%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP171-2 : 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 10:53:41 EDT 2022 % Result : Unsatisfiable 10.02s 10.01s % Output : Proof 10.02s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : GRP171-2 : TPTP v8.1.0. Bugfixed v1.2.1. % 0.03/0.13 % Command : moca.sh %s % 0.12/0.34 % Computer : n007.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 : 300 % 0.12/0.34 % WCLimit : 600 % 0.12/0.34 % DateTime : Mon Jun 13 08:10:09 EDT 2022 % 0.12/0.34 % CPUTime : % 10.02/10.01 % SZS status Unsatisfiable % 10.02/10.01 % SZS output start Proof % 10.02/10.01 The input problem is unsatisfiable because % 10.02/10.01 % 10.02/10.01 [1] the following set of Horn clauses is unsatisfiable: % 10.02/10.01 % 10.02/10.01 multiply(identity, X) = X % 10.02/10.01 multiply(inverse(X), X) = identity % 10.02/10.01 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 10.02/10.01 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.02/10.01 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.02/10.01 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 10.02/10.01 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 10.02/10.01 least_upper_bound(X, X) = X % 10.02/10.01 greatest_lower_bound(X, X) = X % 10.02/10.01 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 10.02/10.01 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 10.02/10.01 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 10.02/10.01 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 10.02/10.01 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 10.02/10.01 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 10.02/10.01 least_upper_bound(identity, a) = a % 10.02/10.01 least_upper_bound(identity, b) = b % 10.02/10.01 greatest_lower_bound(identity, multiply(a, b)) = identity ==> \bottom % 10.02/10.01 % 10.02/10.01 This holds because % 10.02/10.01 % 10.02/10.01 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 10.02/10.01 % 10.02/10.01 E: % 10.02/10.01 f1(greatest_lower_bound(identity, multiply(a, b))) = true__ % 10.02/10.01 f1(identity) = false__ % 10.02/10.01 greatest_lower_bound(X, X) = X % 10.02/10.01 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.02/10.01 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 10.02/10.01 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 10.02/10.01 least_upper_bound(X, X) = X % 10.02/10.01 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.02/10.01 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 10.02/10.01 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 10.02/10.01 least_upper_bound(identity, a) = a % 10.02/10.01 least_upper_bound(identity, b) = b % 10.02/10.01 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 10.02/10.01 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 10.02/10.01 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 10.02/10.01 multiply(identity, X) = X % 10.02/10.01 multiply(inverse(X), X) = identity % 10.02/10.01 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 10.02/10.01 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 10.02/10.01 G: % 10.02/10.01 true__ = false__ % 10.02/10.01 % 10.02/10.01 This holds because % 10.02/10.01 % 10.02/10.01 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 10.02/10.01 % 10.02/10.01 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 