TSTP Solution File: GRP082-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP082-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 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 : n014.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 01:16:54 EDT 2023
% Result : Unsatisfiable 0.21s 0.59s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP082-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n014.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Mon Aug 28 20:04:30 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.59 Command-line arguments: --no-flatten-goal
% 0.21/0.59
% 0.21/0.59 % SZS status Unsatisfiable
% 0.21/0.59
% 0.21/0.67 % SZS output start Proof
% 0.21/0.67 Take the following subset of the input axioms:
% 0.21/0.67 fof(multiply, axiom, ![X, Y]: multiply(X, Y)=inverse(double_divide(Y, X))).
% 0.21/0.67 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)))).
% 0.21/0.67 fof(single_axiom, axiom, ![Z, U, X2, Y2]: double_divide(inverse(X2), inverse(double_divide(inverse(double_divide(X2, double_divide(Y2, Z))), double_divide(U, double_divide(Y2, U)))))=Z).
% 0.21/0.67
% 0.21/0.67 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.67 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.67 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.67 fresh(y, y, x1...xn) = u
% 0.21/0.67 C => fresh(s, t, x1...xn) = v
% 0.21/0.67 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.67 variables of u and v.
% 0.21/0.67 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.67 input problem has no model of domain size 1).
% 0.21/0.67
% 0.21/0.67 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.67
% 0.21/0.67 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 0.21/0.67 Axiom 2 (single_axiom): double_divide(inverse(X), inverse(double_divide(inverse(double_divide(X, double_divide(Y, Z))), double_divide(W, double_divide(Y, W))))) = Z.
% 0.21/0.67
% 0.21/0.67 Lemma 3: double_divide(inverse(X), multiply(double_divide(Y, double_divide(Z, Y)), multiply(double_divide(Z, W), X))) = W.
% 0.21/0.67 Proof:
% 0.21/0.67 double_divide(inverse(X), multiply(double_divide(Y, double_divide(Z, Y)), multiply(double_divide(Z, W), X)))
% 0.21/0.67 = { by axiom 1 (multiply) }
% 0.21/0.67 double_divide(inverse(X), multiply(double_divide(Y, double_divide(Z, Y)), inverse(double_divide(X, double_divide(Z, W)))))
% 0.21/0.67 = { by axiom 1 (multiply) }
% 0.21/0.67 double_divide(inverse(X), inverse(double_divide(inverse(double_divide(X, double_divide(Z, W))), double_divide(Y, double_divide(Z, Y)))))
% 0.21/0.67 = { by axiom 2 (single_axiom) }
% 0.21/0.67 W
% 0.21/0.67
% 0.21/0.67 Lemma 4: double_divide(inverse(X), multiply(double_divide(Y, double_divide(inverse(Z), Y)), multiply(W, X))) = multiply(double_divide(V, double_divide(U, V)), multiply(double_divide(U, W), Z)).
% 0.21/0.67 Proof:
% 0.21/0.67 double_divide(inverse(X), multiply(double_divide(Y, double_divide(inverse(Z), Y)), multiply(W, X)))
% 0.21/0.67 = { by lemma 3 R->L }
% 0.21/0.67 double_divide(inverse(X), multiply(double_divide(Y, double_divide(inverse(Z), Y)), multiply(double_divide(inverse(Z), multiply(double_divide(V, double_divide(U, V)), multiply(double_divide(U, W), Z))), X)))
% 0.21/0.67 = { by lemma 3 }
% 0.21/0.67 multiply(double_divide(V, double_divide(U, V)), multiply(double_divide(U, W), Z))
% 0.21/0.67
% 0.21/0.67 Lemma 5: multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(inverse(Z), W)), Z)) = W.
% 0.21/0.67 Proof:
% 0.21/0.67 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(inverse(Z), W)), Z))
% 0.21/0.67 = { by lemma 4 R->L }
% 0.21/0.67 double_divide(inverse(V), multiply(double_divide(U, double_divide(inverse(Z), U)), multiply(double_divide(inverse(Z), W), V)))
% 0.21/0.67 = { by lemma 3 }
% 0.21/0.67 W
% 0.21/0.67
% 0.21/0.67 Lemma 6: multiply(multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, Z), W)), inverse(W)) = inverse(Z).
% 0.21/0.67 Proof:
% 0.21/0.67 multiply(multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, Z), W)), inverse(W))
% 0.21/0.67 = { by axiom 1 (multiply) }
% 0.21/0.67 inverse(double_divide(inverse(W), multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, Z), W))))
% 0.21/0.67 = { by lemma 3 }
% 0.21/0.67 inverse(Z)
% 0.21/0.67
% 0.21/0.67 Lemma 7: multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(Z, double_divide(W, Z))), V)) = double_divide(W, inverse(V)).
