%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE141+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n015.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 11:39:53 AM UTC 2026
% Result : Theorem 0.95s 0.57s
% Output : Proof 0.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE141+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.12/0.38 % Computer : n015.cluster.edu
% 0.12/0.38 % Model : x86_64 x86_64
% 0.12/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.38 % Memory : 8046.5625MB
% 0.12/0.38 % OS : Linux 6.8.0-71-generic
% 0.12/0.38 % CPULimit : 300
% 0.12/0.38 % WCLimit : 300
% 0.12/0.38 % DateTime : Sun Sep 27 13:15:15 UTC 2026
% 0.12/0.38 % CPUTime :
% 0.12/0.38 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.95/0.57 Command-line arguments: --no-flatten-goal
% 0.95/0.57
% 0.95/0.57 % SZS status Theorem
% 0.95/0.57
% 0.95/0.58 % SZS output start Proof
% 0.95/0.58 Axiom 1 (idempotence): addition(X, X) = X.
% 0.95/0.58 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.95/0.58 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.95/0.58 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.95/0.58 Axiom 5 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 0.95/0.58 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.95/0.58 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.95/0.58 Axiom 8 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.95/0.58 Axiom 9 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 0.95/0.58 Axiom 10 (infty_coinduction): ifeq(leq(X, addition(multiplication(Y, X), Z)), true, leq(X, multiplication(strong_iteration(Y), Z)), true) = true.
% 0.95/0.58
% 0.95/0.58 Lemma 11: addition(X, addition(X, Y)) = addition(X, Y).
% 0.95/0.58 Proof:
% 0.95/0.58 addition(X, addition(X, Y))
% 0.95/0.58 = { by axiom 7 (additive_associativity) }
% 0.95/0.58 addition(addition(X, X), Y)
% 0.95/0.58 = { by axiom 1 (idempotence) }
% 0.95/0.58 addition(X, Y)
% 0.95/0.58
% 0.95/0.58 Lemma 12: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 0.95/0.58 Proof:
% 0.95/0.58 multiplication(addition(X, one), Y)
% 0.95/0.58 = { by axiom 2 (additive_commutativity) R->L }
% 0.95/0.58 multiplication(addition(one, X), Y)
% 0.95/0.58 = { by axiom 8 (distributivity2) }
% 0.95/0.58 addition(multiplication(one, Y), multiplication(X, Y))
% 0.95/0.58 = { by axiom 4 (multiplicative_left_identity) }
% 0.95/0.58 addition(Y, multiplication(X, Y))
% 0.95/0.58
% 0.95/0.58 Goal 1 (goals): tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), leq(strong_iteration(one), multiplication(strong_iteration(one), x0))) = tuple(true, true).
% 0.95/0.58 Proof:
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), leq(strong_iteration(one), multiplication(strong_iteration(one), x0)))
% 0.95/0.58 = { by axiom 1 (idempotence) R->L }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)))
% 0.95/0.58 = { by axiom 6 (ifeq_axiom) R->L }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), ifeq(true, true, leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)), true))
% 0.95/0.58 = { by axiom 9 (order) R->L }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), ifeq(ifeq3(addition(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0))), addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0)), leq(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0))), true), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)), true))
% 0.95/0.58 = { by lemma 11 }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), ifeq(ifeq3(addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0)), addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0)), leq(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0))), true), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)), true))
% 0.95/0.58 = { by axiom 5 (ifeq_axiom) }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), ifeq(leq(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(one, strong_iteration(one)), x0))), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)), true))
% 0.95/0.58 = { by axiom 7 (additive_associativity) }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), ifeq(leq(strong_iteration(one), addition(addition(strong_iteration(one), multiplication(one, strong_iteration(one))), x0)), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)), true))
% 0.95/0.58 = { by lemma 12 R->L }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), ifeq(leq(strong_iteration(one), addition(multiplication(addition(one, one), strong_iteration(one)), x0)), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(one, one)), x0)), true))
% 0.95/0.58 = { by axiom 10 (infty_coinduction) }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), strong_iteration(one)), true)
% 0.95/0.58 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(one), one)), true)
% 0.95/0.58 = { by axiom 1 (idempotence) R->L }
% 0.95/0.58 tuple(leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true)
% 0.95/0.58 = { by axiom 6 (ifeq_axiom) R->L }
% 0.95/0.58 tuple(ifeq(true, true, leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true), true)
% 0.95/0.58 = { by axiom 9 (order) R->L }
% 0.95/0.58 tuple(ifeq(ifeq3(addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one))), addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one)), leq(multiplication(strong_iteration(one), x0_2), addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one))), true), true, leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true), true)
% 0.95/0.58 = { by lemma 11 }
% 0.95/0.58 tuple(ifeq(ifeq3(addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one)), addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one)), leq(multiplication(strong_iteration(one), x0_2), addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one))), true), true, leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true), true)
% 0.95/0.58 = { by axiom 5 (ifeq_axiom) }
% 0.95/0.58 tuple(ifeq(leq(multiplication(strong_iteration(one), x0_2), addition(multiplication(strong_iteration(one), x0_2), addition(multiplication(one, multiplication(strong_iteration(one), x0_2)), one))), true, leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true), true)
% 0.95/0.58 = { by axiom 7 (additive_associativity) }
% 0.95/0.58 tuple(ifeq(leq(multiplication(strong_iteration(one), x0_2), addition(addition(multiplication(strong_iteration(one), x0_2), multiplication(one, multiplication(strong_iteration(one), x0_2))), one)), true, leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true), true)
% 0.95/0.58 = { by lemma 12 R->L }
% 0.95/0.58 tuple(ifeq(leq(multiplication(strong_iteration(one), x0_2), addition(multiplication(addition(one, one), multiplication(strong_iteration(one), x0_2)), one)), true, leq(multiplication(strong_iteration(one), x0_2), multiplication(strong_iteration(addition(one, one)), one)), true), true)
% 0.95/0.58 = { by axiom 10 (infty_coinduction) }
% 0.95/0.58 tuple(true, true)
% 0.95/0.58 % SZS output end Proof
% 0.95/0.58
% 0.95/0.58 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------