TSTP Solution File: KLE046+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE046+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:40 EDT 2023
% Result : Theorem 97.62s 13.28s
% Output : Proof 99.36s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE046+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n012.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Aug 29 12:29:09 EDT 2023
% 0.17/0.35 % CPUTime :
% 97.62/13.28 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 97.62/13.28
% 97.62/13.28 % SZS status Theorem
% 97.62/13.28
% 99.04/13.39 % SZS output start Proof
% 99.04/13.39 Take the following subset of the input axioms:
% 99.04/13.39 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 99.04/13.39 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 99.04/13.39 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 99.04/13.39 fof(goals, conjecture, ![X0, X1]: (leq(multiplication(star(X1), star(X0)), multiplication(star(X0), star(X1))) => leq(star(addition(X0, X1)), multiplication(star(X0), star(X1))))).
% 99.04/13.39 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 99.04/13.39 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 99.04/13.39 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 99.04/13.39 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 99.04/13.39 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 99.04/13.39 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 99.04/13.39 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 99.04/13.39 fof(star_induction_right, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), A2_2) => leq(multiplication(C2, star(B2)), A2_2))).
% 99.04/13.39 fof(star_unfold_left, axiom, ![A3]: leq(addition(one, multiplication(star(A3), A3)), star(A3))).
% 99.04/13.39 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 99.04/13.39
% 99.04/13.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 99.04/13.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 99.04/13.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 99.04/13.39 fresh(y, y, x1...xn) = u
% 99.04/13.39 C => fresh(s, t, x1...xn) = v
% 99.19/13.39 where fresh is a fresh function symbol and x1..xn are the free
% 99.19/13.39 variables of u and v.
% 99.19/13.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 99.19/13.39 input problem has no model of domain size 1).
% 99.19/13.39
% 99.19/13.39 The encoding turns the above axioms into the following unit equations and goals:
% 99.19/13.39
% 99.19/13.39 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 99.19/13.39 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 99.19/13.39 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 99.19/13.39 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 99.19/13.39 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 99.19/13.39 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 99.19/13.39 Axiom 7 (order_1): fresh(X, X, Y, Z) = Z.
% 99.19/13.39 Axiom 8 (order): fresh3(X, X, Y, Z) = true.
% 99.19/13.39 Axiom 9 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 99.19/13.39 Axiom 10 (star_induction_right): fresh2(X, X, Y, Z, W) = true.
% 99.19/13.39 Axiom 11 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 99.19/13.39 Axiom 12 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 99.19/13.39 Axiom 13 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 99.19/13.39 Axiom 14 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 99.19/13.39 Axiom 15 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 99.19/13.39 Axiom 16 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 99.19/13.39 Axiom 17 (goals): leq(multiplication(star(x1), star(x0)), multiplication(star(x0), star(x1))) = true.
% 99.19/13.39 Axiom 18 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 99.19/13.39 Axiom 19 (star_induction_right): fresh2(leq(addition(multiplication(X, Y), Z), X), true, X, Y, Z) = leq(multiplication(Z, star(Y)), X).
% 99.19/13.39
% 99.19/13.39 Lemma 20: leq(X, X) = true.
