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 0.12/0.34 % Computer : n005.cluster.edu 0.12/0.34 % Model : x86_64 x86_64 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.34 % Memory : 8042.1875MB 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.34 % CPULimit : 180 0.12/0.34 % DateTime : Thu Aug 29 12:57:03 EDT 2019 0.12/0.34 % CPUTime : 1.52/1.70 % SZS status Unsatisfiable 1.52/1.70 1.52/1.70 % SZS output start Proof 1.52/1.70 Take the following subset of the input axioms: 1.52/1.72 fof(additive_inverse, axiom, ![X]: n1=add(X, inverse(X))). 1.52/1.72 fof(multiply_add, axiom, ![X, Y]: Y=multiply(add(X, Y), Y)). 1.52/1.72 fof(multiply_add_property, axiom, ![X, Y, Z]: add(multiply(Y, X), multiply(Z, X))=multiply(X, add(Y, Z))). 1.52/1.72 fof(pixley1, axiom, ![X, Y]: Y=pixley(X, X, Y)). 1.52/1.72 fof(pixley2, axiom, ![X, Y]: X=pixley(X, Y, Y)). 1.52/1.72 fof(pixley3, axiom, ![X, Y]: X=pixley(X, Y, X)). 1.52/1.72 fof(pixley_defn, axiom, ![X, Y, Z]: pixley(X, Y, Z)=add(multiply(X, inverse(Y)), add(multiply(X, Z), multiply(inverse(Y), Z)))). 1.52/1.72 fof(prove_add_multiply_property, negated_conjecture, add(a, multiply(b, c))!=multiply(add(a, b), add(a, c))). 1.52/1.72 1.52/1.72 Now clausify the problem and encode Horn clauses using encoding 3 of 1.52/1.72 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 1.52/1.72 We repeatedly replace C & s=t => u=v by the two clauses: 1.52/1.72 fresh(y, y, x1...xn) = u 1.52/1.72 C => fresh(s, t, x1...xn) = v 1.52/1.72 where fresh is a fresh function symbol and x1..xn are the free 1.52/1.72 variables of u and v. 1.52/1.72 A predicate p(X) is encoded as p(X)=true (this is sound, because the 1.52/1.72 input problem has no model of domain size 1). 1.52/1.72 1.52/1.72 The encoding turns the above axioms into the following unit equations and goals: 1.52/1.72 1.52/1.72 Axiom 1 (pixley_defn): pixley(X, Y, Z) = add(multiply(X, inverse(Y)), add(multiply(X, Z), multiply(inverse(Y), Z))). 1.52/1.72 Axiom 2 (multiply_add): X = multiply(add(Y, X), X). 1.52/1.72 Axiom 3 (pixley2): X = pixley(X, Y, Y). 1.52/1.72 Axiom 4 (pixley1): X = pixley(Y, Y, X). 1.52/1.72 Axiom 5 (multiply_add_property): add(multiply(X, Y), multiply(Z, Y)) = multiply(Y, add(X, Z)). 1.52/1.72 Axiom 6 (pixley3): X = pixley(X, Y, X). 1.56/1.74 Axiom 7 (additive_inverse): n1 = add(X, inverse(X)). 1.56/1.74 1.56/1.74 Lemma 8: add(multiply(X, inverse(Y)), multiply(Z, add(X, inverse(Y)))) = pixley(X, Y, Z). 1.56/1.74 Proof: 1.56/1.74 add(multiply(X, inverse(Y)), multiply(Z, add(X, inverse(Y)))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 add(multiply(X, inverse(Y)), add(multiply(X, Z), multiply(inverse(Y), Z))) 1.56/1.74 = { by axiom 1 (pixley_defn) } 1.56/1.74 pixley(X, Y, Z) 1.56/1.74 1.56/1.74 Lemma 9: multiply(n1, inverse(X)) = inverse(X). 1.56/1.74 Proof: 1.56/1.74 multiply(n1, inverse(X)) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 multiply(add(X, inverse(X)), inverse(X)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 inverse(X) 1.56/1.74 1.56/1.74 Lemma 10: multiply(X, add(Y, add(Z, X))) = add(multiply(Y, X), X). 1.56/1.74 Proof: 1.56/1.74 multiply(X, add(Y, add(Z, X))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 add(multiply(Y, X), multiply(add(Z, X), X)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(multiply(Y, X), X) 1.56/1.74 1.56/1.74 Lemma 11: add(multiply(multiply(X, inverse(Y)), multiply(inverse(Y), Z)), multiply(inverse(Y), Z)) = multiply(multiply(inverse(Y), Z), pixley(X, Y, Z)). 1.56/1.74 Proof: 1.56/1.74 add(multiply(multiply(X, inverse(Y)), multiply(inverse(Y), Z)), multiply(inverse(Y), Z)) 1.56/1.74 = { by lemma 10 } 1.56/1.74 multiply(multiply(inverse(Y), Z), add(multiply(X, inverse(Y)), add(multiply(X, Z), multiply(inverse(Y), Z)))) 1.56/1.74 = { by axiom 1 (pixley_defn) } 1.56/1.74 multiply(multiply(inverse(Y), Z), pixley(X, Y, Z)) 1.56/1.74 1.56/1.74 Lemma 12: add(multiply(Y, inverse(Y)), multiply(X, n1)) = X. 1.56/1.74 Proof: 1.56/1.74 add(multiply(Y, inverse(Y)), multiply(X, n1)) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 add(multiply(Y, inverse(Y)), multiply(X, add(Y, inverse(Y)))) 1.56/1.74 = { by lemma 8 } 1.56/1.74 pixley(Y, Y, X) 1.56/1.74 = { by axiom 4 (pixley1) } 1.56/1.74 X 1.56/1.74 1.56/1.74 Lemma 13: add(multiply(X, multiply(Y, n1)), multiply(Y, n1)) = multiply(multiply(Y, n1), add(X, Y)). 1.56/1.74 Proof: 1.56/1.74 add(multiply(X, multiply(Y, n1)), multiply(Y, n1)) 1.56/1.74 = { by lemma 10 } 1.56/1.74 multiply(multiply(Y, n1), add(X, add(multiply(?, inverse(?)), multiply(Y, n1)))) 1.56/1.74 = { by lemma 12 } 1.56/1.74 multiply(multiply(Y, n1), add(X, Y)) 1.56/1.74 1.56/1.74 Lemma 14: multiply(X, add(add(Y, X), Z)) = add(X, multiply(Z, X)). 1.56/1.74 Proof: 1.56/1.74 multiply(X, add(add(Y, X), Z)) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 add(multiply(add(Y, X), X), multiply(Z, X)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(X, multiply(Z, X)) 1.56/1.74 1.56/1.74 Lemma 15: add(multiply(X, n1), multiply(Y, multiply(X, n1))) = multiply(multiply(X, n1), add(X, Y)). 1.56/1.74 Proof: 1.56/1.74 add(multiply(X, n1), multiply(Y, multiply(X, n1))) 1.56/1.74 = { by lemma 14 } 1.56/1.74 multiply(multiply(X, n1), add(add(multiply(?, inverse(?)), multiply(X, n1)), Y)) 1.56/1.74 = { by lemma 12 } 1.56/1.74 multiply(multiply(X, n1), add(X, Y)) 1.56/1.74 1.56/1.74 Lemma 16: add(inverse(X), multiply(Y, inverse(X))) = multiply(inverse(X), add(n1, Y)). 1.56/1.74 Proof: 1.56/1.74 add(inverse(X), multiply(Y, inverse(X))) 1.56/1.74 = { by lemma 9 } 1.56/1.74 add(multiply(n1, inverse(X)), multiply(Y, inverse(X))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 multiply(inverse(X), add(n1, Y)) 1.56/1.74 1.56/1.74 Lemma 17: multiply(inverse(X), add(n1, n1)) = add(inverse(X), inverse(X)). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), add(n1, n1)) 1.56/1.74 = { by lemma 16 } 1.56/1.74 add(inverse(X), multiply(n1, inverse(X))) 1.56/1.74 = { by lemma 9 } 1.56/1.74 add(inverse(X), inverse(X)) 1.56/1.74 1.56/1.74 Lemma 18: multiply(multiply(Y, add(Z, X)), multiply(X, Y)) = multiply(X, Y). 1.56/1.74 Proof: 1.56/1.74 multiply(multiply(Y, add(Z, X)), multiply(X, Y)) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 multiply(add(multiply(Z, Y), multiply(X, Y)), multiply(X, Y)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 multiply(X, Y) 1.56/1.74 1.56/1.74 Lemma 19: multiply(multiply(Y, n1), multiply(inverse(X), Y)) = multiply(inverse(X), Y). 1.56/1.74 Proof: 1.56/1.74 multiply(multiply(Y, n1), multiply(inverse(X), Y)) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 multiply(multiply(Y, add(X, inverse(X))), multiply(inverse(X), Y)) 1.56/1.74 = { by lemma 18 } 1.56/1.74 multiply(inverse(X), Y) 1.56/1.74 1.56/1.74 Lemma 20: add(inverse(X), multiply(Y, add(n1, inverse(X)))) = pixley(n1, X, Y). 