0.06/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33 % Computer : n009.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 : 960
0.12/0.33 % WCLimit : 120
0.12/0.33 % DateTime : Thu Jul 2 08:21:21 EDT 2020
0.12/0.33 % CPUTime :
61.00/8.04 % SZS status Theorem
61.00/8.04
61.00/8.04 % SZS output start Proof
61.00/8.04 Take the following subset of the input axioms:
63.44/8.35 fof(a, conjecture, leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a))).
63.44/8.35 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
63.44/8.35 fof(additive_commutativity, axiom, ![A, B]: addition(B, A)=addition(A, B)).
63.44/8.35 fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
63.44/8.35 fof(left_distributivity, axiom, ![A, B, C]: multiplication(addition(A, B), C)=addition(multiplication(A, C), multiplication(B, C))).
63.44/8.35 fof(multiplicative_associativity, axiom, ![A, B, C]: multiplication(A, multiplication(B, C))=multiplication(multiplication(A, B), C)).
63.44/8.35 fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
63.44/8.35 fof(multiplicative_right_identity, axiom, ![A]: A=multiplication(A, one)).
63.44/8.35 fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
63.44/8.35 fof(right_distributivity, axiom, ![A, B, C]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))).
63.44/8.35 fof(star_unfold_left, axiom, ![A]: leq(addition(one, multiplication(star(A), A)), star(A))).
63.44/8.35 fof(star_unfold_right, axiom, ![A]: leq(addition(one, multiplication(A, star(A))), star(A))).
63.44/8.35
63.44/8.35 Now clausify the problem and encode Horn clauses using encoding 3 of
63.44/8.35 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
63.44/8.35 We repeatedly replace C & s=t => u=v by the two clauses:
63.44/8.35 fresh(y, y, x1...xn) = u
63.44/8.35 C => fresh(s, t, x1...xn) = v
63.44/8.35 where fresh is a fresh function symbol and x1..xn are the free
63.44/8.35 variables of u and v.
63.44/8.35 A predicate p(X) is encoded as p(X)=true (this is sound, because the
63.44/8.35 input problem has no model of domain size 1).
63.44/8.35
63.44/8.35 The encoding turns the above axioms into the following unit equations and goals:
63.44/8.35
63.44/8.35 Axiom 1 (order): fresh3(X, X, Y, Z) = true.
63.44/8.35 Axiom 2 (order_1): fresh(X, X, Y, Z) = Z.
63.44/8.35 Axiom 3 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
63.44/8.35 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
63.44/8.35 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
63.44/8.35 Axiom 6 (multiplicative_right_identity): X = multiplication(X, one).
63.44/8.35 Axiom 7 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
63.44/8.35 Axiom 8 (additive_idempotence): X = addition(X, X).
63.44/8.35 Axiom 9 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
63.44/8.35 Axiom 10 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
63.44/8.35 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
63.44/8.35 Axiom 12 (additive_commutativity): addition(X, Y) = addition(Y, X).
63.44/8.35 Axiom 13 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
63.44/8.35 Axiom 14 (order): fresh3(X, addition(Y, X), Y, X) = leq(Y, X).
63.44/8.35
63.44/8.35 Lemma 15: addition(Y, multiplication(Y, X)) = multiplication(Y, addition(X, one)).
63.44/8.35 Proof:
63.44/8.35 addition(Y, multiplication(Y, X))
63.44/8.35 = { by axiom 6 (multiplicative_right_identity) }
63.44/8.35 addition(multiplication(Y, one), multiplication(Y, X))
63.44/8.35 = { by axiom 10 (right_distributivity) }
63.44/8.35 multiplication(Y, addition(one, X))
63.44/8.35 = { by axiom 12 (additive_commutativity) }
63.44/8.35 multiplication(Y, addition(X, one))
63.44/8.35
63.44/8.35 Lemma 16: addition(one, multiplication(star(X), addition(X, one))) = star(X).
