TSTP Solution File: GRP097-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP097-1 : 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 : n026.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 01:16:58 EDT 2023
% Result : Unsatisfiable 0.21s 0.44s
% 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 : GRP097-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 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 : n026.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 : Mon Aug 28 20:50:50 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.21/0.44 Command-line arguments: --no-flatten-goal
% 0.21/0.44
% 0.21/0.44 % SZS status Unsatisfiable
% 0.21/0.44
% 0.21/0.46 % SZS output start Proof
% 0.21/0.46 Take the following subset of the input axioms:
% 0.21/0.46 fof(multiply, axiom, ![X, Y]: multiply(X, Y)=divide(X, inverse(Y))).
% 0.21/0.46 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)))).
% 0.21/0.46 fof(single_axiom, axiom, ![Z, X2, Y2]: divide(X2, inverse(divide(divide(Y2, Z), divide(X2, Z))))=Y2).
% 0.21/0.46
% 0.21/0.46 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.46 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.46 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.46 fresh(y, y, x1...xn) = u
% 0.21/0.46 C => fresh(s, t, x1...xn) = v
% 0.21/0.46 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.46 variables of u and v.
% 0.21/0.46 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.46 input problem has no model of domain size 1).
% 0.21/0.46
% 0.21/0.46 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.46
% 0.21/0.46 Axiom 1 (multiply): multiply(X, Y) = divide(X, inverse(Y)).
% 0.21/0.46 Axiom 2 (single_axiom): divide(X, inverse(divide(divide(Y, Z), divide(X, Z)))) = Y.
% 0.21/0.46
% 0.21/0.46 Lemma 3: multiply(X, divide(divide(Y, Z), divide(X, Z))) = Y.
% 0.21/0.46 Proof:
% 0.21/0.46 multiply(X, divide(divide(Y, Z), divide(X, Z)))
% 0.21/0.46 = { by axiom 1 (multiply) }
% 0.21/0.46 divide(X, inverse(divide(divide(Y, Z), divide(X, Z))))
% 0.21/0.46 = { by axiom 2 (single_axiom) }
% 0.21/0.46 Y
% 0.21/0.46
% 0.21/0.46 Lemma 4: multiply(X, divide(multiply(Y, Z), multiply(X, Z))) = Y.
% 0.21/0.46 Proof:
% 0.21/0.46 multiply(X, divide(multiply(Y, Z), multiply(X, Z)))
% 0.21/0.46 = { by axiom 1 (multiply) }
% 0.21/0.46 multiply(X, divide(multiply(Y, Z), divide(X, inverse(Z))))
% 0.21/0.46 = { by axiom 1 (multiply) }
% 0.21/0.46 multiply(X, divide(divide(Y, inverse(Z)), divide(X, inverse(Z))))
% 0.21/0.46 = { by lemma 3 }
% 0.21/0.46 Y
% 0.21/0.46
% 0.21/0.46 Lemma 5: multiply(X, divide(Y, Y)) = X.
% 0.21/0.46 Proof:
% 0.21/0.46 multiply(X, divide(Y, Y))
% 0.21/0.46 = { by lemma 3 R->L }
% 0.21/0.46 multiply(X, divide(Y, multiply(X, divide(divide(Y, Z), divide(X, Z)))))
% 0.21/0.46 = { by lemma 3 R->L }
% 0.21/0.46 multiply(X, divide(multiply(X, divide(divide(Y, Z), divide(X, Z))), multiply(X, divide(divide(Y, Z), divide(X, Z)))))
% 0.21/0.46 = { by lemma 4 }
% 0.21/0.46 X
% 0.21/0.46
% 0.21/0.46 Lemma 6: multiply(X, divide(Y, X)) = Y.
% 0.21/0.46 Proof:
% 0.21/0.46 multiply(X, divide(Y, X))
% 0.21/0.46 = { by lemma 5 R->L }
% 0.21/0.46 multiply(X, divide(Y, multiply(X, divide(Z, Z))))
% 0.21/0.46 = { by lemma 5 R->L }
% 0.21/0.46 multiply(X, divide(multiply(Y, divide(Z, Z)), multiply(X, divide(Z, Z))))
% 0.21/0.46 = { by lemma 4 }
% 0.21/0.46 Y
% 0.21/0.46
% 0.21/0.46 Lemma 7: multiply(inverse(X), multiply(Y, X)) = Y.
% 0.21/0.46 Proof:
% 0.21/0.46 multiply(inverse(X), multiply(Y, X))
% 0.21/0.46 = { by axiom 1 (multiply) }
% 0.21/0.46 multiply(inverse(X), divide(Y, inverse(X)))
% 0.21/0.46 = { by lemma 6 }
% 0.21/0.47 Y
% 0.21/0.47
% 0.21/0.47 Lemma 8: multiply(inverse(divide(X, X)), Y) = Y.