1.56/1.74 Proof: 1.56/1.74 add(inverse(X), multiply(Y, add(n1, inverse(X)))) 1.56/1.74 = { by lemma 9 } 1.56/1.74 add(multiply(n1, inverse(X)), multiply(Y, add(n1, inverse(X)))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 add(multiply(n1, inverse(X)), add(multiply(n1, Y), multiply(inverse(X), Y))) 1.56/1.74 = { by axiom 1 (pixley_defn) } 1.56/1.74 pixley(n1, X, Y) 1.56/1.74 1.56/1.74 Lemma 21: add(inverse(n1), multiply(X, n1)) = X. 1.56/1.74 Proof: 1.56/1.74 add(inverse(n1), multiply(X, n1)) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 add(inverse(n1), multiply(X, add(n1, inverse(n1)))) 1.56/1.74 = { by lemma 20 } 1.56/1.74 pixley(n1, n1, X) 1.56/1.74 = { by axiom 4 (pixley1) } 1.56/1.74 X 1.56/1.74 1.56/1.74 Lemma 22: add(X, n1) = add(?, n1). 1.56/1.74 Proof: 1.56/1.74 add(X, n1) 1.56/1.74 = { by lemma 21 } 1.56/1.74 add(inverse(n1), multiply(add(X, n1), n1)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(inverse(n1), n1) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(inverse(n1), multiply(add(?, n1), n1)) 1.56/1.74 = { by lemma 21 } 1.56/1.74 add(?, n1) 1.56/1.74 1.56/1.74 Lemma 23: multiply(n1, add(inverse(X), inverse(X))) = multiply(inverse(X), add(?, n1)). 1.56/1.74 Proof: 1.56/1.74 multiply(n1, add(inverse(X), inverse(X))) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 multiply(multiply(add(n1, n1), n1), add(inverse(X), inverse(X))) 1.56/1.74 = { by lemma 17 } 1.56/1.74 multiply(multiply(add(n1, n1), n1), multiply(inverse(X), add(n1, n1))) 1.56/1.74 = { by lemma 19 } 1.56/1.74 multiply(inverse(X), add(n1, n1)) 1.56/1.74 = { by lemma 22 } 1.56/1.74 multiply(inverse(X), add(?, n1)) 1.56/1.74 1.56/1.74 Lemma 24: multiply(add(Z, X), add(multiply(Y, X), X)) = add(multiply(Y, X), X). 1.56/1.74 Proof: 1.56/1.74 multiply(add(Z, X), add(multiply(Y, X), X)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 multiply(multiply(add(Y, add(Z, X)), add(Z, X)), add(multiply(Y, X), X)) 1.56/1.74 = { by lemma 10 } 1.56/1.74 multiply(multiply(add(Y, add(Z, X)), add(Z, X)), multiply(X, add(Y, add(Z, X)))) 1.56/1.74 = { by lemma 18 } 1.56/1.74 multiply(X, add(Y, add(Z, X))) 1.56/1.74 = { by lemma 10 } 1.56/1.74 add(multiply(Y, X), X) 1.56/1.74 1.56/1.74 Lemma 25: multiply(add(X, Y), add(Y, Y)) = add(Y, Y). 1.56/1.74 Proof: 1.56/1.74 multiply(add(X, Y), add(Y, Y)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 multiply(add(X, Y), add(multiply(add(?, Y), Y), Y)) 1.56/1.74 = { by lemma 24 } 1.56/1.74 add(multiply(add(?, Y), Y), Y) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(Y, Y) 1.56/1.74 1.56/1.74 Lemma 26: multiply(inverse(X), add(?, n1)) = add(inverse(X), inverse(X)). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), add(?, n1)) 1.56/1.74 = { by lemma 23 } 1.56/1.74 multiply(n1, add(inverse(X), inverse(X))) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 multiply(add(X, inverse(X)), add(inverse(X), inverse(X))) 1.56/1.74 = { by lemma 25 } 1.56/1.74 add(inverse(X), inverse(X)) 1.56/1.74 1.56/1.74 Lemma 27: add(inverse(n1), add(inverse(X), inverse(X))) = multiply(inverse(X), n1). 1.56/1.74 Proof: 1.56/1.74 add(inverse(n1), add(inverse(X), inverse(X))) 1.56/1.74 = { by lemma 26 } 1.56/1.74 add(inverse(n1), multiply(inverse(X), add(?, n1))) 1.56/1.74 = { by lemma 23 } 1.56/1.74 add(inverse(n1), multiply(n1, add(inverse(X), inverse(X)))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 add(inverse(n1), add(multiply(inverse(X), n1), multiply(inverse(X), n1))) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(inverse(n1), add(multiply(add(?, multiply(inverse(X), n1)), multiply(inverse(X), n1)), multiply(inverse(X), n1))) 1.56/1.74 = { by lemma 13 } 1.56/1.74 add(inverse(n1), multiply(multiply(inverse(X), n1), add(add(?, multiply(inverse(X), n1)), inverse(X)))) 1.56/1.74 = { by lemma 14 } 1.56/1.74 add(inverse(n1), add(multiply(inverse(X), n1), multiply(inverse(X), multiply(inverse(X), n1)))) 1.56/1.74 = { by lemma 15 } 1.56/1.74 add(inverse(n1), multiply(multiply(inverse(X), n1), add(inverse(X), inverse(X)))) 1.56/1.74 = { by lemma 13 } 1.56/1.74 add(inverse(n1), add(multiply(inverse(X), multiply(inverse(X), n1)), multiply(inverse(X), n1))) 1.56/1.74 = { by lemma 9 } 1.56/1.74 add(inverse(n1), add(multiply(multiply(n1, inverse(X)), multiply(inverse(X), n1)), multiply(inverse(X), n1))) 1.56/1.74 = { by lemma 11 } 1.56/1.74 add(inverse(n1), multiply(multiply(inverse(X), n1), pixley(n1, X, n1))) 1.56/1.74 = { by axiom 6 (pixley3) } 1.56/1.74 add(inverse(n1), multiply(multiply(inverse(X), n1), n1)) 1.56/1.74 = { by lemma 21 } 1.56/1.74 multiply(inverse(X), n1) 1.56/1.74 1.56/1.74 Lemma 28: add(inverse(n1), n1) = add(?, n1). 1.56/1.74 Proof: 1.56/1.74 add(inverse(n1), n1) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 add(inverse(n1), multiply(add(?, n1), n1)) 1.56/1.74 = { by lemma 21 } 1.56/1.74 add(?, n1) 1.56/1.74 1.56/1.74 Lemma 29: add(multiply(Y, inverse(X)), inverse(X)) = multiply(inverse(X), add(Y, n1)). 1.56/1.74 Proof: 1.56/1.74 add(multiply(Y, inverse(X)), inverse(X)) 1.56/1.74 = { by lemma 9 } 1.56/1.74 add(multiply(Y, inverse(X)), multiply(n1, inverse(X))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 multiply(inverse(X), add(Y, n1)) 1.56/1.74 1.56/1.74 Lemma 30: multiply(inverse(X), add(?, n1)) = multiply(inverse(X), n1). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), add(?, n1)) 1.56/1.74 = { by lemma 28 } 1.56/1.74 multiply(inverse(X), add(inverse(n1), n1)) 1.56/1.74 = { by lemma 29 } 1.56/1.74 add(multiply(inverse(n1), inverse(X)), inverse(X)) 1.56/1.74 = { by lemma 10 } 1.56/1.74 multiply(inverse(X), add(inverse(n1), add(inverse(X), inverse(X)))) 1.56/1.74 = { by lemma 21 } 1.56/1.74 multiply(add(inverse(n1), multiply(inverse(X), n1)), add(inverse(n1), add(inverse(X), inverse(X)))) 1.56/1.74 = { by lemma 27 } 1.56/1.74 multiply(add(inverse(n1), multiply(inverse(X), n1)), multiply(inverse(X), n1)) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 multiply(inverse(X), n1) 1.56/1.74 1.56/1.74 Lemma 31: add(inverse(X), inverse(X)) = multiply(inverse(X), n1). 1.56/1.74 Proof: 1.56/1.74 add(inverse(X), inverse(X)) 1.56/1.74 = { by lemma 26 } 1.56/1.74 multiply(inverse(X), add(?, n1)) 1.56/1.74 = { by lemma 30 } 1.56/1.74 multiply(inverse(X), n1) 1.56/1.74 1.56/1.74 Lemma 32: multiply(inverse(X), n1) = inverse(X). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), n1) 1.56/1.74 = { by lemma 27 } 1.56/1.74 add(inverse(n1), add(inverse(X), inverse(X))) 1.56/1.74 = { by lemma 31 } 1.56/1.74 add(inverse(n1), multiply(inverse(X), n1)) 1.56/1.74 = { by lemma 21 } 1.56/1.74 inverse(X) 1.56/1.74 1.56/1.74 Lemma 33: multiply(multiply(Y, X), multiply(multiply(X, n1), Y)) = multiply(multiply(X, n1), Y). 1.56/1.74 Proof: 1.56/1.74 multiply(multiply(Y, X), multiply(multiply(X, n1), Y)) 1.56/1.74 = { by lemma 12 } 1.56/1.74 multiply(multiply(Y, add(multiply(?, inverse(?)), multiply(X, n1))), multiply(multiply(X, n1), Y)) 1.56/1.74 = { by lemma 18 } 1.56/1.74 multiply(multiply(X, n1), Y) 1.56/1.74 1.56/1.74 Lemma 34: add(inverse(n1), inverse(X)) = inverse(X). 