10.02/10.01 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 10.02/10.01 least_upper_bound(Y1, multiply(Y0, Y1)) = multiply(least_upper_bound(Y0, identity), Y1) % 10.02/10.01 least_upper_bound(Y1, multiply(Y2, Y1)) = multiply(least_upper_bound(identity, Y2), Y1) % 10.02/10.01 least_upper_bound(identity, least_upper_bound(Y0, a)) = least_upper_bound(a, Y0) % 10.02/10.01 least_upper_bound(identity, least_upper_bound(Y0, b)) = least_upper_bound(b, Y0) % 10.02/10.01 f1(greatest_lower_bound(identity, multiply(a, b))) -> true__ % 10.02/10.01 f1(identity) -> false__ % 10.02/10.01 greatest_lower_bound(X, X) -> X % 10.02/10.01 greatest_lower_bound(X, least_upper_bound(X, Y)) -> X % 10.02/10.01 greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y0)) -> greatest_lower_bound(Y0, Y1) % 10.02/10.01 greatest_lower_bound(Y0, greatest_lower_bound(Y1, greatest_lower_bound(Y0, Y1))) -> greatest_lower_bound(Y0, Y1) % 10.02/10.01 greatest_lower_bound(Y0, least_upper_bound(Y1, Y0)) -> Y0 % 10.02/10.01 greatest_lower_bound(Y0, least_upper_bound(identity, least_upper_bound(Y0, a))) -> Y0 % 10.02/10.01 greatest_lower_bound(Y0, least_upper_bound(identity, least_upper_bound(Y0, b))) -> Y0 % 10.02/10.01 greatest_lower_bound(Y0, multiply(a, Y0)) -> Y0 % 10.02/10.01 greatest_lower_bound(Y0, multiply(b, Y0)) -> Y0 % 10.02/10.01 greatest_lower_bound(Y1, greatest_lower_bound(X1, Y1)) -> greatest_lower_bound(X1, Y1) % 10.02/10.01 greatest_lower_bound(Y1, greatest_lower_bound(Y1, Y2)) -> greatest_lower_bound(Y1, Y2) % 10.02/10.01 greatest_lower_bound(a, least_upper_bound(identity, least_upper_bound(Y1, a))) -> a % 10.02/10.01 greatest_lower_bound(b, least_upper_bound(identity, least_upper_bound(Y1, b))) -> b % 10.02/10.01 greatest_lower_bound(greatest_lower_bound(X, Y), Z) -> greatest_lower_bound(X, greatest_lower_bound(Y, Z)) % 10.02/10.01 greatest_lower_bound(identity, a) -> identity % 10.02/10.01 greatest_lower_bound(identity, b) -> identity % 10.02/10.01 greatest_lower_bound(identity, greatest_lower_bound(Y0, a)) -> greatest_lower_bound(identity, Y0) % 10.02/10.01 greatest_lower_bound(identity, greatest_lower_bound(Y0, b)) -> greatest_lower_bound(identity, Y0) % 10.02/10.01 greatest_lower_bound(identity, greatest_lower_bound(a, Y2)) -> greatest_lower_bound(identity, Y2) % 10.02/10.01 greatest_lower_bound(identity, greatest_lower_bound(b, Y2)) -> greatest_lower_bound(identity, Y2) % 10.02/10.01 greatest_lower_bound(identity, least_upper_bound(Y0, a)) -> identity % 10.02/10.01 greatest_lower_bound(identity, least_upper_bound(Y0, b)) -> identity % 10.02/10.01 greatest_lower_bound(identity, least_upper_bound(a, X0)) -> identity % 10.02/10.01 greatest_lower_bound(identity, least_upper_bound(b, X0)) -> identity % 10.02/10.01 greatest_lower_bound(identity, multiply(a, a)) -> identity % 10.02/10.01 greatest_lower_bound(identity, multiply(a, b)) -> identity % 10.02/10.01 greatest_lower_bound(identity, multiply(b, a)) -> identity % 10.02/10.01 greatest_lower_bound(identity, multiply(b, b)) -> identity % 10.02/10.01 greatest_lower_bound(least_upper_bound(X0, a), least_upper_bound(a, X0)) -> least_upper_bound(X0, a) % 10.02/10.01 greatest_lower_bound(least_upper_bound(X0, b), least_upper_bound(b, X0)) -> least_upper_bound(X0, b) % 10.02/10.01 