TSTP Solution File: KLE098+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE098+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 : n001.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 05:35:52 EDT 2023
% Result : Theorem 3.44s 0.91s
% Output : Proof 4.31s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE098+1 : 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
% 0.13/0.34 % Computer : n001.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 13:04:07 EDT 2023
% 0.13/0.34 % CPUTime :
% 3.44/0.91 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 3.44/0.91
% 3.44/0.91 % SZS status Theorem
% 3.44/0.91
% 4.31/0.93 % SZS output start Proof
% 4.31/0.93 Take the following subset of the input axioms:
% 4.31/0.93 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 4.31/0.93 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 4.31/0.93 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 4.31/0.93 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 4.31/0.93 fof(domain1, axiom, ![X0]: multiplication(antidomain(X0), X0)=zero).
% 4.31/0.93 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 4.31/0.93 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 4.31/0.93 fof(forward_diamond, axiom, ![X1, X0_2]: forward_diamond(X0_2, X1)=domain(multiplication(X0_2, domain(X1)))).
% 4.31/0.93 fof(goals, conjecture, ![X2, X0_2, X1_2]: (addition(forward_diamond(X0_2, domain(X1_2)), domain(X2))=domain(X2) => addition(multiplication(X0_2, domain(X1_2)), multiplication(domain(X2), X0_2))=multiplication(domain(X2), X0_2))).
% 4.31/0.93 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 4.31/0.93 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 4.31/0.93 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 4.31/0.93 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 4.31/0.93
% 4.31/0.93 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.31/0.93 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.31/0.93 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.31/0.93 fresh(y, y, x1...xn) = u
% 4.31/0.93 C => fresh(s, t, x1...xn) = v
% 4.31/0.93 where fresh is a fresh function symbol and x1..xn are the free
% 4.31/0.93 variables of u and v.
% 4.31/0.93 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.31/0.93 input problem has no model of domain size 1).
% 4.31/0.93
% 4.31/0.93 The encoding turns the above axioms into the following unit equations and goals:
% 4.31/0.93
% 4.31/0.93 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 4.31/0.93 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 4.31/0.93 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 4.31/0.93 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 4.31/0.93 Axiom 5 (additive_identity): addition(X, zero) = X.
% 4.31/0.93 Axiom 6 (domain4): domain(X) = antidomain(antidomain(X)).
% 4.31/0.93 Axiom 7 (domain1): multiplication(antidomain(X), X) = zero.
% 4.31/0.93 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 4.31/0.93 Axiom 9 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 4.31/0.93 Axiom 10 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 4.31/0.93 Axiom 11 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 4.31/0.93 Axiom 12 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 4.31/0.93 Axiom 13 (goals): addition(forward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
% 4.31/0.93
% 4.31/0.93 Lemma 14: addition(domain(X), antidomain(X)) = one.
% 4.31/0.93 Proof:
% 4.31/0.93 addition(domain(X), antidomain(X))
% 4.31/0.93 = { by axiom 6 (domain4) }
% 4.31/0.93 addition(antidomain(antidomain(X)), antidomain(X))
% 4.31/0.93 = { by axiom 10 (domain3) }
% 4.31/0.93 one
% 4.31/0.93
% 4.31/0.93 Lemma 15: addition(one, domain(X)) = one.
% 4.31/0.93 Proof:
% 4.31/0.93 addition(one, domain(X))
% 4.31/0.93 = { by axiom 4 (additive_commutativity) R->L }
% 4.31/0.93 addition(domain(X), one)
% 4.31/0.93 = { by lemma 14 R->L }
% 4.31/0.93 addition(domain(X), addition(domain(X), antidomain(X)))
% 4.31/0.93 = { by axiom 8 (additive_associativity) }
% 4.31/0.93 addition(addition(domain(X), domain(X)), antidomain(X))
% 4.31/0.93 = { by axiom 3 (additive_idempotence) }
% 4.31/0.93 addition(domain(X), antidomain(X))
% 4.31/0.93 = { by lemma 14 }
% 4.31/0.93 one
% 4.31/0.93
% 4.31/0.93 Lemma 16: multiplication(domain(X), X) = X.
% 4.31/0.93 Proof:
% 4.31/0.93 multiplication(domain(X), X)
% 4.31/0.93 = { by axiom 5 (additive_identity) R->L }
% 4.31/0.93 addition(multiplication(domain(X), X), zero)
% 4.31/0.93 = { by axiom 7 (domain1) R->L }
% 4.31/0.93 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 4.31/0.93 = { by axiom 12 (left_distributivity) R->L }
% 4.31/0.93 multiplication(addition(domain(X), antidomain(X)), X)
% 4.31/0.93 = { by lemma 14 }
% 4.31/0.93 multiplication(one, X)
% 4.31/0.93 = { by axiom 2 (multiplicative_left_identity) }
% 4.31/0.93 X
% 4.31/0.93
% 4.31/0.93 Lemma 17: addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Z, Y)).