% 0.21/0.67 Proof:
% 0.21/0.67 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(Z, double_divide(W, Z))), V))
% 0.21/0.67 = { by lemma 4 R->L }
% 0.21/0.67 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(Z, double_divide(W, Z)), inverse(U))))
% 0.21/0.67 = { by lemma 5 R->L }
% 0.21/0.67 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(Z, double_divide(multiply(double_divide(S, double_divide(X2, S)), multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2)), Z)), inverse(U))))
% 0.21/0.67 = { by axiom 1 (multiply) }
% 0.21/0.67 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(Z, double_divide(inverse(double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S)))), Z)), inverse(U))))
% 0.21/0.67 = { by lemma 6 R->L }
% 0.21/0.67 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(Z, double_divide(inverse(double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S)))), Z)), multiply(multiply(double_divide(Z2, double_divide(W2, Z2)), multiply(double_divide(W2, U), V2)), inverse(V2)))))
% 0.21/0.67 = { by lemma 3 R->L }
% 0.21/0.67 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), double_divide(inverse(U2), multiply(double_divide(T2, double_divide(inverse(inverse(V2)), T2)), multiply(double_divide(inverse(inverse(V2)), multiply(double_divide(Z, double_divide(inverse(double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S)))), Z)), multiply(multiply(double_divide(Z2, double_divide(W2, Z2)), multiply(double_divide(W2, U), V2)), inverse(V2)))), U2)))))
% 0.21/0.67 = { by lemma 4 }
% 0.21/0.67 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), double_divide(inverse(U2), multiply(double_divide(T2, double_divide(inverse(inverse(V2)), T2)), multiply(multiply(double_divide(S2, double_divide(X3, S2)), multiply(double_divide(X3, multiply(double_divide(Z2, double_divide(W2, Z2)), multiply(double_divide(W2, U), V2))), double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S))))), U2)))))
% 0.21/0.68 = { by lemma 4 R->L }
% 0.21/0.68 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), double_divide(inverse(U2), multiply(double_divide(T2, double_divide(inverse(inverse(V2)), T2)), multiply(double_divide(inverse(inverse(V2)), multiply(double_divide(inverse(V), double_divide(inverse(double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S)))), inverse(V))), multiply(multiply(double_divide(Z2, double_divide(W2, Z2)), multiply(double_divide(W2, U), V2)), inverse(V2)))), U2)))))
% 0.21/0.68 = { by lemma 3 }
% 0.21/0.68 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(inverse(V), double_divide(inverse(double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S)))), inverse(V))), multiply(multiply(double_divide(Z2, double_divide(W2, Z2)), multiply(double_divide(W2, U), V2)), inverse(V2)))))
% 0.21/0.68 = { by lemma 6 }
% 0.21/0.68 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(inverse(V), double_divide(inverse(double_divide(multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2), double_divide(S, double_divide(X2, S)))), inverse(V))), inverse(U))))
% 0.21/0.68 = { by axiom 1 (multiply) R->L }
% 0.21/0.68 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(inverse(V), double_divide(multiply(double_divide(S, double_divide(X2, S)), multiply(double_divide(X2, double_divide(inverse(Y2), W)), Y2)), inverse(V))), inverse(U))))
% 0.21/0.68 = { by lemma 5 }
% 0.21/0.68 double_divide(inverse(inverse(U)), multiply(double_divide(T, double_divide(inverse(V), T)), multiply(double_divide(inverse(V), double_divide(W, inverse(V))), inverse(U))))
% 0.21/0.68 = { by lemma 3 }
% 0.21/0.68 double_divide(W, inverse(V))
% 0.21/0.68
% 0.21/0.68 Lemma 8: double_divide(Z, double_divide(Y, Z)) = double_divide(X, double_divide(Y, X)).
