TSTP Solution File: KLE045+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE045+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 : n023.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:40 EDT 2023
% Result : Theorem 271.17s 34.65s
% Output : Proof 271.17s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : KLE045+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
% 0.13/0.35 % Computer : n023.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 11:19:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 271.17/34.65 Command-line arguments: --ground-connectedness --complete-subsets
% 271.17/34.65
% 271.17/34.65 % SZS status Theorem
% 271.17/34.65
% 271.17/34.68 % SZS output start Proof
% 271.17/34.68 Take the following subset of the input axioms:
% 271.17/34.68 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 271.17/34.68 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 271.17/34.68 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 271.17/34.68 fof(goals, conjecture, ![X0, X1, X2]: (leq(multiplication(X0, X2), multiplication(X2, X1)) => leq(multiplication(star(X0), X2), multiplication(X2, star(X1))))).
% 271.17/34.68 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 271.17/34.68 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 271.17/34.68 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 271.17/34.68 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 271.17/34.68 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 271.17/34.68 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 271.17/34.68 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 271.17/34.68 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))).
% 271.17/34.68 fof(star_unfold_left, axiom, ![A3]: leq(addition(one, multiplication(star(A3), A3)), star(A3))).
% 271.17/34.68 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 271.17/34.68
% 271.17/34.68 Now clausify the problem and encode Horn clauses using encoding 3 of
% 271.17/34.68 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 271.17/34.68 We repeatedly replace C & s=t => u=v by the two clauses:
% 271.17/34.68 fresh(y, y, x1...xn) = u
% 271.17/34.68 C => fresh(s, t, x1...xn) = v
% 271.17/34.68 where fresh is a fresh function symbol and x1..xn are the free
% 271.17/34.68 variables of u and v.
% 271.17/34.68 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 271.17/34.68 input problem has no model of domain size 1).
% 271.17/34.68
% 271.17/34.68 The encoding turns the above axioms into the following unit equations and goals:
% 271.17/34.68
% 271.17/34.68 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 271.17/34.68 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 271.17/34.68 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 271.17/34.68 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 271.17/34.68 Axiom 5 (order_1): fresh(X, X, Y, Z) = Z.
% 271.17/34.68 Axiom 6 (order): fresh3(X, X, Y, Z) = true.
% 271.17/34.68 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 271.17/34.68 Axiom 8 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 271.17/34.68 Axiom 9 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 271.17/34.68 Axiom 10 (star_induction_right): fresh2(X, X, Y, Z, W) = true.
% 271.17/34.68 Axiom 11 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 271.17/34.68 Axiom 12 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 271.17/34.68 Axiom 13 (goals): leq(multiplication(x0, x2), multiplication(x2, x1)) = true.
% 271.17/34.68 Axiom 14 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 271.17/34.69 Axiom 15 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 271.17/34.69 Axiom 16 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 271.17/34.69 Axiom 17 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 271.17/34.69 Axiom 18 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 271.17/34.69 Axiom 19 (star_induction_right): fresh2(leq(addition(multiplication(X, Y), Z), X), true, X, Y, Z) = leq(multiplication(Z, star(Y)), X).
% 271.17/34.69
% 271.17/34.69 Lemma 20: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 271.17/34.69 Proof:
% 271.17/34.69 multiplication(addition(one, Y), X)
% 271.17/34.69 = { by axiom 15 (left_distributivity) }
% 271.17/34.69 addition(multiplication(one, X), multiplication(Y, X))
% 271.17/34.69 = { by axiom 4 (multiplicative_left_identity) }
% 271.17/34.69 addition(X, multiplication(Y, X))
% 271.17/34.69
% 271.17/34.69 Lemma 21: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 271.17/34.69 Proof:
% 271.17/34.69 addition(one, multiplication(addition(X, one), star(X)))
% 271.17/34.69 = { by axiom 2 (additive_commutativity) R->L }
% 271.17/34.69 addition(one, multiplication(addition(one, X), star(X)))
% 271.17/34.69 = { by lemma 20 }
% 271.17/34.69 addition(one, addition(star(X), multiplication(X, star(X))))
% 271.17/34.69 = { by axiom 2 (additive_commutativity) R->L }
% 271.17/34.69 addition(one, addition(multiplication(X, star(X)), star(X)))
% 271.17/34.69 = { by axiom 7 (additive_associativity) }
% 271.17/34.69 addition(addition(one, multiplication(X, star(X))), star(X))
% 271.17/34.69 = { by axiom 11 (order_1) R->L }
% 271.17/34.69 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 271.17/34.69 = { by axiom 16 (star_unfold_right) }
% 271.17/34.69 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 271.17/34.69 = { by axiom 5 (order_1) }
% 271.17/34.69 star(X)
% 271.17/34.69
% 271.17/34.69 Lemma 22: addition(X, addition(X, Y)) = addition(X, Y).
