TSTP Solution File: HEN011-5 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : HEN011-5 : TPTP v8.1.2. Bugfixed v1.2.1.
% 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 01:57:04 EDT 2023
% Result : Unsatisfiable 0.20s 0.48s
% Output : Proof 0.20s
% 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 : HEN011-5 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n016.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Thu Aug 24 13:57:24 EDT 2023
% 0.17/0.35 % CPUTime :
% 0.20/0.48 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.48
% 0.20/0.48 % SZS status Unsatisfiable
% 0.20/0.48
% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 Take the following subset of the input axioms:
% 0.20/0.49 fof(divide_and_equal, axiom, ![X, Y]: (divide(X, Y)!=zero | (divide(Y, X)!=zero | X=Y))).
% 0.20/0.49 fof(one_inversion_equals_three, axiom, ![X2]: divide(identity, divide(identity, divide(identity, X2)))=divide(identity, X2)).
% 0.20/0.49 fof(property_of_divide2, axiom, ![Z, X2, Y2]: (divide(X2, Y2)!=zero | divide(divide(Z, Y2), divide(Z, X2))=zero)).
% 0.20/0.49 fof(property_of_divide3, axiom, ![X2, Y2, Z2]: (divide(X2, Y2)!=zero | divide(divide(X2, Z2), divide(Y2, Z2))=zero)).
% 0.20/0.49 fof(prove_this, negated_conjecture, divide(divide(identity, a), divide(identity, divide(identity, b)))!=divide(divide(identity, b), divide(identity, divide(identity, a)))).
% 0.20/0.49 fof(quotient_property, axiom, ![X2, Y2, Z2]: divide(divide(divide(X2, Z2), divide(Y2, Z2)), divide(divide(X2, Y2), Z2))=zero).
% 0.20/0.49 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: divide(divide(X2, Y2), X2)=zero).
% 0.20/0.49 fof(x_divide_x_is_zero, axiom, ![X2]: divide(X2, X2)=zero).
% 0.20/0.49 fof(x_divide_zero_is_x, axiom, ![X2]: divide(X2, zero)=X2).
% 0.20/0.49
% 0.20/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.49 fresh(y, y, x1...xn) = u
% 0.20/0.49 C => fresh(s, t, x1...xn) = v
% 0.20/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.49 variables of u and v.
% 0.20/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.49 input problem has no model of domain size 1).
% 0.20/0.49
% 0.20/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.49
% 0.20/0.49 Axiom 1 (x_divide_x_is_zero): divide(X, X) = zero.
% 0.20/0.49 Axiom 2 (x_divide_zero_is_x): divide(X, zero) = X.
% 0.20/0.49 Axiom 3 (divide_and_equal): fresh(X, X, Y, Z) = Z.
% 0.20/0.49 Axiom 4 (divide_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.20/0.49 Axiom 5 (quotient_smaller_than_numerator): divide(divide(X, Y), X) = zero.
% 0.20/0.49 Axiom 6 (property_of_divide2): fresh7(X, X, Y, Z, W) = zero.
% 0.20/0.49 Axiom 7 (property_of_divide3): fresh5(X, X, Y, Z, W) = zero.
% 0.20/0.49 Axiom 8 (divide_and_equal): fresh2(divide(X, Y), zero, Y, X) = fresh(divide(Y, X), zero, Y, X).
% 0.20/0.49 Axiom 9 (one_inversion_equals_three): divide(identity, divide(identity, divide(identity, X))) = divide(identity, X).
% 0.20/0.49 Axiom 10 (property_of_divide2): fresh7(divide(X, Y), zero, X, Y, Z) = divide(divide(Z, Y), divide(Z, X)).
% 0.20/0.49 Axiom 11 (property_of_divide3): fresh5(divide(X, Y), zero, X, Y, Z) = divide(divide(X, Z), divide(Y, Z)).
% 0.20/0.49 Axiom 12 (quotient_property): divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = zero.
% 0.20/0.49
% 0.20/0.49 Lemma 13: divide(divide(X, Y), divide(X, divide(Y, Z))) = zero.
% 0.20/0.49 Proof:
% 0.20/0.49 divide(divide(X, Y), divide(X, divide(Y, Z)))
% 0.20/0.49 = { by axiom 10 (property_of_divide2) R->L }
% 0.20/0.49 fresh7(divide(divide(Y, Z), Y), zero, divide(Y, Z), Y, X)
% 0.20/0.49 = { by axiom 5 (quotient_smaller_than_numerator) }
% 0.20/0.49 fresh7(zero, zero, divide(Y, Z), Y, X)
% 0.20/0.49 = { by axiom 6 (property_of_divide2) }
% 0.20/0.49 zero
% 0.20/0.49
% 0.20/0.49 Lemma 14: divide(divide(divide(X, Y), Z), divide(divide(X, Z), Y)) = zero.
