%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP141-1 : TPTP v8.1.0. Bugfixed v1.2.1. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n022.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 10:53:23 EDT 2022 % Result : Unsatisfiable 10.09s 10.04s % Output : Proof 10.09s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : GRP141-1 : TPTP v8.1.0. Bugfixed v1.2.1. % 0.11/0.13 % Command : moca.sh %s % 0.14/0.34 % Computer : n022.cluster.edu % 0.14/0.34 % Model : x86_64 x86_64 % 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.34 % Memory : 8042.1875MB % 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.34 % CPULimit : 300 % 0.14/0.34 % WCLimit : 600 % 0.14/0.34 % DateTime : Tue Jun 14 02:09:18 EDT 2022 % 0.14/0.34 % CPUTime : % 10.09/10.04 % SZS status Unsatisfiable % 10.09/10.04 % SZS output start Proof % 10.09/10.04 The input problem is unsatisfiable because % 10.09/10.04 % 10.09/10.04 [1] the following set of Horn clauses is unsatisfiable: % 10.09/10.04 % 10.09/10.04 multiply(identity, X) = X % 10.09/10.04 multiply(inverse(X), X) = identity % 10.09/10.04 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 10.09/10.04 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.09/10.04 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.09/10.04 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 10.09/10.04 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 10.09/10.04 least_upper_bound(X, X) = X % 10.09/10.04 greatest_lower_bound(X, X) = X % 10.09/10.04 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 10.09/10.04 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 10.09/10.04 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 10.09/10.04 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 10.09/10.04 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 10.09/10.04 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 10.09/10.04 least_upper_bound(a, c) = a % 10.09/10.04 least_upper_bound(b, c) = b % 10.09/10.04 greatest_lower_bound(greatest_lower_bound(a, b), c) = c ==> \bottom % 10.09/10.04 % 10.09/10.04 This holds because % 10.09/10.04 % 10.09/10.04 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 10.09/10.04 % 10.09/10.04 E: % 10.09/10.04 f1(c) = false__ % 10.09/10.04 f1(greatest_lower_bound(greatest_lower_bound(a, b), c)) = true__ % 10.09/10.04 greatest_lower_bound(X, X) = X % 10.09/10.04 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.09/10.04 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 10.09/10.04 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 10.09/10.04 least_upper_bound(X, X) = X % 10.09/10.04 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.09/10.04 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 10.09/10.04 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 10.09/10.04 least_upper_bound(a, c) = a % 10.09/10.04 least_upper_bound(b, c) = b % 10.09/10.04 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 10.09/10.04 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 10.09/10.04 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 10.09/10.04 multiply(identity, X) = X % 10.09/10.04 multiply(inverse(X), X) = identity % 10.09/10.04 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 10.09/10.04 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 10.09/10.04 G: % 