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(Z, double_divide(Y, Z))
% 0.21/0.68 = { by lemma 5 R->L }
% 0.21/0.68 double_divide(Z, double_divide(Y, multiply(double_divide(X2, double_divide(T, X2)), multiply(double_divide(T, double_divide(inverse(S), Z)), S))))
% 0.21/0.68 = { by lemma 5 R->L }
% 0.21/0.68 double_divide(multiply(double_divide(X2, double_divide(T, X2)), multiply(double_divide(T, double_divide(inverse(S), Z)), S)), double_divide(Y, multiply(double_divide(X2, double_divide(T, X2)), multiply(double_divide(T, double_divide(inverse(S), Z)), S))))
% 0.21/0.68 = { by axiom 1 (multiply) }
% 0.21/0.68 double_divide(multiply(double_divide(X2, double_divide(T, X2)), multiply(double_divide(T, double_divide(inverse(S), Z)), S)), double_divide(Y, inverse(double_divide(multiply(double_divide(T, double_divide(inverse(S), Z)), S), double_divide(X2, double_divide(T, X2))))))
% 0.21/0.68 = { by lemma 7 R->L }
% 0.21/0.68 double_divide(multiply(double_divide(X2, double_divide(T, X2)), multiply(double_divide(T, double_divide(inverse(S), Z)), S)), multiply(double_divide(Y2, double_divide(Z2, Y2)), multiply(double_divide(Z2, double_divide(multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U)), double_divide(Y, multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U))))), double_divide(multiply(double_divide(T, double_divide(inverse(S), Z)), S), double_divide(X2, double_divide(T, X2))))))
% 0.21/0.68 = { by axiom 1 (multiply) }
% 0.21/0.68 double_divide(inverse(double_divide(multiply(double_divide(T, double_divide(inverse(S), Z)), S), double_divide(X2, double_divide(T, X2)))), multiply(double_divide(Y2, double_divide(Z2, Y2)), multiply(double_divide(Z2, double_divide(multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U)), double_divide(Y, multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U))))), double_divide(multiply(double_divide(T, double_divide(inverse(S), Z)), S), double_divide(X2, double_divide(T, X2))))))
% 0.21/0.68 = { by lemma 3 }
% 0.21/0.68 double_divide(multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U)), double_divide(Y, multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U))))
% 0.21/0.68 = { by lemma 5 }
% 0.21/0.68 double_divide(X, double_divide(Y, multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), X)), U))))
% 0.21/0.68 = { by lemma 5 }
% 0.21/0.68 double_divide(X, double_divide(Y, X))
% 0.21/0.68
% 0.21/0.68 Lemma 9: multiply(double_divide(X, Z), Z) = multiply(double_divide(X, Y), Y).
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(double_divide(X, Z), Z)
% 0.21/0.68 = { by axiom 1 (multiply) }
% 0.21/0.68 inverse(double_divide(Z, double_divide(X, Z)))
% 0.21/0.68 = { by lemma 8 }
% 0.21/0.68 inverse(double_divide(Y, double_divide(X, Y)))
% 0.21/0.68 = { by axiom 1 (multiply) R->L }
% 0.21/0.68 multiply(double_divide(X, Y), Y)
% 0.21/0.68
% 0.21/0.68 Lemma 10: multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, Z), Z)) = double_divide(W, multiply(double_divide(W, V), V)).
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, Z), Z))
% 0.21/0.68 = { by lemma 9 }
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(V, double_divide(W, V))), double_divide(V, double_divide(W, V))))
% 0.21/0.68 = { by lemma 7 }
% 0.21/0.68 double_divide(W, inverse(double_divide(V, double_divide(W, V))))
% 0.21/0.68 = { by axiom 1 (multiply) R->L }
% 0.21/0.68 double_divide(W, multiply(double_divide(W, V), V))
% 0.21/0.68
% 0.21/0.68 Lemma 11: double_divide(inverse(X), double_divide(Y, multiply(double_divide(Y, Z), Z))) = X.
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(inverse(X), double_divide(Y, multiply(double_divide(Y, Z), Z)))
% 0.21/0.68 = { by lemma 10 R->L }
% 0.21/0.68 double_divide(inverse(X), multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, X), X)))
% 0.21/0.68 = { by lemma 3 }
% 0.21/0.68 X
% 0.21/0.68
% 0.21/0.68 Lemma 12: multiply(double_divide(X, multiply(double_divide(X, Y), Y)), Z) = Z.
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(double_divide(X, multiply(double_divide(X, Y), Y)), Z)
% 0.21/0.68 = { by lemma 5 R->L }
% 0.21/0.68 multiply(double_divide(X, multiply(double_divide(X, Y), Y)), multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), Z)), U)))
% 0.21/0.68 = { by axiom 1 (multiply) }
% 0.21/0.68 multiply(double_divide(X, multiply(double_divide(X, Y), Y)), inverse(double_divide(multiply(double_divide(V, double_divide(inverse(U), Z)), U), double_divide(W, double_divide(V, W)))))
% 0.21/0.68 = { by axiom 1 (multiply) }
% 0.21/0.68 inverse(double_divide(inverse(double_divide(multiply(double_divide(V, double_divide(inverse(U), Z)), U), double_divide(W, double_divide(V, W)))), double_divide(X, multiply(double_divide(X, Y), Y))))
% 0.21/0.68 = { by lemma 11 }
% 0.21/0.68 inverse(double_divide(multiply(double_divide(V, double_divide(inverse(U), Z)), U), double_divide(W, double_divide(V, W))))
% 0.21/0.68 = { by axiom 1 (multiply) R->L }
% 0.21/0.68 multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(inverse(U), Z)), U))
% 0.21/0.68 = { by lemma 5 }
% 0.21/0.68 Z
% 0.21/0.68
% 0.21/0.68 Lemma 13: double_divide(inverse(X), multiply(double_divide(Y, double_divide(Z, Y)), X)) = multiply(double_divide(Z, W), W).
