TSTP Solution File: GRP062-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP062-1 : TPTP v8.1.2. Released v1.0.0.
% 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 : n011.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 01:16:50 EDT 2023
% Result : Unsatisfiable 0.19s 0.48s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP062-1 : TPTP v8.1.2. Released v1.0.0.
% 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 : n011.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 20:53:41 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.48 Command-line arguments: --no-flatten-goal
% 0.19/0.48
% 0.19/0.48 % SZS status Unsatisfiable
% 0.19/0.48
% 0.19/0.56 % SZS output start Proof
% 0.19/0.56 Take the following subset of the input axioms:
% 0.19/0.56 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)))).
% 0.19/0.56 fof(single_axiom, axiom, ![X, Y, Z, U]: inverse(multiply(X, multiply(Y, multiply(multiply(Z, inverse(Z)), inverse(multiply(U, multiply(X, Y)))))))=U).
% 0.19/0.56
% 0.19/0.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.56 fresh(y, y, x1...xn) = u
% 0.19/0.56 C => fresh(s, t, x1...xn) = v
% 0.19/0.56 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.56 variables of u and v.
% 0.19/0.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.56 input problem has no model of domain size 1).
% 0.19/0.56
% 0.19/0.56 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.56
% 0.19/0.56 Axiom 1 (single_axiom): inverse(multiply(X, multiply(Y, multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(X, Y))))))) = W.
% 0.19/0.56
% 0.19/0.56 Lemma 2: inverse(multiply(X, multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, X)))), multiply(multiply(V, inverse(V)), Z)))) = W.
% 0.19/0.56 Proof:
% 0.19/0.56 inverse(multiply(X, multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, X)))), multiply(multiply(V, inverse(V)), Z))))
% 0.19/0.56 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.56 inverse(multiply(X, multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, X)))), multiply(multiply(V, inverse(V)), inverse(multiply(W, multiply(X, multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, X)))))))))))
% 0.19/0.56 = { by axiom 1 (single_axiom) }
% 0.19/0.56 W
% 0.19/0.56
% 0.19/0.56 Lemma 3: multiply(Y, inverse(Y)) = multiply(X, inverse(X)).
% 0.19/0.56 Proof:
% 0.19/0.56 multiply(Y, inverse(Y))
% 0.19/0.56 = { by lemma 2 R->L }
% 0.19/0.56 inverse(multiply(inverse(multiply(Z, multiply(W, V))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(T, multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, V))))))), multiply(multiply(S, inverse(S)), T))))
% 0.19/0.56 = { by lemma 2 R->L }
% 0.19/0.56 inverse(multiply(inverse(multiply(Z, multiply(W, V))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(T, inverse(multiply(multiply(multiply(X2, inverse(X2)), Z), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(V, multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, V)))), multiply(multiply(X2, inverse(X2)), Z))))), multiply(multiply(Z2, inverse(Z2)), V))))))), multiply(multiply(S, inverse(S)), T))))
% 0.19/0.56 = { by lemma 2 }
% 0.19/0.56 inverse(multiply(inverse(multiply(Z, multiply(W, V))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(T, inverse(multiply(multiply(multiply(X2, inverse(X2)), Z), multiply(multiply(multiply(Y2, inverse(Y2)), W), multiply(multiply(Z2, inverse(Z2)), V))))))), multiply(multiply(S, inverse(S)), T))))
% 0.19/0.56 = { by lemma 2 R->L }
% 0.19/0.56 inverse(multiply(inverse(multiply(Z, multiply(W, V))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(T, inverse(multiply(multiply(multiply(X2, inverse(X2)), Z), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(V, multiply(multiply(multiply(X, inverse(X)), inverse(multiply(Z, multiply(W, V)))), multiply(multiply(X2, inverse(X2)), Z))))), multiply(multiply(Z2, inverse(Z2)), V))))))), multiply(multiply(S, inverse(S)), T))))
% 0.19/0.56 = { by lemma 2 }
% 0.19/0.56 inverse(multiply(inverse(multiply(Z, multiply(W, V))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(T, multiply(multiply(X, inverse(X)), inverse(multiply(Z, multiply(W, V))))))), multiply(multiply(S, inverse(S)), T))))
% 0.19/0.56 = { by lemma 2 }
% 0.19/0.56 multiply(X, inverse(X))
% 0.19/0.56
% 0.19/0.56 Lemma 4: inverse(multiply(multiply(multiply(X, inverse(X)), Y), multiply(multiply(multiply(Z, inverse(Z)), W), multiply(multiply(V, inverse(V)), U)))) = multiply(multiply(T, inverse(T)), inverse(multiply(Y, multiply(W, U)))).
