%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP166-2 : TPTP v8.1.0. Bugfixed v1.2.1. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n027.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 10:53:35 EDT 2022 % Result : Unsatisfiable 87.08s 86.43s % Output : Proof 87.08s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : GRP166-2 : TPTP v8.1.0. Bugfixed v1.2.1. % 0.03/0.13 % Command : moca.sh %s % 0.12/0.32 % Computer : n027.cluster.edu % 0.12/0.32 % Model : x86_64 x86_64 % 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.32 % Memory : 8042.1875MB % 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.32 % CPULimit : 300 % 0.12/0.32 % WCLimit : 600 % 0.12/0.32 % DateTime : Mon Jun 13 19:35:02 EDT 2022 % 0.12/0.32 % CPUTime : % 87.08/86.43 % SZS status Unsatisfiable % 87.08/86.43 % SZS output start Proof % 87.08/86.43 The input problem is unsatisfiable because % 87.08/86.43 % 87.08/86.43 [1] the following set of Horn clauses is unsatisfiable: % 87.08/86.43 % 87.08/86.43 multiply(identity, X) = X % 87.08/86.43 multiply(inverse(X), X) = identity % 87.08/86.43 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 87.08/86.43 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 87.08/86.43 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 87.08/86.43 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 87.08/86.43 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 87.08/86.43 least_upper_bound(X, X) = X % 87.08/86.43 greatest_lower_bound(X, X) = X % 87.08/86.43 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 87.08/86.43 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 87.08/86.43 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 87.08/86.43 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 87.08/86.43 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 87.08/86.43 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 87.08/86.43 greatest_lower_bound(a, identity) = identity % 87.08/86.43 greatest_lower_bound(b, identity) = identity % 87.08/86.43 greatest_lower_bound(a, multiply(a, b)) = a ==> \bottom % 87.08/86.43 % 87.08/86.43 This holds because % 87.08/86.43 % 87.08/86.43 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 87.08/86.43 % 87.08/86.43 E: % 87.08/86.43 f1(a) = false__ % 87.08/86.43 f1(greatest_lower_bound(a, multiply(a, b))) = true__ % 87.08/86.43 greatest_lower_bound(X, X) = X % 87.08/86.43 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 87.08/86.43 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 87.08/86.43 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 87.08/86.43 greatest_lower_bound(a, identity) = identity % 87.08/86.43 greatest_lower_bound(b, identity) = identity % 87.08/86.43 least_upper_bound(X, X) = X % 87.08/86.43 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 87.08/86.43 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 87.08/86.43 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 87.08/86.43 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 87.08/86.43 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 87.08/86.43 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 87.08/86.43 multiply(identity, X) = X % 87.08/86.43 multiply(inverse(X), X) = identity % 87.08/86.43 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 87.08/86.43 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 87.08/86.43 G: % 87.08/86.43 true__ = false__ % 87.08/86.43 % 87.08/86.43 This holds because % 87.08/86.43 % 87.08/86.43 