TSTP Solution File: GRP086-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP086-1 : 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 : n025.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 01:16:55 EDT 2023
% Result : Unsatisfiable 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : GRP086-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n025.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 01:16:55 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.20/0.39 Command-line arguments: --no-flatten-goal
% 0.20/0.39
% 0.20/0.39 % SZS status Unsatisfiable
% 0.20/0.39
% 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(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, ![X, Y, Z]: multiply(X, multiply(multiply(Y, Z), inverse(multiply(X, Z))))=Y).
% 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
% 0.20/0.42 variables of u and v.
% 0.20/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.42 input problem has no model of domain size 1).
% 0.20/0.42
% 0.20/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.42
% 0.20/0.42 Axiom 1 (single_axiom): multiply(X, multiply(multiply(Y, Z), inverse(multiply(X, Z)))) = Y.
% 0.20/0.42
% 0.20/0.42 Lemma 2: multiply(X, multiply(Y, inverse(multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z))))))) = W.
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z)))))))
% 0.20/0.42 = { by axiom 1 (single_axiom) R->L }
% 0.20/0.42 multiply(X, multiply(multiply(W, multiply(multiply(Y, Z), inverse(multiply(W, Z)))), inverse(multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z)))))))
% 0.20/0.42 = { by axiom 1 (single_axiom) }
% 0.20/0.42 W
% 0.20/0.42
% 0.20/0.42 Lemma 3: multiply(X, multiply(Y, inverse(Y))) = X.
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(Y, inverse(Y)))
% 0.20/0.42 = { by axiom 1 (single_axiom) R->L }
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(X, multiply(multiply(Y, Z), inverse(multiply(X, Z)))))))
% 0.20/0.42 = { by lemma 2 }
% 0.20/0.42 X
% 0.20/0.42
% 0.20/0.42 Lemma 4: multiply(X, multiply(Y, inverse(X))) = Y.
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(Y, inverse(X)))
% 0.20/0.42 = { by lemma 3 R->L }
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(X, multiply(Z, inverse(Z))))))
% 0.20/0.42 = { by lemma 3 R->L }
% 0.20/0.42 multiply(X, multiply(multiply(Y, multiply(Z, inverse(Z))), inverse(multiply(X, multiply(Z, inverse(Z))))))
% 0.20/0.42 = { by axiom 1 (single_axiom) }
% 0.20/0.42 Y
% 0.20/0.42
% 0.20/0.42 Lemma 5: multiply(multiply(X, Y), multiply(Z, inverse(X))) = multiply(Z, Y).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(multiply(X, Y), multiply(Z, inverse(X)))
% 0.20/0.42 = { by lemma 2 R->L }
% 0.20/0.42 multiply(multiply(X, Y), multiply(Z, inverse(multiply(multiply(X, Y), multiply(multiply(Z, Y), inverse(multiply(multiply(X, Y), multiply(multiply(multiply(Z, Y), Y), inverse(multiply(X, Y))))))))))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 multiply(multiply(X, Y), multiply(Z, inverse(multiply(multiply(X, Y), multiply(multiply(Z, Y), inverse(multiply(multiply(Z, Y), Y)))))))
% 0.20/0.42 = { by lemma 2 }
% 0.20/0.42 multiply(Z, Y)
% 0.20/0.42
% 0.20/0.42 Lemma 6: multiply(X, multiply(Y, inverse(Z))) = multiply(Y, multiply(X, inverse(Z))).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(Y, inverse(Z)))
% 0.20/0.42 = { by lemma 5 R->L }
% 0.20/0.42 multiply(multiply(Z, multiply(Y, inverse(Z))), multiply(X, inverse(Z)))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 multiply(Y, multiply(X, inverse(Z)))
% 0.20/0.42
% 0.20/0.42 Lemma 7: multiply(X, multiply(multiply(Y, Z), inverse(Y))) = multiply(X, Z).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(multiply(Y, Z), inverse(Y)))
% 0.20/0.42 = { by lemma 6 R->L }
% 0.20/0.42 multiply(multiply(Y, Z), multiply(X, inverse(Y)))
% 0.20/0.42 = { by lemma 5 }
% 0.20/0.42 multiply(X, Z)
% 0.20/0.42
% 0.20/0.42 Lemma 8: multiply(multiply(X, Y), Z) = multiply(multiply(X, Z), Y).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(multiply(X, Y), Z)
% 0.20/0.42 = { by lemma 5 R->L }
% 0.20/0.42 multiply(multiply(X, Z), multiply(multiply(X, Y), inverse(X)))
% 0.20/0.42 = { by lemma 7 }
% 0.20/0.42 multiply(multiply(X, Z), Y)
% 0.20/0.42
% 0.20/0.42 Lemma 9: multiply(X, multiply(multiply(Y, inverse(X)), Z)) = multiply(Y, Z).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(multiply(Y, inverse(X)), Z))
% 0.20/0.42 = { by lemma 8 R->L }
% 0.20/0.42 multiply(X, multiply(multiply(Y, Z), inverse(X)))
% 0.20/0.42 = { by lemma 4 }
% 0.20/0.42 multiply(Y, Z)
% 0.20/0.42
% 0.20/0.42 Lemma 10: multiply(X, inverse(multiply(Y, inverse(Y)))) = X.