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(inverse(X), multiply(double_divide(Y, double_divide(Z, Y)), X))
% 0.21/0.68 = { by lemma 12 R->L }
% 0.21/0.68 double_divide(inverse(X), multiply(double_divide(Y, double_divide(Z, Y)), multiply(double_divide(Z, multiply(double_divide(Z, W), W)), X)))
% 0.21/0.68 = { by lemma 3 }
% 0.21/0.68 multiply(double_divide(Z, W), W)
% 0.21/0.68
% 0.21/0.68 Lemma 14: double_divide(inverse(multiply(double_divide(X, double_divide(inverse(Y), Z)), Y)), Z) = multiply(double_divide(X, W), W).
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(inverse(multiply(double_divide(X, double_divide(inverse(Y), Z)), Y)), Z)
% 0.21/0.68 = { by lemma 5 R->L }
% 0.21/0.68 double_divide(inverse(multiply(double_divide(X, double_divide(inverse(Y), Z)), Y)), multiply(double_divide(V, double_divide(X, V)), multiply(double_divide(X, double_divide(inverse(Y), Z)), Y)))
% 0.21/0.68 = { by lemma 13 }
% 0.21/0.68 multiply(double_divide(X, W), W)
% 0.21/0.68
% 0.21/0.68 Lemma 15: multiply(double_divide(inverse(Z), Z), X) = multiply(double_divide(inverse(X), Y), Y).
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(double_divide(inverse(Z), Z), X)
% 0.21/0.68 = { by lemma 12 R->L }
% 0.21/0.68 multiply(double_divide(inverse(Z), multiply(double_divide(W, multiply(double_divide(W, S), S)), Z)), X)
% 0.21/0.68 = { by lemma 14 R->L }
% 0.21/0.68 multiply(double_divide(inverse(Z), multiply(double_divide(W, double_divide(inverse(multiply(double_divide(W, double_divide(inverse(U), W)), U)), W)), Z)), X)
% 0.21/0.68 = { by lemma 13 }
% 0.21/0.68 multiply(multiply(double_divide(inverse(multiply(double_divide(W, double_divide(inverse(U), W)), U)), T), T), X)
% 0.21/0.68 = { by lemma 13 R->L }
% 0.21/0.68 multiply(double_divide(inverse(X), multiply(double_divide(W, double_divide(inverse(multiply(double_divide(W, double_divide(inverse(U), W)), U)), W)), X)), X)
% 0.21/0.68 = { by lemma 14 }
% 0.21/0.68 multiply(double_divide(inverse(X), multiply(double_divide(W, multiply(double_divide(W, V), V)), X)), X)
% 0.21/0.68 = { by lemma 12 }
% 0.21/0.68 multiply(double_divide(inverse(X), X), X)
% 0.21/0.68 = { by lemma 9 R->L }
% 0.21/0.68 multiply(double_divide(inverse(X), Y), Y)
% 0.21/0.68
% 0.21/0.68 Lemma 16: multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(Z, multiply(double_divide(Z, W), W))), V)) = multiply(double_divide(inverse(U), U), V).
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(Z, multiply(double_divide(Z, W), W))), V))
% 0.21/0.68 = { by axiom 1 (multiply) }
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(Z, inverse(double_divide(W, double_divide(Z, W))))), V))
% 0.21/0.68 = { by lemma 7 R->L }
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, multiply(double_divide(T, double_divide(S, T)), multiply(double_divide(S, double_divide(W, double_divide(Z, W))), double_divide(W, double_divide(Z, W))))), V))
% 0.21/0.68 = { by lemma 9 R->L }
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, multiply(double_divide(T, double_divide(S, T)), multiply(double_divide(S, double_divide(X2, double_divide(inverse(V), X2))), double_divide(X2, double_divide(inverse(V), X2))))), V))
% 0.21/0.68 = { by lemma 7 }
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(inverse(V), inverse(double_divide(X2, double_divide(inverse(V), X2))))), V))
% 0.21/0.68 = { by axiom 1 (multiply) R->L }
% 0.21/0.68 multiply(double_divide(X, double_divide(Y, X)), multiply(double_divide(Y, double_divide(inverse(V), multiply(double_divide(inverse(V), X2), X2))), V))
% 0.21/0.68 = { by lemma 5 }
% 0.21/0.68 multiply(double_divide(inverse(V), X2), X2)
% 0.21/0.68 = { by lemma 15 R->L }
% 0.21/0.68 multiply(double_divide(inverse(U), U), V)
% 0.21/0.68
% 0.21/0.68 Lemma 17: multiply(X, double_divide(Y, multiply(double_divide(Y, Z), Z))) = multiply(double_divide(inverse(X), W), W).