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 87.08/86.43 % 87.08/86.43 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 87.08/86.43 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 87.08/86.43 least_upper_bound(Y0, a) = least_upper_bound(a, Y0) % 87.08/86.43 least_upper_bound(Y0, b) = least_upper_bound(b, Y0) % 87.08/86.43 f1(a) -> false__ % 87.08/86.43 f1(greatest_lower_bound(a, multiply(a, b))) -> true__ % 87.08/86.43 greatest_lower_bound(X, X) -> X % 87.08/86.43 greatest_lower_bound(X, least_upper_bound(X, Y)) -> X % 87.08/86.43 greatest_lower_bound(X1, multiply(inverse(a), X1)) -> multiply(inverse(a), X1) % 87.08/86.43 greatest_lower_bound(X1, multiply(inverse(b), X1)) -> multiply(inverse(b), X1) % 87.08/86.43 greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y0)) -> greatest_lower_bound(Y0, Y1) % 87.08/86.43 greatest_lower_bound(Y0, least_upper_bound(Y1, Y0)) -> Y0 % 87.08/86.43 greatest_lower_bound(Y0, multiply(Y0, a)) -> Y0 % 87.08/86.43 greatest_lower_bound(Y0, multiply(Y0, b)) -> Y0 % 87.08/86.43 greatest_lower_bound(Y1, greatest_lower_bound(Y1, Y2)) -> greatest_lower_bound(Y1, Y2) % 87.08/86.43 greatest_lower_bound(Y1, multiply(a, Y1)) -> Y1 % 87.08/86.43 greatest_lower_bound(Y1, multiply(b, Y1)) -> Y1 % 87.08/86.43 greatest_lower_bound(a, greatest_lower_bound(Y0, identity)) -> greatest_lower_bound(identity, Y0) % 87.08/86.43 greatest_lower_bound(a, greatest_lower_bound(identity, Y2)) -> greatest_lower_bound(identity, Y2) % 87.08/86.43 greatest_lower_bound(a, identity) -> identity % 87.08/86.43 greatest_lower_bound(a, inverse(a)) -> inverse(a) % 87.08/86.43 greatest_lower_bound(a, inverse(b)) -> inverse(b) % 87.08/86.43 greatest_lower_bound(b, greatest_lower_bound(Y0, identity)) -> greatest_lower_bound(identity, Y0) % 87.08/86.43 greatest_lower_bound(b, greatest_lower_bound(identity, Y2)) -> greatest_lower_bound(identity, Y2) % 87.08/86.43 greatest_lower_bound(b, identity) -> identity % 87.08/86.43 greatest_lower_bound(b, inverse(b)) -> inverse(b) % 87.08/86.43 greatest_lower_bound(greatest_lower_bound(X, Y), Z) -> greatest_lower_bound(X, greatest_lower_bound(Y, Z)) % 87.08/86.43 greatest_lower_bound(identity, a) -> identity % 87.08/86.43 greatest_lower_bound(identity, b) -> identity % 87.08/86.43 greatest_lower_bound(identity, greatest_lower_bound(Y0, a)) -> greatest_lower_bound(identity, Y0) % 87.08/86.43 greatest_lower_bound(identity, greatest_lower_bound(Y0, b)) -> greatest_lower_bound(identity, Y0) % 87.08/86.43 greatest_lower_bound(identity, greatest_lower_bound(a, Y0)) -> greatest_lower_bound(identity, Y0) % 87.08/86.43 greatest_lower_bound(identity, greatest_lower_bound(b, Y0)) -> greatest_lower_bound(identity, Y0) % 87.08/86.43 greatest_lower_bound(identity, inverse(a)) -> inverse(a) % 87.08/86.43 greatest_lower_bound(identity, inverse(b)) -> inverse(b) % 87.08/86.43 greatest_lower_bound(identity, least_upper_bound(X1, a)) -> identity % 87.08/86.43 greatest_lower_bound(identity, least_upper_bound(Y0, b)) -> identity % 87.08/86.43 greatest_lower_bound(identity, least_upper_bound(a, X1)) -> identity % 87.08/86.43 greatest_lower_bound(identity, least_upper_bound(b, X1)) -> identity % 87.08/86.43 greatest_lower_bound(identity, multiply(a, a)) -> identity % 87.08/86.43 greatest_lower_bound(identity, multiply(a, b)) -> identity % 87.08/86.43 greatest_lower_bound(identity, multiply(b, a)) -> identity % 87.08/86.43 greatest_lower_bound(inverse(a), b) -> inverse(a) % 87.08/86.43 