0.03/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.12 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.12/0.33 % Computer : n016.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 180 0.12/0.33 % DateTime : Thu Aug 29 11:52:23 EDT 2019 0.12/0.33 % CPUTime : 1.02/1.21 % SZS status Unsatisfiable 1.02/1.21 1.02/1.21 % SZS output start Proof 1.02/1.21 Take the following subset of the input axioms: 1.02/1.21 fof(prove_these_axioms_1, negated_conjecture, multiply(inverse(a1), a1)!=multiply(inverse(b1), b1)). 1.02/1.21 fof(single_axiom, axiom, ![A, B, C, D]: inverse(multiply(A, multiply(B, multiply(multiply(inverse(B), C), inverse(multiply(D, multiply(A, C)))))))=D). 1.02/1.21 1.02/1.21 Now clausify the problem and encode Horn clauses using encoding 3 of 1.02/1.21 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 1.02/1.21 We repeatedly replace C & s=t => u=v by the two clauses: 1.02/1.21 fresh(y, y, x1...xn) = u 1.02/1.21 C => fresh(s, t, x1...xn) = v 1.02/1.21 where fresh is a fresh function symbol and x1..xn are the free 1.02/1.21 variables of u and v. 1.02/1.21 A predicate p(X) is encoded as p(X)=true (this is sound, because the 1.02/1.21 input problem has no model of domain size 1). 1.02/1.21 1.02/1.21 The encoding turns the above axioms into the following unit equations and goals: 1.02/1.21 1.02/1.21 Axiom 1 (single_axiom): inverse(multiply(X, multiply(Y, multiply(multiply(inverse(Y), Z), inverse(multiply(W, multiply(X, Z))))))) = W. 1.02/1.21 1.02/1.21 Lemma 2: inverse(multiply(Y, multiply(multiply(Z, multiply(W, multiply(multiply(inverse(W), V), inverse(multiply(U, multiply(Z, V)))))), multiply(multiply(U, T), inverse(multiply(X, multiply(Y, T))))))) = X. 1.02/1.21 Proof: 1.02/1.21 inverse(multiply(Y, multiply(multiply(Z, multiply(W, multiply(multiply(inverse(W), V), inverse(multiply(U, multiply(Z, V)))))), multiply(multiply(U, T), inverse(multiply(X, multiply(Y, T))))))) 1.02/1.21 = { by axiom 1 (single_axiom) } 1.02/1.21 inverse(multiply(Y, multiply(multiply(Z, multiply(W, multiply(multiply(inverse(W), V), inverse(multiply(U, multiply(Z, V)))))), multiply(multiply(inverse(multiply(Z, multiply(W, multiply(multiply(inverse(W), V), inverse(multiply(U, multiply(Z, V))))))), T), inverse(multiply(X, multiply(Y, T))))))) 1.02/1.21 = { by axiom 1 (single_axiom) } 1.02/1.21 X 1.02/1.21 1.02/1.21 Lemma 3: inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), W), inverse(multiply(V, multiply(X, W))))), V)))) = X. 1.02/1.21 Proof: 1.02/1.21 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), W), inverse(multiply(V, multiply(X, W))))), V)))) 1.02/1.21 = { by axiom 1 (single_axiom) } 1.02/1.21 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), W), inverse(multiply(V, multiply(X, W))))), inverse(multiply(X, multiply(Y, multiply(multiply(inverse(Y), W), inverse(multiply(V, multiply(X, W))))))))))) 1.02/1.21 = { by axiom 1 (single_axiom) } 1.02/1.21 X 1.02/1.21 1.02/1.21 Lemma 4: inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), multiply(multiply(inverse(X), W), inverse(multiply(V, multiply(U, W))))), V)), U)))) = X. 1.02/1.21 Proof: 1.02/1.21 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), multiply(multiply(inverse(X), W), inverse(multiply(V, multiply(U, W))))), V)), U)))) 1.02/1.21 = { by lemma 3 } 1.02/1.21 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), multiply(multiply(inverse(X), W), inverse(multiply(V, multiply(U, W))))), V)), inverse(multiply(X, multiply(Y, multiply(multiply(inverse(Y), multiply(multiply(inverse(X), W), inverse(multiply(V, multiply(U, W))))), V)))))))) 1.02/1.21 = { by axiom 1 (single_axiom) } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 5: multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(X), multiply(?, ?)))))) = X. 1.02/1.22 Proof: 1.02/1.22 multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(X), multiply(?, ?)))))) 