%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE043+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 11:39:42 AM UTC 2026
% Result : Theorem 166.61s 21.47s
% Output : Proof 168.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE043+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.35 % Computer : n015.cluster.edu
% 0.10/0.35 % Model : x86_64 x86_64
% 0.10/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.35 % Memory : 8046.5625MB
% 0.10/0.35 % OS : Linux 6.8.0-71-generic
% 0.10/0.35 % CPULimit : 300
% 0.10/0.35 % WCLimit : 300
% 0.10/0.35 % DateTime : Sun Sep 27 13:10:15 UTC 2026
% 0.10/0.36 % CPUTime :
% 0.10/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 166.61/21.47 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
% 166.61/21.47
% 166.61/21.47 % SZS status Theorem
% 166.61/21.47
% 167.40/21.59 % SZS output start Proof
% 167.40/21.59 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 167.40/21.59 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 167.40/21.59 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 167.40/21.59 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 167.40/21.59 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 167.40/21.59 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 167.40/21.59 Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 167.40/21.59 Axiom 8 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 167.40/21.59 Axiom 9 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 167.40/21.59 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 167.40/21.59 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 167.40/21.59 Axiom 12 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 167.40/21.59 Axiom 13 (order): ifeq3(addition(X, Y), Y, leq(X, Y), true) = true.
% 167.40/21.59 Axiom 14 (order_1): ifeq2(leq(X, Y), true, addition(X, Y), Y) = Y.
% 167.40/21.59 Axiom 15 (star_induction_right): ifeq(leq(addition(multiplication(X, Y), Z), X), true, leq(multiplication(Z, star(Y)), X), true) = true.
% 167.40/21.59 Axiom 16 (star_induction_left): ifeq(leq(addition(multiplication(X, Y), Z), Y), true, leq(multiplication(star(X), Z), Y), true) = true.
% 167.40/21.59
% 167.40/21.59 Lemma 17: addition(one, addition(multiplication(X, star(X)), star(X))) = star(X).
% 167.40/21.59 Proof:
% 167.40/21.59 addition(one, addition(multiplication(X, star(X)), star(X)))
% 167.40/21.59 = { by axiom 6 (additive_associativity) }
% 167.40/21.59 addition(addition(one, multiplication(X, star(X))), star(X))
% 167.40/21.59 = { by axiom 9 (ifeq_axiom) R->L }
% 167.40/21.59 ifeq2(true, true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 167.40/21.59 = { by axiom 12 (star_unfold_right) R->L }
% 167.40/21.59 ifeq2(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(addition(one, multiplication(X, star(X))), star(X)), star(X))
% 167.40/21.59 = { by axiom 14 (order_1) }
% 167.40/21.59 star(X)
% 167.40/21.59
% 167.40/21.59 Lemma 18: addition(one, star(X)) = star(X).
% 167.40/21.59 Proof:
% 167.40/21.59 addition(one, star(X))
% 167.40/21.60 = { by lemma 17 R->L }
% 167.40/21.60 addition(one, addition(one, addition(multiplication(X, star(X)), star(X))))
% 167.40/21.60 = { by axiom 6 (additive_associativity) }
% 167.40/21.60 addition(addition(one, one), addition(multiplication(X, star(X)), star(X)))
% 167.40/21.60 = { by axiom 3 (additive_idempotence) }
% 167.40/21.60 addition(one, addition(multiplication(X, star(X)), star(X)))
% 167.40/21.60 = { by lemma 17 }
% 167.40/21.60 star(X)
% 167.40/21.60
% 167.40/21.60 Lemma 19: addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Z, Y)).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(multiplication(X, Y), multiplication(X, Z))
% 167.40/21.60 = { by axiom 10 (right_distributivity) R->L }
% 167.40/21.60 multiplication(X, addition(Y, Z))
% 167.40/21.60 = { by axiom 4 (additive_commutativity) }
% 167.40/21.60 multiplication(X, addition(Z, Y))
% 167.40/21.60
% 167.40/21.60 Lemma 20: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(multiplication(X, Y), multiplication(Z, Y))
% 167.40/21.60 = { by axiom 11 (left_distributivity) R->L }
% 167.40/21.60 multiplication(addition(X, Z), Y)
% 167.40/21.60 = { by axiom 4 (additive_commutativity) }
% 167.40/21.60 multiplication(addition(Z, X), Y)
% 167.40/21.60
% 167.40/21.60 Lemma 21: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(Y, addition(X, Z))