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(inverse(divide(X, X)), Y)
% 0.21/0.47 = { by lemma 5 R->L }
% 0.21/0.47 multiply(inverse(divide(X, X)), multiply(Y, divide(X, X)))
% 0.21/0.47 = { by lemma 7 }
% 0.21/0.47 Y
% 0.21/0.47
% 0.21/0.47 Lemma 9: divide(multiply(X, Y), Y) = X.
% 0.21/0.47 Proof:
% 0.21/0.47 divide(multiply(X, Y), Y)
% 0.21/0.47 = { by lemma 8 R->L }
% 0.21/0.47 divide(multiply(X, Y), multiply(inverse(divide(Z, Z)), Y))
% 0.21/0.47 = { by lemma 8 R->L }
% 0.21/0.47 multiply(inverse(divide(Z, Z)), divide(multiply(X, Y), multiply(inverse(divide(Z, Z)), Y)))
% 0.21/0.47 = { by lemma 4 }
% 0.21/0.47 X
% 0.21/0.47
% 0.21/0.47 Lemma 10: multiply(X, Y) = multiply(Y, X).
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(X, Y)
% 0.21/0.47 = { by lemma 6 R->L }
% 0.21/0.47 multiply(Y, divide(multiply(X, Y), Y))
% 0.21/0.47 = { by lemma 9 }
% 0.21/0.47 multiply(Y, X)
% 0.21/0.47
% 0.21/0.47 Lemma 11: inverse(divide(X, X)) = divide(Y, Y).
% 0.21/0.47 Proof:
% 0.21/0.47 inverse(divide(X, X))
% 0.21/0.47 = { by lemma 5 R->L }
% 0.21/0.47 multiply(inverse(divide(X, X)), divide(Y, Y))
% 0.21/0.47 = { by lemma 8 }
% 0.21/0.47 divide(Y, Y)
% 0.21/0.47
% 0.21/0.47 Lemma 12: multiply(X, inverse(multiply(X, Y))) = inverse(Y).
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(X, inverse(multiply(X, Y)))
% 0.21/0.47 = { by lemma 10 R->L }
% 0.21/0.47 multiply(inverse(multiply(X, Y)), X)
% 0.21/0.47 = { by lemma 7 R->L }
% 0.21/0.47 multiply(inverse(multiply(X, Y)), multiply(inverse(Y), multiply(X, Y)))
% 0.21/0.47 = { by lemma 7 }
% 0.21/0.47 inverse(Y)
% 0.21/0.47
% 0.21/0.47 Lemma 13: multiply(X, inverse(Y)) = inverse(divide(Y, X)).
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(X, inverse(Y))
% 0.21/0.47 = { by lemma 6 R->L }
% 0.21/0.47 multiply(X, inverse(multiply(X, divide(Y, X))))
% 0.21/0.47 = { by lemma 12 }
% 0.21/0.47 inverse(divide(Y, X))
% 0.21/0.47
% 0.21/0.47 Lemma 14: multiply(X, inverse(divide(X, Y))) = Y.
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(X, inverse(divide(X, Y)))
% 0.21/0.47 = { by lemma 10 R->L }
% 0.21/0.47 multiply(inverse(divide(X, Y)), X)
% 0.21/0.47 = { by lemma 6 R->L }
% 0.21/0.47 multiply(inverse(divide(X, Y)), multiply(Y, divide(X, Y)))
% 0.21/0.47 = { by lemma 7 }
% 0.21/0.47 Y
% 0.21/0.47
% 0.21/0.47 Lemma 15: inverse(divide(X, Y)) = divide(Y, X).
% 0.21/0.47 Proof:
% 0.21/0.47 inverse(divide(X, Y))
% 0.21/0.47 = { by lemma 13 R->L }
% 0.21/0.47 multiply(Y, inverse(X))
% 0.21/0.47 = { by lemma 9 R->L }
% 0.21/0.47 multiply(Y, inverse(divide(multiply(X, divide(Y, X)), divide(Y, X))))
% 0.21/0.47 = { by lemma 6 }
% 0.21/0.47 multiply(Y, inverse(divide(Y, divide(Y, X))))
% 0.21/0.47 = { by lemma 14 }
% 0.21/0.47 divide(Y, X)
% 0.21/0.47
% 0.21/0.47 Lemma 16: multiply(inverse(X), X) = divide(Y, Y).
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(inverse(X), X)
% 0.21/0.47 = { by lemma 8 R->L }
% 0.21/0.47 multiply(inverse(divide(Z, Z)), multiply(inverse(X), X))
% 0.21/0.47 = { by axiom 1 (multiply) }
% 0.21/0.47 multiply(inverse(divide(Z, Z)), divide(inverse(X), inverse(X)))
% 0.21/0.47 = { by lemma 5 }
% 0.21/0.47 inverse(divide(Z, Z))
% 0.21/0.47 = { by lemma 11 }
% 0.21/0.47 divide(Y, Y)
% 0.21/0.47
% 0.21/0.47 Lemma 17: divide(X, divide(inverse(Y), Z)) = multiply(X, multiply(Y, Z)).
