TSTP Solution File: KLE043+2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE043+2 : 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 : n019.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 135.55s 18.06s
% Output : Proof 137.38s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE043+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n019.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.20/0.34 % CPULimit : 300
% 0.20/0.34 % WCLimit : 300
% 0.20/0.34 % DateTime : Tue Aug 29 12:28:57 EDT 2023
% 0.20/0.35 % CPUTime :
% 135.55/18.06 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 135.55/18.06
% 135.55/18.06 % SZS status Theorem
% 135.55/18.07
% 135.55/18.10 % SZS output start Proof
% 135.55/18.10 Take the following subset of the input axioms:
% 135.55/18.11 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 135.55/18.11 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 135.55/18.11 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 135.55/18.11 fof(goals, conjecture, ![X0, X1]: (leq(multiplication(star(X0), X1), addition(X1, multiplication(multiplication(X0, star(X0)), X1))) & leq(addition(X1, multiplication(multiplication(X0, star(X0)), X1)), multiplication(star(X0), X1)))).
% 135.55/18.11 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 135.55/18.11 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 135.55/18.11 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 135.55/18.11 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 135.55/18.11 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 135.55/18.11 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 135.55/18.11 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 135.55/18.11
% 137.38/18.11 Now clausify the problem and encode Horn clauses using encoding 3 of
% 137.38/18.11 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 137.38/18.11 We repeatedly replace C & s=t => u=v by the two clauses:
% 137.38/18.11 fresh(y, y, x1...xn) = u
% 137.38/18.11 C => fresh(s, t, x1...xn) = v
% 137.38/18.11 where fresh is a fresh function symbol and x1..xn are the free
% 137.38/18.11 variables of u and v.
% 137.38/18.11 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 137.38/18.11 input problem has no model of domain size 1).
% 137.38/18.11
% 137.38/18.11 The encoding turns the above axioms into the following unit equations and goals:
% 137.38/18.11
% 137.38/18.11 Axiom 1 (multiplicative_left_identity): multiplication(one, X) = X.
% 137.38/18.11 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 137.38/18.11 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 137.38/18.11 Axiom 4 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 137.38/18.11 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 137.38/18.11 Axiom 6 (order_1): fresh(X, X, Y, Z) = Z.
% 137.38/18.11 Axiom 7 (order): fresh3(X, X, Y, Z) = true.
% 137.38/18.11 Axiom 8 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 137.38/18.11 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 137.38/18.11 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 137.38/18.11 Axiom 11 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 137.38/18.11 Axiom 12 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 137.38/18.11 Axiom 13 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 137.38/18.11 Axiom 14 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 137.38/18.11
% 137.38/18.11 Lemma 15: addition(X, addition(X, Y)) = addition(X, Y).
% 137.38/18.11 Proof:
% 137.38/18.11 addition(X, addition(X, Y))
% 137.38/18.11 = { by axiom 5 (additive_associativity) }
% 137.38/18.11 addition(addition(X, X), Y)
% 137.38/18.11 = { by axiom 2 (additive_idempotence) }
% 137.38/18.11 addition(X, Y)
% 137.38/18.11
% 137.38/18.11 Lemma 16: addition(Z, addition(X, Y)) = addition(X, addition(Y, Z)).
% 137.38/18.11 Proof:
% 137.38/18.11 addition(Z, addition(X, Y))
% 137.38/18.11 = { by axiom 3 (additive_commutativity) R->L }
% 137.38/18.11 addition(addition(X, Y), Z)
% 137.38/18.11 = { by axiom 5 (additive_associativity) R->L }
% 137.38/18.11 addition(X, addition(Y, Z))
% 137.38/18.11
% 137.38/18.11 Lemma 17: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 137.38/18.11 Proof:
% 137.38/18.11 addition(multiplication(X, Y), multiplication(Z, Y))
% 137.38/18.11 = { by axiom 10 (left_distributivity) R->L }
% 137.38/18.11 multiplication(addition(X, Z), Y)
% 137.38/18.11 = { by axiom 3 (additive_commutativity) }
% 137.38/18.11 multiplication(addition(Z, X), Y)
% 137.38/18.11
% 137.38/18.11 Lemma 18: addition(one, addition(multiplication(X, star(X)), star(X))) = star(X).
% 137.38/18.11 Proof:
% 137.38/18.11 addition(one, addition(multiplication(X, star(X)), star(X)))
% 137.38/18.11 = { by axiom 5 (additive_associativity) }
% 137.38/18.11 addition(addition(one, multiplication(X, star(X))), star(X))
% 137.38/18.11 = { by axiom 11 (order_1) R->L }
% 137.38/18.11 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 137.38/18.11 = { by axiom 13 (star_unfold_right) }
% 137.38/18.11 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 137.38/18.11 = { by axiom 6 (order_1) }
% 137.38/18.11 star(X)
% 137.38/18.11
% 137.38/18.11 Lemma 19: addition(multiplication(multiplication(X, star(X)), Y), multiplication(star(X), Y)) = multiplication(star(X), Y).
