TSTP Solution File: GRP502-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP502-1 : TPTP v8.1.2. Released v2.6.0.
% 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 01:18:41 EDT 2023
% Result : Unsatisfiable 0.19s 0.47s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP502-1 : TPTP v8.1.2. Released v2.6.0.
% 0.07/0.13 % 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 19:39:46 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.47 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.47
% 0.19/0.47 % SZS status Unsatisfiable
% 0.19/0.47
% 0.19/0.54 % SZS output start Proof
% 0.19/0.54 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 0.19/0.54 Axiom 2 (single_axiom): double_divide(double_divide(X, inverse(double_divide(Y, Z))), double_divide(inverse(Y), inverse(double_divide(W, double_divide(X, W))))) = Z.
% 0.19/0.54
% 0.19/0.54 Lemma 3: double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, W), W))) = Y.
% 0.19/0.54 Proof:
% 0.19/0.54 double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, W), W)))
% 0.19/0.54 = { by axiom 1 (multiply) }
% 0.19/0.54 double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), inverse(double_divide(W, double_divide(X, W)))))
% 0.19/0.54 = { by axiom 1 (multiply) }
% 0.19/0.54 double_divide(double_divide(X, inverse(double_divide(Z, Y))), double_divide(inverse(Z), inverse(double_divide(W, double_divide(X, W)))))
% 0.19/0.54 = { by axiom 2 (single_axiom) }
% 0.19/0.54 Y
% 0.19/0.54
% 0.19/0.54 Lemma 4: multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(W, X))) = inverse(W).
% 0.19/0.54 Proof:
% 0.19/0.54 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(W, X)))
% 0.19/0.54 = { by axiom 1 (multiply) }
% 0.19/0.54 inverse(double_divide(double_divide(Y, multiply(W, X)), double_divide(inverse(X), multiply(double_divide(Y, Z), Z))))
% 0.19/0.54 = { by lemma 3 }
% 0.19/0.54 inverse(W)
% 0.19/0.54
% 0.19/0.54 Lemma 5: multiply(double_divide(X, Z), Z) = multiply(double_divide(X, Y), Y).
% 0.19/0.54 Proof:
% 0.19/0.54 multiply(double_divide(X, Z), Z)
% 0.19/0.54 = { by lemma 3 R->L }
% 0.19/0.54 double_divide(double_divide(W, multiply(multiply(double_divide(X, Z), Z), inverse(V))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by axiom 1 (multiply) }
% 0.19/0.54 double_divide(double_divide(W, inverse(double_divide(inverse(V), multiply(double_divide(X, Z), Z)))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by lemma 4 R->L }
% 0.19/0.54 double_divide(double_divide(W, multiply(double_divide(inverse(double_divide(X, multiply(T, V))), multiply(double_divide(S, X2), X2)), double_divide(S, multiply(double_divide(inverse(V), multiply(double_divide(X, Z), Z)), double_divide(X, multiply(T, V)))))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by lemma 4 }
% 0.19/0.54 double_divide(double_divide(W, multiply(double_divide(inverse(double_divide(X, multiply(T, V))), multiply(double_divide(S, X2), X2)), double_divide(S, inverse(T)))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by lemma 4 R->L }
% 0.19/0.54 double_divide(double_divide(W, multiply(double_divide(inverse(double_divide(X, multiply(T, V))), multiply(double_divide(S, X2), X2)), double_divide(S, multiply(double_divide(inverse(V), multiply(double_divide(X, Y), Y)), double_divide(X, multiply(T, V)))))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by lemma 4 }
% 0.19/0.54 double_divide(double_divide(W, inverse(double_divide(inverse(V), multiply(double_divide(X, Y), Y)))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by axiom 1 (multiply) R->L }
% 0.19/0.54 double_divide(double_divide(W, multiply(multiply(double_divide(X, Y), Y), inverse(V))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 0.19/0.54 = { by lemma 3 }
% 0.19/0.54 multiply(double_divide(X, Y), Y)
% 0.19/0.54
% 0.19/0.55 Lemma 6: multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, V), V))) = multiply(X, W).
