0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.12 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.12/0.33 % Computer : n025.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 12:56:30 EDT 2019 0.12/0.33 % CPUTime : 0.19/0.48 % SZS status Unsatisfiable 0.19/0.48 0.19/0.48 % SZS output start Proof 0.19/0.48 Take the following subset of the input axioms: 0.19/0.50 fof(prove_these_axioms_3, negated_conjecture, multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3))). 0.19/0.50 fof(single_axiom, axiom, ![B, A, C]: B=multiply(multiply(inverse(multiply(A, inverse(multiply(B, C)))), multiply(A, inverse(C))), inverse(multiply(inverse(C), C)))). 0.19/0.50 0.19/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of 0.19/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.19/0.50 We repeatedly replace C & s=t => u=v by the two clauses: 0.19/0.50 fresh(y, y, x1...xn) = u 0.19/0.50 C => fresh(s, t, x1...xn) = v 0.19/0.50 where fresh is a fresh function symbol and x1..xn are the free 0.19/0.50 variables of u and v. 0.19/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the 0.19/0.50 input problem has no model of domain size 1). 0.19/0.50 0.19/0.50 The encoding turns the above axioms into the following unit equations and goals: 0.19/0.50 0.19/0.50 Axiom 1 (single_axiom): X = multiply(multiply(inverse(multiply(Y, inverse(multiply(X, Z)))), multiply(Y, inverse(Z))), inverse(multiply(inverse(Z), Z))). 0.19/0.50 0.19/0.50 Lemma 2: multiply(inverse(multiply(X, inverse(multiply(Y, Z)))), multiply(X, inverse(Z))) = multiply(inverse(multiply(?, inverse(multiply(Y, Z)))), multiply(?, inverse(Z))). 0.19/0.50 Proof: 0.19/0.50 multiply(inverse(multiply(X, inverse(multiply(Y, Z)))), multiply(X, inverse(Z))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(multiply(inverse(multiply(W, inverse(multiply(multiply(inverse(multiply(X, inverse(multiply(Y, Z)))), multiply(X, inverse(Z))), inverse(multiply(inverse(Z), Z)))))), multiply(W, inverse(inverse(multiply(inverse(Z), Z))))), inverse(multiply(inverse(inverse(multiply(inverse(Z), Z))), inverse(multiply(inverse(Z), Z))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(inverse(multiply(inverse(Z), Z))))), inverse(multiply(inverse(inverse(multiply(inverse(Z), Z))), inverse(multiply(inverse(Z), Z))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(multiply(inverse(multiply(W, inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(Y, Z)))), multiply(?, inverse(Z))), inverse(multiply(inverse(Z), Z)))))), multiply(W, inverse(inverse(multiply(inverse(Z), Z))))), inverse(multiply(inverse(inverse(multiply(inverse(Z), Z))), inverse(multiply(inverse(Z), Z))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(inverse(multiply(?, inverse(multiply(Y, Z)))), multiply(?, inverse(Z))) 0.19/0.50 0.19/0.50 Lemma 3: multiply(multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(inverse(multiply(inverse(Z), Z))))), inverse(multiply(inverse(inverse(multiply(inverse(Z), Z))), inverse(multiply(inverse(Z), Z))))) = multiply(inverse(multiply(?, inverse(multiply(Y, Z)))), multiply(?, inverse(Z))). 0.19/0.50 Proof: 0.19/0.50 multiply(multiply(inverse(multiply(W, inverse(Y))), multiply(W, inverse(inverse(multiply(inverse(Z), Z))))), inverse(multiply(inverse(inverse(multiply(inverse(Z), Z))), inverse(multiply(inverse(Z), Z))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(multiply(inverse(multiply(W, inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(Y, Z)))), multiply(?, inverse(Z))), inverse(multiply(inverse(Z), Z)))))), multiply(W, inverse(inverse(multiply(inverse(Z), Z))))), inverse(multiply(inverse(inverse(multiply(inverse(Z), Z))), inverse(multiply(inverse(Z), Z))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(inverse(multiply(?, inverse(multiply(Y, Z)))), multiply(?, inverse(Z))) 0.19/0.50 0.19/0.50 Lemma 4: multiply(inverse(multiply(?, inverse(multiply(multiply(X, inverse(multiply(inverse(Y), Y))), Y)))), multiply(?, inverse(Y))) = X. 0.19/0.50 Proof: 0.19/0.50 multiply(inverse(multiply(?, inverse(multiply(multiply(X, inverse(multiply(inverse(Y), Y))), Y)))), multiply(?, inverse(Y))) 0.19/0.50 = { by lemma 3 } 0.19/0.50 multiply(multiply(inverse(multiply(?, inverse(multiply(X, inverse(multiply(inverse(Y), Y)))))), multiply(?, inverse(inverse(multiply(inverse(Y), Y))))), inverse(multiply(inverse(inverse(multiply(inverse(Y), Y))), inverse(multiply(inverse(Y), Y))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 X 0.19/0.50 0.19/0.50 Lemma 5: multiply(inverse(multiply(Z, inverse(X))), multiply(Z, inverse(multiply(?, inverse(Y))))) = multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(?, inverse(Y))))). 