1.02/1.22 = { by lemma 3 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(X), ?), inverse(multiply(?, multiply(?, ?))))), inverse(multiply(?, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(X), multiply(?, ?)))))), multiply(multiply(inverse(X), ?), inverse(multiply(?, multiply(?, ?))))))))), ?)))) 1.02/1.22 = { by lemma 2 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(X), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?)))) 1.02/1.22 = { by lemma 4 } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 6: multiply(X, multiply(Y, multiply(multiply(inverse(Y), Z), inverse(multiply(W, multiply(X, Z)))))) = multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(W, multiply(?, ?)))))). 1.02/1.22 Proof: 1.02/1.22 multiply(X, multiply(Y, multiply(multiply(inverse(Y), Z), inverse(multiply(W, multiply(X, Z)))))) 1.02/1.22 = { by lemma 3 } 1.02/1.22 inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), multiply(multiply(inverse(V), multiply(multiply(W, T), inverse(multiply(S, multiply(X2, T))))), inverse(multiply(X2, multiply(multiply(X, multiply(Y, multiply(multiply(inverse(Y), Z), inverse(multiply(W, multiply(X, Z)))))), multiply(multiply(W, T), inverse(multiply(S, multiply(X2, T))))))))), X2)))) 1.02/1.22 = { by lemma 2 } 1.02/1.22 inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), multiply(multiply(inverse(V), multiply(multiply(W, T), inverse(multiply(S, multiply(X2, T))))), S)), X2)))) 1.02/1.22 = { by lemma 2 } 1.02/1.22 inverse(multiply(V, multiply(U, multiply(multiply(inverse(U), multiply(multiply(inverse(V), multiply(multiply(W, T), inverse(multiply(S, multiply(X2, T))))), inverse(multiply(X2, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(W, multiply(?, ?)))))), multiply(multiply(W, T), inverse(multiply(S, multiply(X2, T))))))))), X2)))) 1.02/1.22 = { by lemma 3 } 1.02/1.22 multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(W, multiply(?, ?)))))) 1.02/1.22 1.02/1.22 Lemma 7: inverse(multiply(inverse(Y), multiply(Y, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(?, ?)))))), inverse(multiply(X, multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(?, ?)))))))))))) = X. 1.02/1.22 Proof: 1.02/1.22 inverse(multiply(inverse(Y), multiply(Y, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(?, ?)))))), inverse(multiply(X, multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(?, ?)))))))))))) 1.02/1.22 = { by lemma 6 } 1.02/1.22 inverse(multiply(inverse(Y), multiply(Y, multiply(multiply(inverse(Y), multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(inverse(Y), ?)))))), inverse(multiply(X, multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(?, ?)))))))))))) 1.02/1.22 = { by lemma 6 } 1.02/1.22 inverse(multiply(inverse(Y), multiply(Y, multiply(multiply(inverse(Y), multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(inverse(Y), ?)))))), inverse(multiply(X, multiply(inverse(Y), multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(Z, multiply(inverse(Y), ?)))))))))))) 1.02/1.22 = { by axiom 1 (single_axiom) } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 8: inverse(multiply(inverse(Y), multiply(Y, multiply(Z, inverse(multiply(X, Z)))))) = X. 1.02/1.22 Proof: 1.02/1.22 inverse(multiply(inverse(Y), multiply(Y, multiply(Z, inverse(multiply(X, Z)))))) 1.02/1.22 = { by lemma 5 } 1.02/1.22 inverse(multiply(inverse(Y), multiply(Y, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(Z), multiply(?, ?)))))), inverse(multiply(X, Z)))))) 1.02/1.22 = { by lemma 5 } 1.02/1.22 inverse(multiply(inverse(Y), multiply(Y, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(Z), multiply(?, ?)))))), inverse(multiply(X, multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(Z), multiply(?, ?)))))))))))) 