TSTP Solution File: GRP060-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP060-1 : 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 : n026.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 01:16:49 EDT 2023
% Result : Unsatisfiable 0.19s 0.68s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP060-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % 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 : n026.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 21:45:05 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.68 Command-line arguments: --no-flatten-goal
% 0.19/0.68
% 0.19/0.68 % SZS status Unsatisfiable
% 0.19/0.68
% 0.19/0.80 % SZS output start Proof
% 0.19/0.80 Take the following subset of the input axioms:
% 0.19/0.80 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)))).
% 0.19/0.80 fof(single_axiom, axiom, ![X, Y, Z, U]: multiply(X, inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(U, Y))), X)))))=U).
% 0.19/0.80
% 0.19/0.80 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.80 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.80 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.80 fresh(y, y, x1...xn) = u
% 0.19/0.80 C => fresh(s, t, x1...xn) = v
% 0.19/0.80 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.80 variables of u and v.
% 0.19/0.80 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.80 input problem has no model of domain size 1).
% 0.19/0.80
% 0.19/0.80 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.80
% 0.19/0.80 Axiom 1 (single_axiom): multiply(X, inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(W, Y))), X))))) = W.
% 0.19/0.80
% 0.19/0.80 Lemma 2: multiply(X, inverse(multiply(multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(Z, W))), inverse(V))), multiply(V, multiply(Z, X))))) = W.
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(X, inverse(multiply(multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(Z, W))), inverse(V))), multiply(V, multiply(Z, X)))))
% 0.19/0.80 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.80 multiply(X, inverse(multiply(multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(Z, W))), inverse(V))), multiply(V, multiply(multiply(inverse(V), inverse(multiply(W, multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(Z, W))), inverse(V)))))), X)))))
% 0.19/0.80 = { by axiom 1 (single_axiom) }
% 0.19/0.80 W
% 0.19/0.80
% 0.19/0.80 Lemma 3: multiply(X, inverse(multiply(multiply(Y, multiply(Z, inverse(W))), multiply(W, multiply(V, X))))) = multiply(U, multiply(multiply(inverse(U), inverse(multiply(Z, V))), inverse(Y))).
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(X, inverse(multiply(multiply(Y, multiply(Z, inverse(W))), multiply(W, multiply(V, X)))))
% 0.19/0.80 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.80 multiply(X, inverse(multiply(multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(Z, V))), inverse(Y)))))), inverse(W))), multiply(W, multiply(V, X)))))
% 0.19/0.80 = { by lemma 2 }
% 0.19/0.80 multiply(U, multiply(multiply(inverse(U), inverse(multiply(Z, V))), inverse(Y)))
% 0.19/0.80
% 0.19/0.80 Lemma 4: multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(multiply(Z, W))), Z))), inverse(Y))) = W.
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(multiply(Z, W))), Z))), inverse(Y)))
% 0.19/0.80 = { by lemma 3 R->L }
% 0.19/0.80 multiply(V, inverse(multiply(multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(Z, W))), inverse(U))), multiply(U, multiply(Z, V)))))
% 0.19/0.80 = { by lemma 2 }
% 0.19/0.80 W
% 0.19/0.80
% 0.19/0.80 Lemma 5: multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(Z)), W))), inverse(Y))) = inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(Z, V))), W)))).
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(Z)), W))), inverse(Y)))
% 0.19/0.80 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.80 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(multiply(W, inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(Z, V))), W))))))), W))), inverse(Y)))
% 0.19/0.80 = { by lemma 4 }
% 0.19/0.80 inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(Z, V))), W))))
% 0.19/0.80
% 0.19/0.80 Lemma 6: inverse(multiply(X, multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(multiply(Z, W), X))), Z)))) = W.
% 0.19/0.80 Proof:
% 0.19/0.80 inverse(multiply(X, multiply(Y, multiply(multiply(inverse(Y), inverse(multiply(multiply(Z, W), X))), Z))))
% 0.19/0.80 = { by lemma 5 R->L }
% 0.19/0.80 multiply(V, multiply(multiply(inverse(V), inverse(multiply(multiply(inverse(U), inverse(multiply(Z, W))), Z))), inverse(U)))
% 0.19/0.80 = { by lemma 4 }
% 0.19/0.80 W
% 0.19/0.80
% 0.19/0.80 Lemma 7: multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(multiply(Z, W))), Z))), inverse(Y)) = inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(W, V))), X)))).
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(multiply(Z, W))), Z))), inverse(Y))
% 0.19/0.80 = { by lemma 2 R->L }
% 0.19/0.80 multiply(T, inverse(multiply(multiply(S, multiply(multiply(inverse(S), inverse(multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(Y), inverse(multiply(Z, W))), Z))), inverse(Y))))), inverse(X2))), multiply(X2, multiply(X, T)))))
% 0.19/0.80 = { by lemma 4 }
% 0.19/0.80 multiply(T, inverse(multiply(multiply(S, multiply(multiply(inverse(S), inverse(W)), inverse(X2))), multiply(X2, multiply(X, T)))))
% 0.19/0.80 = { by lemma 3 }
% 0.19/0.80 multiply(Y2, multiply(multiply(inverse(Y2), inverse(multiply(multiply(inverse(S), inverse(W)), X))), inverse(S)))
% 0.19/0.80 = { by lemma 5 }
% 0.19/0.80 inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(W, V))), X))))
% 0.19/0.80
% 0.19/0.80 Lemma 8: multiply(X, multiply(multiply(inverse(X), inverse(multiply(Y, multiply(inverse(Z), inverse(multiply(W, multiply(V, multiply(Y, inverse(Z))))))))), inverse(V))) = W.
