%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP166-3 : TPTP v8.1.0. Bugfixed v1.2.1. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n021.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 3.93s 3.98s % Output : Proof 3.93s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.05/0.13 % Problem : GRP166-3 : TPTP v8.1.0. Bugfixed v1.2.1. % 0.05/0.13 % Command : moca.sh %s % 0.14/0.35 % Computer : n021.cluster.edu % 0.14/0.35 % Model : x86_64 x86_64 % 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.35 % Memory : 8042.1875MB % 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 600 % 0.14/0.35 % DateTime : Tue Jun 14 01:49:12 EDT 2022 % 0.14/0.35 % CPUTime : % 3.93/3.98 % SZS status Unsatisfiable % 3.93/3.98 % SZS output start Proof % 3.93/3.98 The input problem is unsatisfiable because % 3.93/3.98 % 3.93/3.98 [1] the following set of Horn clauses is unsatisfiable: % 3.93/3.98 % 3.93/3.98 multiply(identity, X) = X % 3.93/3.98 multiply(inverse(X), X) = identity % 3.93/3.98 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 3.93/3.98 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 3.93/3.98 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 3.93/3.98 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 3.93/3.98 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 3.93/3.98 least_upper_bound(X, X) = X % 3.93/3.98 greatest_lower_bound(X, X) = X % 3.93/3.98 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 3.93/3.98 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 3.93/3.98 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 3.93/3.98 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 3.93/3.98 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 3.93/3.98 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 3.93/3.98 least_upper_bound(a, identity) = a % 3.93/3.98 least_upper_bound(b, identity) = b % 3.93/3.98 least_upper_bound(a, multiply(b, a)) = multiply(b, a) ==> \bottom % 3.93/3.98 % 3.93/3.98 This holds because % 3.93/3.98 % 3.93/3.98 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 3.93/3.98 % 3.93/3.98 E: % 3.93/3.98 f1(least_upper_bound(a, multiply(b, a))) = true__ % 3.93/3.98 f1(multiply(b, a)) = false__ % 3.93/3.98 greatest_lower_bound(X, X) = X % 3.93/3.98 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 3.93/3.98 greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z) % 3.93/3.98 greatest_lower_bound(X, least_upper_bound(X, Y)) = X % 3.93/3.98 least_upper_bound(X, X) = X % 3.93/3.98 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 3.93/3.98 least_upper_bound(X, greatest_lower_bound(X, Y)) = X % 3.93/3.98 least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z) % 3.93/3.98 least_upper_bound(a, identity) = a % 3.93/3.98 least_upper_bound(b, identity) = b % 3.93/3.98 multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)) % 3.93/3.98 multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)) % 3.93/3.98 