%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE145+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n010.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 11:39:54 AM UTC 2026
% Result : Theorem 238.89s 30.53s
% Output : Proof 239.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE145+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n010.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 13:12:46 UTC 2026
% 0.08/0.35 % CPUTime :
% 0.08/0.35 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 238.89/30.53 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-regeneralise
% 238.89/30.53
% 238.89/30.53 % SZS status Theorem
% 238.89/30.53
% 238.89/30.58 % SZS output start Proof
% 238.89/30.58 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 238.89/30.58 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 238.89/30.58 Axiom 3 (idempotence): addition(X, X) = X.
% 238.89/30.58 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 238.89/30.58 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 238.89/30.58 Axiom 6 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 238.89/30.58 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 238.89/30.58 Axiom 8 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 238.89/30.58 Axiom 9 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 238.89/30.58 Axiom 10 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 238.89/30.58 Axiom 11 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 238.89/30.58 Axiom 12 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 238.89/30.58 Axiom 13 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 238.89/30.58 Axiom 14 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 238.89/30.58 Axiom 15 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 238.89/30.58 Axiom 16 (infty_coinduction): ifeq(leq(X, addition(multiplication(Y, X), Z)), true, leq(X, multiplication(strong_iteration(Y), Z)), true) = true.
% 238.89/30.58 Axiom 17 (star_induction1): ifeq(leq(addition(multiplication(X, Y), Z), Y), true, leq(multiplication(star(X), Z), Y), true) = true.
% 238.89/30.58
% 238.89/30.58 Lemma 18: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 238.89/30.58 Proof:
% 238.89/30.58 addition(one, multiplication(X, strong_iteration(X)))
% 238.89/30.58 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.58 addition(multiplication(X, strong_iteration(X)), one)
% 238.89/30.58 = { by axiom 6 (infty_unfold1) R->L }
% 238.89/30.58 strong_iteration(X)
% 238.89/30.58
% 238.89/30.58 Lemma 19: addition(X, addition(X, Y)) = addition(X, Y).
% 238.89/30.58 Proof:
% 238.89/30.58 addition(X, addition(X, Y))
% 238.89/30.58 = { by axiom 7 (additive_associativity) }
% 238.89/30.58 addition(addition(X, X), Y)
% 238.89/30.58 = { by axiom 3 (idempotence) }
% 238.89/30.58 addition(X, Y)
% 238.89/30.58
% 238.89/30.58 Lemma 20: addition(one, strong_iteration(X)) = strong_iteration(X).
% 238.89/30.58 Proof:
% 238.89/30.58 addition(one, strong_iteration(X))
% 238.89/30.58 = { by lemma 18 R->L }
% 238.89/30.58 addition(one, addition(one, multiplication(X, strong_iteration(X))))
% 238.89/30.58 = { by lemma 19 }
% 238.89/30.58 addition(one, multiplication(X, strong_iteration(X)))
% 238.89/30.58 = { by lemma 18 }
% 238.89/30.58 strong_iteration(X)
% 238.89/30.58
% 238.89/30.58 Lemma 21: addition(X, addition(Y, X)) = addition(Y, X).
% 238.89/30.58 Proof:
% 238.89/30.58 addition(X, addition(Y, X))
% 238.89/30.58 = { by lemma 19 R->L }
% 238.89/30.58 addition(X, addition(Y, addition(Y, X)))
% 238.89/30.58 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.58 addition(X, addition(Y, addition(X, Y)))
% 238.89/30.58 = { by axiom 7 (additive_associativity) }
% 238.89/30.58 addition(addition(X, Y), addition(X, Y))
% 238.89/30.58 = { by axiom 3 (idempotence) }
% 238.89/30.58 addition(X, Y)
% 238.89/30.58 = { by axiom 4 (additive_commutativity) }
% 238.89/30.58 addition(Y, X)
% 238.89/30.58
% 238.89/30.58 Lemma 22: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 238.89/30.58 Proof:
% 238.89/30.58 addition(X, multiplication(Y, X))
% 238.89/30.58 = { by axiom 2 (multiplicative_left_identity) R->L }
% 238.89/30.58 addition(multiplication(one, X), multiplication(Y, X))
% 238.89/30.58 = { by axiom 13 (distributivity2) R->L }
% 238.89/30.58 multiplication(addition(one, Y), X)
% 238.89/30.58 = { by axiom 4 (additive_commutativity) }
% 238.89/30.58 multiplication(addition(Y, one), X)
% 238.89/30.58
% 238.89/30.58 Lemma 23: addition(one, addition(X, multiplication(Y, strong_iteration(Y)))) = addition(X, strong_iteration(Y)).