0.19/0.50 Proof: 0.19/0.50 multiply(inverse(multiply(Z, inverse(X))), multiply(Z, inverse(multiply(?, inverse(Y))))) 0.19/0.50 = { by lemma 4 } 0.19/0.50 multiply(inverse(multiply(Z, inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(X, inverse(multiply(inverse(Y), Y))), Y)))), multiply(?, inverse(Y)))))), multiply(Z, inverse(multiply(?, inverse(Y))))) 0.19/0.50 = { by lemma 2 } 0.19/0.50 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(X, inverse(multiply(inverse(Y), Y))), Y)))), multiply(?, inverse(Y)))))), multiply(?, inverse(multiply(?, inverse(Y))))) 0.19/0.50 = { by lemma 4 } 0.19/0.50 multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(?, inverse(Y))))) 0.19/0.50 0.19/0.50 Lemma 6: inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(Y))), inverse(multiply(inverse(Y), Y))), Y)))) = X. 0.19/0.50 Proof: 0.19/0.50 inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(Y))), inverse(multiply(inverse(Y), Y))), Y)))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.50 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(Y))), inverse(multiply(inverse(Y), Y))), Y)))), multiply(?, inverse(Y)))))), multiply(?, inverse(multiply(?, inverse(Y))))), inverse(multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(Y))))) 0.19/0.50 = { by lemma 4 } 0.19/0.50 multiply(multiply(inverse(multiply(?, inverse(multiply(X, multiply(?, inverse(Y)))))), multiply(?, inverse(multiply(?, inverse(Y))))), inverse(multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(Y))))) 0.19/0.50 = { by axiom 1 (single_axiom) } 0.19/0.51 X 0.19/0.51 0.19/0.51 Lemma 7: multiply(inverse(multiply(W, inverse(X))), multiply(W, Y)) = multiply(inverse(multiply(?, inverse(X))), multiply(?, Y)). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(multiply(W, inverse(X))), multiply(W, Y)) 0.19/0.51 = { by lemma 6 } 0.19/0.51 multiply(inverse(multiply(W, inverse(X))), multiply(W, inverse(multiply(?, inverse(multiply(multiply(multiply(Y, multiply(?, inverse(Z))), inverse(multiply(inverse(Z), Z))), Z)))))) 0.19/0.51 = { by lemma 5 } 0.19/0.51 multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(multiply(?, inverse(multiply(multiply(multiply(Y, multiply(?, inverse(Z))), inverse(multiply(inverse(Z), Z))), Z)))))) 0.19/0.51 = { by lemma 6 } 0.19/0.51 multiply(inverse(multiply(?, inverse(X))), multiply(?, Y)) 0.19/0.51 0.19/0.51 Lemma 8: multiply(inverse(Z), multiply(multiply(inverse(multiply(?, inverse(multiply(Z, X)))), multiply(?, inverse(X))), Y)) = multiply(inverse(multiply(?, inverse(multiply(inverse(X), X)))), multiply(?, Y)). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(Z), multiply(multiply(inverse(multiply(?, inverse(multiply(Z, X)))), multiply(?, inverse(X))), Y)) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 multiply(inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(Z, X)))), multiply(?, inverse(X))), inverse(multiply(inverse(X), X)))), multiply(multiply(inverse(multiply(?, inverse(multiply(Z, X)))), multiply(?, inverse(X))), Y)) 0.19/0.51 = { by lemma 7 } 0.19/0.51 multiply(inverse(multiply(?, inverse(multiply(inverse(X), X)))), multiply(?, Y)) 0.19/0.51 0.19/0.51 Lemma 9: multiply(inverse(Y), Y) = multiply(inverse(?), ?). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(Y), Y) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 multiply(inverse(Y), multiply(multiply(inverse(multiply(?, inverse(multiply(Y, X)))), multiply(?, inverse(X))), inverse(multiply(inverse(X), X)))) 0.19/0.51 = { by lemma 8 } 0.19/0.51 multiply(inverse(multiply(?, inverse(multiply(inverse(X), X)))), multiply(?, inverse(multiply(inverse(X), X)))) 0.19/0.51 = { by lemma 8 } 0.19/0.51 multiply(inverse(?), multiply(multiply(inverse(multiply(?, inverse(multiply(?, X)))), multiply(?, inverse(X))), inverse(multiply(inverse(X), X)))) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 multiply(inverse(?), ?) 