% 0.19/0.55 Proof:
% 0.19/0.55 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, V), V)))
% 0.19/0.55 = { by lemma 5 }
% 0.19/0.55 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, X), X)))
% 0.19/0.55 = { by lemma 4 }
% 0.19/0.55 inverse(double_divide(W, X))
% 0.19/0.55 = { by axiom 1 (multiply) R->L }
% 0.19/0.55 multiply(X, W)
% 0.19/0.55
% 0.19/0.55 Lemma 7: multiply(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), V), V) = multiply(Y, X).
% 0.19/0.55 Proof:
% 0.19/0.55 multiply(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), V), V)
% 0.19/0.55 = { by lemma 5 }
% 0.19/0.55 multiply(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), double_divide(inverse(Z), multiply(double_divide(X, U), U))), double_divide(inverse(Z), multiply(double_divide(X, U), U)))
% 0.19/0.55 = { by lemma 3 }
% 0.19/0.55 multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), double_divide(inverse(Z), multiply(double_divide(X, U), U)))
% 0.19/0.55 = { by lemma 6 }
% 0.19/0.55 multiply(Y, X)
% 0.19/0.55
% 0.19/0.55 Lemma 8: double_divide(double_divide(X, multiply(double_divide(Y, Z), Z)), double_divide(inverse(W), multiply(double_divide(X, V), V))) = double_divide(Y, W).
% 0.19/0.55 Proof:
% 0.19/0.55 double_divide(double_divide(X, multiply(double_divide(Y, Z), Z)), double_divide(inverse(W), multiply(double_divide(X, V), V)))
% 0.19/0.55 = { by lemma 5 }
% 0.19/0.55 double_divide(double_divide(X, multiply(double_divide(Y, W), W)), double_divide(inverse(W), multiply(double_divide(X, V), V)))
% 0.19/0.55 = { by lemma 3 }
% 0.19/0.55 double_divide(Y, W)
% 0.19/0.55
% 0.19/0.55 Lemma 9: multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, U), U))) = multiply(double_divide(Y, X), V).
% 0.19/0.55 Proof:
% 0.19/0.55 multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, U), U)))
% 0.19/0.55 = { by lemma 5 }
% 0.19/0.55 multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, double_divide(Y, X)), double_divide(Y, X))))
% 0.19/0.55 = { by axiom 1 (multiply) }
% 0.19/0.55 multiply(double_divide(inverse(double_divide(Y, X)), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, double_divide(Y, X)), double_divide(Y, X))))
% 0.19/0.55 = { by lemma 4 }
% 0.19/0.55 inverse(double_divide(V, double_divide(Y, X)))
% 0.19/0.55 = { by axiom 1 (multiply) R->L }
% 0.19/0.55 multiply(double_divide(Y, X), V)
% 0.19/0.55
% 0.19/0.55 Lemma 10: multiply(double_divide(W, double_divide(Y, W)), Z) = multiply(double_divide(X, double_divide(Y, X)), Z).
% 0.19/0.55 Proof:
% 0.19/0.55 multiply(double_divide(W, double_divide(Y, W)), Z)
% 0.19/0.55 = { by lemma 9 R->L }
% 0.19/0.55 multiply(double_divide(multiply(double_divide(Y, W), W), multiply(double_divide(V, U), U)), double_divide(V, multiply(double_divide(Z, T), T)))
% 0.19/0.55 = { by lemma 5 }
% 0.19/0.55 multiply(double_divide(multiply(double_divide(Y, X), X), multiply(double_divide(V, U), U)), double_divide(V, multiply(double_divide(Z, T), T)))
% 0.19/0.55 = { by lemma 9 }
% 0.19/0.55 multiply(double_divide(X, double_divide(Y, X)), Z)
% 0.19/0.55
% 0.19/0.55 Lemma 11: double_divide(Z, double_divide(Y, Z)) = double_divide(X, double_divide(Y, X)).
% 0.19/0.55 Proof:
% 0.19/0.55 double_divide(Z, double_divide(Y, Z))
% 0.19/0.55 = { by lemma 3 R->L }
% 0.19/0.55 double_divide(double_divide(W, multiply(double_divide(Z, double_divide(Y, Z)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 0.19/0.55 = { by lemma 10 }
% 0.19/0.55 double_divide(double_divide(W, multiply(double_divide(X, double_divide(Y, X)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 0.19/0.55 = { by lemma 3 }
% 0.19/0.55 double_divide(X, double_divide(Y, X))
% 0.19/0.55
% 0.19/0.55 Lemma 12: double_divide(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), W) = double_divide(V, double_divide(double_divide(Y, multiply(W, X)), V)).
