TSTP Solution File: HEN011-4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : HEN011-4 : 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 : n019.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 0.21s 0.50s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : HEN011-4 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n019.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Thu Aug 24 13:30:58 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.21/0.50 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.50
% 0.21/0.50 % SZS status Unsatisfiable
% 0.21/0.50
% 0.21/0.50 % SZS output start Proof
% 0.21/0.50 Take the following subset of the input axioms:
% 0.21/0.51 fof(less_equal_and_equal, axiom, ![X, Y]: (~less_equal(X, Y) | (~less_equal(Y, X) | X=Y))).
% 0.21/0.51 fof(one_inversion_equals_three, axiom, ![X2]: divide(identity, divide(identity, divide(identity, X2)))=divide(identity, X2)).
% 0.21/0.51 fof(property_of_divide1, axiom, ![Z, X2, Y2]: (~less_equal(divide(X2, Y2), Z) | less_equal(divide(X2, Z), Y2))).
% 0.21/0.51 fof(prove_this, negated_conjecture, divide(divide(identity, a), divide(identity, divide(identity, b)))!=divide(divide(identity, b), divide(identity, divide(identity, a)))).
% 0.21/0.51 fof(quotient_property, axiom, ![X2, Y2, Z2]: less_equal(divide(divide(X2, Z2), divide(Y2, Z2)), divide(divide(X2, Y2), Z2))).
% 0.21/0.51 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: less_equal(divide(X2, Y2), X2)).
% 0.21/0.51 fof(transitivity_of_less_equal, axiom, ![X2, Y2, Z2]: (~less_equal(X2, Y2) | (~less_equal(Y2, Z2) | less_equal(X2, Z2)))).
% 0.21/0.51
% 0.21/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.51 fresh(y, y, x1...xn) = u
% 0.21/0.51 C => fresh(s, t, x1...xn) = v
% 0.21/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.51 variables of u and v.
% 0.21/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.51 input problem has no model of domain size 1).
% 0.21/0.51
% 0.21/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.51
% 0.21/0.51 Axiom 1 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.21/0.51 Axiom 2 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.21/0.51 Axiom 3 (transitivity_of_less_equal): fresh3(X, X, Y, Z) = true.
% 0.21/0.51 Axiom 4 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.21/0.51 Axiom 5 (property_of_divide1): fresh8(X, X, Y, Z, W) = true.
% 0.21/0.51 Axiom 6 (transitivity_of_less_equal): fresh4(X, X, Y, Z, W) = less_equal(Y, W).
% 0.21/0.51 Axiom 7 (one_inversion_equals_three): divide(identity, divide(identity, divide(identity, X))) = divide(identity, X).
% 0.21/0.51 Axiom 8 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.21/0.51 Axiom 9 (transitivity_of_less_equal): fresh4(less_equal(X, Y), true, Z, X, Y) = fresh3(less_equal(Z, X), true, Z, Y).
% 0.21/0.51 Axiom 10 (property_of_divide1): fresh8(less_equal(divide(X, Y), Z), true, X, Y, Z) = less_equal(divide(X, Z), Y).
% 0.21/0.51 Axiom 11 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.21/0.51
% 0.21/0.51 Lemma 12: less_equal(divide(divide(identity, X), divide(identity, Y)), divide(Y, divide(identity, divide(identity, X)))) = true.
% 0.21/0.51 Proof:
% 0.21/0.51 less_equal(divide(divide(identity, X), divide(identity, Y)), divide(Y, divide(identity, divide(identity, X))))
% 0.21/0.51 = { by axiom 10 (property_of_divide1) R->L }
% 0.21/0.51 fresh8(less_equal(divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(identity, Y)), true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 6 (transitivity_of_less_equal) R->L }
% 0.21/0.51 fresh8(fresh4(true, true, divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(divide(identity, Y), divide(identity, divide(identity, X))), divide(identity, Y)), true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 1 (quotient_smaller_than_numerator) R->L }
% 0.21/0.51 fresh8(fresh4(less_equal(divide(divide(identity, Y), divide(identity, divide(identity, X))), divide(identity, Y)), true, divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(divide(identity, Y), divide(identity, divide(identity, X))), divide(identity, Y)), true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 9 (transitivity_of_less_equal) }
% 0.21/0.51 fresh8(fresh3(less_equal(divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(divide(identity, Y), divide(identity, divide(identity, X)))), true, divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(identity, Y)), true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 7 (one_inversion_equals_three) R->L }
% 0.21/0.51 fresh8(fresh3(less_equal(divide(divide(identity, divide(identity, divide(identity, X))), divide(Y, divide(identity, divide(identity, X)))), divide(divide(identity, Y), divide(identity, divide(identity, X)))), true, divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(identity, Y)), true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 11 (quotient_property) }
% 0.21/0.51 fresh8(fresh3(true, true, divide(divide(identity, X), divide(Y, divide(identity, divide(identity, X)))), divide(identity, Y)), true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 3 (transitivity_of_less_equal) }
% 0.21/0.51 fresh8(true, true, divide(identity, X), divide(Y, divide(identity, divide(identity, X))), divide(identity, Y))
% 0.21/0.51 = { by axiom 5 (property_of_divide1) }
% 0.21/0.51 true
% 0.21/0.51
% 0.21/0.51 Goal 1 (prove_this): divide(divide(identity, a), divide(identity, divide(identity, b))) = divide(divide(identity, b), divide(identity, divide(identity, a))).
% 0.21/0.51 Proof:
% 0.21/0.51 divide(divide(identity, a), divide(identity, divide(identity, b)))
% 0.21/0.51 = { by axiom 2 (less_equal_and_equal) R->L }
% 0.21/0.51 fresh(true, true, divide(divide(identity, b), divide(identity, divide(identity, a))), divide(divide(identity, a), divide(identity, divide(identity, b))))
% 0.21/0.51 = { by lemma 12 R->L }
% 0.21/0.51 fresh(less_equal(divide(divide(identity, b), divide(identity, divide(identity, a))), divide(divide(identity, a), divide(identity, divide(identity, b)))), true, divide(divide(identity, b), divide(identity, divide(identity, a))), divide(divide(identity, a), divide(identity, divide(identity, b))))
% 0.21/0.51 = { by axiom 8 (less_equal_and_equal) R->L }
% 0.21/0.51 fresh2(less_equal(divide(divide(identity, a), divide(identity, divide(identity, b))), divide(divide(identity, b), divide(identity, divide(identity, a)))), true, divide(divide(identity, b), divide(identity, divide(identity, a))), divide(divide(identity, a), divide(identity, divide(identity, b))))
% 0.21/0.51 = { by lemma 12 }
% 0.21/0.51 fresh2(true, true, divide(divide(identity, b), divide(identity, divide(identity, a))), divide(divide(identity, a), divide(identity, divide(identity, b))))
% 0.21/0.51 = { by axiom 4 (less_equal_and_equal) }
% 0.21/0.51 divide(divide(identity, b), divide(identity, divide(identity, a)))
% 0.21/0.51 % SZS output end Proof
% 0.21/0.51
% 0.21/0.51 RESULT: Unsatisfiable (the axioms are contradictory).
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