TSTP Solution File: KLE080+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE080+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 : n012.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 05:35:48 EDT 2023
% Result : Theorem 0.19s 0.71s
% Output : Proof 0.19s
% 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 : KLE080+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 : n012.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 11:21:39 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.71 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.71
% 0.19/0.71 % SZS status Theorem
% 0.19/0.71
% 0.19/0.72 % SZS output start Proof
% 0.19/0.72 Take the following subset of the input axioms:
% 0.19/0.72 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.19/0.72 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.19/0.72 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 0.19/0.72 fof(domain2, axiom, ![X1, X0_2]: domain(multiplication(X0_2, X1))=domain(multiplication(X0_2, domain(X1)))).
% 0.19/0.72 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 0.19/0.72 fof(domain4, axiom, domain(zero)=zero).
% 0.19/0.73 fof(domain5, axiom, ![X1_2, X0_2]: domain(addition(X0_2, X1_2))=addition(domain(X0_2), domain(X1_2))).
% 0.19/0.73 fof(goals, conjecture, ![X0_2]: (![X1_2]: (addition(domain(X1_2), antidomain(X1_2))=one & multiplication(domain(X1_2), antidomain(X1_2))=zero) => antidomain(antidomain(X0_2))=domain(X0_2))).
% 0.19/0.73 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 0.19/0.73 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 0.19/0.73 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.19/0.73 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.19/0.73 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.19/0.73
% 0.19/0.73 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.73 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.73 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.73 fresh(y, y, x1...xn) = u
% 0.19/0.73 C => fresh(s, t, x1...xn) = v
% 0.19/0.73 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.73 variables of u and v.
% 0.19/0.73 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.73 input problem has no model of domain size 1).
% 0.19/0.73
% 0.19/0.73 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.73
% 0.19/0.73 Axiom 1 (domain4): domain(zero) = zero.
% 0.19/0.73 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.73 Axiom 3 (additive_identity): addition(X, zero) = X.
% 0.19/0.73 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.19/0.73 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 0.19/0.73 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.19/0.73 Axiom 7 (domain3): addition(domain(X), one) = one.
% 0.19/0.73 Axiom 8 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.19/0.73 Axiom 9 (goals): addition(domain(X), antidomain(X)) = one.
% 0.19/0.73 Axiom 10 (goals_1): multiplication(domain(X), antidomain(X)) = zero.
% 0.19/0.73 Axiom 11 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.19/0.73 Axiom 12 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.19/0.73 Axiom 13 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.19/0.73 Axiom 14 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.19/0.73
% 0.19/0.73 Lemma 15: addition(one, domain(X)) = one.
% 0.19/0.73 Proof:
% 0.19/0.73 addition(one, domain(X))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.73 addition(domain(X), one)
% 0.19/0.73 = { by axiom 7 (domain3) }
% 0.19/0.73 one
% 0.19/0.73
% 0.19/0.73 Lemma 16: domain(one) = one.
% 0.19/0.73 Proof:
% 0.19/0.73 domain(one)
% 0.19/0.73 = { by axiom 4 (multiplicative_right_identity) R->L }
% 0.19/0.73 multiplication(domain(one), one)
% 0.19/0.73 = { by axiom 12 (domain1) R->L }
% 0.19/0.73 addition(one, multiplication(domain(one), one))
% 0.19/0.73 = { by axiom 4 (multiplicative_right_identity) }
% 0.19/0.73 addition(one, domain(one))
% 0.19/0.73 = { by lemma 15 }
% 0.19/0.73 one
% 0.19/0.73
% 0.19/0.73 Lemma 17: domain(domain(X)) = domain(X).
% 0.19/0.73 Proof:
% 0.19/0.73 domain(domain(X))
% 0.19/0.73 = { by axiom 6 (multiplicative_left_identity) R->L }
% 0.19/0.73 domain(multiplication(one, domain(X)))
% 0.19/0.73 = { by axiom 11 (domain2) R->L }
% 0.19/0.73 domain(multiplication(one, X))
% 0.19/0.73 = { by axiom 6 (multiplicative_left_identity) }
% 0.19/0.73 domain(X)
% 0.19/0.73
% 0.19/0.73 Lemma 18: addition(antidomain(X), domain(X)) = one.
% 0.19/0.73 Proof:
% 0.19/0.73 addition(antidomain(X), domain(X))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.73 addition(domain(X), antidomain(X))
% 0.19/0.73 = { by axiom 9 (goals) }
% 0.19/0.73 one
% 0.19/0.73
% 0.19/0.73 Lemma 19: domain(antidomain(antidomain(X))) = domain(X).
