TSTP Solution File: KLE160+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE160+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 05:36:07 EDT 2023
% Result : Theorem 253.58s 32.46s
% Output : Proof 253.58s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE160+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n016.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 11:31:01 EDT 2023
% 0.12/0.34 % CPUTime :
% 253.58/32.46 Command-line arguments: --flatten
% 253.58/32.46
% 253.58/32.46 % SZS status Theorem
% 253.58/32.46
% 253.58/32.49 % SZS output start Proof
% 253.58/32.49 Take the following subset of the input axioms:
% 253.58/32.49 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 253.58/32.49 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 253.58/32.49 fof(distributivity1, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 253.58/32.49 fof(distributivity2, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 253.58/32.49 fof(goals, conjecture, ![X0, X1, X2]: (leq(multiplication(X0, X1), multiplication(X1, X2)) => leq(multiplication(star(X0), X1), multiplication(X1, star(X2))))).
% 253.58/32.49 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 253.58/32.49 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 253.58/32.49 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 253.58/32.49 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 253.58/32.49 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 253.58/32.49 fof(star_induction1, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, C2), B2), C2) => leq(multiplication(star(A2_2), B2), C2))).
% 253.58/32.49 fof(star_unfold1, axiom, ![A3]: addition(one, multiplication(A3, star(A3)))=star(A3)).
% 253.58/32.49 fof(star_unfold2, axiom, ![A3]: addition(one, multiplication(star(A3), A3))=star(A3)).
% 253.58/32.49
% 253.58/32.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 253.58/32.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 253.58/32.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 253.58/32.49 fresh(y, y, x1...xn) = u
% 253.58/32.49 C => fresh(s, t, x1...xn) = v
% 253.58/32.49 where fresh is a fresh function symbol and x1..xn are the free
% 253.58/32.49 variables of u and v.
% 253.58/32.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 253.58/32.49 input problem has no model of domain size 1).
% 253.58/32.49
% 253.58/32.49 The encoding turns the above axioms into the following unit equations and goals:
% 253.58/32.49
% 253.58/32.49 Axiom 1 (idempotence): addition(X, X) = X.
% 253.58/32.49 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 253.58/32.49 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 253.58/32.49 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 253.58/32.49 Axiom 5 (order_1): fresh(X, X, Y, Z) = Z.
% 253.58/32.49 Axiom 6 (order): fresh5(X, X, Y, Z) = true.
% 253.58/32.49 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 253.58/32.49 Axiom 8 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 253.58/32.49 Axiom 9 (star_induction1): fresh3(X, X, Y, Z, W) = true.
% 253.58/32.49 Axiom 10 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 253.58/32.49 Axiom 11 (star_unfold2): addition(one, multiplication(star(X), X)) = star(X).
% 253.58/32.49 Axiom 12 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 253.58/32.49 Axiom 13 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 253.58/32.49 Axiom 14 (goals): leq(multiplication(x0, x1), multiplication(x1, x2)) = true.
% 253.58/32.49 Axiom 15 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 253.58/32.49 Axiom 16 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 253.58/32.49 Axiom 17 (star_induction1): fresh3(leq(addition(multiplication(X, Y), Z), Y), true, X, Z, Y) = leq(multiplication(star(X), Z), Y).
% 253.58/32.49
% 253.58/32.49 Lemma 18: addition(X, addition(X, Y)) = addition(X, Y).
% 253.58/32.49 Proof:
% 253.58/32.49 addition(X, addition(X, Y))
% 253.58/32.49 = { by axiom 7 (additive_associativity) }
% 253.58/32.49 addition(addition(X, X), Y)
% 253.58/32.49 = { by axiom 1 (idempotence) }
% 253.58/32.49 addition(X, Y)
% 253.58/32.49
% 253.58/32.49 Lemma 19: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 253.58/32.49 Proof:
% 253.58/32.49 multiplication(X, addition(Y, one))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) R->L }
% 253.58/32.49 multiplication(X, addition(one, Y))
% 253.58/32.49 = { by axiom 15 (distributivity1) }
% 253.58/32.49 addition(multiplication(X, one), multiplication(X, Y))
% 253.58/32.49 = { by axiom 3 (multiplicative_right_identity) }
% 253.58/32.49 addition(X, multiplication(X, Y))
% 253.58/32.49
% 253.58/32.49 Lemma 20: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 253.58/32.49 Proof:
% 253.58/32.49 multiplication(addition(one, Y), X)
% 253.58/32.49 = { by axiom 16 (distributivity2) }
% 253.58/32.49 addition(multiplication(one, X), multiplication(Y, X))
% 253.58/32.49 = { by axiom 4 (multiplicative_left_identity) }
% 253.58/32.49 addition(X, multiplication(Y, X))
% 253.58/32.49
% 253.58/32.49 Lemma 21: multiplication(multiplication(X, star(Y)), Y) = multiplication(multiplication(X, Y), star(Y)).