greatest_lower_bound(least_upper_bound(X0, a), least_upper_bound(a, X0)) -> least_upper_bound(X0, a) % 87.08/86.43 greatest_lower_bound(least_upper_bound(X0, b), least_upper_bound(b, X0)) -> least_upper_bound(X0, b) % 87.08/86.43 greatest_lower_bound(least_upper_bound(X0, identity), least_upper_bound(a, X0)) -> least_upper_bound(X0, identity) % 87.08/86.43 greatest_lower_bound(least_upper_bound(X0, identity), least_upper_bound(b, X0)) -> least_upper_bound(X0, identity) % 87.08/86.43 greatest_lower_bound(least_upper_bound(identity, X0), least_upper_bound(a, X0)) -> least_upper_bound(identity, X0) % 87.08/86.43 greatest_lower_bound(least_upper_bound(identity, X0), least_upper_bound(b, X0)) -> least_upper_bound(identity, X0) % 87.08/86.43 greatest_lower_bound(multiply(Y0, a), multiply(Y0, identity)) -> Y0 % 87.08/86.43 inverse(identity) -> identity % 87.08/86.43 inverse(inverse(Y1)) -> Y1 % 87.08/86.43 least_upper_bound(X, X) -> X % 87.08/86.43 least_upper_bound(X, greatest_lower_bound(X, Y)) -> X % 87.08/86.43 least_upper_bound(Y0, greatest_lower_bound(Y1, Y0)) -> Y0 % 87.08/86.43 least_upper_bound(Y0, least_upper_bound(Y1, Y0)) -> least_upper_bound(Y0, Y1) % 87.08/86.43 least_upper_bound(Y1, least_upper_bound(X1, Y1)) -> least_upper_bound(X1, Y1) % 87.08/86.43 least_upper_bound(Y1, least_upper_bound(Y1, Y2)) -> least_upper_bound(Y1, Y2) % 87.08/86.43 least_upper_bound(Y1, multiply(Y1, a)) -> multiply(Y1, a) % 87.08/86.43 least_upper_bound(Y1, multiply(Y1, b)) -> multiply(Y1, b) % 87.08/86.43 least_upper_bound(Y1, multiply(a, Y1)) -> multiply(a, Y1) % 87.08/86.43 least_upper_bound(Y1, multiply(b, Y1)) -> multiply(b, Y1) % 87.08/86.43 least_upper_bound(a, greatest_lower_bound(Y0, identity)) -> a % 87.08/86.43 least_upper_bound(a, greatest_lower_bound(identity, X0)) -> a % 87.08/86.43 least_upper_bound(a, identity) -> a % 87.08/86.43 least_upper_bound(a, inverse(a)) -> a % 87.08/86.43 least_upper_bound(a, inverse(b)) -> a % 87.08/86.43 least_upper_bound(a, least_upper_bound(X0, identity)) -> least_upper_bound(a, X0) % 87.08/86.43 least_upper_bound(a, least_upper_bound(identity, Y2)) -> least_upper_bound(a, Y2) % 87.08/86.43 least_upper_bound(b, greatest_lower_bound(Y0, identity)) -> b % 87.08/86.43 least_upper_bound(b, greatest_lower_bound(identity, X0)) -> b % 87.08/86.43 least_upper_bound(b, identity) -> b % 87.08/86.43 least_upper_bound(b, inverse(b)) -> b % 87.08/86.43 least_upper_bound(b, least_upper_bound(X0, identity)) -> least_upper_bound(b, X0) % 87.08/86.43 least_upper_bound(b, least_upper_bound(identity, Y0)) -> least_upper_bound(b, Y0) % 87.08/86.43 least_upper_bound(greatest_lower_bound(X0, a), greatest_lower_bound(identity, X0)) -> greatest_lower_bound(X0, a) % 87.08/86.43 least_upper_bound(greatest_lower_bound(X0, b), greatest_lower_bound(identity, X0)) -> greatest_lower_bound(X0, b) % 87.08/86.43 least_upper_bound(greatest_lower_bound(X0, identity), greatest_lower_bound(identity, X0)) -> greatest_lower_bound(X0, identity) % 87.08/86.43 least_upper_bound(greatest_lower_bound(Y0, identity), a) -> a % 87.08/86.43 least_upper_bound(greatest_lower_bound(Y0, identity), b) -> b % 87.08/86.43 least_upper_bound(greatest_lower_bound(Y0, identity), greatest_lower_bound(Y0, a)) -> greatest_lower_bound(Y0, a) % 87.08/86.43 least_upper_bound(greatest_lower_bound(Y0, identity), greatest_lower_bound(Y0, b)) -> greatest_lower_bound(Y0, b) % 87.08/86.43 least_upper_bound(greatest_lower_bound(Y0, identity), greatest_lower_bound(a, Y0)) -> greatest_lower_bound(a, Y0) % 