%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE148+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:39:55 AM UTC 2026
% Result : Theorem 2.28s 0.79s
% Output : Proof 3.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE148+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n013.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sun Sep 27 13:11:51 UTC 2026
% 0.08/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 2.28/0.79 Command-line arguments: --flatten --complete-subsets
% 2.28/0.79
% 2.28/0.79 % SZS status Theorem
% 2.28/0.79
% 3.05/0.81 % SZS output start Proof
% 3.05/0.81 Axiom 1 (idempotence): addition(X, X) = X.
% 3.05/0.81 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.05/0.81 Axiom 3 (additive_identity): addition(X, zero) = X.
% 3.05/0.81 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 3.05/0.81 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 3.05/0.81 Axiom 6 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 3.05/0.81 Axiom 7 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 3.05/0.81 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 3.05/0.81 Axiom 9 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 3.05/0.81 Axiom 10 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 3.05/0.81 Axiom 11 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.05/0.81 Axiom 12 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 3.05/0.81 Axiom 13 (goals_1): ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), true, multiplication(x0_2, x1_2), zero) = zero.
% 3.05/0.81
% 3.05/0.81 Lemma 14: addition(X, addition(X, Y)) = addition(X, Y).
% 3.05/0.81 Proof:
% 3.05/0.81 addition(X, addition(X, Y))
% 3.05/0.81 = { by axiom 8 (additive_associativity) }
% 3.05/0.81 addition(addition(X, X), Y)
% 3.05/0.81 = { by axiom 1 (idempotence) }
% 3.05/0.81 addition(X, Y)
% 3.05/0.81
% 3.05/0.81 Lemma 15: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 3.05/0.81 Proof:
% 3.05/0.81 multiplication(X, addition(Y, one))
% 3.05/0.81 = { by axiom 2 (additive_commutativity) R->L }
% 3.05/0.81 multiplication(X, addition(one, Y))
% 3.05/0.81 = { by axiom 11 (distributivity1) }
% 3.05/0.81 addition(multiplication(X, one), multiplication(X, Y))
% 3.05/0.81 = { by axiom 4 (multiplicative_right_identity) }
% 3.05/0.81 addition(X, multiplication(X, Y))
% 3.05/0.81
% 3.05/0.81 Lemma 16: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 3.05/0.81 Proof:
% 3.05/0.81 addition(one, multiplication(X, strong_iteration(X)))
% 3.05/0.81 = { by axiom 2 (additive_commutativity) R->L }
% 3.05/0.81 addition(multiplication(X, strong_iteration(X)), one)
% 3.05/0.81 = { by axiom 9 (infty_unfold1) R->L }
% 3.05/0.81 strong_iteration(X)
% 3.05/0.81
% 3.05/0.81 Lemma 17: addition(multiplication(X, strong_iteration(Y)), X) = multiplication(X, strong_iteration(Y)).
% 3.05/0.81 Proof:
% 3.05/0.81 addition(multiplication(X, strong_iteration(Y)), X)
% 3.05/0.81 = { by axiom 2 (additive_commutativity) R->L }
% 3.05/0.81 addition(X, multiplication(X, strong_iteration(Y)))
% 3.05/0.81 = { by lemma 15 R->L }
% 3.05/0.81 multiplication(X, addition(strong_iteration(Y), one))
% 3.05/0.81 = { by axiom 2 (additive_commutativity) R->L }
% 3.05/0.81 multiplication(X, addition(one, strong_iteration(Y)))
% 3.05/0.81 = { by lemma 16 R->L }
% 3.05/0.81 multiplication(X, addition(one, addition(one, multiplication(Y, strong_iteration(Y)))))
% 3.05/0.81 = { by lemma 14 }
% 3.05/0.81 multiplication(X, addition(one, multiplication(Y, strong_iteration(Y))))
% 3.05/0.81 = { by lemma 16 }
% 3.05/0.81 multiplication(X, strong_iteration(Y))
% 3.05/0.81
% 3.05/0.81 Goal 1 (goals): tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), x0_2), leq(x0, multiplication(x0, strong_iteration(x1)))) = tuple(true, true).