% 99.19/13.39 Proof:
% 99.19/13.39 leq(X, X)
% 99.19/13.39 = { by axiom 14 (order) R->L }
% 99.19/13.39 fresh3(addition(X, X), X, X, X)
% 99.19/13.39 = { by axiom 3 (additive_idempotence) }
% 99.19/13.39 fresh3(X, X, X, X)
% 99.19/13.39 = { by axiom 8 (order) }
% 99.19/13.39 true
% 99.19/13.39
% 99.19/13.39 Lemma 21: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 99.19/13.39 Proof:
% 99.19/13.39 addition(X, multiplication(Y, X))
% 99.19/13.39 = { by axiom 2 (multiplicative_left_identity) R->L }
% 99.19/13.40 addition(multiplication(one, X), multiplication(Y, X))
% 99.19/13.40 = { by axiom 12 (left_distributivity) R->L }
% 99.19/13.40 multiplication(addition(one, Y), X)
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 multiplication(addition(Y, one), X)
% 99.19/13.40
% 99.19/13.40 Lemma 22: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(one, multiplication(addition(X, one), star(X)))
% 99.19/13.40 = { by lemma 21 R->L }
% 99.19/13.40 addition(one, addition(star(X), multiplication(X, star(X))))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(one, addition(multiplication(X, star(X)), star(X)))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 addition(addition(one, multiplication(X, star(X))), star(X))
% 99.19/13.40 = { by axiom 13 (order_1) R->L }
% 99.19/13.40 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 99.19/13.40 = { by axiom 15 (star_unfold_right) }
% 99.19/13.40 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 99.19/13.40 = { by axiom 7 (order_1) }
% 99.19/13.40 star(X)
% 99.19/13.40
% 99.19/13.40 Lemma 23: addition(X, addition(X, Y)) = addition(X, Y).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(X, addition(X, Y))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 addition(addition(X, X), Y)
% 99.19/13.40 = { by axiom 3 (additive_idempotence) }
% 99.19/13.40 addition(X, Y)
% 99.19/13.40
% 99.19/13.40 Lemma 24: addition(one, star(X)) = star(X).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(one, star(X))
% 99.19/13.40 = { by lemma 22 R->L }
% 99.19/13.40 addition(one, addition(one, multiplication(addition(X, one), star(X))))
% 99.19/13.40 = { by lemma 23 }
% 99.19/13.40 addition(one, multiplication(addition(X, one), star(X)))
% 99.19/13.40 = { by lemma 22 }
% 99.19/13.40 star(X)
% 99.19/13.40
% 99.19/13.40 Lemma 25: addition(X, addition(Y, X)) = addition(Y, X).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(X, addition(Y, X))
% 99.19/13.40 = { by lemma 23 R->L }
% 99.19/13.40 addition(X, addition(Y, addition(Y, X)))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(X, addition(Y, addition(X, Y)))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 addition(addition(X, Y), addition(X, Y))
% 99.19/13.40 = { by axiom 3 (additive_idempotence) }
% 99.19/13.40 addition(X, Y)
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 addition(Y, X)
% 99.19/13.40
% 99.19/13.40 Lemma 26: addition(X, addition(Y, Z)) = addition(Y, addition(X, Z)).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(X, addition(Y, Z))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(addition(Y, Z), X)
% 99.19/13.40 = { by axiom 6 (additive_associativity) R->L }
% 99.19/13.40 addition(Y, addition(Z, X))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 addition(Y, addition(X, Z))
% 99.19/13.40
% 99.19/13.40 Lemma 27: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(X, multiplication(X, Y))
% 99.19/13.40 = { by axiom 1 (multiplicative_right_identity) R->L }
% 99.19/13.40 addition(multiplication(X, one), multiplication(X, Y))
% 99.19/13.40 = { by axiom 11 (right_distributivity) R->L }
% 99.19/13.40 multiplication(X, addition(one, Y))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 multiplication(X, addition(Y, one))
% 99.19/13.40
% 99.19/13.40 Lemma 28: addition(Z, addition(X, Y)) = addition(X, addition(Y, Z)).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(Z, addition(X, Y))
% 99.19/13.40 = { by lemma 26 R->L }
% 99.19/13.40 addition(X, addition(Z, Y))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 addition(X, addition(Y, Z))
% 99.19/13.40
% 99.19/13.40 Lemma 29: leq(X, addition(X, Y)) = true.