0.19/0.51 0.19/0.51 Lemma 10: multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(X))), inverse(multiply(inverse(?), ?))) = inverse(X). 0.19/0.51 Proof: 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(X))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(X), X)))), multiply(?, inverse(X))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(X), X)))), multiply(?, inverse(X))), inverse(multiply(inverse(X), X))) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 inverse(X) 0.19/0.51 0.19/0.51 Lemma 11: multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))) = inverse(multiply(inverse(?), ?)). 0.19/0.51 Proof: 0.19/0.51 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(?, inverse(multiply(?, ?)))), multiply(?, inverse(?))), inverse(multiply(inverse(?), ?)))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 8 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(?), ?)))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 10 } 0.19/0.51 inverse(multiply(inverse(?), ?)) 0.19/0.51 0.19/0.51 Lemma 12: multiply(inverse(multiply(W, X)), multiply(W, Z)) = multiply(inverse(multiply(?, X)), multiply(?, Z)). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(multiply(W, X)), multiply(W, Z)) 0.19/0.51 = { by lemma 6 } 0.19/0.51 multiply(inverse(multiply(W, inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(Y))), inverse(multiply(inverse(Y), Y))), Y)))))), multiply(W, Z)) 0.19/0.51 = { by lemma 7 } 0.19/0.51 multiply(inverse(multiply(?, inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(Y))), inverse(multiply(inverse(Y), Y))), Y)))))), multiply(?, Z)) 0.19/0.51 = { by lemma 6 } 0.19/0.51 multiply(inverse(multiply(?, X)), multiply(?, Z)) 0.19/0.51 0.19/0.51 Lemma 13: multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(X), Y)) = multiply(inverse(multiply(?, X)), multiply(?, Y)). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(X), Y)) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(inverse(multiply(inverse(X), X)), multiply(inverse(X), Y)) 0.19/0.51 = { by lemma 12 } 0.19/0.51 multiply(inverse(multiply(?, X)), multiply(?, Y)) 0.19/0.51 0.19/0.51 Lemma 14: multiply(multiply(inverse(multiply(?, inverse(multiply(X, Y)))), multiply(?, inverse(Y))), inverse(multiply(inverse(?), ?))) = X. 0.19/0.51 Proof: 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(X, Y)))), multiply(?, inverse(Y))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 13 } 0.19/0.51 multiply(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(inverse(multiply(X, Y))), inverse(Y))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(inverse(inverse(multiply(X, Y))), inverse(multiply(X, Y)))), multiply(inverse(inverse(multiply(X, Y))), inverse(Y))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(inverse(inverse(multiply(X, Y))), inverse(multiply(X, Y)))), multiply(inverse(inverse(multiply(X, Y))), inverse(Y))), inverse(multiply(inverse(Y), Y))) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 X 0.19/0.51 0.19/0.51 Lemma 15: inverse(multiply(inverse(?), ?)) = multiply(inverse(?), ?). 0.19/0.51 Proof: 0.19/0.51 inverse(multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 11 } 0.19/0.51 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(inverse(multiply(inverse(?), ?))))), multiply(?, inverse(inverse(multiply(inverse(?), ?))))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 11 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))))), multiply(?, inverse(inverse(multiply(inverse(?), ?))))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 14 } 0.19/0.51 multiply(inverse(?), ?) 0.19/0.51 0.19/0.51 Lemma 16: multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(?), ?)) = multiply(inverse(multiply(?, X)), multiply(?, Y)). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(Y), Y)) 0.19/0.51 = { by lemma 12 } 0.19/0.51 multiply(inverse(multiply(?, X)), multiply(?, Y)) 0.19/0.51 0.19/0.51 Lemma 17: multiply(inverse(multiply(multiply(inverse(?), ?), X)), multiply(inverse(?), ?)) = multiply(inverse(multiply(?, X)), multiply(?, multiply(inverse(?), ?))). 