TSTP Solution File: KLE041+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE041+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:38 EDT 2023
% Result : Theorem 39.71s 5.41s
% Output : Proof 40.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE041+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n016.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
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 12:41:46 EDT 2023
% 0.12/0.34 % CPUTime :
% 39.71/5.41 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 39.71/5.41
% 39.71/5.41 % SZS status Theorem
% 39.71/5.41
% 40.31/5.42 % SZS output start Proof
% 40.31/5.42 Take the following subset of the input axioms:
% 40.31/5.43 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 40.31/5.43 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 40.31/5.43 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 40.31/5.43 fof(goals, conjecture, ![X0, X1]: (leq(X0, X1) => leq(star(X0), star(X1)))).
% 40.31/5.43 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 40.31/5.43 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 40.31/5.43 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 40.31/5.43 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 40.31/5.43 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 40.31/5.43 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 40.31/5.43 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 40.31/5.43
% 40.31/5.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 40.31/5.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 40.31/5.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 40.31/5.43 fresh(y, y, x1...xn) = u
% 40.31/5.43 C => fresh(s, t, x1...xn) = v
% 40.31/5.43 where fresh is a fresh function symbol and x1..xn are the free
% 40.31/5.43 variables of u and v.
% 40.31/5.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 40.31/5.43 input problem has no model of domain size 1).
% 40.31/5.43
% 40.31/5.43 The encoding turns the above axioms into the following unit equations and goals:
% 40.31/5.43
% 40.31/5.43 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 40.31/5.43 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 40.31/5.43 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 40.31/5.43 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 40.31/5.43 Axiom 5 (goals): leq(x0, x1) = true.
% 40.31/5.43 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 40.31/5.43 Axiom 7 (order_1): fresh(X, X, Y, Z) = Z.
% 40.31/5.43 Axiom 8 (order): fresh3(X, X, Y, Z) = true.
% 40.31/5.43 Axiom 9 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 40.31/5.43 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 40.31/5.43 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 40.31/5.43 Axiom 12 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 40.31/5.43 Axiom 13 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 40.31/5.43 Axiom 14 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 40.31/5.43 Axiom 15 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 40.31/5.43
% 40.31/5.43 Lemma 16: addition(x0, x1) = x1.
% 40.31/5.43 Proof:
% 40.31/5.43 addition(x0, x1)
% 40.31/5.43 = { by axiom 12 (order_1) R->L }
% 40.31/5.43 fresh(leq(x0, x1), true, x0, x1)
% 40.31/5.43 = { by axiom 5 (goals) }
% 40.31/5.43 fresh(true, true, x0, x1)
% 40.31/5.43 = { by axiom 7 (order_1) }
% 40.31/5.43 x1
% 40.31/5.43
% 40.31/5.43 Lemma 17: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 40.31/5.43 Proof:
% 40.31/5.43 addition(X, multiplication(Y, X))
% 40.31/5.43 = { by axiom 2 (multiplicative_left_identity) R->L }
% 40.31/5.43 addition(multiplication(one, X), multiplication(Y, X))
% 40.31/5.43 = { by axiom 11 (left_distributivity) R->L }
% 40.31/5.43 multiplication(addition(one, Y), X)
% 40.31/5.43 = { by axiom 4 (additive_commutativity) }
% 40.31/5.43 multiplication(addition(Y, one), X)
% 40.31/5.43
% 40.31/5.43 Lemma 18: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 40.31/5.43 Proof:
% 40.31/5.43 addition(one, multiplication(addition(X, one), star(X)))
% 40.31/5.43 = { by lemma 17 R->L }
% 40.31/5.43 addition(one, addition(star(X), multiplication(X, star(X))))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) R->L }
% 40.31/5.43 addition(one, addition(multiplication(X, star(X)), star(X)))
% 40.31/5.43 = { by axiom 6 (additive_associativity) }
% 40.31/5.43 addition(addition(one, multiplication(X, star(X))), star(X))
% 40.31/5.43 = { by axiom 12 (order_1) R->L }
% 40.31/5.43 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 40.31/5.43 = { by axiom 14 (star_unfold_right) }
% 40.31/5.43 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 40.31/5.43 = { by axiom 7 (order_1) }
% 40.31/5.43 star(X)
% 40.31/5.43
% 40.31/5.43 Lemma 19: addition(X, addition(X, Y)) = addition(X, Y).