% 99.19/13.40 Proof:
% 99.19/13.40 leq(X, addition(X, Y))
% 99.19/13.40 = { by axiom 14 (order) R->L }
% 99.19/13.40 fresh3(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 99.19/13.40 = { by lemma 23 }
% 99.19/13.40 fresh3(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 99.19/13.40 = { by axiom 8 (order) }
% 99.19/13.40 true
% 99.19/13.40
% 99.19/13.40 Lemma 30: addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Z, Y)).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(multiplication(X, Y), multiplication(X, Z))
% 99.19/13.40 = { by axiom 11 (right_distributivity) R->L }
% 99.19/13.40 multiplication(X, addition(Y, Z))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 multiplication(X, addition(Z, Y))
% 99.19/13.40
% 99.19/13.40 Lemma 31: addition(multiplication(X, star(Y)), X) = multiplication(X, star(Y)).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(multiplication(X, star(Y)), X)
% 99.19/13.40 = { by axiom 1 (multiplicative_right_identity) R->L }
% 99.19/13.40 addition(multiplication(X, star(Y)), multiplication(X, one))
% 99.19/13.40 = { by lemma 30 }
% 99.19/13.40 multiplication(X, addition(one, star(Y)))
% 99.19/13.40 = { by lemma 24 }
% 99.19/13.40 multiplication(X, star(Y))
% 99.19/13.40
% 99.19/13.40 Lemma 32: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(multiplication(X, Y), multiplication(Z, Y))
% 99.19/13.40 = { by axiom 12 (left_distributivity) R->L }
% 99.19/13.40 multiplication(addition(X, Z), Y)
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 multiplication(addition(Z, X), Y)
% 99.19/13.40
% 99.19/13.40 Lemma 33: addition(multiplication(star(X), Y), Y) = multiplication(star(X), Y).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(multiplication(star(X), Y), Y)
% 99.19/13.40 = { by axiom 2 (multiplicative_left_identity) R->L }
% 99.19/13.40 addition(multiplication(star(X), Y), multiplication(one, Y))
% 99.19/13.40 = { by lemma 32 }
% 99.19/13.40 multiplication(addition(one, star(X)), Y)
% 99.19/13.40 = { by lemma 24 }
% 99.19/13.40 multiplication(star(X), Y)
% 99.19/13.40
% 99.19/13.40 Lemma 34: addition(one, multiplication(star(X), addition(X, one))) = star(X).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(one, multiplication(star(X), addition(X, one)))
% 99.19/13.40 = { by lemma 27 R->L }
% 99.19/13.40 addition(one, addition(star(X), multiplication(star(X), X)))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(one, addition(multiplication(star(X), X), star(X)))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 addition(addition(one, multiplication(star(X), X)), star(X))
% 99.19/13.40 = { by axiom 13 (order_1) R->L }
% 99.19/13.40 fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
% 99.19/13.40 = { by axiom 16 (star_unfold_left) }
% 99.19/13.40 fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
% 99.19/13.40 = { by axiom 7 (order_1) }
% 99.19/13.40 star(X)
% 99.19/13.40
% 99.19/13.40 Lemma 35: multiplication(star(X), addition(X, one)) = star(X).
% 99.19/13.40 Proof:
% 99.19/13.40 multiplication(star(X), addition(X, one))
% 99.19/13.40 = { by axiom 3 (additive_idempotence) R->L }
% 99.19/13.40 multiplication(star(X), addition(X, addition(one, one)))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 multiplication(star(X), addition(addition(X, one), one))
% 99.19/13.40 = { by lemma 27 R->L }
% 99.19/13.40 addition(star(X), multiplication(star(X), addition(X, one)))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(multiplication(star(X), addition(X, one)), star(X))
% 99.19/13.40 = { by lemma 34 R->L }
% 99.19/13.40 addition(multiplication(star(X), addition(X, one)), addition(one, multiplication(star(X), addition(X, one))))
% 99.19/13.40 = { by lemma 25 }
% 99.19/13.40 addition(one, multiplication(star(X), addition(X, one)))
% 99.19/13.40 = { by lemma 34 }
% 99.19/13.40 star(X)
% 99.19/13.40
% 99.19/13.40 Lemma 36: multiplication(addition(X, one), star(X)) = star(X).