% 0.19/0.56 Proof:
% 0.19/0.57 inverse(multiply(multiply(multiply(X, inverse(X)), Y), multiply(multiply(multiply(Z, inverse(Z)), W), multiply(multiply(V, inverse(V)), U))))
% 0.19/0.57 = { by lemma 2 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(X, inverse(X)), Y), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(U, multiply(multiply(multiply(T, inverse(T)), inverse(multiply(Y, multiply(W, U)))), multiply(multiply(X, inverse(X)), Y))))), multiply(multiply(V, inverse(V)), U))))
% 0.19/0.57 = { by lemma 2 }
% 0.19/0.57 multiply(multiply(T, inverse(T)), inverse(multiply(Y, multiply(W, U))))
% 0.19/0.57
% 0.19/0.57 Lemma 5: inverse(multiply(X, multiply(inverse(X), multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W)))))))) = Z.
% 0.19/0.57 Proof:
% 0.19/0.57 inverse(multiply(X, multiply(inverse(X), multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 inverse(multiply(X, multiply(inverse(X), multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(X, inverse(X))))))))
% 0.19/0.57 = { by axiom 1 (single_axiom) }
% 0.19/0.57 Z
% 0.19/0.57
% 0.19/0.57 Lemma 6: multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(Y, W)))) = inverse(W).
% 0.19/0.57 Proof:
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(Y, W))))
% 0.19/0.57 = { by lemma 4 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(V, inverse(V)), inverse(multiply(Y, multiply(Z, inverse(Z))))), multiply(multiply(multiply(U, inverse(U)), Y), multiply(multiply(T, inverse(T)), W))))
% 0.19/0.57 = { by lemma 5 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(V, inverse(V)), inverse(multiply(Y, multiply(Z, inverse(Z))))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(W, multiply(inverse(W), multiply(multiply(V, inverse(V)), inverse(multiply(Y, multiply(Z, inverse(Z))))))))), multiply(multiply(T, inverse(T)), W))))
% 0.19/0.57 = { by lemma 2 }
% 0.19/0.57 inverse(W)
% 0.19/0.57
% 0.19/0.57 Lemma 7: multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(W, inverse(W))))) = inverse(inverse(Y)).
% 0.19/0.57 Proof:
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(W, inverse(W)))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(Y, inverse(Y)))))
% 0.19/0.57 = { by lemma 6 }
% 0.19/0.57 inverse(inverse(Y))
% 0.19/0.57
% 0.19/0.57 Lemma 8: inverse(multiply(inverse(X), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))), multiply(multiply(V, inverse(V)), Z)))) = X.
% 0.19/0.57 Proof:
% 0.19/0.57 inverse(multiply(inverse(X), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))), multiply(multiply(V, inverse(V)), Z))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 inverse(multiply(inverse(X), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(X, inverse(X))))), multiply(multiply(V, inverse(V)), Z))))
% 0.19/0.57 = { by lemma 2 }
% 0.19/0.57 X
% 0.19/0.57
% 0.19/0.57 Lemma 9: inverse(multiply(inverse(X), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(Z, inverse(Z))), multiply(W, inverse(W))))), multiply(V, inverse(V))))) = X.
% 0.19/0.57 Proof:
% 0.19/0.57 inverse(multiply(inverse(X), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(Z, inverse(Z))), multiply(W, inverse(W))))), multiply(V, inverse(V)))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 inverse(multiply(inverse(X), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(Z, inverse(Z))), multiply(W, inverse(W))))), multiply(multiply(Z, inverse(Z)), inverse(multiply(Z, inverse(Z)))))))
% 0.19/0.57 = { by lemma 8 }
% 0.19/0.57 X
% 0.19/0.57
% 0.19/0.57 Lemma 10: inverse(multiply(multiply(multiply(X, inverse(X)), Y), multiply(multiply(Z, inverse(Z)), multiply(multiply(W, inverse(W)), V)))) = multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(inverse(multiply(T, inverse(T))), V)))).
