TSTP Solution File: KLE044+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE044+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 : n025.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 55.23s 7.53s
% Output : Proof 55.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE044+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
% 0.13/0.35 % Computer : n025.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 12:46:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 55.23/7.53 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 55.23/7.53
% 55.23/7.53 % SZS status Theorem
% 55.23/7.53
% 55.23/7.55 % SZS output start Proof
% 55.23/7.55 Take the following subset of the input axioms:
% 55.23/7.55 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 55.23/7.55 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 55.23/7.55 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 55.23/7.55 fof(goals, conjecture, ![X0]: (leq(star(addition(one, X0)), star(X0)) & leq(star(X0), star(addition(one, X0))))).
% 55.23/7.55 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 55.23/7.55 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 55.23/7.55 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 55.23/7.55 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 55.23/7.55 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 55.23/7.55 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 55.23/7.55 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 55.23/7.55
% 55.23/7.55 Now clausify the problem and encode Horn clauses using encoding 3 of
% 55.23/7.55 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 55.23/7.55 We repeatedly replace C & s=t => u=v by the two clauses:
% 55.23/7.55 fresh(y, y, x1...xn) = u
% 55.23/7.55 C => fresh(s, t, x1...xn) = v
% 55.23/7.55 where fresh is a fresh function symbol and x1..xn are the free
% 55.23/7.55 variables of u and v.
% 55.23/7.55 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 55.23/7.55 input problem has no model of domain size 1).
% 55.23/7.55
% 55.23/7.55 The encoding turns the above axioms into the following unit equations and goals:
% 55.23/7.55
% 55.23/7.55 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 55.23/7.55 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 55.23/7.55 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 55.23/7.55 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 55.23/7.55 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 55.23/7.55 Axiom 6 (order_1): fresh(X, X, Y, Z) = Z.
% 55.23/7.55 Axiom 7 (order): fresh3(X, X, Y, Z) = true.
% 55.23/7.55 Axiom 8 (star_induction_left): fresh4(X, X, Y, Z, W) = true.
% 55.23/7.55 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 55.23/7.55 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 55.23/7.55 Axiom 11 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 55.23/7.55 Axiom 12 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 55.23/7.55 Axiom 13 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 55.23/7.55 Axiom 14 (star_induction_left): fresh4(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 55.23/7.55
% 55.23/7.55 Lemma 15: addition(X, addition(X, Y)) = addition(X, Y).
% 55.23/7.55 Proof:
% 55.23/7.55 addition(X, addition(X, Y))
% 55.23/7.55 = { by axiom 5 (additive_associativity) }
% 55.23/7.55 addition(addition(X, X), Y)
% 55.23/7.55 = { by axiom 1 (additive_idempotence) }
% 55.23/7.55 addition(X, Y)
% 55.23/7.55
% 55.23/7.55 Lemma 16: addition(X, addition(Y, X)) = addition(Y, X).
% 55.23/7.55 Proof:
% 55.23/7.55 addition(X, addition(Y, X))
% 55.23/7.55 = { by lemma 15 R->L }
% 55.23/7.55 addition(X, addition(Y, addition(Y, X)))
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 addition(X, addition(Y, addition(X, Y)))
% 55.23/7.55 = { by axiom 5 (additive_associativity) }
% 55.23/7.55 addition(addition(X, Y), addition(X, Y))
% 55.23/7.55 = { by axiom 1 (additive_idempotence) }
% 55.23/7.55 addition(X, Y)
% 55.23/7.55 = { by axiom 2 (additive_commutativity) }
% 55.23/7.55 addition(Y, X)
% 55.23/7.55
% 55.23/7.55 Lemma 17: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 55.23/7.55 Proof:
% 55.23/7.55 multiplication(addition(X, one), Y)
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 multiplication(addition(one, X), Y)
% 55.23/7.55 = { by axiom 10 (left_distributivity) }
% 55.23/7.55 addition(multiplication(one, Y), multiplication(X, Y))
% 55.23/7.55 = { by axiom 4 (multiplicative_left_identity) }
% 55.23/7.55 addition(Y, multiplication(X, Y))
% 55.23/7.55
% 55.23/7.55 Lemma 18: addition(one, addition(star(X), multiplication(X, star(X)))) = star(X).