% 99.19/13.40 Proof:
% 99.19/13.40 multiplication(addition(X, one), star(X))
% 99.19/13.40 = { by axiom 3 (additive_idempotence) R->L }
% 99.19/13.40 multiplication(addition(X, addition(one, one)), star(X))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 multiplication(addition(addition(X, one), one), star(X))
% 99.19/13.40 = { by lemma 21 R->L }
% 99.19/13.40 addition(star(X), multiplication(addition(X, one), star(X)))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(multiplication(addition(X, one), star(X)), star(X))
% 99.19/13.40 = { by lemma 22 R->L }
% 99.19/13.40 addition(multiplication(addition(X, one), star(X)), addition(one, multiplication(addition(X, one), star(X))))
% 99.19/13.40 = { by lemma 25 }
% 99.19/13.40 addition(one, multiplication(addition(X, one), star(X)))
% 99.19/13.40 = { by lemma 22 }
% 99.19/13.40 star(X)
% 99.19/13.40
% 99.19/13.40 Lemma 37: addition(multiplication(X, Y), addition(Z, multiplication(X, W))) = addition(Z, multiplication(X, addition(Y, W))).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(multiplication(X, Y), addition(Z, multiplication(X, W)))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 addition(multiplication(X, Y), addition(multiplication(X, W), Z))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 addition(addition(multiplication(X, Y), multiplication(X, W)), Z)
% 99.19/13.40 = { by axiom 11 (right_distributivity) R->L }
% 99.19/13.40 addition(multiplication(X, addition(Y, W)), Z)
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 addition(Z, multiplication(X, addition(Y, W)))
% 99.19/13.40
% 99.19/13.40 Lemma 38: addition(multiplication(Y, Z), multiplication(addition(Y, one), X)) = addition(X, multiplication(Y, addition(Z, X))).
% 99.19/13.40 Proof:
% 99.19/13.40 addition(multiplication(Y, Z), multiplication(addition(Y, one), X))
% 99.19/13.40 = { by lemma 21 R->L }
% 99.19/13.40 addition(multiplication(Y, Z), addition(X, multiplication(Y, X)))
% 99.19/13.40 = { by lemma 37 }
% 99.19/13.40 addition(X, multiplication(Y, addition(Z, X)))
% 99.19/13.40
% 99.19/13.40 Lemma 39: leq(multiplication(X, Y), multiplication(addition(X, Z), Y)) = true.
% 99.19/13.40 Proof:
% 99.19/13.40 leq(multiplication(X, Y), multiplication(addition(X, Z), Y))
% 99.19/13.40 = { by axiom 12 (left_distributivity) }
% 99.19/13.40 leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y)))
% 99.19/13.40 = { by lemma 29 }
% 99.19/13.40 true
% 99.19/13.40
% 99.19/13.40 Lemma 40: multiplication(multiplication(X, star(Y)), addition(one, addition(Z, Y))) = multiplication(multiplication(X, star(Y)), addition(Z, one)).
% 99.19/13.40 Proof:
% 99.19/13.40 multiplication(multiplication(X, star(Y)), addition(one, addition(Z, Y)))
% 99.19/13.40 = { by axiom 5 (multiplicative_associativity) R->L }
% 99.19/13.40 multiplication(X, multiplication(star(Y), addition(one, addition(Z, Y))))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 multiplication(X, multiplication(star(Y), addition(addition(Z, Y), one)))
% 99.19/13.40 = { by lemma 27 R->L }
% 99.19/13.40 multiplication(X, addition(star(Y), multiplication(star(Y), addition(Z, Y))))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 multiplication(X, addition(multiplication(star(Y), addition(Z, Y)), star(Y)))
% 99.19/13.40 = { by lemma 34 R->L }
% 99.19/13.40 multiplication(X, addition(multiplication(star(Y), addition(Z, Y)), addition(one, multiplication(star(Y), addition(Y, one)))))
% 99.19/13.40 = { by lemma 37 }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(addition(Z, Y), addition(Y, one)))))
% 99.19/13.40 = { by lemma 26 }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(Y, addition(addition(Z, Y), one)))))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(Y, addition(one, addition(Z, Y))))))
% 99.19/13.40 = { by axiom 6 (additive_associativity) }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(addition(Y, one), addition(Z, Y)))))
% 99.19/13.40 = { by axiom 11 (right_distributivity) }
% 99.19/13.40 multiplication(X, addition(one, addition(multiplication(star(Y), addition(Y, one)), multiplication(star(Y), addition(Z, Y)))))
% 99.19/13.40 = { by lemma 35 }
% 99.19/13.40 multiplication(X, addition(one, addition(star(Y), multiplication(star(Y), addition(Z, Y)))))
% 99.19/13.40 = { by lemma 27 }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(addition(Z, Y), one))))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(one, addition(Z, Y)))))
% 99.19/13.40 = { by lemma 28 }
% 99.19/13.40 multiplication(X, addition(one, multiplication(star(Y), addition(Z, addition(Y, one)))))
% 99.19/13.40 = { by lemma 37 R->L }
% 99.19/13.40 multiplication(X, addition(multiplication(star(Y), Z), addition(one, multiplication(star(Y), addition(Y, one)))))
% 99.19/13.40 = { by lemma 34 }
% 99.19/13.40 multiplication(X, addition(multiplication(star(Y), Z), star(Y)))
% 99.19/13.40 = { by axiom 4 (additive_commutativity) }
% 99.19/13.40 multiplication(X, addition(star(Y), multiplication(star(Y), Z)))
% 99.19/13.40 = { by lemma 27 }
% 99.19/13.40 multiplication(X, multiplication(star(Y), addition(Z, one)))
% 99.19/13.40 = { by axiom 5 (multiplicative_associativity) }
% 99.19/13.40 multiplication(multiplication(X, star(Y)), addition(Z, one))
% 99.19/13.40
% 99.19/13.40 Lemma 41: fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X) = leq(multiplication(star(Y), X), Z).