10.09/10.04 true__ = false__ % 10.09/10.04 % 10.09/10.04 This holds because % 10.09/10.04 % 10.09/10.04 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 10.09/10.04 % 10.09/10.04 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.09/10.04 greatest_lower_bound(Y0, Y1) = greatest_lower_bound(Y1, greatest_lower_bound(least_upper_bound(X1, Y1), Y0)) % 10.09/10.04 greatest_lower_bound(Y0, Y1) = greatest_lower_bound(Y1, greatest_lower_bound(least_upper_bound(Y1, X1), Y0)) % 10.09/10.04 greatest_lower_bound(Y1, greatest_lower_bound(Y0, Y2)) = greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y2)) % 10.09/10.04 greatest_lower_bound(Y2, greatest_lower_bound(Y0, Y1)) = greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y2)) % 10.09/10.04 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.09/10.04 least_upper_bound(Y1, multiply(Y0, Y1)) = multiply(least_upper_bound(Y0, identity), Y1) % 10.09/10.04 least_upper_bound(Y1, multiply(Y2, Y1)) = multiply(least_upper_bound(identity, Y2), Y1) % 10.09/10.04 least_upper_bound(c, least_upper_bound(Y0, b)) = least_upper_bound(b, Y0) % 10.09/10.04 f1(c) -> false__ % 10.09/10.04 f1(greatest_lower_bound(c, greatest_lower_bound(a, b))) -> true__ % 10.09/10.04 f1(greatest_lower_bound(greatest_lower_bound(a, b), c)) -> true__ % 10.09/10.04 greatest_lower_bound(X, X) -> X % 10.09/10.04 greatest_lower_bound(X, least_upper_bound(X, Y)) -> X % 10.09/10.04 greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y0)) -> greatest_lower_bound(Y0, Y1) % 10.09/10.04 greatest_lower_bound(Y0, greatest_lower_bound(Y1, greatest_lower_bound(Y0, Y1))) -> greatest_lower_bound(Y0, Y1) % 10.09/10.04 greatest_lower_bound(Y0, least_upper_bound(Y1, Y0)) -> Y0 % 10.09/10.04 greatest_lower_bound(Y0, multiply(least_upper_bound(X1, identity), Y0)) -> Y0 % 10.09/10.04 greatest_lower_bound(Y0, multiply(least_upper_bound(identity, X1), Y0)) -> Y0 % 10.09/10.04 greatest_lower_bound(Y1, greatest_lower_bound(Y0, least_upper_bound(Y1, X1))) -> greatest_lower_bound(Y0, Y1) % 10.09/10.04 greatest_lower_bound(Y1, greatest_lower_bound(Y1, Y2)) -> greatest_lower_bound(Y1, Y2) % 10.09/10.04 greatest_lower_bound(Y1, greatest_lower_bound(least_upper_bound(Y1, Y2), Y0)) -> greatest_lower_bound(Y1, Y0) % 10.09/10.04 greatest_lower_bound(Y1, multiply(greatest_lower_bound(X1, identity), Y1)) -> multiply(greatest_lower_bound(X1, identity), Y1) % 10.09/10.04 greatest_lower_bound(Y1, multiply(greatest_lower_bound(identity, X1), Y1)) -> multiply(greatest_lower_bound(identity, X1), Y1) % 10.09/10.04 greatest_lower_bound(a, c) -> c % 10.09/10.04 greatest_lower_bound(a, greatest_lower_bound(X0, c)) -> greatest_lower_bound(X0, c) % 10.09/10.04 greatest_lower_bound(a, greatest_lower_bound(c, X0)) -> greatest_lower_bound(c, X0) % 10.09/10.04 greatest_lower_bound(a, greatest_lower_bound(least_upper_bound(false__, a), greatest_lower_bound(X0, c))) -> greatest_lower_bound(X0, c) % 10.09/10.04 greatest_lower_bound(a, greatest_lower_bound(least_upper_bound(false__, a), greatest_lower_bound(c, X0))) -> greatest_lower_bound(c, X0) % 10.09/10.04 greatest_lower_bound(b, least_upper_bound(c, least_upper_bound(Y1, b))) -> b % 10.09/10.04 greatest_lower_bound(c, b) -> c % 10.09/10.04 greatest_lower_bound(c, greatest_lower_bound(Y0, b)) -> greatest_lower_bound(c, Y0) % 10.09/10.04 