1.56/1.74 Proof: 1.56/1.74 add(inverse(n1), inverse(X)) 1.56/1.74 = { by lemma 32 } 1.56/1.74 add(inverse(n1), multiply(inverse(X), n1)) 1.56/1.74 = { by lemma 21 } 1.56/1.74 inverse(X) 1.56/1.74 1.56/1.74 Lemma 35: add(?, n1) = n1. 1.56/1.74 Proof: 1.56/1.74 add(?, n1) 1.56/1.74 = { by lemma 28 } 1.56/1.74 add(inverse(n1), n1) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 add(inverse(n1), add(inverse(n1), inverse(inverse(n1)))) 1.56/1.74 = { by lemma 34 } 1.56/1.74 add(inverse(n1), inverse(inverse(n1))) 1.56/1.74 = { by axiom 7 (additive_inverse) } 1.56/1.74 n1 1.56/1.74 1.56/1.74 Lemma 36: multiply(inverse(X), pixley(Y, X, n1)) = inverse(X). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), pixley(Y, X, n1)) 1.56/1.74 = { by lemma 32 } 1.56/1.74 multiply(multiply(inverse(X), n1), pixley(Y, X, n1)) 1.56/1.74 = { by lemma 31 } 1.56/1.74 multiply(add(inverse(X), inverse(X)), pixley(Y, X, n1)) 1.56/1.74 = { by lemma 17 } 1.56/1.74 multiply(multiply(inverse(X), add(n1, n1)), pixley(Y, X, n1)) 1.56/1.74 = { by lemma 8 } 1.56/1.74 multiply(multiply(inverse(X), add(n1, n1)), add(multiply(Y, inverse(X)), multiply(n1, add(Y, inverse(X))))) 1.56/1.74 = { by lemma 35 } 1.56/1.74 multiply(multiply(inverse(X), add(n1, n1)), add(multiply(Y, inverse(X)), multiply(add(?, n1), add(Y, inverse(X))))) 1.56/1.74 = { by lemma 22 } 1.56/1.74 multiply(multiply(inverse(X), add(n1, n1)), add(multiply(Y, inverse(X)), multiply(add(n1, n1), add(Y, inverse(X))))) 1.56/1.74 = { by axiom 5 (multiply_add_property) } 1.56/1.74 multiply(multiply(inverse(X), add(n1, n1)), add(multiply(Y, inverse(X)), add(multiply(Y, add(n1, n1)), multiply(inverse(X), add(n1, n1))))) 1.56/1.74 = { by lemma 10 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), multiply(inverse(X), add(n1, n1))), multiply(inverse(X), add(n1, n1))) 1.56/1.74 = { by lemma 22 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), multiply(inverse(X), add(?, n1))), multiply(inverse(X), add(n1, n1))) 1.56/1.74 = { by lemma 30 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), multiply(inverse(X), n1)), multiply(inverse(X), add(n1, n1))) 1.56/1.74 = { by lemma 32 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), inverse(X)), multiply(inverse(X), add(n1, n1))) 1.56/1.74 = { by lemma 22 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), inverse(X)), multiply(inverse(X), add(?, n1))) 1.56/1.74 = { by lemma 30 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), inverse(X)), multiply(inverse(X), n1)) 1.56/1.74 = { by lemma 32 } 1.56/1.74 add(multiply(multiply(Y, inverse(X)), inverse(X)), inverse(X)) 1.56/1.74 = { by lemma 29 } 1.56/1.74 multiply(inverse(X), add(multiply(Y, inverse(X)), n1)) 1.56/1.74 = { by lemma 22 } 1.56/1.74 multiply(inverse(X), add(?, n1)) 1.56/1.74 = { by lemma 30 } 1.56/1.74 multiply(inverse(X), n1) 1.56/1.74 = { by lemma 32 } 1.56/1.74 inverse(X) 1.56/1.74 1.56/1.74 Lemma 37: multiply(inverse(n1), X) = inverse(n1). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(n1), X) 1.56/1.74 = { by axiom 3 (pixley2) } 1.56/1.74 multiply(inverse(n1), pixley(X, n1, n1)) 1.56/1.74 = { by lemma 36 } 1.56/1.74 inverse(n1) 1.56/1.74 1.56/1.74 Lemma 38: multiply(X, inverse(X)) = inverse(n1). 1.56/1.74 Proof: 1.56/1.74 multiply(X, inverse(X)) 1.56/1.74 = { by lemma 34 } 1.56/1.74 multiply(X, add(inverse(n1), inverse(X))) 1.56/1.74 = { by axiom 2 (multiply_add) } 1.56/1.74 multiply(add(multiply(inverse(n1), inverse(X)), multiply(X, add(inverse(n1), inverse(X)))), multiply(X, add(inverse(n1), inverse(X)))) 1.56/1.74 = { by lemma 8 } 1.56/1.74 multiply(pixley(inverse(n1), X, X), multiply(X, add(inverse(n1), inverse(X)))) 1.56/1.74 = { by axiom 3 (pixley2) } 1.56/1.74 multiply(inverse(n1), multiply(X, add(inverse(n1), inverse(X)))) 1.56/1.74 = { by lemma 37 } 1.56/1.74 inverse(n1) 1.56/1.74 1.56/1.74 Lemma 39: multiply(inverse(X), X) = inverse(n1). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), X) 1.56/1.74 = { by lemma 32 } 1.56/1.74 multiply(multiply(inverse(X), n1), X) 1.56/1.74 = { by lemma 33 } 1.56/1.74 multiply(multiply(X, inverse(X)), multiply(multiply(inverse(X), n1), X)) 1.56/1.74 = { by lemma 38 } 1.56/1.74 multiply(inverse(n1), multiply(multiply(inverse(X), n1), X)) 1.56/1.74 = { by lemma 37 } 1.56/1.74 inverse(n1) 1.56/1.74 1.56/1.74 Lemma 40: multiply(inverse(X), add(inverse(X), Y)) = multiply(inverse(X), add(n1, Y)). 1.56/1.74 Proof: 1.56/1.74 multiply(inverse(X), add(inverse(X), Y)) 1.56/1.74 = { by lemma 9 } 1.56/1.74 multiply(multiply(n1, inverse(X)), add(inverse(X), Y)) 1.56/1.74 = { by lemma 32 } 1.56/1.74 multiply(multiply(n1, inverse(X)), add(multiply(inverse(X), n1), Y)) 1.56/1.74 = { by lemma 30 } 1.58/1.74 multiply(multiply(n1, inverse(X)), add(multiply(inverse(X), add(?, n1)), Y)) 1.58/1.74 = { by axiom 5 (multiply_add_property) } 1.58/1.74 multiply(multiply(n1, inverse(X)), add(add(multiply(?, inverse(X)), multiply(n1, inverse(X))), Y)) 1.58/1.74 = { by lemma 14 } 1.58/1.74 add(multiply(n1, inverse(X)), multiply(Y, multiply(n1, inverse(X)))) 1.58/1.74 = { by lemma 9 } 1.58/1.74 add(inverse(X), multiply(Y, multiply(n1, inverse(X)))) 1.58/1.74 = { by lemma 9 } 1.58/1.74 add(inverse(X), multiply(Y, inverse(X))) 1.58/1.74 = { by lemma 16 } 1.58/1.74 multiply(inverse(X), add(n1, Y)) 1.58/1.74 1.58/1.74 Lemma 41: inverse(inverse(X)) = X. 1.58/1.74 Proof: 1.58/1.74 inverse(inverse(X)) 1.58/1.74 = { by axiom 3 (pixley2) } 1.58/1.74 pixley(inverse(inverse(X)), X, X) 1.58/1.74 = { by lemma 8 } 1.58/1.74 add(multiply(inverse(inverse(X)), inverse(X)), multiply(X, add(inverse(inverse(X)), inverse(X)))) 1.58/1.74 = { by lemma 39 } 1.58/1.74 add(inverse(n1), multiply(X, add(inverse(inverse(X)), inverse(X)))) 1.58/1.74 = { by lemma 32 } 1.58/1.74 add(inverse(n1), multiply(X, add(inverse(inverse(X)), multiply(inverse(X), n1)))) 1.58/1.74 = { by axiom 7 (additive_inverse) } 1.58/1.74 add(inverse(n1), multiply(X, add(inverse(inverse(X)), multiply(inverse(X), add(inverse(X), inverse(inverse(X))))))) 1.58/1.74 = { by lemma 40 } 1.58/1.74 add(inverse(n1), multiply(X, add(inverse(inverse(X)), multiply(inverse(X), add(n1, inverse(inverse(X))))))) 1.58/1.74 = { by lemma 20 } 1.58/1.74 add(inverse(n1), multiply(X, pixley(n1, inverse(X), inverse(X)))) 1.58/1.74 = { by axiom 3 (pixley2) } 1.58/1.74 add(inverse(n1), multiply(X, n1)) 1.58/1.74 = { by lemma 21 } 1.58/1.74 X 1.58/1.74 1.58/1.74 Lemma 42: add(X, X) = X. 1.58/1.74 Proof: 1.58/1.74 add(X, X) 1.58/1.74 = { by lemma 41 } 1.58/1.74 add(inverse(inverse(X)), X) 1.58/1.74 = { by lemma 41 } 1.58/1.74 add(inverse(inverse(X)), inverse(inverse(X))) 1.58/1.74 = { by lemma 25 } 1.58/1.74 multiply(add(inverse(X), inverse(inverse(X))), add(inverse(inverse(X)), inverse(inverse(X)))) 1.58/1.74 = { by axiom 7 (additive_inverse) } 1.58/1.74 multiply(n1, add(inverse(inverse(X)), inverse(inverse(X)))) 1.58/1.74 = { by lemma 23 } 1.58/1.74 multiply(inverse(inverse(X)), add(?, n1)) 1.58/1.74 = { by lemma 30 } 1.58/1.74 multiply(inverse(inverse(X)), n1) 1.58/1.74 = { by lemma 32 } 1.58/1.74 inverse(inverse(X)) 1.58/1.74 = { by lemma 41 } 1.58/1.74 X 1.58/1.74 1.58/1.74 Lemma 43: multiply(X, Y) = multiply(Y, X). 