TSTP Solution File: KLE042+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE042+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 : n016.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 05:35:38 EDT 2023
% Result : Theorem 87.12s 11.31s
% Output : Proof 87.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : KLE042+1 : TPTP v8.1.2. Released v4.0.0.
% 0.03/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n016.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 11:28:45 EDT 2023
% 0.13/0.34 % CPUTime :
% 87.12/11.31 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 87.12/11.31
% 87.12/11.31 % SZS status Theorem
% 87.12/11.31
% 87.67/11.35 % SZS output start Proof
% 87.67/11.35 Take the following subset of the input axioms:
% 87.67/11.35 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 87.67/11.35 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 87.67/11.35 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 87.67/11.35 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 87.67/11.35 fof(goals, conjecture, ![X0, X1]: multiplication(star(multiplication(X0, X1)), X0)=multiplication(X0, star(multiplication(X1, X0)))).
% 87.67/11.35 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 87.67/11.35 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 87.67/11.35 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 87.67/11.35 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 87.67/11.35 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 87.67/11.35 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 87.67/11.35 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 87.67/11.35 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))).
% 87.67/11.35 fof(star_unfold_left, axiom, ![A3]: leq(addition(one, multiplication(star(A3), A3)), star(A3))).
% 87.67/11.35 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 87.67/11.35
% 87.67/11.35 Now clausify the problem and encode Horn clauses using encoding 3 of
% 87.67/11.35 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 87.67/11.35 We repeatedly replace C & s=t => u=v by the two clauses:
% 87.67/11.35 fresh(y, y, x1...xn) = u
% 87.67/11.35 C => fresh(s, t, x1...xn) = v
% 87.67/11.35 where fresh is a fresh function symbol and x1..xn are the free
% 87.67/11.35 variables of u and v.
% 87.67/11.35 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 87.67/11.35 input problem has no model of domain size 1).
% 87.67/11.35
% 87.67/11.35 The encoding turns the above axioms into the following unit equations and goals:
% 87.67/11.35
% 87.67/11.35 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 87.67/11.35 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 87.67/11.35 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 87.67/11.35 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 87.67/11.35 Axiom 5 (additive_identity): addition(X, zero) = X.
% 87.67/11.35 Axiom 6 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 87.67/11.35 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 87.67/11.35 Axiom 8 (order_1): fresh(X, X, Y, Z) = Z.
% 87.67/11.35 Axiom 9 (order): fresh3(X, X, Y, Z) = true.
% 87.67/11.35 Axiom 10 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 87.67/11.35 Axiom 11 (star_induction_right): fresh2(X, X, Y, Z, W) = true.
% 87.67/11.35 Axiom 12 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 87.67/11.35 Axiom 13 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 87.67/11.35 Axiom 14 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 87.67/11.35 Axiom 15 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 87.67/11.35 Axiom 16 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 87.67/11.35 Axiom 17 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 87.67/11.35 Axiom 18 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 87.67/11.35 Axiom 19 (star_induction_right): fresh2(leq(addition(multiplication(X, Y), Z), X), true, X, Y, Z) = leq(multiplication(Z, star(Y)), X).
% 87.67/11.35
% 87.67/11.35 Lemma 20: addition(zero, X) = X.
% 87.67/11.35 Proof:
% 87.67/11.35 addition(zero, X)
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 addition(X, zero)
% 87.67/11.35 = { by axiom 5 (additive_identity) }
% 87.67/11.35 X
% 87.67/11.35
% 87.67/11.35 Lemma 21: leq(X, X) = true.