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(X, multiply(multiply(inverse(X), inverse(multiply(Y, multiply(inverse(Z), inverse(multiply(W, multiply(V, multiply(Y, inverse(Z))))))))), inverse(V)))
% 0.19/0.80 = { by lemma 3 R->L }
% 0.19/0.80 multiply(U, inverse(multiply(multiply(V, multiply(Y, inverse(Z))), multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(W, multiply(V, multiply(Y, inverse(Z)))))), U)))))
% 0.19/0.80 = { by axiom 1 (single_axiom) }
% 0.19/0.80 W
% 0.19/0.80
% 0.19/0.80 Lemma 9: multiply(V, inverse(multiply(Y, multiply(Z, multiply(W, V))))) = multiply(X, inverse(multiply(Y, multiply(Z, multiply(W, X))))).
% 0.19/0.80 Proof:
% 0.19/0.80 multiply(V, inverse(multiply(Y, multiply(Z, multiply(W, V)))))
% 0.19/0.80 = { by lemma 8 R->L }
% 0.19/0.80 multiply(V, inverse(multiply(multiply(S, multiply(multiply(inverse(S), inverse(multiply(W, multiply(inverse(T), inverse(multiply(Y, multiply(Z, multiply(W, inverse(T))))))))), inverse(Z))), multiply(Z, multiply(W, V)))))
% 0.19/0.80 = { by lemma 2 }
% 0.19/0.80 multiply(inverse(T), inverse(multiply(Y, multiply(Z, multiply(W, inverse(T))))))
% 0.19/0.80 = { by lemma 2 R->L }
% 0.19/0.80 multiply(X, inverse(multiply(multiply(U, multiply(multiply(inverse(U), inverse(multiply(W, multiply(inverse(T), inverse(multiply(Y, multiply(Z, multiply(W, inverse(T))))))))), inverse(Z))), multiply(Z, multiply(W, X)))))
% 0.19/0.81 = { by lemma 8 }
% 0.19/0.81 multiply(X, inverse(multiply(Y, multiply(Z, multiply(W, X)))))
% 0.19/0.81
% 0.19/0.81 Lemma 10: multiply(inverse(X), inverse(multiply(multiply(Y, inverse(multiply(Z, multiply(X, multiply(W, Y))))), Z))) = W.
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(inverse(X), inverse(multiply(multiply(Y, inverse(multiply(Z, multiply(X, multiply(W, Y))))), Z)))
% 0.19/0.81 = { by lemma 9 }
% 0.19/0.81 multiply(inverse(X), inverse(multiply(multiply(inverse(V), inverse(multiply(Z, multiply(X, multiply(W, inverse(V)))))), Z)))
% 0.19/0.81 = { by lemma 8 R->L }
% 0.19/0.81 multiply(inverse(X), inverse(multiply(multiply(inverse(V), inverse(multiply(Z, multiply(X, multiply(W, inverse(V)))))), multiply(U, multiply(multiply(inverse(U), inverse(multiply(W, multiply(inverse(V), inverse(multiply(Z, multiply(X, multiply(W, inverse(V))))))))), inverse(X))))))
% 0.19/0.81 = { by axiom 1 (single_axiom) }
% 0.19/0.81 W
% 0.19/0.81
% 0.19/0.81 Lemma 11: multiply(multiply(inverse(X), inverse(multiply(Y, multiply(Z, inverse(multiply(W, multiply(V, multiply(Y, Z)))))))), inverse(V)) = inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(W, U))), X)))).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, multiply(Z, inverse(multiply(W, multiply(V, multiply(Y, Z)))))))), inverse(V))
% 0.19/0.81 = { by lemma 10 R->L }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(V), inverse(multiply(multiply(Z, inverse(multiply(W, multiply(V, multiply(Y, Z))))), W))), multiply(Z, inverse(multiply(W, multiply(V, multiply(Y, Z)))))))), inverse(V))
% 0.19/0.81 = { by lemma 7 }
% 0.19/0.81 inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(W, U))), X))))
% 0.19/0.81
% 0.19/0.81 Lemma 12: inverse(multiply(X, multiply(W, multiply(inverse(W), Z)))) = inverse(multiply(X, multiply(Y, multiply(inverse(Y), Z)))).