% 0.21/0.47 Proof:
% 0.21/0.47 divide(X, divide(inverse(Y), Z))
% 0.21/0.47 = { by lemma 15 R->L }
% 0.21/0.47 divide(X, inverse(divide(Z, inverse(Y))))
% 0.21/0.47 = { by lemma 13 R->L }
% 0.21/0.47 divide(X, multiply(inverse(Y), inverse(Z)))
% 0.21/0.47 = { by lemma 7 R->L }
% 0.21/0.47 divide(X, multiply(inverse(Y), inverse(multiply(inverse(Y), multiply(Z, Y)))))
% 0.21/0.47 = { by lemma 12 }
% 0.21/0.47 divide(X, inverse(multiply(Z, Y)))
% 0.21/0.47 = { by lemma 10 }
% 0.21/0.47 divide(X, inverse(multiply(Y, Z)))
% 0.21/0.47 = { by axiom 1 (multiply) R->L }
% 0.21/0.47 multiply(X, multiply(Y, Z))
% 0.21/0.47
% 0.21/0.47 Lemma 18: multiply(X, divide(Y, Z)) = divide(X, divide(Z, Y)).
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(X, divide(Y, Z))
% 0.21/0.47 = { by lemma 15 R->L }
% 0.21/0.47 multiply(X, inverse(divide(Z, Y)))
% 0.21/0.47 = { by lemma 13 R->L }
% 0.21/0.47 multiply(X, multiply(Y, inverse(Z)))
% 0.21/0.47 = { by lemma 10 }
% 0.21/0.47 multiply(X, multiply(inverse(Z), Y))
% 0.21/0.47 = { by lemma 17 R->L }
% 0.21/0.47 divide(X, divide(inverse(inverse(Z)), Y))
% 0.21/0.47 = { by lemma 5 R->L }
% 0.21/0.47 divide(X, divide(multiply(inverse(inverse(Z)), divide(W, W)), Y))
% 0.21/0.47 = { by lemma 11 R->L }
% 0.21/0.47 divide(X, divide(multiply(inverse(inverse(Z)), inverse(divide(V, V))), Y))
% 0.21/0.47 = { by lemma 12 R->L }
% 0.21/0.47 divide(X, divide(multiply(inverse(inverse(Z)), multiply(Z, inverse(multiply(Z, divide(V, V))))), Y))
% 0.21/0.47 = { by lemma 5 }
% 0.21/0.47 divide(X, divide(multiply(inverse(inverse(Z)), multiply(Z, inverse(Z))), Y))
% 0.21/0.47 = { by lemma 7 }
% 0.21/0.47 divide(X, divide(Z, Y))
% 0.21/0.47
% 0.21/0.47 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4)) = tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4)).
% 0.21/0.47 Proof:
% 0.21/0.47 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.21/0.47 = { by lemma 16 }
% 0.21/0.47 tuple(divide(X, X), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.21/0.47 = { by lemma 16 }
% 0.21/0.47 tuple(divide(X, X), multiply(divide(Y, Y), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.21/0.48 = { by lemma 10 }
% 0.21/0.48 tuple(divide(X, X), multiply(a2, divide(Y, Y)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.21/0.48 = { by lemma 5 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.21/0.48 = { by lemma 10 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 0.21/0.48 = { by lemma 10 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(b3, a3)), multiply(a4, b4))
% 0.21/0.48 = { by lemma 17 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, divide(c3, divide(inverse(b3), a3)), multiply(a4, b4))
% 0.21/0.48 = { by lemma 18 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, divide(a3, inverse(b3))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 15 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, inverse(divide(inverse(b3), a3))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 13 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(a3, inverse(inverse(b3)))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 4 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(a3, inverse(multiply(a3, divide(multiply(inverse(b3), divide(c3, inverse(b3))), multiply(a3, divide(c3, inverse(b3)))))))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 6 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(a3, inverse(multiply(a3, divide(c3, multiply(a3, divide(c3, inverse(b3)))))))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 18 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(a3, inverse(divide(a3, divide(multiply(a3, divide(c3, inverse(b3))), c3))))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 18 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, multiply(a3, inverse(divide(a3, divide(divide(a3, divide(inverse(b3), c3)), c3))))), multiply(a4, b4))
% 0.21/0.48 = { by lemma 14 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(c3, divide(divide(a3, divide(inverse(b3), c3)), c3)), multiply(a4, b4))
% 0.21/0.48 = { by lemma 6 }
% 0.21/0.48 tuple(divide(X, X), a2, divide(a3, divide(inverse(b3), c3)), multiply(a4, b4))
% 0.21/0.48 = { by lemma 17 }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.21/0.48 = { by lemma 10 R->L }
% 0.21/0.48 tuple(divide(X, X), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.21/0.48 = { by lemma 16 R->L }
% 0.21/0.48 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.21/0.48 % SZS output end Proof
% 0.21/0.48
% 0.21/0.48 RESULT: Unsatisfiable (the axioms are contradictory).
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