% 40.31/5.43 Proof:
% 40.31/5.43 addition(X, addition(X, Y))
% 40.31/5.43 = { by axiom 6 (additive_associativity) }
% 40.31/5.43 addition(addition(X, X), Y)
% 40.31/5.43 = { by axiom 3 (additive_idempotence) }
% 40.31/5.43 addition(X, Y)
% 40.31/5.43
% 40.31/5.43 Lemma 20: addition(one, star(X)) = star(X).
% 40.31/5.43 Proof:
% 40.31/5.43 addition(one, star(X))
% 40.31/5.43 = { by lemma 18 R->L }
% 40.31/5.43 addition(one, addition(one, multiplication(addition(X, one), star(X))))
% 40.31/5.43 = { by lemma 19 }
% 40.31/5.43 addition(one, multiplication(addition(X, one), star(X)))
% 40.31/5.43 = { by lemma 18 }
% 40.31/5.43 star(X)
% 40.31/5.43
% 40.31/5.43 Lemma 21: multiplication(addition(X, one), star(X)) = star(X).
% 40.31/5.43 Proof:
% 40.31/5.43 multiplication(addition(X, one), star(X))
% 40.31/5.43 = { by axiom 3 (additive_idempotence) R->L }
% 40.31/5.43 multiplication(addition(X, addition(one, one)), star(X))
% 40.31/5.43 = { by axiom 6 (additive_associativity) }
% 40.31/5.43 multiplication(addition(addition(X, one), one), star(X))
% 40.31/5.43 = { by lemma 17 R->L }
% 40.31/5.43 addition(star(X), multiplication(addition(X, one), star(X)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) R->L }
% 40.31/5.43 addition(multiplication(addition(X, one), star(X)), star(X))
% 40.31/5.43 = { by lemma 18 R->L }
% 40.31/5.43 addition(multiplication(addition(X, one), star(X)), addition(one, multiplication(addition(X, one), star(X))))
% 40.31/5.43 = { by lemma 19 R->L }
% 40.31/5.43 addition(multiplication(addition(X, one), star(X)), addition(one, addition(one, multiplication(addition(X, one), star(X)))))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) R->L }
% 40.31/5.43 addition(multiplication(addition(X, one), star(X)), addition(one, addition(multiplication(addition(X, one), star(X)), one)))
% 40.31/5.43 = { by axiom 6 (additive_associativity) }
% 40.31/5.43 addition(addition(multiplication(addition(X, one), star(X)), one), addition(multiplication(addition(X, one), star(X)), one))
% 40.31/5.43 = { by axiom 3 (additive_idempotence) }
% 40.31/5.43 addition(multiplication(addition(X, one), star(X)), one)
% 40.31/5.43 = { by axiom 4 (additive_commutativity) }
% 40.31/5.43 addition(one, multiplication(addition(X, one), star(X)))
% 40.31/5.43 = { by lemma 18 }
% 40.31/5.43 star(X)
% 40.31/5.43
% 40.31/5.43 Goal 1 (goals_1): leq(star(x0), star(x1)) = true.