% 0.19/0.81 Proof:
% 0.19/0.81 inverse(multiply(X, multiply(W, multiply(inverse(W), Z))))
% 0.19/0.81 = { by lemma 6 R->L }
% 0.19/0.81 inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(Z, inverse(multiply(X, multiply(W, multiply(inverse(W), Z))))), Y2))), Z))))
% 0.19/0.81 = { by lemma 7 R->L }
% 0.19/0.81 multiply(multiply(inverse(Z), inverse(multiply(multiply(inverse(T), inverse(multiply(inverse(W), multiply(Z, inverse(multiply(X, multiply(W, multiply(inverse(W), Z)))))))), inverse(W)))), inverse(T))
% 0.19/0.81 = { by lemma 11 }
% 0.19/0.81 multiply(multiply(inverse(Z), inverse(inverse(multiply(S, multiply(X2, multiply(multiply(inverse(X2), inverse(multiply(X, S))), T)))))), inverse(T))
% 0.19/0.81 = { by lemma 11 R->L }
% 0.19/0.81 multiply(multiply(inverse(Z), inverse(multiply(multiply(inverse(T), inverse(multiply(inverse(Y), multiply(Z, inverse(multiply(X, multiply(Y, multiply(inverse(Y), Z)))))))), inverse(Y)))), inverse(T))
% 0.19/0.81 = { by lemma 7 }
% 0.19/0.81 inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(Z, inverse(multiply(X, multiply(Y, multiply(inverse(Y), Z))))), V))), Z))))
% 0.19/0.81 = { by lemma 6 }
% 0.19/0.81 inverse(multiply(X, multiply(Y, multiply(inverse(Y), Z))))
% 0.19/0.81
% 0.19/0.81 Lemma 13: multiply(Z, multiply(inverse(Z), Y)) = multiply(X, multiply(inverse(X), Y)).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(Z, multiply(inverse(Z), Y))
% 0.19/0.81 = { by lemma 2 R->L }
% 0.19/0.81 multiply(W, inverse(multiply(multiply(V, multiply(multiply(inverse(V), inverse(multiply(U, multiply(Z, multiply(inverse(Z), Y))))), inverse(T))), multiply(T, multiply(U, W)))))
% 0.19/0.81 = { by lemma 12 }
% 0.19/0.81 multiply(W, inverse(multiply(multiply(V, multiply(multiply(inverse(V), inverse(multiply(U, multiply(X, multiply(inverse(X), Y))))), inverse(T))), multiply(T, multiply(U, W)))))
% 0.19/0.81 = { by lemma 2 }
% 0.19/0.81 multiply(X, multiply(inverse(X), Y))
% 0.19/0.81
% 0.19/0.81 Lemma 14: multiply(multiply(inverse(X), inverse(multiply(W, Z))), W) = multiply(multiply(inverse(X), inverse(multiply(Y, Z))), Y).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(W, Z))), W)
% 0.19/0.81 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))))), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))
% 0.19/0.81 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(X2, inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2))))))), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))
% 0.19/0.81 = { by lemma 2 R->L }
% 0.19/0.81 multiply(W2, inverse(multiply(multiply(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))), multiply(multiply(inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))))))))), inverse(multiply(inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2)))), multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(X2, inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2))))))), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))))), inverse(V2))), multiply(V2, multiply(inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2)))), W2)))))
% 0.19/0.81 = { by axiom 1 (single_axiom) }
% 0.19/0.81 multiply(W2, inverse(multiply(multiply(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))), multiply(X2, inverse(V2))), multiply(V2, multiply(inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2)))), W2)))))
% 0.19/0.81 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.81 multiply(W2, inverse(multiply(multiply(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))), multiply(multiply(inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))))))))), inverse(multiply(inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2)))), multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), inverse(multiply(X2, inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2))))))), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))))), inverse(V2))), multiply(V2, multiply(inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2)))), W2)))))
% 0.19/0.81 = { by lemma 2 }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), inverse(multiply(X2, inverse(multiply(Y2, multiply(Z2, multiply(multiply(inverse(Z2), inverse(multiply(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))), Y2))), X2))))))), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))
% 0.19/0.81 = { by axiom 1 (single_axiom) }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S))))), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))
% 0.19/0.81 = { by lemma 6 }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), V), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(S, V), U))), S)))))))))))
% 0.19/0.81 = { by lemma 6 }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), V), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(W, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(W, Z)))), V)))))))
% 0.19/0.81 = { by lemma 13 R->L }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), V), inverse(multiply(Z, multiply(X, multiply(multiply(inverse(X), inverse(multiply(Y, Z))), multiply(inverse(multiply(inverse(X), inverse(multiply(Y, Z)))), V)))))))
% 0.19/0.81 = { by axiom 1 (single_axiom) }
% 0.19/0.81 multiply(multiply(inverse(X), inverse(multiply(Y, Z))), Y)
% 0.19/0.81
% 0.19/0.81 Lemma 15: multiply(inverse(X), inverse(multiply(multiply(Y, inverse(multiply(Z, multiply(W, multiply(inverse(W), Y))))), Z))) = inverse(X).
% 0.19/0.81 Proof:
% 0.19/0.81 multiply(inverse(X), inverse(multiply(multiply(Y, inverse(multiply(Z, multiply(W, multiply(inverse(W), Y))))), Z)))
% 0.19/0.81 = { by lemma 12 }
% 0.19/0.82 multiply(inverse(X), inverse(multiply(multiply(Y, inverse(multiply(Z, multiply(X, multiply(inverse(X), Y))))), Z)))
% 0.19/0.82 = { by lemma 10 }
% 0.19/0.82 inverse(X)
% 0.19/0.82
% 0.19/0.82 Lemma 16: multiply(Y, inverse(Y)) = multiply(X, inverse(X)).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(Y, inverse(Y))
% 0.19/0.82 = { by lemma 15 R->L }
% 0.19/0.82 multiply(Y, multiply(inverse(Y), inverse(multiply(multiply(Z, inverse(multiply(W, multiply(V, multiply(inverse(V), Z))))), W))))
% 0.19/0.82 = { by lemma 13 }
% 0.19/0.82 multiply(X, multiply(inverse(X), inverse(multiply(multiply(Z, inverse(multiply(W, multiply(V, multiply(inverse(V), Z))))), W))))
% 0.19/0.82 = { by lemma 15 }
% 0.19/0.82 multiply(X, inverse(X))
% 0.19/0.82
% 0.19/0.82 Lemma 17: multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(Y, inverse(Y)), Z))), multiply(Z, W))) = W.