0.19/0.51 Proof: 0.19/0.51 multiply(inverse(multiply(multiply(inverse(?), ?), X)), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 15 } 0.19/0.51 multiply(inverse(multiply(inverse(multiply(inverse(?), ?)), X)), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 16 } 0.19/0.51 multiply(inverse(multiply(?, X)), multiply(?, multiply(inverse(?), ?))) 0.19/0.51 0.19/0.51 Lemma 18: multiply(multiply(inverse(multiply(?, inverse(multiply(X, multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) = X. 0.19/0.51 Proof: 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(X, multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 17 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(inverse(?), ?)), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 15 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), inverse(multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 11 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 15 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))), inverse(multiply(inverse(?), ?))) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.19/0.51 = { by lemma 13 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))), inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?)))) 0.19/0.51 = { by axiom 1 (single_axiom) } 0.19/0.51 X 0.19/0.51 0.19/0.51 Lemma 19: inverse(multiply(multiply(inverse(?), ?), X)) = inverse(X). 0.19/0.51 Proof: 0.19/0.51 inverse(multiply(multiply(inverse(?), ?), X)) 0.19/0.51 = { by lemma 18 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(multiply(inverse(?), ?), X)), multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 17 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(?, X)), multiply(?, multiply(inverse(?), ?)))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 13 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(X), multiply(inverse(?), ?)))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 11 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?))), multiply(inverse(X), multiply(inverse(?), ?)))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(X), multiply(inverse(?), ?))))), multiply(?, inverse(multiply(inverse(X), multiply(inverse(?), ?))))), inverse(multiply(inverse(?), ?))), multiply(inverse(X), multiply(inverse(?), ?)))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 9 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(X), multiply(inverse(?), ?))))), multiply(?, inverse(multiply(inverse(X), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(inverse(X), multiply(inverse(?), ?))), multiply(inverse(X), multiply(inverse(?), ?))))), multiply(inverse(X), multiply(inverse(?), ?)))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 6 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(X), multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 18 } 0.19/0.51 inverse(X) 0.19/0.51 0.19/0.51 Lemma 20: multiply(multiply(inverse(inverse(multiply(X, multiply(inverse(?), ?)))), multiply(inverse(?), ?)), multiply(inverse(?), ?)) = X. 0.19/0.51 Proof: 0.19/0.51 multiply(multiply(inverse(inverse(multiply(X, multiply(inverse(?), ?)))), multiply(inverse(?), ?)), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 19 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(inverse(?), ?)), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 15 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), inverse(multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 11 } 0.19/0.51 multiply(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(X, multiply(inverse(?), ?))))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), ?)))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 12 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(X, multiply(inverse(?), ?))))), multiply(?, inverse(multiply(inverse(?), ?)))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 15 } 0.19/0.51 multiply(multiply(inverse(multiply(?, inverse(multiply(X, multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?)) 