TSTP Solution File: KLE159+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE159+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 : n026.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 05:36:07 EDT 2023
% Result : Theorem 202.67s 26.51s
% Output : Proof 202.67s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : KLE159+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n026.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Tue Aug 29 12:27:35 EDT 2023
% 0.15/0.36 % CPUTime :
% 202.67/26.51 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 202.67/26.51
% 202.67/26.51 % SZS status Theorem
% 202.67/26.51
% 202.67/26.54 % SZS output start Proof
% 202.67/26.54 Take the following subset of the input axioms:
% 202.67/26.54 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 202.67/26.54 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 202.67/26.54 fof(distributivity1, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 202.67/26.54 fof(distributivity2, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 202.67/26.54 fof(goals, conjecture, ![X0, X1, X2]: (leq(multiplication(X0, X1), multiplication(X2, X0)) => leq(multiplication(X0, star(X1)), multiplication(star(X2), X0)))).
% 202.67/26.54 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 202.67/26.54 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 202.67/26.54 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 202.67/26.54 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 202.67/26.54 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 202.67/26.54 fof(star_induction1, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, C2), B2), C2) => leq(multiplication(star(A2_2), B2), C2))).
% 202.67/26.54 fof(star_induction2, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(C2, A2_2), B2), C2) => leq(multiplication(B2, star(A2_2)), C2))).
% 202.67/26.54 fof(star_unfold1, axiom, ![A3]: addition(one, multiplication(A3, star(A3)))=star(A3)).
% 202.67/26.54
% 202.67/26.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 202.67/26.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 202.67/26.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 202.67/26.54 fresh(y, y, x1...xn) = u
% 202.67/26.54 C => fresh(s, t, x1...xn) = v
% 202.67/26.54 where fresh is a fresh function symbol and x1..xn are the free
% 202.67/26.54 variables of u and v.
% 202.67/26.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 202.67/26.54 input problem has no model of domain size 1).
% 202.67/26.54
% 202.67/26.54 The encoding turns the above axioms into the following unit equations and goals:
% 202.67/26.54
% 202.67/26.54 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 202.67/26.54 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 202.67/26.54 Axiom 3 (idempotence): addition(X, X) = X.
% 202.67/26.54 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 202.67/26.54 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 202.67/26.54 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 202.67/26.54 Axiom 7 (order_1): fresh(X, X, Y, Z) = Z.
% 202.67/26.54 Axiom 8 (order): fresh5(X, X, Y, Z) = true.
% 202.67/26.54 Axiom 9 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 202.67/26.54 Axiom 10 (star_induction1): fresh3(X, X, Y, Z, W) = true.
% 202.67/26.54 Axiom 11 (star_induction2): fresh2(X, X, Y, Z, W) = true.
% 202.67/26.54 Axiom 12 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 202.67/26.54 Axiom 13 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 202.67/26.54 Axiom 14 (goals): leq(multiplication(x0, x1), multiplication(x2, x0)) = true.
% 202.67/26.54 Axiom 15 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 202.67/26.54 Axiom 16 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 202.67/26.54 Axiom 17 (star_induction1): fresh3(leq(addition(multiplication(X, Y), Z), Y), true, X, Z, Y) = leq(multiplication(star(X), Z), Y).
% 202.67/26.54 Axiom 18 (star_induction2): fresh2(leq(addition(multiplication(X, Y), Z), X), true, Y, Z, X) = leq(multiplication(Z, star(Y)), X).
% 202.67/26.54
% 202.67/26.54 Lemma 19: addition(one, addition(X, multiplication(Y, star(Y)))) = addition(X, star(Y)).
% 202.67/26.54 Proof:
% 202.67/26.54 addition(one, addition(X, multiplication(Y, star(Y))))
% 202.67/26.54 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.54 addition(one, addition(multiplication(Y, star(Y)), X))
% 202.67/26.54 = { by axiom 6 (additive_associativity) }
% 202.67/26.54 addition(addition(one, multiplication(Y, star(Y))), X)
% 202.67/26.54 = { by axiom 9 (star_unfold1) }
% 202.67/26.54 addition(star(Y), X)
% 202.67/26.54 = { by axiom 4 (additive_commutativity) }
% 202.67/26.54 addition(X, star(Y))
% 202.67/26.54
% 202.67/26.54 Lemma 20: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 202.67/26.54 Proof:
% 202.67/26.54 addition(X, multiplication(X, Y))
% 202.67/26.54 = { by axiom 1 (multiplicative_right_identity) R->L }
% 202.67/26.54 addition(multiplication(X, one), multiplication(X, Y))
% 202.67/26.54 = { by axiom 12 (distributivity1) R->L }
% 202.67/26.54 multiplication(X, addition(one, Y))
% 202.67/26.54 = { by axiom 4 (additive_commutativity) }
% 202.67/26.54 multiplication(X, addition(Y, one))
% 202.67/26.54
% 202.67/26.54 Lemma 21: addition(X, addition(X, Y)) = addition(X, Y).