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(X, double_divide(Y, multiply(double_divide(Y, Z), Z)))
% 0.21/0.68 = { by lemma 11 R->L }
% 0.21/0.68 multiply(double_divide(inverse(X), double_divide(Y, multiply(double_divide(Y, Z), Z))), double_divide(Y, multiply(double_divide(Y, Z), Z)))
% 0.21/0.68 = { by lemma 9 R->L }
% 0.21/0.68 multiply(double_divide(inverse(X), W), W)
% 0.21/0.68
% 0.21/0.68 Lemma 18: double_divide(X, multiply(double_divide(X, Y), Y)) = double_divide(inverse(Z), Z).
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(X, multiply(double_divide(X, Y), Y))
% 0.21/0.68 = { by lemma 3 R->L }
% 0.21/0.68 double_divide(inverse(double_divide(inverse(Z), Z)), multiply(double_divide(W, double_divide(V, W)), multiply(double_divide(V, double_divide(X, multiply(double_divide(X, Y), Y))), double_divide(inverse(Z), Z))))
% 0.21/0.68 = { by lemma 16 }
% 0.21/0.68 double_divide(inverse(double_divide(inverse(Z), Z)), multiply(double_divide(inverse(double_divide(inverse(Z), Z)), double_divide(inverse(Z), Z)), double_divide(inverse(Z), Z)))
% 0.21/0.68 = { by lemma 17 R->L }
% 0.21/0.68 double_divide(inverse(double_divide(inverse(Z), Z)), multiply(double_divide(inverse(Z), Z), double_divide(U, multiply(double_divide(U, T), T))))
% 0.21/0.68 = { by lemma 16 R->L }
% 0.21/0.68 double_divide(inverse(double_divide(inverse(Z), Z)), multiply(double_divide(S, double_divide(X2, S)), multiply(double_divide(X2, double_divide(U, multiply(double_divide(U, T), T))), double_divide(U, multiply(double_divide(U, T), T)))))
% 0.21/0.68 = { by lemma 10 }
% 0.21/0.68 double_divide(inverse(double_divide(inverse(Z), Z)), double_divide(Y2, multiply(double_divide(Y2, Z2), Z2)))
% 0.21/0.68 = { by lemma 11 }
% 0.21/0.68 double_divide(inverse(Z), Z)
% 0.21/0.68
% 0.21/0.68 Lemma 19: double_divide(X, double_divide(inverse(inverse(Y)), X)) = Y.
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(X, double_divide(inverse(inverse(Y)), X))
% 0.21/0.68 = { by lemma 8 }
% 0.21/0.68 double_divide(inverse(Y), double_divide(inverse(inverse(Y)), inverse(Y)))
% 0.21/0.68 = { by lemma 18 R->L }
% 0.21/0.68 double_divide(inverse(Y), double_divide(Z, multiply(double_divide(Z, W), W)))
% 0.21/0.68 = { by lemma 11 }
% 0.21/0.68 Y
% 0.21/0.68
% 0.21/0.68 Lemma 20: multiply(double_divide(inverse(X), X), Y) = Y.
% 0.21/0.68 Proof:
% 0.21/0.68 multiply(double_divide(inverse(X), X), Y)
% 0.21/0.68 = { by lemma 18 R->L }
% 0.21/0.68 multiply(double_divide(Z, multiply(double_divide(Z, W), W)), Y)
% 0.21/0.68 = { by lemma 12 }
% 0.21/0.68 Y
% 0.21/0.68
% 0.21/0.68 Lemma 21: double_divide(inverse(X), multiply(Y, X)) = inverse(Y).
% 0.21/0.68 Proof:
% 0.21/0.68 double_divide(inverse(X), multiply(Y, X))
% 0.21/0.68 = { by lemma 19 R->L }
% 0.21/0.68 double_divide(inverse(X), multiply(double_divide(Z, double_divide(inverse(inverse(Y)), Z)), X))
% 0.21/0.68 = { by lemma 13 }
% 0.21/0.68 multiply(double_divide(inverse(inverse(Y)), inverse(Y)), inverse(Y))
% 0.21/0.68 = { by lemma 20 }
% 0.21/0.68 inverse(Y)
% 0.21/0.68
% 0.21/0.68 Lemma 22: multiply(inverse(X), X) = double_divide(inverse(Y), Y).