% 167.40/21.60 = { by axiom 4 (additive_commutativity) R->L }
% 167.40/21.60 addition(addition(X, Z), Y)
% 167.40/21.60 = { by axiom 6 (additive_associativity) R->L }
% 167.40/21.60 addition(X, addition(Z, Y))
% 167.40/21.60 = { by axiom 4 (additive_commutativity) }
% 167.40/21.60 addition(X, addition(Y, Z))
% 167.40/21.60
% 167.40/21.60 Lemma 22: addition(X, addition(Y, X)) = addition(Y, X).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(X, addition(Y, X))
% 167.40/21.60 = { by lemma 21 }
% 167.40/21.60 addition(Y, addition(X, X))
% 167.40/21.60 = { by axiom 3 (additive_idempotence) }
% 167.40/21.60 addition(Y, X)
% 167.40/21.60
% 167.40/21.60 Lemma 23: addition(multiplication(X, star(X)), star(X)) = star(X).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(multiplication(X, star(X)), star(X))
% 167.40/21.60 = { by axiom 3 (additive_idempotence) R->L }
% 167.40/21.60 addition(multiplication(X, star(X)), addition(star(X), star(X)))
% 167.40/21.60 = { by axiom 6 (additive_associativity) }
% 167.40/21.60 addition(addition(multiplication(X, star(X)), star(X)), star(X))
% 167.40/21.60 = { by lemma 17 R->L }
% 167.40/21.60 addition(addition(multiplication(X, star(X)), star(X)), addition(one, addition(multiplication(X, star(X)), star(X))))
% 167.40/21.60 = { by lemma 22 }
% 167.40/21.60 addition(one, addition(multiplication(X, star(X)), star(X)))
% 167.40/21.60 = { by lemma 17 }
% 167.40/21.60 star(X)
% 167.40/21.60
% 167.40/21.60 Lemma 24: multiplication(addition(X, one), star(X)) = star(X).
% 167.40/21.60 Proof:
% 167.40/21.60 multiplication(addition(X, one), star(X))
% 167.40/21.60 = { by axiom 4 (additive_commutativity) R->L }
% 167.40/21.60 multiplication(addition(one, X), star(X))
% 167.40/21.60 = { by lemma 20 R->L }
% 167.40/21.60 addition(multiplication(X, star(X)), multiplication(one, star(X)))
% 167.40/21.60 = { by axiom 2 (multiplicative_left_identity) }
% 167.40/21.60 addition(multiplication(X, star(X)), star(X))
% 167.40/21.60 = { by lemma 23 }
% 167.40/21.60 star(X)
% 167.40/21.60
% 167.40/21.60 Lemma 25: addition(X, star(X)) = star(X).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(X, star(X))
% 167.40/21.60 = { by lemma 18 R->L }
% 167.40/21.60 addition(X, addition(one, star(X)))
% 167.40/21.60 = { by axiom 6 (additive_associativity) }
% 167.40/21.60 addition(addition(X, one), star(X))
% 167.40/21.60 = { by axiom 9 (ifeq_axiom) R->L }
% 167.40/21.60 ifeq2(true, true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by axiom 13 (order) R->L }
% 167.40/21.60 ifeq2(ifeq3(addition(addition(X, one), multiplication(addition(X, one), star(X))), multiplication(addition(X, one), star(X)), leq(addition(X, one), multiplication(addition(X, one), star(X))), true), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by axiom 4 (additive_commutativity) R->L }
% 167.40/21.60 ifeq2(ifeq3(addition(multiplication(addition(X, one), star(X)), addition(X, one)), multiplication(addition(X, one), star(X)), leq(addition(X, one), multiplication(addition(X, one), star(X))), true), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by axiom 1 (multiplicative_right_identity) R->L }
% 167.40/21.60 ifeq2(ifeq3(addition(multiplication(addition(X, one), star(X)), multiplication(addition(X, one), one)), multiplication(addition(X, one), star(X)), leq(addition(X, one), multiplication(addition(X, one), star(X))), true), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by lemma 19 }
% 167.40/21.60 ifeq2(ifeq3(multiplication(addition(X, one), addition(one, star(X))), multiplication(addition(X, one), star(X)), leq(addition(X, one), multiplication(addition(X, one), star(X))), true), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by lemma 18 }
% 167.40/21.60 ifeq2(ifeq3(multiplication(addition(X, one), star(X)), multiplication(addition(X, one), star(X)), leq(addition(X, one), multiplication(addition(X, one), star(X))), true), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by axiom 8 (ifeq_axiom) }
% 167.40/21.60 ifeq2(leq(addition(X, one), multiplication(addition(X, one), star(X))), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by lemma 24 }
% 167.40/21.60 ifeq2(leq(addition(X, one), star(X)), true, addition(addition(X, one), star(X)), star(X))
% 167.40/21.60 = { by axiom 14 (order_1) }
% 167.40/21.60 star(X)
% 167.40/21.60
% 167.40/21.60 Lemma 26: addition(star(X), multiplication(X, star(X))) = star(X).