% 0.19/0.55 Proof:
% 0.19/0.55 double_divide(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), W)
% 0.19/0.55 = { by lemma 3 R->L }
% 0.19/0.55 double_divide(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(double_divide(Y, multiply(W, X)), double_divide(inverse(X), multiply(double_divide(Y, Z), Z))))
% 0.19/0.55 = { by lemma 11 R->L }
% 0.19/0.55 double_divide(V, double_divide(double_divide(Y, multiply(W, X)), V))
% 0.19/0.55
% 0.19/0.55 Lemma 13: double_divide(double_divide(X, Y), double_divide(Z, double_divide(X, Z))) = double_divide(W, double_divide(Y, W)).
% 0.19/0.55 Proof:
% 0.19/0.55 double_divide(double_divide(X, Y), double_divide(Z, double_divide(X, Z)))
% 0.19/0.55 = { by lemma 11 }
% 0.19/0.55 double_divide(double_divide(X, Y), double_divide(Y, double_divide(X, Y)))
% 0.19/0.55 = { by lemma 11 R->L }
% 0.19/0.55 double_divide(W, double_divide(Y, W))
% 0.19/0.55
% 0.19/0.55 Lemma 14: double_divide(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), V), double_divide(X, Y)) = double_divide(U, double_divide(V, U)).
% 0.19/0.55 Proof:
% 0.19/0.55 double_divide(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), V), double_divide(X, Y))
% 0.19/0.55 = { by lemma 8 R->L }
% 0.19/0.55 double_divide(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), V), double_divide(double_divide(inverse(Z), multiply(double_divide(X, T), T)), double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W))))
% 0.19/0.55 = { by lemma 12 }
% 0.19/0.55 double_divide(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), V), double_divide(S, double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), W), W)), Z)), S)))
% 0.19/0.55 = { by lemma 13 }
% 0.19/0.55 double_divide(U, double_divide(V, U))
% 0.19/0.55
% 0.19/0.55 Lemma 15: double_divide(double_divide(X, Y), double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))) = Y.
% 0.19/0.55 Proof:
% 0.19/0.55 double_divide(double_divide(X, Y), double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V)))
% 0.19/0.55 = { by lemma 8 R->L }
% 0.19/0.55 double_divide(double_divide(U, multiply(double_divide(double_divide(X, Y), double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2))))), double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2)))))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by axiom 1 (multiply) }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2)))), double_divide(double_divide(X, Y), double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2)))))))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 3 R->L }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2)))), double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2))), double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2)))))))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 12 R->L }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(double_divide(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W)), double_divide(double_divide(T, multiply(double_divide(X, Y), S)), double_divide(inverse(S), multiply(double_divide(T, X2), X2)))), double_divide(double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2)), double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))))))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 14 }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(double_divide(inverse(double_divide(X, Y)), double_divide(multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W), inverse(double_divide(X, Y)))), double_divide(double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2)), double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))))))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 13 }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(U2, double_divide(double_divide(multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(double_divide(double_divide(T, multiply(double_divide(X, Y), S)), multiply(double_divide(inverse(double_divide(inverse(S), multiply(double_divide(T, X2), X2))), multiply(double_divide(inverse(Y2), Z2), Z2)), Y2)), W2), W2))), W), inverse(double_divide(X, Y))), U2)))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 7 }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(U2, double_divide(double_divide(multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), multiply(double_divide(inverse(S), multiply(double_divide(T, X2), X2)), double_divide(T, multiply(double_divide(X, Y), S))))), W), inverse(double_divide(X, Y))), U2)))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 4 }
