0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.13/0.34 % Computer : n004.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 : 180 0.13/0.34 % DateTime : Thu Aug 29 12:38:08 EDT 2019 0.13/0.34 % CPUTime : 0.19/0.47 % SZS status Unsatisfiable 0.19/0.47 0.19/0.47 % SZS output start Proof 0.19/0.47 Take the following subset of the input axioms: 0.19/0.47 fof(prove_these_axioms_2, negated_conjecture, a2!=multiply(multiply(inverse(b2), b2), a2)). 0.19/0.47 fof(single_axiom, axiom, ![A, B, C]: multiply(A, inverse(multiply(multiply(multiply(inverse(B), B), inverse(multiply(inverse(multiply(A, inverse(B))), C))), B)))=C). 0.19/0.47 0.19/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of 0.19/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.19/0.47 We repeatedly replace C & s=t => u=v by the two clauses: 0.19/0.47 fresh(y, y, x1...xn) = u 0.19/0.47 C => fresh(s, t, x1...xn) = v 0.19/0.47 where fresh is a fresh function symbol and x1..xn are the free 0.19/0.47 variables of u and v. 0.19/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the 0.19/0.47 input problem has no model of domain size 1). 0.19/0.47 0.19/0.47 The encoding turns the above axioms into the following unit equations and goals: 0.19/0.47 0.19/0.48 Axiom 1 (single_axiom): multiply(X, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(X, inverse(Y))), Z))), Y))) = Z. 0.19/0.48 0.19/0.48 Lemma 2: multiply(inverse(multiply(Y, inverse(Z))), multiply(Y, inverse(multiply(multiply(multiply(inverse(Z), Z), inverse(X)), Z)))) = X. 0.19/0.48 Proof: 0.19/0.48 multiply(inverse(multiply(Y, inverse(Z))), multiply(Y, inverse(multiply(multiply(multiply(inverse(Z), Z), inverse(X)), Z)))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(Y, inverse(Z))), multiply(Y, inverse(multiply(multiply(multiply(inverse(Z), Z), inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), X))), ?))))), Z)))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(Y, inverse(Z))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), X))), ?))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 X 0.19/0.48 0.19/0.48 Lemma 3: multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(multiply(inverse(Y), Y), inverse(X))), Z))), X) = multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(Z, Y)))). 0.19/0.48 Proof: 0.19/0.48 multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(multiply(inverse(Y), Y), inverse(X))), Z))), X) 0.19/0.48 = { by lemma 2 } 0.19/0.48 multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(multiply(inverse(Y), Y), inverse(X))), Z))), X))), Y)))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(Z, Y)))) 0.19/0.48 0.19/0.48 Lemma 4: multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(X, Y)))))) = X. 0.19/0.48 Proof: 0.19/0.48 multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(X, Y)))))) 0.19/0.48 = { by lemma 3 } 0.19/0.48 multiply(multiply(inverse(Y), Y), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(Y), Y), inverse(?))), X))), ?))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 X 0.19/0.48 0.19/0.48 Lemma 5: inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(?))), multiply(inverse(multiply(X, inverse(Y))), Z)))), ?)) = Z. 0.19/0.48 Proof: 0.19/0.48 inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(?))), multiply(inverse(multiply(X, inverse(Y))), Z)))), ?)) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(X, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(?))), multiply(inverse(multiply(X, inverse(Y))), Z)))), ?))))), Y))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(X, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(X, inverse(Y))), Z))), Y))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 Z 0.19/0.48 0.19/0.48 Lemma 6: multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(Z, X))))))) = multiply(inverse(multiply(multiply(inverse(X), X), inverse(Y))), Z). 0.19/0.48 Proof: 0.19/0.48 multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(Z, X))))))) 0.19/0.48 = { by lemma 3 } 0.19/0.48 multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(multiply(inverse(X), X), inverse(Y))), Z))), Y)))) 0.19/0.48 = { by lemma 2 } 0.19/0.48 multiply(inverse(multiply(multiply(inverse(X), X), inverse(Y))), Z) 0.19/0.48 0.19/0.48 Lemma 7: multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(Z))) = multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(Z))). 0.19/0.48 Proof: 0.19/0.48 multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(Z))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(multiply(inverse(multiply(?, inverse(X))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(X))), inverse(?))), Z))), ?)))))