% 4.31/0.93 Proof:
% 4.31/0.93 addition(multiplication(X, Y), multiplication(X, Z))
% 4.31/0.93 = { by axiom 11 (right_distributivity) R->L }
% 4.31/0.93 multiplication(X, addition(Y, Z))
% 4.31/0.93 = { by axiom 4 (additive_commutativity) }
% 4.31/0.93 multiplication(X, addition(Z, Y))
% 4.31/0.93
% 4.31/0.93 Goal 1 (goals_1): addition(multiplication(x0, domain(x1)), multiplication(domain(x2), x0)) = multiplication(domain(x2), x0).
% 4.31/0.93 Proof:
% 4.31/0.93 addition(multiplication(x0, domain(x1)), multiplication(domain(x2), x0))
% 4.31/0.93 = { by axiom 4 (additive_commutativity) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(x0, domain(x1)))
% 4.31/0.93 = { by axiom 2 (multiplicative_left_identity) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(one, multiplication(x0, domain(x1))))
% 4.31/0.93 = { by lemma 15 R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(one, domain(x2)), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 12 (left_distributivity) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), addition(multiplication(one, multiplication(x0, domain(x1))), multiplication(domain(x2), multiplication(x0, domain(x1)))))
% 4.31/0.93 = { by axiom 2 (multiplicative_left_identity) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), addition(multiplication(x0, domain(x1)), multiplication(domain(x2), multiplication(x0, domain(x1)))))
% 4.31/0.93 = { by axiom 4 (additive_commutativity) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), addition(multiplication(domain(x2), multiplication(x0, domain(x1))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by lemma 16 R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), addition(multiplication(domain(x2), multiplication(x0, domain(x1))), multiplication(domain(multiplication(x0, domain(x1))), multiplication(x0, domain(x1)))))
% 4.31/0.93 = { by axiom 12 (left_distributivity) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, domain(x1)))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 6 (domain4) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(antidomain(x1))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by lemma 16 R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(multiplication(domain(antidomain(x1)), antidomain(x1)))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 6 (domain4) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(multiplication(antidomain(antidomain(antidomain(x1))), antidomain(x1)))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 6 (domain4) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(multiplication(antidomain(domain(x1)), antidomain(x1)))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 5 (additive_identity) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(addition(multiplication(antidomain(domain(x1)), antidomain(x1)), zero))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 7 (domain1) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(addition(multiplication(antidomain(domain(x1)), antidomain(x1)), multiplication(antidomain(domain(x1)), domain(x1))))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 11 (right_distributivity) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(multiplication(antidomain(domain(x1)), addition(antidomain(x1), domain(x1))))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 4 (additive_commutativity) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(multiplication(antidomain(domain(x1)), addition(domain(x1), antidomain(x1))))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by lemma 14 }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(multiplication(antidomain(domain(x1)), one))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 1 (multiplicative_right_identity) }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, antidomain(antidomain(domain(x1)))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 6 (domain4) R->L }
% 4.31/0.93 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), domain(multiplication(x0, domain(domain(x1))))), multiplication(x0, domain(x1))))
% 4.31/0.93 = { by axiom 9 (forward_diamond) R->L }
% 4.31/0.94 addition(multiplication(domain(x2), x0), multiplication(addition(domain(x2), forward_diamond(x0, domain(x1))), multiplication(x0, domain(x1))))
% 4.31/0.94 = { by axiom 4 (additive_commutativity) R->L }
% 4.31/0.94 addition(multiplication(domain(x2), x0), multiplication(addition(forward_diamond(x0, domain(x1)), domain(x2)), multiplication(x0, domain(x1))))
% 4.31/0.94 = { by axiom 13 (goals) }
% 4.31/0.94 addition(multiplication(domain(x2), x0), multiplication(domain(x2), multiplication(x0, domain(x1))))
% 4.31/0.94 = { by lemma 17 }
% 4.31/0.94 multiplication(domain(x2), addition(multiplication(x0, domain(x1)), x0))
% 4.31/0.94 = { by axiom 1 (multiplicative_right_identity) R->L }
% 4.31/0.94 multiplication(domain(x2), addition(multiplication(x0, domain(x1)), multiplication(x0, one)))
% 4.31/0.94 = { by lemma 17 }
% 4.31/0.94 multiplication(domain(x2), multiplication(x0, addition(one, domain(x1))))
% 4.31/0.94 = { by lemma 15 }
% 4.31/0.94 multiplication(domain(x2), multiplication(x0, one))
% 4.31/0.94 = { by axiom 1 (multiplicative_right_identity) }
% 4.31/0.94 multiplication(domain(x2), x0)
% 4.31/0.94 % SZS output end Proof
% 4.31/0.94
% 4.31/0.94 RESULT: Theorem (the conjecture is true).
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