% 0.19/0.57 Proof:
% 0.19/0.57 inverse(multiply(multiply(multiply(X, inverse(X)), Y), multiply(multiply(Z, inverse(Z)), multiply(multiply(W, inverse(W)), V))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 inverse(multiply(multiply(multiply(X, inverse(X)), Y), multiply(multiply(multiply(T, inverse(T)), inverse(multiply(T, inverse(T)))), multiply(multiply(W, inverse(W)), V))))
% 0.19/0.57 = { by lemma 4 }
% 0.19/0.57 multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(inverse(multiply(T, inverse(T))), V))))
% 0.19/0.57
% 0.19/0.57 Lemma 11: multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, inverse(Y))), multiply(Z, inverse(Z))))) = inverse(multiply(W, inverse(W))).
% 0.19/0.57 Proof:
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(Y, inverse(Y))), multiply(Z, inverse(Z)))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(multiply(V, multiply(inverse(V), multiply(multiply(U, inverse(U)), inverse(multiply(multiply(W, inverse(W)), multiply(T, inverse(T))))))), inverse(multiply(V, multiply(inverse(V), multiply(multiply(U, inverse(U)), inverse(multiply(multiply(W, inverse(W)), multiply(T, inverse(T)))))))))), multiply(Z, inverse(Z)))))
% 0.19/0.57 = { by lemma 5 }
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(multiply(V, multiply(inverse(V), multiply(multiply(U, inverse(U)), inverse(multiply(multiply(W, inverse(W)), multiply(T, inverse(T))))))), multiply(W, inverse(W)))), multiply(Z, inverse(Z)))))
% 0.19/0.57 = { by lemma 7 }
% 0.19/0.57 inverse(inverse(multiply(V, multiply(inverse(V), multiply(multiply(U, inverse(U)), inverse(multiply(multiply(W, inverse(W)), multiply(T, inverse(T)))))))))
% 0.19/0.57 = { by lemma 5 }
% 0.19/0.57 inverse(multiply(W, inverse(W)))
% 0.19/0.57
% 0.19/0.57 Lemma 12: inverse(multiply(inverse(X), multiply(inverse(multiply(Y, inverse(Y))), multiply(Z, inverse(Z))))) = X.
% 0.19/0.57 Proof:
% 0.19/0.57 inverse(multiply(inverse(X), multiply(inverse(multiply(Y, inverse(Y))), multiply(Z, inverse(Z)))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 inverse(multiply(inverse(X), multiply(inverse(multiply(Y, inverse(Y))), multiply(multiply(W, inverse(W)), inverse(multiply(W, inverse(W)))))))
% 0.19/0.57 = { by lemma 11 R->L }
% 0.19/0.57 inverse(multiply(inverse(X), multiply(multiply(multiply(V, inverse(V)), inverse(multiply(inverse(multiply(W, inverse(W))), multiply(U, inverse(U))))), multiply(multiply(W, inverse(W)), inverse(multiply(W, inverse(W)))))))
% 0.19/0.57 = { by lemma 8 }
% 0.19/0.57 X
% 0.19/0.57
% 0.19/0.57 Lemma 13: inverse(multiply(multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y))), multiply(multiply(Z, inverse(Z)), multiply(W, inverse(W))))) = multiply(V, inverse(V)).