% 271.17/34.69 Proof:
% 271.17/34.69 addition(X, addition(X, Y))
% 271.17/34.69 = { by axiom 7 (additive_associativity) }
% 271.17/34.69 addition(addition(X, X), Y)
% 271.17/34.69 = { by axiom 1 (additive_idempotence) }
% 271.17/34.69 addition(X, Y)
% 271.17/34.69
% 271.17/34.69 Lemma 23: addition(one, star(X)) = star(X).
% 271.17/34.69 Proof:
% 271.17/34.69 addition(one, star(X))
% 271.17/34.69 = { by lemma 21 R->L }
% 271.17/34.69 addition(one, addition(one, multiplication(addition(X, one), star(X))))
% 271.17/34.69 = { by lemma 22 }
% 271.17/34.69 addition(one, multiplication(addition(X, one), star(X)))
% 271.17/34.69 = { by lemma 21 }
% 271.17/34.69 star(X)
% 271.17/34.69
% 271.17/34.69 Lemma 24: leq(X, addition(X, Y)) = true.
% 271.17/34.69 Proof:
% 271.17/34.69 leq(X, addition(X, Y))
% 271.17/34.69 = { by axiom 12 (order) R->L }
% 271.17/34.69 fresh3(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 271.17/34.69 = { by lemma 22 }
% 271.17/34.69 fresh3(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 271.17/34.69 = { by axiom 6 (order) }
% 271.17/34.69 true
% 271.17/34.69
% 271.17/34.69 Lemma 25: addition(star(X), one) = star(X).
% 271.17/34.69 Proof:
% 271.17/34.69 addition(star(X), one)
% 271.17/34.69 = { by axiom 2 (additive_commutativity) R->L }
% 271.17/34.69 addition(one, star(X))
% 271.17/34.69 = { by axiom 11 (order_1) R->L }
% 271.17/34.69 fresh(leq(one, star(X)), true, one, star(X))
% 271.17/34.69 = { by axiom 5 (order_1) R->L }
% 271.17/34.69 fresh(leq(one, fresh(true, true, addition(one, multiplication(X, star(X))), star(X))), true, one, star(X))
% 271.17/34.69 = { by axiom 16 (star_unfold_right) R->L }
% 271.17/34.69 fresh(leq(one, fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))), true, one, star(X))
% 271.17/34.69 = { by axiom 11 (order_1) }
% 271.17/34.69 fresh(leq(one, addition(addition(one, multiplication(X, star(X))), star(X))), true, one, star(X))
% 271.17/34.69 = { by axiom 7 (additive_associativity) R->L }
% 271.17/34.69 fresh(leq(one, addition(one, addition(multiplication(X, star(X)), star(X)))), true, one, star(X))
% 271.17/34.69 = { by axiom 2 (additive_commutativity) }
% 271.17/34.69 fresh(leq(one, addition(one, addition(star(X), multiplication(X, star(X))))), true, one, star(X))
% 271.17/34.69 = { by lemma 24 }
% 271.17/34.69 fresh(true, true, one, star(X))
% 271.17/34.69 = { by axiom 5 (order_1) }
% 271.17/34.69 star(X)
% 271.17/34.69
% 271.17/34.69 Lemma 26: addition(multiplication(X, star(Y)), X) = multiplication(X, star(Y)).
% 271.17/34.69 Proof:
% 271.17/34.69 addition(multiplication(X, star(Y)), X)
% 271.17/34.69 = { by axiom 3 (multiplicative_right_identity) R->L }
% 271.17/34.69 addition(multiplication(X, star(Y)), multiplication(X, one))
% 271.17/34.69 = { by axiom 14 (right_distributivity) R->L }
% 271.17/34.69 multiplication(X, addition(star(Y), one))
% 271.17/34.69 = { by lemma 25 }
% 271.17/34.69 multiplication(X, star(Y))
% 271.17/34.69
% 271.17/34.69 Goal 1 (goals_1): leq(multiplication(star(x0), x2), multiplication(x2, star(x1))) = true.