% 87.67/11.35 Proof:
% 87.67/11.35 leq(X, X)
% 87.67/11.35 = { by axiom 15 (order) R->L }
% 87.67/11.35 fresh3(addition(X, X), X, X, X)
% 87.67/11.35 = { by axiom 3 (additive_idempotence) }
% 87.67/11.35 fresh3(X, X, X, X)
% 87.67/11.35 = { by axiom 9 (order) }
% 87.67/11.35 true
% 87.67/11.35
% 87.67/11.35 Lemma 22: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(X, multiplication(Y, X))
% 87.67/11.35 = { by axiom 2 (multiplicative_left_identity) R->L }
% 87.67/11.35 addition(multiplication(one, X), multiplication(Y, X))
% 87.67/11.35 = { by axiom 13 (left_distributivity) R->L }
% 87.67/11.35 multiplication(addition(one, Y), X)
% 87.67/11.35 = { by axiom 4 (additive_commutativity) }
% 87.67/11.35 multiplication(addition(Y, one), X)
% 87.67/11.35
% 87.67/11.35 Lemma 23: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(one, multiplication(addition(X, one), star(X)))
% 87.67/11.35 = { by lemma 22 R->L }
% 87.67/11.35 addition(one, addition(star(X), multiplication(X, star(X))))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 addition(one, addition(multiplication(X, star(X)), star(X)))
% 87.67/11.35 = { by axiom 7 (additive_associativity) }
% 87.67/11.35 addition(addition(one, multiplication(X, star(X))), star(X))
% 87.67/11.35 = { by axiom 14 (order_1) R->L }
% 87.67/11.35 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 87.67/11.35 = { by axiom 16 (star_unfold_right) }
% 87.67/11.35 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 87.67/11.35 = { by axiom 8 (order_1) }
% 87.67/11.35 star(X)
% 87.67/11.35
% 87.67/11.35 Lemma 24: addition(X, addition(X, Y)) = addition(X, Y).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(X, addition(X, Y))
% 87.67/11.35 = { by axiom 7 (additive_associativity) }
% 87.67/11.35 addition(addition(X, X), Y)
% 87.67/11.35 = { by axiom 3 (additive_idempotence) }
% 87.67/11.35 addition(X, Y)
% 87.67/11.35
% 87.67/11.35 Lemma 25: addition(one, star(X)) = star(X).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(one, star(X))
% 87.67/11.35 = { by lemma 23 R->L }
% 87.67/11.35 addition(one, addition(one, multiplication(addition(X, one), star(X))))
% 87.67/11.35 = { by lemma 24 }
% 87.67/11.35 addition(one, multiplication(addition(X, one), star(X)))
% 87.67/11.35 = { by lemma 23 }
% 87.67/11.35 star(X)
% 87.67/11.35
% 87.67/11.35 Lemma 26: addition(X, addition(Y, X)) = addition(Y, X).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(X, addition(Y, X))
% 87.67/11.35 = { by lemma 24 R->L }
% 87.67/11.35 addition(X, addition(Y, addition(Y, X)))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 addition(X, addition(Y, addition(X, Y)))
% 87.67/11.35 = { by axiom 7 (additive_associativity) }
% 87.67/11.35 addition(addition(X, Y), addition(X, Y))
% 87.67/11.35 = { by axiom 3 (additive_idempotence) }
% 87.67/11.35 addition(X, Y)
% 87.67/11.35 = { by axiom 4 (additive_commutativity) }
% 87.67/11.35 addition(Y, X)
% 87.67/11.35
% 87.67/11.35 Lemma 27: fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X) = leq(multiplication(star(Y), X), Z).
% 87.67/11.35 Proof:
% 87.67/11.35 fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X)
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 fresh4(leq(addition(multiplication(Y, Z), X), Z), true, Y, Z, X)
% 87.67/11.35 = { by axiom 18 (star_induction_left) }
% 87.67/11.35 leq(multiplication(star(Y), X), Z)
% 87.67/11.35
% 87.67/11.35 Lemma 28: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(X, multiplication(X, Y))
% 87.67/11.35 = { by axiom 1 (multiplicative_right_identity) R->L }
% 87.67/11.35 addition(multiplication(X, one), multiplication(X, Y))
% 87.67/11.35 = { by axiom 12 (right_distributivity) R->L }
% 87.67/11.35 multiplication(X, addition(one, Y))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) }
% 87.67/11.35 multiplication(X, addition(Y, one))
% 87.67/11.35
% 87.67/11.35 Lemma 29: addition(one, multiplication(star(X), addition(X, one))) = star(X).