% 0.19/0.73 Proof:
% 0.19/0.73 domain(antidomain(antidomain(X)))
% 0.19/0.73 = { by axiom 6 (multiplicative_left_identity) R->L }
% 0.19/0.73 domain(multiplication(one, antidomain(antidomain(X))))
% 0.19/0.73 = { by lemma 16 R->L }
% 0.19/0.73 domain(multiplication(domain(one), antidomain(antidomain(X))))
% 0.19/0.73 = { by axiom 9 (goals) R->L }
% 0.19/0.73 domain(multiplication(domain(addition(domain(X), antidomain(X))), antidomain(antidomain(X))))
% 0.19/0.73 = { by axiom 8 (domain5) }
% 0.19/0.73 domain(multiplication(addition(domain(domain(X)), domain(antidomain(X))), antidomain(antidomain(X))))
% 0.19/0.73 = { by lemma 17 }
% 0.19/0.73 domain(multiplication(addition(domain(X), domain(antidomain(X))), antidomain(antidomain(X))))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.73 domain(multiplication(addition(domain(antidomain(X)), domain(X)), antidomain(antidomain(X))))
% 0.19/0.73 = { by axiom 14 (left_distributivity) }
% 0.19/0.73 domain(addition(multiplication(domain(antidomain(X)), antidomain(antidomain(X))), multiplication(domain(X), antidomain(antidomain(X)))))
% 0.19/0.73 = { by axiom 10 (goals_1) }
% 0.19/0.73 domain(addition(zero, multiplication(domain(X), antidomain(antidomain(X)))))
% 0.19/0.73 = { by axiom 10 (goals_1) R->L }
% 0.19/0.73 domain(addition(multiplication(domain(X), antidomain(X)), multiplication(domain(X), antidomain(antidomain(X)))))
% 0.19/0.73 = { by axiom 13 (right_distributivity) R->L }
% 0.19/0.73 domain(multiplication(domain(X), addition(antidomain(X), antidomain(antidomain(X)))))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) }
% 0.19/0.73 domain(multiplication(domain(X), addition(antidomain(antidomain(X)), antidomain(X))))
% 0.19/0.73 = { by axiom 11 (domain2) }
% 0.19/0.73 domain(multiplication(domain(X), domain(addition(antidomain(antidomain(X)), antidomain(X)))))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.73 domain(multiplication(domain(X), domain(addition(antidomain(X), antidomain(antidomain(X))))))
% 0.19/0.73 = { by axiom 8 (domain5) }
% 0.19/0.73 domain(multiplication(domain(X), addition(domain(antidomain(X)), domain(antidomain(antidomain(X))))))
% 0.19/0.73 = { by lemma 17 R->L }
% 0.19/0.73 domain(multiplication(domain(X), addition(domain(domain(antidomain(X))), domain(antidomain(antidomain(X))))))
% 0.19/0.73 = { by axiom 8 (domain5) R->L }
% 0.19/0.73 domain(multiplication(domain(X), domain(addition(domain(antidomain(X)), antidomain(antidomain(X))))))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) }
% 0.19/0.73 domain(multiplication(domain(X), domain(addition(antidomain(antidomain(X)), domain(antidomain(X))))))
% 0.19/0.73 = { by lemma 18 }
% 0.19/0.73 domain(multiplication(domain(X), domain(one)))
% 0.19/0.73 = { by lemma 16 }
% 0.19/0.73 domain(multiplication(domain(X), one))
% 0.19/0.73 = { by axiom 4 (multiplicative_right_identity) }
% 0.19/0.73 domain(domain(X))
% 0.19/0.73 = { by lemma 17 }
% 0.19/0.73 domain(X)
% 0.19/0.73
% 0.19/0.73 Lemma 20: multiplication(domain(X), addition(X, Y)) = addition(X, multiplication(domain(X), Y)).