% 0.19/0.57 Proof:
% 0.19/0.57 inverse(multiply(multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y))), multiply(multiply(Z, inverse(Z)), multiply(W, inverse(W)))))
% 0.19/0.57 = { by lemma 3 }
% 0.19/0.57 inverse(multiply(multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y))), multiply(multiply(Z, inverse(Z)), multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))))))
% 0.19/0.57 = { by lemma 10 }
% 0.19/0.57 multiply(multiply(T, inverse(T)), inverse(multiply(multiply(Y, inverse(Y)), multiply(inverse(multiply(S, inverse(S))), inverse(multiply(U, inverse(U)))))))
% 0.19/0.57 = { by lemma 10 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))), multiply(multiply(Y2, inverse(Y2)), inverse(multiply(U, inverse(U)))))))
% 0.19/0.57 = { by lemma 12 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))), multiply(multiply(Y2, inverse(Y2)), inverse(inverse(multiply(inverse(multiply(U, inverse(U))), multiply(inverse(multiply(Z2, inverse(Z2))), multiply(W2, inverse(W2))))))))))
% 0.19/0.57 = { by lemma 11 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))), multiply(multiply(Y2, inverse(Y2)), inverse(inverse(multiply(multiply(multiply(V2, inverse(V2)), inverse(multiply(inverse(multiply(U2, inverse(U2))), multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y))))))), multiply(inverse(multiply(Z2, inverse(Z2))), multiply(W2, inverse(W2))))))))))
% 0.19/0.57 = { by lemma 4 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))), multiply(multiply(Y2, inverse(Y2)), inverse(inverse(multiply(inverse(multiply(multiply(multiply(U2, inverse(U2)), inverse(multiply(U2, inverse(U2)))), multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y))))))), multiply(inverse(multiply(Z2, inverse(Z2))), multiply(W2, inverse(W2))))))))))
% 0.19/0.57 = { by lemma 3 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))), multiply(multiply(Y2, inverse(Y2)), inverse(inverse(multiply(inverse(multiply(multiply(V, inverse(V)), multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y))))))), multiply(inverse(multiply(Z2, inverse(Z2))), multiply(W2, inverse(W2))))))))))
% 0.19/0.57 = { by lemma 12 }
% 0.19/0.57 inverse(multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))), multiply(multiply(Y2, inverse(Y2)), inverse(multiply(multiply(V, inverse(V)), multiply(multiply(multiply(X2, inverse(X2)), multiply(Y, inverse(Y))), multiply(multiply(Y, inverse(Y)), inverse(multiply(Y, inverse(Y)))))))))))
% 0.19/0.57 = { by axiom 1 (single_axiom) }
% 0.19/0.57 multiply(V, inverse(V))
% 0.19/0.57
% 0.19/0.57 Lemma 14: multiply(multiply(X, inverse(X)), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))), multiply(V, inverse(V)))) = inverse(multiply(multiply(U, inverse(U)), Z)).
% 0.19/0.57 Proof:
% 0.19/0.57 multiply(multiply(X, inverse(X)), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))), multiply(V, inverse(V))))
% 0.19/0.57 = { by lemma 8 R->L }
% 0.19/0.57 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))), multiply(V, inverse(V)))), multiply(multiply(multiply(Y, inverse(Y)), inverse(multiply(Z, multiply(W, inverse(W))))), multiply(multiply(U, inverse(U)), Z)))))
% 0.19/0.57 = { by lemma 6 }
% 0.19/0.57 inverse(multiply(multiply(U, inverse(U)), Z))
% 0.19/0.57
% 0.19/0.57 Lemma 15: multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y))) = multiply(Z, inverse(Z)).
% 0.19/0.57 Proof:
% 0.19/0.57 multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y)))
% 0.19/0.57 = { by lemma 2 R->L }
% 0.19/0.57 inverse(multiply(multiply(multiply(W, inverse(W)), multiply(V, inverse(V))), multiply(multiply(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))))), inverse(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), multiply(multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y))), multiply(multiply(W, inverse(W)), multiply(V, inverse(V))))))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))), inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))))))))
% 0.19/0.58 = { by lemma 9 R->L }
% 0.19/0.58 inverse(multiply(multiply(multiply(W, inverse(W)), multiply(V, inverse(V))), multiply(multiply(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))))), inverse(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(multiply(inverse(multiply(multiply(multiply(X, inverse(X)), multiply(Y, inverse(Y))), multiply(multiply(W, inverse(W)), multiply(V, inverse(V))))), multiply(multiply(multiply(T, inverse(T)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(S, inverse(S))))), multiply(X2, inverse(X2)))))))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))), inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))))))))
% 0.19/0.58 = { by lemma 13 }
% 0.19/0.58 inverse(multiply(multiply(multiply(W, inverse(W)), multiply(V, inverse(V))), multiply(multiply(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))))), inverse(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(multiply(multiply(Y2, inverse(Y2)), multiply(multiply(multiply(T, inverse(T)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(S, inverse(S))))), multiply(X2, inverse(X2)))))))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))), inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))))))))
% 0.19/0.58 = { by lemma 14 }
% 0.19/0.58 inverse(multiply(multiply(multiply(W, inverse(W)), multiply(V, inverse(V))), multiply(multiply(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))))), inverse(multiply(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))), inverse(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U))))))))), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))), inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))))))))
% 0.19/0.58 = { by lemma 13 }
% 0.19/0.58 multiply(Z, inverse(Z))
% 0.19/0.58
% 0.19/0.58 Lemma 16: inverse(multiply(X, inverse(X))) = multiply(Y, inverse(Y)).