% 87.67/11.35 Proof:
% 87.67/11.35 addition(one, multiplication(star(X), addition(X, one)))
% 87.67/11.35 = { by lemma 28 R->L }
% 87.67/11.35 addition(one, addition(star(X), multiplication(star(X), X)))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 addition(one, addition(multiplication(star(X), X), star(X)))
% 87.67/11.35 = { by axiom 7 (additive_associativity) }
% 87.67/11.35 addition(addition(one, multiplication(star(X), X)), star(X))
% 87.67/11.35 = { by axiom 14 (order_1) R->L }
% 87.67/11.35 fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
% 87.67/11.35 = { by axiom 17 (star_unfold_left) }
% 87.67/11.35 fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
% 87.67/11.35 = { by axiom 8 (order_1) }
% 87.67/11.35 star(X)
% 87.67/11.35
% 87.67/11.35 Lemma 30: multiplication(star(X), addition(X, one)) = star(X).
% 87.67/11.35 Proof:
% 87.67/11.35 multiplication(star(X), addition(X, one))
% 87.67/11.35 = { by axiom 3 (additive_idempotence) R->L }
% 87.67/11.35 multiplication(star(X), addition(X, addition(one, one)))
% 87.67/11.35 = { by axiom 7 (additive_associativity) }
% 87.67/11.35 multiplication(star(X), addition(addition(X, one), one))
% 87.67/11.35 = { by lemma 28 R->L }
% 87.67/11.35 addition(star(X), multiplication(star(X), addition(X, one)))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 addition(multiplication(star(X), addition(X, one)), star(X))
% 87.67/11.35 = { by lemma 29 R->L }
% 87.67/11.35 addition(multiplication(star(X), addition(X, one)), addition(one, multiplication(star(X), addition(X, one))))
% 87.67/11.35 = { by lemma 26 }
% 87.67/11.35 addition(one, multiplication(star(X), addition(X, one)))
% 87.67/11.35 = { by lemma 29 }
% 87.67/11.35 star(X)
% 87.67/11.35
% 87.67/11.35 Lemma 31: fresh2(leq(multiplication(X, Y), X), true, X, Y, multiplication(X, Y)) = leq(multiplication(X, multiplication(Y, star(Y))), X).
% 87.67/11.35 Proof:
% 87.67/11.35 fresh2(leq(multiplication(X, Y), X), true, X, Y, multiplication(X, Y))
% 87.67/11.35 = { by axiom 3 (additive_idempotence) R->L }
% 87.67/11.35 fresh2(leq(addition(multiplication(X, Y), multiplication(X, Y)), X), true, X, Y, multiplication(X, Y))
% 87.67/11.35 = { by axiom 19 (star_induction_right) }
% 87.67/11.35 leq(multiplication(multiplication(X, Y), star(Y)), X)
% 87.67/11.35 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.35 leq(multiplication(X, multiplication(Y, star(Y))), X)
% 87.67/11.35
% 87.67/11.35 Lemma 32: star(addition(one, X)) = star(X).
% 87.67/11.35 Proof:
% 87.67/11.35 star(addition(one, X))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 star(addition(X, one))
% 87.67/11.35 = { by axiom 8 (order_1) R->L }
% 87.67/11.35 fresh(true, true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 10 (star_induction_left) R->L }
% 87.67/11.35 fresh(fresh4(true, true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 9 (order) R->L }
% 87.67/11.35 fresh(fresh4(fresh3(addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one)))), addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one)))), multiplication(addition(X, one), star(addition(X, one))), addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one))))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by lemma 24 R->L }
% 87.67/11.35 fresh(fresh4(fresh3(addition(multiplication(addition(X, one), star(addition(X, one))), addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one))))), addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one)))), multiplication(addition(X, one), star(addition(X, one))), addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one))))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 15 (order) }
% 87.67/11.35 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), addition(multiplication(addition(X, one), star(addition(X, one))), multiplication(one, star(addition(X, one))))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 13 (left_distributivity) R->L }
% 87.67/11.35 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), multiplication(addition(addition(X, one), one), star(addition(X, one)))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 3 (additive_idempotence) R->L }
% 87.67/11.35 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), multiplication(addition(addition(X, one), addition(one, one)), star(addition(X, one)))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 7 (additive_associativity) }
% 87.67/11.35 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), multiplication(addition(addition(addition(X, one), one), one), star(addition(X, one)))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by lemma 22 R->L }
