TSTP Solution File: KLE044+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE044+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 : n004.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 05:35:39 EDT 2023
% Result : Theorem 19.01s 2.82s
% Output : Proof 19.96s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.14 % Problem : KLE044+1 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.36 % Computer : n004.cluster.edu
% 0.12/0.36 % Model : x86_64 x86_64
% 0.12/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.36 % Memory : 8042.1875MB
% 0.12/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.36 % CPULimit : 300
% 0.12/0.36 % WCLimit : 300
% 0.12/0.36 % DateTime : Tue Aug 29 11:45:52 EDT 2023
% 0.12/0.36 % CPUTime :
% 19.01/2.82 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 19.01/2.82
% 19.01/2.82 % SZS status Theorem
% 19.01/2.82
% 19.57/2.87 % SZS output start Proof
% 19.57/2.87 Take the following subset of the input axioms:
% 19.57/2.87 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 19.57/2.87 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 19.57/2.87 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 19.57/2.87 fof(goals, conjecture, ![X0]: star(addition(one, X0))=star(X0)).
% 19.57/2.87 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 19.57/2.87 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 19.57/2.87 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 19.57/2.87 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 19.57/2.87 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 19.57/2.87 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 19.57/2.87 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 19.57/2.87 fof(star_unfold_left, axiom, ![A3]: leq(addition(one, multiplication(star(A3), A3)), star(A3))).
% 19.57/2.87 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 19.57/2.87
% 19.57/2.87 Now clausify the problem and encode Horn clauses using encoding 3 of
% 19.57/2.87 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 19.57/2.87 We repeatedly replace C & s=t => u=v by the two clauses:
% 19.57/2.87 fresh(y, y, x1...xn) = u
% 19.57/2.87 C => fresh(s, t, x1...xn) = v
% 19.57/2.87 where fresh is a fresh function symbol and x1..xn are the free
% 19.57/2.87 variables of u and v.
% 19.57/2.87 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 19.57/2.87 input problem has no model of domain size 1).
% 19.57/2.87
% 19.57/2.87 The encoding turns the above axioms into the following unit equations and goals:
% 19.57/2.87
% 19.57/2.87 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 19.57/2.87 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 19.57/2.87 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 19.57/2.87 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 19.57/2.87 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 19.57/2.87 Axiom 6 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 19.57/2.87 Axiom 7 (order_1): fresh(X, X, Y, Z) = Z.
% 19.57/2.87 Axiom 8 (order): fresh3(X, X, Y, Z) = true.
% 19.57/2.87 Axiom 9 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 19.57/2.87 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 19.57/2.87 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 19.57/2.87 Axiom 12 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 19.57/2.87 Axiom 13 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 19.57/2.87 Axiom 14 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 19.57/2.87 Axiom 15 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 19.57/2.88 Axiom 16 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 19.57/2.88
% 19.57/2.88 Lemma 17: leq(X, X) = true.
% 19.57/2.88 Proof:
% 19.57/2.88 leq(X, X)
% 19.57/2.88 = { by axiom 13 (order) R->L }
% 19.57/2.88 fresh3(addition(X, X), X, X, X)
% 19.57/2.88 = { by axiom 1 (additive_idempotence) }
% 19.57/2.88 fresh3(X, X, X, X)
% 19.57/2.88 = { by axiom 8 (order) }
% 19.57/2.88 true
% 19.57/2.88
% 19.57/2.88 Lemma 18: addition(X, addition(X, Y)) = addition(X, Y).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(X, addition(X, Y))
% 19.57/2.88 = { by axiom 5 (additive_associativity) }
% 19.57/2.88 addition(addition(X, X), Y)
% 19.57/2.88 = { by axiom 1 (additive_idempotence) }
% 19.57/2.88 addition(X, Y)
% 19.57/2.88
% 19.57/2.88 Lemma 19: addition(X, addition(Y, X)) = addition(Y, X).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(X, addition(Y, X))
% 19.57/2.88 = { by lemma 18 R->L }
% 19.57/2.88 addition(X, addition(Y, addition(Y, X)))