0.19/0.51 = { by lemma 18 } 0.19/0.52 X 0.19/0.52 0.19/0.52 Lemma 21: inverse(multiply(X, multiply(inverse(?), ?))) = inverse(X). 0.19/0.52 Proof: 0.19/0.52 inverse(multiply(X, multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 19 } 0.19/0.52 inverse(multiply(multiply(inverse(?), ?), multiply(X, multiply(inverse(?), ?)))) 0.19/0.52 = { by lemma 19 } 0.19/0.52 inverse(multiply(multiply(inverse(?), ?), multiply(multiply(inverse(?), ?), multiply(X, multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 15 } 0.19/0.52 inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(inverse(?), ?), multiply(X, multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 15 } 0.19/0.52 inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(inverse(?), ?)), multiply(X, multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 13 } 0.19/0.52 inverse(multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(X, multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 20 } 0.19/0.52 inverse(multiply(multiply(inverse(inverse(multiply(multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(X, multiply(inverse(?), ?)))), multiply(inverse(?), ?)))), multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 15 } 0.19/0.52 inverse(multiply(multiply(inverse(inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, multiply(X, multiply(inverse(?), ?)))), multiply(inverse(?), ?)))), multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 6 } 0.19/0.52 inverse(multiply(multiply(inverse(inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(multiply(?, inverse(multiply(multiply(multiply(multiply(X, multiply(inverse(?), ?)), multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?)))))), multiply(inverse(?), ?)))), multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 15 } 0.19/0.52 inverse(multiply(multiply(inverse(inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(?), ?)))), multiply(?, inverse(multiply(?, inverse(multiply(multiply(multiply(multiply(X, multiply(inverse(?), ?)), multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?)))))), inverse(multiply(inverse(?), ?))))), multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 10 } 0.19/0.52 inverse(multiply(multiply(inverse(inverse(inverse(multiply(?, inverse(multiply(multiply(multiply(multiply(X, multiply(inverse(?), ?)), multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?)))))), multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 6 } 0.19/0.52 inverse(multiply(multiply(inverse(inverse(multiply(X, multiply(inverse(?), ?)))), multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 20 } 0.19/0.52 inverse(X) 0.19/0.52 0.19/0.52 Lemma 22: inverse(multiply(inverse(multiply(?, X)), multiply(?, Y))) = inverse(multiply(inverse(X), Y)). 0.19/0.52 Proof: 0.19/0.52 inverse(multiply(inverse(multiply(?, X)), multiply(?, Y))) 0.19/0.52 = { by lemma 13 } 0.19/0.52 inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(X), Y))) 0.19/0.52 = { by lemma 15 } 0.19/0.52 inverse(multiply(multiply(inverse(?), ?), multiply(inverse(X), Y))) 0.19/0.52 = { by lemma 19 } 0.19/0.52 inverse(multiply(inverse(X), Y)) 0.19/0.52 0.19/0.52 Lemma 23: inverse(inverse(X)) = X. 0.19/0.52 Proof: 0.19/0.52 inverse(inverse(X)) 0.19/0.52 = { by lemma 6 } 0.19/0.52 inverse(inverse(inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?)))))) 0.19/0.52 = { by lemma 21 } 0.19/0.52 inverse(inverse(inverse(multiply(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?))), multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 21 } 0.19/0.52 inverse(multiply(inverse(inverse(multiply(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?))), multiply(inverse(?), ?)))), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 22 } 0.19/0.52 inverse(multiply(inverse(multiply(?, inverse(multiply(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?))), multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?)))) 0.19/0.52 = { by lemma 21 } 0.19/0.52 inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?))), multiply(inverse(?), ?))))), multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 18 } 0.19/0.52 inverse(multiply(?, inverse(multiply(multiply(multiply(X, multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))), ?)))) 