% 0.19/0.58 Proof:
% 0.19/0.58 inverse(multiply(X, inverse(X)))
% 0.19/0.58 = { by lemma 15 R->L }
% 0.19/0.58 inverse(multiply(multiply(Z, inverse(Z)), multiply(W, inverse(W))))
% 0.19/0.58 = { by lemma 14 R->L }
% 0.19/0.58 multiply(multiply(V, inverse(V)), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(multiply(W, inverse(W)), multiply(T, inverse(T))))), multiply(S, inverse(S))))
% 0.19/0.58 = { by lemma 15 }
% 0.19/0.58 multiply(multiply(V, inverse(V)), multiply(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))), multiply(S, inverse(S))))
% 0.19/0.58 = { by lemma 15 }
% 0.19/0.58 multiply(multiply(V, inverse(V)), multiply(X2, inverse(X2)))
% 0.19/0.58 = { by lemma 15 }
% 0.19/0.58 multiply(Y, inverse(Y))
% 0.19/0.58
% 0.19/0.58 Lemma 17: inverse(inverse(X)) = X.
% 0.19/0.58 Proof:
% 0.19/0.58 inverse(inverse(X))
% 0.19/0.58 = { by lemma 7 R->L }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), multiply(W, inverse(W)))))
% 0.19/0.58 = { by lemma 16 R->L }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(multiply(V, inverse(V))))))
% 0.19/0.58 = { by lemma 16 R->L }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(inverse(multiply(multiply(U, inverse(U)), inverse(multiply(U, inverse(U)))))))))
% 0.19/0.58 = { by lemma 14 R->L }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(multiply(multiply(T, inverse(T)), multiply(multiply(multiply(S, inverse(S)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(X2, inverse(X2))))), multiply(Y2, inverse(Y2))))))))
% 0.19/0.58 = { by lemma 16 R->L }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(multiply(inverse(multiply(Z2, inverse(Z2))), multiply(multiply(multiply(S, inverse(S)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(X2, inverse(X2))))), multiply(Y2, inverse(Y2))))))))
% 0.19/0.58 = { by lemma 5 R->L }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(multiply(inverse(multiply(X, multiply(inverse(X), multiply(multiply(W2, inverse(W2)), inverse(multiply(inverse(multiply(Z2, inverse(Z2))), multiply(V2, inverse(V2)))))))), multiply(multiply(multiply(S, inverse(S)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(X2, inverse(X2))))), multiply(Y2, inverse(Y2))))))))
% 0.19/0.58 = { by lemma 11 }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(multiply(inverse(multiply(X, multiply(inverse(X), inverse(multiply(U2, inverse(U2)))))), multiply(multiply(multiply(S, inverse(S)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(X2, inverse(X2))))), multiply(Y2, inverse(Y2))))))))
% 0.19/0.58 = { by lemma 16 }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), inverse(multiply(inverse(multiply(X, multiply(inverse(X), multiply(T2, inverse(T2))))), multiply(multiply(multiply(S, inverse(S)), inverse(multiply(inverse(multiply(U, inverse(U))), multiply(X2, inverse(X2))))), multiply(Y2, inverse(Y2))))))))
% 0.19/0.58 = { by lemma 9 }
% 0.19/0.58 multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(multiply(X, multiply(Z, inverse(Z)))), multiply(X, multiply(inverse(X), multiply(T2, inverse(T2)))))))
% 0.19/0.58 = { by lemma 6 }
% 0.19/0.58 inverse(multiply(inverse(X), multiply(T2, inverse(T2))))
% 0.19/0.58 = { by lemma 15 R->L }
% 0.19/0.58 inverse(multiply(inverse(X), multiply(multiply(multiply(S2, inverse(S2)), inverse(multiply(S2, inverse(S2)))), multiply(X3, inverse(X3)))))
% 0.19/0.58 = { by lemma 15 R->L }
% 0.19/0.58 inverse(multiply(inverse(X), multiply(multiply(multiply(S2, inverse(S2)), inverse(multiply(S2, inverse(S2)))), multiply(multiply(Y3, inverse(Y3)), multiply(Z3, inverse(Z3))))))
% 0.19/0.58 = { by lemma 15 R->L }
% 0.19/0.58 inverse(multiply(inverse(X), multiply(multiply(multiply(S2, inverse(S2)), inverse(multiply(multiply(Z3, inverse(Z3)), multiply(W3, inverse(W3))))), multiply(multiply(Y3, inverse(Y3)), multiply(Z3, inverse(Z3))))))
% 0.19/0.58 = { by lemma 8 }
% 0.19/0.58 X
% 0.19/0.58
% 0.19/0.58 Lemma 18: multiply(inverse(X), X) = multiply(Y, inverse(Y)).