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(Y, inverse(Y)), Z))), multiply(Z, W)))
% 0.19/0.82 = { by lemma 16 }
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(multiply(Z, W), inverse(multiply(Z, W))), Z))), multiply(Z, W)))
% 0.19/0.82 = { by lemma 6 R->L }
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(multiply(Z, W), inverse(multiply(Z, W))), Z))), inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(T, multiply(Z, W)), V))), T))))))
% 0.19/0.82 = { by lemma 3 R->L }
% 0.19/0.82 multiply(S, inverse(multiply(multiply(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(T, multiply(Z, W)), V))), T))), multiply(multiply(multiply(Z, W), inverse(multiply(Z, W))), inverse(X2))), multiply(X2, multiply(Z, S)))))
% 0.19/0.82 = { by lemma 6 R->L }
% 0.19/0.82 multiply(S, inverse(multiply(multiply(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(T, multiply(Z, W)), V))), T))), multiply(multiply(inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(T, multiply(Z, W)), V))), T)))), inverse(multiply(Z, W))), inverse(X2))), multiply(X2, multiply(Z, S)))))
% 0.19/0.82 = { by lemma 2 }
% 0.19/0.82 W
% 0.19/0.82
% 0.19/0.82 Lemma 18: multiply(X, multiply(multiply(inverse(X), inverse(multiply(Y, Z))), Y)) = inverse(Z).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(Y, Z))), Y))
% 0.19/0.82 = { by lemma 14 }
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(multiply(Z, inverse(Z)), Z))), multiply(Z, inverse(Z))))
% 0.19/0.82 = { by lemma 17 }
% 0.19/0.82 inverse(Z)
% 0.19/0.82
% 0.19/0.82 Lemma 19: multiply(X, inverse(multiply(Y, inverse(Y)))) = X.
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(X, inverse(multiply(Y, inverse(Y))))
% 0.19/0.82 = { by lemma 18 R->L }
% 0.19/0.82 multiply(X, inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(X, Y))), X)))))
% 0.19/0.82 = { by axiom 1 (single_axiom) }
% 0.19/0.82 X
% 0.19/0.82
% 0.19/0.82 Lemma 20: multiply(X, multiply(inverse(X), Y)) = inverse(inverse(Y)).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(X, multiply(inverse(X), Y))
% 0.19/0.82 = { by lemma 19 R->L }
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(Y, inverse(Y)))), Y))
% 0.19/0.82 = { by lemma 18 }
% 0.19/0.82 inverse(inverse(Y))
% 0.19/0.82
% 0.19/0.82 Lemma 21: multiply(multiply(X, inverse(multiply(W, Z))), W) = multiply(multiply(X, inverse(multiply(Y, Z))), Y).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(multiply(X, inverse(multiply(W, Z))), W)
% 0.19/0.82 = { by lemma 6 R->L }
% 0.19/0.82 multiply(multiply(inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(T, X), V))), T)))), inverse(multiply(W, Z))), W)
% 0.19/0.82 = { by lemma 14 }
% 0.19/0.82 multiply(multiply(inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(T, X), V))), T)))), inverse(multiply(Y, Z))), Y)
% 0.19/0.82 = { by lemma 6 }
% 0.19/0.82 multiply(multiply(X, inverse(multiply(Y, Z))), Y)
% 0.19/0.82
% 0.19/0.82 Lemma 22: multiply(multiply(multiply(X, Y), inverse(multiply(Z, Y))), Z) = multiply(multiply(W, inverse(W)), X).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(multiply(multiply(X, Y), inverse(multiply(Z, Y))), Z)
% 0.19/0.82 = { by lemma 21 }
% 0.19/0.82 multiply(multiply(multiply(X, Y), inverse(multiply(X, Y))), X)
% 0.19/0.82 = { by lemma 16 R->L }
% 0.19/0.82 multiply(multiply(W, inverse(W)), X)
% 0.19/0.82
% 0.19/0.82 Lemma 23: multiply(multiply(X, inverse(X)), Y) = multiply(Y, multiply(Z, inverse(Z))).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(multiply(X, inverse(X)), Y)
% 0.19/0.82 = { by lemma 22 R->L }
% 0.19/0.82 multiply(multiply(multiply(Y, inverse(multiply(Z, inverse(Z)))), inverse(multiply(multiply(Z, inverse(Z)), inverse(multiply(Z, inverse(Z)))))), multiply(Z, inverse(Z)))
% 0.19/0.82 = { by lemma 19 }
% 0.19/0.82 multiply(multiply(Y, inverse(multiply(multiply(Z, inverse(Z)), inverse(multiply(Z, inverse(Z)))))), multiply(Z, inverse(Z)))
% 0.19/0.82 = { by lemma 19 }
% 0.19/0.82 multiply(Y, multiply(Z, inverse(Z)))
% 0.19/0.82
% 0.19/0.82 Lemma 24: multiply(multiply(Z, inverse(Z)), X) = multiply(multiply(X, inverse(Y)), Y).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(multiply(Z, inverse(Z)), X)
% 0.19/0.82 = { by lemma 22 R->L }
% 0.19/0.82 multiply(multiply(multiply(X, inverse(Y)), inverse(multiply(Y, inverse(Y)))), Y)
% 0.19/0.82 = { by lemma 19 }