0.19/0.52 = { by lemma 6 } 0.19/0.52 X 0.19/0.52 0.19/0.52 Lemma 24: multiply(inverse(?), ?) = multiply(?, inverse(?)). 0.19/0.52 Proof: 0.19/0.52 multiply(inverse(?), ?) 0.19/0.52 = { by lemma 9 } 0.19/0.52 multiply(inverse(inverse(?)), inverse(?)) 0.19/0.52 = { by lemma 23 } 0.19/0.52 multiply(?, inverse(?)) 0.19/0.52 0.19/0.52 Lemma 25: multiply(X, multiply(?, inverse(?))) = X. 0.19/0.52 Proof: 0.19/0.52 multiply(X, multiply(?, inverse(?))) 0.19/0.52 = { by lemma 24 } 0.19/0.52 multiply(X, multiply(inverse(?), ?)) 0.19/0.52 = { by lemma 23 } 0.19/0.52 inverse(inverse(multiply(X, multiply(inverse(?), ?)))) 0.19/0.52 = { by lemma 21 } 0.19/0.52 inverse(inverse(X)) 0.19/0.52 = { by lemma 23 } 0.19/0.52 X 0.19/0.52 0.19/0.52 Lemma 26: inverse(multiply(inverse(Y), X)) = multiply(inverse(X), Y). 0.19/0.52 Proof: 0.19/0.52 inverse(multiply(inverse(Y), X)) 0.19/0.52 = { by lemma 22 } 0.19/0.52 inverse(multiply(inverse(multiply(?, Y)), multiply(?, X))) 0.19/0.52 = { by lemma 16 } 0.19/0.52 inverse(multiply(inverse(multiply(inverse(X), Y)), multiply(inverse(?), ?))) 0.19/0.52 = { by lemma 21 } 0.19/0.52 inverse(inverse(multiply(inverse(X), Y))) 0.19/0.52 = { by lemma 23 } 0.19/0.52 multiply(inverse(X), Y) 0.19/0.52 0.19/0.52 Lemma 27: multiply(inverse(X), inverse(Y)) = inverse(multiply(Y, X)). 0.19/0.52 Proof: 0.19/0.52 multiply(inverse(X), inverse(Y)) 0.19/0.52 = { by lemma 26 } 0.19/0.52 inverse(multiply(inverse(inverse(Y)), X)) 0.19/0.52 = { by lemma 23 } 0.19/0.52 inverse(multiply(Y, X)) 0.19/0.52 0.19/0.52 Lemma 28: inverse(multiply(X, inverse(Y))) = multiply(Y, inverse(X)). 0.19/0.52 Proof: 0.19/0.52 inverse(multiply(X, inverse(Y))) 0.19/0.52 = { by lemma 27 } 0.19/0.52 multiply(inverse(inverse(Y)), inverse(X)) 0.19/0.52 = { by lemma 23 } 0.19/0.52 multiply(Y, inverse(X)) 0.19/0.52 0.19/0.52 Lemma 29: multiply(inverse(multiply(?, inverse(multiply(X, Y)))), multiply(?, inverse(Y))) = multiply(inverse(multiply(?, inverse(multiply(X, ?)))), multiply(?, inverse(?))). 0.19/0.52 Proof: 0.19/0.52 multiply(inverse(multiply(?, inverse(multiply(X, Y)))), multiply(?, inverse(Y))) 0.19/0.52 = { by axiom 1 (single_axiom) } 0.19/0.52 multiply(multiply(inverse(multiply(Z, inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(X, Y)))), multiply(?, inverse(Y))), inverse(multiply(inverse(?), ?)))))), multiply(Z, inverse(inverse(multiply(inverse(?), ?))))), inverse(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 14 } 0.19/0.52 multiply(multiply(inverse(multiply(Z, inverse(X))), multiply(Z, inverse(inverse(multiply(inverse(?), ?))))), inverse(multiply(inverse(inverse(multiply(inverse(?), ?))), inverse(multiply(inverse(?), ?))))) 0.19/0.52 = { by lemma 3 } 0.19/0.52 multiply(inverse(multiply(?, inverse(multiply(X, ?)))), multiply(?, inverse(?))) 0.19/0.52 0.19/0.52 Lemma 30: inverse(multiply(inverse(inverse(multiply(X, Y))), inverse(Y))) = inverse(multiply(inverse(inverse(multiply(X, ?))), inverse(?))). 0.19/0.52 Proof: 0.19/0.52 inverse(multiply(inverse(inverse(multiply(X, Y))), inverse(Y))) 0.19/0.52 = { by lemma 22 } 0.19/0.52 inverse(multiply(inverse(multiply(?, inverse(multiply(X, Y)))), multiply(?, inverse(Y)))) 0.19/0.52 = { by lemma 29 } 0.19/0.52 inverse(multiply(inverse(multiply(?, inverse(multiply(X, ?)))), multiply(?, inverse(?)))) 0.19/0.52 = { by lemma 22 } 0.19/0.52 inverse(multiply(inverse(inverse(multiply(X, ?))), inverse(?))) 0.19/0.52 0.19/0.52 Lemma 31: multiply(X, inverse(multiply(Y, X))) = multiply(?, inverse(multiply(Y, ?))). 0.19/0.52 Proof: 0.19/0.52 multiply(X, inverse(multiply(Y, X))) 0.19/0.52 = { by lemma 28 } 0.19/0.52 inverse(multiply(multiply(Y, X), inverse(X))) 0.19/0.52 = { by lemma 27 } 0.19/0.52 multiply(inverse(inverse(X)), inverse(multiply(Y, X))) 0.19/0.52 = { by lemma 26 } 0.19/0.52 inverse(multiply(inverse(inverse(multiply(Y, X))), inverse(X))) 0.19/0.52 = { by lemma 30 } 0.19/0.52 inverse(multiply(inverse(inverse(multiply(Y, ?))), inverse(?))) 0.19/0.52 = { by lemma 26 } 0.19/0.52 multiply(inverse(inverse(?)), inverse(multiply(Y, ?))) 0.19/0.52 = { by lemma 27 } 0.19/0.52 inverse(multiply(multiply(Y, ?), inverse(?))) 