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 addition(X, addition(Y, addition(X, Y)))
% 19.57/2.88 = { by axiom 5 (additive_associativity) }
% 19.57/2.88 addition(addition(X, Y), addition(X, Y))
% 19.57/2.88 = { by axiom 1 (additive_idempotence) }
% 19.57/2.88 addition(X, Y)
% 19.57/2.88 = { by axiom 2 (additive_commutativity) }
% 19.57/2.88 addition(Y, X)
% 19.57/2.88
% 19.57/2.88 Lemma 20: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 19.57/2.88 Proof:
% 19.57/2.88 multiplication(X, addition(one, Y))
% 19.57/2.88 = { by axiom 10 (right_distributivity) }
% 19.57/2.88 addition(multiplication(X, one), multiplication(X, Y))
% 19.57/2.88 = { by axiom 3 (multiplicative_right_identity) }
% 19.57/2.88 addition(X, multiplication(X, Y))
% 19.57/2.88
% 19.57/2.88 Lemma 21: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 19.57/2.88 Proof:
% 19.57/2.88 multiplication(addition(X, one), Y)
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 multiplication(addition(one, X), Y)
% 19.57/2.88 = { by axiom 11 (left_distributivity) }
% 19.57/2.88 addition(multiplication(one, Y), multiplication(X, Y))
% 19.57/2.88 = { by axiom 4 (multiplicative_left_identity) }
% 19.57/2.88 addition(Y, multiplication(X, Y))
% 19.57/2.88
% 19.57/2.88 Lemma 22: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 19.57/2.88 Proof:
% 19.57/2.88 multiplication(addition(one, Y), X)
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 multiplication(addition(Y, one), X)
% 19.57/2.88 = { by lemma 21 }
% 19.57/2.88 addition(X, multiplication(Y, X))
% 19.57/2.88
% 19.57/2.88 Lemma 23: addition(one, addition(star(X), multiplication(X, star(X)))) = star(X).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(one, addition(star(X), multiplication(X, star(X))))
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 addition(one, addition(multiplication(X, star(X)), star(X)))
% 19.57/2.88 = { by axiom 5 (additive_associativity) }
% 19.57/2.88 addition(addition(one, multiplication(X, star(X))), star(X))
% 19.57/2.88 = { by axiom 12 (order_1) R->L }
% 19.57/2.88 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 19.57/2.88 = { by axiom 14 (star_unfold_right) }
% 19.57/2.88 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 19.57/2.88 = { by axiom 7 (order_1) }
% 19.57/2.88 star(X)
% 19.57/2.88
% 19.57/2.88 Lemma 24: addition(star(X), multiplication(X, star(X))) = star(X).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(star(X), multiplication(X, star(X)))
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 addition(multiplication(X, star(X)), star(X))
% 19.57/2.88 = { by lemma 19 R->L }
% 19.57/2.88 addition(star(X), addition(multiplication(X, star(X)), star(X)))
% 19.57/2.88 = { by axiom 5 (additive_associativity) }
% 19.57/2.88 addition(addition(star(X), multiplication(X, star(X))), star(X))
% 19.57/2.88 = { by lemma 23 R->L }
% 19.57/2.88 addition(addition(star(X), multiplication(X, star(X))), addition(one, addition(star(X), multiplication(X, star(X)))))
% 19.57/2.88 = { by lemma 19 }
% 19.57/2.88 addition(one, addition(star(X), multiplication(X, star(X))))
% 19.57/2.88 = { by lemma 23 }
% 19.57/2.88 star(X)
% 19.57/2.88
% 19.57/2.88 Lemma 25: addition(X, multiplication(X, star(Y))) = multiplication(X, star(Y)).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(X, multiplication(X, star(Y)))
% 19.57/2.88 = { by lemma 24 R->L }
% 19.57/2.88 addition(X, multiplication(X, addition(star(Y), multiplication(Y, star(Y)))))
% 19.57/2.88 = { by lemma 20 R->L }
% 19.57/2.88 multiplication(X, addition(one, addition(star(Y), multiplication(Y, star(Y)))))
% 19.57/2.88 = { by lemma 23 }
% 19.57/2.88 multiplication(X, star(Y))
% 19.57/2.88
% 19.57/2.88 Lemma 26: addition(one, addition(star(X), multiplication(star(X), X))) = star(X).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(one, addition(star(X), multiplication(star(X), X)))
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 addition(one, addition(multiplication(star(X), X), star(X)))
% 19.57/2.88 = { by axiom 5 (additive_associativity) }
% 19.57/2.88 addition(addition(one, multiplication(star(X), X)), star(X))
% 19.57/2.88 = { by axiom 12 (order_1) R->L }
% 19.57/2.88 fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
% 19.57/2.88 = { by axiom 15 (star_unfold_left) }
% 19.57/2.88 fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
% 19.57/2.88 = { by axiom 7 (order_1) }
% 19.57/2.88 star(X)
% 19.57/2.88
% 19.57/2.88 Lemma 27: addition(star(X), multiplication(star(X), X)) = star(X).