63.44/8.35 Proof:
63.44/8.35 addition(one, multiplication(star(X), addition(X, one)))
63.44/8.35 = { by lemma 15 }
63.44/8.35 addition(one, addition(star(X), multiplication(star(X), X)))
63.44/8.35 = { by axiom 12 (additive_commutativity) }
63.44/8.35 addition(one, addition(multiplication(star(X), X), star(X)))
63.44/8.35 = { by axiom 5 (additive_associativity) }
63.44/8.35 addition(addition(one, multiplication(star(X), X)), star(X))
63.44/8.35 = { by axiom 13 (order_1) }
63.44/8.35 fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
63.44/8.35 = { by axiom 3 (star_unfold_left) }
63.44/8.35 fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
63.44/8.35 = { by axiom 2 (order_1) }
63.44/8.35 star(X)
63.44/8.35
63.44/8.35 Lemma 17: addition(X, addition(X, Y)) = addition(X, Y).
63.44/8.35 Proof:
63.44/8.35 addition(X, addition(X, Y))
63.44/8.35 = { by axiom 5 (additive_associativity) }
63.44/8.35 addition(addition(X, X), Y)
63.44/8.35 = { by axiom 8 (additive_idempotence) }
63.44/8.35 addition(X, Y)
63.44/8.35
63.44/8.35 Lemma 18: leq(X, addition(X, Y)) = true.
63.44/8.35 Proof:
63.44/8.35 leq(X, addition(X, Y))
63.44/8.35 = { by axiom 14 (order) }
63.44/8.35 fresh3(addition(X, Y), addition(X, addition(X, Y)), X, addition(X, Y))
63.44/8.35 = { by lemma 17 }
63.44/8.35 fresh3(addition(X, Y), addition(X, Y), X, addition(X, Y))
63.44/8.35 = { by axiom 1 (order) }
63.44/8.35 true
63.44/8.35
63.44/8.35 Lemma 19: leq(Y, addition(X, Y)) = true.
63.44/8.35 Proof:
63.44/8.35 leq(Y, addition(X, Y))
63.44/8.35 = { by axiom 12 (additive_commutativity) }
63.44/8.35 leq(Y, addition(Y, X))
63.44/8.35 = { by lemma 18 }
63.44/8.35 true
63.44/8.35
63.44/8.35 Lemma 20: multiplication(star(X), addition(X, one)) = star(X).
63.44/8.35 Proof:
63.44/8.35 multiplication(star(X), addition(X, one))
63.44/8.35 = { by axiom 8 (additive_idempotence) }
63.44/8.35 multiplication(star(X), addition(X, addition(one, one)))
63.44/8.35 = { by axiom 5 (additive_associativity) }
63.44/8.35 multiplication(star(X), addition(addition(X, one), one))
63.44/8.35 = { by lemma 15 }
63.44/8.35 addition(star(X), multiplication(star(X), addition(X, one)))
63.44/8.35 = { by axiom 12 (additive_commutativity) }
63.44/8.35 addition(multiplication(star(X), addition(X, one)), star(X))
63.44/8.35 = { by axiom 13 (order_1) }
63.44/8.36 fresh(leq(multiplication(star(X), addition(X, one)), star(X)), true, multiplication(star(X), addition(X, one)), star(X))
63.44/8.36 = { by lemma 16 }
63.44/8.36 fresh(leq(multiplication(star(X), addition(X, one)), addition(one, multiplication(star(X), addition(X, one)))), true, multiplication(star(X), addition(X, one)), star(X))
63.44/8.36 = { by lemma 19 }
63.44/8.36 fresh(true, true, multiplication(star(X), addition(X, one)), star(X))
63.44/8.36 = { by axiom 2 (order_1) }
63.44/8.36 star(X)
63.44/8.36
63.44/8.36 Lemma 21: addition(one, star(X)) = star(X).
63.44/8.36 Proof:
63.44/8.36 addition(one, star(X))
63.44/8.36 = { by lemma 16 }
63.44/8.36 addition(one, addition(one, multiplication(star(X), addition(X, one))))
63.44/8.36 = { by lemma 17 }
63.44/8.36 addition(one, multiplication(star(X), addition(X, one)))
63.44/8.36 = { by lemma 16 }
63.44/8.36 star(X)
63.44/8.36
63.44/8.36 Lemma 22: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
63.44/8.36 Proof:
63.44/8.36 addition(X, multiplication(Y, X))
63.44/8.36 = { by axiom 4 (multiplicative_left_identity) }
63.44/8.36 addition(multiplication(one, X), multiplication(Y, X))
63.44/8.36 = { by axiom 11 (left_distributivity) }
63.44/8.36 multiplication(addition(one, Y), X)
63.44/8.36 = { by axiom 12 (additive_commutativity) }
63.44/8.36 multiplication(addition(Y, one), X)
63.44/8.36
63.44/8.36 Lemma 23: multiplication(addition(X, one), star(X)) = star(X).