% 202.67/26.54 Proof:
% 202.67/26.54 addition(X, addition(X, Y))
% 202.67/26.54 = { by axiom 6 (additive_associativity) }
% 202.67/26.54 addition(addition(X, X), Y)
% 202.67/26.54 = { by axiom 3 (idempotence) }
% 202.67/26.54 addition(X, Y)
% 202.67/26.54
% 202.67/26.54 Lemma 22: addition(one, star(X)) = star(X).
% 202.67/26.54 Proof:
% 202.67/26.54 addition(one, star(X))
% 202.67/26.54 = { by axiom 9 (star_unfold1) R->L }
% 202.67/26.54 addition(one, addition(one, multiplication(X, star(X))))
% 202.67/26.54 = { by lemma 21 }
% 202.67/26.54 addition(one, multiplication(X, star(X)))
% 202.67/26.54 = { by axiom 9 (star_unfold1) }
% 202.67/26.54 star(X)
% 202.67/26.54
% 202.67/26.54 Lemma 23: addition(X, star(X)) = star(X).
% 202.67/26.54 Proof:
% 202.67/26.54 addition(X, star(X))
% 202.67/26.54 = { by lemma 19 R->L }
% 202.67/26.54 addition(one, addition(X, multiplication(X, star(X))))
% 202.67/26.54 = { by lemma 20 }
% 202.67/26.54 addition(one, multiplication(X, addition(star(X), one)))
% 202.67/26.54 = { by axiom 4 (additive_commutativity) }
% 202.67/26.54 addition(one, multiplication(X, addition(one, star(X))))
% 202.67/26.54 = { by lemma 22 }
% 202.67/26.54 addition(one, multiplication(X, star(X)))
% 202.67/26.54 = { by axiom 9 (star_unfold1) }
% 202.67/26.54 star(X)
% 202.67/26.54
% 202.67/26.54 Lemma 24: leq(X, addition(X, Y)) = true.
% 202.67/26.54 Proof:
% 202.67/26.54 leq(X, addition(X, Y))
% 202.67/26.54 = { by axiom 16 (order) R->L }
% 202.67/26.54 fresh5(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 202.67/26.54 = { by lemma 21 }
% 202.67/26.54 fresh5(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 202.67/26.54 = { by axiom 8 (order) }
% 202.67/26.54 true
% 202.67/26.54
% 202.67/26.54 Lemma 25: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 202.67/26.54 Proof:
% 202.67/26.54 addition(multiplication(X, Y), multiplication(Z, Y))
% 202.67/26.54 = { by axiom 13 (distributivity2) R->L }
% 202.67/26.54 multiplication(addition(X, Z), Y)
% 202.67/26.54 = { by axiom 4 (additive_commutativity) }
% 202.67/26.55 multiplication(addition(Z, X), Y)
% 202.67/26.55
% 202.67/26.55 Goal 1 (goals_1): leq(multiplication(x0, star(x1)), multiplication(star(x2), x0)) = true.
% 202.67/26.55 Proof:
% 202.67/26.55 leq(multiplication(x0, star(x1)), multiplication(star(x2), x0))
% 202.67/26.55 = { by axiom 7 (order_1) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(true, true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 11 (star_induction2) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(true, true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 24 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), addition(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), multiplication(star(x2), x0)))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 5 (multiplicative_associativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), addition(multiplication(star(x2), multiplication(x0, x1)), multiplication(star(x2), multiplication(star(x2), x0)))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 12 (distributivity1) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(x0, x1), multiplication(star(x2), x0)))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 23 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(x0, x1), multiplication(addition(x2, star(x2)), x0)))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 25 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(x0, x1), addition(multiplication(star(x2), x0), multiplication(x2, x0))))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(addition(multiplication(star(x2), x0), multiplication(x2, x0)), multiplication(x0, x1)))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 6 (additive_associativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(star(x2), x0), addition(multiplication(x2, x0), multiplication(x0, x1))))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(star(x2), x0), addition(multiplication(x0, x1), multiplication(x2, x0))))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 15 (order_1) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(star(x2), x0), fresh(leq(multiplication(x0, x1), multiplication(x2, x0)), true, multiplication(x0, x1), multiplication(x2, x0))))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 14 (goals) }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(star(x2), x0), fresh(true, true, multiplication(x0, x1), multiplication(x2, x0))))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 7 (order_1) }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), addition(multiplication(star(x2), x0), multiplication(x2, x0)))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 25 }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), multiplication(addition(x2, star(x2)), x0))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 23 }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), multiplication(star(x2), x0))), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 5 (multiplicative_associativity) }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(multiplication(star(x2), star(x2)), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 22 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(multiplication(star(x2), addition(one, star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(multiplication(star(x2), addition(star(x2), one)), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 20 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(star(x2), multiplication(star(x2), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(multiplication(star(x2), star(x2)), star(x2)), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 19 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, addition(multiplication(star(x2), star(x2)), multiplication(x2, star(x2)))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 13 (distributivity2) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, multiplication(addition(star(x2), x2), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, multiplication(addition(x2, star(x2)), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 