% 0.19/0.58 Proof:
% 0.19/0.58 multiply(inverse(X), X)
% 0.19/0.58 = { by lemma 17 R->L }
% 0.19/0.58 multiply(inverse(X), inverse(inverse(X)))
% 0.19/0.58 = { by lemma 3 R->L }
% 0.19/0.58 multiply(Y, inverse(Y))
% 0.19/0.58
% 0.19/0.58 Lemma 19: multiply(X, multiply(inverse(X), inverse(inverse(Y)))) = multiply(Y, multiply(Z, inverse(Z))).
% 0.19/0.58 Proof:
% 0.19/0.58 multiply(X, multiply(inverse(X), inverse(inverse(Y))))
% 0.19/0.58 = { by lemma 8 R->L }
% 0.19/0.58 inverse(multiply(inverse(multiply(X, multiply(inverse(X), inverse(inverse(Y))))), multiply(multiply(multiply(W, inverse(W)), inverse(multiply(V, multiply(U, inverse(U))))), multiply(multiply(T, inverse(T)), V))))
% 0.19/0.58 = { by lemma 7 R->L }
% 0.19/0.58 inverse(multiply(inverse(multiply(X, multiply(inverse(X), multiply(multiply(S, inverse(S)), inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(X2, inverse(X2)))))))), multiply(multiply(multiply(W, inverse(W)), inverse(multiply(V, multiply(U, inverse(U))))), multiply(multiply(T, inverse(T)), V))))
% 0.19/0.58 = { by lemma 5 }
% 0.19/0.58 inverse(multiply(inverse(multiply(Y, multiply(Z, inverse(Z)))), multiply(multiply(multiply(W, inverse(W)), inverse(multiply(V, multiply(U, inverse(U))))), multiply(multiply(T, inverse(T)), V))))
% 0.19/0.58 = { by lemma 8 }
% 0.19/0.58 multiply(Y, multiply(Z, inverse(Z)))
% 0.19/0.58
% 0.19/0.58 Lemma 20: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.19/0.58 Proof:
% 0.19/0.58 multiply(inverse(X), multiply(X, Y))
% 0.19/0.58 = { by lemma 17 R->L }
% 0.19/0.58 multiply(inverse(X), multiply(X, inverse(inverse(Y))))
% 0.19/0.58 = { by lemma 17 R->L }
% 0.19/0.58 multiply(inverse(X), multiply(inverse(inverse(X)), inverse(inverse(Y))))
% 0.19/0.58 = { by lemma 19 }
% 0.19/0.58 multiply(Y, multiply(Z, inverse(Z)))
% 0.19/0.58 = { by lemma 19 R->L }
% 0.19/0.58 multiply(inverse(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T))))), multiply(inverse(inverse(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T)))))), inverse(inverse(Y))))
% 0.19/0.58 = { by lemma 17 }
% 0.19/0.58 multiply(inverse(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T))))), multiply(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T)))), inverse(inverse(Y))))
% 0.19/0.58 = { by lemma 17 }
% 0.19/0.58 multiply(inverse(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T))))), multiply(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T)))), Y))
% 0.19/0.58 = { by lemma 12 }
% 0.19/0.58 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(inverse(multiply(T, inverse(T))), multiply(T, inverse(T)))), Y))
% 0.19/0.58 = { by lemma 18 }
% 0.19/0.58 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(S, inverse(S))), Y))
% 0.19/0.58 = { by lemma 16 R->L }
% 0.19/0.58 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), inverse(multiply(X2, inverse(X2)))), Y))
% 0.19/0.59 = { by lemma 8 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(inverse(multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), inverse(multiply(X2, inverse(X2))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by lemma 6 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(multiply(multiply(U2, inverse(U2)), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(T2, inverse(T2)))), multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), inverse(multiply(X2, inverse(X2)))))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by lemma 8 