% 137.38/18.11 Proof:
% 137.38/18.11 addition(multiplication(multiplication(X, star(X)), Y), multiplication(star(X), Y))
% 137.38/18.11 = { by axiom 3 (additive_commutativity) R->L }
% 137.38/18.11 addition(multiplication(star(X), Y), multiplication(multiplication(X, star(X)), Y))
% 137.38/18.11 = { by lemma 17 }
% 137.38/18.11 multiplication(addition(multiplication(X, star(X)), star(X)), Y)
% 137.38/18.11 = { by axiom 2 (additive_idempotence) R->L }
% 137.38/18.11 multiplication(addition(multiplication(X, star(X)), addition(star(X), star(X))), Y)
% 137.38/18.11 = { by axiom 5 (additive_associativity) }
% 137.38/18.11 multiplication(addition(addition(multiplication(X, star(X)), star(X)), star(X)), Y)
% 137.38/18.11 = { by lemma 18 R->L }
% 137.38/18.11 multiplication(addition(addition(multiplication(X, star(X)), star(X)), addition(one, addition(multiplication(X, star(X)), star(X)))), Y)
% 137.38/18.11 = { by lemma 15 R->L }
% 137.38/18.11 multiplication(addition(addition(multiplication(X, star(X)), star(X)), addition(one, addition(one, addition(multiplication(X, star(X)), star(X))))), Y)
% 137.38/18.11 = { by axiom 3 (additive_commutativity) R->L }
% 137.38/18.11 multiplication(addition(addition(multiplication(X, star(X)), star(X)), addition(one, addition(addition(multiplication(X, star(X)), star(X)), one))), Y)
% 137.38/18.11 = { by axiom 5 (additive_associativity) }
% 137.38/18.11 multiplication(addition(addition(addition(multiplication(X, star(X)), star(X)), one), addition(addition(multiplication(X, star(X)), star(X)), one)), Y)
% 137.38/18.11 = { by axiom 2 (additive_idempotence) }
% 137.38/18.11 multiplication(addition(addition(multiplication(X, star(X)), star(X)), one), Y)
% 137.38/18.11 = { by axiom 3 (additive_commutativity) }
% 137.38/18.11 multiplication(addition(one, addition(multiplication(X, star(X)), star(X))), Y)
% 137.38/18.11 = { by lemma 18 }
% 137.38/18.11 multiplication(star(X), Y)
% 137.38/18.11
% 137.38/18.11 Lemma 20: addition(addition(X, multiplication(multiplication(Y, star(Y)), X)), multiplication(star(Y), X)) = multiplication(star(Y), X).
% 137.38/18.11 Proof:
% 137.38/18.11 addition(addition(X, multiplication(multiplication(Y, star(Y)), X)), multiplication(star(Y), X))
% 137.38/18.11 = { by axiom 3 (additive_commutativity) R->L }
% 137.38/18.11 addition(multiplication(star(Y), X), addition(X, multiplication(multiplication(Y, star(Y)), X)))
% 137.38/18.11 = { by lemma 16 }
% 137.38/18.11 addition(X, addition(multiplication(multiplication(Y, star(Y)), X), multiplication(star(Y), X)))
% 137.38/18.11 = { by lemma 19 }
% 137.38/18.11 addition(X, multiplication(star(Y), X))
% 137.38/18.11 = { by axiom 3 (additive_commutativity) R->L }
% 137.38/18.12 addition(multiplication(star(Y), X), X)
% 137.38/18.12 = { by axiom 1 (multiplicative_left_identity) R->L }
% 137.38/18.12 addition(multiplication(star(Y), X), multiplication(one, X))
% 137.38/18.12 = { by lemma 17 }
% 137.38/18.12 multiplication(addition(one, star(Y)), X)
% 137.38/18.12 = { by lemma 18 R->L }
% 137.38/18.12 multiplication(addition(one, addition(one, addition(multiplication(Y, star(Y)), star(Y)))), X)
% 137.38/18.12 = { by lemma 15 }
% 137.38/18.12 multiplication(addition(one, addition(multiplication(Y, star(Y)), star(Y))), X)
% 137.38/18.12 = { by lemma 18 }
% 137.38/18.12 multiplication(star(Y), X)
% 137.38/18.12
% 137.38/18.12 Goal 1 (goals): tuple(leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)))) = tuple(true, true).