63.44/8.36 Proof:
63.44/8.36 multiplication(addition(X, one), star(X))
63.44/8.36 = { by lemma 22 }
63.44/8.36 addition(star(X), multiplication(X, star(X)))
63.44/8.36 = { by lemma 21 }
63.44/8.36 addition(addition(one, star(X)), multiplication(X, star(X)))
63.44/8.36 = { by axiom 5 (additive_associativity) }
63.44/8.36 addition(one, addition(star(X), multiplication(X, star(X))))
63.44/8.36 = { by axiom 12 (additive_commutativity) }
63.44/8.36 addition(one, addition(multiplication(X, star(X)), star(X)))
63.44/8.36 = { by axiom 5 (additive_associativity) }
63.44/8.36 addition(addition(one, multiplication(X, star(X))), star(X))
63.44/8.36 = { by axiom 13 (order_1) }
63.44/8.36 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
63.44/8.36 = { by axiom 7 (star_unfold_right) }
63.44/8.36 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
63.44/8.36 = { by axiom 2 (order_1) }
63.44/8.36 star(X)
63.44/8.36
63.44/8.36 Lemma 24: addition(X, star(X)) = star(X).
63.44/8.36 Proof:
63.44/8.36 addition(X, star(X))
63.44/8.36 = { by lemma 21 }
63.44/8.36 addition(X, addition(one, star(X)))
63.44/8.36 = { by axiom 5 (additive_associativity) }
63.44/8.36 addition(addition(X, one), star(X))
63.44/8.36 = { by axiom 13 (order_1) }
63.44/8.36 fresh(leq(addition(X, one), star(X)), true, addition(X, one), star(X))
63.44/8.36 = { by lemma 23 }
63.44/8.36 fresh(leq(addition(X, one), multiplication(addition(X, one), star(X))), true, addition(X, one), star(X))
63.44/8.36 = { by lemma 16 }
63.44/8.36 fresh(leq(addition(X, one), multiplication(addition(X, one), addition(one, multiplication(star(X), addition(X, one))))), true, addition(X, one), star(X))
63.44/8.36 = { by axiom 12 (additive_commutativity) }
63.44/8.36 fresh(leq(addition(X, one), multiplication(addition(X, one), addition(multiplication(star(X), addition(X, one)), one))), true, addition(X, one), star(X))
63.44/8.36 = { by lemma 15 }
63.44/8.36 fresh(leq(addition(X, one), addition(addition(X, one), multiplication(addition(X, one), multiplication(star(X), addition(X, one))))), true, addition(X, one), star(X))
63.44/8.36 = { by lemma 18 }
63.44/8.36 fresh(true, true, addition(X, one), star(X))
63.44/8.36 = { by axiom 2 (order_1) }
65.83/8.66 star(X)
65.83/8.66
65.83/8.66 Goal 1 (a): leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)) = true.
