TSTP Solution File: GRP109-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP109-1 : 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 : n013.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 01:17:01 EDT 2023
% Result : Unsatisfiable 0.20s 0.42s
% 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 : GRP109-1 : 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 : n013.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 23:42:47 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.42 Command-line arguments: --no-flatten-goal
% 0.20/0.42
% 0.20/0.42 % SZS status Unsatisfiable
% 0.20/0.42
% 0.20/0.43 % SZS output start Proof
% 0.20/0.43 Take the following subset of the input axioms:
% 0.20/0.43 fof(multiply, axiom, ![X, Y]: multiply(X, Y)=inverse(double_divide(Y, X))).
% 0.20/0.43 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)))).
% 0.20/0.43 fof(single_axiom, axiom, ![Z, X2, Y2]: double_divide(inverse(double_divide(X2, inverse(double_divide(inverse(Y2), double_divide(X2, Z))))), Z)=Y2).
% 0.20/0.43
% 0.20/0.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.43 fresh(y, y, x1...xn) = u
% 0.20/0.43 C => fresh(s, t, x1...xn) = v
% 0.20/0.43 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.43 variables of u and v.
% 0.20/0.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.43 input problem has no model of domain size 1).
% 0.20/0.43
% 0.20/0.43 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.43
% 0.20/0.43 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 0.20/0.43 Axiom 2 (single_axiom): double_divide(inverse(double_divide(X, inverse(double_divide(inverse(Y), double_divide(X, Z))))), Z) = Y.
% 0.20/0.43
% 0.20/0.43 Lemma 3: double_divide(multiply(multiply(double_divide(X, Y), inverse(Z)), X), Y) = Z.
% 0.20/0.43 Proof:
% 0.20/0.43 double_divide(multiply(multiply(double_divide(X, Y), inverse(Z)), X), Y)
% 0.20/0.43 = { by axiom 1 (multiply) }
% 0.20/0.43 double_divide(multiply(inverse(double_divide(inverse(Z), double_divide(X, Y))), X), Y)
% 0.20/0.43 = { by axiom 1 (multiply) }
% 0.20/0.43 double_divide(inverse(double_divide(X, inverse(double_divide(inverse(Z), double_divide(X, Y))))), Y)
% 0.20/0.43 = { by axiom 2 (single_axiom) }
% 0.20/0.43 Z
% 0.20/0.43
% 0.20/0.43 Lemma 4: multiply(X, multiply(multiply(double_divide(Y, X), inverse(Z)), Y)) = inverse(Z).
% 0.20/0.43 Proof:
% 0.20/0.43 multiply(X, multiply(multiply(double_divide(Y, X), inverse(Z)), Y))
% 0.20/0.43 = { by axiom 1 (multiply) }
% 0.20/0.43 inverse(double_divide(multiply(multiply(double_divide(Y, X), inverse(Z)), Y), X))
% 0.20/0.43 = { by lemma 3 }
% 0.20/0.43 inverse(Z)
% 0.20/0.43
% 0.20/0.43 Lemma 5: double_divide(inverse(X), multiply(X, inverse(Y))) = Y.
% 0.20/0.43 Proof:
% 0.20/0.43 double_divide(inverse(X), multiply(X, inverse(Y)))
% 0.20/0.43 = { by lemma 4 R->L }
% 0.20/0.43 double_divide(multiply(multiply(X, inverse(Y)), multiply(multiply(double_divide(Z, multiply(X, inverse(Y))), inverse(X)), Z)), multiply(X, inverse(Y)))
% 0.20/0.43 = { by lemma 3 R->L }
% 0.20/0.43 double_divide(multiply(multiply(double_divide(multiply(multiply(double_divide(Z, multiply(X, inverse(Y))), inverse(X)), Z), multiply(X, inverse(Y))), inverse(Y)), multiply(multiply(double_divide(Z, multiply(X, inverse(Y))), inverse(X)), Z)), multiply(X, inverse(Y)))
% 0.20/0.43 = { by lemma 3 }
% 0.20/0.43 Y
% 0.20/0.43
% 0.20/0.43 Lemma 6: multiply(multiply(X, inverse(Y)), inverse(X)) = inverse(Y).
% 0.20/0.43 Proof:
% 0.20/0.43 multiply(multiply(X, inverse(Y)), inverse(X))
% 0.20/0.43 = { by axiom 1 (multiply) }
% 0.20/0.43 inverse(double_divide(inverse(X), multiply(X, inverse(Y))))
% 0.20/0.43 = { by lemma 5 }
% 0.20/0.43 inverse(Y)
% 0.20/0.43
% 0.20/0.43 Lemma 7: double_divide(inverse(multiply(X, inverse(Y))), inverse(Y)) = X.
% 0.20/0.43 Proof:
% 0.20/0.43 double_divide(inverse(multiply(X, inverse(Y))), inverse(Y))
% 0.20/0.43 = { by lemma 6 R->L }
% 0.20/0.43 double_divide(inverse(multiply(X, inverse(Y))), multiply(multiply(X, inverse(Y)), inverse(X)))
% 0.20/0.44 = { by lemma 5 }
% 0.20/0.44 X
% 0.20/0.44
% 0.20/0.44 Lemma 8: multiply(multiply(X, multiply(Y, Z)), inverse(X)) = multiply(Y, Z).