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(?, inverse(X))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(X))), inverse(?))), Z))), ?))))), X))))))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(multiply(multiply(inverse(X), X), inverse(Z)), X))))))) 0.19/0.48 = { by lemma 6 } 0.19/0.48 multiply(inverse(multiply(multiply(inverse(X), X), inverse(Y))), multiply(multiply(inverse(X), X), inverse(Z))) 0.19/0.48 = { by lemma 6 } 0.19/0.48 multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(multiply(multiply(inverse(X), X), inverse(Z)), X))))))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(?, inverse(X))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(X))), inverse(?))), Z))), ?))))), X))))))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(X))), inverse(?))), Z))), ?)))))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(Z))) 0.19/0.48 0.19/0.48 Lemma 8: multiply(inverse(multiply(X, Z)), multiply(X, inverse(Y))) = multiply(inverse(multiply(?, Z)), multiply(?, inverse(Y))). 0.19/0.48 Proof: 0.19/0.48 multiply(inverse(multiply(X, Z)), multiply(X, inverse(Y))) 0.19/0.48 = { by lemma 5 } 0.19/0.48 multiply(inverse(multiply(X, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(V, inverse(W))), inverse(?))), multiply(inverse(multiply(V, inverse(W))), Z)))), ?)))), multiply(X, inverse(Y))) 0.19/0.48 = { by lemma 7 } 0.19/0.48 multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(V, inverse(W))), inverse(?))), multiply(inverse(multiply(V, inverse(W))), Z)))), ?)))), multiply(?, inverse(Y))) 0.19/0.48 = { by lemma 5 } 0.19/0.48 multiply(inverse(multiply(?, Z)), multiply(?, inverse(Y))) 0.19/0.48 0.19/0.48 Lemma 9: multiply(inverse(multiply(V, X)), multiply(V, W)) = multiply(inverse(multiply(?, X)), multiply(?, W)). 0.19/0.48 Proof: 0.19/0.48 multiply(inverse(multiply(V, X)), multiply(V, W)) 0.19/0.48 = { by lemma 5 } 0.19/0.48 multiply(inverse(multiply(V, X)), multiply(V, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), multiply(inverse(multiply(Y, inverse(Z))), W)))), ?)))) 0.19/0.48 = { by lemma 8 } 0.19/0.48 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), multiply(inverse(multiply(Y, inverse(Z))), W)))), ?)))) 0.19/0.48 = { by lemma 5 } 0.19/0.48 multiply(inverse(multiply(?, X)), multiply(?, W)) 0.19/0.48 0.19/0.48 Lemma 10: multiply(inverse(multiply(?, W)), multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(V, X))))))) = multiply(inverse(multiply(multiply(inverse(X), X), W)), V). 0.19/0.48 Proof: 0.19/0.48 multiply(inverse(multiply(?, W)), multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(V, X))))))) 0.19/0.48 = { by lemma 5 } 0.19/0.48 multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(?))), inverse(?))), multiply(inverse(multiply(?, inverse(?))), W)))), ?)))), multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(V, X))))))) 0.19/0.48 = { by lemma 6 } 0.19/0.48 multiply(inverse(multiply(multiply(inverse(X), X), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(?))), inverse(?))), multiply(inverse(multiply(?, inverse(?))), W)))), ?)))), V) 0.19/0.48 = { by lemma 5 } 0.19/0.48 multiply(inverse(multiply(multiply(inverse(X), X), W)), V) 0.19/0.48 0.19/0.48 Lemma 11: inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)) = inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)). 0.19/0.48 Proof: 0.19/0.48 inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(Y, inverse(multiply(multiply(multiply(inverse(Z), Z), inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X))))), Z))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(Y, inverse(multiply(multiply(multiply(inverse(Z), Z), inverse(W)), Z))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.48 multiply(Y, inverse(multiply(multiply(multiply(inverse(Z), Z), inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?))))), Z))) 0.19/0.48 = { by axiom 1 (single_axiom) } 0.19/0.49 inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)) 0.19/0.49 0.19/0.49 Lemma 12: inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(Z), inverse(X))), Y))), X)) = inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(Z), inverse(?))), Y))), ?)). 0.19/0.49 Proof: 0.19/0.49 inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(Z), inverse(X))), Y))), X)) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(V, inverse(multiply(multiply(multiply(inverse(W), W), inverse(multiply(inverse(multiply(V, inverse(W))), Z))), W)))), inverse(X))), Y))), X)) 0.19/0.49 = { by lemma 11 } 0.19/0.49 inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(V, inverse(multiply(multiply(multiply(inverse(W), W), inverse(multiply(inverse(multiply(V, inverse(W))), Z))), W)))), inverse(?))), Y))), ?)) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(Z), inverse(?))), Y))), ?)) 0.19/0.49 0.19/0.49 Lemma 13: multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(Y), inverse(X))), Z))), X) = multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(Y), inverse(?))), Z))), ?). 