% 0.19/0.82 multiply(multiply(X, inverse(Y)), Y)
% 0.19/0.82
% 0.19/0.82 Lemma 25: multiply(inverse(X), multiply(Y, inverse(Y))) = inverse(X).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(inverse(X), multiply(Y, inverse(Y)))
% 0.19/0.82 = { by lemma 23 R->L }
% 0.19/0.82 multiply(multiply(Z, inverse(Z)), inverse(X))
% 0.19/0.82 = { by lemma 24 }
% 0.19/0.82 multiply(multiply(inverse(X), inverse(multiply(multiply(X, inverse(X)), multiply(X, inverse(X))))), multiply(multiply(X, inverse(X)), multiply(X, inverse(X))))
% 0.19/0.82 = { by lemma 16 }
% 0.19/0.82 multiply(multiply(inverse(X), inverse(multiply(multiply(multiply(multiply(X, inverse(X)), multiply(X, inverse(X))), inverse(multiply(multiply(X, inverse(X)), multiply(X, inverse(X))))), multiply(X, inverse(X))))), multiply(multiply(X, inverse(X)), multiply(X, inverse(X))))
% 0.19/0.82 = { by lemma 6 R->L }
% 0.19/0.82 multiply(multiply(inverse(X), inverse(multiply(multiply(multiply(multiply(X, inverse(X)), multiply(X, inverse(X))), inverse(multiply(multiply(X, inverse(X)), multiply(X, inverse(X))))), multiply(X, inverse(X))))), inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), inverse(multiply(multiply(U, multiply(multiply(X, inverse(X)), multiply(X, inverse(X)))), W))), U)))))
% 0.19/0.82 = { by lemma 6 R->L }
% 0.19/0.82 multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), inverse(multiply(multiply(U, multiply(multiply(X, inverse(X)), multiply(X, inverse(X)))), W))), U)))), inverse(multiply(multiply(X, inverse(X)), multiply(X, inverse(X))))), multiply(X, inverse(X))))), inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), inverse(multiply(multiply(U, multiply(multiply(X, inverse(X)), multiply(X, inverse(X)))), W))), U)))))
% 0.19/0.82 = { by lemma 7 }
% 0.19/0.82 inverse(multiply(T, multiply(S, multiply(multiply(inverse(S), inverse(multiply(multiply(X, inverse(X)), T))), X))))
% 0.19/0.82 = { by lemma 6 }
% 0.19/0.82 inverse(X)
% 0.19/0.82
% 0.19/0.82 Lemma 26: multiply(X, multiply(Y, inverse(inverse(Z)))) = multiply(multiply(X, Y), Z).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(X, multiply(Y, inverse(inverse(Z))))
% 0.19/0.82 = { by lemma 19 R->L }
% 0.19/0.82 multiply(multiply(X, inverse(multiply(multiply(Y, inverse(inverse(Z))), inverse(multiply(Y, inverse(inverse(Z))))))), multiply(Y, inverse(inverse(Z))))
% 0.19/0.82 = { by lemma 21 R->L }
% 0.19/0.82 multiply(multiply(X, inverse(multiply(Z, inverse(multiply(Y, inverse(inverse(Z))))))), Z)
% 0.19/0.82 = { by lemma 18 R->L }
% 0.19/0.82 multiply(multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(multiply(inverse(V), multiply(Z, inverse(multiply(Y, inverse(inverse(Z)))))))), inverse(V)))), Z)
% 0.19/0.82 = { by lemma 20 R->L }
% 0.19/0.82 multiply(multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(multiply(inverse(V), multiply(Z, inverse(multiply(Y, multiply(V, multiply(inverse(V), Z)))))))), inverse(V)))), Z)
% 0.19/0.82 = { by lemma 3 R->L }
% 0.19/0.82 multiply(multiply(X, multiply(U, inverse(multiply(multiply(V, multiply(inverse(V), inverse(T))), multiply(T, multiply(multiply(Z, inverse(multiply(Y, multiply(V, multiply(inverse(V), Z))))), U)))))), Z)
% 0.19/0.82 = { by lemma 9 }
% 0.19/0.82 multiply(multiply(X, multiply(U, inverse(multiply(multiply(V, multiply(inverse(V), inverse(T))), multiply(T, multiply(multiply(inverse(T), inverse(multiply(Y, multiply(V, multiply(inverse(V), inverse(T)))))), U)))))), Z)
% 0.19/0.82 = { by axiom 1 (single_axiom) }
% 0.19/0.82 multiply(multiply(X, Y), Z)
% 0.19/0.82
% 0.19/0.82 Lemma 27: multiply(X, multiply(inverse(X), multiply(Y, inverse(Y)))) = inverse(inverse(multiply(Z, inverse(Z)))).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(X, multiply(inverse(X), multiply(Y, inverse(Y))))
% 0.19/0.82 = { by lemma 23 R->L }
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(inverse(X))), inverse(X)))
% 0.19/0.82 = { by lemma 19 R->L }
% 0.19/0.82 multiply(X, multiply(multiply(inverse(X), inverse(multiply(inverse(X), inverse(multiply(Z, inverse(Z)))))), inverse(X)))
% 0.19/0.82 = { by lemma 18 }
% 0.19/0.82 inverse(inverse(multiply(Z, inverse(Z))))
% 0.19/0.82
% 0.19/0.82 Lemma 28: multiply(multiply(inverse(X), inverse(inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(W, Y))), V)))))), inverse(V)) = inverse(multiply(W, multiply(U, multiply(inverse(U), X)))).