multiply(greatest_lower_bound(Y, Z), X) = greatest_lower_bound(multiply(Y, X), multiply(Z, X)) % 3.93/3.98 multiply(identity, X) = X % 3.93/3.98 multiply(inverse(X), X) = identity % 3.93/3.98 multiply(least_upper_bound(Y, Z), X) = least_upper_bound(multiply(Y, X), multiply(Z, X)) % 3.93/3.98 multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)) % 3.93/3.98 G: % 3.93/3.98 true__ = false__ % 3.93/3.98 % 3.93/3.98 This holds because % 3.93/3.98 % 3.93/3.98 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 3.93/3.98 % 3.93/3.98 greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X) % 3.93/3.98 greatest_lower_bound(X1, multiply(X0, X1)) = multiply(greatest_lower_bound(identity, inverse(X0)), multiply(X0, X1)) % 3.93/3.98 greatest_lower_bound(Y0, Y1) = greatest_lower_bound(Y1, greatest_lower_bound(least_upper_bound(X1, Y1), Y0)) % 3.93/3.98 greatest_lower_bound(Y0, multiply(X0, Y0)) = multiply(greatest_lower_bound(identity, greatest_lower_bound(X0, identity)), Y0) % 3.93/3.98 greatest_lower_bound(Y0, multiply(greatest_lower_bound(X1, identity), Y1)) = greatest_lower_bound(Y1, greatest_lower_bound(Y0, multiply(X1, Y1))) % 3.93/3.98 greatest_lower_bound(Y0, multiply(greatest_lower_bound(X1, identity), Y1)) = greatest_lower_bound(Y1, greatest_lower_bound(multiply(X1, Y1), Y0)) % 3.93/3.98 greatest_lower_bound(Y0, multiply(greatest_lower_bound(identity, X1), Y1)) = greatest_lower_bound(Y1, greatest_lower_bound(Y0, multiply(X1, Y1))) % 3.93/3.98 greatest_lower_bound(Y0, multiply(greatest_lower_bound(identity, X1), Y1)) = greatest_lower_bound(Y1, greatest_lower_bound(multiply(X1, Y1), Y0)) % 3.93/3.98 greatest_lower_bound(Y1, greatest_lower_bound(Y0, Y2)) = greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y2)) % 3.93/3.98 greatest_lower_bound(Y1, multiply(Y0, Y1)) = multiply(greatest_lower_bound(Y0, identity), Y1) % 3.93/3.98 greatest_lower_bound(Y1, multiply(Y2, Y1)) = multiply(greatest_lower_bound(identity, Y2), Y1) % 3.93/3.98 greatest_lower_bound(Y2, greatest_lower_bound(Y0, Y1)) = greatest_lower_bound(Y0, greatest_lower_bound(Y1, Y2)) % 3.93/3.98 greatest_lower_bound(identity, multiply(Y0, Y1)) = multiply(greatest_lower_bound(Y0, inverse(Y1)), Y1) % 3.93/3.98 greatest_lower_bound(identity, multiply(inverse(Y1), Y2)) = multiply(inverse(Y1), greatest_lower_bound(Y1, Y2)) % 3.93/3.98 greatest_lower_bound(identity, multiply(inverse(Y2), Y1)) = multiply(inverse(Y2), greatest_lower_bound(Y1, Y2)) % 3.93/3.98 greatest_lower_bound(multiply(Y0, Y1), identity) = multiply(greatest_lower_bound(Y0, inverse(Y1)), Y1) % 3.93/3.98 least_upper_bound(X, Y) = least_upper_bound(Y, X) % 3.93/3.98 least_upper_bound(X1, multiply(X0, X1)) = multiply(least_upper_bound(identity, inverse(X0)), multiply(X0, X1)) % 3.93/3.98 least_upper_bound(Y0, Y1) = least_upper_bound(Y1, least_upper_bound(greatest_lower_bound(X1, Y1), Y0)) % 3.93/3.98 least_upper_bound(Y0, a) = least_upper_bound(identity, least_upper_bound(a, Y0)) % 3.93/3.98 