TSTP Solution File: GRP089-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP089-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 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 : n009.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 01:16:56 EDT 2023
% Result : Unsatisfiable 0.13s 0.40s
% 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
% 0.10/0.12 % Problem : GRP089-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.10/0.13 % 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 : n009.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 : Mon Aug 28 22:13:20 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.13/0.40 Command-line arguments: --no-flatten-goal
% 0.13/0.40
% 0.13/0.40 % SZS status Unsatisfiable
% 0.13/0.40
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 Take the following subset of the input axioms:
% 0.20/0.41 fof(inverse, axiom, ![X, Z]: inverse(X)=divide(divide(Z, Z), X)).
% 0.20/0.41 fof(multiply, axiom, ![Y, X2, Z2]: multiply(X2, Y)=divide(X2, divide(divide(Z2, Z2), Y))).
% 0.20/0.41 fof(prove_these_axioms, negated_conjecture, multiply(inverse(a1), a1)!=multiply(inverse(b1), b1) | (multiply(multiply(inverse(b2), b2), a2)!=a2 | (multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3)) | multiply(a4, b4)!=multiply(b4, a4)))).
% 0.20/0.41 fof(single_axiom, axiom, ![X2, Y2, Z2]: divide(X2, divide(divide(X2, Y2), divide(Z2, Y2)))=Z2).
% 0.20/0.41
% 0.20/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.41 fresh(y, y, x1...xn) = u
% 0.20/0.41 C => fresh(s, t, x1...xn) = v
% 0.20/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.41 variables of u and v.
% 0.20/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.41 input problem has no model of domain size 1).
% 0.20/0.41
% 0.20/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.41
% 0.20/0.41 Axiom 1 (inverse): inverse(X) = divide(divide(Y, Y), X).
% 0.20/0.41 Axiom 2 (multiply): multiply(X, Y) = divide(X, divide(divide(Z, Z), Y)).
% 0.20/0.41 Axiom 3 (single_axiom): divide(X, divide(divide(X, Y), divide(Z, Y))) = Z.
% 0.20/0.41
% 0.20/0.41 Lemma 4: divide(X, inverse(Y)) = multiply(X, Y).
% 0.20/0.41 Proof:
% 0.20/0.41 divide(X, inverse(Y))
% 0.20/0.41 = { by axiom 1 (inverse) }
% 0.20/0.41 divide(X, divide(divide(Z, Z), Y))
% 0.20/0.41 = { by axiom 2 (multiply) R->L }
% 0.20/0.41 multiply(X, Y)
% 0.20/0.41
% 0.20/0.41 Lemma 5: multiply(X, divide(Y, X)) = Y.
% 0.20/0.41 Proof:
% 0.20/0.41 multiply(X, divide(Y, X))
% 0.20/0.41 = { by lemma 4 R->L }
% 0.20/0.41 divide(X, inverse(divide(Y, X)))
% 0.20/0.41 = { by axiom 1 (inverse) }
% 0.20/0.41 divide(X, divide(divide(X, X), divide(Y, X)))
% 0.20/0.41 = { by axiom 3 (single_axiom) }
% 0.20/0.41 Y
% 0.20/0.41
% 0.20/0.41 Lemma 6: divide(Y, Y) = divide(X, X).
% 0.20/0.41 Proof:
% 0.20/0.41 divide(Y, Y)
% 0.20/0.41 = { by lemma 5 R->L }
% 0.20/0.41 multiply(Z, divide(divide(Y, Y), Z))
% 0.20/0.41 = { by axiom 1 (inverse) R->L }
% 0.20/0.41 multiply(Z, inverse(Z))
% 0.20/0.41 = { by axiom 1 (inverse) }
% 0.20/0.41 multiply(Z, divide(divide(X, X), Z))
% 0.20/0.41 = { by lemma 5 }
% 0.20/0.41 divide(X, X)
% 0.20/0.41
% 0.20/0.41 Lemma 7: divide(X, divide(Y, Y)) = X.