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(multiply(inverse(X), inverse(inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(W, Y))), V)))))), inverse(V))
% 0.19/0.82 = { by lemma 11 R->L }
% 0.19/0.82 multiply(multiply(inverse(X), inverse(multiply(multiply(inverse(V), inverse(multiply(inverse(U), multiply(X, inverse(multiply(W, multiply(U, multiply(inverse(U), X)))))))), inverse(U)))), inverse(V))
% 0.19/0.82 = { by lemma 7 }
% 0.19/0.82 inverse(multiply(T, multiply(S, multiply(multiply(inverse(S), inverse(multiply(multiply(X, inverse(multiply(W, multiply(U, multiply(inverse(U), X))))), T))), X))))
% 0.19/0.82 = { by lemma 6 }
% 0.19/0.82 inverse(multiply(W, multiply(U, multiply(inverse(U), X))))
% 0.19/0.82
% 0.19/0.82 Lemma 29: inverse(multiply(multiply(X, Y), multiply(Z, multiply(inverse(Z), W)))) = multiply(multiply(inverse(W), inverse(Y)), inverse(X)).
% 0.19/0.82 Proof:
% 0.19/0.82 inverse(multiply(multiply(X, Y), multiply(Z, multiply(inverse(Z), W))))
% 0.19/0.82 = { by lemma 28 R->L }
% 0.19/0.82 multiply(multiply(inverse(W), inverse(inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), inverse(multiply(multiply(X, Y), V))), X)))))), inverse(X))
% 0.19/0.82 = { by lemma 6 }
% 0.19/0.82 multiply(multiply(inverse(W), inverse(Y)), inverse(X))
% 0.19/0.82
% 0.19/0.82 Lemma 30: multiply(multiply(X, inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(W)), X))))), Y) = W.
% 0.19/0.82 Proof:
% 0.19/0.82 multiply(multiply(X, inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(W)), X))))), Y)
% 0.19/0.82 = { by lemma 9 }
% 0.19/0.82 multiply(multiply(inverse(V), inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), inverse(W)), inverse(V)))))), Y)
% 0.19/0.82 = { by lemma 14 }
% 0.19/0.82 multiply(multiply(inverse(V), inverse(multiply(inverse(U), multiply(Z, multiply(multiply(inverse(Z), inverse(W)), inverse(V)))))), inverse(U))
% 0.19/0.82 = { by lemma 29 R->L }
% 0.19/0.82 multiply(multiply(inverse(V), inverse(multiply(inverse(U), multiply(Z, inverse(multiply(multiply(V, W), multiply(U, multiply(inverse(U), Z)))))))), inverse(U))
% 0.19/0.82 = { by lemma 11 }
% 0.19/0.82 inverse(multiply(T, multiply(S, multiply(multiply(inverse(S), inverse(multiply(multiply(V, W), T))), V))))
% 0.19/0.82 = { by lemma 6 }
% 0.19/0.82 W
% 0.19/0.82
% 0.19/0.82 Lemma 31: inverse(multiply(X, inverse(X))) = multiply(Y, inverse(Y)).
% 0.19/0.82 Proof:
% 0.19/0.82 inverse(multiply(X, inverse(X)))
% 0.19/0.82 = { by lemma 30 R->L }
% 0.19/0.82 inverse(multiply(multiply(X, inverse(multiply(Z, multiply(W, multiply(multiply(inverse(W), inverse(multiply(X, inverse(X)))), X))))), Z))
% 0.19/0.82 = { by lemma 18 }
% 0.19/0.82 inverse(multiply(multiply(X, inverse(multiply(Z, inverse(inverse(X))))), Z))
% 0.19/0.82 = { by lemma 20 R->L }
% 0.19/0.82 inverse(multiply(multiply(X, inverse(multiply(Z, multiply(V, multiply(inverse(V), X))))), Z))
% 0.19/0.82 = { by lemma 2 R->L }
% 0.19/0.82 multiply(U, inverse(multiply(multiply(T, multiply(multiply(inverse(T), inverse(multiply(inverse(V), inverse(multiply(multiply(X, inverse(multiply(Z, multiply(V, multiply(inverse(V), X))))), Z))))), inverse(S))), multiply(S, multiply(inverse(V), U)))))
% 0.19/0.82 = { by lemma 10 }
% 0.19/0.82 multiply(U, inverse(multiply(multiply(T, multiply(multiply(inverse(T), inverse(inverse(V))), inverse(S))), multiply(S, multiply(inverse(V), U)))))
% 0.19/0.82 = { by lemma 25 R->L }
% 0.19/0.82 multiply(U, inverse(multiply(multiply(T, multiply(multiply(inverse(T), inverse(multiply(inverse(V), multiply(Y, inverse(Y))))), inverse(S))), multiply(S, multiply(inverse(V), U)))))
% 0.19/0.82 = { by lemma 2 }
% 0.19/0.82 multiply(Y, inverse(Y))
% 0.19/0.82
% 0.19/0.82 Lemma 32: inverse(inverse(X)) = X.