% 3.05/0.81 Proof:
% 3.05/0.81 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), x0_2), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.81 = { by axiom 3 (additive_identity) R->L }
% 3.05/0.81 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, zero)), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.81 = { by axiom 5 (left_annihilation) R->L }
% 3.05/0.81 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(zero, strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.81 = { by axiom 13 (goals_1) R->L }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), true, multiplication(x0_2, x1_2), zero), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 12 (order) R->L }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), ifeq3(addition(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), addition(x0, multiplication(x0, strong_iteration(x1))), leq(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), true), multiplication(x0_2, x1_2), zero), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 14 }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), ifeq3(addition(x0, multiplication(x0, strong_iteration(x1))), addition(x0, multiplication(x0, strong_iteration(x1))), leq(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), true), multiplication(x0_2, x1_2), zero), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 6 (ifeq_axiom) }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), leq(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), multiplication(x0_2, x1_2), zero), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 2 (additive_commutativity) }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), leq(x0, addition(multiplication(x0, strong_iteration(x1)), x0)), multiplication(x0_2, x1_2), zero), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 17 }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(ifeq2(leq(x0, multiplication(x0, strong_iteration(x1))), leq(x0, multiplication(x0, strong_iteration(x1))), multiplication(x0_2, x1_2), zero), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 7 (ifeq_axiom) }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(multiplication(x0_2, x1_2), strong_iteration(x1_2)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 10 (multiplicative_associativity) R->L }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), addition(x0_2, multiplication(x0_2, multiplication(x1_2, strong_iteration(x1_2))))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 15 R->L }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, addition(multiplication(x1_2, strong_iteration(x1_2)), one))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 2 (additive_commutativity) }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, addition(one, multiplication(x1_2, strong_iteration(x1_2))))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 16 }
% 3.05/0.82 tuple(leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 6 (ifeq_axiom) R->L }
% 3.05/0.82 tuple(ifeq3(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2)), leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), true), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 1 (idempotence) R->L }
% 3.05/0.82 tuple(ifeq3(addition(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), multiplication(x0_2, strong_iteration(x1_2)), leq(multiplication(x0_2, strong_iteration(x1_2)), multiplication(x0_2, strong_iteration(x1_2))), true), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 12 (order) }
% 3.05/0.82 tuple(true, leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 12 (order) R->L }
% 3.05/0.82 tuple(ifeq3(addition(X, addition(X, multiplication(X, strong_iteration(Y)))), addition(X, multiplication(X, strong_iteration(Y))), leq(X, addition(X, multiplication(X, strong_iteration(Y)))), true), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 14 }
% 3.05/0.82 tuple(ifeq3(addition(X, multiplication(X, strong_iteration(Y))), addition(X, multiplication(X, strong_iteration(Y))), leq(X, addition(X, multiplication(X, strong_iteration(Y)))), true), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 6 (ifeq_axiom) }
% 3.05/0.82 tuple(leq(X, addition(X, multiplication(X, strong_iteration(Y)))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by axiom 2 (additive_commutativity) }
% 3.05/0.82 tuple(leq(X, addition(multiplication(X, strong_iteration(Y)), X)), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 17 }
% 3.05/0.82 tuple(leq(X, multiplication(X, strong_iteration(Y))), leq(x0, multiplication(x0, strong_iteration(x1))))
% 3.05/0.82 = { by lemma 17 R->L }
% 3.05/0.82 tuple(leq(X, multiplication(X, strong_iteration(Y))), leq(x0, addition(multiplication(x0, strong_iteration(x1)), x0)))
% 3.05/0.82 = { by axiom 2 (additive_commutativity) R->L }
% 3.05/0.82 tuple(leq(X, multiplication(X, strong_iteration(Y))), leq(x0, addition(x0, multiplication(x0, strong_iteration(x1)))))
% 3.05/0.82 = { by axiom 6 (ifeq_axiom) R->L }
% 3.05/0.82 tuple(leq(X, multiplication(X, strong_iteration(Y))), ifeq3(addition(x0, multiplication(x0, strong_iteration(x1))), addition(x0, multiplication(x0, strong_iteration(x1))), leq(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), true))
% 3.05/0.82 = { by lemma 14 R->L }
% 3.05/0.82 tuple(leq(X, multiplication(X, strong_iteration(Y))), ifeq3(addition(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), addition(x0, multiplication(x0, strong_iteration(x1))), leq(x0, addition(x0, multiplication(x0, strong_iteration(x1)))), true))
% 3.05/0.82 = { by axiom 12 (order) }
% 3.05/0.82 tuple(leq(X, multiplication(X, strong_iteration(Y))), true)
% 3.05/0.82 = { by lemma 17 R->L }
% 3.05/0.82 tuple(leq(X, addition(multiplication(X, strong_iteration(Y)), X)), true)
% 3.05/0.82 = { by axiom 2 (additive_commutativity) R->L }
% 3.05/0.82 tuple(leq(X, addition(X, multiplication(X, strong_iteration(Y)))), true)
% 3.05/0.82 = { by axiom 6 (ifeq_axiom) R->L }
% 3.05/0.82 tuple(ifeq3(addition(X, multiplication(X, strong_iteration(Y))), addition(X, multiplication(X, strong_iteration(Y))), leq(X, addition(X, multiplication(X, strong_iteration(Y)))), true), true)
% 3.05/0.82 = { by lemma 14 R->L }
% 3.05/0.82 tuple(ifeq3(addition(X, addition(X, multiplication(X, strong_iteration(Y)))), addition(X, multiplication(X, strong_iteration(Y))), leq(X, addition(X, multiplication(X, strong_iteration(Y)))), true), true)
% 3.05/0.82 = { by axiom 12 (order) }
% 3.05/0.82 tuple(true, true)
% 3.05/0.82 % SZS output end Proof
% 3.05/0.82
% 3.05/0.82 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------