% 0.19/0.55 double_divide(double_divide(U, inverse(double_divide(U2, double_divide(double_divide(multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), inverse(double_divide(X, Y)))), W), inverse(double_divide(X, Y))), U2)))), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by axiom 1 (multiply) R->L }
% 0.19/0.55 double_divide(double_divide(U, multiply(double_divide(double_divide(multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), inverse(double_divide(X, Y)))), W), inverse(double_divide(X, Y))), U2), U2)), double_divide(inverse(double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V))), multiply(double_divide(U, V2), V2)))
% 0.19/0.55 = { by lemma 8 }
% 0.19/0.55 double_divide(double_divide(multiply(double_divide(inverse(double_divide(X, Y)), double_divide(inverse(W), inverse(double_divide(X, Y)))), W), inverse(double_divide(X, Y))), double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V)))
% 0.19/0.55 = { by lemma 10 R->L }
% 0.19/0.55 double_divide(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), inverse(double_divide(X, Y))), double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V)))
% 0.19/0.55 = { by axiom 1 (multiply) R->L }
% 0.19/0.55 double_divide(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), multiply(Y, X)), double_divide(inverse(X), multiply(double_divide(multiply(double_divide(Z, double_divide(inverse(W), Z)), W), V), V)))
% 0.19/0.55 = { by lemma 3 }
% 0.19/0.55 Y
% 0.19/0.55
% 0.19/0.56 Lemma 16: double_divide(X, multiply(double_divide(Y, double_divide(inverse(Z), Y)), Z)) = multiply(double_divide(X, W), W).
% 0.19/0.56 Proof:
% 0.19/0.56 double_divide(X, multiply(double_divide(Y, double_divide(inverse(Z), Y)), Z))
% 0.19/0.56 = { by lemma 8 R->L }
% 0.19/0.56 double_divide(double_divide(multiply(double_divide(Y, double_divide(inverse(Z), Y)), Z), multiply(double_divide(X, W), W)), double_divide(inverse(multiply(double_divide(Y, double_divide(inverse(Z), Y)), Z)), multiply(double_divide(multiply(double_divide(Y, double_divide(inverse(Z), Y)), Z), V), V)))
% 0.19/0.56 = { by lemma 15 }
% 0.19/0.56 multiply(double_divide(X, W), W)
% 0.19/0.56
% 0.19/0.56 Lemma 17: double_divide(multiply(Z, W), double_divide(W, Z)) = double_divide(multiply(X, Y), double_divide(Y, X)).
% 0.19/0.56 Proof:
% 0.19/0.56 double_divide(multiply(Z, W), double_divide(W, Z))
% 0.19/0.56 = { by lemma 7 R->L }
% 0.19/0.56 double_divide(multiply(double_divide(double_divide(W, multiply(double_divide(inverse(Z), multiply(double_divide(inverse(Z2), W2), W2)), Z2)), V2), V2), double_divide(W, Z))
% 0.19/0.56 = { by lemma 16 R->L }
% 0.19/0.56 double_divide(double_divide(double_divide(W, multiply(double_divide(inverse(Z), multiply(double_divide(inverse(Z2), W2), W2)), Z2)), multiply(double_divide(S, double_divide(inverse(X2), S)), X2)), double_divide(W, Z))
% 0.19/0.56 = { by lemma 14 }
% 0.19/0.56 double_divide(Y2, double_divide(multiply(double_divide(S, double_divide(inverse(X2), S)), X2), Y2))
% 0.19/0.56 = { by lemma 14 R->L }
% 0.19/0.56 double_divide(double_divide(double_divide(Y, multiply(double_divide(inverse(X), multiply(double_divide(inverse(V), U), U)), V)), multiply(double_divide(S, double_divide(inverse(X2), S)), X2)), double_divide(Y, X))
% 0.19/0.56 = { by lemma 16 }
% 0.19/0.56 double_divide(multiply(double_divide(double_divide(Y, multiply(double_divide(inverse(X), multiply(double_divide(inverse(V), U), U)), V)), T), T), double_divide(Y, X))
% 0.19/0.56 = { by lemma 7 }
% 0.19/0.56 double_divide(multiply(X, Y), double_divide(Y, X))
% 0.19/0.56
% 0.19/0.56 Lemma 18: double_divide(multiply(X, Y), double_divide(Y, X)) = double_divide(inverse(Z), Z).
