%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : RNG024-6 : TPTP v8.1.0. Released v1.0.0. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n025.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 600s % DateTime : Mon Jul 18 20:36:20 EDT 2022 % Result : Unsatisfiable 1.71s 1.88s % Output : Proof 1.71s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.11 % Problem : RNG024-6 : TPTP v8.1.0. Released v1.0.0. % 0.07/0.12 % Command : moca.sh %s % 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 : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Mon May 30 09:15:12 EDT 2022 % 0.12/0.33 % CPUTime : % 1.71/1.88 % SZS status Unsatisfiable % 1.71/1.88 % SZS output start Proof % 1.71/1.88 The input problem is unsatisfiable because % 1.71/1.88 % 1.71/1.88 [1] the following set of Horn clauses is unsatisfiable: % 1.71/1.88 % 1.71/1.88 add(additive_identity, X) = X % 1.71/1.88 add(X, additive_identity) = X % 1.71/1.88 multiply(additive_identity, X) = additive_identity % 1.71/1.88 multiply(X, additive_identity) = additive_identity % 1.71/1.88 add(additive_inverse(X), X) = additive_identity % 1.71/1.88 add(X, additive_inverse(X)) = additive_identity % 1.71/1.88 additive_inverse(additive_inverse(X)) = X % 1.71/1.88 multiply(X, add(Y, Z)) = add(multiply(X, Y), multiply(X, Z)) % 1.71/1.88 multiply(add(X, Y), Z) = add(multiply(X, Z), multiply(Y, Z)) % 1.71/1.88 add(X, Y) = add(Y, X) % 1.71/1.88 add(X, add(Y, Z)) = add(add(X, Y), Z) % 1.71/1.88 multiply(multiply(X, Y), Y) = multiply(X, multiply(Y, Y)) % 1.71/1.88 multiply(multiply(X, X), Y) = multiply(X, multiply(X, Y)) % 1.71/1.88 associator(X, Y, Z) = add(multiply(multiply(X, Y), Z), additive_inverse(multiply(X, multiply(Y, Z)))) % 1.71/1.88 commutator(X, Y) = add(multiply(Y, X), additive_inverse(multiply(X, Y))) % 1.71/1.88 associator(x, y, y) = additive_identity ==> \bottom % 1.71/1.88 % 1.71/1.88 This holds because % 1.71/1.88 % 1.71/1.88 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 1.71/1.88 % 1.71/1.88 E: % 1.71/1.88 add(X, Y) = add(Y, X) % 1.71/1.88 add(X, add(Y, Z)) = add(add(X, Y), Z) % 1.71/1.88 add(X, additive_identity) = X % 1.71/1.88 add(X, additive_inverse(X)) = additive_identity % 1.71/1.88 add(additive_identity, X) = X % 1.71/1.88 add(additive_inverse(X), X) = additive_identity % 1.71/1.88 additive_inverse(additive_inverse(X)) = X % 1.71/1.88 associator(X, Y, Z) = add(multiply(multiply(X, Y), Z), additive_inverse(multiply(X, multiply(Y, Z)))) % 1.71/1.88 commutator(X, Y) = add(multiply(Y, X), additive_inverse(multiply(X, Y))) % 1.71/1.88 f1(additive_identity) = false__ % 1.71/1.88 f1(associator(x, y, y)) = true__ % 1.71/1.88 multiply(X, add(Y, Z)) = add(multiply(X, Y), multiply(X, Z)) % 1.71/1.88 multiply(X, additive_identity) = additive_identity % 1.71/1.88 multiply(add(X, Y), Z) = add(multiply(X, Z), multiply(Y, Z)) % 1.71/1.88 multiply(additive_identity, X) = additive_identity % 1.71/1.88 multiply(multiply(X, X), Y) = multiply(X, multiply(X, Y)) % 1.71/1.88 multiply(multiply(X, Y), Y) = multiply(X, multiply(Y, Y)) % 1.71/1.88 G: % 1.71/1.88 true__ = false__ % 1.71/1.88 % 1.71/1.88 This holds because % 1.71/1.88 % 1.71/1.88 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 1.71/1.88 % 1.71/1.88 add(X, Y) = add(Y, X) % 1.71/1.88 add(Y1, add(Y0, Y2)) = add(Y0, add(Y1, Y2)) % 1.71/1.88 add(Y2, add(Y0, Y1)) = add(Y0, add(Y1, Y2)) % 1.71/1.88 add(multiply(X0, multiply(Y2, Y2)), multiply(multiply(X0, Y2), Y1)) = multiply(multiply(X0, Y2), add(Y1, Y2)) % 1.71/1.88 multiply(Y2, add(Y1, Y1)) = multiply(add(Y2, Y2), Y1) % 1.71/1.88 add(X, additive_identity) -> X % 1.71/1.88 add(X, additive_inverse(X)) -> additive_identity % 1.71/1.88 add(Y0, add(Y1, additive_inverse(add(Y0, Y1)))) -> additive_identity % 1.71/1.88 add(Y0, add(additive_inverse(Y0), Y2)) -> Y2 % 1.71/1.88 add(add(X, Y), Z) -> add(X, add(Y, Z)) % 1.71/1.88 add(additive_identity, X) -> X % 1.71/1.88 add(additive_inverse(X), X) -> additive_identity % 1.71/1.88 add(additive_inverse(Y1), add(Y1, Y2)) -> Y2 % 1.71/1.88 add(multiply(X, Y), multiply(X, Z)) -> multiply(X, add(Y, Z)) % 1.71/1.88 add(multiply(X, Z), multiply(Y, Z)) -> multiply(add(X, Y), Z) % 1.71/1.88 add(multiply(X0, add(X1, X2)), Y2) -> add(multiply(X0, X1), add(multiply(X0, X2), Y2)) % 1.71/1.88 add(multiply(X0, multiply(X0, Y1)), multiply(Y2, Y1)) -> multiply(add(multiply(X0, X0), Y2), Y1) % 1.71/1.88 add(multiply(X0, multiply(Y1, Y1)), multiply(Y2, Y1)) -> multiply(add(multiply(X0, Y1), Y2), Y1) % 1.71/1.88 add(multiply(X0, multiply(Y1, Y1)), multiply(multiply(X0, Y1), Y2)) -> multiply(multiply(X0, Y1), add(Y1, Y2)) % 1.71/1.88 add(multiply(Y0, Y1), multiply(X0, multiply(X0, Y1))) -> multiply(add(Y0, multiply(X0, X0)), Y1) % 1.71/1.88 add(multiply(Y0, Y1), multiply(X0, multiply(Y1, Y1))) -> multiply(add(Y0, multiply(X0, Y1)), Y1) % 1.71/1.88 add(multiply(add(X0, X2), X1), Y2) -> add(multiply(X0, X1), add(multiply(X2, X1), Y2)) % 1.71/1.88 additive_inverse(additive_identity) -> additive_identity % 1.71/1.88 additive_inverse(additive_inverse(X)) -> X % 1.71/1.88 associator(X, Y, Z) -> add(multiply(multiply(X, Y), Z), additive_inverse(multiply(X, multiply(Y, Z)))) % 1.71/1.88 commutator(X, Y) -> add(multiply(Y, X), additive_inverse(multiply(X, Y))) % 1.71/1.88 f1(additive_identity) -> false__ % 1.71/1.88 f1(associator(x, y, y)) -> true__ % 1.71/1.88 multiply(X, additive_identity) -> additive_identity % 1.71/1.88 multiply(additive_identity, X) -> additive_identity % 1.71/1.88 multiply(multiply(X, X), Y) -> multiply(X, multiply(X, Y)) % 1.71/1.88 multiply(multiply(X, Y), Y) -> multiply(X, multiply(Y, Y)) % 1.71/1.88 multiply(multiply(X0, multiply(X0, Y1)), Y1) -> multiply(X0, multiply(X0, multiply(Y1, Y1))) % 1.71/1.88 multiply(multiply(X0, multiply(X0, multiply(X0, X0))), Y1) -> multiply(X0, multiply(X0, multiply(X0, multiply(X0, Y1)))) % 1.71/1.88 multiply(multiply(X0, multiply(Y1, Y1)), Y1) -> multiply(multiply(X0, Y1), multiply(Y1, Y1)) % 1.71/1.88 true__ -> false__ % 1.71/1.88 with the LPO induced by % 1.71/1.88 y > x > f1 > commutator > associator > additive_inverse > add > multiply > additive_identity > true__ > false__ % 1.71/1.88 % 1.71/1.88 % SZS output end Proof % 1.71/1.88 %------------------------------------------------------------------------------