% 0.21/0.68 Proof:
% 0.21/0.69 multiply(inverse(X), X)
% 0.21/0.69 = { by lemma 20 R->L }
% 0.21/0.69 multiply(inverse(X), multiply(double_divide(inverse(Z), Z), X))
% 0.21/0.69 = { by lemma 15 }
% 0.21/0.69 multiply(inverse(X), multiply(double_divide(inverse(X), W), W))
% 0.21/0.69 = { by lemma 17 R->L }
% 0.21/0.69 multiply(inverse(X), multiply(X, double_divide(V, multiply(double_divide(V, U), U))))
% 0.21/0.69 = { by lemma 21 R->L }
% 0.21/0.69 multiply(double_divide(inverse(double_divide(V, multiply(double_divide(V, U), U))), multiply(X, double_divide(V, multiply(double_divide(V, U), U)))), multiply(X, double_divide(V, multiply(double_divide(V, U), U))))
% 0.21/0.69 = { by lemma 15 R->L }
% 0.21/0.69 multiply(double_divide(inverse(T), T), double_divide(V, multiply(double_divide(V, U), U)))
% 0.21/0.69 = { by lemma 20 }
% 0.21/0.69 double_divide(V, multiply(double_divide(V, U), U))
% 0.21/0.69 = { by lemma 18 }
% 0.21/0.69 double_divide(inverse(Y), Y)
% 0.21/0.69
% 0.21/0.69 Lemma 23: inverse(inverse(X)) = X.
% 0.21/0.69 Proof:
% 0.21/0.69 inverse(inverse(X))
% 0.21/0.69 = { by lemma 21 R->L }
% 0.21/0.69 double_divide(inverse(X), multiply(inverse(X), X))
% 0.21/0.69 = { by lemma 22 }
% 0.21/0.69 double_divide(inverse(X), double_divide(inverse(inverse(X)), inverse(X)))
% 0.21/0.69 = { by lemma 19 }
% 0.21/0.69 X
% 0.21/0.69
% 0.21/0.69 Lemma 24: inverse(multiply(X, Y)) = double_divide(Y, X).
% 0.21/0.69 Proof:
% 0.21/0.69 inverse(multiply(X, Y))
% 0.21/0.69 = { by axiom 1 (multiply) }
% 0.21/0.69 inverse(inverse(double_divide(Y, X)))
% 0.21/0.69 = { by lemma 23 }
% 0.21/0.69 double_divide(Y, X)
% 0.21/0.69
% 0.21/0.69 Lemma 25: multiply(multiply(double_divide(X, Y), Y), X) = double_divide(inverse(Z), Z).
% 0.21/0.69 Proof:
% 0.21/0.69 multiply(multiply(double_divide(X, Y), Y), X)
% 0.21/0.69 = { by lemma 14 R->L }
% 0.21/0.69 multiply(double_divide(inverse(multiply(double_divide(X, double_divide(inverse(W), X)), W)), X), X)
% 0.21/0.69 = { by lemma 13 R->L }
% 0.21/0.69 double_divide(inverse(Z), multiply(double_divide(X, double_divide(inverse(multiply(double_divide(X, double_divide(inverse(W), X)), W)), X)), Z))
% 0.21/0.69 = { by lemma 14 }
% 0.21/0.69 double_divide(inverse(Z), multiply(double_divide(X, multiply(double_divide(X, V), V)), Z))
% 0.21/0.69 = { by lemma 12 }
% 0.21/0.69 double_divide(inverse(Z), Z)
% 0.21/0.69
% 0.21/0.69 Lemma 26: double_divide(double_divide(inverse(X), X), Y) = inverse(Y).
% 0.21/0.69 Proof:
% 0.21/0.69 double_divide(double_divide(inverse(X), X), Y)
% 0.21/0.69 = { by lemma 20 R->L }
% 0.21/0.69 double_divide(double_divide(inverse(X), X), multiply(double_divide(inverse(Z), Z), Y))
% 0.21/0.69 = { by lemma 15 }
% 0.21/0.69 double_divide(double_divide(inverse(X), X), multiply(double_divide(inverse(Y), W), W))
% 0.21/0.69 = { by lemma 25 R->L }
% 0.21/0.69 double_divide(multiply(multiply(double_divide(V, U), U), V), multiply(double_divide(inverse(Y), W), W))
% 0.21/0.69 = { by axiom 1 (multiply) }
% 0.21/0.69 double_divide(inverse(double_divide(V, multiply(double_divide(V, U), U))), multiply(double_divide(inverse(Y), W), W))
% 0.21/0.69 = { by lemma 17 R->L }
% 0.21/0.69 double_divide(inverse(double_divide(V, multiply(double_divide(V, U), U))), multiply(Y, double_divide(V, multiply(double_divide(V, U), U))))
% 0.21/0.69 = { by lemma 21 }
% 0.21/0.69 inverse(Y)
% 0.21/0.69
% 0.21/0.69 Lemma 27: double_divide(inverse(X), X) = double_divide(Y, inverse(Y)).