1.02/1.22 = { by lemma 7 } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 9: multiply(X, multiply(Y, multiply(multiply(inverse(Y), multiply(Z, inverse(multiply(W, Z)))), W))) = X. 1.02/1.22 Proof: 1.02/1.22 multiply(X, multiply(Y, multiply(multiply(inverse(Y), multiply(Z, inverse(multiply(W, Z)))), W))) 1.02/1.22 = { by lemma 8 } 1.02/1.22 multiply(X, multiply(Y, multiply(multiply(inverse(Y), multiply(Z, inverse(multiply(W, Z)))), inverse(multiply(inverse(X), multiply(X, multiply(Z, inverse(multiply(W, Z))))))))) 1.02/1.22 = { by lemma 6 } 1.02/1.22 multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(inverse(X), multiply(?, ?)))))) 1.02/1.22 = { by lemma 5 } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 10: inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(inverse(Y), inverse(multiply(W, X)))), W)))) = X. 1.02/1.22 Proof: 1.02/1.22 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(inverse(Y), inverse(multiply(W, X)))), W)))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?))), inverse(multiply(W, X)))), W)))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(multiply(inverse(Y), multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?))), inverse(multiply(W, multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?))))))), W)))) 1.02/1.22 = { by lemma 3 } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 11: inverse(multiply(Y, multiply(Z, multiply(inverse(Z), inverse(multiply(X, Y)))))) = X. 1.02/1.22 Proof: 1.02/1.22 inverse(multiply(Y, multiply(Z, multiply(inverse(Z), inverse(multiply(X, Y)))))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?))), inverse(multiply(X, Y)))))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 inverse(multiply(Y, multiply(Z, multiply(multiply(inverse(Z), multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?))), inverse(multiply(X, multiply(Y, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?))))))))) 1.02/1.22 = { by axiom 1 (single_axiom) } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 12: multiply(X, multiply(inverse(X), inverse(multiply(Y, Z)))) = multiply(?, multiply(inverse(?), inverse(multiply(Y, Z)))). 1.02/1.22 Proof: 1.02/1.22 multiply(X, multiply(inverse(X), inverse(multiply(Y, Z)))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), multiply(inverse(W), inverse(multiply(Z, multiply(X, multiply(inverse(X), inverse(multiply(Y, Z)))))))), Z)))) 1.02/1.22 = { by lemma 11 } 1.02/1.22 inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), multiply(inverse(W), Y)), Z)))) 1.02/1.22 = { by lemma 11 } 1.02/1.22 inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), multiply(inverse(W), inverse(multiply(Z, multiply(?, multiply(inverse(?), inverse(multiply(Y, Z)))))))), Z)))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(Y, Z)))) 1.02/1.22 1.02/1.22 Lemma 13: multiply(X, multiply(inverse(X), Y)) = multiply(?, multiply(inverse(?), Y)). 1.02/1.22 Proof: 1.02/1.22 multiply(X, multiply(inverse(X), Y)) 1.02/1.22 = { by axiom 1 (single_axiom) } 1.02/1.22 multiply(X, multiply(inverse(X), inverse(multiply(V, multiply(W, multiply(multiply(inverse(W), Z), inverse(multiply(Y, multiply(V, Z))))))))) 1.02/1.22 = { by lemma 12 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(V, multiply(W, multiply(multiply(inverse(W), Z), inverse(multiply(Y, multiply(V, Z))))))))) 1.02/1.22 = { by axiom 1 (single_axiom) } 1.02/1.22 multiply(?, multiply(inverse(?), Y)) 1.02/1.22 1.02/1.22 Lemma 14: multiply(X, inverse(X)) = multiply(?, inverse(?)). 