% 167.40/21.60 Proof:
% 167.40/21.60 addition(star(X), multiplication(X, star(X)))
% 167.40/21.60 = { by axiom 4 (additive_commutativity) R->L }
% 167.40/21.60 addition(multiplication(X, star(X)), star(X))
% 167.40/21.60 = { by axiom 3 (additive_idempotence) R->L }
% 167.40/21.60 addition(multiplication(X, star(X)), addition(star(X), star(X)))
% 167.40/21.60 = { by axiom 6 (additive_associativity) }
% 167.40/21.60 addition(addition(multiplication(X, star(X)), star(X)), star(X))
% 167.40/21.60 = { by lemma 17 R->L }
% 167.40/21.60 addition(addition(multiplication(X, star(X)), star(X)), addition(one, addition(multiplication(X, star(X)), star(X))))
% 168.21/21.60 = { by lemma 22 }
% 168.21/21.60 addition(one, addition(multiplication(X, star(X)), star(X)))
% 168.21/21.60 = { by lemma 17 }
% 168.21/21.60 star(X)
% 168.21/21.60
% 168.21/21.60 Lemma 27: multiplication(star(X), star(star(X))) = star(star(X)).
% 168.21/21.60 Proof:
% 168.21/21.60 multiplication(star(X), star(star(X)))
% 168.21/21.60 = { by lemma 23 R->L }
% 168.21/21.60 multiplication(star(X), star(addition(multiplication(X, star(X)), star(X))))
% 168.21/21.60 = { by lemma 17 R->L }
% 168.21/21.60 multiplication(addition(one, addition(multiplication(X, star(X)), star(X))), star(addition(multiplication(X, star(X)), star(X))))
% 168.21/21.60 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.60 multiplication(addition(addition(multiplication(X, star(X)), star(X)), one), star(addition(multiplication(X, star(X)), star(X))))
% 168.21/21.60 = { by lemma 24 }
% 168.21/21.60 star(addition(multiplication(X, star(X)), star(X)))
% 168.21/21.60 = { by lemma 23 }
% 168.21/21.60 star(star(X))
% 168.21/21.60
% 168.21/21.60 Lemma 28: ifeq2(leq(X, Y), true, addition(Y, X), Y) = Y.
% 168.21/21.60 Proof:
% 168.21/21.60 ifeq2(leq(X, Y), true, addition(Y, X), Y)
% 168.21/21.60 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.60 ifeq2(leq(X, Y), true, addition(X, Y), Y)
% 168.21/21.60 = { by axiom 14 (order_1) }
% 168.21/21.60 Y
% 168.21/21.60
% 168.21/21.60 Lemma 29: star(star(X)) = star(X).