least_upper_bound(Y0, b) = least_upper_bound(identity, least_upper_bound(b, Y0)) % 3.93/3.98 least_upper_bound(Y0, least_upper_bound(a, X0)) = least_upper_bound(a, least_upper_bound(Y0, least_upper_bound(identity, X0))) % 3.93/3.98 least_upper_bound(Y0, least_upper_bound(a, X0)) = least_upper_bound(identity, least_upper_bound(X0, least_upper_bound(Y0, a))) % 3.93/3.98 least_upper_bound(Y0, least_upper_bound(a, X0)) = least_upper_bound(identity, least_upper_bound(a, least_upper_bound(X0, Y0))) % 3.93/3.98 least_upper_bound(Y0, least_upper_bound(b, X0)) = least_upper_bound(b, least_upper_bound(Y0, least_upper_bound(identity, X0))) % 3.93/3.98 least_upper_bound(Y0, least_upper_bound(b, X0)) = least_upper_bound(identity, least_upper_bound(X0, least_upper_bound(Y0, b))) % 3.93/3.98 least_upper_bound(Y0, least_upper_bound(b, X0)) = least_upper_bound(identity, least_upper_bound(b, least_upper_bound(X0, Y0))) % 3.93/3.98 least_upper_bound(Y0, multiply(X0, Y0)) = multiply(least_upper_bound(identity, least_upper_bound(X0, identity)), Y0) % 3.93/3.98 least_upper_bound(Y0, multiply(X0, multiply(X1, Y0))) = multiply(least_upper_bound(identity, multiply(X0, X1)), Y0) % 3.93/3.98 least_upper_bound(Y0, multiply(least_upper_bound(X1, identity), Y1)) = least_upper_bound(Y1, least_upper_bound(Y0, multiply(X1, Y1))) % 3.93/3.98 least_upper_bound(Y0, multiply(least_upper_bound(X1, identity), Y1)) = least_upper_bound(Y1, least_upper_bound(multiply(X1, Y1), Y0)) % 3.93/3.98 least_upper_bound(Y0, multiply(least_upper_bound(identity, X1), Y1)) = least_upper_bound(Y1, least_upper_bound(Y0, multiply(X1, Y1))) % 3.93/3.98 least_upper_bound(Y0, multiply(least_upper_bound(identity, X1), Y1)) = least_upper_bound(Y1, least_upper_bound(multiply(X1, Y1), Y0)) % 3.93/3.98 least_upper_bound(Y1, least_upper_bound(Y0, Y2)) = least_upper_bound(Y0, least_upper_bound(Y1, Y2)) % 3.93/3.98 least_upper_bound(Y1, multiply(Y0, Y1)) = multiply(least_upper_bound(Y0, identity), Y1) % 3.93/3.98 least_upper_bound(Y1, multiply(Y2, Y1)) = multiply(least_upper_bound(identity, Y2), Y1) % 3.93/3.98 least_upper_bound(Y2, least_upper_bound(Y0, Y1)) = least_upper_bound(Y0, least_upper_bound(Y1, Y2)) % 3.93/3.98 least_upper_bound(a, least_upper_bound(X2, least_upper_bound(identity, X1))) = least_upper_bound(a, least_upper_bound(X1, X2)) % 3.93/3.98 least_upper_bound(b, least_upper_bound(X2, least_upper_bound(identity, X1))) = least_upper_bound(b, least_upper_bound(X1, X2)) % 3.93/3.98 least_upper_bound(identity, least_upper_bound(Y0, a)) = least_upper_bound(a, Y0) % 3.93/3.98 least_upper_bound(identity, least_upper_bound(Y0, b)) = least_upper_bound(b, Y0) % 3.93/3.98 least_upper_bound(identity, multiply(Y0, Y1)) = multiply(least_upper_bound(Y0, inverse(Y1)), Y1) % 3.93/3.98 least_upper_bound(identity, multiply(inverse(Y1), Y2)) = multiply(inverse(Y1), least_upper_bound(Y1, Y2)) % 3.93/3.98 least_upper_bound(identity, multiply(inverse(Y2), Y1)) = multiply(inverse(Y2), least_upper_bound(Y1, Y2)) % 