% 0.20/0.49 Proof:
% 0.20/0.49 divide(divide(divide(X, Y), Z), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.49 divide(fresh2(zero, zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by lemma 13 R->L }
% 0.20/0.49 divide(fresh2(divide(divide(divide(divide(divide(X, Y), Y), divide(Z, Y)), divide(divide(X, Y), Z)), divide(divide(divide(divide(X, Y), Y), divide(Z, Y)), divide(divide(divide(X, Y), Z), Y))), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 12 (quotient_property) }
% 0.20/0.49 divide(fresh2(divide(divide(divide(divide(divide(X, Y), Y), divide(Z, Y)), divide(divide(X, Y), Z)), zero), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 2 (x_divide_zero_is_x) }
% 0.20/0.49 divide(fresh2(divide(divide(divide(divide(X, Y), Y), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.49 divide(fresh2(divide(divide(fresh2(zero, zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 12 (quotient_property) R->L }
% 0.20/0.49 divide(fresh2(divide(divide(fresh2(divide(divide(divide(X, Y), divide(Y, Y)), divide(divide(X, Y), Y)), zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 1 (x_divide_x_is_zero) }
% 0.20/0.49 divide(fresh2(divide(divide(fresh2(divide(divide(divide(X, Y), zero), divide(divide(X, Y), Y)), zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 2 (x_divide_zero_is_x) }
% 0.20/0.49 divide(fresh2(divide(divide(fresh2(divide(divide(X, Y), divide(divide(X, Y), Y)), zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 8 (divide_and_equal) }
% 0.20/0.49 divide(fresh2(divide(divide(fresh(divide(divide(divide(X, Y), Y), divide(X, Y)), zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 11 (property_of_divide3) R->L }
% 0.20/0.49 divide(fresh2(divide(divide(fresh(fresh5(divide(divide(X, Y), X), zero, divide(X, Y), X, Y), zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 5 (quotient_smaller_than_numerator) }
% 0.20/0.49 divide(fresh2(divide(divide(fresh(fresh5(zero, zero, divide(X, Y), X, Y), zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 7 (property_of_divide3) }
% 0.20/0.49 divide(fresh2(divide(divide(fresh(zero, zero, divide(divide(X, Y), Y), divide(X, Y)), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 3 (divide_and_equal) }
% 0.20/0.49 divide(fresh2(divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Y), Z)), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 8 (divide_and_equal) }
% 0.20/0.49 divide(fresh(divide(divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by lemma 13 }
% 0.20/0.49 divide(fresh(zero, zero, divide(divide(X, Y), Z), divide(divide(X, Y), divide(Z, Y))), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 3 (divide_and_equal) }
% 0.20/0.49 divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y))
% 0.20/0.49 = { by axiom 12 (quotient_property) }
% 0.20/0.49 zero
% 0.20/0.49
% 0.20/0.49 Lemma 15: divide(divide(X, Y), Z) = divide(divide(X, Z), Y).
% 0.20/0.49 Proof:
% 0.20/0.49 divide(divide(X, Y), Z)
% 0.20/0.49 = { by axiom 3 (divide_and_equal) R->L }
% 0.20/0.49 fresh(zero, zero, divide(divide(X, Z), Y), divide(divide(X, Y), Z))
% 0.20/0.50 = { by lemma 14 R->L }
% 0.20/0.50 fresh(divide(divide(divide(X, Z), Y), divide(divide(X, Y), Z)), zero, divide(divide(X, Z), Y), divide(divide(X, Y), Z))
% 0.20/0.50 = { by axiom 8 (divide_and_equal) R->L }
% 0.20/0.50 fresh2(divide(divide(divide(X, Y), Z), divide(divide(X, Z), Y)), zero, divide(divide(X, Z), Y), divide(divide(X, Y), Z))
% 0.20/0.50 = { by lemma 14 }
% 0.20/0.50 fresh2(zero, zero, divide(divide(X, Z), Y), divide(divide(X, Y), Z))
% 0.20/0.50 = { by axiom 4 (divide_and_equal) }
% 0.20/0.50 divide(divide(X, Z), Y)
% 0.20/0.50
% 0.20/0.50 Lemma 16: divide(divide(identity, X), divide(identity, divide(identity, Y))) = divide(divide(identity, X), Y).
% 0.20/0.50 Proof:
% 0.20/0.50 divide(divide(identity, X), divide(identity, divide(identity, Y)))
% 0.20/0.50 = { by lemma 15 R->L }
% 0.20/0.50 divide(divide(identity, divide(identity, divide(identity, Y))), X)
% 0.20/0.50 = { by axiom 9 (one_inversion_equals_three) }
% 0.20/0.50 divide(divide(identity, Y), X)
% 0.20/0.50 = { by lemma 15 }
% 0.20/0.50 divide(divide(identity, X), Y)
% 0.20/0.50
% 0.20/0.50 Goal 1 (prove_this): divide(divide(identity, a), divide(identity, divide(identity, b))) = divide(divide(identity, b), divide(identity, divide(identity, a))).
% 0.20/0.50 Proof:
% 0.20/0.50 divide(divide(identity, a), divide(identity, divide(identity, b)))
% 0.20/0.50 = { by lemma 16 }
% 0.20/0.50 divide(divide(identity, a), b)
% 0.20/0.50 = { by lemma 15 R->L }
% 0.20/0.50 divide(divide(identity, b), a)
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 divide(divide(identity, b), divide(identity, divide(identity, a)))
% 0.20/0.50 % SZS output end Proof
% 0.20/0.50
% 0.20/0.50 RESULT: Unsatisfiable (the axioms are contradictory).
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