% 253.58/32.49 Proof:
% 253.58/32.49 multiplication(multiplication(X, star(Y)), Y)
% 253.58/32.49 = { by axiom 8 (multiplicative_associativity) R->L }
% 253.58/32.49 multiplication(X, multiplication(star(Y), Y))
% 253.58/32.49 = { by axiom 10 (star_unfold1) R->L }
% 253.58/32.49 multiplication(X, multiplication(addition(one, multiplication(Y, star(Y))), Y))
% 253.58/32.49 = { by lemma 20 }
% 253.58/32.49 multiplication(X, addition(Y, multiplication(multiplication(Y, star(Y)), Y)))
% 253.58/32.49 = { by axiom 8 (multiplicative_associativity) R->L }
% 253.58/32.49 multiplication(X, addition(Y, multiplication(Y, multiplication(star(Y), Y))))
% 253.58/32.49 = { by axiom 15 (distributivity1) }
% 253.58/32.49 addition(multiplication(X, Y), multiplication(X, multiplication(Y, multiplication(star(Y), Y))))
% 253.58/32.49 = { by axiom 8 (multiplicative_associativity) }
% 253.58/32.49 addition(multiplication(X, Y), multiplication(multiplication(X, Y), multiplication(star(Y), Y)))
% 253.58/32.49 = { by lemma 19 R->L }
% 253.58/32.49 multiplication(multiplication(X, Y), addition(multiplication(star(Y), Y), one))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) }
% 253.58/32.49 multiplication(multiplication(X, Y), addition(one, multiplication(star(Y), Y)))
% 253.58/32.49 = { by axiom 11 (star_unfold2) }
% 253.58/32.49 multiplication(multiplication(X, Y), star(Y))
% 253.58/32.49
% 253.58/32.49 Lemma 22: addition(multiplication(X, star(Y)), multiplication(multiplication(X, star(Y)), Y)) = multiplication(X, star(Y)).
% 253.58/32.49 Proof:
% 253.58/32.49 addition(multiplication(X, star(Y)), multiplication(multiplication(X, star(Y)), Y))
% 253.58/32.49 = { by axiom 8 (multiplicative_associativity) R->L }
% 253.58/32.49 addition(multiplication(X, star(Y)), multiplication(X, multiplication(star(Y), Y)))
% 253.58/32.49 = { by axiom 15 (distributivity1) R->L }
% 253.58/32.49 multiplication(X, addition(star(Y), multiplication(star(Y), Y)))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) R->L }
% 253.58/32.49 multiplication(X, addition(multiplication(star(Y), Y), star(Y)))
% 253.58/32.49 = { by axiom 11 (star_unfold2) R->L }
% 253.58/32.49 multiplication(X, addition(multiplication(star(Y), Y), addition(one, multiplication(star(Y), Y))))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) R->L }
% 253.58/32.49 multiplication(X, addition(multiplication(star(Y), Y), addition(multiplication(star(Y), Y), one)))
% 253.58/32.49 = { by lemma 18 }
% 253.58/32.49 multiplication(X, addition(multiplication(star(Y), Y), one))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) }
% 253.58/32.49 multiplication(X, addition(one, multiplication(star(Y), Y)))
% 253.58/32.49 = { by axiom 11 (star_unfold2) }
% 253.58/32.49 multiplication(X, star(Y))
% 253.58/32.49
% 253.58/32.49 Goal 1 (goals_1): leq(multiplication(star(x0), x1), multiplication(x1, star(x2))) = true.