% 0.19/0.56 Proof:
% 0.19/0.56 double_divide(multiply(X, Y), double_divide(Y, X))
% 0.19/0.56 = { by lemma 17 }
% 0.19/0.56 double_divide(multiply(double_divide(inverse(W), multiply(double_divide(V, U), U)), double_divide(V, multiply(Z, W))), double_divide(double_divide(V, multiply(Z, W)), double_divide(inverse(W), multiply(double_divide(V, U), U))))
% 0.19/0.56 = { by lemma 3 }
% 0.19/0.56 double_divide(multiply(double_divide(inverse(W), multiply(double_divide(V, U), U)), double_divide(V, multiply(Z, W))), Z)
% 0.19/0.56 = { by lemma 4 }
% 0.19/0.56 double_divide(inverse(Z), Z)
% 0.19/0.56
% 0.19/0.56 Lemma 19: multiply(double_divide(X, Y), multiply(Y, X)) = multiply(Z, inverse(Z)).
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(double_divide(X, Y), multiply(Y, X))
% 0.19/0.56 = { by axiom 1 (multiply) }
% 0.19/0.56 inverse(double_divide(multiply(Y, X), double_divide(X, Y)))
% 0.19/0.56 = { by lemma 18 }
% 0.19/0.56 inverse(double_divide(inverse(Z), Z))
% 0.19/0.56 = { by axiom 1 (multiply) R->L }
% 0.19/0.56 multiply(Z, inverse(Z))
% 0.19/0.56
% 0.19/0.56 Lemma 20: double_divide(inverse(Y), Y) = double_divide(inverse(X), X).
% 0.19/0.56 Proof:
% 0.19/0.56 double_divide(inverse(Y), Y)
% 0.19/0.56 = { by lemma 4 R->L }
% 0.19/0.56 double_divide(multiply(double_divide(inverse(U), multiply(double_divide(T, S), S)), double_divide(T, multiply(Y, U))), Y)
% 0.19/0.56 = { by lemma 3 R->L }
% 0.19/0.56 double_divide(multiply(double_divide(inverse(U), multiply(double_divide(T, S), S)), double_divide(T, multiply(Y, U))), double_divide(double_divide(T, multiply(Y, U)), double_divide(inverse(U), multiply(double_divide(T, S), S))))
% 0.19/0.56 = { by lemma 17 R->L }
% 0.19/0.56 double_divide(multiply(double_divide(inverse(Z), multiply(double_divide(W, V), V)), double_divide(W, multiply(X, Z))), double_divide(double_divide(W, multiply(X, Z)), double_divide(inverse(Z), multiply(double_divide(W, V), V))))
% 0.19/0.56 = { by lemma 3 }
% 0.19/0.56 double_divide(multiply(double_divide(inverse(Z), multiply(double_divide(W, V), V)), double_divide(W, multiply(X, Z))), X)
% 0.19/0.56 = { by lemma 4 }
% 0.19/0.56 double_divide(inverse(X), X)
% 0.19/0.56
% 0.19/0.56 Lemma 21: multiply(double_divide(inverse(Z), Z), X) = multiply(X, double_divide(inverse(Y), Y)).
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(double_divide(inverse(Z), Z), X)
% 0.19/0.56 = { by lemma 20 }
% 0.19/0.56 multiply(double_divide(inverse(X), X), X)
% 0.19/0.56 = { by lemma 5 R->L }
% 0.19/0.56 multiply(double_divide(inverse(X), multiply(double_divide(W, V), V)), multiply(double_divide(W, V), V))
% 0.19/0.56 = { by lemma 16 R->L }
% 0.19/0.56 multiply(double_divide(inverse(X), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(double_divide(inverse(Y), Y), double_divide(inverse(double_divide(inverse(Y), Y)), double_divide(inverse(Y), Y))), double_divide(inverse(Y), Y))))
% 0.19/0.56 = { by lemma 20 }
% 0.19/0.56 multiply(double_divide(inverse(X), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(double_divide(inverse(Y), Y), double_divide(inverse(double_divide(inverse(Y), Y)), double_divide(inverse(Y), Y))), double_divide(inverse(double_divide(inverse(Y), Y)), double_divide(inverse(Y), Y)))))
% 0.19/0.56 = { by lemma 6 }
% 0.19/0.56 multiply(X, double_divide(inverse(Y), Y))
% 0.19/0.56
% 0.19/0.56 Lemma 22: double_divide(X, double_divide(inverse(multiply(Y, Z)), X)) = double_divide(Z, Y).