23 }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, multiplication(star(x2), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 9 (star_unfold1) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, multiplication(star(x2), addition(one, multiplication(x2, star(x2))))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, multiplication(star(x2), addition(multiplication(x2, star(x2)), one))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by lemma 20 R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, addition(star(x2), multiplication(star(x2), multiplication(x2, star(x2))))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, addition(multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 15 (order_1) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(leq(multiplication(star(x2), multiplication(x2, star(x2))), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.55 = { by axiom 17 (star_induction1) R->L }
% 202.67/26.55 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(leq(addition(multiplication(x2, star(x2)), multiplication(x2, star(x2))), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 3 (idempotence) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(leq(multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 16 (order) R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(multiplication(x2, star(x2)), star(x2)), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 9 (star_unfold1) R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(multiplication(x2, star(x2)), addition(one, multiplication(x2, star(x2)))), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by lemma 21 R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(multiplication(x2, star(x2)), addition(one, addition(one, multiplication(x2, star(x2))))), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 4 (additive_commutativity) R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(multiplication(x2, star(x2)), addition(one, addition(multiplication(x2, star(x2)), one))), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 6 (additive_associativity) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(addition(multiplication(x2, star(x2)), one), addition(multiplication(x2, star(x2)), one)), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 3 (idempotence) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(multiplication(x2, star(x2)), one), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 4 (additive_commutativity) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(addition(one, multiplication(x2, star(x2))), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 9 (star_unfold1) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(fresh5(star(x2), star(x2), multiplication(x2, star(x2)), star(x2)), true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 8 (order) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(fresh3(true, true, x2, multiplication(x2, star(x2)), star(x2)), true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 10 (star_induction1) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, fresh(true, true, multiplication(star(x2), multiplication(x2, star(x2))), star(x2))), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 7 (order_1) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(addition(one, star(x2)), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by lemma 22 }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 3 (idempotence) R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(fresh2(leq(addition(multiplication(multiplication(star(x2), x0), x1), multiplication(multiplication(star(x2), x0), x1)), multiplication(star(x2), x0)), true, x1, multiplication(multiplication(star(x2), x0), x1), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 18 (star_induction2) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(leq(multiplication(multiplication(multiplication(star(x2), x0), x1), star(x1)), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 5 (multiplicative_associativity) R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), fresh(leq(multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)), true, multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 15 (order_1) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), addition(multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1))), multiplication(star(x2), x0)))
% 202.67/26.56 = { by axiom 4 (additive_commutativity) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), addition(multiplication(star(x2), x0), multiplication(multiplication(star(x2), x0), multiplication(x1, star(x1)))))
% 202.67/26.56 = { by lemma 20 }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(multiplication(star(x2), x0), addition(multiplication(x1, star(x1)), one)))
% 202.67/26.56 = { by axiom 4 (additive_commutativity) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(multiplication(star(x2), x0), addition(one, multiplication(x1, star(x1)))))
% 202.67/26.56 = { by axiom 9 (star_unfold1) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(multiplication(star(x2), x0), star(x1)))
% 202.67/26.56 = { by lemma 22 R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(multiplication(addition(one, star(x2)), x0), star(x1)))
% 202.67/26.56 = { by lemma 25 R->L }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(addition(multiplication(star(x2), x0), multiplication(one, x0)), star(x1)))
% 202.67/26.56 = { by axiom 2 (multiplicative_left_identity) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(addition(multiplication(star(x2), x0), x0), star(x1)))
% 202.67/26.56 = { by axiom 4 (additive_commutativity) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), multiplication(addition(x0, multiplication(star(x2), x0)), star(x1)))
% 202.67/26.56 = { by axiom 13 (distributivity2) }
% 202.67/26.56 leq(multiplication(x0, star(x1)), addition(multiplication(x0, star(x1)), multiplication(multiplication(star(x2), x0), star(x1))))
% 202.67/26.56 = { by lemma 24 }
% 202.67/26.56 true
% 202.67/26.56 % SZS output end Proof
% 202.67/26.56
% 202.67/26.56 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------