%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE143+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n017.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 17.91s 2.72s
% Output : Proof 17.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE143+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n017.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sun Sep 27 13:07:19 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 17.91/2.72 Command-line arguments: --flatten --complete-subsets
% 17.91/2.72
% 17.91/2.72 % SZS status Theorem
% 17.91/2.72
% 17.91/2.73 % SZS output start Proof
% 17.91/2.73 Axiom 1 (idempotence): addition(X, X) = X.
% 17.91/2.73 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 17.91/2.73 Axiom 3 (additive_identity): addition(X, zero) = X.
% 17.91/2.73 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 17.91/2.73 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 17.91/2.73 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 17.91/2.73 Axiom 7 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 17.91/2.73 Axiom 8 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 17.91/2.73 Axiom 9 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 17.91/2.73 Axiom 10 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 17.91/2.73 Axiom 11 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 17.91/2.73 Axiom 12 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 17.91/2.73 Axiom 13 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 17.91/2.73 Axiom 14 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 17.91/2.73 Axiom 15 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 17.91/2.73 Axiom 16 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 17.91/2.73 Axiom 17 (infty_coinduction): ifeq(leq(X, addition(multiplication(Y, X), Z)), true, leq(X, multiplication(strong_iteration(Y), Z)), true) = true.
% 17.91/2.73
% 17.91/2.73 Lemma 18: addition(X, addition(X, Y)) = addition(X, Y).
% 17.91/2.73 Proof:
% 17.91/2.73 addition(X, addition(X, Y))
% 17.91/2.73 = { by axiom 10 (additive_associativity) }
% 17.91/2.73 addition(addition(X, X), Y)
% 17.91/2.73 = { by axiom 1 (idempotence) }
% 17.91/2.73 addition(X, Y)
% 17.91/2.73
% 17.91/2.73 Lemma 19: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 17.91/2.73 Proof:
% 17.91/2.73 multiplication(X, addition(Y, one))
% 17.91/2.73 = { by axiom 2 (additive_commutativity) R->L }
% 17.91/2.73 multiplication(X, addition(one, Y))
% 17.91/2.73 = { by axiom 13 (distributivity1) }
% 17.91/2.73 addition(multiplication(X, one), multiplication(X, Y))
% 17.91/2.73 = { by axiom 4 (multiplicative_right_identity) }
% 17.91/2.73 addition(X, multiplication(X, Y))
% 17.91/2.73
% 17.91/2.73 Lemma 20: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 17.91/2.73 Proof:
% 17.91/2.73 addition(one, multiplication(X, strong_iteration(X)))
% 17.91/2.73 = { by axiom 2 (additive_commutativity) R->L }
% 17.91/2.73 addition(multiplication(X, strong_iteration(X)), one)
% 17.91/2.73 = { by axiom 11 (infty_unfold1) R->L }
% 17.91/2.73 strong_iteration(X)
% 17.91/2.73
% 17.91/2.73 Lemma 21: addition(X, multiplication(X, multiplication(Y, strong_iteration(Y)))) = multiplication(X, strong_iteration(Y)).
% 17.91/2.73 Proof:
% 17.91/2.73 addition(X, multiplication(X, multiplication(Y, strong_iteration(Y))))
% 17.91/2.73 = { by lemma 19 R->L }
% 17.91/2.73 multiplication(X, addition(multiplication(Y, strong_iteration(Y)), one))
% 17.91/2.73 = { by axiom 2 (additive_commutativity) }
% 17.91/2.73 multiplication(X, addition(one, multiplication(Y, strong_iteration(Y))))
% 17.91/2.73 = { by lemma 20 }
% 17.91/2.73 multiplication(X, strong_iteration(Y))
% 17.91/2.73
% 17.91/2.73 Goal 1 (goals): multiplication(strong_iteration(x0), strong_iteration(x0)) = strong_iteration(x0).