87.08/86.43 least_upper_bound(greatest_lower_bound(Y0, identity), greatest_lower_bound(b, Y0)) -> greatest_lower_bound(b, Y0) % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, X0), greatest_lower_bound(a, X0)) -> greatest_lower_bound(a, X0) % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, X0), greatest_lower_bound(b, X0)) -> greatest_lower_bound(b, X0) % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, Y0), a) -> a % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, Y0), b) -> b % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, Y0), greatest_lower_bound(Y0, a)) -> greatest_lower_bound(Y0, a) % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, Y0), greatest_lower_bound(Y0, b)) -> greatest_lower_bound(Y0, b) % 87.08/86.43 least_upper_bound(greatest_lower_bound(identity, Y0), greatest_lower_bound(Y0, identity)) -> greatest_lower_bound(Y0, identity) % 87.08/86.43 least_upper_bound(identity, a) -> a % 87.08/86.43 least_upper_bound(identity, b) -> b % 87.08/86.43 least_upper_bound(identity, greatest_lower_bound(a, b)) -> greatest_lower_bound(a, b) % 87.08/86.43 least_upper_bound(identity, greatest_lower_bound(b, a)) -> greatest_lower_bound(b, a) % 87.08/86.43 least_upper_bound(identity, inverse(a)) -> identity % 87.08/86.43 least_upper_bound(identity, inverse(b)) -> identity % 87.08/86.43 least_upper_bound(identity, least_upper_bound(Y0, a)) -> least_upper_bound(Y0, a) % 87.08/86.43 least_upper_bound(identity, least_upper_bound(Y0, b)) -> least_upper_bound(Y0, b) % 87.08/86.43 least_upper_bound(identity, least_upper_bound(a, Y2)) -> least_upper_bound(a, Y2) % 87.08/86.43 least_upper_bound(identity, least_upper_bound(b, Y2)) -> least_upper_bound(b, Y2) % 87.08/86.43 least_upper_bound(identity, multiply(a, a)) -> multiply(a, a) % 87.08/86.43 least_upper_bound(identity, multiply(a, b)) -> multiply(a, b) % 87.08/86.43 least_upper_bound(identity, multiply(b, a)) -> multiply(b, a) % 87.08/86.43 least_upper_bound(inverse(a), b) -> b % 87.08/86.43 least_upper_bound(least_upper_bound(X, Y), Z) -> least_upper_bound(X, least_upper_bound(Y, Z)) % 87.08/86.43 multiply(X, greatest_lower_bound(Y, Z)) -> greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 87.08/86.43 multiply(X, least_upper_bound(Y, Z)) -> least_upper_bound(multiply(X, Y), multiply(X, Z)) % 87.08/86.43 multiply(X0, inverse(X0)) -> identity % 87.08/86.43 multiply(X0, multiply(inverse(X0), Y1)) -> Y1 % 87.08/86.43 multiply(Y0, identity) -> Y0 % 87.08/86.43 multiply(Y0, multiply(Y1, inverse(multiply(Y0, Y1)))) -> identity % 87.08/86.43 multiply(greatest_lower_bound(Y, Z), X) -> greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 87.08/86.43 multiply(greatest_lower_bound(Y0, identity), Y1) -> greatest_lower_bound(Y1, multiply(Y0, Y1)) % 87.08/86.43 multiply(identity, X) -> X % 87.08/86.43 multiply(inverse(X), X) -> identity % 87.08/86.43 multiply(inverse(Y1), multiply(Y1, Y2)) -> Y2 % 87.08/86.43 multiply(least_upper_bound(Y, Z), X) -> least_upper_bound(multiply(Y, X), multiply(Z, X)) % 87.08/86.43 multiply(multiply(X, Y), Z) -> multiply(X, multiply(Y, Z)) % 87.08/86.43 true__ -> false__ % 87.08/86.43 with the LPO induced by % 87.08/86.43 b > a > inverse > identity > multiply > least_upper_bound > greatest_lower_bound > f1 > true__ > false__ % 87.08/86.43 % 87.08/86.43 % SZS output end Proof % 87.08/86.43 %------------------------------------------------------------------------------