% 0.20/0.44 Proof:
% 0.20/0.44 multiply(multiply(X, multiply(Y, Z)), inverse(X))
% 0.20/0.44 = { by axiom 1 (multiply) }
% 0.20/0.44 multiply(multiply(X, inverse(double_divide(Z, Y))), inverse(X))
% 0.20/0.44 = { by lemma 6 }
% 0.20/0.44 inverse(double_divide(Z, Y))
% 0.20/0.44 = { by axiom 1 (multiply) R->L }
% 0.20/0.44 multiply(Y, Z)
% 0.20/0.44
% 0.20/0.44 Lemma 9: multiply(X, multiply(Y, inverse(X))) = Y.
% 0.20/0.44 Proof:
% 0.20/0.44 multiply(X, multiply(Y, inverse(X)))
% 0.20/0.44 = { by lemma 7 R->L }
% 0.20/0.44 double_divide(inverse(multiply(multiply(X, multiply(Y, inverse(X))), inverse(X))), inverse(X))
% 0.20/0.44 = { by lemma 8 }
% 0.20/0.44 double_divide(inverse(multiply(Y, inverse(X))), inverse(X))
% 0.20/0.44 = { by lemma 7 }
% 0.20/0.44 Y
% 0.20/0.44
% 0.20/0.44 Lemma 10: multiply(Y, multiply(X, Z)) = multiply(X, multiply(Y, Z)).
% 0.20/0.44 Proof:
% 0.20/0.44 multiply(Y, multiply(X, Z))
% 0.20/0.44 = { by lemma 9 R->L }
% 0.20/0.44 multiply(Y, multiply(multiply(double_divide(Z, Y), multiply(X, inverse(double_divide(Z, Y)))), Z))
% 0.20/0.44 = { by axiom 1 (multiply) }
% 0.20/0.44 multiply(Y, multiply(multiply(double_divide(Z, Y), inverse(double_divide(inverse(double_divide(Z, Y)), X))), Z))
% 0.20/0.44 = { by lemma 4 }
% 0.20/0.44 inverse(double_divide(inverse(double_divide(Z, Y)), X))
% 0.20/0.44 = { by axiom 1 (multiply) R->L }
% 0.20/0.44 multiply(X, inverse(double_divide(Z, Y)))
% 0.20/0.44 = { by axiom 1 (multiply) R->L }
% 0.20/0.44 multiply(X, multiply(Y, Z))
% 0.20/0.44
% 0.20/0.44 Lemma 11: multiply(X, multiply(Y, inverse(Y))) = X.
% 0.20/0.44 Proof:
% 0.20/0.44 multiply(X, multiply(Y, inverse(Y)))
% 0.20/0.44 = { by lemma 10 }
% 0.20/0.44 multiply(Y, multiply(X, inverse(Y)))
% 0.20/0.44 = { by lemma 9 }
% 0.20/0.44 X
% 0.20/0.44
% 0.20/0.44 Lemma 12: multiply(Y, X) = multiply(X, Y).
% 0.20/0.44 Proof:
% 0.20/0.44 multiply(Y, X)
% 0.20/0.44 = { by lemma 11 R->L }
% 0.20/0.44 multiply(Y, multiply(X, multiply(Z, inverse(Z))))
% 0.20/0.44 = { by lemma 10 R->L }
% 0.20/0.44 multiply(X, multiply(Y, multiply(Z, inverse(Z))))
% 0.20/0.44 = { by lemma 11 }
% 0.20/0.44 multiply(X, Y)
% 0.20/0.44
% 0.20/0.44 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)).
% 0.20/0.44 Proof:
% 0.20/0.44 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.20/0.44 = { by lemma 12 R->L }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.20/0.44 = { by lemma 12 R->L }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), multiply(a2, multiply(inverse(b2), b2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.20/0.44 = { by lemma 12 R->L }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), multiply(a2, multiply(b2, inverse(b2))), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.20/0.44 = { by lemma 11 }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), a2, multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.20/0.44 = { by lemma 12 R->L }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), a2, multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 0.20/0.44 = { by lemma 10 R->L }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), a2, multiply(a3, multiply(c3, b3)), multiply(a4, b4))
% 0.20/0.44 = { by lemma 12 R->L }
% 0.20/0.44 tuple(multiply(a1, inverse(a1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.20/0.44 = { by lemma 11 R->L }
% 0.20/0.44 tuple(multiply(multiply(a1, multiply(b1, inverse(b1))), inverse(a1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.20/0.44 = { by lemma 8 }
% 0.20/0.44 tuple(multiply(b1, inverse(b1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.20/0.44 = { by lemma 12 }
% 0.20/0.44 tuple(multiply(b1, inverse(b1)), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.20/0.44 = { by lemma 12 }
% 0.20/0.44 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.20/0.44 % SZS output end Proof
% 0.20/0.44
% 0.20/0.44 RESULT: Unsatisfiable (the axioms are contradictory).
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