0.19/0.52 = { by lemma 28 } 0.19/0.52 multiply(?, inverse(multiply(Y, ?))) 0.19/0.52 0.19/0.52 Lemma 32: multiply(multiply(X, ?), inverse(?)) = X. 0.19/0.52 Proof: 0.19/0.52 multiply(multiply(X, ?), inverse(?)) 0.19/0.52 = { by lemma 25 } 0.19/0.52 multiply(multiply(multiply(X, ?), inverse(?)), multiply(?, inverse(?))) 0.19/0.52 = { by lemma 25 } 0.19/0.52 multiply(multiply(multiply(multiply(X, ?), inverse(?)), multiply(?, inverse(?))), multiply(?, inverse(?))) 0.19/0.52 = { by lemma 28 } 0.19/0.52 multiply(multiply(inverse(multiply(?, inverse(multiply(X, ?)))), multiply(?, inverse(?))), multiply(?, inverse(?))) 0.19/0.52 = { by lemma 24 } 0.19/0.52 multiply(multiply(inverse(multiply(?, inverse(multiply(X, ?)))), multiply(?, inverse(?))), multiply(inverse(?), ?)) 0.19/0.52 = { by lemma 26 } 0.19/0.52 multiply(multiply(inverse(multiply(?, inverse(multiply(X, ?)))), multiply(?, inverse(?))), inverse(multiply(inverse(?), ?))) 0.19/0.52 = { by axiom 1 (single_axiom) } 0.19/0.52 X 0.19/0.52 0.19/0.52 Lemma 33: multiply(inverse(X), multiply(?, inverse(?))) = multiply(inverse(multiply(?, X)), ?). 0.19/0.52 Proof: 0.19/0.52 multiply(inverse(X), multiply(?, inverse(?))) 0.19/0.52 = { by lemma 24 } 0.19/0.52 multiply(inverse(X), multiply(inverse(?), ?)) 0.19/0.52 = { by lemma 26 } 0.19/0.52 inverse(multiply(inverse(multiply(inverse(?), ?)), X)) 0.19/0.52 = { by lemma 22 } 0.19/0.52 inverse(multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, X))) 0.19/0.52 = { by lemma 21 } 0.19/0.52 inverse(multiply(inverse(?), multiply(?, X))) 0.19/0.52 = { by lemma 26 } 0.19/0.52 multiply(inverse(multiply(?, X)), ?) 0.19/0.52 0.19/0.52 Lemma 34: multiply(inverse(multiply(?, X)), ?) = inverse(X). 0.19/0.52 Proof: 0.19/0.52 multiply(inverse(multiply(?, X)), ?) 0.19/0.52 = { by lemma 33 } 0.19/0.52 multiply(inverse(X), multiply(?, inverse(?))) 0.19/0.52 = { by lemma 24 } 0.19/0.52 multiply(inverse(X), multiply(inverse(?), ?)) 0.19/0.52 = { by lemma 26 } 0.19/0.52 inverse(multiply(inverse(multiply(inverse(?), ?)), X)) 0.19/0.52 = { by lemma 15 } 0.19/0.52 inverse(multiply(multiply(inverse(?), ?), X)) 0.19/0.52 = { by lemma 19 } 0.36/0.54 inverse(X) 0.36/0.54 0.36/0.54 Goal 1 (prove_these_axioms_3): multiply(multiply(a3, b3), c3) = multiply(a3, multiply(b3, c3)). 0.36/0.54 Proof: 0.36/0.54 multiply(multiply(a3, b3), c3) 0.36/0.54 = { by lemma 32 } 0.36/0.54 multiply(multiply(a3, multiply(multiply(b3, ?), inverse(?))), c3) 0.36/0.54 = { by lemma 28 } 0.36/0.54 multiply(multiply(a3, inverse(multiply(?, inverse(multiply(b3, ?))))), c3) 0.36/0.54 = { by lemma 31 } 0.36/0.54 multiply(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))), c3) 0.36/0.54 = { by lemma 23 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), c3) 0.36/0.54 = { by lemma 32 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), multiply(multiply(c3, ?), inverse(?))) 0.36/0.54 = { by lemma 28 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), inverse(multiply(?, inverse(multiply(c3, ?))))) 0.36/0.54 = { by lemma 27 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), multiply(inverse(inverse(multiply(c3, ?))), inverse(?))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), inverse(multiply(inverse(inverse(?)), inverse(multiply(c3, ?))))) 0.36/0.54 = { by lemma 22 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), inverse(multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(multiply(c3, ?)))))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), multiply(inverse(multiply(?, inverse(multiply(c3, ?)))), multiply(?, inverse(?)))) 0.36/0.54 = { by lemma 29 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), multiply(inverse(multiply(?, inverse(multiply(c3, inverse(multiply(b3, c3)))))), multiply(?, inverse(inverse(multiply(b3, c3)))))) 0.36/0.54 = { by lemma 2 } 0.36/0.54 multiply(inverse(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3))))))), multiply(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3)))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 inverse(multiply(inverse(multiply(inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3)))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))), inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3)))))))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 inverse(multiply(inverse(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3)))))))), inverse(multiply(a3, inverse(multiply(c3, inverse(multiply(b3, c3)))))))) 0.36/0.54 = { by lemma 30 } 0.36/0.54 inverse(multiply(inverse(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?))), inverse(?))