% 253.58/32.49 Proof:
% 253.58/32.49 leq(multiplication(star(x0), x1), multiplication(x1, star(x2)))
% 253.58/32.49 = { by axiom 5 (order_1) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(leq(multiplication(x0, x1), multiplication(x1, x2)), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 14 (goals) }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(true, leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 9 (star_induction1) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(x0, x1), multiplication(x1, x2)), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 14 (goals) }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(true, leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 6 (order) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(fresh5(addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2))), addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2))), multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by lemma 18 R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(fresh5(addition(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2)))), addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2))), multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 13 (order) }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(multiplication(x0, x1), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by lemma 22 R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(addition(multiplication(x1, star(x2)), multiplication(multiplication(x1, star(x2)), x2)), multiplication(multiplication(x0, x1), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 7 (additive_associativity) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), addition(multiplication(multiplication(x1, star(x2)), x2), multiplication(multiplication(x0, x1), star(x2))))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by lemma 21 }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), addition(multiplication(multiplication(x1, x2), star(x2)), multiplication(multiplication(x0, x1), star(x2))))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 16 (distributivity2) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(addition(multiplication(x1, x2), multiplication(x0, x1)), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 2 (additive_commutativity) }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(addition(multiplication(x0, x1), multiplication(x1, x2)), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 12 (order_1) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(fresh(leq(multiplication(x0, x1), multiplication(x1, x2)), true, multiplication(x0, x1), multiplication(x1, x2)), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 14 (goals) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(fresh(leq(multiplication(x0, x1), multiplication(x1, x2)), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(x0, x1), multiplication(x1, x2)), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 5 (order_1) }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(multiplication(x1, x2), star(x2)))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by lemma 21 R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), addition(multiplication(x1, star(x2)), multiplication(multiplication(x1, star(x2)), x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by lemma 22 }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(multiplication(x0, x1), star(x2)), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 8 (multiplicative_associativity) R->L }
% 253.58/32.49 leq(multiplication(star(x0), x1), fresh(fresh3(leq(multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.49 = { by axiom 1 (idempotence) R->L }
% 253.58/32.50 leq(multiplication(star(x0), x1), fresh(fresh3(leq(addition(multiplication(x0, multiplication(x1, star(x2))), multiplication(x0, multiplication(x1, star(x2)))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 14 (goals) }
% 253.58/32.50 leq(multiplication(star(x0), x1), fresh(fresh3(leq(addition(multiplication(x0, multiplication(x1, star(x2))), multiplication(x0, multiplication(x1, star(x2)))), multiplication(x1, star(x2))), true, x0, multiplication(x0, multiplication(x1, star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 17 (star_induction1) }
% 253.58/32.50 leq(multiplication(star(x0), x1), fresh(leq(multiplication(star(x0), multiplication(x0, multiplication(x1, star(x2)))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 8 (multiplicative_associativity) }
% 253.58/32.50 leq(multiplication(star(x0), x1), fresh(leq(multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))), leq(multiplication(x0, x1), multiplication(x1, x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 14 (goals) }
% 253.58/32.50 leq(multiplication(star(x0), x1), fresh(leq(multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))), true, multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 12 (order_1) }
% 253.58/32.50 leq(multiplication(star(x0), x1), addition(multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2))), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 2 (additive_commutativity) }
% 253.58/32.50 leq(multiplication(star(x0), x1), addition(multiplication(x1, star(x2)), multiplication(star(x0), multiplication(multiplication(x0, x1), star(x2)))))
% 253.58/32.50 = { by axiom 8 (multiplicative_associativity) R->L }
% 253.58/32.50 leq(multiplication(star(x0), x1), addition(multiplication(x1, star(x2)), multiplication(star(x0), multiplication(x0, multiplication(x1, star(x2))))))
% 253.58/32.50 = { by axiom 8 (multiplicative_associativity) }
% 253.58/32.50 leq(multiplication(star(x0), x1), addition(multiplication(x1, star(x2)), multiplication(multiplication(star(x0), x0), multiplication(x1, star(x2)))))
% 253.58/32.50 = { by lemma 20 R->L }
% 253.58/32.50 leq(multiplication(star(x0), x1), multiplication(addition(one, multiplication(star(x0), x0)), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 11 (star_unfold2) }
% 253.58/32.50 leq(multiplication(star(x0), x1), multiplication(star(x0), multiplication(x1, star(x2))))
% 253.58/32.50 = { by axiom 8 (multiplicative_associativity) }
% 253.58/32.50 leq(multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by axiom 13 (order) R->L }
% 253.58/32.50 fresh5(addition(multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2))), multiplication(multiplication(star(x0), x1), star(x2)), multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by lemma 19 R->L }
% 253.58/32.50 fresh5(multiplication(multiplication(star(x0), x1), addition(star(x2), one)), multiplication(multiplication(star(x0), x1), star(x2)), multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by axiom 2 (additive_commutativity) R->L }
% 253.58/32.50 fresh5(multiplication(multiplication(star(x0), x1), addition(one, star(x2))), multiplication(multiplication(star(x0), x1), star(x2)), multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by axiom 10 (star_unfold1) R->L }
% 253.58/32.50 fresh5(multiplication(multiplication(star(x0), x1), addition(one, addition(one, multiplication(x2, star(x2))))), multiplication(multiplication(star(x0), x1), star(x2)), multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by lemma 18 }
% 253.58/32.50 fresh5(multiplication(multiplication(star(x0), x1), addition(one, multiplication(x2, star(x2)))), multiplication(multiplication(star(x0), x1), star(x2)), multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by axiom 10 (star_unfold1) }
% 253.58/32.50 fresh5(multiplication(multiplication(star(x0), x1), star(x2)), multiplication(multiplication(star(x0), x1), star(x2)), multiplication(star(x0), x1), multiplication(multiplication(star(x0), x1), star(x2)))
% 253.58/32.50 = { by axiom 6 (order) }
% 253.58/32.50 true
% 253.58/32.50 % SZS output end Proof
% 253.58/32.50
% 253.58/32.50 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------