greatest_lower_bound(c, greatest_lower_bound(b, Y2)) -> greatest_lower_bound(c, Y2) % 10.09/10.04 greatest_lower_bound(c, least_upper_bound(Y0, b)) -> c % 10.09/10.04 greatest_lower_bound(c, least_upper_bound(b, X0)) -> c % 10.09/10.04 greatest_lower_bound(greatest_lower_bound(X, Y), Z) -> greatest_lower_bound(X, greatest_lower_bound(Y, Z)) % 10.09/10.04 greatest_lower_bound(identity, greatest_lower_bound(least_upper_bound(false__, identity), inverse(least_upper_bound(Y1, identity)))) -> inverse(least_upper_bound(Y1, identity)) % 10.09/10.04 greatest_lower_bound(identity, greatest_lower_bound(least_upper_bound(false__, identity), inverse(least_upper_bound(identity, Y1)))) -> inverse(least_upper_bound(identity, Y1)) % 10.09/10.04 greatest_lower_bound(least_upper_bound(X0, b), least_upper_bound(b, X0)) -> least_upper_bound(X0, b) % 10.09/10.04 greatest_lower_bound(least_upper_bound(X0, c), least_upper_bound(a, X0)) -> least_upper_bound(X0, c) % 10.09/10.04 greatest_lower_bound(least_upper_bound(X0, c), least_upper_bound(b, X0)) -> least_upper_bound(X0, c) % 10.09/10.04 greatest_lower_bound(least_upper_bound(a, X0), least_upper_bound(c, X0)) -> least_upper_bound(c, X0) % 10.09/10.04 greatest_lower_bound(least_upper_bound(c, X0), least_upper_bound(b, X0)) -> least_upper_bound(c, X0) % 10.09/10.04 greatest_lower_bound(least_upper_bound(false__, a), c) -> c % 10.09/10.04 greatest_lower_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, greatest_lower_bound(Y, Z)) % 10.09/10.04 greatest_lower_bound(multiply(Y, X), multiply(Z, X)) -> multiply(greatest_lower_bound(Y, Z), X) % 10.09/10.04 inverse(identity) -> identity % 10.09/10.04 inverse(inverse(Y0)) -> Y0 % 10.09/10.04 inverse(inverse(identity)) -> identity % 10.09/10.04 least_upper_bound(X, X) -> X % 10.09/10.04 least_upper_bound(X, greatest_lower_bound(X, Y)) -> X % 10.09/10.04 least_upper_bound(Y0, greatest_lower_bound(X1, greatest_lower_bound(X2, Y0))) -> Y0 % 10.09/10.04 least_upper_bound(Y0, greatest_lower_bound(X1, greatest_lower_bound(Y0, X2))) -> Y0 % 10.09/10.04 least_upper_bound(Y0, greatest_lower_bound(Y1, Y0)) -> Y0 % 10.09/10.04 least_upper_bound(Y0, least_upper_bound(Y1, Y0)) -> least_upper_bound(Y0, Y1) % 10.09/10.04 least_upper_bound(Y0, least_upper_bound(Y1, least_upper_bound(Y0, Y1))) -> least_upper_bound(Y0, Y1) % 10.09/10.04 least_upper_bound(Y1, least_upper_bound(Y1, Y2)) -> least_upper_bound(Y1, Y2) % 10.09/10.04 least_upper_bound(Y1, multiply(greatest_lower_bound(X1, identity), Y1)) -> Y1 % 10.09/10.04 least_upper_bound(Y1, multiply(greatest_lower_bound(identity, X1), Y1)) -> Y1 % 10.09/10.04 least_upper_bound(Y1, multiply(least_upper_bound(X1, identity), Y1)) -> multiply(least_upper_bound(X1, identity), Y1) % 10.09/10.04 least_upper_bound(Y1, multiply(least_upper_bound(identity, X1), Y1)) -> multiply(least_upper_bound(identity, X1), Y1) % 10.09/10.04 least_upper_bound(Y2, least_upper_bound(X1, greatest_lower_bound(Y1, Y2))) -> least_upper_bound(Y2, X1) % 10.09/10.04 least_upper_bound(a, c) -> a % 10.09/10.04 least_upper_bound(a, greatest_lower_bound(Y0, c)) -> a % 10.09/10.04 least_upper_bound(a, greatest_lower_bound(c, X1)) -> a % 10.09/10.04 least_upper_bound(a, least_upper_bound(Y0, c)) -> least_upper_bound(a, Y0) % 10.09/10.04 least_upper_bound(a, least_upper_bound(c, Y2)) -> least_upper_bound(a, Y2) % 