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 multiply(inverse(inverse(?)), inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?))) 0.36/0.54 = { by lemma 27 } 0.36/0.54 inverse(multiply(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?), inverse(?))) 0.36/0.54 = { by lemma 28 } 0.36/0.54 multiply(?, inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?))) 0.36/0.54 = { by lemma 23 } 0.36/0.54 multiply(?, inverse(inverse(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?))))) 0.36/0.54 = { by lemma 34 } 0.36/0.54 multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?)))), ?))) 0.36/0.54 = { by lemma 34 } 0.36/0.54 multiply(?, inverse(multiply(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?))))), ?), ?))) 0.36/0.54 = { by lemma 33 } 0.36/0.54 multiply(?, inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), ?)))), multiply(?, inverse(?))), ?))) 0.36/0.54 = { by lemma 29 } 0.36/0.54 multiply(?, inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), ?))) 0.36/0.54 = { by lemma 2 } 0.36/0.54 multiply(?, inverse(multiply(multiply(inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(?, multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(multiply(inverse(multiply(?, inverse(multiply(?, multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), ?))) 0.36/0.54 = { by axiom 1 (single_axiom) } 0.36/0.54 multiply(?, inverse(multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(?, inverse(multiply(?, multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), ?))) 0.36/0.54 = { by lemma 31 } 0.36/0.54 multiply(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))), inverse(multiply(multiply(inverse(?), multiply(multiply(inverse(multiply(?, inverse(multiply(?, multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3))))))))) 0.36/0.54 = { by axiom 1 (single_axiom) } 0.36/0.54 multiply(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))), inverse(multiply(multiply(inverse(multiply(multiply(inverse(multiply(?, inverse(multiply(?, multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(multiply(inverse(multiply(?, inverse(multiply(?, multiply(a3, inverse(inverse(multiply(b3, c3)))))))), multiply(?, inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))), inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3))))))))) 0.36/0.54 = { by axiom 1 (single_axiom) } 0.36/0.54 multiply(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))), inverse(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))))) 0.36/0.54 = { by lemma 27 } 0.36/0.54 inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3))))))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 multiply(inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3)))))), multiply(a3, inverse(inverse(multiply(b3, c3))))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 multiply(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(a3, inverse(inverse(multiply(b3, c3))))) 0.36/0.54 = { by lemma 9 } 0.36/0.54 multiply(multiply(inverse(?), ?), multiply(a3, inverse(inverse(multiply(b3, c3))))) 0.36/0.54 = { by lemma 15 } 0.36/0.54 multiply(inverse(multiply(inverse(?), ?)), multiply(a3, inverse(inverse(multiply(b3, c3))))) 0.36/0.54 = { by lemma 26 } 0.36/0.54 inverse(multiply(inverse(multiply(a3, inverse(inverse(multiply(b3, c3))))), multiply(inverse(?), ?))) 0.36/0.54 = { by lemma 21 } 0.36/0.54 inverse(inverse(multiply(a3, inverse(inverse(multiply(b3, c3)))))) 0.36/0.54 = { by lemma 23 } 0.36/0.54 multiply(a3, inverse(inverse(multiply(b3, c3)))) 0.36/0.54 = { by lemma 23 } 0.36/0.54 multiply(a3, multiply(b3, c3)) 0.36/0.54 % SZS output end Proof 0.36/0.54 0.36/0.54 RESULT: Unsatisfiable (the axioms are contradictory). 0.36/0.54 EOF