% 0.21/0.69 Proof:
% 0.21/0.69 double_divide(inverse(X), X)
% 0.21/0.69 = { by lemma 18 R->L }
% 0.21/0.69 double_divide(double_divide(inverse(Z), Z), multiply(double_divide(double_divide(inverse(Z), Z), Y), Y))
% 0.21/0.69 = { by lemma 26 }
% 0.21/0.69 double_divide(double_divide(inverse(Z), Z), multiply(inverse(Y), Y))
% 0.21/0.69 = { by lemma 26 }
% 0.21/0.69 inverse(multiply(inverse(Y), Y))
% 0.21/0.69 = { by lemma 24 }
% 0.21/0.69 double_divide(Y, inverse(Y))
% 0.21/0.69
% 0.21/0.69 Lemma 28: double_divide(X, double_divide(Y, X)) = Y.
% 0.21/0.69 Proof:
% 0.21/0.69 double_divide(X, double_divide(Y, X))
% 0.21/0.69 = { by lemma 8 }
% 0.21/0.69 double_divide(double_divide(inverse(Z), Z), double_divide(Y, double_divide(inverse(Z), Z)))
% 0.21/0.69 = { by lemma 26 }
% 0.21/0.69 inverse(double_divide(Y, double_divide(inverse(Z), Z)))
% 0.21/0.69 = { by axiom 1 (multiply) R->L }
% 0.21/0.69 multiply(double_divide(inverse(Z), Z), Y)
% 0.21/0.69 = { by lemma 20 }
% 0.21/0.69 Y
% 0.21/0.69
% 0.21/0.69 Lemma 29: multiply(inverse(X), Y) = double_divide(X, inverse(Y)).
% 0.21/0.69 Proof:
% 0.21/0.69 multiply(inverse(X), Y)
% 0.21/0.69 = { by lemma 28 R->L }
% 0.21/0.69 double_divide(double_divide(inverse(Y), multiply(inverse(X), Y)), double_divide(multiply(inverse(X), Y), double_divide(inverse(Y), multiply(inverse(X), Y))))
% 0.21/0.69 = { by lemma 28 }
% 0.21/0.69 double_divide(double_divide(inverse(Y), multiply(inverse(X), Y)), inverse(Y))
% 0.21/0.69 = { by lemma 21 }
% 0.21/0.69 double_divide(inverse(inverse(X)), inverse(Y))
% 0.21/0.69 = { by lemma 23 }
% 0.21/0.69 double_divide(X, inverse(Y))
% 0.21/0.69
% 0.21/0.69 Lemma 30: multiply(X, double_divide(Y, inverse(Y))) = X.
% 0.21/0.69 Proof:
% 0.21/0.69 multiply(X, double_divide(Y, inverse(Y)))
% 0.21/0.69 = { by lemma 27 R->L }
% 0.21/0.69 multiply(X, double_divide(inverse(Z), Z))
% 0.21/0.69 = { by lemma 25 R->L }
% 0.21/0.69 multiply(X, multiply(multiply(double_divide(W, V), V), W))
% 0.21/0.69 = { by axiom 1 (multiply) }
% 0.21/0.69 multiply(X, inverse(double_divide(W, multiply(double_divide(W, V), V))))
% 0.21/0.69 = { by lemma 20 R->L }
% 0.21/0.69 multiply(multiply(double_divide(inverse(U), U), X), inverse(double_divide(W, multiply(double_divide(W, V), V))))
% 0.21/0.69 = { by lemma 15 }
% 0.21/0.69 multiply(multiply(double_divide(inverse(X), T), T), inverse(double_divide(W, multiply(double_divide(W, V), V))))
% 0.21/0.69 = { by lemma 17 R->L }
% 0.21/0.69 multiply(multiply(X, double_divide(W, multiply(double_divide(W, V), V))), inverse(double_divide(W, multiply(double_divide(W, V), V))))
% 0.21/0.69 = { by lemma 19 R->L }
% 0.21/0.69 multiply(multiply(double_divide(S, double_divide(inverse(inverse(X)), S)), double_divide(W, multiply(double_divide(W, V), V))), inverse(double_divide(W, multiply(double_divide(W, V), V))))
% 0.21/0.69 = { by lemma 12 R->L }
% 0.21/0.69 multiply(multiply(double_divide(S, double_divide(inverse(inverse(X)), S)), multiply(double_divide(inverse(inverse(X)), multiply(double_divide(inverse(inverse(X)), inverse(X)), inverse(X))), double_divide(W, multiply(double_divide(W, V), V)))), inverse(double_divide(W, multiply(double_divide(W, V), V))))
% 0.21/0.69 = { by lemma 6 }
% 0.21/0.69 inverse(multiply(double_divide(inverse(inverse(X)), inverse(X)), inverse(X)))
% 0.21/0.69 = { by lemma 20 }
% 0.21/0.69 inverse(inverse(X))
% 0.21/0.69 = { by lemma 23 }
% 0.21/0.69 X
% 0.21/0.69
% 0.21/0.69 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3)) = tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3))).