0.19/0.49 Proof: 0.19/0.49 multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(Y), inverse(X))), Z))), X) 0.19/0.49 = { by lemma 2 } 0.19/0.49 multiply(inverse(multiply(W, inverse(V))), multiply(W, inverse(multiply(multiply(multiply(inverse(V), V), inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(Y), inverse(X))), Z))), X))), V)))) 0.19/0.49 = { by lemma 12 } 0.19/0.49 multiply(inverse(multiply(W, inverse(V))), multiply(W, inverse(multiply(multiply(multiply(inverse(V), V), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(Y), inverse(?))), Z))), ?))), V)))) 0.19/0.49 = { by lemma 2 } 0.19/0.49 multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(Y), inverse(?))), Z))), ?) 0.19/0.49 0.19/0.49 Lemma 14: multiply(inverse(multiply(multiply(inverse(multiply(?, X)), multiply(?, X)), V)), U) = multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), V)), U). 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(multiply(multiply(inverse(multiply(?, X)), multiply(?, X)), V)), U) 0.19/0.49 = { by lemma 9 } 0.19/0.49 multiply(inverse(multiply(multiply(inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)), multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)), V)), U) 0.19/0.49 = { by lemma 10 } 0.19/0.49 multiply(inverse(multiply(?, V)), multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)))), multiply(?, inverse(multiply(U, multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)))))))) 0.19/0.49 = { by lemma 11 } 0.19/0.49 multiply(inverse(multiply(?, V)), multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)))), multiply(?, inverse(multiply(U, multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(X))), W))), X)))))))) 0.19/0.49 = { by lemma 13 } 0.19/0.49 multiply(inverse(multiply(?, V)), multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)))), multiply(?, inverse(multiply(U, multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)))))))) 0.19/0.49 = { by lemma 10 } 0.19/0.49 multiply(inverse(multiply(multiply(inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)), multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(Y, inverse(Z))), inverse(?))), W))), ?)), V)), U) 0.19/0.49 = { by lemma 9 } 0.19/0.49 multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), V)), U) 0.19/0.49 0.19/0.49 Lemma 15: inverse(multiply(multiply(inverse(multiply(?, X)), multiply(?, X)), Y)) = inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), Y)). 0.19/0.49 Proof: 0.19/0.49 inverse(multiply(multiply(inverse(multiply(?, X)), multiply(?, X)), Y)) 0.19/0.49 = { by lemma 4 } 0.19/0.49 multiply(multiply(inverse(Z), Z), inverse(multiply(inverse(multiply(?, inverse(Z))), multiply(?, inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, X)), multiply(?, X)), Y)), Z)))))) 0.19/0.49 = { by lemma 14 } 0.19/0.49 multiply(multiply(inverse(Z), Z), inverse(multiply(inverse(multiply(?, inverse(Z))), multiply(?, inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), Y)), Z)))))) 0.19/0.49 = { by lemma 4 } 0.19/0.49 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), Y)) 0.19/0.49 0.19/0.49 Lemma 16: multiply(inverse(multiply(?, X)), multiply(?, X)) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(multiply(?, X)), multiply(?, X)) 0.19/0.49 = { by lemma 4 } 0.19/0.49 multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(multiply(inverse(multiply(?, X)), multiply(?, X)), Y)))))) 0.19/0.49 = { by lemma 15 } 0.19/0.49 multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), Y)))))) 0.19/0.49 = { by lemma 4 } 0.19/0.49 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.19/0.49 0.19/0.49 Lemma 17: multiply(inverse(X), X) = multiply(inverse(?), ?). 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(X), X) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), X))), Y)))), X) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), X))), Y)))), multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), X))), Y)))) 0.19/0.49 = { by lemma 16 } 0.19/0.49 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.19/0.49 = { by lemma 16 } 0.19/0.49 multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), ?))), Y)))), multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), ?))), Y)))) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 multiply(inverse(?), multiply(?, inverse(multiply(multiply(multiply(inverse(Y), Y), inverse(multiply(inverse(multiply(?, inverse(Y))), ?))), Y)))) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 multiply(inverse(?), ?) 0.19/0.49 0.19/0.49 Lemma 18: multiply(inverse(multiply(W, inverse(?))), multiply(W, Z)) = multiply(inverse(multiply(?, inverse(?))), multiply(?, Z)). 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(multiply(W, inverse(?))), multiply(W, Z)) 0.19/0.49 = { by lemma 5 } 0.19/0.49 multiply(inverse(multiply(W, inverse(?))), multiply(W, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(?))), multiply(inverse(multiply(X, inverse(Y))), Z)))), ?)))) 