% 168.21/21.60 Proof:
% 168.21/21.60 star(star(X))
% 168.21/21.60 = { by lemma 25 R->L }
% 168.21/21.61 addition(star(X), star(star(X)))
% 168.21/21.61 = { by axiom 9 (ifeq_axiom) R->L }
% 168.21/21.61 ifeq2(true, true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 15 (star_induction_right) R->L }
% 168.21/21.61 ifeq2(ifeq(leq(addition(multiplication(star(X), star(X)), multiplication(star(X), star(X))), star(X)), true, leq(multiplication(multiplication(star(X), star(X)), star(star(X))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 3 (additive_idempotence) }
% 168.21/21.61 ifeq2(ifeq(leq(multiplication(star(X), star(X)), star(X)), true, leq(multiplication(multiplication(star(X), star(X)), star(star(X))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 5 (multiplicative_associativity) R->L }
% 168.21/21.61 ifeq2(ifeq(leq(multiplication(star(X), star(X)), star(X)), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 7 (ifeq_axiom) R->L }
% 168.21/21.61 ifeq2(ifeq(ifeq(true, true, leq(multiplication(star(X), star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 13 (order) R->L }
% 168.21/21.61 ifeq2(ifeq(ifeq(ifeq3(addition(star(X), star(X)), star(X), leq(star(X), star(X)), true), true, leq(multiplication(star(X), star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 3 (additive_idempotence) }
% 168.21/21.61 ifeq2(ifeq(ifeq(ifeq3(star(X), star(X), leq(star(X), star(X)), true), true, leq(multiplication(star(X), star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.61 ifeq2(ifeq(ifeq(leq(star(X), star(X)), true, leq(multiplication(star(X), star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by lemma 26 R->L }
% 168.21/21.61 ifeq2(ifeq(ifeq(leq(addition(star(X), multiplication(X, star(X))), star(X)), true, leq(multiplication(star(X), star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.61 ifeq2(ifeq(ifeq(leq(addition(multiplication(X, star(X)), star(X)), star(X)), true, leq(multiplication(star(X), star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 16 (star_induction_left) }
% 168.21/21.61 ifeq2(ifeq(true, true, leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by axiom 7 (ifeq_axiom) }
% 168.21/21.61 ifeq2(leq(multiplication(star(X), multiplication(star(X), star(star(X)))), star(X)), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by lemma 27 }
% 168.21/21.61 ifeq2(leq(multiplication(star(X), star(star(X))), star(X)), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by lemma 27 }
% 168.21/21.61 ifeq2(leq(star(star(X)), star(X)), true, addition(star(X), star(star(X))), star(X))
% 168.21/21.61 = { by lemma 28 }
% 168.21/21.61 star(X)
% 168.21/21.61
% 168.21/21.61 Lemma 30: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 168.21/21.61 Proof:
% 168.21/21.61 addition(X, multiplication(X, Y))
% 168.21/21.61 = { by axiom 1 (multiplicative_right_identity) R->L }
% 168.21/21.61 addition(multiplication(X, one), multiplication(X, Y))
% 168.21/21.61 = { by axiom 10 (right_distributivity) R->L }
% 168.21/21.61 multiplication(X, addition(one, Y))
% 168.21/21.61 = { by axiom 4 (additive_commutativity) }
% 168.21/21.61 multiplication(X, addition(Y, one))
% 168.21/21.61
% 168.21/21.61 Lemma 31: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 168.21/21.61 Proof:
% 168.21/21.61 addition(X, multiplication(Y, X))
% 168.21/21.61 = { by axiom 2 (multiplicative_left_identity) R->L }
% 168.21/21.61 addition(multiplication(one, X), multiplication(Y, X))
% 168.21/21.61 = { by axiom 11 (left_distributivity) R->L }
% 168.21/21.61 multiplication(addition(one, Y), X)
% 168.21/21.61 = { by axiom 4 (additive_commutativity) }
% 168.21/21.61 multiplication(addition(Y, one), X)
% 168.21/21.61
% 168.21/21.61 Lemma 32: addition(Z, addition(X, Y)) = addition(X, addition(Y, Z)).
% 168.21/21.61 Proof:
% 168.21/21.61 addition(Z, addition(X, Y))
% 168.21/21.61 = { by lemma 21 }
% 168.21/21.61 addition(X, addition(Z, Y))
% 168.21/21.61 = { by axiom 4 (additive_commutativity) }
% 168.21/21.61 addition(X, addition(Y, Z))
% 168.21/21.61
% 168.21/21.61 Lemma 33: addition(X, multiplication(star(Y), X)) = multiplication(star(Y), X).