% 0.19/0.56 Proof:
% 0.19/0.56 double_divide(X, double_divide(inverse(multiply(Y, Z)), X))
% 0.19/0.56 = { by lemma 11 }
% 0.19/0.56 double_divide(multiply(Y, Z), double_divide(inverse(multiply(Y, Z)), multiply(Y, Z)))
% 0.19/0.56 = { by lemma 3 R->L }
% 0.19/0.56 double_divide(double_divide(W, multiply(double_divide(multiply(Y, Z), double_divide(inverse(multiply(Y, Z)), multiply(Y, Z))), double_divide(inverse(multiply(Y, Z)), multiply(Y, Z)))), double_divide(inverse(double_divide(inverse(multiply(Y, Z)), multiply(Y, Z))), multiply(double_divide(W, V), V)))
% 0.19/0.56 = { by lemma 5 }
% 0.19/0.56 double_divide(double_divide(W, multiply(double_divide(multiply(Y, Z), double_divide(Z, Y)), double_divide(Z, Y))), double_divide(inverse(double_divide(inverse(multiply(Y, Z)), multiply(Y, Z))), multiply(double_divide(W, V), V)))
% 0.19/0.56 = { by lemma 18 }
% 0.19/0.56 double_divide(double_divide(W, multiply(double_divide(inverse(U), U), double_divide(Z, Y))), double_divide(inverse(double_divide(inverse(multiply(Y, Z)), multiply(Y, Z))), multiply(double_divide(W, V), V)))
% 0.19/0.56 = { by lemma 21 }
% 0.19/0.56 double_divide(double_divide(W, multiply(double_divide(Z, Y), double_divide(inverse(multiply(Y, Z)), multiply(Y, Z)))), double_divide(inverse(double_divide(inverse(multiply(Y, Z)), multiply(Y, Z))), multiply(double_divide(W, V), V)))
% 0.19/0.56 = { by lemma 3 }
% 0.19/0.56 double_divide(Z, Y)
% 0.19/0.56
% 0.19/0.56 Lemma 23: multiply(X, multiply(Y, inverse(Y))) = X.
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(X, multiply(Y, inverse(Y)))
% 0.19/0.56 = { by lemma 19 R->L }
% 0.19/0.56 multiply(X, multiply(double_divide(Z, W), multiply(W, Z)))
% 0.19/0.56 = { by lemma 22 R->L }
% 0.19/0.56 multiply(X, multiply(double_divide(V, double_divide(inverse(multiply(W, Z)), V)), multiply(W, Z)))
% 0.19/0.56 = { by lemma 15 R->L }
% 0.19/0.56 double_divide(double_divide(multiply(double_divide(V, double_divide(inverse(multiply(W, Z)), V)), multiply(W, Z)), multiply(X, multiply(double_divide(V, double_divide(inverse(multiply(W, Z)), V)), multiply(W, Z)))), double_divide(inverse(multiply(double_divide(V, double_divide(inverse(multiply(W, Z)), V)), multiply(W, Z))), multiply(double_divide(multiply(double_divide(V, double_divide(inverse(multiply(W, Z)), V)), multiply(W, Z)), U), U)))
% 0.19/0.56 = { by lemma 3 }
% 0.19/0.56 X
% 0.19/0.56
% 0.19/0.56 Lemma 24: double_divide(inverse(X), X) = multiply(Y, inverse(Y)).
% 0.19/0.56 Proof:
% 0.19/0.56 double_divide(inverse(X), X)
% 0.19/0.56 = { by lemma 23 R->L }
% 0.19/0.56 multiply(double_divide(inverse(X), X), multiply(X, inverse(X)))
% 0.19/0.56 = { by lemma 19 }
% 0.19/0.56 multiply(Y, inverse(Y))
% 0.19/0.56
% 0.19/0.56 Lemma 25: multiply(multiply(X, inverse(X)), Y) = Y.