greatest_lower_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, greatest_lower_bound(Y, Z)) % 10.02/10.01 greatest_lower_bound(multiply(Y, X), multiply(Z, X)) -> multiply(greatest_lower_bound(Y, Z), X) % 10.02/10.01 least_upper_bound(X, X) -> X % 10.02/10.01 least_upper_bound(X, greatest_lower_bound(X, Y)) -> X % 10.02/10.01 least_upper_bound(Y0, greatest_lower_bound(Y1, Y0)) -> Y0 % 10.02/10.01 least_upper_bound(Y0, least_upper_bound(Y1, Y0)) -> least_upper_bound(Y0, Y1) % 10.02/10.01 least_upper_bound(Y0, least_upper_bound(Y1, least_upper_bound(Y0, Y1))) -> least_upper_bound(Y0, Y1) % 10.02/10.01 least_upper_bound(Y1, least_upper_bound(X1, Y1)) -> least_upper_bound(X1, Y1) % 10.02/10.01 least_upper_bound(Y1, least_upper_bound(Y1, Y2)) -> least_upper_bound(Y1, Y2) % 10.02/10.01 least_upper_bound(Y1, multiply(a, Y1)) -> multiply(a, Y1) % 10.02/10.01 least_upper_bound(Y1, multiply(b, Y1)) -> multiply(b, Y1) % 10.02/10.01 least_upper_bound(greatest_lower_bound(X0, a), greatest_lower_bound(identity, X0)) -> greatest_lower_bound(X0, a) % 10.02/10.01 least_upper_bound(greatest_lower_bound(X0, b), greatest_lower_bound(identity, X0)) -> greatest_lower_bound(X0, b) % 10.02/10.01 least_upper_bound(greatest_lower_bound(identity, X0), greatest_lower_bound(a, X0)) -> greatest_lower_bound(a, X0) % 10.02/10.01 least_upper_bound(greatest_lower_bound(identity, X0), greatest_lower_bound(b, X0)) -> greatest_lower_bound(b, X0) % 10.02/10.01 least_upper_bound(identity, a) -> a % 10.02/10.01 least_upper_bound(identity, b) -> b % 10.02/10.01 least_upper_bound(identity, least_upper_bound(Y0, a)) -> least_upper_bound(Y0, a) % 10.02/10.01 least_upper_bound(identity, least_upper_bound(Y0, b)) -> least_upper_bound(Y0, b) % 10.02/10.01 least_upper_bound(identity, least_upper_bound(a, Y2)) -> least_upper_bound(a, Y2) % 10.02/10.01 least_upper_bound(identity, least_upper_bound(b, Y2)) -> least_upper_bound(b, Y2) % 10.02/10.01 least_upper_bound(identity, multiply(a, a)) -> multiply(a, a) % 10.02/10.01 least_upper_bound(identity, multiply(a, b)) -> multiply(a, b) % 10.02/10.01 least_upper_bound(identity, multiply(b, a)) -> multiply(b, a) % 10.02/10.01 least_upper_bound(identity, multiply(b, b)) -> multiply(b, b) % 10.02/10.01 least_upper_bound(least_upper_bound(X, Y), Z) -> least_upper_bound(X, least_upper_bound(Y, Z)) % 10.02/10.01 least_upper_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, least_upper_bound(Y, Z)) % 10.02/10.01 least_upper_bound(multiply(Y, X), multiply(Z, X)) -> multiply(least_upper_bound(Y, Z), X) % 10.02/10.01 multiply(identity, X) -> X % 10.02/10.01 multiply(inverse(X), X) -> identity % 10.02/10.01 multiply(inverse(Y1), multiply(Y1, Y2)) -> Y2 % 10.02/10.01 multiply(inverse(identity), Y1) -> Y1 % 10.02/10.01 multiply(inverse(inverse(X0)), X1) -> multiply(X0, X1) % 10.02/10.01 multiply(inverse(inverse(Y1)), identity) -> Y1 % 10.02/10.01 multiply(least_upper_bound(identity, inverse(Y0)), Y0) -> least_upper_bound(Y0, identity) % 10.02/10.01 multiply(multiply(X, Y), Z) -> multiply(X, multiply(Y, Z)) % 10.02/10.01 true__ -> false__ % 10.02/10.01 with the LPO induced by % 10.02/10.01 a > f1 > b > inverse > identity > least_upper_bound > greatest_lower_bound > multiply > true__ > false__ % 10.02/10.01 % 10.02/10.01 % SZS output end Proof % 10.02/10.01 %------------------------------------------------------------------------------