% 0.21/0.69 Proof:
% 0.21/0.69 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3))
% 0.21/0.69 = { by lemma 22 }
% 0.21/0.69 tuple(double_divide(inverse(X), X), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3))
% 0.21/0.69 = { by lemma 22 }
% 0.21/0.69 tuple(double_divide(inverse(X), X), multiply(double_divide(inverse(Y), Y), a2), multiply(multiply(a3, b3), c3))
% 0.21/0.69 = { by lemma 20 }
% 0.21/0.69 tuple(double_divide(inverse(X), X), a2, multiply(multiply(a3, b3), c3))
% 0.21/0.69 = { by lemma 27 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(multiply(a3, b3), c3))
% 0.21/0.69 = { by lemma 28 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(double_divide(W, double_divide(multiply(a3, b3), W)), c3))
% 0.21/0.69 = { by axiom 1 (multiply) }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(double_divide(W, double_divide(inverse(double_divide(b3, a3)), W)), c3))
% 0.21/0.69 = { by lemma 28 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(inverse(double_divide(b3, a3)), c3))
% 0.21/0.69 = { by lemma 29 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(double_divide(b3, a3), inverse(c3)))
% 0.21/0.69 = { by lemma 28 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(double_divide(b3, a3), double_divide(V, double_divide(inverse(c3), V))))
% 0.21/0.69 = { by lemma 24 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, inverse(multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3))))
% 0.21/0.69 = { by lemma 28 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3)), double_divide(inverse(multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3))), multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3)))))
% 0.21/0.69 = { by lemma 18 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3)), double_divide(U, multiply(double_divide(U, T), T))))
% 0.21/0.69 = { by lemma 30 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(multiply(multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3)), double_divide(S, inverse(S))), double_divide(U, multiply(double_divide(U, T), T))))
% 0.21/0.69 = { by axiom 1 (multiply) }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(inverse(double_divide(double_divide(S, inverse(S)), multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3)))), double_divide(U, multiply(double_divide(U, T), T))))
% 0.21/0.69 = { by lemma 11 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(double_divide(S, inverse(S)), multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3))))
% 0.21/0.69 = { by lemma 29 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(multiply(inverse(S), S), multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3))))
% 0.21/0.69 = { by axiom 1 (multiply) }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(inverse(double_divide(S, inverse(S))), multiply(double_divide(V, double_divide(inverse(c3), V)), double_divide(b3, a3))))
% 0.21/0.69 = { by lemma 30 R->L }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, double_divide(inverse(double_divide(S, inverse(S))), multiply(double_divide(V, double_divide(inverse(c3), V)), multiply(double_divide(b3, a3), double_divide(S, inverse(S))))))
% 0.21/0.69 = { by lemma 4 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(double_divide(X2, double_divide(a3, X2)), multiply(double_divide(a3, double_divide(b3, a3)), c3)))
% 0.21/0.69 = { by lemma 28 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(a3, multiply(double_divide(a3, double_divide(b3, a3)), c3)))
% 0.21/0.69 = { by lemma 28 }
% 0.21/0.69 tuple(double_divide(Z, inverse(Z)), a2, multiply(a3, multiply(b3, c3)))
% 0.21/0.69 = { by lemma 27 R->L }
% 0.21/0.69 tuple(double_divide(inverse(Y2), Y2), a2, multiply(a3, multiply(b3, c3)))
% 0.21/0.69 = { by lemma 22 R->L }
% 0.21/0.69 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)))
% 0.21/0.69 % SZS output end Proof
% 0.21/0.69
% 0.21/0.69 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------