% 99.19/13.40 Proof:
% 99.19/13.40 fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X)
% 99.19/13.40 = { by axiom 4 (additive_commutativity) R->L }
% 99.19/13.40 fresh4(leq(addition(multiplication(Y, Z), X), Z), true, Y, Z, X)
% 99.19/13.40 = { by axiom 18 (star_induction_left) }
% 99.19/13.40 leq(multiplication(star(Y), X), Z)
% 99.19/13.40
% 99.19/13.40 Lemma 42: fresh4(leq(multiplication(addition(X, one), Y), Y), true, X, Y, Y) = leq(multiplication(star(X), Y), Y).
% 99.19/13.40 Proof:
% 99.19/13.40 fresh4(leq(multiplication(addition(X, one), Y), Y), true, X, Y, Y)
% 99.19/13.40 = { by lemma 21 R->L }
% 99.19/13.40 fresh4(leq(addition(Y, multiplication(X, Y)), Y), true, X, Y, Y)
% 99.19/13.40 = { by lemma 41 }
% 99.19/13.40 leq(multiplication(star(X), Y), Y)
% 99.19/13.40
% 99.19/13.41 Lemma 43: fresh2(leq(multiplication(X, Y), X), true, X, Y, multiplication(X, Y)) = leq(multiplication(X, multiplication(Y, star(Y))), X).
% 99.19/13.41 Proof:
% 99.19/13.41 fresh2(leq(multiplication(X, Y), X), true, X, Y, multiplication(X, Y))
% 99.19/13.41 = { by axiom 3 (additive_idempotence) R->L }
% 99.19/13.41 fresh2(leq(addition(multiplication(X, Y), multiplication(X, Y)), X), true, X, Y, multiplication(X, Y))
% 99.19/13.41 = { by axiom 19 (star_induction_right) }
% 99.19/13.41 leq(multiplication(multiplication(X, Y), star(Y)), X)
% 99.19/13.41 = { by axiom 5 (multiplicative_associativity) R->L }
% 99.19/13.41 leq(multiplication(X, multiplication(Y, star(Y))), X)
% 99.19/13.41
% 99.19/13.41 Goal 1 (goals_1): leq(star(addition(x0, x1)), multiplication(star(x0), star(x1))) = true.