% 40.31/5.43 Proof:
% 40.31/5.43 leq(star(x0), star(x1))
% 40.31/5.43 = { by axiom 7 (order_1) R->L }
% 40.31/5.43 leq(star(x0), fresh(true, true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 9 (star_induction_left) R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(true, true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 8 (order) R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(fresh3(star(addition(x0, x1)), star(addition(x0, x1)), star(addition(x0, x1)), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 3 (additive_idempotence) R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(fresh3(addition(star(addition(x0, x1)), star(addition(x0, x1))), star(addition(x0, x1)), star(addition(x0, x1)), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 13 (order) }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(star(addition(x0, x1)), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(star(addition(x1, x0)), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by lemma 21 R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(multiplication(addition(addition(x1, x0), one), star(addition(x1, x0))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 6 (additive_associativity) R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(multiplication(addition(x1, addition(x0, one)), star(addition(x1, x0))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) R->L }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(multiplication(addition(x1, addition(one, x0)), star(addition(x1, x0))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 6 (additive_associativity) }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(multiplication(addition(addition(x1, one), x0), star(addition(x1, x0))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(multiplication(addition(x0, addition(x1, one)), star(addition(x1, x0))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(multiplication(addition(x0, addition(x1, one)), star(addition(x0, x1))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 11 (left_distributivity) }
% 40.31/5.43 leq(star(x0), fresh(fresh4(leq(addition(multiplication(x0, star(addition(x0, x1))), multiplication(addition(x1, one), star(addition(x0, x1)))), star(addition(x0, x1))), true, x0, star(addition(x0, x1)), multiplication(addition(x1, one), star(addition(x0, x1)))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 15 (star_induction_left) }
% 40.31/5.43 leq(star(x0), fresh(leq(multiplication(star(x0), multiplication(addition(x1, one), star(addition(x0, x1)))), star(addition(x0, x1))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by lemma 16 }
% 40.31/5.43 leq(star(x0), fresh(leq(multiplication(star(x0), multiplication(addition(x1, one), star(x1))), star(addition(x0, x1))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by lemma 21 }
% 40.31/5.43 leq(star(x0), fresh(leq(multiplication(star(x0), star(x1)), star(addition(x0, x1))), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by lemma 16 }
% 40.31/5.43 leq(star(x0), fresh(leq(multiplication(star(x0), star(x1)), star(x1)), true, multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 12 (order_1) }
% 40.31/5.43 leq(star(x0), addition(multiplication(star(x0), star(x1)), star(x1)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) }
% 40.31/5.43 leq(star(x0), addition(star(x1), multiplication(star(x0), star(x1))))
% 40.31/5.43 = { by lemma 17 }
% 40.31/5.43 leq(star(x0), multiplication(addition(star(x0), one), star(x1)))
% 40.31/5.43 = { by axiom 4 (additive_commutativity) }
% 40.31/5.43 leq(star(x0), multiplication(addition(one, star(x0)), star(x1)))
% 40.31/5.43 = { by lemma 20 }
% 40.31/5.43 leq(star(x0), multiplication(star(x0), star(x1)))
% 40.31/5.43 = { by axiom 1 (multiplicative_right_identity) R->L }
% 40.31/5.43 leq(multiplication(star(x0), one), multiplication(star(x0), star(x1)))
% 40.31/5.43 = { by lemma 20 R->L }
% 40.31/5.43 leq(multiplication(star(x0), one), multiplication(star(x0), addition(one, star(x1))))
% 40.31/5.43 = { by axiom 10 (right_distributivity) }
% 40.31/5.43 leq(multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), star(x1))))
% 40.31/5.43 = { by axiom 13 (order) R->L }
% 40.31/5.43 fresh3(addition(multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), star(x1)))), addition(multiplication(star(x0), one), multiplication(star(x0), star(x1))), multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), star(x1))))
% 40.31/5.43 = { by lemma 19 }
% 40.31/5.43 fresh3(addition(multiplication(star(x0), one), multiplication(star(x0), star(x1))), addition(multiplication(star(x0), one), multiplication(star(x0), star(x1))), multiplication(star(x0), one), addition(multiplication(star(x0), one), multiplication(star(x0), star(x1))))
% 40.31/5.43 = { by axiom 8 (order) }
% 40.31/5.43 true
% 40.31/5.43 % SZS output end Proof
% 40.31/5.43
% 40.31/5.43 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------