%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP174-1 : TPTP v8.1.0. Bugfixed v1.2.1. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n007.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 : Sat Jul 16 10:53:43 EDT 2022 % Result : Unsatisfiable 10.30s 10.30s % Output : Proof 10.30s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : GRP174-1 : TPTP v8.1.0. Bugfixed v1.2.1. % 0.00/0.12 % Command : moca.sh %s % 0.12/0.33 % Computer : n007.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 : Tue Jun 14 07:51:39 EDT 2022 % 0.12/0.33 % CPUTime : % 10.30/10.30 % SZS status Unsatisfiable % 10.30/10.30 % SZS output start Proof % 10.30/10.30 The input problem is unsatisfiable because % 10.30/10.30 % 10.30/10.30 [1] the following set of Horn clauses is unsatisfiable: % 10.30/10.30 % 10.30/10.30 multiply(identity, X) = X % 10.30/10.30 multiply(inverse(X), X) = identity % 10.30/10.30 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 10.30/10.30 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.30/10.30 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.30/10.30 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 10.30/10.30 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 10.30/10.30 least_upper_bound(X, X) = X % 10.30/10.30 greatest_lower_bound(X, X) = X % 10.30/10.30 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 10.30/10.30 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 10.30/10.30 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 10.30/10.30 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 10.30/10.30 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 10.30/10.30 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 10.30/10.30 greatest_lower_bound(identity, a) = a % 10.30/10.30 greatest_lower_bound(identity, inverse(a)) = inverse(a) % 10.30/10.30 identity = a ==> \bottom % 10.30/10.30 % 10.30/10.30 This holds because % 10.30/10.30 % 10.30/10.30 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 10.30/10.30 % 10.30/10.30 E: % 10.30/10.30 f1(a) = false__ % 10.30/10.30 f1(identity) = true__ % 10.30/10.30 greatest_lower_bound(X, X) = X % 10.30/10.30 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.30/10.30 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 10.30/10.30 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 10.30/10.30 greatest_lower_bound(identity, a) = a % 10.30/10.30 greatest_lower_bound(identity, inverse(a)) = inverse(a) % 10.30/10.30 least_upper_bound(X, X) = X % 10.30/10.30 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.30/10.30 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 10.30/10.30 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 10.30/10.30 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 10.30/10.30 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 10.30/10.30 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 10.30/10.30 multiply(identity, X) = X % 10.30/10.30 multiply(inverse(X), X) = identity % 10.30/10.30 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 10.30/10.30 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 10.30/10.30 G: % 10.30/10.30 true__ = false__ % 10.30/10.30 % 10.30/10.30 This holds because % 10.30/10.30 % 10.30/10.30 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 10.30/10.30 % 10.30/10.30 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.30/10.30 greatest_lower_bound(Y1, multiply(Y0, Y1)) = multiply(greatest_lower_bound(Y0, identity), Y1) % 10.30/10.30 greatest_lower_bound(Y1, multiply(Y2, Y1)) = multiply(greatest_lower_bound(identity, Y2), Y1) % 10.30/10.30 greatest_lower_bound(identity, greatest_lower_bound(Y0, a)) = greatest_lower_bound(a, Y0) % 10.30/10.30 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.30/10.30 least_upper_bound(Y1, multiply(Y0, Y1)) = multiply(least_upper_bound(Y0, identity), Y1) % 10.30/10.30 least_upper_bound(Y1, multiply(Y2, Y1)) = multiply(least_upper_bound(identity, Y2), Y1) % 10.30/10.30 a -> identity % 10.30/10.30 f1(a) -> false__ % 10.30/10.30 f1(identity) -> true__ % 10.30/10.30 greatest_lower_bound(X, X) -> X % 10.30/10.30 greatest_lower_bound(X, least_upper_bound(X, Y)) -> X % 10.30/10.30 greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y0)) -> greatest_lower_bound(Y0, Y1) % 10.30/10.30 greatest_lower_bound(Y0, greatest_lower_bound(Y1, greatest_lower_bound(Y0, Y1))) -> greatest_lower_bound(Y0, Y1) % 10.30/10.30 greatest_lower_bound(Y0, least_upper_bound(Y1, Y0)) -> Y0 % 10.30/10.30 greatest_lower_bound(Y0, multiply(least_upper_bound(identity, X1), Y0)) -> Y0 % 10.30/10.30 greatest_lower_bound(Y1, greatest_lower_bound(Y1, Y2)) -> greatest_lower_bound(Y1, Y2) % 10.30/10.30 greatest_lower_bound(Y1, multiply(a, Y1)) -> multiply(a, Y1) % 10.30/10.30 greatest_lower_bound(greatest_lower_bound(X, Y), Z) -> greatest_lower_bound(X, greatest_lower_bound(Y, Z)) % 10.30/10.30 greatest_lower_bound(identity, a) -> a % 10.30/10.30 greatest_lower_bound(identity, greatest_lower_bound(Y0, a)) -> greatest_lower_bound(Y0, a) % 10.30/10.30 greatest_lower_bound(identity, greatest_lower_bound(a, Y2)) -> greatest_lower_bound(a, Y2) % 10.30/10.30 greatest_lower_bound(identity, inverse(a)) -> inverse(a) % 10.30/10.30 greatest_lower_bound(least_upper_bound(X0, identity), least_upper_bound(identity, X0)) -> least_upper_bound(X0, identity) % 10.30/10.30 greatest_lower_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, greatest_lower_bound(Y, Z)) % 10.30/10.30 greatest_lower_bound(multiply(Y, X), multiply(Z, X)) -> multiply(greatest_lower_bound(Y, Z), X) % 10.30/10.30 inverse(a) -> identity % 10.30/10.30 inverse(identity) -> identity % 10.30/10.30 inverse(inverse(Y0)) -> Y0 % 10.30/10.30 inverse(inverse(identity)) -> identity % 10.30/10.30 least_upper_bound(X, X) -> X % 10.30/10.30 least_upper_bound(X, greatest_lower_bound(X, Y)) -> X % 10.30/10.30 least_upper_bound(Y0, greatest_lower_bound(Y1, Y0)) -> Y0 % 10.30/10.30 least_upper_bound(Y0, least_upper_bound(Y1, Y0)) -> least_upper_bound(Y0, Y1) % 10.30/10.30 least_upper_bound(Y0, least_upper_bound(Y1, least_upper_bound(Y0, Y1))) -> least_upper_bound(Y0, Y1) % 10.30/10.30 least_upper_bound(Y1, least_upper_bound(Y1, Y2)) -> least_upper_bound(Y1, Y2) % 10.30/10.30 least_upper_bound(Y1, multiply(a, Y1)) -> Y1 % 10.30/10.30 least_upper_bound(Y1, multiply(inverse(a), Y1)) -> Y1 % 10.30/10.30 least_upper_bound(a, greatest_lower_bound(identity, greatest_lower_bound(Y1, a))) -> a % 10.30/10.30 least_upper_bound(greatest_lower_bound(X0, a), greatest_lower_bound(a, X0)) -> greatest_lower_bound(X0, a) % 10.30/10.30 least_upper_bound(greatest_lower_bound(Y0, identity), greatest_lower_bound(identity, Y0)) -> greatest_lower_bound(Y0, identity) % 10.30/10.30 least_upper_bound(identity, a) -> identity % 10.30/10.30 least_upper_bound(identity, greatest_lower_bound(Y0, a)) -> identity % 10.30/10.30 least_upper_bound(identity, greatest_lower_bound(a, X0)) -> identity % 10.30/10.30 least_upper_bound(identity, inverse(X0)) -> multiply(least_upper_bound(identity, X0), inverse(X0)) % 10.30/10.30 least_upper_bound(identity, inverse(a)) -> identity % 10.30/10.30 least_upper_bound(identity, least_upper_bound(Y0, a)) -> least_upper_bound(identity, Y0) % 10.30/10.30 least_upper_bound(identity, least_upper_bound(Y0, inverse(a))) -> least_upper_bound(identity, Y0) % 10.30/10.30 least_upper_bound(identity, least_upper_bound(a, Y2)) -> least_upper_bound(identity, Y2) % 10.30/10.30 least_upper_bound(identity, least_upper_bound(inverse(a), Y2)) -> least_upper_bound(identity, Y2) % 10.30/10.30 least_upper_bound(least_upper_bound(X, Y), Z) -> least_upper_bound(X, least_upper_bound(Y, Z)) % 10.30/10.30 least_upper_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, least_upper_bound(Y, Z)) % 10.30/10.30 least_upper_bound(multiply(Y, X), multiply(Z, X)) -> multiply(least_upper_bound(Y, Z), X) % 10.30/10.30 multiply(X0, multiply(inverse(X0), Y1)) -> Y1 % 10.30/10.30 multiply(Y0, inverse(Y0)) -> identity % 10.30/10.30 multiply(Y0, inverse(inverse(inverse(Y0)))) -> identity % 10.30/10.30 multiply(Y0, multiply(Y1, inverse(multiply(Y0, Y1)))) -> identity % 10.30/10.30 multiply(Y1, identity) -> Y1 % 10.30/10.30 multiply(identity, X) -> X % 10.30/10.30 multiply(inverse(X), X) -> identity % 10.30/10.30 multiply(inverse(Y1), multiply(Y1, Y2)) -> Y2 % 10.30/10.30 multiply(inverse(identity), Y1) -> Y1 % 10.30/10.30 multiply(inverse(inverse(X0)), X1) -> multiply(X0, X1) % 10.30/10.30 multiply(inverse(inverse(Y1)), identity) -> Y1 % 10.30/10.30 multiply(least_upper_bound(identity, inverse(Y0)), Y0) -> least_upper_bound(Y0, identity) % 10.30/10.30 multiply(least_upper_bound(identity, inverse(identity)), Y0) -> Y0 % 10.30/10.30 multiply(multiply(X, Y), Z) -> multiply(X, multiply(Y, Z)) % 10.30/10.30 true__ -> false__ % 10.30/10.30 with the LPO induced by % 10.30/10.30 f1 > a > inverse > identity > least_upper_bound > greatest_lower_bound > multiply > true__ > false__ % 10.30/10.30 % 10.30/10.30 % SZS output end Proof % 10.30/10.30 %------------------------------------------------------------------------------