% 99.19/13.41 Proof:
% 99.19/13.41 leq(star(addition(x0, x1)), multiplication(star(x0), star(x1)))
% 99.19/13.41 = { by axiom 7 (order_1) R->L }
% 99.19/13.41 leq(star(addition(x0, x1)), fresh(true, true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.19/13.41 = { by axiom 10 (star_induction_right) R->L }
% 99.19/13.41 leq(star(addition(x0, x1)), fresh(fresh2(true, true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.19/13.41 = { by axiom 8 (order) R->L }
% 99.19/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.19/13.41 = { by lemma 21 R->L }
% 99.19/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), addition(star(x0), multiplication(addition(one, addition(x0, x1)), star(x0)))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.19/13.41 = { by lemma 35 R->L }
% 99.19/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), addition(multiplication(star(x0), addition(x0, one)), multiplication(addition(one, addition(x0, x1)), star(x0)))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.19/13.41 = { by lemma 34 R->L }
% 99.19/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), addition(multiplication(star(x0), addition(x0, one)), multiplication(addition(one, addition(x0, x1)), addition(one, multiplication(star(x0), addition(x0, one)))))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.19/13.41 = { by lemma 38 R->L }
% 99.34/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), addition(multiplication(addition(one, addition(x0, x1)), one), multiplication(addition(addition(one, addition(x0, x1)), one), multiplication(star(x0), addition(x0, one))))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.34/13.41 = { by axiom 1 (multiplicative_right_identity) }
% 99.34/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), addition(addition(one, addition(x0, x1)), multiplication(addition(addition(one, addition(x0, x1)), one), multiplication(star(x0), addition(x0, one))))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.34/13.41 = { by lemma 35 }
% 99.34/13.41 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(multiplication(multiplication(star(x0), star(x1)), addition(addition(one, addition(x0, x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.34/13.42 = { by axiom 11 (right_distributivity) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(fresh3(addition(multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0))), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 14 (order) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by lemma 40 }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, addition(x0, x1)), one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 6 (additive_associativity) R->L }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(addition(x0, x1), one)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 1 (multiplicative_right_identity) R->L }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(addition(x0, x1), multiplication(one, one))), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 3 (additive_idempotence) R->L }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(addition(x0, x1), multiplication(one, addition(one, one)))), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(multiplication(one, addition(one, one)), addition(x0, x1))), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 6 (additive_associativity) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(one, multiplication(one, addition(one, one))), addition(x0, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by lemma 27 }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(multiplication(one, addition(addition(one, one), one)), addition(x0, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 4 (additive_commutativity) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(multiplication(one, addition(one, addition(one, one))), addition(x0, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by lemma 23 }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(multiplication(one, addition(one, one)), addition(x0, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 3 (additive_idempotence) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(multiplication(one, one), addition(x0, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 1 (multiplicative_right_identity) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by lemma 28 }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(x0, addition(x1, one)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(x0, addition(one, x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 6 (additive_associativity) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(x0, one), x1), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 12 (left_distributivity) }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), addition(multiplication(addition(x0, one), star(x0)), multiplication(x1, star(x0))))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by lemma 36 }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), addition(star(x0), multiplication(x1, star(x0))))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by lemma 21 }
% 99.36/13.42 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), multiplication(addition(x1, one), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.42 = { by axiom 5 (multiplicative_associativity) }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(multiplication(star(x0), star(x1)), addition(x1, one)), star(x0))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 5 (multiplicative_associativity) R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), multiplication(star(x1), addition(x1, one))), star(x0))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 35 }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), star(x0))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 24 R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), addition(one, star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(multiplication(star(x0), star(x1)), addition(star(x0), one))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 27 R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), addition(multiplication(star(x0), star(x1)), multiplication(multiplication(star(x0), star(x1)), star(x0)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), addition(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 13 (order_1) R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 24 R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, star(x0)), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 7 (order_1) R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, fresh(true, true, multiplication(star(x0), star(x0)), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 9 (star_induction_left) R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, fresh(fresh4(true, true, x0, star(x0), star(x0)), true, multiplication(star(x0), star(x0)), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 20 R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, fresh(fresh4(leq(star(x0), star(x0)), true, x0, star(x0), star(x0)), true, multiplication(star(x0), star(x0)), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 36 R->L }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, fresh(fresh4(leq(multiplication(addition(x0, one), star(x0)), star(x0)), true, x0, star(x0), star(x0)), true, multiplication(star(x0), star(x0)), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 42 }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, fresh(leq(multiplication(star(x0), star(x0)), star(x0)), true, multiplication(star(x0), star(x0)), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 13 (order_1) }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, addition(multiplication(star(x0), star(x0)), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by axiom 4 (additive_commutativity) }