% 87.67/11.35 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), addition(star(addition(X, one)), multiplication(addition(addition(X, one), one), star(addition(X, one))))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.35 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.35 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), addition(multiplication(addition(addition(X, one), one), star(addition(X, one))), star(addition(X, one)))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 23 R->L }
% 87.67/11.36 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), addition(multiplication(addition(addition(X, one), one), star(addition(X, one))), addition(one, multiplication(addition(addition(X, one), one), star(addition(X, one)))))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 26 }
% 87.67/11.36 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), addition(one, multiplication(addition(addition(X, one), one), star(addition(X, one))))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 23 }
% 87.67/11.36 fresh(fresh4(leq(multiplication(addition(X, one), star(addition(X, one))), star(addition(X, one))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 22 R->L }
% 87.67/11.36 fresh(fresh4(leq(addition(star(addition(X, one)), multiplication(X, star(addition(X, one)))), star(addition(X, one))), true, X, star(addition(X, one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 27 }
% 87.67/11.36 fresh(leq(multiplication(star(X), star(addition(X, one))), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 25 R->L }
% 87.67/11.36 fresh(leq(multiplication(star(X), addition(one, star(addition(X, one)))), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 fresh(leq(multiplication(star(X), addition(star(addition(X, one)), one)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 28 R->L }
% 87.67/11.36 fresh(leq(addition(star(X), multiplication(star(X), star(addition(X, one)))), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 fresh(leq(addition(multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 14 (order_1) R->L }
% 87.67/11.36 fresh(leq(fresh(leq(multiplication(star(X), star(addition(X, one))), star(X)), true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 30 R->L }
% 87.67/11.36 fresh(leq(fresh(leq(multiplication(multiplication(star(X), addition(X, one)), star(addition(X, one))), star(X)), true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.36 fresh(leq(fresh(leq(multiplication(star(X), multiplication(addition(X, one), star(addition(X, one)))), star(X)), true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 31 R->L }
% 87.67/11.36 fresh(leq(fresh(fresh2(leq(multiplication(star(X), addition(X, one)), star(X)), true, star(X), addition(X, one), multiplication(star(X), addition(X, one))), true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 30 }
% 87.67/11.36 fresh(leq(fresh(fresh2(leq(star(X), star(X)), true, star(X), addition(X, one), multiplication(star(X), addition(X, one))), true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by lemma 21 }
% 87.67/11.36 fresh(leq(fresh(fresh2(true, true, star(X), addition(X, one), multiplication(star(X), addition(X, one))), true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 11 (star_induction_right) }
% 87.67/11.36 fresh(leq(fresh(true, true, multiplication(star(X), star(addition(X, one))), star(X)), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 8 (order_1) }
% 87.67/11.36 fresh(leq(star(X), star(addition(X, one))), true, star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 14 (order_1) }
% 87.67/11.36 addition(star(X), star(addition(X, one)))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 addition(star(addition(X, one)), star(X))
% 87.67/11.36 = { by axiom 14 (order_1) R->L }
% 87.67/11.36 fresh(leq(star(addition(X, one)), star(X)), true, star(addition(X, one)), star(X))
% 87.67/11.36 = { by axiom 1 (multiplicative_right_identity) R->L }
% 87.67/11.36 fresh(leq(multiplication(star(addition(X, one)), one), star(X)), true, star(addition(X, one)), star(X))
% 87.67/11.36 = { by lemma 27 R->L }
% 87.67/11.36 fresh(fresh4(leq(addition(one, multiplication(addition(X, one), star(X))), star(X)), true, addition(X, one), star(X), one), true, star(addition(X, one)), star(X))
% 87.67/11.36 = { by lemma 23 }
% 87.67/11.36 fresh(fresh4(leq(star(X), star(X)), true, addition(X, one), star(X), one), true, star(addition(X, one)), star(X))
% 87.67/11.36 = { by lemma 21 }
% 87.67/11.36 fresh(fresh4(true, true, addition(X, one), star(X), one), true, star(addition(X, one)), star(X))
% 87.67/11.36 = { by axiom 10 (star_induction_left) }
% 87.67/11.36 fresh(true, true, star(addition(X, one)), star(X))
% 87.67/11.36 = { by axiom 8 (order_1) }
% 87.67/11.36 star(X)
% 87.67/11.36
% 87.67/11.36 Lemma 33: addition(multiplication(X, Y), multiplication(Z, multiplication(W, Y))) = multiplication(addition(X, multiplication(Z, W)), Y).