% 271.17/34.69 Proof:
% 271.17/34.69 leq(multiplication(star(x0), x2), multiplication(x2, star(x1)))
% 271.17/34.69 = { by axiom 5 (order_1) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(true, true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 9 (star_induction_left) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(true, true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 6 (order) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(fresh3(multiplication(x2, star(x1)), multiplication(x2, star(x1)), multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 1 (additive_idempotence) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(fresh3(addition(multiplication(x2, star(x1)), multiplication(x2, star(x1))), multiplication(x2, star(x1)), multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 12 (order) }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 5 (order_1) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(true, true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 10 (star_induction_right) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(true, true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by lemma 24 R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), addition(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, star(x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 2 (additive_commutativity) }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), addition(multiplication(x2, star(x1)), multiplication(multiplication(x2, star(x1)), x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 8 (multiplicative_associativity) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), addition(multiplication(x2, star(x1)), multiplication(x2, multiplication(star(x1), x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 14 (right_distributivity) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, addition(star(x1), multiplication(star(x1), x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by lemma 23 R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, addition(addition(one, star(x1)), multiplication(star(x1), x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 7 (additive_associativity) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, addition(one, addition(star(x1), multiplication(star(x1), x1))))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 2 (additive_commutativity) }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, addition(one, addition(multiplication(star(x1), x1), star(x1))))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 7 (additive_associativity) }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, addition(addition(one, multiplication(star(x1), x1)), star(x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 11 (order_1) R->L }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, fresh(leq(addition(one, multiplication(star(x1), x1)), star(x1)), true, addition(one, multiplication(star(x1), x1)), star(x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.69 = { by axiom 17 (star_unfold_left) }
% 271.17/34.69 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, fresh(true, true, addition(one, multiplication(star(x1), x1)), star(x1)))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 5 (order_1) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by lemma 26 R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(multiplication(addition(multiplication(x2, star(x1)), x2), x1), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 15 (left_distributivity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, x1)), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 5 (order_1) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(multiplication(x2, star(x1)), x1), fresh(true, true, multiplication(x0, x2), multiplication(x2, x1))), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 13 (goals) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(multiplication(x2, star(x1)), x1), fresh(leq(multiplication(x0, x2), multiplication(x2, x1)), true, multiplication(x0, x2), multiplication(x2, x1))), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 11 (order_1) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(multiplication(x2, star(x1)), x1), addition(multiplication(x0, x2), multiplication(x2, x1))), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 2 (additive_commutativity) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(multiplication(x2, star(x1)), x1), addition(multiplication(x2, x1), multiplication(x0, x2))), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 7 (additive_associativity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(addition(multiplication(multiplication(x2, star(x1)), x1), multiplication(x2, x1)), multiplication(x0, x2)), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 15 (left_distributivity) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(addition(multiplication(x2, star(x1)), x2), x1), multiplication(x0, x2)), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 2 (additive_commutativity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(x0, x2), multiplication(addition(multiplication(x2, star(x1)), x2), x1)), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by lemma 26 }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(x0, x2), multiplication(multiplication(x2, star(x1)), x1)), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 2 (additive_commutativity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(fresh2(leq(addition(multiplication(multiplication(x2, star(x1)), x1), multiplication(x0, x2)), multiplication(x2, star(x1))), true, multiplication(x2, star(x1)), x1, multiplication(x0, x2)), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 19 (star_induction_right) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(leq(multiplication(multiplication(x0, x2), star(x1)), multiplication(x2, star(x1))), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 8 (multiplicative_associativity) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(fresh(leq(multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 11 (order_1) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(fresh4(leq(addition(multiplication(x0, multiplication(x2, star(x1))), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, x0, multiplication(x2, star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 18 (star_induction_left) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(leq(multiplication(star(x0), multiplication(x2, star(x1))), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 8 (multiplicative_associativity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), fresh(leq(multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))), true, multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 11 (order_1) }
% 271.17/34.70 leq(multiplication(star(x0), x2), addition(multiplication(multiplication(star(x0), x2), star(x1)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 2 (additive_commutativity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), addition(multiplication(x2, star(x1)), multiplication(multiplication(star(x0), x2), star(x1))))
% 271.17/34.70 = { by axiom 8 (multiplicative_associativity) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), addition(multiplication(x2, star(x1)), multiplication(star(x0), multiplication(x2, star(x1)))))
% 271.17/34.70 = { by lemma 20 R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), multiplication(addition(one, star(x0)), multiplication(x2, star(x1))))
% 271.17/34.70 = { by lemma 23 }
% 271.17/34.70 leq(multiplication(star(x0), x2), multiplication(star(x0), multiplication(x2, star(x1))))
% 271.17/34.70 = { by axiom 8 (multiplicative_associativity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), multiplication(multiplication(star(x0), x2), star(x1)))
% 271.17/34.70 = { by lemma 25 R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), multiplication(multiplication(star(x0), x2), addition(star(x1), one)))
% 271.17/34.70 = { by axiom 2 (additive_commutativity) R->L }
% 271.17/34.70 leq(multiplication(star(x0), x2), multiplication(multiplication(star(x0), x2), addition(one, star(x1))))
% 271.17/34.70 = { by axiom 14 (right_distributivity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), addition(multiplication(multiplication(star(x0), x2), one), multiplication(multiplication(star(x0), x2), star(x1))))
% 271.17/34.70 = { by axiom 3 (multiplicative_right_identity) }
% 271.17/34.70 leq(multiplication(star(x0), x2), addition(multiplication(star(x0), x2), multiplication(multiplication(star(x0), x2), star(x1))))
% 271.17/34.70 = { by lemma 24 }
% 271.17/34.70 true
% 271.17/34.70 % SZS output end Proof
% 271.17/34.70
% 271.17/34.70 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------