% 99.36/13.43 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, addition(star(x0), multiplication(star(x0), star(x0)))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.43 = { by lemma 27 }
% 99.36/13.44 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, multiplication(star(x0), addition(star(x0), one))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.44 = { by axiom 4 (additive_commutativity) }
% 99.36/13.44 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, multiplication(star(x0), addition(one, star(x0)))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.44 = { by lemma 24 }
% 99.36/13.44 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(one, multiplication(star(x0), star(x0))), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.44 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.44 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(addition(multiplication(star(x0), star(x0)), one), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.44 = { by lemma 21 R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(star(x1), multiplication(multiplication(star(x0), star(x0)), star(x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 5 (multiplicative_associativity) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(star(x1), multiplication(star(x0), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 33 R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(star(x1), multiplication(star(x0), addition(multiplication(star(x0), star(x1)), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 38 R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), multiplication(star(x0), star(x1))), multiplication(addition(star(x0), one), star(x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), multiplication(star(x0), star(x1))), multiplication(addition(one, star(x0)), star(x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 32 R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), multiplication(star(x0), star(x1))), addition(multiplication(star(x0), star(x1)), multiplication(one, star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 26 }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), star(x1)), addition(multiplication(star(x0), multiplication(star(x0), star(x1))), multiplication(one, star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 2 (multiplicative_left_identity) }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), star(x1)), addition(multiplication(star(x0), multiplication(star(x0), star(x1))), star(x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), star(x1)), addition(star(x1), multiplication(star(x0), multiplication(star(x0), star(x1)))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 6 (additive_associativity) }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(addition(multiplication(star(x0), star(x1)), star(x1)), multiplication(star(x0), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 33 }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), star(x1)), multiplication(star(x0), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 30 }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), addition(multiplication(star(x0), star(x1)), star(x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 33 }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), multiplication(star(x0), star(x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 7 (order_1) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), fresh(true, true, multiplication(star(x1), star(x0)), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 17 (goals) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), fresh(leq(multiplication(star(x1), star(x0)), multiplication(star(x0), star(x1))), true, multiplication(star(x1), star(x0)), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 13 (order_1) }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), addition(multiplication(star(x1), star(x0)), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 11 (right_distributivity) }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(star(x0), multiplication(star(x1), star(x0))), multiplication(star(x0), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 5 (multiplicative_associativity) }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(x0)), addition(multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), multiplication(star(x0), star(x1))))), true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 29 }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), fresh(true, true, multiplication(multiplication(star(x0), star(x1)), star(x0)), multiplication(star(x0), star(x1)))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 7 (order_1) }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(x0, one)), multiplication(star(x0), star(x1))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 40 R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(fresh2(leq(multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1))), multiplication(star(x0), star(x1))), true, multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, addition(x0, x1)))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by lemma 43 }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), star(addition(one, addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), star(addition(addition(x0, x1), one)))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 7 (order_1) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(true, true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.45 = { by axiom 9 (star_induction_left) R->L }