% 87.67/11.36 Proof:
% 87.67/11.36 addition(multiplication(X, Y), multiplication(Z, multiplication(W, Y)))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) }
% 87.67/11.36 addition(multiplication(X, Y), multiplication(multiplication(Z, W), Y))
% 87.67/11.36 = { by axiom 13 (left_distributivity) R->L }
% 87.67/11.36 multiplication(addition(X, multiplication(Z, W)), Y)
% 87.67/11.36
% 87.67/11.36 Lemma 34: addition(one, addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y))) = star(multiplication(X, Y)).
% 87.67/11.36 Proof:
% 87.67/11.36 addition(one, addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y)))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 addition(one, addition(multiplication(multiplication(star(multiplication(X, Y)), X), Y), star(multiplication(X, Y))))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.36 addition(one, addition(multiplication(star(multiplication(X, Y)), multiplication(X, Y)), star(multiplication(X, Y))))
% 87.67/11.36 = { by axiom 7 (additive_associativity) }
% 87.67/11.36 addition(addition(one, multiplication(star(multiplication(X, Y)), multiplication(X, Y))), star(multiplication(X, Y)))
% 87.67/11.36 = { by axiom 14 (order_1) R->L }
% 87.67/11.36 fresh(leq(addition(one, multiplication(star(multiplication(X, Y)), multiplication(X, Y))), star(multiplication(X, Y))), true, addition(one, multiplication(star(multiplication(X, Y)), multiplication(X, Y))), star(multiplication(X, Y)))
% 87.67/11.36 = { by axiom 17 (star_unfold_left) }
% 87.67/11.36 fresh(true, true, addition(one, multiplication(star(multiplication(X, Y)), multiplication(X, Y))), star(multiplication(X, Y)))
% 87.67/11.36 = { by axiom 8 (order_1) }
% 87.67/11.36 star(multiplication(X, Y))
% 87.67/11.36
% 87.67/11.36 Lemma 35: multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X))) = multiplication(star(multiplication(X, Y)), X).
% 87.67/11.36 Proof:
% 87.67/11.36 multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), multiplication(X, addition(one, multiplication(Y, X))))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), multiplication(X, addition(multiplication(Y, X), one)))
% 87.67/11.36 = { by lemma 28 R->L }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), addition(X, multiplication(X, multiplication(Y, X))))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), addition(X, multiplication(multiplication(X, Y), X)))
% 87.67/11.36 = { by lemma 22 }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), multiplication(addition(multiplication(X, Y), one), X))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) }
% 87.67/11.36 multiplication(multiplication(star(multiplication(X, Y)), addition(multiplication(X, Y), one)), X)
% 87.67/11.36 = { by lemma 28 R->L }
% 87.67/11.36 multiplication(addition(star(multiplication(X, Y)), multiplication(star(multiplication(X, Y)), multiplication(X, Y))), X)
% 87.67/11.36 = { by lemma 33 R->L }
% 87.67/11.36 addition(multiplication(star(multiplication(X, Y)), X), multiplication(star(multiplication(X, Y)), multiplication(multiplication(X, Y), X)))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.36 addition(multiplication(star(multiplication(X, Y)), X), multiplication(star(multiplication(X, Y)), multiplication(X, multiplication(Y, X))))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) }
% 87.67/11.36 addition(multiplication(star(multiplication(X, Y)), X), multiplication(multiplication(star(multiplication(X, Y)), X), multiplication(Y, X)))
% 87.67/11.36 = { by lemma 33 }
% 87.67/11.36 multiplication(addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y)), X)
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 multiplication(addition(multiplication(multiplication(star(multiplication(X, Y)), X), Y), star(multiplication(X, Y))), X)
% 87.67/11.36 = { by lemma 26 R->L }
% 87.67/11.36 multiplication(addition(star(multiplication(X, Y)), addition(multiplication(multiplication(star(multiplication(X, Y)), X), Y), star(multiplication(X, Y)))), X)
% 87.67/11.36 = { by axiom 7 (additive_associativity) }
% 87.67/11.36 multiplication(addition(addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y)), star(multiplication(X, Y))), X)
% 87.67/11.36 = { by lemma 34 R->L }
% 87.67/11.36 multiplication(addition(addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y)), addition(one, addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y)))), X)
% 87.67/11.36 = { by lemma 26 }
% 87.67/11.36 multiplication(addition(one, addition(star(multiplication(X, Y)), multiplication(multiplication(star(multiplication(X, Y)), X), Y))), X)
% 87.67/11.36 = { by lemma 34 }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), X)
% 87.67/11.36
% 87.67/11.36 Lemma 36: multiplication(multiplication(star(multiplication(X, Y)), X), star(multiplication(Y, X))) = multiplication(star(multiplication(X, Y)), X).
