%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE142+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 0.61s 0.55s
% Output : Proof 0.61s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE142+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.36 % Computer : n005.cluster.edu
% 0.08/0.36 % Model : x86_64 x86_64
% 0.08/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.36 % Memory : 8046.5625MB
% 0.08/0.36 % OS : Linux 6.8.0-71-generic
% 0.08/0.36 % CPULimit : 300
% 0.08/0.36 % WCLimit : 300
% 0.08/0.36 % DateTime : Sun Sep 27 13:11:47 UTC 2026
% 0.08/0.36 % CPUTime :
% 0.08/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.61/0.55 Command-line arguments: --no-flatten-goal
% 0.61/0.55
% 0.61/0.55 % SZS status Theorem
% 0.61/0.55
% 0.61/0.56 % SZS output start Proof
% 0.61/0.56 Axiom 1 (idempotence): addition(X, X) = X.
% 0.61/0.56 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.61/0.56 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.61/0.56 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.61/0.56 Axiom 5 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 0.61/0.56 Axiom 6 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 0.61/0.56 Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 0.61/0.56 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.61/0.56 Axiom 9 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 0.61/0.56 Axiom 10 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.61/0.56 Axiom 11 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 0.61/0.56 Axiom 12 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 0.61/0.56 Axiom 13 (infty_coinduction): ifeq(leq(X, addition(multiplication(Y, X), Z)), true, leq(X, multiplication(strong_iteration(Y), Z)), true) = true.
% 0.61/0.56
% 0.61/0.56 Lemma 14: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 0.61/0.56 Proof:
% 0.61/0.56 addition(one, multiplication(X, strong_iteration(X)))
% 0.61/0.56 = { by axiom 2 (additive_commutativity) R->L }
% 0.61/0.56 addition(multiplication(X, strong_iteration(X)), one)
% 0.61/0.56 = { by axiom 9 (infty_unfold1) R->L }
% 0.61/0.56 strong_iteration(X)
% 0.61/0.56
% 0.61/0.56 Lemma 15: addition(X, addition(X, Y)) = addition(X, Y).
% 0.61/0.56 Proof:
% 0.61/0.56 addition(X, addition(X, Y))
% 0.61/0.56 = { by axiom 8 (additive_associativity) }
% 0.61/0.56 addition(addition(X, X), Y)
% 0.61/0.56 = { by axiom 1 (idempotence) }
% 0.61/0.56 addition(X, Y)
% 0.61/0.56
% 0.61/0.56 Lemma 16: addition(one, strong_iteration(X)) = strong_iteration(X).
% 0.61/0.56 Proof:
% 0.61/0.56 addition(one, strong_iteration(X))
% 0.61/0.56 = { by lemma 14 R->L }
% 0.61/0.56 addition(one, addition(one, multiplication(X, strong_iteration(X))))
% 0.61/0.56 = { by lemma 15 }
% 0.61/0.56 addition(one, multiplication(X, strong_iteration(X)))
% 0.61/0.56 = { by lemma 14 }
% 0.61/0.56 strong_iteration(X)
% 0.61/0.56
% 0.61/0.56 Lemma 17: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 0.61/0.56 Proof:
% 0.61/0.56 multiplication(addition(X, one), Y)
% 0.61/0.56 = { by axiom 2 (additive_commutativity) R->L }
% 0.61/0.56 multiplication(addition(one, X), Y)
% 0.61/0.56 = { by axiom 10 (distributivity2) }
% 0.61/0.56 addition(multiplication(one, Y), multiplication(X, Y))
% 0.61/0.56 = { by axiom 4 (multiplicative_left_identity) }
% 0.61/0.56 addition(Y, multiplication(X, Y))
% 0.61/0.56
% 0.61/0.56 Goal 1 (goals): strong_iteration(strong_iteration(x0)) = strong_iteration(one).