% 17.91/2.73 Proof:
% 17.91/2.73 multiplication(strong_iteration(x0), strong_iteration(x0))
% 17.91/2.73 = { by lemma 20 R->L }
% 17.91/2.73 multiplication(strong_iteration(x0), addition(one, multiplication(x0, strong_iteration(x0))))
% 17.91/2.73 = { by lemma 18 R->L }
% 17.91/2.73 multiplication(strong_iteration(x0), addition(one, addition(one, multiplication(x0, strong_iteration(x0)))))
% 17.91/2.73 = { by lemma 20 }
% 17.91/2.73 multiplication(strong_iteration(x0), addition(one, strong_iteration(x0)))
% 17.91/2.73 = { by axiom 2 (additive_commutativity) }
% 17.91/2.73 multiplication(strong_iteration(x0), addition(strong_iteration(x0), one))
% 17.91/2.73 = { by lemma 19 }
% 17.91/2.73 addition(strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)))
% 17.91/2.73 = { by axiom 2 (additive_commutativity) }
% 17.91/2.73 addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 8 (ifeq_axiom) R->L }
% 17.91/2.74 ifeq2(true, true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 17 (infty_coinduction) R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 2 (additive_commutativity) R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(strong_iteration(zero), multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0))))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 20 R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, multiplication(zero, strong_iteration(zero))), multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0))))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 18 R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, addition(one, multiplication(zero, strong_iteration(zero)))), multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0))))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 20 }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, strong_iteration(zero)), multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0))))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 10 (additive_associativity) R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(one, addition(strong_iteration(zero), multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0)))))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 2 (additive_commutativity) R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(one, addition(multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0))), strong_iteration(zero)))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 10 (additive_associativity) }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, multiplication(x0, multiplication(strong_iteration(x0), strong_iteration(x0)))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 12 (multiplicative_associativity) }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, multiplication(multiplication(x0, strong_iteration(x0)), strong_iteration(x0))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 21 R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, addition(multiplication(x0, strong_iteration(x0)), multiplication(multiplication(x0, strong_iteration(x0)), multiplication(x0, strong_iteration(x0))))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 18 R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, addition(multiplication(x0, strong_iteration(x0)), addition(multiplication(x0, strong_iteration(x0)), multiplication(multiplication(x0, strong_iteration(x0)), multiplication(x0, strong_iteration(x0)))))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 19 R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(one, addition(multiplication(x0, strong_iteration(x0)), multiplication(multiplication(x0, strong_iteration(x0)), addition(multiplication(x0, strong_iteration(x0)), one)))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 10 (additive_associativity) }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(addition(one, multiplication(x0, strong_iteration(x0))), multiplication(multiplication(x0, strong_iteration(x0)), addition(multiplication(x0, strong_iteration(x0)), one))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 19 }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(addition(one, multiplication(x0, strong_iteration(x0))), addition(multiplication(x0, strong_iteration(x0)), multiplication(multiplication(x0, strong_iteration(x0)), multiplication(x0, strong_iteration(x0))))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 6 (multiplicative_left_identity) R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(addition(one, multiplication(x0, strong_iteration(x0))), addition(multiplication(one, multiplication(x0, strong_iteration(x0))), multiplication(multiplication(x0, strong_iteration(x0)), multiplication(x0, strong_iteration(x0))))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 14 (distributivity2) R->L }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(addition(addition(one, multiplication(x0, strong_iteration(x0))), multiplication(addition(one, multiplication(x0, strong_iteration(x0))), multiplication(x0, strong_iteration(x0)))), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 21 }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(multiplication(addition(one, multiplication(x0, strong_iteration(x0))), strong_iteration(x0)), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 20 }
% 17.91/2.74 ifeq2(ifeq(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero))), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 7 (ifeq_axiom) R->L }
% 17.91/2.74 ifeq2(ifeq(ifeq3(addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero)), addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero)), leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero))), true), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 18 R->L }
% 17.91/2.74 ifeq2(ifeq(ifeq3(addition(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero))), addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero)), leq(multiplication(strong_iteration(x0), strong_iteration(x0)), addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(zero))), true), true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 15 (order) }
% 17.91/2.74 ifeq2(ifeq(true, true, leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 9 (ifeq_axiom) }
% 17.91/2.74 ifeq2(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(zero))), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by lemma 20 R->L }
% 17.91/2.74 ifeq2(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), addition(one, multiplication(zero, strong_iteration(zero))))), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 5 (left_annihilation) }
% 17.91/2.74 ifeq2(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), addition(one, zero))), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 3 (additive_identity) }
% 17.91/2.74 ifeq2(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), one)), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 4 (multiplicative_right_identity) }
% 17.91/2.74 ifeq2(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, addition(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0))
% 17.91/2.74 = { by axiom 16 (order_1) }
% 17.91/2.74 strong_iteration(x0)
% 17.91/2.74 % SZS output end Proof
% 17.91/2.74
% 17.91/2.74 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------