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(multiply(X, inverse(X)), Y)
% 0.19/0.56 = { by lemma 24 R->L }
% 0.19/0.56 multiply(double_divide(inverse(Z), Z), Y)
% 0.19/0.56 = { by lemma 21 }
% 0.19/0.56 multiply(Y, double_divide(inverse(W), W))
% 0.19/0.56 = { by lemma 24 }
% 0.19/0.56 multiply(Y, multiply(V, inverse(V)))
% 0.19/0.56 = { by lemma 23 }
% 0.19/0.56 Y
% 0.19/0.56
% 0.19/0.56 Lemma 26: multiply(inverse(X), X) = multiply(Y, inverse(Y)).
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(inverse(X), X)
% 0.19/0.56 = { by lemma 3 R->L }
% 0.19/0.56 multiply(inverse(X), double_divide(double_divide(Z, multiply(X, W)), double_divide(inverse(W), multiply(double_divide(Z, V), V))))
% 0.19/0.56 = { by lemma 25 R->L }
% 0.19/0.56 multiply(inverse(X), double_divide(double_divide(Z, multiply(X, W)), multiply(multiply(U, inverse(U)), double_divide(inverse(W), multiply(double_divide(Z, V), V)))))
% 0.19/0.56 = { by lemma 24 R->L }
% 0.19/0.56 multiply(inverse(X), double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))))
% 0.19/0.56 = { by lemma 25 R->L }
% 0.19/0.56 multiply(inverse(X), double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(multiply(T, inverse(T)), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))))
% 0.19/0.56 = { by lemma 24 R->L }
% 0.19/0.56 multiply(inverse(X), double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))))
% 0.19/0.56 = { by lemma 25 R->L }
% 0.19/0.56 multiply(inverse(X), multiply(multiply(S, inverse(S)), double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V))))))
% 0.19/0.56 = { by axiom 1 (multiply) }
% 0.19/0.56 multiply(inverse(X), inverse(double_divide(double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), multiply(S, inverse(S)))))
% 0.19/0.56 = { by lemma 19 R->L }
% 0.19/0.56 multiply(inverse(X), inverse(double_divide(double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), multiply(double_divide(X2, Y2), multiply(Y2, X2)))))
% 0.19/0.56 = { by lemma 22 R->L }
% 0.19/0.56 multiply(inverse(X), inverse(double_divide(double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), multiply(double_divide(Z2, double_divide(inverse(multiply(Y2, X2)), Z2)), multiply(Y2, X2)))))
% 0.19/0.56 = { by lemma 16 }
% 0.19/0.56 multiply(inverse(X), inverse(multiply(double_divide(double_divide(double_divide(Z, multiply(X, W)), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), multiply(double_divide(inverse(double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), double_divide(inverse(W), multiply(double_divide(Z, V), V)))), W2), W2)))
% 0.19/0.56 = { by lemma 7 }
% 0.19/0.56 multiply(inverse(X), inverse(multiply(double_divide(inverse(W), multiply(double_divide(Z, V), V)), double_divide(Z, multiply(X, W)))))
% 0.19/0.56 = { by lemma 4 }
% 0.19/0.56 multiply(inverse(X), inverse(inverse(X)))
% 0.19/0.56 = { by axiom 1 (multiply) }
% 0.19/0.56 inverse(double_divide(inverse(inverse(X)), inverse(X)))
% 0.19/0.56 = { by lemma 20 R->L }
% 0.19/0.56 inverse(double_divide(inverse(Y), Y))
% 0.19/0.56 = { by axiom 1 (multiply) R->L }
% 0.19/0.56 multiply(Y, inverse(Y))
% 0.19/0.56
% 0.19/0.56 Goal 1 (prove_these_axioms_1): multiply(inverse(a1), a1) = multiply(inverse(b1), b1).
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(inverse(a1), a1)
% 0.19/0.56 = { by lemma 26 }
% 0.19/0.56 multiply(X, inverse(X))
% 0.19/0.56 = { by lemma 26 R->L }
% 0.19/0.56 multiply(inverse(b1), b1)
% 0.19/0.56 % SZS output end Proof
% 0.19/0.56
% 0.19/0.56 RESULT: Unsatisfiable (the axioms are contradictory).
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