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, inverse(multiply(Y, inverse(Y))))
% 0.20/0.42 = { by lemma 9 R->L }
% 0.20/0.42 multiply(Y, multiply(multiply(X, inverse(Y)), inverse(multiply(Y, inverse(Y)))))
% 0.20/0.42 = { by axiom 1 (single_axiom) }
% 0.20/0.42 X
% 0.20/0.42
% 0.20/0.42 Lemma 11: multiply(Y, X) = multiply(X, Y).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(Y, X)
% 0.20/0.42 = { by lemma 10 R->L }
% 0.20/0.42 multiply(Y, multiply(X, inverse(multiply(Z, inverse(Z)))))
% 0.20/0.42 = { by lemma 6 }
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(Z, inverse(Z)))))
% 0.20/0.42 = { by lemma 10 }
% 0.20/0.42 multiply(X, Y)
% 0.20/0.42
% 0.20/0.42 Lemma 12: multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z)))) = multiply(X, multiply(Y, inverse(W))).
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z))))
% 0.20/0.42 = { by lemma 2 R->L }
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(X, multiply(multiply(Y, multiply(V, inverse(V))), inverse(multiply(multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z)))), multiply(V, inverse(V)))))))))
% 0.20/0.42 = { by lemma 3 }
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(X, multiply(Y, inverse(multiply(multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z)))), multiply(V, inverse(V)))))))))
% 0.20/0.42 = { by lemma 3 }
% 0.20/0.42 multiply(X, multiply(Y, inverse(multiply(X, multiply(Y, inverse(multiply(X, multiply(multiply(Y, Z), inverse(multiply(W, Z))))))))))
% 0.20/0.42 = { by lemma 2 }
% 0.20/0.42 multiply(X, multiply(Y, inverse(W)))
% 0.20/0.42
% 0.20/0.42 Lemma 13: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.20/0.42 Proof:
% 0.20/0.42 multiply(inverse(X), multiply(X, Y))
% 0.20/0.42 = { by lemma 11 }
% 0.20/0.42 multiply(inverse(X), multiply(Y, X))
% 0.20/0.42 = { by lemma 11 }
% 0.20/0.42 multiply(multiply(Y, X), inverse(X))
% 0.20/0.42 = { by lemma 8 R->L }
% 0.20/0.42 multiply(multiply(Y, inverse(X)), X)
% 0.20/0.42 = { by lemma 2 R->L }
% 0.20/0.42 multiply(multiply(Y, inverse(X)), multiply(Z, multiply(multiply(multiply(Y, inverse(X)), inverse(Z)), inverse(multiply(Z, multiply(multiply(multiply(multiply(Y, inverse(X)), inverse(Z)), W), inverse(multiply(X, W))))))))
% 0.20/0.42 = { by lemma 9 }
% 0.20/0.42 multiply(multiply(Y, inverse(X)), multiply(multiply(Y, inverse(X)), inverse(multiply(Z, multiply(multiply(multiply(multiply(Y, inverse(X)), inverse(Z)), W), inverse(multiply(X, W)))))))
% 0.20/0.42 = { by lemma 12 }
% 0.20/0.42 multiply(multiply(Y, inverse(X)), multiply(multiply(Y, inverse(X)), inverse(multiply(Z, multiply(multiply(multiply(Y, inverse(X)), inverse(Z)), inverse(X))))))
% 0.20/0.42 = { by lemma 9 }
% 0.20/0.42 multiply(multiply(Y, inverse(X)), multiply(multiply(Y, inverse(X)), inverse(multiply(multiply(Y, inverse(X)), inverse(X)))))
% 0.20/0.42 = { by axiom 1 (single_axiom) }
% 0.20/0.42 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 11 R->L }
% 0.20/0.42 tuple(multiply(a2, multiply(inverse(b2), b2)), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 R->L }
% 0.20/0.43 tuple(multiply(a2, multiply(b2, inverse(b2))), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.43 = { by lemma 3 }
% 0.20/0.43 tuple(a2, multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 R->L }
% 0.20/0.43 tuple(a2, multiply(c3, multiply(a3, b3)), multiply(inverse(a1), a1), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 R->L }
% 0.20/0.43 tuple(a2, multiply(c3, multiply(a3, b3)), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 }
% 0.20/0.43 tuple(a2, multiply(multiply(a3, b3), c3), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 8 R->L }
% 0.20/0.43 tuple(a2, multiply(multiply(a3, c3), b3), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 R->L }
% 0.20/0.43 tuple(a2, multiply(b3, multiply(a3, c3)), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 }
% 0.20/0.43 tuple(a2, multiply(b3, multiply(c3, a3)), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 5 R->L }
% 0.20/0.43 tuple(a2, multiply(multiply(inverse(c3), multiply(c3, a3)), multiply(b3, inverse(inverse(c3)))), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 13 }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, inverse(inverse(c3)))), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 13 R->L }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, inverse(multiply(inverse(X), multiply(X, inverse(c3)))))), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 12 R->L }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, inverse(multiply(inverse(X), multiply(multiply(X, Y), inverse(multiply(c3, Y))))))), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 13 R->L }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, multiply(inverse(X), multiply(X, inverse(multiply(inverse(X), multiply(multiply(X, Y), inverse(multiply(c3, Y))))))))), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 2 }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(a1, inverse(a1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 9 R->L }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(b1, multiply(multiply(a1, inverse(b1)), inverse(a1))), multiply(a4, b4))
% 0.20/0.43 = { by lemma 7 }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(b1, inverse(b1)), multiply(a4, b4))
% 0.20/0.43 = { by lemma 11 }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(b1, inverse(b1)), multiply(b4, a4))
% 0.20/0.43 = { by lemma 11 }
% 0.20/0.43 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1), multiply(b4, a4))
% 0.20/0.43 % SZS output end Proof
% 0.20/0.43
% 0.20/0.43 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------