1.58/1.74 Proof: 1.58/1.74 multiply(X, Y) 1.58/1.74 = { by lemma 42 } 1.58/1.74 add(multiply(X, Y), multiply(X, Y)) 1.58/1.74 = { by axiom 5 (multiply_add_property) } 1.58/1.74 multiply(Y, add(X, X)) 1.58/1.74 = { by lemma 42 } 1.58/1.74 multiply(Y, X) 1.58/1.74 1.58/1.74 Lemma 44: multiply(X, n1) = X. 1.58/1.74 Proof: 1.58/1.74 multiply(X, n1) 1.58/1.74 = { by lemma 41 } 1.58/1.74 multiply(inverse(inverse(X)), n1) 1.58/1.74 = { by lemma 32 } 1.58/1.74 inverse(inverse(X)) 1.58/1.74 = { by lemma 41 } 1.58/1.74 X 1.58/1.74 1.58/1.74 Lemma 45: multiply(n1, X) = X. 1.58/1.74 Proof: 1.58/1.74 multiply(n1, X) 1.58/1.74 = { by lemma 41 } 1.58/1.74 multiply(n1, inverse(inverse(X))) 1.58/1.74 = { by lemma 9 } 1.58/1.74 inverse(inverse(X)) 1.58/1.74 = { by lemma 41 } 1.58/1.74 X 1.58/1.74 1.58/1.74 Lemma 46: add(multiply(X, Y), X) = X. 1.58/1.74 Proof: 1.58/1.74 add(multiply(X, Y), X) 1.58/1.74 = { by lemma 43 } 1.58/1.74 add(multiply(Y, X), X) 1.58/1.74 = { by lemma 41 } 1.58/1.74 add(multiply(Y, inverse(inverse(X))), X) 1.58/1.74 = { by lemma 41 } 1.58/1.74 add(multiply(Y, inverse(inverse(X))), inverse(inverse(X))) 1.58/1.74 = { by lemma 29 } 1.58/1.74 multiply(inverse(inverse(X)), add(Y, n1)) 1.58/1.74 = { by lemma 41 } 1.58/1.74 multiply(X, add(Y, n1)) 1.58/1.74 = { by lemma 22 } 1.58/1.74 multiply(X, add(?, n1)) 1.58/1.74 = { by lemma 35 } 1.58/1.74 multiply(X, n1) 1.58/1.74 = { by lemma 44 } 1.58/1.74 X 1.58/1.74 1.58/1.74 Lemma 47: add(n1, X) = n1. 1.58/1.74 Proof: 1.58/1.74 add(n1, X) 1.58/1.74 = { by lemma 41 } 1.58/1.74 add(n1, inverse(inverse(X))) 1.58/1.74 = { by lemma 46 } 1.58/1.74 add(multiply(add(n1, inverse(inverse(X))), inverse(inverse(X))), add(n1, inverse(inverse(X)))) 1.58/1.74 = { by axiom 2 (multiply_add) } 1.58/1.74 add(inverse(inverse(X)), add(n1, inverse(inverse(X)))) 1.58/1.74 = { by lemma 45 } 1.58/1.74 add(inverse(inverse(X)), multiply(n1, add(n1, inverse(inverse(X))))) 1.58/1.74 = { by lemma 20 } 1.58/1.74 pixley(n1, inverse(X), n1) 1.58/1.74 = { by axiom 6 (pixley3) } 1.58/1.74 n1 1.58/1.74 1.58/1.74 Lemma 48: add(X, multiply(X, Y)) = X. 1.58/1.74 Proof: 1.58/1.74 add(X, multiply(X, Y)) 1.58/1.74 = { by lemma 45 } 1.58/1.74 add(multiply(n1, X), multiply(X, Y)) 1.58/1.74 = { by lemma 43 } 1.58/1.74 add(multiply(n1, X), multiply(Y, X)) 1.58/1.74 = { by axiom 5 (multiply_add_property) } 1.58/1.74 multiply(X, add(n1, Y)) 1.58/1.74 = { by lemma 47 } 1.58/1.74 multiply(X, n1) 1.58/1.74 = { by lemma 44 } 1.58/1.74 X 1.58/1.74 1.58/1.74 Lemma 49: add(X, multiply(Y, X)) = X. 1.58/1.74 Proof: 1.58/1.74 add(X, multiply(Y, X)) 1.58/1.74 = { by lemma 43 } 1.58/1.74 add(X, multiply(X, Y)) 1.58/1.74 = { by lemma 48 } 1.58/1.75 X 1.58/1.75 1.58/1.75 Lemma 50: multiply(add(Y, X), multiply(Y, inverse(Z))) = multiply(Y, inverse(Z)). 1.58/1.75 Proof: 1.58/1.75 multiply(add(Y, X), multiply(Y, inverse(Z))) 1.58/1.75 = { by lemma 43 } 1.58/1.75 multiply(add(Y, X), multiply(inverse(Z), Y)) 1.58/1.75 = { by lemma 43 } 1.58/1.75 multiply(multiply(inverse(Z), Y), add(Y, X)) 1.58/1.75 = { by lemma 44 } 1.58/1.75 multiply(multiply(inverse(Z), Y), add(multiply(Y, n1), X)) 1.58/1.75 = { by axiom 5 (multiply_add_property) } 1.58/1.75 add(multiply(multiply(Y, n1), multiply(inverse(Z), Y)), multiply(X, multiply(inverse(Z), Y))) 1.58/1.75 = { by lemma 19 } 1.58/1.75 add(multiply(inverse(Z), Y), multiply(X, multiply(inverse(Z), Y))) 1.58/1.75 = { by lemma 49 } 1.58/1.75 multiply(inverse(Z), Y) 1.58/1.75 = { by lemma 43 } 1.58/1.75 multiply(Y, inverse(Z)) 1.58/1.75 1.58/1.75 Lemma 51: multiply(multiply(X, inverse(Y)), add(add(X, Z), W)) = multiply(X, inverse(Y)). 1.58/1.75 Proof: 1.58/1.75 multiply(multiply(X, inverse(Y)), add(add(X, Z), W)) 1.58/1.75 = { by axiom 5 (multiply_add_property) } 1.58/1.75 add(multiply(add(X, Z), multiply(X, inverse(Y))), multiply(W, multiply(X, inverse(Y)))) 1.58/1.75 = { by lemma 50 } 1.58/1.75 add(multiply(X, inverse(Y)), multiply(W, multiply(X, inverse(Y)))) 1.58/1.75 = { by lemma 49 } 1.58/1.75 multiply(X, inverse(Y)) 1.58/1.75 1.58/1.75 Lemma 52: add(multiply(X, inverse(Z)), add(X, Y)) = add(X, Y). 1.58/1.75 Proof: 1.58/1.75 add(multiply(X, inverse(Z)), add(X, Y)) 1.58/1.75 = { by lemma 51 } 1.58/1.75 add(multiply(multiply(X, inverse(Z)), add(add(X, Y), add(X, Y))), add(X, Y)) 1.58/1.75 = { by lemma 42 } 1.58/1.75 add(multiply(multiply(X, inverse(Z)), add(add(X, Y), add(X, Y))), add(add(X, Y), add(X, Y))) 1.58/1.75 = { by lemma 25 } 1.58/1.75 add(multiply(multiply(X, inverse(Z)), add(add(X, Y), add(X, Y))), multiply(add(?, add(X, Y)), add(add(X, Y), add(X, Y)))) 1.58/1.75 = { by axiom 5 (multiply_add_property) } 1.58/1.75 multiply(add(add(X, Y), add(X, Y)), add(multiply(X, inverse(Z)), add(?, add(X, Y)))) 1.58/1.75 = { by lemma 42 } 1.58/1.75 multiply(add(X, Y), add(multiply(X, inverse(Z)), add(?, add(X, Y)))) 1.58/1.75 = { by lemma 10 } 1.58/1.75 add(multiply(multiply(X, inverse(Z)), add(X, Y)), add(X, Y)) 1.58/1.75 = { by lemma 43 } 1.58/1.75 add(multiply(add(X, Y), multiply(X, inverse(Z))), add(X, Y)) 1.58/1.75 = { by lemma 46 } 1.58/1.75 add(X, Y) 1.58/1.75 1.58/1.75 Lemma 53: add(multiply(X, Z), add(X, Y)) = add(X, Y). 1.58/1.75 Proof: 1.58/1.75 add(multiply(X, Z), add(X, Y)) 1.58/1.75 = { by lemma 41 } 1.58/1.75 add(multiply(X, inverse(inverse(Z))), add(X, Y)) 1.58/1.75 = { by lemma 52 } 1.58/1.75 add(X, Y) 1.58/1.75 1.58/1.75 Lemma 54: add(add(X, Y), X) = add(X, Y). 1.58/1.75 Proof: 1.58/1.75 add(add(X, Y), X) 1.58/1.75 = { by lemma 48 } 1.58/1.75 add(add(X, Y), add(X, multiply(X, Y))) 1.58/1.75 = { by lemma 41 } 1.58/1.75 add(add(X, Y), add(inverse(inverse(X)), multiply(X, Y))) 1.58/1.75 = { by lemma 43 } 1.58/1.75 add(add(X, Y), add(inverse(inverse(X)), multiply(Y, X))) 1.58/1.75 = { by lemma 41 } 1.58/1.75 add(add(X, Y), add(inverse(inverse(X)), multiply(Y, inverse(inverse(X))))) 1.58/1.75 = { by lemma 16 } 1.58/1.75 add(add(X, Y), multiply(inverse(inverse(X)), add(n1, Y))) 1.58/1.75 = { by lemma 40 } 1.58/1.75 add(add(X, Y), multiply(inverse(inverse(X)), add(inverse(inverse(X)), Y))) 1.58/1.75 = { by lemma 41 } 1.58/1.75 add(add(X, Y), multiply(X, add(inverse(inverse(X)), Y))) 1.58/1.75 = { by lemma 41 } 1.58/1.75 add(add(X, Y), multiply(X, add(X, Y))) 1.58/1.75 = { by lemma 49 } 1.58/1.75 add(X, Y) 1.58/1.75 1.58/1.75 Lemma 55: multiply(X, add(Y, inverse(X))) = multiply(X, Y). 