65.83/8.66 Proof:
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a))
65.83/8.66 = { by lemma 20 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), multiplication(star(a), addition(a, one)))
65.83/8.66 = { by lemma 15 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(star(a), a)))
65.83/8.66 = { by axiom 2 (order_1) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(true, true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 18 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(addition(one, multiplication(a, a)), a), addition(multiplication(addition(one, multiplication(a, a)), a), multiplication(addition(one, multiplication(a, a)), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(addition(multiplication(a, a), one), a), addition(multiplication(addition(one, multiplication(a, a)), a), multiplication(addition(one, multiplication(a, a)), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 22 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(addition(a, multiplication(multiplication(a, a), a)), addition(multiplication(addition(one, multiplication(a, a)), a), multiplication(addition(one, multiplication(a, a)), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 9 (multiplicative_associativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(addition(a, multiplication(a, multiplication(a, a))), addition(multiplication(addition(one, multiplication(a, a)), a), multiplication(addition(one, multiplication(a, a)), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 15 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(multiplication(a, a), one)), addition(multiplication(addition(one, multiplication(a, a)), a), multiplication(addition(one, multiplication(a, a)), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), addition(multiplication(addition(one, multiplication(a, a)), a), multiplication(addition(one, multiplication(a, a)), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 10 (right_distributivity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(one, multiplication(a, a)), addition(a, star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(multiplication(a, a), one), addition(a, star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 24 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(multiplication(a, a), one), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 22 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), addition(star(a), multiplication(multiplication(a, a), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 23 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), addition(multiplication(addition(a, one), star(a)), multiplication(multiplication(a, a), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 11 (left_distributivity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(addition(a, one), multiplication(a, a)), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 5 (additive_associativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(a, addition(one, multiplication(a, a))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(a, addition(multiplication(a, a), one)), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 5 (additive_associativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(addition(a, multiplication(a, a)), one), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(one, addition(a, multiplication(a, a))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 15 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(one, multiplication(a, addition(a, one))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 8 (additive_idempotence) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(one, multiplication(a, addition(a, addition(one, one)))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 5 (additive_associativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(one, multiplication(a, addition(addition(a, one), one))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 15 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(one, addition(a, multiplication(a, addition(a, one)))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 5 (additive_associativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(addition(one, a), multiplication(a, addition(a, one))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 15 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(addition(one, a), addition(a, multiplication(a, a))), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 22 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(addition(one, a), multiplication(addition(a, one), a)), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(addition(one, a), multiplication(addition(one, a), a)), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by lemma 15 }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(multiplication(addition(one, a), addition(a, one)), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 12 (additive_commutativity) }
65.83/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(multiplication(addition(a, one), addition(a, one)), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
65.83/8.66 = { by axiom 9 (multiplicative_associativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(a, one), multiplication(addition(a, one), star(a)))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
66.03/8.66 = { by lemma 23 }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), multiplication(addition(a, one), star(a))), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
66.03/8.66 = { by lemma 23 }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(fresh(leq(multiplication(a, addition(one, multiplication(a, a))), star(a)), true, multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
66.03/8.66 = { by axiom 13 (order_1) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(multiplication(a, addition(one, multiplication(a, a))), star(a)), a)))
66.03/8.66 = { by axiom 12 (additive_commutativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(multiplication(a, addition(multiplication(a, a), one)), star(a)), a)))
66.03/8.66 = { by lemma 15 }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(addition(a, multiplication(a, multiplication(a, a))), star(a)), a)))
66.03/8.66 = { by axiom 5 (additive_associativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(a, addition(multiplication(a, multiplication(a, a)), star(a))), a)))
66.03/8.66 = { by axiom 12 (additive_commutativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(a, addition(star(a), multiplication(a, multiplication(a, a)))), a)))
66.03/8.66 = { by axiom 5 (additive_associativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(addition(a, star(a)), multiplication(a, multiplication(a, a))), a)))
66.03/8.66 = { by lemma 24 }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(star(a), multiplication(a, multiplication(a, a))), a)))
66.03/8.66 = { by axiom 12 (additive_commutativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(addition(multiplication(a, multiplication(a, a)), star(a)), a)))
66.03/8.66 = { by axiom 11 (left_distributivity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), addition(multiplication(multiplication(a, multiplication(a, a)), a), multiplication(star(a), a))))
66.03/8.66 = { by axiom 12 (additive_commutativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), addition(multiplication(star(a), a), multiplication(multiplication(a, multiplication(a, a)), a))))
66.03/8.66 = { by axiom 5 (additive_associativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(addition(star(a), multiplication(star(a), a)), multiplication(multiplication(a, multiplication(a, a)), a)))
66.03/8.66 = { by lemma 15 }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(multiplication(star(a), addition(a, one)), multiplication(multiplication(a, multiplication(a, a)), a)))
66.03/8.66 = { by lemma 20 }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(multiplication(a, multiplication(a, a)), a)))
66.03/8.66 = { by axiom 9 (multiplicative_associativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(a, multiplication(multiplication(a, a), a))))
66.03/8.66 = { by axiom 9 (multiplicative_associativity) }
66.03/8.66 leq(multiplication(a, multiplication(a, multiplication(a, a))), addition(star(a), multiplication(a, multiplication(a, multiplication(a, a)))))
66.03/8.66 = { by lemma 19 }
66.03/8.66 true
66.03/8.66 % SZS output end Proof
66.03/8.66
66.03/8.66 RESULT: Theorem (the conjecture is true).
66.03/8.68 EOF