% 87.67/11.36 Proof:
% 87.67/11.36 multiplication(multiplication(star(multiplication(X, Y)), X), star(multiplication(Y, X)))
% 87.67/11.36 = { by lemma 32 R->L }
% 87.67/11.36 multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X))))
% 87.67/11.36 = { by lemma 25 R->L }
% 87.67/11.36 multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, star(addition(one, multiplication(Y, X)))))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 multiplication(multiplication(star(multiplication(X, Y)), X), addition(star(addition(one, multiplication(Y, X))), one))
% 87.67/11.36 = { by lemma 28 R->L }
% 87.67/11.36 addition(multiplication(star(multiplication(X, Y)), X), multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 addition(multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by axiom 14 (order_1) R->L }
% 87.67/11.36 fresh(leq(multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X)), true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by lemma 35 R->L }
% 87.67/11.36 fresh(leq(multiplication(multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X))), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X)), true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.36 fresh(leq(multiplication(multiplication(star(multiplication(X, Y)), X), multiplication(addition(one, multiplication(Y, X)), star(addition(one, multiplication(Y, X))))), multiplication(star(multiplication(X, Y)), X)), true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by lemma 31 R->L }
% 87.67/11.36 fresh(fresh2(leq(multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X))), multiplication(star(multiplication(X, Y)), X)), true, multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)), multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)))), true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by lemma 35 }
% 87.67/11.36 fresh(fresh2(leq(multiplication(star(multiplication(X, Y)), X), multiplication(star(multiplication(X, Y)), X)), true, multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)), multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)))), true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by lemma 21 }
% 87.67/11.36 fresh(fresh2(true, true, multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)), multiplication(multiplication(star(multiplication(X, Y)), X), addition(one, multiplication(Y, X)))), true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by axiom 11 (star_induction_right) }
% 87.67/11.36 fresh(true, true, multiplication(multiplication(star(multiplication(X, Y)), X), star(addition(one, multiplication(Y, X)))), multiplication(star(multiplication(X, Y)), X))
% 87.67/11.36 = { by axiom 8 (order_1) }
% 87.67/11.36 multiplication(star(multiplication(X, Y)), X)
% 87.67/11.36
% 87.67/11.36 Lemma 37: addition(one, addition(star(multiplication(X, Y)), multiplication(X, multiplication(Y, star(multiplication(X, Y)))))) = star(multiplication(X, Y)).