% 168.21/21.61 Proof:
% 168.21/21.61 addition(X, multiplication(star(Y), X))
% 168.21/21.61 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.61 addition(multiplication(star(Y), X), X)
% 168.21/21.61 = { by axiom 2 (multiplicative_left_identity) R->L }
% 168.21/21.61 addition(multiplication(star(Y), X), multiplication(one, X))
% 168.21/21.61 = { by lemma 20 }
% 168.21/21.61 multiplication(addition(one, star(Y)), X)
% 168.21/21.61 = { by lemma 18 }
% 168.21/21.61 multiplication(star(Y), X)
% 168.21/21.61
% 168.21/21.61 Lemma 34: addition(one, multiplication(X, star(X))) = star(multiplication(X, star(X))).
% 168.21/21.61 Proof:
% 168.21/21.61 addition(one, multiplication(X, star(X)))
% 168.21/21.61 = { by lemma 28 R->L }
% 168.21/21.61 ifeq2(leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 7 (ifeq_axiom) R->L }
% 168.21/21.61 ifeq2(ifeq(true, true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 13 (order) R->L }
% 168.21/21.61 ifeq2(ifeq(ifeq3(addition(addition(one, multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), addition(one, multiplication(X, star(X))), leq(addition(one, multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 3 (additive_idempotence) }
% 168.21/21.61 ifeq2(ifeq(ifeq3(addition(one, multiplication(X, star(X))), addition(one, multiplication(X, star(X))), leq(addition(one, multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 14 (order_1) R->L }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 7 (ifeq_axiom) R->L }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(ifeq(true, true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 13 (order) R->L }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(ifeq(ifeq3(addition(multiplication(X, star(X)), star(X)), star(X), leq(multiplication(X, star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by lemma 23 }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(ifeq(ifeq3(star(X), star(X), leq(multiplication(X, star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(ifeq(leq(multiplication(X, star(X)), star(X)), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.61 = { by axiom 3 (additive_idempotence) R->L }
% 168.21/21.61 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(ifeq(leq(addition(multiplication(X, star(X)), multiplication(X, star(X))), star(X)), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 16 (star_induction_left) }
% 168.21/21.62 ifeq2(ifeq(leq(addition(one, multiplication(X, ifeq2(true, true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 9 (ifeq_axiom) }
% 168.21/21.62 ifeq2(ifeq(leq(addition(one, multiplication(X, addition(multiplication(star(X), multiplication(X, star(X))), star(X)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) }
% 168.21/21.62 ifeq2(ifeq(leq(addition(one, multiplication(X, addition(star(X), multiplication(star(X), multiplication(X, star(X)))))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by lemma 30 }
% 168.21/21.62 ifeq2(ifeq(leq(addition(one, multiplication(X, multiplication(star(X), addition(multiplication(X, star(X)), one)))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) }
% 168.21/21.62 ifeq2(ifeq(leq(addition(one, multiplication(X, multiplication(star(X), addition(one, multiplication(X, star(X)))))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 5 (multiplicative_associativity) }
% 168.21/21.62 ifeq2(ifeq(leq(addition(one, multiplication(multiplication(X, star(X)), addition(one, multiplication(X, star(X))))), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.62 ifeq2(ifeq(leq(addition(multiplication(multiplication(X, star(X)), addition(one, multiplication(X, star(X)))), one), addition(one, multiplication(X, star(X)))), true, leq(star(multiplication(X, star(X))), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 1 (multiplicative_right_identity) R->L }
% 168.21/21.62 ifeq2(ifeq(leq(addition(multiplication(multiplication(X, star(X)), addition(one, multiplication(X, star(X)))), one), addition(one, multiplication(X, star(X)))), true, leq(multiplication(star(multiplication(X, star(X))), one), addition(one, multiplication(X, star(X)))), true), true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 16 (star_induction_left) }
% 168.21/21.62 ifeq2(true, true, addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X)))), addition(one, multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 9 (ifeq_axiom) }
% 168.21/21.62 addition(addition(one, multiplication(X, star(X))), star(multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 6 (additive_associativity) R->L }
% 168.21/21.62 addition(one, addition(multiplication(X, star(X)), star(multiplication(X, star(X)))))
% 168.21/21.62 = { by lemma 26 R->L }
% 168.21/21.62 addition(one, addition(multiplication(X, star(X)), addition(star(multiplication(X, star(X))), multiplication(multiplication(X, star(X)), star(multiplication(X, star(X)))))))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.62 addition(one, addition(multiplication(X, star(X)), addition(multiplication(multiplication(X, star(X)), star(multiplication(X, star(X)))), star(multiplication(X, star(X))))))