1.58/1.75 Proof: 1.58/1.75 multiply(X, add(Y, inverse(X))) 1.58/1.75 = { by axiom 5 (multiply_add_property) } 1.58/1.75 add(multiply(Y, X), multiply(inverse(X), X)) 1.58/1.75 = { by lemma 39 } 1.58/1.75 add(multiply(Y, X), inverse(n1)) 1.58/1.75 = { by lemma 37 } 1.58/1.75 add(multiply(Y, X), multiply(inverse(n1), multiply(multiply(inverse(add(?, multiply(Y, X))), n1), multiply(Y, X)))) 1.58/1.75 = { by lemma 37 } 1.58/1.75 add(multiply(Y, X), multiply(multiply(inverse(n1), multiply(multiply(Y, X), inverse(add(?, multiply(Y, X))))), multiply(multiply(inverse(add(?, multiply(Y, X))), n1), multiply(Y, X)))) 1.58/1.75 = { by lemma 39 } 1.58/1.75 add(multiply(Y, X), multiply(multiply(multiply(inverse(add(?, multiply(Y, X))), add(?, multiply(Y, X))), multiply(multiply(Y, X), inverse(add(?, multiply(Y, X))))), multiply(multiply(inverse(add(?, multiply(Y, X))), n1), multiply(Y, X)))) 1.58/1.75 = { by lemma 18 } 1.58/1.75 add(multiply(Y, X), multiply(multiply(multiply(Y, X), inverse(add(?, multiply(Y, X)))), multiply(multiply(inverse(add(?, multiply(Y, X))), n1), multiply(Y, X)))) 1.58/1.75 = { by lemma 33 } 1.58/1.75 add(multiply(Y, X), multiply(multiply(inverse(add(?, multiply(Y, X))), n1), multiply(Y, X))) 1.58/1.75 = { by lemma 32 } 1.58/1.75 add(multiply(Y, X), multiply(inverse(add(?, multiply(Y, X))), multiply(Y, X))) 1.58/1.75 = { by lemma 14 } 1.58/1.75 multiply(multiply(Y, X), add(add(?, multiply(Y, X)), inverse(add(?, multiply(Y, X))))) 1.58/1.75 = { by axiom 7 (additive_inverse) } 1.58/1.75 multiply(multiply(Y, X), n1) 1.58/1.75 = { by lemma 44 } 1.58/1.75 multiply(Y, X) 1.58/1.75 = { by lemma 43 } 1.58/1.75 multiply(X, Y) 1.58/1.75 1.58/1.75 Lemma 56: add(inverse(Y), multiply(Z, add(add(X, inverse(Y)), inverse(Y)))) = pixley(add(X, inverse(Y)), Y, Z). 1.58/1.75 Proof: 1.58/1.75 add(inverse(Y), multiply(Z, add(add(X, inverse(Y)), inverse(Y)))) 1.58/1.75 = { by axiom 2 (multiply_add) } 1.58/1.75 add(multiply(add(X, inverse(Y)), inverse(Y)), multiply(Z, add(add(X, inverse(Y)), inverse(Y)))) 1.58/1.75 = { by lemma 8 } 1.58/1.75 pixley(add(X, inverse(Y)), Y, Z) 1.58/1.75 1.58/1.75 Lemma 57: add(inverse(Y), multiply(X, Y)) = add(X, inverse(Y)). 1.58/1.75 Proof: 1.58/1.75 add(inverse(Y), multiply(X, Y)) 1.58/1.75 = { by lemma 43 } 1.58/1.75 add(inverse(Y), multiply(Y, X)) 1.58/1.75 = { by lemma 55 } 1.58/1.75 add(inverse(Y), multiply(Y, add(X, inverse(Y)))) 1.58/1.75 = { by lemma 55 } 1.58/1.75 add(inverse(Y), multiply(Y, add(add(X, inverse(Y)), inverse(Y)))) 1.58/1.75 = { by lemma 56 } 1.58/1.75 pixley(add(X, inverse(Y)), Y, Y) 1.58/1.75 = { by axiom 3 (pixley2) } 1.58/1.75 add(X, inverse(Y)) 1.58/1.75 1.58/1.75 Lemma 58: multiply(multiply(multiply(inverse(X), X), Y), multiply(inverse(X), X)) = multiply(inverse(X), X). 1.58/1.75 Proof: 1.58/1.75 multiply(multiply(multiply(inverse(X), X), Y), multiply(inverse(X), X)) 1.58/1.75 = { by axiom 3 (pixley2) } 1.58/1.75 multiply(multiply(multiply(inverse(X), X), pixley(Y, X, X)), multiply(inverse(X), X)) 1.58/1.75 = { by lemma 11 } 1.58/1.75 multiply(add(multiply(multiply(Y, inverse(X)), multiply(inverse(X), X)), multiply(inverse(X), X)), multiply(inverse(X), X)) 1.58/1.75 = { by axiom 2 (multiply_add) } 1.58/1.76 multiply(inverse(X), X) 1.58/1.76 1.58/1.76 Lemma 59: add(inverse(n1), X) = X. 1.58/1.76 Proof: 1.58/1.76 add(inverse(n1), X) 1.58/1.76 = { by lemma 38 } 1.58/1.76 add(multiply(X, inverse(X)), X) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(X)), X) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(inverse(inverse(X)))), X) 1.58/1.76 = { by lemma 32 } 1.58/1.76 add(multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(X))), n1)), X) 1.58/1.76 = { by lemma 31 } 1.58/1.76 add(multiply(inverse(inverse(X)), add(inverse(inverse(inverse(X))), inverse(inverse(inverse(X))))), X) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(add(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), X) 1.58/1.76 = { by lemma 25 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 58 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 33 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by axiom 3 (pixley2) } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), pixley(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), inverse(inverse(X)), inverse(inverse(X)))), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 11 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(add(multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), inverse(inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 58 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(add(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(inverse(inverse(X)), add(inverse(inverse(inverse(X))), inverse(inverse(inverse(X))))), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 31 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(X))), n1)), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 32 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(inverse(inverse(X)), inverse(inverse(inverse(X)))), multiply(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 58 } 1.58/1.76 add(multiply(add(?, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), add(multiply(multiply(inverse(inverse(X)), inverse(inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))), X) 1.58/1.76 = { by lemma 24 } 1.58/1.76 add(add(multiply(multiply(inverse(inverse(X)), inverse(inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), X) 1.58/1.76 = { by lemma 11 } 1.58/1.76 add(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), pixley(inverse(inverse(X)), inverse(inverse(X)), inverse(inverse(X)))), X) 1.58/1.76 = { by axiom 6 (pixley3) } 1.58/1.76 add(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), inverse(inverse(X))), X) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), inverse(inverse(X))), inverse(inverse(X))) 1.58/1.76 = { by lemma 29 } 1.58/1.76 multiply(inverse(inverse(X)), add(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), n1)) 1.58/1.76 = { by lemma 22 } 1.58/1.76 multiply(inverse(inverse(X)), add(?, n1)) 1.58/1.76 = { by lemma 30 } 1.58/1.76 multiply(inverse(inverse(X)), n1) 1.58/1.76 = { by lemma 32 } 1.58/1.76 inverse(inverse(X)) 1.58/1.76 = { by lemma 41 } 1.58/1.76 X 1.58/1.76 1.58/1.76 Lemma 60: multiply(X, add(inverse(n1), Y)) = add(inverse(n1), multiply(Y, X)). 1.58/1.76 Proof: 1.58/1.76 multiply(X, add(inverse(n1), Y)) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(multiply(inverse(n1), X), multiply(Y, X)) 1.58/1.76 = { by lemma 37 } 1.58/1.76 add(inverse(n1), multiply(Y, X)) 1.58/1.76 1.58/1.76 Lemma 61: pixley(inverse(n1), X, Y) = multiply(Y, inverse(X)). 1.58/1.76 Proof: 1.58/1.76 pixley(inverse(n1), X, Y) 1.58/1.76 = { by lemma 8 } 1.58/1.76 add(multiply(inverse(n1), inverse(X)), multiply(Y, add(inverse(n1), inverse(X)))) 1.58/1.76 = { by lemma 59 } 1.58/1.76 add(multiply(inverse(n1), inverse(X)), multiply(Y, inverse(X))) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 multiply(inverse(X), add(inverse(n1), Y)) 1.58/1.76 = { by lemma 60 } 1.58/1.76 add(inverse(n1), multiply(Y, inverse(X))) 1.58/1.76 = { by lemma 59 } 1.58/1.76 multiply(Y, inverse(X)) 1.58/1.76 1.58/1.76 Lemma 62: add(inverse(n1), add(inverse(n1), multiply(inverse(X), Y))) = pixley(inverse(n1), X, Y). 