% 0.20/0.41 Proof:
% 0.20/0.41 divide(X, divide(Y, Y))
% 0.20/0.41 = { by lemma 6 }
% 0.20/0.41 divide(X, divide(divide(X, Z), divide(X, Z)))
% 0.20/0.41 = { by axiom 3 (single_axiom) }
% 0.20/0.41 X
% 0.20/0.41
% 0.20/0.41 Lemma 8: multiply(divide(X, X), Y) = inverse(inverse(Y)).
% 0.20/0.41 Proof:
% 0.20/0.41 multiply(divide(X, X), Y)
% 0.20/0.41 = { by lemma 4 R->L }
% 0.20/0.41 divide(divide(X, X), inverse(Y))
% 0.20/0.41 = { by axiom 1 (inverse) R->L }
% 0.20/0.41 inverse(inverse(Y))
% 0.20/0.41
% 0.20/0.41 Lemma 9: inverse(inverse(X)) = X.
% 0.20/0.41 Proof:
% 0.20/0.41 inverse(inverse(X))
% 0.20/0.41 = { by lemma 8 R->L }
% 0.20/0.41 multiply(divide(Y, Y), X)
% 0.20/0.41 = { by lemma 7 R->L }
% 0.20/0.41 multiply(divide(Y, Y), divide(X, divide(Y, Y)))
% 0.20/0.41 = { by lemma 5 }
% 0.20/0.41 X
% 0.20/0.42
% 0.20/0.42 Lemma 10: inverse(divide(inverse(X), divide(Y, X))) = Y.
% 0.20/0.42 Proof:
% 0.20/0.42 inverse(divide(inverse(X), divide(Y, X)))
% 0.20/0.42 = { by axiom 1 (inverse) }
% 0.20/0.42 inverse(divide(divide(divide(Z, Z), X), divide(Y, X)))
% 0.20/0.42 = { by axiom 1 (inverse) }
% 0.20/0.42 divide(divide(Z, Z), divide(divide(divide(Z, Z), X), divide(Y, X)))
% 0.20/0.42 = { by axiom 3 (single_axiom) }
% 0.20/0.42 Y
% 0.20/0.42
% 0.20/0.42 Lemma 11: divide(inverse(divide(X, Y)), Y) = inverse(X).
% 0.20/0.42 Proof:
% 0.20/0.42 divide(inverse(divide(X, Y)), Y)
% 0.20/0.42 = { by lemma 10 R->L }
% 0.20/0.42 inverse(divide(inverse(divide(X, Y)), divide(divide(inverse(divide(X, Y)), Y), divide(X, Y))))
% 0.20/0.42 = { by axiom 3 (single_axiom) }
% 0.20/0.42 inverse(X)
% 0.20/0.42
% 0.20/0.42 Lemma 12: divide(X, multiply(X, Y)) = inverse(Y).
% 0.20/0.42 Proof:
% 0.20/0.42 divide(X, multiply(X, Y))
% 0.20/0.42 = { by lemma 10 R->L }
% 0.20/0.42 divide(inverse(divide(inverse(inverse(Y)), divide(X, inverse(Y)))), multiply(X, Y))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 divide(inverse(divide(inverse(inverse(Y)), multiply(X, Y))), multiply(X, Y))
% 0.20/0.42 = { by lemma 9 }
% 0.20/0.42 divide(inverse(divide(Y, multiply(X, Y))), multiply(X, Y))
% 0.20/0.42 = { by lemma 11 }
% 0.20/0.42 inverse(Y)
% 0.20/0.42
% 0.20/0.42 Lemma 13: divide(inverse(X), Y) = inverse(multiply(Y, X)).