% 168.21/21.62 = { by lemma 21 }
% 168.21/21.62 addition(one, addition(multiplication(multiplication(X, star(X)), star(multiplication(X, star(X)))), addition(multiplication(X, star(X)), star(multiplication(X, star(X))))))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.62 addition(one, addition(multiplication(multiplication(X, star(X)), star(multiplication(X, star(X)))), addition(star(multiplication(X, star(X))), multiplication(X, star(X)))))
% 168.21/21.62 = { by axiom 6 (additive_associativity) }
% 168.21/21.62 addition(one, addition(addition(multiplication(multiplication(X, star(X)), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), multiplication(X, star(X))))
% 168.21/21.62 = { by axiom 6 (additive_associativity) }
% 168.21/21.62 addition(addition(one, addition(multiplication(multiplication(X, star(X)), star(multiplication(X, star(X)))), star(multiplication(X, star(X))))), multiplication(X, star(X)))
% 168.21/21.62 = { by lemma 17 }
% 168.21/21.62 addition(star(multiplication(X, star(X))), multiplication(X, star(X)))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) }
% 168.21/21.62 addition(multiplication(X, star(X)), star(multiplication(X, star(X))))
% 168.21/21.62 = { by lemma 25 }
% 168.21/21.62 star(multiplication(X, star(X)))
% 168.21/21.62
% 168.21/21.62 Lemma 35: addition(multiplication(multiplication(X, star(X)), Y), multiplication(star(X), Y)) = multiplication(star(X), Y).
% 168.21/21.62 Proof:
% 168.21/21.62 addition(multiplication(multiplication(X, star(X)), Y), multiplication(star(X), Y))
% 168.21/21.62 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.62 addition(multiplication(star(X), Y), multiplication(multiplication(X, star(X)), Y))
% 168.21/21.62 = { by lemma 20 }
% 168.21/21.62 multiplication(addition(multiplication(X, star(X)), star(X)), Y)
% 168.21/21.62 = { by lemma 23 }
% 168.21/21.62 multiplication(star(X), Y)
% 168.21/21.62
% 168.21/21.62 Lemma 36: addition(Y, multiplication(multiplication(X, star(X)), Y)) = multiplication(star(X), Y).
% 168.21/21.62 Proof:
% 168.21/21.62 addition(Y, multiplication(multiplication(X, star(X)), Y))
% 168.21/21.62 = { by lemma 31 }
% 168.21/21.62 multiplication(addition(multiplication(X, star(X)), one), Y)
% 168.21/21.62 = { by axiom 4 (additive_commutativity) }
% 168.21/21.62 multiplication(addition(one, multiplication(X, star(X))), Y)
% 168.21/21.62 = { by lemma 34 }
% 168.21/21.62 multiplication(star(multiplication(X, star(X))), Y)
% 168.21/21.62 = { by axiom 14 (order_1) R->L }
% 168.21/21.62 multiplication(ifeq2(leq(star(X), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.62 = { by axiom 14 (order_1) R->L }
% 168.21/21.62 multiplication(ifeq2(leq(ifeq2(leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.62 = { by axiom 7 (ifeq_axiom) R->L }
% 168.21/21.62 multiplication(ifeq2(leq(ifeq2(ifeq(true, true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.62 = { by axiom 13 (order) R->L }
% 168.21/21.62 multiplication(ifeq2(leq(ifeq2(ifeq(ifeq3(addition(multiplication(X, star(X)), star(X)), star(X), leq(multiplication(X, star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.62 = { by lemma 23 }
% 168.21/21.63 multiplication(ifeq2(leq(ifeq2(ifeq(ifeq3(star(X), star(X), leq(multiplication(X, star(X)), star(X)), true), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.63 multiplication(ifeq2(leq(ifeq2(ifeq(leq(multiplication(X, star(X)), star(X)), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 3 (additive_idempotence) R->L }
% 168.21/21.63 multiplication(ifeq2(leq(ifeq2(ifeq(leq(addition(multiplication(X, star(X)), multiplication(X, star(X))), star(X)), true, leq(multiplication(star(X), multiplication(X, star(X))), star(X)), true), true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 16 (star_induction_left) }
% 168.21/21.63 multiplication(ifeq2(leq(ifeq2(true, true, addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 9 (ifeq_axiom) }
% 168.21/21.63 multiplication(ifeq2(leq(addition(multiplication(star(X), multiplication(X, star(X))), star(X)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 4 (additive_commutativity) }
% 168.21/21.63 multiplication(ifeq2(leq(addition(star(X), multiplication(star(X), multiplication(X, star(X)))), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 30 }
% 168.21/21.63 multiplication(ifeq2(leq(multiplication(star(X), addition(multiplication(X, star(X)), one)), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 4 (additive_commutativity) }
% 168.21/21.63 multiplication(ifeq2(leq(multiplication(star(X), addition(one, multiplication(X, star(X)))), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 34 }
% 168.21/21.63 multiplication(ifeq2(leq(multiplication(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 29 R->L }
% 168.21/21.63 multiplication(ifeq2(leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(multiplication(X, star(X)))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 29 R->L }
% 168.21/21.63 multiplication(ifeq2(leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 7 (ifeq_axiom) R->L }