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(multiply(multiply(U2, inverse(U2)), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(T2, inverse(T2)))), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), inverse(multiply(X2, inverse(X2)))))), multiply(multiply(multiply(S2, inverse(S2)), inverse(multiply(X3, multiply(Y3, inverse(Y3))))), multiply(multiply(Z3, inverse(Z3)), X3))))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by lemma 11 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(multiply(multiply(U2, inverse(U2)), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(T2, inverse(T2)))), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), multiply(multiply(W3, inverse(W3)), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))))), multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))))))))))), multiply(multiply(multiply(S2, inverse(S2)), inverse(multiply(X3, multiply(Y3, inverse(Y3))))), multiply(multiply(Z3, inverse(Z3)), X3))))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by axiom 1 (single_axiom) }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(multiply(multiply(U2, inverse(U2)), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(T2, inverse(T2)))), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))))), multiply(multiply(multiply(S2, inverse(S2)), inverse(multiply(X3, multiply(Y3, inverse(Y3))))), multiply(multiply(Z3, inverse(Z3)), X3))))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by lemma 8 }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(multiply(multiply(U2, inverse(U2)), inverse(multiply(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(T2, inverse(T2)))), multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by lemma 6 }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(inverse(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))))), multiply(multiply(multiply(Y2, inverse(Y2)), inverse(multiply(Z2, multiply(W2, inverse(W2))))), multiply(multiply(V2, inverse(V2)), Z2)))), Y))
% 0.19/0.59 = { by lemma 8 }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), Y))
% 0.19/0.59 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))))
% 0.19/0.59 = { by lemma 17 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))), inverse(inverse(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))))))
% 0.19/0.59 = { by lemma 18 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(inverse(multiply(V3, inverse(V3))), multiply(V3, inverse(V3))))
% 0.19/0.59 = { by lemma 11 R->L }
% 0.19/0.59 multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(multiply(multiply(U3, inverse(U3)), inverse(multiply(inverse(multiply(T3, inverse(T3))), multiply(S3, inverse(S3))))), multiply(V3, inverse(V3))))
% 0.19/0.59 = { by lemma 17 R->L }
% 0.19/0.59 inverse(inverse(multiply(inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V))))))), multiply(multiply(multiply(U3, inverse(U3)), inverse(multiply(inverse(multiply(T3, inverse(T3))), multiply(S3, inverse(S3))))), multiply(V3, inverse(V3))))))
% 0.19/0.59 = { by lemma 9 }
% 0.19/0.59 inverse(multiply(W, multiply(V, multiply(multiply(U, inverse(U)), inverse(multiply(Y, multiply(W, V)))))))
% 0.19/0.59 = { by axiom 1 (single_axiom) }
% 0.19/0.59 Y
% 0.19/0.59
% 0.19/0.59 Lemma 21: multiply(multiply(X, inverse(X)), inverse(Y)) = inverse(Y).
% 0.19/0.59 Proof:
% 0.19/0.59 multiply(multiply(X, inverse(X)), inverse(Y))
% 0.19/0.59 = { by lemma 20 R->L }
% 0.19/0.59 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(inverse(Z)), multiply(inverse(Z), Y))))
% 0.19/0.59 = { by lemma 20 R->L }
% 0.19/0.59 multiply(multiply(X, inverse(X)), inverse(multiply(inverse(multiply(inverse(Z), multiply(Z, inverse(Z)))), multiply(inverse(Z), Y))))
% 0.19/0.59 = { by lemma 6 }
% 0.19/0.59 inverse(Y)
% 0.19/0.59
% 0.19/0.59 Lemma 22: multiply(multiply(X, inverse(X)), Y) = Y.