% 19.57/2.88 Proof:
% 19.57/2.88 addition(star(X), multiplication(star(X), X))
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 addition(multiplication(star(X), X), star(X))
% 19.57/2.88 = { by lemma 19 R->L }
% 19.57/2.88 addition(star(X), addition(multiplication(star(X), X), star(X)))
% 19.57/2.88 = { by axiom 5 (additive_associativity) }
% 19.57/2.88 addition(addition(star(X), multiplication(star(X), X)), star(X))
% 19.57/2.88 = { by lemma 26 R->L }
% 19.57/2.88 addition(addition(star(X), multiplication(star(X), X)), addition(one, addition(star(X), multiplication(star(X), X))))
% 19.57/2.88 = { by lemma 19 }
% 19.57/2.88 addition(one, addition(star(X), multiplication(star(X), X)))
% 19.57/2.88 = { by lemma 26 }
% 19.57/2.88 star(X)
% 19.57/2.88
% 19.57/2.88 Lemma 28: fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X) = leq(multiplication(star(Y), X), Z).
% 19.57/2.88 Proof:
% 19.57/2.88 fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X)
% 19.57/2.88 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.88 fresh4(leq(addition(multiplication(Y, Z), X), Z), true, Y, Z, X)
% 19.57/2.88 = { by axiom 16 (star_induction_left) }
% 19.57/2.89 leq(multiplication(star(Y), X), Z)
% 19.57/2.89
% 19.57/2.89 Goal 1 (goals): star(addition(one, x0)) = star(x0).
% 19.57/2.89 Proof:
% 19.57/2.89 star(addition(one, x0))
% 19.57/2.89 = { by axiom 7 (order_1) R->L }
% 19.57/2.89 fresh(true, true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 9 (star_induction_left) R->L }
% 19.57/2.89 fresh(fresh4(true, true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 8 (order) R->L }
% 19.57/2.89 fresh(fresh4(fresh3(addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))), addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))), addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by lemma 18 R->L }
% 19.57/2.89 fresh(fresh4(fresh3(addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))), addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 13 (order) }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 5 (additive_associativity) R->L }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(x0, star(addition(one, x0))), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by lemma 21 R->L }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(x0, star(addition(one, x0))), multiplication(addition(multiplication(x0, star(addition(one, x0))), one), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by lemma 27 R->L }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(x0, star(addition(one, x0))), multiplication(addition(multiplication(x0, star(addition(one, x0))), one), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 3 (multiplicative_right_identity) R->L }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(multiplication(x0, star(addition(one, x0))), one), multiplication(addition(multiplication(x0, star(addition(one, x0))), one), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by lemma 21 }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(multiplication(x0, star(addition(one, x0))), one), addition(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))), multiplication(multiplication(x0, star(addition(one, x0))), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0)))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(multiplication(x0, star(addition(one, x0))), one), addition(multiplication(multiplication(x0, star(addition(one, x0))), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0)))), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 5 (additive_associativity) }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(addition(multiplication(multiplication(x0, star(addition(one, x0))), one), multiplication(multiplication(x0, star(addition(one, x0))), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))))), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 10 (right_distributivity) R->L }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(multiplication(multiplication(x0, star(addition(one, x0))), addition(one, addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))))), addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 2 (additive_commutativity) }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))), multiplication(multiplication(x0, star(addition(one, x0))), addition(one, addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0)))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by lemma 26 }
% 19.57/2.89 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.89 = { by axiom 5 (additive_associativity) R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), addition(multiplication(star(addition(one, x0)), addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 18 R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), addition(multiplication(star(addition(one, x0)), addition(one, addition(one, x0))), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 20 }
% 19.57/2.90 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), addition(addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), addition(one, x0))), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 27 }
% 19.57/2.90 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 18 }