% 238.89/30.58 Proof:
% 238.89/30.58 addition(one, addition(X, multiplication(Y, strong_iteration(Y))))
% 238.89/30.58 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.58 addition(one, addition(multiplication(Y, strong_iteration(Y)), X))
% 238.89/30.58 = { by axiom 7 (additive_associativity) }
% 238.89/30.58 addition(addition(one, multiplication(Y, strong_iteration(Y))), X)
% 238.89/30.58 = { by lemma 18 }
% 238.89/30.58 addition(strong_iteration(Y), X)
% 238.89/30.58 = { by axiom 4 (additive_commutativity) }
% 238.89/30.58 addition(X, strong_iteration(Y))
% 238.89/30.58
% 238.89/30.58 Lemma 24: multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))) = strong_iteration(multiplication(X, strong_iteration(X))).
% 238.89/30.58 Proof:
% 238.89/30.59 multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))
% 238.89/30.59 = { by axiom 15 (order_1) R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by lemma 18 R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(one, multiplication(addition(one, multiplication(X, strong_iteration(X))), strong_iteration(multiplication(X, strong_iteration(X))))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(one, multiplication(addition(multiplication(X, strong_iteration(X)), one), strong_iteration(multiplication(X, strong_iteration(X))))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(multiplication(addition(multiplication(X, strong_iteration(X)), one), strong_iteration(multiplication(X, strong_iteration(X)))), one), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 13 (distributivity2) }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(addition(multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X)))), multiplication(one, strong_iteration(multiplication(X, strong_iteration(X))))), one), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 7 (additive_associativity) R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X)))), addition(multiplication(one, strong_iteration(multiplication(X, strong_iteration(X)))), one)), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(addition(multiplication(one, strong_iteration(multiplication(X, strong_iteration(X)))), one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X))))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 7 (additive_associativity) R->L }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(multiplication(one, strong_iteration(multiplication(X, strong_iteration(X)))), addition(one, multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X)))))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by lemma 18 }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(multiplication(one, strong_iteration(multiplication(X, strong_iteration(X)))), strong_iteration(multiplication(X, strong_iteration(X)))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 4 (additive_commutativity) }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, addition(strong_iteration(multiplication(X, strong_iteration(X))), multiplication(one, strong_iteration(multiplication(X, strong_iteration(X))))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by lemma 22 }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, multiplication(addition(one, one), strong_iteration(multiplication(X, strong_iteration(X)))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 3 (idempotence) }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, multiplication(one, strong_iteration(multiplication(X, strong_iteration(X)))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 2 (multiplicative_left_identity) }
% 238.89/30.59 ifeq2(leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 8 (ifeq_axiom) R->L }
% 238.89/30.59 ifeq2(ifeq(true, true, leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true), true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 14 (order) R->L }
% 238.89/30.59 ifeq2(ifeq(ifeq3(addition(one, addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X))))))), addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X)))))), leq(one, addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X))))))), true), true, leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true), true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by lemma 19 }
% 238.89/30.59 ifeq2(ifeq(ifeq3(addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X)))))), addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X)))))), leq(one, addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X))))))), true), true, leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true), true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 9 (ifeq_axiom) }
% 238.89/30.59 ifeq2(ifeq(leq(one, addition(one, addition(multiplication(X, one), multiplication(multiplication(X, strong_iteration(X)), strong_iteration(multiplication(X, strong_iteration(X))))))), true, leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true), true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by lemma 23 }
% 238.89/30.59 ifeq2(ifeq(leq(one, addition(multiplication(X, one), strong_iteration(multiplication(X, strong_iteration(X))))), true, leq(one, multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X))))), true), true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 16 (infty_coinduction) }
% 238.89/30.59 ifeq2(true, true, strong_iteration(multiplication(X, strong_iteration(X))), multiplication(strong_iteration(X), strong_iteration(multiplication(X, strong_iteration(X)))))
% 238.89/30.59 = { by axiom 10 (ifeq_axiom) }
% 238.89/30.59 strong_iteration(multiplication(X, strong_iteration(X)))
% 238.89/30.59
% 238.89/30.59 Goal 1 (goals): tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), star(strong_iteration(x0)))) = tuple(true, true).