% 168.21/21.63 multiplication(ifeq2(ifeq(true, true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 13 (order) R->L }
% 168.21/21.63 multiplication(ifeq2(ifeq(ifeq3(addition(multiplication(star(multiplication(X, star(X))), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X)))), leq(multiplication(star(multiplication(X, star(X))), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 23 }
% 168.21/21.63 multiplication(ifeq2(ifeq(ifeq3(star(star(multiplication(X, star(X)))), star(star(multiplication(X, star(X)))), leq(multiplication(star(multiplication(X, star(X))), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.63 multiplication(ifeq2(ifeq(leq(multiplication(star(multiplication(X, star(X))), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 34 R->L }
% 168.21/21.63 multiplication(ifeq2(ifeq(leq(multiplication(addition(one, multiplication(X, star(X))), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.63 multiplication(ifeq2(ifeq(leq(multiplication(addition(multiplication(X, star(X)), one), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by lemma 31 R->L }
% 168.21/21.63 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.63 = { by axiom 14 (order_1) R->L }
% 168.21/21.63 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 7 (ifeq_axiom) R->L }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(ifeq(true, true, leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 13 (order) R->L }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(ifeq(ifeq3(addition(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X))))), leq(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by lemma 35 }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(ifeq(ifeq3(multiplication(star(X), star(star(multiplication(X, star(X))))), multiplication(star(X), star(star(multiplication(X, star(X))))), leq(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(ifeq(leq(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true, leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 3 (additive_idempotence) R->L }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(ifeq(leq(addition(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true, leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 5 (multiplicative_associativity) R->L }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(ifeq(leq(addition(multiplication(X, multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true, leq(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), true), true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 16 (star_induction_left) }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(ifeq2(true, true, addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 9 (ifeq_axiom) }
% 168.21/21.64 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(addition(multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), multiplication(star(X), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.64 = { by axiom 4 (additive_commutativity) }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(addition(multiplication(star(X), star(star(multiplication(X, star(X))))), multiplication(star(X), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by lemma 19 }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 4 (additive_commutativity) }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X)))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 5 (multiplicative_associativity) R->L }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(X, multiplication(star(X), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by lemma 33 R->L }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(X, addition(star(star(multiplication(X, star(X)))), multiplication(star(X), star(star(multiplication(X, star(X)))))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), multiplication(X, addition(multiplication(star(X), star(star(multiplication(X, star(X))))), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 10 (right_distributivity) }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), addition(multiplication(X, multiplication(star(X), star(star(multiplication(X, star(X)))))), multiplication(X, star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 5 (multiplicative_associativity) }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(star(star(multiplication(X, star(X)))), addition(multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))), multiplication(X, star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by lemma 32 R->L }
% 168.21/21.65 multiplication(ifeq2(ifeq(leq(addition(multiplication(X, star(star(multiplication(X, star(X))))), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true, leq(multiplication(star(X), addition(star(star(multiplication(X, star(X)))), multiplication(multiplication(X, star(X)), star(star(multiplication(X, star(X))))))), star(star(multiplication(X, star(X))))), true), true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 16 (star_induction_left) }
