C: $i] : refl(((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C)) <=> ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C)))),
% 0.69/0.87 inference(bind,[status(th)],[])).
% 0.69/0.87 tff(311,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C)) <=> ![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(quant_intro,[status(thm)],[310])).
% 0.69/0.87 tff(312,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C)) <=> ![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(313,plain,
% 0.69/0.87 (^[A: $i, B: $i, C: $i] : rewrite((leq(addition(multiplication(A, C), B), C) => leq(multiplication(star(A), B), C)) <=> ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C)))),
% 0.69/0.87 inference(bind,[status(th)],[])).
% 0.69/0.87 tff(314,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : (leq(addition(multiplication(A, C), B), C) => leq(multiplication(star(A), B), C)) <=> ![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(quant_intro,[status(thm)],[313])).
% 0.69/0.87 tff(315,axiom,(![A: $i, B: $i, C: $i] : (leq(addition(multiplication(A, C), B), C) => leq(multiplication(star(A), B), C))), file('/export/starexec/sandbox2/benchmark/Axioms/KLE004+0.ax','star_induction1')).
% 0.69/0.87 tff(316,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[315, 314])).
% 0.69/0.87 tff(317,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[316, 312])).
% 0.69/0.87 tff(318,plain,(
% 0.69/0.87 ![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(skolemize,[status(sab)],[317])).
% 0.69/0.87 tff(319,plain,
% 0.69/0.87 (![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))),
% 0.69/0.87 inference(modus_ponens,[status(thm)],[318, 311])).
% 0.69/0.87 tff(320,plain,
% 0.69/0.87 (((~![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))) | ((~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))) | leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)))) <=> ((~![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))) | (~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))) | leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.87 inference(rewrite,[status(thm)],[])).
% 0.69/0.87 tff(321,plain,
% 0.69/0.87 ((~![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))) | ((~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))) | leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0)))),
% 0.69/0.88 inference(quant_inst,[status(thm)],[])).
% 0.69/0.88 tff(322,plain,
% 0.69/0.88 ((~![A: $i, B: $i, C: $i] : ((~leq(addition(multiplication(A, C), B), C)) | leq(multiplication(star(A), B), C))) | (~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))) | leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.88 inference(modus_ponens,[status(thm)],[321, 320])).
% 0.69/0.88 tff(323,plain,
% 0.69/0.88 ((~leq(addition(multiplication(X0!0, strong_iteration(X0!0)), strong_iteration(X0!0)), strong_iteration(X0!0))) | leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.88 inference(unit_resolution,[status(thm)],[322, 319])).
% 0.69/0.88 tff(324,plain,
% 0.69/0.88 (leq(multiplication(star(X0!0), strong_iteration(X0!0)), strong_iteration(X0!0))),
% 0.69/0.88 inference(unit_resolution,[status(thm)],[323, 309])).
% 0.69/0.88 tff(325,plain,
% 0.69/0.88 ($false),
% 0.69/0.88 inference(unit_resolution,[status(thm)],[324, 304])).
% 0.69/0.88 % SZS output end Proof
% 0.69/0.88 % E exiting
%------------------------------------------------------------------------------