1.02/1.22 Proof: 1.02/1.22 multiply(X, inverse(X)) 1.02/1.22 = { by lemma 9 } 1.02/1.22 multiply(X, multiply(inverse(X), multiply(W, multiply(multiply(inverse(W), multiply(Z, inverse(multiply(Y, Z)))), Y)))) 1.02/1.22 = { by lemma 13 } 1.02/1.22 multiply(?, multiply(inverse(?), multiply(W, multiply(multiply(inverse(W), multiply(Z, inverse(multiply(Y, Z)))), Y)))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 multiply(?, inverse(?)) 1.02/1.22 1.02/1.22 Lemma 15: inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), multiply(inverse(W), Y)), Z)))) = multiply(?, multiply(inverse(?), inverse(multiply(Y, Z)))). 1.02/1.22 Proof: 1.02/1.22 inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), multiply(inverse(W), Y)), Z)))) 1.02/1.22 = { by lemma 11 } 1.02/1.22 inverse(multiply(W, multiply(V, multiply(multiply(inverse(V), multiply(inverse(W), inverse(multiply(Z, multiply(?, multiply(inverse(?), inverse(multiply(Y, Z)))))))), Z)))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(Y, Z)))) 1.02/1.22 1.02/1.22 Lemma 16: multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(X, Y)), X)))) = Y. 1.02/1.22 Proof: 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(X, Y)), X)))) 1.02/1.22 = { by lemma 15 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(inverse(?), inverse(multiply(X, Y)))), X)))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 Y 1.02/1.22 1.02/1.22 Lemma 17: multiply(X, multiply(Y, inverse(multiply(Z, Y)))) = multiply(X, multiply(?, inverse(multiply(Z, ?)))). 1.02/1.22 Proof: 1.02/1.22 multiply(X, multiply(Y, inverse(multiply(Z, Y)))) 1.02/1.22 = { by lemma 16 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(inverse(X), multiply(X, multiply(Y, inverse(multiply(Z, Y)))))), inverse(X))))) 1.02/1.22 = { by lemma 8 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(Z, inverse(X))))) 1.02/1.22 = { by lemma 8 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(inverse(X), multiply(X, multiply(?, inverse(multiply(Z, ?)))))), inverse(X))))) 1.02/1.22 = { by lemma 16 } 1.02/1.22 multiply(X, multiply(?, inverse(multiply(Z, ?)))) 1.02/1.22 1.02/1.22 Lemma 18: multiply(X, inverse(multiply(Y, X))) = multiply(?, inverse(multiply(Y, ?))). 1.02/1.22 Proof: 1.02/1.22 multiply(X, inverse(multiply(Y, X))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 inverse(multiply(Z, multiply(W, multiply(multiply(inverse(W), multiply(inverse(Z), inverse(multiply(V, multiply(X, inverse(multiply(Y, X))))))), V)))) 1.02/1.22 = { by lemma 17 } 1.02/1.22 inverse(multiply(Z, multiply(W, multiply(multiply(inverse(W), multiply(inverse(Z), inverse(multiply(V, multiply(?, inverse(multiply(Y, ?))))))), V)))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 multiply(?, inverse(multiply(Y, ?))) 1.02/1.22 1.02/1.22 Lemma 19: multiply(X, multiply(?, inverse(multiply(inverse(multiply(inverse(X), Y)), ?)))) = Y. 1.02/1.22 Proof: 1.02/1.22 multiply(X, multiply(?, inverse(multiply(inverse(multiply(inverse(X), Y)), ?)))) 1.02/1.22 = { by lemma 16 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(inverse(X), multiply(X, multiply(?, inverse(multiply(inverse(multiply(inverse(X), Y)), ?)))))), inverse(X))))) 1.02/1.22 = { by lemma 8 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(inverse(X), Y)), inverse(X))))) 1.02/1.22 = { by lemma 16 } 1.02/1.22 Y 1.02/1.22 1.02/1.22 Lemma 20: multiply(?, multiply(inverse(?), inverse(inverse(X)))) = multiply(X, multiply(?, inverse(?))). 1.02/1.22 Proof: 1.02/1.22 multiply(?, multiply(inverse(?), inverse(inverse(X)))) 1.02/1.22 = { by lemma 13 } 1.02/1.22 multiply(X, multiply(inverse(X), inverse(inverse(X)))) 1.02/1.22 = { by lemma 14 } 1.02/1.22 multiply(X, multiply(?, inverse(?))) 1.02/1.22 1.02/1.22 Lemma 21: multiply(inverse(?), inverse(multiply(inverse(multiply(X, Y)), X))) = multiply(?, multiply(inverse(?), inverse(multiply(inverse(Y), ?)))). 