% 0.19/0.73 Proof:
% 0.19/0.73 multiplication(domain(X), addition(X, Y))
% 0.19/0.73 = { by axiom 13 (right_distributivity) }
% 0.19/0.73 addition(multiplication(domain(X), X), multiplication(domain(X), Y))
% 0.19/0.73 = { by axiom 12 (domain1) R->L }
% 0.19/0.73 addition(addition(X, multiplication(domain(X), X)), multiplication(domain(X), Y))
% 0.19/0.73 = { by axiom 6 (multiplicative_left_identity) R->L }
% 0.19/0.73 addition(addition(multiplication(one, X), multiplication(domain(X), X)), multiplication(domain(X), Y))
% 0.19/0.73 = { by axiom 14 (left_distributivity) R->L }
% 0.19/0.73 addition(multiplication(addition(one, domain(X)), X), multiplication(domain(X), Y))
% 0.19/0.73 = { by lemma 15 }
% 0.19/0.73 addition(multiplication(one, X), multiplication(domain(X), Y))
% 0.19/0.73 = { by axiom 6 (multiplicative_left_identity) }
% 0.19/0.73 addition(X, multiplication(domain(X), Y))
% 0.19/0.73
% 0.19/0.73 Lemma 21: addition(antidomain(antidomain(X)), multiplication(domain(X), domain(antidomain(X)))) = domain(X).
% 0.19/0.73 Proof:
% 0.19/0.73 addition(antidomain(antidomain(X)), multiplication(domain(X), domain(antidomain(X))))
% 0.19/0.73 = { by lemma 19 R->L }
% 0.19/0.73 addition(antidomain(antidomain(X)), multiplication(domain(antidomain(antidomain(X))), domain(antidomain(X))))
% 0.19/0.73 = { by lemma 20 R->L }
% 0.19/0.73 multiplication(domain(antidomain(antidomain(X))), addition(antidomain(antidomain(X)), domain(antidomain(X))))
% 0.19/0.73 = { by lemma 18 }
% 0.19/0.73 multiplication(domain(antidomain(antidomain(X))), one)
% 0.19/0.73 = { by axiom 4 (multiplicative_right_identity) }
% 0.19/0.73 domain(antidomain(antidomain(X)))
% 0.19/0.73 = { by lemma 19 }
% 0.19/0.73 domain(X)
% 0.19/0.73
% 0.19/0.73 Goal 1 (goals_2): antidomain(antidomain(x0)) = domain(x0).
% 0.19/0.73 Proof:
% 0.19/0.73 antidomain(antidomain(x0))
% 0.19/0.73 = { by axiom 3 (additive_identity) R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), zero)
% 0.19/0.73 = { by axiom 5 (left_annihilation) R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), multiplication(zero, domain(x0)))
% 0.19/0.73 = { by axiom 1 (domain4) R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), multiplication(domain(zero), domain(x0)))
% 0.19/0.73 = { by axiom 10 (goals_1) R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), multiplication(domain(multiplication(domain(x0), antidomain(x0))), domain(x0)))
% 0.19/0.73 = { by axiom 11 (domain2) }
% 0.19/0.73 addition(antidomain(antidomain(x0)), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), domain(x0)))
% 0.19/0.73 = { by lemma 21 R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), addition(antidomain(antidomain(x0)), multiplication(domain(x0), domain(antidomain(x0))))))
% 0.19/0.73 = { by axiom 2 (additive_commutativity) R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), addition(multiplication(domain(x0), domain(antidomain(x0))), antidomain(antidomain(x0)))))
% 0.19/0.73 = { by lemma 20 }
% 0.19/0.73 addition(antidomain(antidomain(x0)), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(domain(multiplication(domain(x0), domain(antidomain(x0)))), antidomain(antidomain(x0)))))
% 0.19/0.73 = { by axiom 11 (domain2) R->L }
% 0.19/0.73 addition(antidomain(antidomain(x0)), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(domain(multiplication(domain(x0), antidomain(x0))), antidomain(antidomain(x0)))))
% 0.19/0.73 = { by axiom 10 (goals_1) }
% 0.19/0.73 addition(antidomain(antidomain(x0)), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(domain(zero), antidomain(antidomain(x0)))))
% 0.19/0.73 = { by axiom 1 (domain4) }
% 0.19/0.74 addition(antidomain(antidomain(x0)), addition(multiplication(domain(x0), domain(antidomain(x0))), multiplication(zero, antidomain(antidomain(x0)))))
% 0.19/0.74 = { by axiom 5 (left_annihilation) }
% 0.19/0.74 addition(antidomain(antidomain(x0)), addition(multiplication(domain(x0), domain(antidomain(x0))), zero))
% 0.19/0.74 = { by axiom 3 (additive_identity) }
% 0.19/0.74 addition(antidomain(antidomain(x0)), multiplication(domain(x0), domain(antidomain(x0))))
% 0.19/0.74 = { by lemma 21 }
% 0.19/0.74 domain(x0)
% 0.19/0.74 % SZS output end Proof
% 0.19/0.74
% 0.19/0.74 RESULT: Theorem (the conjecture is true).
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