10.09/10.04 least_upper_bound(b, c) -> b % 10.09/10.04 least_upper_bound(b, greatest_lower_bound(Y0, c)) -> b % 10.09/10.04 least_upper_bound(b, greatest_lower_bound(c, X1)) -> b % 10.09/10.04 least_upper_bound(b, least_upper_bound(Y0, c)) -> least_upper_bound(b, Y0) % 10.09/10.04 least_upper_bound(b, least_upper_bound(c, Y2)) -> least_upper_bound(b, Y2) % 10.09/10.04 least_upper_bound(c, b) -> b % 10.09/10.04 least_upper_bound(c, least_upper_bound(Y0, b)) -> least_upper_bound(Y0, b) % 10.09/10.04 least_upper_bound(c, least_upper_bound(b, Y2)) -> least_upper_bound(b, Y2) % 10.09/10.04 least_upper_bound(greatest_lower_bound(X0, b), greatest_lower_bound(c, X0)) -> greatest_lower_bound(X0, b) % 10.09/10.04 least_upper_bound(greatest_lower_bound(X1, X2), greatest_lower_bound(X1, greatest_lower_bound(X2, Y1))) -> greatest_lower_bound(X1, X2) % 10.09/10.04 least_upper_bound(greatest_lower_bound(X1, X2), greatest_lower_bound(X1, greatest_lower_bound(Y1, X2))) -> greatest_lower_bound(X1, X2) % 10.09/10.04 least_upper_bound(greatest_lower_bound(Y0, c), b) -> b % 10.09/10.04 least_upper_bound(greatest_lower_bound(c, X0), greatest_lower_bound(b, X0)) -> greatest_lower_bound(b, X0) % 10.09/10.04 least_upper_bound(greatest_lower_bound(c, Y0), b) -> b % 10.09/10.04 least_upper_bound(identity, inverse(X0)) -> multiply(least_upper_bound(identity, X0), inverse(X0)) % 10.09/10.04 least_upper_bound(identity, inverse(Y1)) -> multiply(least_upper_bound(Y1, identity), inverse(Y1)) % 10.09/10.04 least_upper_bound(identity, inverse(greatest_lower_bound(Y1, identity))) -> inverse(greatest_lower_bound(Y1, identity)) % 10.09/10.04 least_upper_bound(identity, inverse(greatest_lower_bound(identity, Y1))) -> inverse(greatest_lower_bound(identity, Y1)) % 10.09/10.04 least_upper_bound(identity, inverse(identity)) -> identity % 10.09/10.04 least_upper_bound(least_upper_bound(X, Y), Z) -> least_upper_bound(X, least_upper_bound(Y, Z)) % 10.09/10.04 least_upper_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, least_upper_bound(Y, Z)) % 10.09/10.04 least_upper_bound(multiply(Y, X), multiply(Z, X)) -> multiply(least_upper_bound(Y, Z), X) % 10.09/10.04 multiply(X0, multiply(inverse(X0), Y1)) -> Y1 % 10.09/10.04 multiply(Y0, inverse(Y0)) -> identity % 10.09/10.04 multiply(Y0, multiply(Y1, inverse(multiply(Y0, Y1)))) -> identity % 10.09/10.04 multiply(Y1, identity) -> Y1 % 10.09/10.04 multiply(identity, X) -> X % 10.09/10.04 multiply(inverse(X), X) -> identity % 10.09/10.04 multiply(inverse(Y1), multiply(Y1, Y2)) -> Y2 % 10.09/10.04 multiply(inverse(identity), Y1) -> Y1 % 10.09/10.04 multiply(inverse(inverse(X0)), X1) -> multiply(X0, X1) % 10.09/10.04 multiply(inverse(inverse(Y1)), identity) -> Y1 % 10.09/10.04 multiply(least_upper_bound(identity, inverse(Y0)), Y0) -> least_upper_bound(Y0, identity) % 10.09/10.04 multiply(least_upper_bound(identity, inverse(identity)), Y0) -> Y0 % 10.09/10.04 multiply(multiply(X, Y), Z) -> multiply(X, multiply(Y, Z)) % 10.09/10.04 true__ -> false__ % 10.09/10.04 with the LPO induced by % 10.09/10.04 f1 > b > c > a > inverse > identity > least_upper_bound > greatest_lower_bound > multiply > true__ > false__ % 10.09/10.04 % 10.09/10.04 % SZS output end Proof % 10.09/10.04 %------------------------------------------------------------------------------