1.02/1.22 Proof: 1.02/1.22 multiply(inverse(?), inverse(multiply(inverse(multiply(X, Y)), X))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(inverse(?), inverse(multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(X, Y)), X))))))), ?)))) 1.02/1.22 = { by lemma 16 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(inverse(?), inverse(Y))), ?)))) 1.02/1.22 = { by lemma 15 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(Y), ?)))) 1.02/1.22 1.02/1.22 Lemma 22: multiply(multiply(inverse(Y), X), multiply(?, inverse(?))) = inverse(multiply(inverse(X), Y)). 1.02/1.22 Proof: 1.02/1.22 multiply(multiply(inverse(Y), X), multiply(?, inverse(?))) 1.02/1.22 = { by lemma 20 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(inverse(multiply(inverse(Y), X))))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(inverse(Y), X)), multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?)))))) 1.02/1.22 = { by lemma 15 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(inverse(?), inverse(multiply(inverse(Y), X)))), multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(?, ?)))), ?)))))) 1.02/1.22 = { by lemma 9 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(inverse(?), multiply(inverse(?), inverse(multiply(inverse(Y), X))))))) 1.02/1.22 = { by lemma 19 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(inverse(?), multiply(inverse(?), inverse(multiply(inverse(multiply(X, multiply(?, inverse(multiply(inverse(multiply(inverse(X), Y)), ?))))), X))))))) 1.02/1.22 = { by lemma 21 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(inverse(?), multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(X), Y)), ?)))), ?)))))))) 1.02/1.22 = { by lemma 16 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(inverse(X), Y)), ?)))))) 1.02/1.22 = { by lemma 11 } 1.02/1.22 inverse(multiply(inverse(X), Y)) 1.02/1.22 1.02/1.22 Lemma 23: inverse(inverse(multiply(multiply(?, inverse(?)), X))) = X. 1.02/1.22 Proof: 1.02/1.22 inverse(inverse(multiply(multiply(?, inverse(?)), X))) 1.02/1.22 = { by lemma 19 } 1.02/1.22 inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(inverse(?), inverse(multiply(multiply(?, inverse(?)), X)))), ?))))) 1.02/1.22 = { by lemma 22 } 1.02/1.22 inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(inverse(?), inverse(multiply(multiply(?, inverse(?)), X)))), multiply(?, inverse(?)))))) 1.02/1.22 = { by lemma 10 } 1.02/1.22 X 1.02/1.22 1.02/1.22 Lemma 24: multiply(?, inverse(multiply(?, ?))) = inverse(?). 1.02/1.22 Proof: 1.02/1.22 multiply(?, inverse(multiply(?, ?))) 1.02/1.22 = { by lemma 18 } 1.02/1.22 multiply(inverse(?), inverse(multiply(?, inverse(?)))) 1.02/1.22 = { by lemma 14 } 1.02/1.22 multiply(inverse(?), inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(?)))), inverse(inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(?)))))))) 1.02/1.22 = { by lemma 23 } 1.02/1.22 multiply(inverse(?), inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(?)))), multiply(?, inverse(?))))) 1.02/1.22 = { by lemma 21 } 1.02/1.22 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, inverse(?))), ?)))) 1.02/1.22 = { by lemma 16 } 1.06/1.23 inverse(?) 1.06/1.23 1.06/1.23 Lemma 25: multiply(X, multiply(?, inverse(?))) = X. 1.06/1.23 Proof: 1.06/1.23 multiply(X, multiply(?, inverse(?))) 1.06/1.23 = { by lemma 24 } 1.06/1.23 multiply(X, multiply(?, multiply(?, inverse(multiply(?, ?))))) 1.06/1.23 = { by lemma 19 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(?, ?))))) 1.06/1.23 = { by lemma 8 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(inverse(?), multiply(?, multiply(?, inverse(multiply(?, ?)))))), ?))))) 1.06/1.23 = { by lemma 24 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(inverse(?), multiply(?, inverse(?)))), ?))))) 1.06/1.23 = { by lemma 4 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(inverse(?), inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(?, inverse(?))), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?)))))), ?))))) 