% 99.36/13.45 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(fresh4(true, true, addition(x0, x1), star(addition(addition(x0, x1), one)), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 39 R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(fresh4(leq(multiplication(addition(addition(x0, x1), one), star(addition(addition(x0, x1), one))), multiplication(addition(addition(addition(x0, x1), one), one), star(addition(addition(x0, x1), one)))), true, addition(x0, x1), star(addition(addition(x0, x1), one)), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 36 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(fresh4(leq(multiplication(addition(addition(x0, x1), one), star(addition(addition(x0, x1), one))), star(addition(addition(x0, x1), one))), true, addition(x0, x1), star(addition(addition(x0, x1), one)), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 42 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 24 R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(multiplication(star(addition(x0, x1)), addition(one, star(addition(addition(x0, x1), one)))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(multiplication(star(addition(x0, x1)), addition(star(addition(addition(x0, x1), one)), one)), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 27 R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(addition(star(addition(x0, x1)), multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one)))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(addition(multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 13 (order_1) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(leq(multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 35 R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(leq(multiplication(multiplication(star(addition(x0, x1)), addition(addition(x0, x1), one)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 5 (multiplicative_associativity) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(leq(multiplication(star(addition(x0, x1)), multiplication(addition(addition(x0, x1), one), star(addition(addition(x0, x1), one)))), star(addition(x0, x1))), true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 43 R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(fresh2(leq(multiplication(star(addition(x0, x1)), addition(addition(x0, x1), one)), star(addition(x0, x1))), true, star(addition(x0, x1)), addition(addition(x0, x1), one), multiplication(star(addition(x0, x1)), addition(addition(x0, x1), one))), true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 35 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(fresh2(leq(star(addition(x0, x1)), star(addition(x0, x1))), true, star(addition(x0, x1)), addition(addition(x0, x1), one), multiplication(star(addition(x0, x1)), addition(addition(x0, x1), one))), true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 20 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(fresh2(true, true, star(addition(x0, x1)), addition(addition(x0, x1), one), multiplication(star(addition(x0, x1)), addition(addition(x0, x1), one))), true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 10 (star_induction_right) }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(fresh(true, true, multiplication(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), star(addition(x0, x1))), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 7 (order_1) }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(star(addition(x0, x1)), star(addition(addition(x0, x1), one))), true, star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 13 (order_1) }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), addition(star(addition(x0, x1)), star(addition(addition(x0, x1), one))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), addition(star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 13 (order_1) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(star(addition(addition(x0, x1), one)), star(addition(x0, x1))), true, star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 1 (multiplicative_right_identity) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(leq(multiplication(star(addition(addition(x0, x1), one)), one), star(addition(x0, x1))), true, star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 41 R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(fresh4(leq(addition(one, multiplication(addition(addition(x0, x1), one), star(addition(x0, x1)))), star(addition(x0, x1))), true, addition(addition(x0, x1), one), star(addition(x0, x1)), one), true, star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 22 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(fresh4(leq(star(addition(x0, x1)), star(addition(x0, x1))), true, addition(addition(x0, x1), one), star(addition(x0, x1)), one), true, star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 20 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(fresh4(true, true, addition(addition(x0, x1), one), star(addition(x0, x1)), one), true, star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 9 (star_induction_left) }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), fresh(true, true, star(addition(addition(x0, x1), one)), star(addition(x0, x1))))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 7 (order_1) }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(one, addition(x0, x1)), star(addition(x0, x1)))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), multiplication(addition(addition(x0, x1), one), star(addition(x0, x1)))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by lemma 36 }
% 99.36/13.46 leq(star(addition(x0, x1)), fresh(leq(multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))), true, multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 13 (order_1) }
% 99.36/13.46 leq(star(addition(x0, x1)), addition(multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))), multiplication(star(x0), star(x1))))
% 99.36/13.46 = { by axiom 4 (additive_commutativity) }
% 99.36/13.46 leq(star(addition(x0, x1)), addition(multiplication(star(x0), star(x1)), multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1)))))
% 99.36/13.46 = { by lemma 27 }
% 99.36/13.46 leq(star(addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(star(addition(x0, x1)), one)))
% 99.36/13.46 = { by axiom 4 (additive_commutativity) }
% 99.36/13.46 leq(star(addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), addition(one, star(addition(x0, x1)))))
% 99.36/13.46 = { by lemma 24 }
% 99.36/13.46 leq(star(addition(x0, x1)), multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))))
% 99.36/13.47 = { by axiom 2 (multiplicative_left_identity) R->L }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(multiplication(star(x0), star(x1)), star(addition(x0, x1))))
% 99.36/13.47 = { by lemma 31 R->L }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(addition(multiplication(star(x0), star(x1)), star(x0)), star(addition(x0, x1))))
% 99.36/13.47 = { by lemma 24 R->L }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(addition(multiplication(star(x0), star(x1)), addition(one, star(x0))), star(addition(x0, x1))))
% 99.36/13.47 = { by axiom 4 (additive_commutativity) R->L }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(addition(multiplication(star(x0), star(x1)), addition(star(x0), one)), star(addition(x0, x1))))
% 99.36/13.47 = { by axiom 6 (additive_associativity) }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(addition(addition(multiplication(star(x0), star(x1)), star(x0)), one), star(addition(x0, x1))))
% 99.36/13.47 = { by lemma 31 }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(addition(multiplication(star(x0), star(x1)), one), star(addition(x0, x1))))
% 99.36/13.47 = { by axiom 4 (additive_commutativity) }
% 99.36/13.47 leq(multiplication(one, star(addition(x0, x1))), multiplication(addition(one, multiplication(star(x0), star(x1))), star(addition(x0, x1))))
% 99.36/13.47 = { by lemma 39 }
% 99.36/13.47 true
% 99.36/13.47 % SZS output end Proof
% 99.36/13.47
% 99.36/13.47 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------