% 0.20/0.42 Proof:
% 0.20/0.42 divide(inverse(X), Y)
% 0.20/0.42 = { by lemma 5 R->L }
% 0.20/0.42 divide(inverse(X), multiply(inverse(X), divide(Y, inverse(X))))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 divide(inverse(X), multiply(inverse(X), multiply(Y, X)))
% 0.20/0.42 = { by lemma 12 }
% 0.20/0.42 inverse(multiply(Y, X))
% 0.20/0.42
% 0.20/0.42 Lemma 14: multiply(X, Y) = multiply(Y, X).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, Y)
% 0.20/0.42 = { by axiom 3 (single_axiom) R->L }
% 0.20/0.42 divide(Y, divide(divide(Y, divide(Z, Z)), divide(multiply(X, Y), divide(Z, Z))))
% 0.20/0.42 = { by lemma 7 }
% 0.20/0.42 divide(Y, divide(Y, divide(multiply(X, Y), divide(Z, Z))))
% 0.20/0.42 = { by lemma 7 }
% 0.20/0.42 divide(Y, divide(Y, multiply(X, Y)))
% 0.20/0.42 = { by lemma 9 R->L }
% 0.20/0.42 divide(Y, inverse(inverse(divide(Y, multiply(X, Y)))))
% 0.20/0.42 = { by lemma 5 R->L }
% 0.20/0.42 divide(Y, inverse(multiply(multiply(X, Y), divide(inverse(divide(Y, multiply(X, Y))), multiply(X, Y)))))
% 0.20/0.42 = { by lemma 11 }
% 0.20/0.42 divide(Y, inverse(multiply(multiply(X, Y), inverse(Y))))
% 0.20/0.42 = { by lemma 13 R->L }
% 0.20/0.42 divide(Y, divide(inverse(inverse(Y)), multiply(X, Y)))
% 0.20/0.42 = { by lemma 12 R->L }
% 0.20/0.42 divide(Y, divide(inverse(divide(X, multiply(X, Y))), multiply(X, Y)))
% 0.20/0.42 = { by lemma 11 }
% 0.20/0.42 divide(Y, inverse(X))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 multiply(Y, X)
% 0.20/0.42
% 0.20/0.42 Lemma 15: multiply(inverse(X), X) = divide(Y, Y).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(inverse(X), X)
% 0.20/0.42 = { by lemma 4 R->L }
% 0.20/0.42 divide(inverse(X), inverse(X))
% 0.20/0.42 = { by lemma 6 R->L }
% 0.20/0.42 divide(Y, Y)
% 0.20/0.42
% 0.20/0.42 Goal 1 (prove_these_axioms): tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4)) = tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1), multiply(b4, a4)).
% 0.20/0.42 Proof:
% 0.20/0.42 tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.42 = { by lemma 15 }
% 0.20/0.42 tuple(multiply(divide(X, X), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.42 = { by lemma 8 }
% 0.20/0.42 tuple(inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.42 = { by lemma 15 }
% 0.20/0.42 tuple(inverse(inverse(a2)), multiply(multiply(a3, b3), c3), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 9 }
% 0.20/0.42 tuple(a2, multiply(multiply(a3, b3), c3), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 14 }
% 0.20/0.42 tuple(a2, multiply(c3, multiply(a3, b3)), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 4 R->L }
% 0.20/0.42 tuple(a2, divide(c3, inverse(multiply(a3, b3))), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 13 R->L }
% 0.20/0.42 tuple(a2, divide(c3, divide(inverse(b3), a3)), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 5 R->L }
% 0.20/0.42 tuple(a2, multiply(a3, divide(divide(c3, divide(inverse(b3), a3)), a3)), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by axiom 3 (single_axiom) R->L }
% 0.20/0.42 tuple(a2, multiply(a3, divide(c3, divide(divide(c3, divide(inverse(b3), a3)), divide(divide(divide(c3, divide(inverse(b3), a3)), a3), divide(inverse(b3), a3))))), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by axiom 3 (single_axiom) }
% 0.20/0.42 tuple(a2, multiply(a3, divide(c3, inverse(b3))), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 tuple(a2, multiply(a3, multiply(c3, b3)), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 14 }
% 0.20/0.42 tuple(a2, multiply(a3, multiply(b3, c3)), divide(Y, Y), multiply(a4, b4))
% 0.20/0.42 = { by lemma 14 R->L }
% 0.20/0.42 tuple(a2, multiply(a3, multiply(b3, c3)), divide(Y, Y), multiply(b4, a4))
% 0.20/0.42 = { by lemma 15 R->L }
% 0.20/0.42 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1), multiply(b4, a4))
% 0.20/0.42 % SZS output end Proof
% 0.20/0.42
% 0.20/0.42 RESULT: Unsatisfiable (the axioms are contradictory).
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