1.58/1.76 Proof: 1.58/1.76 add(inverse(n1), add(inverse(n1), multiply(inverse(X), Y))) 1.58/1.76 = { by lemma 37 } 1.58/1.76 add(multiply(inverse(n1), inverse(X)), add(inverse(n1), multiply(inverse(X), Y))) 1.58/1.76 = { by lemma 60 } 1.58/1.76 add(multiply(inverse(n1), inverse(X)), multiply(Y, add(inverse(n1), inverse(X)))) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(multiply(inverse(n1), inverse(X)), add(multiply(inverse(n1), Y), multiply(inverse(X), Y))) 1.58/1.76 = { by axiom 1 (pixley_defn) } 1.58/1.76 pixley(inverse(n1), X, Y) 1.58/1.76 1.58/1.76 Lemma 63: add(X, Y) = add(Y, X). 1.58/1.76 Proof: 1.58/1.76 add(X, Y) 1.58/1.76 = { by lemma 54 } 1.58/1.76 add(add(X, Y), X) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(add(X, Y), inverse(inverse(X))) 1.58/1.76 = { by lemma 57 } 1.58/1.76 add(inverse(inverse(X)), multiply(add(X, Y), inverse(X))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(inverse(inverse(X)), multiply(add(inverse(inverse(X)), Y), inverse(X))) 1.58/1.76 = { by lemma 61 } 1.58/1.76 add(inverse(inverse(X)), pixley(inverse(n1), X, add(inverse(inverse(X)), Y))) 1.58/1.76 = { by lemma 62 } 1.58/1.76 add(inverse(inverse(X)), add(inverse(n1), add(inverse(n1), multiply(inverse(X), add(inverse(inverse(X)), Y))))) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(inverse(inverse(X)), add(inverse(n1), add(inverse(n1), add(multiply(inverse(inverse(X)), inverse(X)), multiply(Y, inverse(X)))))) 1.58/1.76 = { by lemma 39 } 1.58/1.76 add(inverse(inverse(X)), add(inverse(n1), add(inverse(n1), add(inverse(n1), multiply(Y, inverse(X)))))) 1.58/1.76 = { by lemma 59 } 1.58/1.76 add(inverse(inverse(X)), add(inverse(n1), add(inverse(n1), multiply(Y, inverse(X))))) 1.58/1.76 = { by lemma 43 } 1.58/1.76 add(inverse(inverse(X)), add(inverse(n1), add(inverse(n1), multiply(inverse(X), Y)))) 1.58/1.76 = { by lemma 62 } 1.58/1.76 add(inverse(inverse(X)), pixley(inverse(n1), X, Y)) 1.58/1.76 = { by lemma 61 } 1.58/1.76 add(inverse(inverse(X)), multiply(Y, inverse(X))) 1.58/1.76 = { by lemma 57 } 1.58/1.76 add(Y, inverse(inverse(X))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(Y, X) 1.58/1.76 1.58/1.76 Lemma 64: pixley(X, Y, n1) = add(X, inverse(Y)). 1.58/1.76 Proof: 1.58/1.76 pixley(X, Y, n1) 1.58/1.76 = { by lemma 41 } 1.58/1.76 pixley(inverse(inverse(X)), Y, n1) 1.58/1.76 = { by lemma 35 } 1.58/1.76 pixley(inverse(inverse(X)), Y, add(?, n1)) 1.58/1.76 = { by axiom 1 (pixley_defn) } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(Y)), add(multiply(inverse(inverse(X)), add(?, n1)), multiply(inverse(Y), add(?, n1)))) 1.58/1.76 = { by lemma 30 } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(Y)), add(multiply(inverse(inverse(X)), n1), multiply(inverse(Y), add(?, n1)))) 1.58/1.76 = { by lemma 32 } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(Y)), add(inverse(inverse(X)), multiply(inverse(Y), add(?, n1)))) 1.58/1.76 = { by lemma 30 } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(Y)), add(inverse(inverse(X)), multiply(inverse(Y), n1))) 1.58/1.76 = { by lemma 32 } 1.58/1.76 add(multiply(inverse(inverse(X)), inverse(Y)), add(inverse(inverse(X)), inverse(Y))) 1.58/1.76 = { by lemma 52 } 1.58/1.76 add(inverse(inverse(X)), inverse(Y)) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(X, inverse(Y)) 1.58/1.76 1.58/1.76 Lemma 65: multiply(add(X, inverse(Y)), add(Z, inverse(Y))) = add(inverse(Y), multiply(X, add(Z, inverse(Y)))). 1.58/1.76 Proof: 1.58/1.76 multiply(add(X, inverse(Y)), add(Z, inverse(Y))) 1.58/1.76 = { by lemma 43 } 1.58/1.76 multiply(add(Z, inverse(Y)), add(X, inverse(Y))) 1.58/1.76 = { by lemma 64 } 1.58/1.76 multiply(pixley(Z, Y, n1), add(X, inverse(Y))) 1.58/1.76 = { by lemma 63 } 1.58/1.76 multiply(pixley(Z, Y, n1), add(inverse(Y), X)) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(multiply(inverse(Y), pixley(Z, Y, n1)), multiply(X, pixley(Z, Y, n1))) 1.58/1.76 = { by lemma 36 } 1.58/1.76 add(inverse(Y), multiply(X, pixley(Z, Y, n1))) 1.58/1.76 = { by lemma 64 } 1.58/1.76 add(inverse(Y), multiply(X, add(Z, inverse(Y)))) 1.58/1.76 1.58/1.76 Lemma 66: multiply(add(X, Y), add(X, Z)) = add(X, multiply(Y, add(X, Z))). 1.58/1.76 Proof: 1.58/1.76 multiply(add(X, Y), add(X, Z)) 1.58/1.76 = { by lemma 63 } 1.58/1.76 multiply(add(Y, X), add(X, Z)) 1.58/1.76 = { by lemma 41 } 1.58/1.76 multiply(add(Y, inverse(inverse(X))), add(X, Z)) 1.58/1.76 = { by lemma 63 } 1.58/1.76 multiply(add(Y, inverse(inverse(X))), add(Z, X)) 1.58/1.76 = { by lemma 41 } 1.58/1.76 multiply(add(Y, inverse(inverse(X))), add(Z, inverse(inverse(X)))) 1.58/1.76 = { by lemma 65 } 1.58/1.76 add(inverse(inverse(X)), multiply(Y, add(Z, inverse(inverse(X))))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(X, multiply(Y, add(Z, inverse(inverse(X))))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(X, multiply(Y, add(Z, X))) 1.58/1.76 = { by lemma 63 } 1.58/1.76 add(X, multiply(Y, add(X, Z))) 1.58/1.76 1.58/1.76 Lemma 67: add(inverse(Y), multiply(Z, add(X, inverse(Y)))) = pixley(X, Y, add(Z, inverse(Y))). 1.58/1.76 Proof: 1.58/1.76 add(inverse(Y), multiply(Z, add(X, inverse(Y)))) 1.58/1.76 = { by lemma 53 } 1.58/1.76 add(multiply(inverse(Y), X), add(inverse(Y), multiply(Z, add(X, inverse(Y))))) 1.58/1.76 = { by lemma 43 } 1.58/1.76 add(multiply(X, inverse(Y)), add(inverse(Y), multiply(Z, add(X, inverse(Y))))) 1.58/1.76 = { by lemma 65 } 1.58/1.76 add(multiply(X, inverse(Y)), multiply(add(Z, inverse(Y)), add(X, inverse(Y)))) 1.58/1.76 = { by lemma 8 } 1.58/1.76 pixley(X, Y, add(Z, inverse(Y))) 1.58/1.76 1.58/1.76 Lemma 68: pixley(X, inverse(Y), add(Y, Z)) = add(Y, multiply(Z, add(X, Y))). 1.58/1.76 Proof: 1.58/1.76 pixley(X, inverse(Y), add(Y, Z)) 1.58/1.76 = { by lemma 63 } 1.58/1.76 pixley(X, inverse(Y), add(Z, Y)) 1.58/1.76 = { by lemma 41 } 1.58/1.76 pixley(X, inverse(Y), add(Z, inverse(inverse(Y)))) 1.58/1.76 = { by lemma 67 } 1.58/1.76 add(inverse(inverse(Y)), multiply(Z, add(X, inverse(inverse(Y))))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(Y, multiply(Z, add(X, inverse(inverse(Y))))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 add(Y, multiply(Z, add(X, Y))) 1.58/1.76 1.58/1.76 Lemma 69: pixley(Z, inverse(Y), add(X, Y)) = add(Y, multiply(Z, add(X, Y))). 1.58/1.76 Proof: 1.58/1.76 pixley(Z, inverse(Y), add(X, Y)) 1.58/1.76 = { by lemma 41 } 1.58/1.76 pixley(Z, inverse(Y), add(X, inverse(inverse(Y)))) 1.58/1.76 = { by lemma 67 } 1.58/1.76 add(inverse(inverse(Y)), multiply(X, add(Z, inverse(inverse(Y))))) 1.58/1.76 = { by lemma 65 } 1.58/1.76 multiply(add(X, inverse(inverse(Y))), add(Z, inverse(inverse(Y)))) 1.58/1.76 = { by lemma 64 } 1.58/1.76 multiply(pixley(X, inverse(Y), n1), add(Z, inverse(inverse(Y)))) 1.58/1.76 = { by axiom 5 (multiply_add_property) } 1.58/1.76 add(multiply(Z, pixley(X, inverse(Y), n1)), multiply(inverse(inverse(Y)), pixley(X, inverse(Y), n1))) 1.58/1.76 = { by lemma 64 } 1.58/1.76 add(multiply(Z, add(X, inverse(inverse(Y)))), multiply(inverse(inverse(Y)), pixley(X, inverse(Y), n1))) 1.58/1.76 = { by lemma 36 } 1.58/1.76 add(multiply(Z, add(X, inverse(inverse(Y)))), inverse(inverse(Y))) 1.58/1.76 = { by lemma 63 } 1.58/1.76 add(inverse(inverse(Y)), multiply(Z, add(X, inverse(inverse(Y))))) 1.58/1.76 = { by lemma 67 } 1.58/1.76 pixley(X, inverse(Y), add(Z, inverse(inverse(Y)))) 1.58/1.76 = { by lemma 41 } 1.58/1.76 pixley(X, inverse(Y), add(Z, Y)) 1.58/1.76 = { by lemma 63 } 1.58/1.76 pixley(X, inverse(Y), add(Y, Z)) 1.58/1.76 = { by lemma 68 } 1.58/1.76 add(Y, multiply(Z, add(X, Y))) 1.58/1.76 1.58/1.76 Lemma 70: multiply(multiply(inverse(X), add(Y, n1)), inverse(X)) = inverse(X). 