3.93/3.98 least_upper_bound(multiply(Y0, Y1), identity) = multiply(least_upper_bound(Y0, inverse(Y1)), Y1) % 3.93/3.98 multiply(greatest_lower_bound(X1, greatest_lower_bound(X2, identity)), Y1) = greatest_lower_bound(Y1, multiply(greatest_lower_bound(X1, X2), Y1)) % 3.93/3.98 multiply(greatest_lower_bound(X1, greatest_lower_bound(identity, X2)), Y1) = greatest_lower_bound(Y1, multiply(greatest_lower_bound(X1, X2), Y1)) % 3.93/3.98 multiply(least_upper_bound(X1, least_upper_bound(X2, identity)), Y1) = least_upper_bound(Y1, multiply(least_upper_bound(X1, X2), Y1)) % 3.93/3.98 f1(least_upper_bound(a, multiply(b, a))) -> true__ % 3.93/3.99 f1(multiply(b, a)) -> false__ % 3.93/3.99 greatest_lower_bound(X, X) -> X % 3.93/3.99 greatest_lower_bound(X, least_upper_bound(X, Y)) -> X % 3.93/3.99 greatest_lower_bound(Y0, greatest_lower_bound(Y1, greatest_lower_bound(Y0, Y1))) -> greatest_lower_bound(Y0, Y1) % 3.93/3.99 greatest_lower_bound(Y0, greatest_lower_bound(Y1, least_upper_bound(X1, greatest_lower_bound(Y0, Y1)))) -> greatest_lower_bound(Y0, Y1) % 3.93/3.99 greatest_lower_bound(Y0, greatest_lower_bound(Y1, least_upper_bound(greatest_lower_bound(Y0, Y1), X1))) -> greatest_lower_bound(Y0, Y1) % 3.93/3.99 greatest_lower_bound(Y0, greatest_lower_bound(Y2, least_upper_bound(Y0, Y1))) -> greatest_lower_bound(Y0, Y2) % 3.93/3.99 greatest_lower_bound(Y0, greatest_lower_bound(least_upper_bound(X1, Y0), Y2)) -> greatest_lower_bound(Y0, Y2) % 3.93/3.99 greatest_lower_bound(Y0, greatest_lower_bound(least_upper_bound(Y0, X1), Y2)) -> greatest_lower_bound(Y0, Y2) % 3.93/3.99 greatest_lower_bound(Y0, least_upper_bound(X1, least_upper_bound(X2, Y0))) -> Y0 % 3.93/3.99 greatest_lower_bound(Y0, least_upper_bound(X1, least_upper_bound(Y0, X2))) -> Y0 % 3.93/3.99 greatest_lower_bound(Y0, least_upper_bound(Y1, Y0)) -> Y0 % 3.93/3.99 greatest_lower_bound(Y0, multiply(greatest_lower_bound(X1, identity), Y0)) -> greatest_lower_bound(Y0, multiply(X1, Y0)) % 3.93/3.99 greatest_lower_bound(Y0, multiply(greatest_lower_bound(identity, X1), Y0)) -> greatest_lower_bound(Y0, multiply(X1, Y0)) % 3.93/3.99 greatest_lower_bound(Y0, multiply(least_upper_bound(X1, identity), Y0)) -> Y0 % 3.93/3.99 greatest_lower_bound(Y0, multiply(least_upper_bound(identity, X1), Y0)) -> Y0 % 3.93/3.99 greatest_lower_bound(Y1, greatest_lower_bound(Y0, least_upper_bound(X1, Y1))) -> greatest_lower_bound(Y0, Y1) % 3.93/3.99 greatest_lower_bound(Y1, greatest_lower_bound(Y0, least_upper_bound(Y1, X1))) -> greatest_lower_bound(Y0, Y1) % 3.93/3.99 greatest_lower_bound(Y1, greatest_lower_bound(Y1, Y2)) -> greatest_lower_bound(Y1, Y2) % 3.93/3.99 greatest_lower_bound(Y1, multiply(greatest_lower_bound(identity, X1), Y1)) -> multiply(greatest_lower_bound(identity, X1), Y1) % 3.93/3.99 greatest_lower_bound(greatest_lower_bound(X, Y), Z) -> greatest_lower_bound(X, greatest_lower_bound(Y, Z)) % 3.93/3.99 greatest_lower_bound(identity, a) -> identity % 3.93/3.99 greatest_lower_bound(identity, b) -> identity % 3.93/3.99 greatest_lower_bound(identity, greatest_lower_bound(a, Y2)) -> greatest_lower_bound(identity, Y2) % 3.93/3.99 greatest_lower_bound(identity, greatest_lower_bound(b, Y2)) -> greatest_lower_bound(identity, Y2) % 3.93/3.99 greatest_lower_bound(identity, multiply(least_upper_bound(X0, inverse(X1)), X1)) -> identity % 3.93/3.99 greatest_lower_bound(least_upper_bound(X0, X1), least_upper_bound(X0, least_upper_bound(X1, Y1))) -> least_upper_bound(X0, X1) % 3.93/3.99 greatest_lower_bound(least_upper_bound(X1, X2), least_upper_bound(X1, least_upper_bound(Y1, X2))) -> least_upper_bound(X1, X2) % 3.93/3.99 greatest_lower_bound(least_upper_bound(X1, X2), least_upper_bound(X2, least_upper_bound(Y1, X1))) -> least_upper_bound(X1, X2) % 3.93/3.99 greatest_lower_bound(least_upper_bound(Y1, X2), least_upper_bound(greatest_lower_bound(Y1, X1), X2)) -> least_upper_bound(greatest_lower_bound(Y1, X1), X2) % 3.93/3.99 greatest_lower_bound(least_upper_bound(identity, X0), least_upper_bound(a, X0)) -> least_upper_bound(identity, X0) % 3.93/3.99 greatest_lower_bound(least_upper_bound(identity, X0), least_upper_bound(b, X0)) -> least_upper_bound(identity, X0) % 3.93/3.99 greatest_lower_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, greatest_lower_bound(Y, Z)) % 3.93/3.99 greatest_lower_bound(multiply(X0, greatest_lower_bound(X1, X2)), Y2) -> greatest_lower_bound(multiply(X0, X1), greatest_lower_bound(multiply(X0, X2), Y2)) % 3.93/3.99 greatest_lower_bound(multiply(X0, multiply(X1, Y1)), multiply(Y2, Y1)) -> multiply(greatest_lower_bound(multiply(X0, X1), Y2), Y1) % 3.93/3.99 greatest_lower_bound(multiply(Y, X), multiply(Z, X)) -> multiply(greatest_lower_bound(Y, Z), X) % 3.93/3.99 greatest_lower_bound(multiply(Y0, Y1), multiply(X0, multiply(X1, Y1))) -> multiply(greatest_lower_bound(Y0, multiply(X0, X1)), Y1) % 3.93/3.99 greatest_lower_bound(multiply(greatest_lower_bound(X0, X2), X1), Y2) -> greatest_lower_bound(multiply(X0, X1), greatest_lower_bound(multiply(X2, X1), Y2)) % 3.93/3.99 least_upper_bound(X, X) -> X % 3.93/3.99 least_upper_bound(X, greatest_lower_bound(X, Y)) -> X % 3.93/3.99 least_upper_bound(Y0, greatest_lower_bound(X1, greatest_lower_bound(X2, Y0))) -> Y0 % 3.93/3.99 least_upper_bound(Y0, greatest_lower_bound(X1, greatest_lower_bound(Y0, X2))) -> Y0 % 3.93/3.99 least_upper_bound(Y0, greatest_lower_bound(Y1, Y0)) -> Y0 % 3.93/3.99 least_upper_bound(Y0, least_upper_bound(Y1, greatest_lower_bound(X1, least_upper_bound(Y0, Y1)))) -> least_upper_bound(Y0, Y1) % 3.93/3.99 least_upper_bound(Y0, least_upper_bound(Y1, greatest_lower_bound(least_upper_bound(Y0, Y1), X1))) -> least_upper_bound(Y0, Y1) % 3.93/3.99 least_upper_bound(Y0, least_upper_bound(Y1, least_upper_bound(Y0, Y1))) -> least_upper_bound(Y0, Y1) % 3.93/3.99 least_upper_bound(Y0, least_upper_bound(Y2, greatest_lower_bound(Y0, Y1))) -> least_upper_bound(Y0, Y2) % 3.93/3.99 