% 168.21/21.65 multiplication(ifeq2(true, true, addition(star(X), star(multiplication(X, star(X)))), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 9 (ifeq_axiom) }
% 168.21/21.65 multiplication(addition(star(X), star(multiplication(X, star(X)))), Y)
% 168.21/21.65 = { by axiom 1 (multiplicative_right_identity) R->L }
% 168.21/21.65 multiplication(multiplication(addition(star(X), star(multiplication(X, star(X)))), one), Y)
% 168.21/21.65 = { by lemma 34 R->L }
% 168.21/21.65 multiplication(multiplication(addition(star(X), addition(one, multiplication(X, star(X)))), one), Y)
% 168.21/21.65 = { by axiom 11 (left_distributivity) }
% 168.21/21.65 multiplication(addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), Y)
% 168.21/21.65 = { by axiom 9 (ifeq_axiom) R->L }
% 168.21/21.65 multiplication(ifeq2(true, true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by axiom 13 (order) R->L }
% 168.21/21.65 multiplication(ifeq2(ifeq3(addition(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), multiplication(star(X), one), leq(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), true), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by axiom 4 (additive_commutativity) R->L }
% 168.21/21.65 multiplication(ifeq2(ifeq3(addition(multiplication(star(X), one), addition(one, multiplication(multiplication(X, star(X)), one))), multiplication(star(X), one), leq(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), true), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by lemma 32 }
% 168.21/21.65 multiplication(ifeq2(ifeq3(addition(one, addition(multiplication(multiplication(X, star(X)), one), multiplication(star(X), one))), multiplication(star(X), one), leq(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), true), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by lemma 35 }
% 168.21/21.65 multiplication(ifeq2(ifeq3(addition(one, multiplication(star(X), one)), multiplication(star(X), one), leq(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), true), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by lemma 33 }
% 168.21/21.65 multiplication(ifeq2(ifeq3(multiplication(star(X), one), multiplication(star(X), one), leq(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), true), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by axiom 8 (ifeq_axiom) }
% 168.21/21.65 multiplication(ifeq2(leq(addition(one, multiplication(multiplication(X, star(X)), one)), multiplication(star(X), one)), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by lemma 31 }
% 168.21/21.65 multiplication(ifeq2(leq(multiplication(addition(multiplication(X, star(X)), one), one), multiplication(star(X), one)), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by axiom 4 (additive_commutativity) }
% 168.21/21.65 multiplication(ifeq2(leq(multiplication(addition(one, multiplication(X, star(X))), one), multiplication(star(X), one)), true, addition(multiplication(star(X), one), multiplication(addition(one, multiplication(X, star(X))), one)), multiplication(star(X), one)), Y)
% 168.21/21.65 = { by lemma 28 }
% 168.21/21.65 multiplication(multiplication(star(X), one), Y)
% 168.21/21.65 = { by axiom 1 (multiplicative_right_identity) }
% 168.21/21.66 multiplication(star(X), Y)
% 168.21/21.66
% 168.21/21.66 Goal 1 (goals): tuple(leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)))) = tuple(true, true).
% 168.21/21.66 Proof:
% 168.21/21.66 tuple(leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 168.21/21.66 = { by lemma 36 R->L }
% 168.21/21.66 tuple(leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1))), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 168.21/21.66 = { by axiom 8 (ifeq_axiom) R->L }
% 168.21/21.66 tuple(ifeq3(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1)), leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1))), true), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 168.21/21.66 = { by axiom 3 (additive_idempotence) R->L }
% 168.21/21.66 tuple(ifeq3(addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1))), addition(x1, multiplication(multiplication(x0, star(x0)), x1)), leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1))), true), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 168.21/21.66 = { by axiom 13 (order) }
% 168.21/21.66 tuple(true, leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 168.21/21.66 = { by lemma 36 R->L }
% 168.21/21.66 tuple(true, leq(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 168.21/21.66 = { by axiom 8 (ifeq_axiom) R->L }
% 168.21/21.66 tuple(true, ifeq3(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), leq(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true))
% 168.21/21.66 = { by axiom 3 (additive_idempotence) R->L }
% 168.21/21.66 tuple(true, ifeq3(addition(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), leq(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true))
% 168.21/21.66 = { by axiom 13 (order) }
% 168.21/21.66 tuple(true, true)
% 168.21/21.66 % SZS output end Proof
% 168.21/21.66
% 168.21/21.66 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------