% 0.19/0.82 Proof:
% 0.19/0.82 inverse(inverse(X))
% 0.19/0.83 = { by lemma 20 R->L }
% 0.19/0.83 multiply(X, multiply(inverse(X), X))
% 0.19/0.83 = { by lemma 25 R->L }
% 0.19/0.83 multiply(X, multiply(multiply(inverse(X), multiply(Y, inverse(Y))), X))
% 0.19/0.83 = { by lemma 26 R->L }
% 0.19/0.83 multiply(X, multiply(inverse(X), multiply(multiply(Y, inverse(Y)), inverse(inverse(X)))))
% 0.19/0.83 = { by lemma 23 }
% 0.19/0.83 multiply(X, multiply(inverse(X), multiply(inverse(inverse(X)), multiply(Z, inverse(Z)))))
% 0.19/0.83 = { by lemma 27 }
% 0.19/0.83 multiply(X, inverse(inverse(multiply(W, inverse(W)))))
% 0.19/0.83 = { by lemma 31 }
% 0.19/0.83 multiply(X, inverse(multiply(V, inverse(V))))
% 0.19/0.83 = { by lemma 19 }
% 0.19/0.83 X
% 0.19/0.83
% 0.19/0.83 Lemma 33: multiply(inverse(X), X) = multiply(Y, inverse(Y)).
% 0.19/0.83 Proof:
% 0.19/0.83 multiply(inverse(X), X)
% 0.19/0.83 = { by lemma 25 R->L }
% 0.19/0.83 multiply(multiply(inverse(X), multiply(Z, inverse(Z))), X)
% 0.19/0.83 = { by lemma 26 R->L }
% 0.19/0.83 multiply(inverse(X), multiply(multiply(Z, inverse(Z)), inverse(inverse(X))))
% 0.19/0.83 = { by lemma 23 }
% 0.19/0.83 multiply(inverse(X), multiply(inverse(inverse(X)), multiply(W, inverse(W))))
% 0.19/0.83 = { by lemma 27 }
% 0.19/0.83 inverse(inverse(multiply(V, inverse(V))))
% 0.19/0.83 = { by lemma 31 }
% 0.19/0.83 inverse(multiply(U, inverse(U)))
% 0.19/0.83 = { by lemma 31 }
% 0.19/0.83 multiply(Y, inverse(Y))
% 0.19/0.83
% 0.19/0.83 Lemma 34: multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 0.19/0.83 Proof:
% 0.19/0.83 multiply(multiply(X, Y), Z)
% 0.19/0.83 = { by lemma 26 R->L }
% 0.19/0.83 multiply(X, multiply(Y, inverse(inverse(Z))))
% 0.19/0.83 = { by lemma 32 }
% 0.19/0.83 multiply(X, multiply(Y, Z))
% 0.19/0.83
% 0.19/0.83 Lemma 35: multiply(X, multiply(Y, inverse(Y))) = X.
% 0.19/0.83 Proof:
% 0.19/0.83 multiply(X, multiply(Y, inverse(Y)))
% 0.19/0.83 = { by lemma 33 R->L }
% 0.19/0.83 multiply(X, multiply(inverse(Z), Z))
% 0.19/0.83 = { by lemma 32 R->L }
% 0.19/0.83 multiply(X, multiply(inverse(inverse(inverse(Z))), Z))
% 0.19/0.83 = { by lemma 20 R->L }
% 0.19/0.83 multiply(X, multiply(inverse(multiply(Z, multiply(inverse(Z), Z))), Z))
% 0.19/0.83 = { by lemma 34 R->L }
% 0.19/0.83 multiply(multiply(X, inverse(multiply(Z, multiply(inverse(Z), Z)))), Z)
% 0.19/0.83 = { by lemma 33 }
% 0.19/0.83 multiply(multiply(X, inverse(multiply(Z, multiply(W, inverse(W))))), Z)
% 0.19/0.83 = { by lemma 30 R->L }
% 0.19/0.83 multiply(multiply(V, inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(multiply(X, inverse(multiply(Z, multiply(W, inverse(W))))), Z))), V))))), U)
% 0.19/0.83 = { by axiom 1 (single_axiom) R->L }
% 0.19/0.83 multiply(multiply(V, inverse(multiply(U, multiply(T, multiply(multiply(inverse(T), inverse(multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(multiply(multiply(inverse(T), inverse(multiply(multiply(X, inverse(multiply(Z, multiply(W, inverse(W))))), Z))), X))), inverse(T)))))), V))))), U)
% 0.19/0.83 = { by lemma 30 }
% 0.19/0.83 multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(multiply(multiply(inverse(T), inverse(multiply(multiply(X, inverse(multiply(Z, multiply(W, inverse(W))))), Z))), X))), inverse(T))))
% 0.19/0.83 = { by lemma 6 R->L }
% 0.19/0.83 multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(multiply(multiply(inverse(T), inverse(multiply(multiply(X, inverse(multiply(Z, multiply(W, inverse(W))))), Z))), inverse(multiply(S, multiply(X2, multiply(multiply(inverse(X2), inverse(multiply(multiply(Y2, X), S))), Y2))))))), inverse(T))))
% 0.19/0.83 = { by lemma 6 R->L }
% 0.19/0.83 multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(multiply(multiply(inverse(T), inverse(multiply(multiply(inverse(multiply(S, multiply(X2, multiply(multiply(inverse(X2), inverse(multiply(multiply(Y2, X), S))), Y2)))), inverse(multiply(Z, multiply(W, inverse(W))))), Z))), inverse(multiply(S, multiply(X2, multiply(multiply(inverse(X2), inverse(multiply(multiply(Y2, X), S))), Y2))))))), inverse(T))))
% 0.19/0.83 = { by lemma 7 }
% 0.19/0.83 multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(inverse(multiply(Z2, multiply(W2, multiply(multiply(inverse(W2), inverse(multiply(multiply(W, inverse(W)), Z2))), T)))))), inverse(T))))
% 0.19/0.83 = { by lemma 28 }
% 0.19/0.83 multiply(X, multiply(W, inverse(multiply(multiply(W, inverse(W)), multiply(V2, multiply(inverse(V2), W))))))
% 0.19/0.83 = { by lemma 29 }
% 0.19/0.83 multiply(X, multiply(W, multiply(multiply(inverse(W), inverse(inverse(W))), inverse(W))))
% 0.19/0.83 = { by lemma 23 }
% 0.19/0.83 multiply(X, multiply(W, multiply(inverse(W), multiply(U2, inverse(U2)))))
% 0.19/0.83 = { by lemma 27 }
% 0.19/0.83 multiply(X, inverse(inverse(multiply(T2, inverse(T2)))))
% 0.19/0.83 = { by lemma 31 }
% 0.19/0.83 multiply(X, inverse(multiply(S2, inverse(S2))))
% 0.19/0.83 = { by lemma 19 }
% 0.19/0.83 X
% 0.19/0.83
% 0.19/0.83 Lemma 36: multiply(X, multiply(inverse(X), Y)) = Y.