least_upper_bound(Y0, least_upper_bound(greatest_lower_bound(Y0, X1), Y2)) -> least_upper_bound(Y0, Y2) % 3.93/3.99 least_upper_bound(Y0, least_upper_bound(greatest_lower_bound(Y1, Y0), Y2)) -> least_upper_bound(Y0, Y2) % 3.93/3.99 least_upper_bound(Y0, multiply(greatest_lower_bound(X1, identity), Y0)) -> Y0 % 3.93/3.99 least_upper_bound(Y0, multiply(greatest_lower_bound(identity, X1), Y0)) -> Y0 % 3.93/3.99 least_upper_bound(Y0, multiply(least_upper_bound(X1, identity), Y0)) -> least_upper_bound(Y0, multiply(X1, Y0)) % 3.93/3.99 least_upper_bound(Y0, multiply(least_upper_bound(identity, X1), Y0)) -> least_upper_bound(Y0, multiply(X1, Y0)) % 3.93/3.99 least_upper_bound(Y1, least_upper_bound(Y0, greatest_lower_bound(X1, Y1))) -> least_upper_bound(Y0, Y1) % 3.93/3.99 least_upper_bound(Y1, least_upper_bound(Y0, greatest_lower_bound(Y1, X1))) -> least_upper_bound(Y0, Y1) % 3.93/3.99 least_upper_bound(Y1, least_upper_bound(Y1, Y2)) -> least_upper_bound(Y1, Y2) % 3.93/3.99 least_upper_bound(Y1, multiply(a, Y1)) -> multiply(a, Y1) % 3.93/3.99 least_upper_bound(Y1, multiply(b, Y1)) -> multiply(b, Y1) % 3.93/3.99 least_upper_bound(Y1, multiply(least_upper_bound(identity, X1), Y1)) -> multiply(least_upper_bound(identity, X1), Y1) % 3.93/3.99 least_upper_bound(a, greatest_lower_bound(X1, identity)) -> a % 3.93/3.99 least_upper_bound(a, greatest_lower_bound(identity, X1)) -> a % 3.93/3.99 least_upper_bound(a, identity) -> a % 3.93/3.99 least_upper_bound(a, least_upper_bound(X0, least_upper_bound(identity, least_upper_bound(X0, a)))) -> least_upper_bound(identity, least_upper_bound(a, X0)) % 3.93/3.99 least_upper_bound(a, least_upper_bound(X1, least_upper_bound(X2, identity))) -> least_upper_bound(a, least_upper_bound(X1, X2)) % 3.93/3.99 least_upper_bound(a, least_upper_bound(X1, least_upper_bound(identity, X2))) -> least_upper_bound(a, least_upper_bound(X1, X2)) % 3.93/3.99 least_upper_bound(a, least_upper_bound(identity, Y2)) -> least_upper_bound(a, Y2) % 3.93/3.99 least_upper_bound(a, multiply(least_upper_bound(X1, identity), identity)) -> least_upper_bound(a, multiply(X1, identity)) % 3.93/3.99 least_upper_bound(a, multiply(least_upper_bound(identity, X1), identity)) -> least_upper_bound(a, multiply(X1, identity)) % 3.93/3.99 least_upper_bound(b, greatest_lower_bound(X1, identity)) -> b % 3.93/3.99 least_upper_bound(b, greatest_lower_bound(identity, X1)) -> b % 3.93/3.99 least_upper_bound(b, identity) -> b % 3.93/3.99 least_upper_bound(b, least_upper_bound(X1, least_upper_bound(X2, identity))) -> least_upper_bound(b, least_upper_bound(X1, X2)) % 3.93/3.99 least_upper_bound(b, least_upper_bound(X1, least_upper_bound(identity, X2))) -> least_upper_bound(b, least_upper_bound(X1, X2)) % 3.93/3.99 least_upper_bound(b, least_upper_bound(identity, Y2)) -> least_upper_bound(b, Y2) % 3.93/3.99 least_upper_bound(b, multiply(least_upper_bound(X1, identity), identity)) -> least_upper_bound(b, multiply(X1, identity)) % 