0.19/0.49 = { by lemma 2 } 0.19/0.49 multiply(inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(?))), multiply(inverse(multiply(X, inverse(Y))), Z)) 0.19/0.49 = { by lemma 2 } 0.19/0.49 multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(X, inverse(Y))), inverse(?))), multiply(inverse(multiply(X, inverse(Y))), Z)))), ?)))) 0.19/0.49 = { by lemma 5 } 0.19/0.49 multiply(inverse(multiply(?, inverse(?))), multiply(?, Z)) 0.19/0.49 0.19/0.49 Lemma 19: multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), X) = X. 0.19/0.49 Proof: 0.19/0.49 multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), X) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(?), ?))), X) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(multiply(multiply(inverse(X), X), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(X))), multiply(multiply(inverse(?), ?), inverse(X))))), X) 0.19/0.49 = { by lemma 3 } 0.19/0.49 multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(X)), ?)))) 0.19/0.49 = { by lemma 2 } 0.19/0.49 X 0.19/0.49 0.19/0.49 Lemma 20: multiply(inverse(multiply(?, X)), multiply(?, Y)) = multiply(inverse(X), Y). 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(multiply(?, X)), multiply(?, Y)) 0.19/0.49 = { by lemma 9 } 0.19/0.49 multiply(inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), X)), multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), Y)) 0.19/0.49 = { by lemma 19 } 0.19/0.49 multiply(inverse(X), multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), Y)) 0.19/0.49 = { by lemma 19 } 0.19/0.49 multiply(inverse(X), Y) 0.19/0.49 0.19/0.49 Lemma 21: multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(X), Y)) = multiply(inverse(X), Y). 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(X), Y)) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(inverse(multiply(inverse(X), X)), multiply(inverse(X), Y)) 0.19/0.49 = { by lemma 9 } 0.19/0.49 multiply(inverse(multiply(?, X)), multiply(?, Y)) 0.19/0.49 = { by lemma 20 } 0.19/0.49 multiply(inverse(X), Y) 0.19/0.49 0.19/0.49 Lemma 22: multiply(inverse(multiply(inverse(?), ?)), X) = X. 0.19/0.49 Proof: 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), X) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(?), inverse(?))), X))), ?)))) 0.19/0.49 = { by lemma 21 } 0.19/0.49 multiply(inverse(?), inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(?), inverse(?))), X))), ?))) 0.19/0.49 = { by axiom 1 (single_axiom) } 0.19/0.49 X 0.19/0.49 0.19/0.49 Lemma 23: multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))) = inverse(multiply(inverse(?), ?)). 0.19/0.49 Proof: 0.19/0.49 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))) 0.19/0.49 = { by lemma 20 } 0.19/0.49 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(inverse(?), ?))) 0.19/0.49 = { by lemma 9 } 0.19/0.49 multiply(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?)), inverse(multiply(inverse(?), ?))) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?)), inverse(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))))) 0.19/0.49 = { by lemma 20 } 0.19/0.49 multiply(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?)), inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(?), ?)))))) 0.19/0.49 = { by lemma 21 } 0.19/0.49 multiply(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?)), inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.19/0.49 = { by lemma 4 } 0.19/0.49 inverse(multiply(inverse(?), ?)) 0.19/0.49 0.19/0.49 Goal 1 (prove_these_axioms_2): a2 = multiply(multiply(inverse(b2), b2), a2). 0.19/0.49 Proof: 0.19/0.49 a2 0.19/0.49 = { by lemma 22 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), a2) 0.19/0.49 = { by lemma 21 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(inverse(?), ?)), a2)) 0.19/0.49 = { by lemma 23 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), a2)) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))), a2)) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(inverse(inverse(multiply(inverse(?), ?)))), inverse(inverse(multiply(inverse(?), ?)))))), a2)) 0.19/0.49 = { by lemma 20 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(multiply(?, inverse(inverse(multiply(inverse(?), ?))))), multiply(?, inverse(inverse(multiply(inverse(?), ?))))))), a2)) 0.19/0.49 = { by lemma 23 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(multiply(?, inverse(inverse(multiply(inverse(?), ?))))), multiply(?, inverse(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))))))), a2)) 0.19/0.49 = { by lemma 4 } 0.19/0.49 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(inverse(?), ?), a2)) 0.19/0.49 = { by lemma 22 } 0.19/0.49 multiply(multiply(inverse(?), ?), a2) 0.19/0.49 = { by lemma 17 } 0.19/0.49 multiply(multiply(inverse(b2), b2), a2) 0.19/0.49 % SZS output end Proof 0.19/0.49 0.19/0.49 RESULT: Unsatisfiable (the axioms are contradictory). 0.19/0.50 EOF