% 0.19/0.83 Proof:
% 0.19/0.83 multiply(X, multiply(inverse(X), Y))
% 0.19/0.83 = { by lemma 34 R->L }
% 0.19/0.83 multiply(multiply(X, inverse(X)), Y)
% 0.19/0.83 = { by lemma 24 }
% 0.19/0.83 multiply(multiply(Y, inverse(Z)), Z)
% 0.19/0.83 = { by lemma 35 R->L }
% 0.19/0.83 multiply(multiply(Y, inverse(multiply(Z, multiply(W, inverse(W))))), Z)
% 0.19/0.83 = { by lemma 34 }
% 0.19/0.83 multiply(Y, multiply(inverse(multiply(Z, multiply(W, inverse(W)))), Z))
% 0.19/0.83 = { by lemma 35 }
% 0.19/0.83 multiply(Y, multiply(inverse(Z), Z))
% 0.19/0.83 = { by lemma 33 }
% 0.19/0.83 multiply(Y, multiply(V, inverse(V)))
% 0.19/0.83 = { by lemma 35 }
% 0.19/0.83 Y
% 0.19/0.83
% 0.19/0.83 Goal 1 (prove_these_axioms): tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1)) = tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1)).
% 0.19/0.83 Proof:
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 0.19/0.83 = { by lemma 34 }
% 0.19/0.83 tuple(multiply(inverse(b2), multiply(b2, a2)), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 0.19/0.83 = { by lemma 34 }
% 0.19/0.83 tuple(multiply(inverse(b2), multiply(b2, a2)), multiply(a3, multiply(b3, c3)), multiply(inverse(a1), a1))
% 0.19/0.83 = { by lemma 33 }
% 0.19/0.83 tuple(multiply(inverse(b2), multiply(b2, a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 36 R->L }
% 0.19/0.83 tuple(multiply(inverse(b2), multiply(multiply(Y, inverse(Y)), multiply(inverse(multiply(Y, inverse(Y))), multiply(b2, a2)))), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 34 R->L }
% 0.19/0.83 tuple(multiply(inverse(b2), multiply(multiply(Y, inverse(Y)), multiply(multiply(inverse(multiply(Y, inverse(Y))), b2), a2))), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 34 R->L }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(inverse(multiply(Y, inverse(Y))), b2), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 35 R->L }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y)))), b2), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 36 R->L }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(Z, multiply(inverse(Z), multiply(inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y)))), b2))), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 34 R->L }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y))))), b2)), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 2 R->L }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y))))), multiply(multiply(Y, inverse(Y)), inverse(multiply(multiply(W, multiply(multiply(inverse(W), inverse(multiply(inverse(V), b2))), inverse(V))), multiply(V, multiply(inverse(V), multiply(Y, inverse(Y))))))))), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 20 }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y))))), multiply(multiply(Y, inverse(Y)), inverse(multiply(multiply(W, multiply(multiply(inverse(W), inverse(multiply(inverse(V), b2))), inverse(V))), inverse(inverse(multiply(Y, inverse(Y))))))))), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 18 }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y))))), multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(b2), inverse(inverse(multiply(Y, inverse(Y))))))))), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 32 }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(multiply(Z, multiply(multiply(inverse(Z), inverse(multiply(multiply(Y, inverse(Y)), multiply(Y, inverse(Y))))), multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(b2), multiply(Y, inverse(Y))))))), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 17 }
% 0.19/0.83 tuple(multiply(multiply(inverse(b2), multiply(Y, inverse(Y))), multiply(inverse(multiply(inverse(b2), multiply(Y, inverse(Y)))), a2)), multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 36 }
% 0.19/0.83 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(X, inverse(X)))
% 0.19/0.83 = { by lemma 33 R->L }
% 0.19/0.83 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1))
% 0.19/0.83 % SZS output end Proof
% 0.19/0.83
% 0.19/0.83 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------