% 137.38/18.12 Proof:
% 137.38/18.12 tuple(leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by lemma 20 R->L }
% 137.38/18.12 tuple(leq(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1))), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 12 (order) R->L }
% 137.38/18.12 tuple(fresh3(addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1))), addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1))), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by lemma 15 }
% 137.38/18.12 tuple(fresh3(addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1)), addition(x1, multiplication(multiplication(x0, star(x0)), x1)), addition(addition(x1, multiplication(multiplication(x0, star(x0)), x1)), multiplication(star(x0), x1))), leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 7 (order) }
% 137.38/18.12 tuple(true, leq(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 6 (order_1) R->L }
% 137.38/18.12 tuple(true, leq(fresh(true, true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 8 (star_induction_left) R->L }
% 137.38/18.12 tuple(true, leq(fresh(fresh4(true, true, x0_2, multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 7 (order) R->L }
% 137.38/18.12 tuple(true, leq(fresh(fresh4(fresh3(multiplication(star(x0_2), x1_2), multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2), multiplication(star(x0_2), x1_2)), true, x0_2, multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by lemma 19 R->L }
% 137.38/18.12 tuple(true, leq(fresh(fresh4(fresh3(addition(multiplication(multiplication(x0_2, star(x0_2)), x1_2), multiplication(star(x0_2), x1_2)), multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2), multiplication(star(x0_2), x1_2)), true, x0_2, multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 12 (order) }
% 137.38/18.12 tuple(true, leq(fresh(fresh4(leq(multiplication(multiplication(x0_2, star(x0_2)), x1_2), multiplication(star(x0_2), x1_2)), true, x0_2, multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 2 (additive_idempotence) R->L }
% 137.38/18.12 tuple(true, leq(fresh(fresh4(leq(addition(multiplication(multiplication(x0_2, star(x0_2)), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), true, x0_2, multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 4 (multiplicative_associativity) R->L }
% 137.38/18.12 tuple(true, leq(fresh(fresh4(leq(addition(multiplication(x0_2, multiplication(star(x0_2), x1_2)), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), true, x0_2, multiplication(star(x0_2), x1_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 14 (star_induction_left) }
% 137.38/18.12 tuple(true, leq(fresh(leq(multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), true, multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 11 (order_1) }
% 137.38/18.12 tuple(true, leq(addition(multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 3 (additive_commutativity) }
% 137.38/18.12 tuple(true, leq(addition(multiplication(star(x0_2), x1_2), multiplication(star(x0_2), multiplication(multiplication(x0_2, star(x0_2)), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 9 (right_distributivity) R->L }
% 137.38/18.12 tuple(true, leq(multiplication(star(x0_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 14 (star_induction_left) R->L }
% 137.38/18.12 tuple(true, fresh4(leq(addition(multiplication(x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by lemma 16 }
% 137.38/18.12 tuple(true, fresh4(leq(addition(x1_2, addition(multiplication(multiplication(x0_2, star(x0_2)), x1_2), multiplication(x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 4 (multiplicative_associativity) R->L }
% 137.38/18.12 tuple(true, fresh4(leq(addition(x1_2, addition(multiplication(x0_2, multiplication(star(x0_2), x1_2)), multiplication(x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 9 (right_distributivity) R->L }
% 137.38/18.12 tuple(true, fresh4(leq(addition(x1_2, multiplication(x0_2, addition(multiplication(star(x0_2), x1_2), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 3 (additive_commutativity) }
% 137.38/18.12 tuple(true, fresh4(leq(addition(x1_2, multiplication(x0_2, addition(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), multiplication(star(x0_2), x1_2)))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by lemma 20 }
% 137.38/18.12 tuple(true, fresh4(leq(addition(x1_2, multiplication(x0_2, multiplication(star(x0_2), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.12 = { by axiom 4 (multiplicative_associativity) }
% 137.38/18.13 tuple(true, fresh4(leq(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.13 = { by axiom 12 (order) R->L }
% 137.38/18.13 tuple(true, fresh4(fresh3(addition(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.13 = { by axiom 2 (additive_idempotence) }
% 137.38/18.13 tuple(true, fresh4(fresh3(addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))), true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.13 = { by axiom 7 (order) }
% 137.38/18.13 tuple(true, fresh4(true, true, x0_2, addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2)), addition(x1_2, multiplication(multiplication(x0_2, star(x0_2)), x1_2))))
% 137.38/18.13 = { by axiom 8 (star_induction_left) }
% 137.38/18.13 tuple(true, true)
% 137.38/18.13 % SZS output end Proof
% 137.38/18.13
% 137.38/18.13 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------