% 87.67/11.36 Proof:
% 87.67/11.36 addition(one, addition(star(multiplication(X, Y)), multiplication(X, multiplication(Y, star(multiplication(X, Y))))))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 addition(one, addition(multiplication(X, multiplication(Y, star(multiplication(X, Y)))), star(multiplication(X, Y))))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) }
% 87.67/11.36 addition(one, addition(multiplication(multiplication(X, Y), star(multiplication(X, Y))), star(multiplication(X, Y))))
% 87.67/11.36 = { by axiom 7 (additive_associativity) }
% 87.67/11.36 addition(addition(one, multiplication(multiplication(X, Y), star(multiplication(X, Y)))), star(multiplication(X, Y)))
% 87.67/11.36 = { by axiom 14 (order_1) R->L }
% 87.67/11.36 fresh(leq(addition(one, multiplication(multiplication(X, Y), star(multiplication(X, Y)))), star(multiplication(X, Y))), true, addition(one, multiplication(multiplication(X, Y), star(multiplication(X, Y)))), star(multiplication(X, Y)))
% 87.67/11.36 = { by axiom 16 (star_unfold_right) }
% 87.67/11.36 fresh(true, true, addition(one, multiplication(multiplication(X, Y), star(multiplication(X, Y)))), star(multiplication(X, Y)))
% 87.67/11.36 = { by axiom 8 (order_1) }
% 87.67/11.36 star(multiplication(X, Y))
% 87.67/11.36
% 87.67/11.36 Lemma 38: multiplication(addition(one, multiplication(X, Y)), multiplication(X, star(multiplication(Y, X)))) = multiplication(X, star(multiplication(Y, X))).
% 87.67/11.36 Proof:
% 87.67/11.36 multiplication(addition(one, multiplication(X, Y)), multiplication(X, star(multiplication(Y, X))))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 multiplication(addition(multiplication(X, Y), one), multiplication(X, star(multiplication(Y, X))))
% 87.67/11.36 = { by lemma 22 R->L }
% 87.67/11.36 addition(multiplication(X, star(multiplication(Y, X))), multiplication(multiplication(X, Y), multiplication(X, star(multiplication(Y, X)))))
% 87.67/11.36 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.36 addition(multiplication(X, star(multiplication(Y, X))), multiplication(X, multiplication(Y, multiplication(X, star(multiplication(Y, X))))))
% 87.67/11.36 = { by axiom 12 (right_distributivity) R->L }
% 87.67/11.36 multiplication(X, addition(star(multiplication(Y, X)), multiplication(Y, multiplication(X, star(multiplication(Y, X))))))
% 87.67/11.36 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.36 multiplication(X, addition(multiplication(Y, multiplication(X, star(multiplication(Y, X)))), star(multiplication(Y, X))))
% 87.67/11.36 = { by lemma 26 R->L }
% 87.67/11.36 multiplication(X, addition(star(multiplication(Y, X)), addition(multiplication(Y, multiplication(X, star(multiplication(Y, X)))), star(multiplication(Y, X)))))
% 87.67/11.36 = { by axiom 7 (additive_associativity) }
% 87.67/11.36 multiplication(X, addition(addition(star(multiplication(Y, X)), multiplication(Y, multiplication(X, star(multiplication(Y, X))))), star(multiplication(Y, X))))
% 87.67/11.36 = { by lemma 37 R->L }
% 87.67/11.36 multiplication(X, addition(addition(star(multiplication(Y, X)), multiplication(Y, multiplication(X, star(multiplication(Y, X))))), addition(one, addition(star(multiplication(Y, X)), multiplication(Y, multiplication(X, star(multiplication(Y, X))))))))
% 87.67/11.37 = { by lemma 26 }
% 87.67/11.37 multiplication(X, addition(one, addition(star(multiplication(Y, X)), multiplication(Y, multiplication(X, star(multiplication(Y, X)))))))
% 87.67/11.37 = { by lemma 37 }
% 87.67/11.37 multiplication(X, star(multiplication(Y, X)))
% 87.67/11.37
% 87.67/11.37 Goal 1 (goals): multiplication(star(multiplication(x0, x1)), x0) = multiplication(x0, star(multiplication(x1, x0))).