1.58/1.76 Proof: 1.58/1.76 multiply(multiply(inverse(X), add(Y, n1)), inverse(X)) 1.58/1.76 = { by lemma 9 } 1.58/1.76 multiply(multiply(inverse(X), add(Y, n1)), multiply(n1, inverse(X))) 1.58/1.76 = { by lemma 18 } 1.58/1.76 multiply(n1, inverse(X)) 1.58/1.76 = { by lemma 9 } 1.58/1.76 inverse(X) 1.58/1.76 1.58/1.77 Lemma 71: multiply(inverse(X), add(add(inverse(X), Z), W)) = inverse(X). 1.58/1.77 Proof: 1.58/1.77 multiply(inverse(X), add(add(inverse(X), Z), W)) 1.58/1.77 = { by lemma 70 } 1.58/1.77 multiply(multiply(multiply(inverse(X), add(?, n1)), inverse(X)), add(add(inverse(X), Z), W)) 1.58/1.77 = { by lemma 32 } 1.58/1.77 multiply(multiply(multiply(inverse(X), add(?, n1)), inverse(X)), add(add(multiply(inverse(X), n1), Z), W)) 1.58/1.77 = { by lemma 30 } 1.58/1.77 multiply(multiply(multiply(inverse(X), add(?, n1)), inverse(X)), add(add(multiply(inverse(X), add(?, n1)), Z), W)) 1.58/1.77 = { by lemma 51 } 1.58/1.77 multiply(multiply(inverse(X), add(?, n1)), inverse(X)) 1.58/1.77 = { by lemma 70 } 1.58/1.77 inverse(X) 1.58/1.77 1.58/1.77 Lemma 72: add(inverse(X), multiply(inverse(Z), add(Y, inverse(X)))) = pixley(add(Y, inverse(X)), Z, inverse(X)). 1.58/1.77 Proof: 1.58/1.77 add(inverse(X), multiply(inverse(Z), add(Y, inverse(X)))) 1.58/1.77 = { by lemma 63 } 1.58/1.77 add(inverse(X), multiply(inverse(Z), add(inverse(X), Y))) 1.58/1.77 = { by lemma 43 } 1.58/1.77 add(inverse(X), multiply(add(inverse(X), Y), inverse(Z))) 1.58/1.77 = { by lemma 63 } 1.58/1.77 add(multiply(add(inverse(X), Y), inverse(Z)), inverse(X)) 1.58/1.77 = { by lemma 71 } 1.58/1.77 add(multiply(add(inverse(X), Y), inverse(Z)), multiply(inverse(X), add(add(inverse(X), Y), inverse(Z)))) 1.58/1.77 = { by lemma 8 } 1.58/1.77 pixley(add(inverse(X), Y), Z, inverse(X)) 1.58/1.77 = { by lemma 63 } 1.58/1.77 pixley(add(Y, inverse(X)), Z, inverse(X)) 1.58/1.77 1.58/1.77 Lemma 73: add(add(X, Y), Y) = add(X, Y). 1.58/1.77 Proof: 1.58/1.77 add(add(X, Y), Y) 1.58/1.77 = { by axiom 2 (multiply_add) } 1.58/1.77 add(add(X, Y), multiply(add(X, Y), Y)) 1.58/1.77 = { by lemma 48 } 1.58/1.77 add(X, Y) 1.58/1.77 1.58/1.77 Lemma 74: pixley(add(X, inverse(Y)), Y, Z) = pixley(X, Y, add(Z, inverse(Y))). 1.58/1.77 Proof: 1.58/1.77 pixley(add(X, inverse(Y)), Y, Z) 1.58/1.77 = { by lemma 41 } 1.58/1.77 pixley(add(X, inverse(Y)), Y, inverse(inverse(Z))) 1.58/1.77 = { by lemma 56 } 1.58/1.77 add(inverse(Y), multiply(inverse(inverse(Z)), add(add(X, inverse(Y)), inverse(Y)))) 1.58/1.77 = { by lemma 72 } 1.58/1.77 pixley(add(add(X, inverse(Y)), inverse(Y)), inverse(Z), inverse(Y)) 1.58/1.77 = { by lemma 73 } 1.58/1.77 pixley(add(X, inverse(Y)), inverse(Z), inverse(Y)) 1.58/1.77 = { by lemma 72 } 1.58/1.77 add(inverse(Y), multiply(inverse(inverse(Z)), add(X, inverse(Y)))) 1.58/1.77 = { by lemma 67 } 1.58/1.77 pixley(X, Y, add(inverse(inverse(Z)), inverse(Y))) 1.58/1.77 = { by lemma 63 } 1.58/1.77 pixley(X, Y, add(inverse(Y), inverse(inverse(Z)))) 1.58/1.77 = { by lemma 41 } 1.58/1.77 pixley(X, Y, add(inverse(Y), Z)) 1.58/1.77 = { by lemma 63 } 1.58/1.77 pixley(X, Y, add(Z, inverse(Y))) 1.58/1.77 1.58/1.77 Lemma 75: add(X, multiply(Y, add(X, inverse(Z)))) = pixley(add(X, Y), Z, X). 1.58/1.77 Proof: 1.58/1.77 add(X, multiply(Y, add(X, inverse(Z)))) 1.58/1.77 = { by lemma 63 } 1.58/1.77 add(X, multiply(Y, add(inverse(Z), X))) 1.58/1.77 = { by lemma 73 } 1.58/1.77 add(X, multiply(Y, add(add(inverse(Z), X), X))) 1.58/1.77 = { by lemma 41 } 1.58/1.77 add(X, multiply(Y, add(add(inverse(Z), inverse(inverse(X))), X))) 1.58/1.77 = { by lemma 68 } 1.58/1.77 pixley(add(inverse(Z), inverse(inverse(X))), inverse(X), add(X, Y)) 1.58/1.77 = { by lemma 74 } 1.58/1.77 pixley(inverse(Z), inverse(X), add(add(X, Y), inverse(inverse(X)))) 1.58/1.77 = { by lemma 41 } 1.58/1.77 pixley(inverse(Z), inverse(X), add(add(X, Y), X)) 1.58/1.77 = { by lemma 69 } 1.58/1.77 add(X, multiply(inverse(Z), add(add(X, Y), X))) 1.58/1.77 = { by lemma 54 } 1.58/1.77 add(X, multiply(inverse(Z), add(X, Y))) 1.58/1.77 = { by lemma 43 } 1.58/1.77 add(X, multiply(add(X, Y), inverse(Z))) 1.58/1.77 = { by lemma 63 } 1.58/1.77 add(multiply(add(X, Y), inverse(Z)), X) 1.58/1.77 = { by lemma 41 } 1.58/1.77 add(multiply(add(X, Y), inverse(Z)), inverse(inverse(X))) 1.58/1.77 = { by lemma 71 } 1.58/1.77 add(multiply(add(X, Y), inverse(Z)), multiply(inverse(inverse(X)), add(add(inverse(inverse(X)), Y), inverse(Z)))) 1.58/1.77 = { by lemma 41 } 1.58/1.77 add(multiply(add(X, Y), inverse(Z)), multiply(X, add(add(inverse(inverse(X)), Y), inverse(Z)))) 1.58/1.77 = { by lemma 41 } 1.58/1.77 add(multiply(add(X, Y), inverse(Z)), multiply(X, add(add(X, Y), inverse(Z)))) 1.58/1.77 = { by lemma 8 } 1.58/1.77 pixley(add(X, Y), Z, X) 1.58/1.77 1.58/1.77 Lemma 76: multiply(multiply(X, Y), multiply(X, multiply(Y, Z))) = multiply(X, multiply(Y, Z)). 1.58/1.77 Proof: 1.58/1.77 multiply(multiply(X, Y), multiply(X, multiply(Y, Z))) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(X, multiply(Y, Z)), multiply(X, Y)) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(X, multiply(Z, Y)), multiply(X, Y)) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(multiply(Z, Y), X), multiply(X, Y)) 1.58/1.77 = { by lemma 49 } 1.58/1.77 multiply(multiply(multiply(Z, Y), X), multiply(X, add(Y, multiply(Z, Y)))) 1.58/1.77 = { by lemma 14 } 1.58/1.77 multiply(multiply(multiply(Z, Y), X), multiply(X, multiply(Y, add(add(?, Y), Z)))) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(X, multiply(Y, add(add(?, Y), Z))), multiply(multiply(Z, Y), X)) 1.58/1.77 = { by axiom 5 (multiply_add_property) } 1.58/1.77 multiply(multiply(X, add(multiply(add(?, Y), Y), multiply(Z, Y))), multiply(multiply(Z, Y), X)) 1.58/1.77 = { by lemma 18 } 1.58/1.77 multiply(multiply(Z, Y), X) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(X, multiply(Z, Y)) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(X, multiply(Y, Z)) 1.58/1.77 1.58/1.77 Lemma 77: multiply(X, multiply(Z, add(Y, X))) = multiply(X, Z). 