% 238.89/30.59 Proof:
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), star(strong_iteration(x0))))
% 238.89/30.59 = { by axiom 11 (star_unfold1) R->L }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), addition(one, multiplication(strong_iteration(x0), star(strong_iteration(x0))))))
% 238.89/30.59 = { by lemma 21 R->L }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), addition(multiplication(strong_iteration(x0), star(strong_iteration(x0))), addition(one, multiplication(strong_iteration(x0), star(strong_iteration(x0)))))))
% 238.89/30.59 = { by axiom 11 (star_unfold1) }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), addition(multiplication(strong_iteration(x0), star(strong_iteration(x0))), star(strong_iteration(x0)))))
% 238.89/30.59 = { by axiom 4 (additive_commutativity) }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), addition(star(strong_iteration(x0)), multiplication(strong_iteration(x0), star(strong_iteration(x0))))))
% 238.89/30.59 = { by lemma 22 }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), multiplication(addition(strong_iteration(x0), one), star(strong_iteration(x0)))))
% 238.89/30.59 = { by axiom 4 (additive_commutativity) }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), multiplication(addition(one, strong_iteration(x0)), star(strong_iteration(x0)))))
% 238.89/30.59 = { by lemma 20 }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))
% 238.89/30.59 = { by axiom 1 (multiplicative_right_identity) R->L }
% 238.89/30.59 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))
% 238.89/30.60 = { by axiom 11 (star_unfold1) R->L }
% 238.89/30.60 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), addition(one, multiplication(strong_iteration(x0), star(strong_iteration(x0)))))))
% 238.89/30.60 = { by axiom 12 (distributivity1) }
% 238.89/30.60 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), leq(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))))
% 238.89/30.60 = { by axiom 9 (ifeq_axiom) R->L }
% 238.89/30.60 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), ifeq3(addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0))))), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0))))), leq(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))), true))
% 238.89/30.60 = { by lemma 19 R->L }
% 238.89/30.60 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), ifeq3(addition(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0))))), leq(multiplication(strong_iteration(x0), one), addition(multiplication(strong_iteration(x0), one), multiplication(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))), true))
% 238.89/30.60 = { by axiom 14 (order) }
% 238.89/30.60 tuple(leq(star(strong_iteration(x0_2)), strong_iteration(x0_2)), true)
% 238.89/30.60 = { by lemma 18 R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true)
% 238.89/30.60 = { by axiom 15 (order_1) R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(x0_2)), true, addition(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(x0_2)), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 4 (additive_commutativity) }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(x0_2)), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by lemma 24 R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 8 (ifeq_axiom) R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(true, true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 14 (order) R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(ifeq3(addition(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 3 (idempotence) }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(ifeq3(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 9 (ifeq_axiom) }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by lemma 24 R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by lemma 18 R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), addition(one, multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 5 (multiplicative_associativity) R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), addition(one, multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))))), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 238.89/30.60 = { by axiom 4 (additive_commutativity) R->L }
% 238.89/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))), one)), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 239.66/30.60 = { by axiom 1 (multiplicative_right_identity) R->L }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(ifeq(leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), addition(multiplication(x0_2, multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))), one)), true, leq(multiplication(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), multiplication(strong_iteration(x0_2), one)), true), true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 239.66/30.60 = { by axiom 16 (infty_coinduction) }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), ifeq2(true, true, addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(x0_2))), true)