% 87.67/11.37 Proof:
% 87.67/11.37 multiplication(star(multiplication(x0, x1)), x0)
% 87.67/11.37 = { by lemma 20 R->L }
% 87.67/11.37 addition(zero, multiplication(star(multiplication(x0, x1)), x0))
% 87.67/11.37 = { by lemma 36 R->L }
% 87.67/11.37 addition(zero, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))))
% 87.67/11.37 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.37 addition(multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), zero)
% 87.67/11.37 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.67/11.37 addition(multiplication(star(multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0)))), zero)
% 87.67/11.37 = { by lemma 25 R->L }
% 87.67/11.37 addition(multiplication(addition(one, star(multiplication(x0, x1))), multiplication(x0, star(multiplication(x1, x0)))), zero)
% 87.67/11.37 = { by axiom 4 (additive_commutativity) R->L }
% 87.67/11.37 addition(multiplication(addition(star(multiplication(x0, x1)), one), multiplication(x0, star(multiplication(x1, x0)))), zero)
% 87.67/11.37 = { by lemma 22 R->L }
% 87.67/11.37 addition(addition(multiplication(x0, star(multiplication(x1, x0))), multiplication(star(multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), zero)
% 87.67/11.37 = { by axiom 7 (additive_associativity) R->L }
% 87.86/11.37 addition(multiplication(x0, star(multiplication(x1, x0))), addition(multiplication(star(multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0)))), zero))
% 87.86/11.37 = { by axiom 4 (additive_commutativity) }
% 87.86/11.37 addition(multiplication(x0, star(multiplication(x1, x0))), addition(zero, multiplication(star(multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))))
% 87.86/11.37 = { by axiom 6 (multiplicative_associativity) }
% 87.86/11.37 addition(multiplication(x0, star(multiplication(x1, x0))), addition(zero, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0)))))
% 87.86/11.37 = { by lemma 36 }
% 87.86/11.37 addition(multiplication(x0, star(multiplication(x1, x0))), addition(zero, multiplication(star(multiplication(x0, x1)), x0)))
% 87.86/11.37 = { by axiom 4 (additive_commutativity) R->L }
% 87.86/11.37 addition(addition(zero, multiplication(star(multiplication(x0, x1)), x0)), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 7 (additive_associativity) R->L }
% 87.86/11.37 addition(zero, addition(multiplication(star(multiplication(x0, x1)), x0), multiplication(x0, star(multiplication(x1, x0)))))
% 87.86/11.37 = { by lemma 20 }
% 87.86/11.37 addition(multiplication(star(multiplication(x0, x1)), x0), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by lemma 36 R->L }
% 87.86/11.37 addition(multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 14 (order_1) R->L }
% 87.86/11.37 fresh(leq(multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0)))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 6 (multiplicative_associativity) R->L }
% 87.86/11.37 fresh(leq(multiplication(star(multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0)))), multiplication(x0, star(multiplication(x1, x0)))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by lemma 32 R->L }
% 87.86/11.37 fresh(leq(multiplication(star(addition(one, multiplication(x0, x1))), multiplication(x0, star(multiplication(x1, x0)))), multiplication(x0, star(multiplication(x1, x0)))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by lemma 38 R->L }
% 87.86/11.37 fresh(leq(multiplication(star(addition(one, multiplication(x0, x1))), multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), multiplication(x0, star(multiplication(x1, x0)))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 18 (star_induction_left) R->L }
% 87.86/11.37 fresh(fresh4(leq(addition(multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0)))), multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), multiplication(x0, star(multiplication(x1, x0)))), true, addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))), multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 3 (additive_idempotence) }
% 87.86/11.37 fresh(fresh4(leq(multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0)))), multiplication(x0, star(multiplication(x1, x0)))), true, addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))), multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by lemma 38 }
% 87.86/11.37 fresh(fresh4(leq(multiplication(x0, star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0)))), true, addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))), multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by lemma 21 }
% 87.86/11.37 fresh(fresh4(true, true, addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))), multiplication(addition(one, multiplication(x0, x1)), multiplication(x0, star(multiplication(x1, x0))))), true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 10 (star_induction_left) }
% 87.86/11.37 fresh(true, true, multiplication(multiplication(star(multiplication(x0, x1)), x0), star(multiplication(x1, x0))), multiplication(x0, star(multiplication(x1, x0))))
% 87.86/11.37 = { by axiom 8 (order_1) }
% 87.86/11.37 multiplication(x0, star(multiplication(x1, x0)))
% 87.86/11.37 % SZS output end Proof
% 87.86/11.37
% 87.86/11.37 RESULT: Theorem (the conjecture is true).
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