% 55.23/7.55 Proof:
% 55.23/7.55 addition(one, addition(star(X), multiplication(X, star(X))))
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 addition(one, addition(multiplication(X, star(X)), star(X)))
% 55.23/7.55 = { by axiom 5 (additive_associativity) }
% 55.23/7.55 addition(addition(one, multiplication(X, star(X))), star(X))
% 55.23/7.55 = { by axiom 11 (order_1) R->L }
% 55.23/7.55 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 55.23/7.55 = { by axiom 13 (star_unfold_right) }
% 55.23/7.55 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 55.23/7.55 = { by axiom 6 (order_1) }
% 55.23/7.55 star(X)
% 55.23/7.55
% 55.23/7.55 Lemma 19: addition(star(X), multiplication(X, star(X))) = star(X).
% 55.23/7.55 Proof:
% 55.23/7.55 addition(star(X), multiplication(X, star(X)))
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 addition(multiplication(X, star(X)), star(X))
% 55.23/7.55 = { by lemma 16 R->L }
% 55.23/7.55 addition(star(X), addition(multiplication(X, star(X)), star(X)))
% 55.23/7.55 = { by axiom 5 (additive_associativity) }
% 55.23/7.55 addition(addition(star(X), multiplication(X, star(X))), star(X))
% 55.23/7.55 = { by lemma 18 R->L }
% 55.23/7.55 addition(addition(star(X), multiplication(X, star(X))), addition(one, addition(star(X), multiplication(X, star(X)))))
% 55.23/7.55 = { by lemma 16 }
% 55.23/7.55 addition(one, addition(star(X), multiplication(X, star(X))))
% 55.23/7.55 = { by lemma 18 }
% 55.23/7.55 star(X)
% 55.23/7.55
% 55.23/7.55 Lemma 20: addition(X, multiplication(X, star(Y))) = multiplication(X, star(Y)).
% 55.23/7.55 Proof:
% 55.23/7.55 addition(X, multiplication(X, star(Y)))
% 55.23/7.55 = { by lemma 19 R->L }
% 55.23/7.55 addition(X, multiplication(X, addition(star(Y), multiplication(Y, star(Y)))))
% 55.23/7.55 = { by axiom 3 (multiplicative_right_identity) R->L }
% 55.23/7.55 addition(multiplication(X, one), multiplication(X, addition(star(Y), multiplication(Y, star(Y)))))
% 55.23/7.55 = { by axiom 9 (right_distributivity) R->L }
% 55.23/7.55 multiplication(X, addition(one, addition(star(Y), multiplication(Y, star(Y)))))
% 55.23/7.55 = { by lemma 18 }
% 55.23/7.55 multiplication(X, star(Y))
% 55.23/7.55
% 55.23/7.55 Lemma 21: fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X) = leq(multiplication(star(Y), X), Z).
% 55.23/7.55 Proof:
% 55.23/7.55 fresh4(leq(addition(X, multiplication(Y, Z)), Z), true, Y, Z, X)
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 fresh4(leq(addition(multiplication(Y, Z), X), Z), true, Y, Z, X)
% 55.23/7.55 = { by axiom 14 (star_induction_left) }
% 55.23/7.55 leq(multiplication(star(Y), X), Z)
% 55.23/7.55
% 55.23/7.55 Goal 1 (goals): tuple(leq(star(addition(one, x0_2)), star(x0_2)), leq(star(x0), star(addition(one, x0)))) = tuple(true, true).