% 0.61/0.56 Proof:
% 0.61/0.56 strong_iteration(strong_iteration(x0))
% 0.61/0.56 = { by lemma 16 R->L }
% 0.61/0.56 addition(one, strong_iteration(strong_iteration(x0)))
% 0.61/0.56 = { by axiom 12 (order_1) R->L }
% 0.61/0.56 addition(one, ifeq2(leq(strong_iteration(one), strong_iteration(strong_iteration(x0))), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.56 = { by lemma 14 R->L }
% 0.61/0.56 addition(one, ifeq2(leq(strong_iteration(one), strong_iteration(addition(one, multiplication(x0, strong_iteration(x0))))), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.56 = { by axiom 2 (additive_commutativity) R->L }
% 0.61/0.57 addition(one, ifeq2(leq(strong_iteration(one), strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one))), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.61/0.57 addition(one, ifeq2(leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 7 (ifeq_axiom) R->L }
% 0.61/0.57 addition(one, ifeq2(ifeq(true, true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 11 (order) R->L }
% 0.61/0.57 addition(one, ifeq2(ifeq(ifeq3(addition(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one))), addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one)), leq(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one))), true), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by lemma 15 }
% 0.61/0.57 addition(one, ifeq2(ifeq(ifeq3(addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one)), addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one)), leq(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one))), true), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 5 (ifeq_axiom) }
% 0.61/0.57 addition(one, ifeq2(ifeq(leq(strong_iteration(one), addition(strong_iteration(one), addition(multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one)), one))), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 8 (additive_associativity) }
% 0.61/0.57 addition(one, ifeq2(ifeq(leq(strong_iteration(one), addition(addition(strong_iteration(one), multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(one))), one)), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by lemma 17 R->L }
% 0.61/0.57 addition(one, ifeq2(ifeq(leq(strong_iteration(one), addition(multiplication(addition(multiplication(x0, strong_iteration(x0)), one), strong_iteration(one)), one)), true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)), true), true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 13 (infty_coinduction) }
% 0.61/0.57 addition(one, ifeq2(true, true, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 6 (ifeq_axiom) }
% 0.61/0.57 addition(one, addition(strong_iteration(one), strong_iteration(strong_iteration(x0))))
% 0.61/0.57 = { by axiom 8 (additive_associativity) }
% 0.61/0.57 addition(addition(one, strong_iteration(one)), strong_iteration(strong_iteration(x0)))
% 0.61/0.57 = { by lemma 16 }
% 0.61/0.57 addition(strong_iteration(one), strong_iteration(strong_iteration(x0)))
% 0.61/0.57 = { by axiom 2 (additive_commutativity) }
% 0.61/0.57 addition(strong_iteration(strong_iteration(x0)), strong_iteration(one))
% 0.61/0.57 = { by axiom 6 (ifeq_axiom) R->L }
% 0.61/0.57 ifeq2(true, true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 13 (infty_coinduction) R->L }
% 0.61/0.57 ifeq2(ifeq(leq(strong_iteration(strong_iteration(x0)), addition(multiplication(addition(one, one), strong_iteration(strong_iteration(x0))), one)), true, leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by lemma 17 }
% 0.61/0.57 ifeq2(ifeq(leq(strong_iteration(strong_iteration(x0)), addition(addition(strong_iteration(strong_iteration(x0)), multiplication(one, strong_iteration(strong_iteration(x0)))), one)), true, leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 8 (additive_associativity) R->L }
% 0.61/0.57 ifeq2(ifeq(leq(strong_iteration(strong_iteration(x0)), addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one))), true, leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 5 (ifeq_axiom) R->L }
% 0.61/0.57 ifeq2(ifeq(ifeq3(addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one)), addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one)), leq(strong_iteration(strong_iteration(x0)), addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one))), true), true, leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by lemma 15 R->L }
% 0.61/0.57 ifeq2(ifeq(ifeq3(addition(strong_iteration(strong_iteration(x0)), addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one))), addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one)), leq(strong_iteration(strong_iteration(x0)), addition(strong_iteration(strong_iteration(x0)), addition(multiplication(one, strong_iteration(strong_iteration(x0))), one))), true), true, leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 11 (order) }
% 0.61/0.57 ifeq2(ifeq(true, true, leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 7 (ifeq_axiom) }
% 0.61/0.57 ifeq2(leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(addition(one, one)), one)), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 1 (idempotence) }
% 0.61/0.57 ifeq2(leq(strong_iteration(strong_iteration(x0)), multiplication(strong_iteration(one), one)), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 3 (multiplicative_right_identity) }
% 0.61/0.57 ifeq2(leq(strong_iteration(strong_iteration(x0)), strong_iteration(one)), true, addition(strong_iteration(strong_iteration(x0)), strong_iteration(one)), strong_iteration(one))
% 0.61/0.57 = { by axiom 12 (order_1) }
% 0.61/0.57 strong_iteration(one)
% 0.61/0.57 % SZS output end Proof
% 0.61/0.57
% 0.61/0.57 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------