1.58/1.77 Proof: 1.58/1.77 multiply(X, multiply(Z, add(Y, X))) 1.58/1.77 = { by lemma 76 } 1.58/1.77 multiply(multiply(X, Z), multiply(X, multiply(Z, add(Y, X)))) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(X, Z), multiply(multiply(Z, add(Y, X)), X)) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(multiply(Z, add(Y, X)), X), multiply(X, Z)) 1.58/1.77 = { by lemma 18 } 1.58/1.77 multiply(multiply(multiply(Z, add(Y, X)), X), multiply(multiply(Z, add(Y, X)), multiply(X, Z))) 1.58/1.77 = { by lemma 76 } 1.58/1.77 multiply(multiply(Z, add(Y, X)), multiply(X, Z)) 1.58/1.77 = { by lemma 18 } 1.58/1.77 multiply(X, Z) 1.58/1.77 1.58/1.77 Lemma 78: multiply(multiply(X, Z), multiply(Y, Z)) = multiply(X, multiply(Y, Z)). 1.58/1.77 Proof: 1.58/1.77 multiply(multiply(X, Z), multiply(Y, Z)) 1.58/1.77 = { by lemma 41 } 1.58/1.77 multiply(multiply(X, inverse(inverse(Z))), multiply(Y, Z)) 1.58/1.77 = { by lemma 41 } 1.58/1.77 multiply(multiply(X, inverse(inverse(Z))), multiply(Y, inverse(inverse(Z)))) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(Y, inverse(inverse(Z))), multiply(X, inverse(inverse(Z)))) 1.58/1.77 = { by lemma 32 } 1.58/1.77 multiply(multiply(Y, inverse(inverse(Z))), multiply(X, multiply(inverse(inverse(Z)), n1))) 1.58/1.77 = { by lemma 47 } 1.58/1.77 multiply(multiply(Y, inverse(inverse(Z))), multiply(X, multiply(inverse(inverse(Z)), add(n1, Y)))) 1.58/1.77 = { by lemma 16 } 1.58/1.77 multiply(multiply(Y, inverse(inverse(Z))), multiply(X, add(inverse(inverse(Z)), multiply(Y, inverse(inverse(Z)))))) 1.58/1.77 = { by lemma 77 } 1.58/1.77 multiply(multiply(Y, inverse(inverse(Z))), X) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(X, multiply(Y, inverse(inverse(Z)))) 1.58/1.77 = { by lemma 41 } 1.58/1.77 multiply(X, multiply(Y, Z)) 1.58/1.77 1.58/1.77 Lemma 79: multiply(multiply(X, Z), Y) = multiply(X, multiply(Y, Z)). 1.58/1.77 Proof: 1.58/1.77 multiply(multiply(X, Z), Y) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(Y, multiply(X, Z)) 1.58/1.77 = { by lemma 78 } 1.58/1.77 multiply(multiply(Y, Z), multiply(X, Z)) 1.58/1.77 = { by lemma 43 } 1.58/1.77 multiply(multiply(X, Z), multiply(Y, Z)) 1.58/1.77 = { by lemma 78 } 1.58/1.79 multiply(X, multiply(Y, Z)) 1.58/1.79 1.58/1.79 Goal 1 (prove_add_multiply_property): add(a, multiply(b, c)) = multiply(add(a, b), add(a, c)). 1.58/1.79 Proof: 1.58/1.79 add(a, multiply(b, c)) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, multiply(c, b)) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, multiply(b, c)) 1.58/1.79 = { by lemma 41 } 1.58/1.79 add(a, multiply(b, inverse(inverse(c)))) 1.58/1.79 = { by lemma 50 } 1.58/1.79 add(a, multiply(add(b, a), multiply(b, inverse(inverse(c))))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, multiply(multiply(b, inverse(inverse(c))), add(b, a))) 1.58/1.79 = { by lemma 73 } 1.58/1.79 add(a, multiply(multiply(b, inverse(inverse(c))), add(add(b, a), a))) 1.58/1.79 = { by lemma 41 } 1.58/1.79 add(a, multiply(multiply(b, inverse(inverse(c))), add(add(b, inverse(inverse(a))), a))) 1.58/1.79 = { by lemma 68 } 1.58/1.79 pixley(add(b, inverse(inverse(a))), inverse(a), add(a, multiply(b, inverse(inverse(c))))) 1.58/1.79 = { by lemma 74 } 1.58/1.79 pixley(b, inverse(a), add(add(a, multiply(b, inverse(inverse(c)))), inverse(inverse(a)))) 1.58/1.79 = { by lemma 41 } 1.58/1.79 pixley(b, inverse(a), add(add(a, multiply(b, inverse(inverse(c)))), a)) 1.58/1.79 = { by lemma 69 } 1.58/1.79 add(a, multiply(b, add(add(a, multiply(b, inverse(inverse(c)))), a))) 1.58/1.79 = { by lemma 54 } 1.58/1.79 add(a, multiply(b, add(a, multiply(b, inverse(inverse(c)))))) 1.58/1.79 = { by lemma 45 } 1.58/1.79 add(a, multiply(b, multiply(n1, add(a, multiply(b, inverse(inverse(c))))))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, multiply(b, multiply(n1, add(a, multiply(inverse(inverse(c)), b))))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, multiply(b, multiply(add(a, multiply(inverse(inverse(c)), b)), n1))) 1.58/1.79 = { by lemma 79 } 1.58/1.79 add(a, multiply(multiply(b, n1), add(a, multiply(inverse(inverse(c)), b)))) 1.58/1.79 = { by axiom 5 (multiply_add_property) } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(inverse(inverse(c)), b), multiply(b, n1)))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(inverse(inverse(c)), b), multiply(n1, b)))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, b), multiply(inverse(inverse(c)), b)))) 1.58/1.79 = { by lemma 44 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), b)))) 1.58/1.79 = { by lemma 44 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(b, n1))))) 1.58/1.79 = { by lemma 41 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(b, inverse(inverse(n1))))))) 1.58/1.79 = { by lemma 50 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(add(b, n1), multiply(b, inverse(inverse(n1)))))))) 1.58/1.79 = { by lemma 41 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(add(b, n1), multiply(b, n1)))))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(multiply(b, n1), add(b, n1)))))) 1.58/1.79 = { by lemma 15 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), multiply(inverse(inverse(c)), add(multiply(b, n1), multiply(n1, multiply(b, n1))))))) 1.58/1.79 = { by lemma 77 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, multiply(b, n1)), inverse(inverse(c))))) 1.58/1.79 = { by lemma 44 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(multiply(n1, b), inverse(inverse(c))))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(n1, b)))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, add(multiply(a, multiply(b, n1)), multiply(inverse(inverse(c)), multiply(b, n1)))) 1.58/1.79 = { by axiom 5 (multiply_add_property) } 1.58/1.79 add(a, multiply(multiply(b, n1), add(a, inverse(inverse(c))))) 1.58/1.79 = { by lemma 79 } 1.58/1.79 add(a, multiply(b, multiply(add(a, inverse(inverse(c))), n1))) 1.58/1.79 = { by lemma 43 } 1.58/1.79 add(a, multiply(b, multiply(n1, add(a, inverse(inverse(c)))))) 1.58/1.79 = { by lemma 45 } 1.58/1.79 add(a, multiply(b, add(a, inverse(inverse(c))))) 1.58/1.79 = { by lemma 52 } 1.58/1.79 add(multiply(a, inverse(inverse(c))), add(a, multiply(b, add(a, inverse(inverse(c)))))) 1.58/1.79 = { by lemma 66 } 1.58/1.79 add(multiply(a, inverse(inverse(c))), multiply(add(a, b), add(a, inverse(inverse(c))))) 1.58/1.79 = { by lemma 8 } 1.58/1.79 pixley(a, inverse(c), add(a, b)) 1.58/1.79 = { by axiom 1 (pixley_defn) } 1.58/1.79 add(multiply(a, inverse(inverse(c))), add(multiply(a, add(a, b)), multiply(inverse(inverse(c)), add(a, b)))) 1.58/1.79 = { by lemma 41 } 1.58/1.79 add(multiply(a, c), add(multiply(a, add(a, b)), multiply(inverse(inverse(c)), add(a, b)))) 1.58/1.79 = { by axiom 5 (multiply_add_property) } 1.58/1.79 add(multiply(a, c), multiply(add(a, b), add(a, inverse(inverse(c))))) 1.58/1.79 = { by lemma 41 } 1.58/1.79 add(multiply(a, c), multiply(add(a, b), add(a, c))) 1.58/1.79 = { by lemma 66 } 1.58/1.79 add(multiply(a, c), add(a, multiply(b, add(a, c)))) 1.58/1.79 = { by lemma 53 } 1.58/1.79 add(a, multiply(b, add(a, c))) 1.58/1.79 = { by lemma 66 } 1.58/1.79 multiply(add(a, b), add(a, c)) 1.58/1.79 % SZS output end Proof 1.58/1.79 1.58/1.79 RESULT: Unsatisfiable (the axioms are contradictory). 1.58/1.79 EOF