% 239.66/30.60 = { by axiom 10 (ifeq_axiom) }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(strong_iteration(x0_2), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))), true)
% 239.66/30.60 = { by axiom 4 (additive_commutativity) R->L }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(x0_2))), true)
% 239.66/30.60 = { by lemma 23 R->L }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, addition(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), multiplication(x0_2, strong_iteration(x0_2))))), true)
% 239.66/30.60 = { by axiom 4 (additive_commutativity) }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, addition(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))), true)
% 239.66/30.60 = { by lemma 23 R->L }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, addition(one, addition(multiplication(x0_2, strong_iteration(x0_2)), multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))))), true)
% 239.66/30.60 = { by axiom 1 (multiplicative_right_identity) R->L }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, addition(one, addition(multiplication(multiplication(x0_2, strong_iteration(x0_2)), one), multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))))), true)
% 239.66/30.60 = { by axiom 12 (distributivity1) R->L }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, addition(one, multiplication(multiplication(x0_2, strong_iteration(x0_2)), addition(one, strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))))), true)
% 239.66/30.60 = { by lemma 20 }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, addition(one, multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))))), true)
% 239.66/30.60 = { by lemma 18 }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), addition(one, strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))), true)
% 239.66/30.60 = { by lemma 20 }
% 239.66/30.60 tuple(leq(star(addition(one, multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true)
% 239.66/30.60 = { by axiom 4 (additive_commutativity) R->L }
% 239.66/30.61 tuple(leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true)
% 239.66/30.61 = { by axiom 8 (ifeq_axiom) R->L }
% 239.66/30.61 tuple(ifeq(true, true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 14 (order) R->L }
% 239.66/30.61 tuple(ifeq(ifeq3(addition(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 3 (idempotence) }
% 239.66/30.61 tuple(ifeq(ifeq3(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 9 (ifeq_axiom) }
% 239.66/30.61 tuple(ifeq(leq(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by lemma 20 R->L }
% 239.66/30.61 tuple(ifeq(leq(addition(one, strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by lemma 18 R->L }
% 239.66/30.61 tuple(ifeq(leq(addition(one, addition(one, multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by lemma 21 R->L }
% 239.66/30.61 tuple(ifeq(leq(addition(one, addition(multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), addition(one, multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by lemma 18 }
% 239.66/30.61 tuple(ifeq(leq(addition(one, addition(multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 4 (additive_commutativity) }
% 239.66/30.61 tuple(ifeq(leq(addition(one, addition(strong_iteration(multiplication(x0_2, strong_iteration(x0_2))), multiplication(multiplication(x0_2, strong_iteration(x0_2)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by lemma 22 }
% 239.66/30.61 tuple(ifeq(leq(addition(one, multiplication(addition(multiplication(x0_2, strong_iteration(x0_2)), one), strong_iteration(multiplication(x0_2, strong_iteration(x0_2))))), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 4 (additive_commutativity) R->L }
% 239.66/30.61 tuple(ifeq(leq(addition(multiplication(addition(multiplication(x0_2, strong_iteration(x0_2)), one), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), one), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 1 (multiplicative_right_identity) R->L }
% 239.66/30.61 tuple(ifeq(leq(addition(multiplication(addition(multiplication(x0_2, strong_iteration(x0_2)), one), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), one), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true, leq(multiplication(star(addition(multiplication(x0_2, strong_iteration(x0_2)), one)), one), strong_iteration(multiplication(x0_2, strong_iteration(x0_2)))), true), true)
% 239.66/30.61 = { by axiom 17 (star_induction1) }
% 239.66/30.61 tuple(true, true)
% 239.66/30.61 % SZS output end Proof
% 239.66/30.61
% 239.66/30.61 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------