1.06/1.23 = { by lemma 23 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(inverse(?), inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, inverse(?)))))), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?)))))), ?))))) 1.06/1.23 = { by lemma 14 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(inverse(?), inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(inverse(multiply(?, inverse(?)))), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?)))))), ?))))) 1.06/1.23 = { by lemma 4 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(inverse(?), inverse(multiply(?, inverse(?))))), ?))))) 1.06/1.23 = { by lemma 18 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(multiply(?, inverse(multiply(?, ?)))), ?))))) 1.06/1.23 = { by lemma 24 } 1.06/1.23 multiply(X, multiply(?, multiply(multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(inverse(inverse(?)), ?)), ?)))), inverse(multiply(inverse(inverse(?)), ?))))) 1.06/1.23 = { by lemma 9 } 1.06/1.24 X 1.06/1.24 1.06/1.24 Lemma 26: multiply(inverse(X), X) = multiply(?, inverse(?)). 1.06/1.24 Proof: 1.06/1.24 multiply(inverse(X), X) 1.06/1.24 = { by lemma 25 } 1.06/1.24 multiply(inverse(multiply(X, multiply(?, inverse(?)))), X) 1.06/1.24 = { by lemma 14 } 1.06/1.24 multiply(inverse(multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))))))), X) 1.06/1.24 = { by lemma 25 } 1.06/1.24 multiply(multiply(inverse(multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))))))), X), multiply(?, inverse(?))) 1.06/1.24 = { by lemma 22 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))))))) 1.06/1.24 = { by lemma 4 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?))))))))) 1.06/1.24 = { by lemma 11 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))), ?)))), multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))), ?)))))))))), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?))))))))) 1.06/1.24 = { by lemma 8 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))), ?)))), multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))))))), ?), inverse(multiply(?, multiply(?, ?))))), ?)), ?))))))))) 1.06/1.24 = { by lemma 4 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))), ?)))), multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))))))))))) 1.06/1.24 = { by lemma 19 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(multiply(?, inverse(?)), multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?))))))), multiply(?, inverse(?)))))))))))) 1.06/1.24 = { by lemma 25 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(multiply(?, inverse(?)), multiply(?, multiply(inverse(?), inverse(inverse(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))))))))))))) 1.06/1.24 = { by lemma 20 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(multiply(?, inverse(?)), multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), multiply(?, inverse(?))))))))) 1.06/1.24 = { by lemma 25 } 1.06/1.24 inverse(multiply(inverse(X), multiply(X, multiply(multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))), inverse(multiply(multiply(?, inverse(?)), multiply(?, multiply(?, multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, ?)))))))))))) 1.06/1.24 = { by lemma 7 } 1.06/1.24 multiply(?, inverse(?)) 1.06/1.24 1.06/1.24 Goal 1 (prove_these_axioms_1): multiply(inverse(a1), a1) = multiply(inverse(b1), b1). 1.06/1.24 Proof: 1.06/1.24 multiply(inverse(a1), a1) 1.06/1.24 = { by lemma 26 } 1.06/1.24 multiply(?, inverse(?)) 1.06/1.24 = { by lemma 26 } 1.06/1.24 multiply(inverse(b1), b1) 1.06/1.24 % SZS output end Proof 1.06/1.24 1.06/1.24 RESULT: Unsatisfiable (the axioms are contradictory). 1.06/1.24 EOF