% 19.57/2.90 fresh(fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by axiom 5 (additive_associativity) R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, addition(x0, multiplication(x0, star(addition(one, x0))))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 25 }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(multiplication(x0, star(addition(one, x0))), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by axiom 6 (multiplicative_associativity) R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, multiplication(star(addition(one, x0)), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 23 R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, multiplication(addition(one, addition(star(addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0))))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 22 }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, addition(star(addition(one, x0)), multiplication(addition(star(addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0)))), star(addition(one, x0)))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 24 }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, addition(star(addition(one, x0)), multiplication(star(addition(one, x0)), star(addition(one, x0)))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by axiom 2 (additive_commutativity) R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, addition(multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by axiom 12 (order_1) R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, fresh(leq(multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))), true, multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 28 R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, fresh(fresh4(leq(addition(star(addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0)))), star(addition(one, x0))), true, addition(one, x0), star(addition(one, x0)), star(addition(one, x0))), true, multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 24 }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, fresh(fresh4(leq(star(addition(one, x0)), star(addition(one, x0))), true, addition(one, x0), star(addition(one, x0)), star(addition(one, x0))), true, multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 17 }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, fresh(fresh4(true, true, addition(one, x0), star(addition(one, x0)), star(addition(one, x0))), true, multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by axiom 9 (star_induction_left) }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, fresh(true, true, multiplication(star(addition(one, x0)), star(addition(one, x0))), star(addition(one, x0))))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by axiom 7 (order_1) }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), addition(star(addition(one, x0)), multiplication(x0, star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 22 R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(addition(one, x0), multiplication(addition(one, x0), star(addition(one, x0))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 25 }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), multiplication(addition(one, x0), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 18 R->L }
% 19.57/2.90 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), multiplication(addition(one, addition(one, x0)), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.90 = { by lemma 22 }
% 19.57/2.91 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), addition(star(addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0))))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.91 = { by lemma 24 }
% 19.57/2.91 fresh(fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), star(addition(one, x0))), true, x0, star(addition(one, x0)), one), true, star(x0), star(addition(one, x0)))
% 19.57/2.91 = { by lemma 28 }
% 19.57/2.91 fresh(leq(multiplication(star(x0), one), star(addition(one, x0))), true, star(x0), star(addition(one, x0)))
% 19.57/2.91 = { by axiom 3 (multiplicative_right_identity) }
% 19.57/2.91 fresh(leq(star(x0), star(addition(one, x0))), true, star(x0), star(addition(one, x0)))
% 19.57/2.91 = { by axiom 12 (order_1) }
% 19.96/2.91 addition(star(x0), star(addition(one, x0)))
% 19.96/2.91 = { by axiom 2 (additive_commutativity) }
% 19.96/2.91 addition(star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by axiom 12 (order_1) R->L }
% 19.96/2.91 fresh(leq(star(addition(one, x0)), star(x0)), true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by axiom 3 (multiplicative_right_identity) R->L }
% 19.96/2.91 fresh(leq(multiplication(star(addition(one, x0)), one), star(x0)), true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by lemma 28 R->L }
% 19.96/2.91 fresh(fresh4(leq(addition(one, multiplication(addition(one, x0), star(x0))), star(x0)), true, addition(one, x0), star(x0), one), true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by lemma 22 }
% 19.96/2.91 fresh(fresh4(leq(addition(one, addition(star(x0), multiplication(x0, star(x0)))), star(x0)), true, addition(one, x0), star(x0), one), true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by lemma 23 }
% 19.96/2.91 fresh(fresh4(leq(star(x0), star(x0)), true, addition(one, x0), star(x0), one), true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by lemma 17 }
% 19.96/2.91 fresh(fresh4(true, true, addition(one, x0), star(x0), one), true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by axiom 9 (star_induction_left) }
% 19.96/2.91 fresh(true, true, star(addition(one, x0)), star(x0))
% 19.96/2.91 = { by axiom 7 (order_1) }
% 19.96/2.91 star(x0)
% 19.96/2.91 % SZS output end Proof
% 19.96/2.91
% 19.96/2.91 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------