% 55.23/7.55 Proof:
% 55.23/7.55 tuple(leq(star(addition(one, x0_2)), star(x0_2)), leq(star(x0), star(addition(one, x0))))
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 tuple(leq(star(addition(x0_2, one)), star(x0_2)), leq(star(x0), star(addition(one, x0))))
% 55.23/7.55 = { by axiom 3 (multiplicative_right_identity) R->L }
% 55.23/7.55 tuple(leq(multiplication(star(addition(x0_2, one)), one), star(x0_2)), leq(star(x0), star(addition(one, x0))))
% 55.23/7.55 = { by lemma 21 R->L }
% 55.23/7.55 tuple(fresh4(leq(addition(one, multiplication(addition(x0_2, one), star(x0_2))), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.55 = { by lemma 17 }
% 55.23/7.55 tuple(fresh4(leq(addition(one, addition(star(x0_2), multiplication(x0_2, star(x0_2)))), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.55 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.55 tuple(fresh4(leq(addition(one, addition(multiplication(x0_2, star(x0_2)), star(x0_2))), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.55 = { by axiom 5 (additive_associativity) }
% 55.23/7.55 tuple(fresh4(leq(addition(addition(one, multiplication(x0_2, star(x0_2))), star(x0_2)), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 11 (order_1) R->L }
% 55.23/7.56 tuple(fresh4(leq(fresh(leq(addition(one, multiplication(x0_2, star(x0_2))), star(x0_2)), true, addition(one, multiplication(x0_2, star(x0_2))), star(x0_2)), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 13 (star_unfold_right) }
% 55.23/7.56 tuple(fresh4(leq(fresh(true, true, addition(one, multiplication(x0_2, star(x0_2))), star(x0_2)), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 6 (order_1) }
% 55.23/7.56 tuple(fresh4(leq(star(x0_2), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 12 (order) R->L }
% 55.23/7.56 tuple(fresh4(fresh3(addition(star(x0_2), star(x0_2)), star(x0_2), star(x0_2), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 1 (additive_idempotence) }
% 55.23/7.56 tuple(fresh4(fresh3(star(x0_2), star(x0_2), star(x0_2), star(x0_2)), true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 7 (order) }
% 55.23/7.56 tuple(fresh4(true, true, addition(x0_2, one), star(x0_2), one), leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 8 (star_induction_left) }
% 55.23/7.56 tuple(true, leq(star(x0), star(addition(one, x0))))
% 55.23/7.56 = { by axiom 3 (multiplicative_right_identity) R->L }
% 55.23/7.56 tuple(true, leq(multiplication(star(x0), one), star(addition(one, x0))))
% 55.23/7.56 = { by lemma 21 R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), star(addition(one, x0))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by lemma 19 R->L }
% 55.23/7.56 tuple(true, 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))
% 55.23/7.56 = { by lemma 17 R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(one, multiplication(x0, star(addition(one, x0)))), multiplication(addition(addition(one, x0), one), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 2 (additive_commutativity) }
% 55.23/7.56 tuple(true, 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))
% 55.23/7.56 = { by lemma 15 }
% 55.23/7.56 tuple(true, 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))
% 55.23/7.56 = { by lemma 20 R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(one, addition(x0, multiplication(x0, star(addition(one, x0))))), multiplication(addition(one, x0), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 5 (additive_associativity) }
% 55.23/7.56 tuple(true, fresh4(leq(addition(addition(one, x0), multiplication(x0, star(addition(one, x0)))), multiplication(addition(one, x0), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by lemma 20 R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(addition(one, x0), 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))
% 55.23/7.56 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(addition(one, x0), multiplication(addition(one, x0), star(addition(one, x0))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 4 (multiplicative_left_identity) R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(one, addition(one, x0)), multiplication(addition(one, x0), star(addition(one, x0))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(one, addition(one, x0)), multiplication(addition(x0, one), star(addition(one, x0))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 10 (left_distributivity) }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(one, addition(one, x0)), addition(multiplication(x0, star(addition(one, x0))), multiplication(one, star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 2 (additive_commutativity) R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(one, addition(one, x0)), addition(multiplication(one, star(addition(one, x0))), multiplication(x0, star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 5 (additive_associativity) }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(addition(multiplication(one, addition(one, x0)), multiplication(one, star(addition(one, x0)))), multiplication(x0, star(addition(one, x0))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 9 (right_distributivity) R->L }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(one, addition(addition(one, x0), star(addition(one, x0)))), multiplication(x0, star(addition(one, x0))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 2 (additive_commutativity) }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(x0, star(addition(one, x0))), multiplication(one, addition(addition(one, x0), star(addition(one, x0)))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 4 (multiplicative_left_identity) }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(multiplication(x0, star(addition(one, x0))), addition(addition(one, x0), star(addition(one, x0))))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 5 (additive_associativity) }
% 55.23/7.56 tuple(true, fresh4(leq(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 12 (order) R->L }
% 55.23/7.56 tuple(true, fresh4(fresh3(addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0)))), addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0))), addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by lemma 15 }
% 55.23/7.56 tuple(true, fresh4(fresh3(addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0))), addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0))), addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), addition(addition(multiplication(x0, star(addition(one, x0))), addition(one, x0)), star(addition(one, x0)))), true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 7 (order) }
% 55.23/7.56 tuple(true, fresh4(true, true, x0, star(addition(one, x0)), one))
% 55.23/7.56 = { by axiom 8 (star_induction_left) }
% 55.23/7.56 tuple(true, true)
% 55.23/7.56 % SZS output end Proof
% 55.23/7.56
% 55.23/7.56 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------