% 0.19/0.59 Proof:
% 0.19/0.59 multiply(multiply(X, inverse(X)), Y)
% 0.19/0.59 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.59 multiply(multiply(X, inverse(X)), inverse(multiply(Z, multiply(W, multiply(multiply(V, inverse(V)), inverse(multiply(Y, multiply(Z, W))))))))
% 0.19/0.59 = { by lemma 21 }
% 0.19/0.59 inverse(multiply(Z, multiply(W, multiply(multiply(V, inverse(V)), inverse(multiply(Y, multiply(Z, W)))))))
% 0.19/0.59 = { by axiom 1 (single_axiom) }
% 0.19/0.59 Y
% 0.19/0.59
% 0.19/0.59 Lemma 23: inverse(multiply(X, inverse(multiply(Y, X)))) = Y.
% 0.19/0.59 Proof:
% 0.19/0.59 inverse(multiply(X, inverse(multiply(Y, X))))
% 0.19/0.59 = { by lemma 22 R->L }
% 0.19/0.59 inverse(multiply(X, inverse(multiply(Y, multiply(multiply(Z, inverse(Z)), X)))))
% 0.19/0.59 = { by lemma 21 R->L }
% 0.19/0.59 inverse(multiply(X, multiply(multiply(W, inverse(W)), inverse(multiply(Y, multiply(multiply(Z, inverse(Z)), X))))))
% 0.19/0.59 = { by lemma 22 R->L }
% 0.19/0.59 inverse(multiply(multiply(Z, inverse(Z)), multiply(X, multiply(multiply(W, inverse(W)), inverse(multiply(Y, multiply(multiply(Z, inverse(Z)), X)))))))
% 0.19/0.59 = { by axiom 1 (single_axiom) }
% 0.19/0.59 Y
% 0.19/0.59
% 0.19/0.59 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))).
% 0.19/0.59 Proof:
% 0.19/0.59 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3))
% 0.19/0.59 = { by lemma 18 }
% 0.19/0.59 tuple(multiply(X, inverse(X)), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3))
% 0.19/0.59 = { by lemma 18 }
% 0.19/0.59 tuple(multiply(X, inverse(X)), multiply(multiply(Y, inverse(Y)), a2), multiply(multiply(a3, b3), c3))
% 0.19/0.59 = { by lemma 22 }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, multiply(multiply(a3, b3), c3))
% 0.19/0.59 = { by lemma 8 R->L }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(multiply(a3, b3), c3)), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.59 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(a3, multiply(b3, multiply(multiply(T, inverse(T)), inverse(multiply(inverse(multiply(multiply(a3, b3), c3)), multiply(a3, b3))))))), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.59 = { by lemma 23 R->L }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(a3, multiply(b3, multiply(multiply(T, inverse(T)), inverse(multiply(inverse(multiply(multiply(a3, b3), c3)), inverse(multiply(c3, inverse(multiply(multiply(a3, b3), c3)))))))))), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.59 = { by lemma 17 R->L }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(a3, multiply(b3, multiply(multiply(T, inverse(T)), inverse(inverse(inverse(multiply(inverse(multiply(multiply(a3, b3), c3)), inverse(multiply(c3, inverse(multiply(multiply(a3, b3), c3)))))))))))), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.59 = { by lemma 23 }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(a3, multiply(b3, multiply(multiply(T, inverse(T)), inverse(inverse(c3)))))), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.59 = { by lemma 21 }
% 0.19/0.59 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(a3, multiply(b3, inverse(inverse(c3))))), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.60 = { by lemma 17 }
% 0.19/0.60 tuple(multiply(X, inverse(X)), a2, inverse(multiply(inverse(multiply(a3, multiply(b3, c3))), multiply(multiply(multiply(Z, inverse(Z)), inverse(multiply(W, multiply(V, inverse(V))))), multiply(multiply(U, inverse(U)), W)))))
% 0.19/0.60 = { by lemma 8 }
% 0.19/0.60 tuple(multiply(X, inverse(X)), a2, multiply(a3, multiply(b3, c3)))
% 0.19/0.60 = { by lemma 18 R->L }
% 0.19/0.60 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)))
% 0.19/0.60 % SZS output end Proof
% 0.19/0.60
% 0.19/0.60 RESULT: Unsatisfiable (the axioms are contradictory).
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