3.93/3.99 least_upper_bound(b, multiply(least_upper_bound(identity, X1), identity)) -> least_upper_bound(b, multiply(X1, identity)) % 3.93/3.99 least_upper_bound(greatest_lower_bound(X0, X1), greatest_lower_bound(X0, greatest_lower_bound(X1, Y1))) -> greatest_lower_bound(X0, X1) % 3.93/3.99 least_upper_bound(greatest_lower_bound(X1, X2), greatest_lower_bound(X1, greatest_lower_bound(Y1, X2))) -> greatest_lower_bound(X1, X2) % 3.93/3.99 least_upper_bound(greatest_lower_bound(X1, X2), greatest_lower_bound(X2, greatest_lower_bound(Y1, X1))) -> greatest_lower_bound(X1, X2) % 3.93/3.99 least_upper_bound(greatest_lower_bound(Y1, X2), greatest_lower_bound(least_upper_bound(Y1, X1), X2)) -> greatest_lower_bound(least_upper_bound(Y1, X1), X2) % 3.93/3.99 least_upper_bound(identity, a) -> a % 3.93/3.99 least_upper_bound(identity, b) -> b % 3.93/3.99 least_upper_bound(identity, least_upper_bound(Y0, a)) -> least_upper_bound(Y0, a) % 3.93/3.99 least_upper_bound(identity, least_upper_bound(Y0, b)) -> least_upper_bound(Y0, b) % 3.93/3.99 least_upper_bound(identity, least_upper_bound(a, Y0)) -> least_upper_bound(a, Y0) % 3.93/3.99 least_upper_bound(identity, least_upper_bound(a, least_upper_bound(X0, Y2))) -> least_upper_bound(a, least_upper_bound(X0, Y2)) % 3.93/3.99 least_upper_bound(identity, least_upper_bound(b, Y0)) -> least_upper_bound(b, Y0) % 3.93/3.99 least_upper_bound(identity, least_upper_bound(b, least_upper_bound(X0, Y2))) -> least_upper_bound(b, least_upper_bound(X0, Y2)) % 3.93/3.99 least_upper_bound(identity, multiply(greatest_lower_bound(X0, inverse(X1)), X1)) -> identity % 3.93/3.99 least_upper_bound(least_upper_bound(X, Y), Z) -> least_upper_bound(X, least_upper_bound(Y, Z)) % 3.93/3.99 least_upper_bound(multiply(X, Y), multiply(X, Z)) -> multiply(X, least_upper_bound(Y, Z)) % 3.93/3.99 least_upper_bound(multiply(X0, least_upper_bound(X1, X2)), Y2) -> least_upper_bound(multiply(X0, X1), least_upper_bound(multiply(X0, X2), Y2)) % 3.93/3.99 least_upper_bound(multiply(X0, multiply(X1, Y1)), multiply(Y2, Y1)) -> multiply(least_upper_bound(multiply(X0, X1), Y2), Y1) % 3.93/3.99 least_upper_bound(multiply(Y, X), multiply(Z, X)) -> multiply(least_upper_bound(Y, Z), X) % 3.93/3.99 least_upper_bound(multiply(Y0, Y1), multiply(X0, multiply(X1, Y1))) -> multiply(least_upper_bound(Y0, multiply(X0, X1)), Y1) % 3.93/3.99 least_upper_bound(multiply(least_upper_bound(X0, X2), X1), Y2) -> least_upper_bound(multiply(X0, X1), least_upper_bound(multiply(X2, X1), Y2)) % 3.93/3.99 multiply(greatest_lower_bound(inverse(Y1), Y2), Y1) -> greatest_lower_bound(identity, multiply(Y2, Y1)) % 3.93/3.99 multiply(identity, X) -> X % 3.93/3.99 multiply(inverse(X), X) -> identity % 3.93/3.99 multiply(inverse(Y1), multiply(Y1, Y2)) -> Y2 % 3.93/3.99 multiply(inverse(greatest_lower_bound(X0, identity)), greatest_lower_bound(Y1, multiply(X0, Y1))) -> Y1 % 3.93/3.99 multiply(inverse(greatest_lower_bound(identity, X0)), greatest_lower_bound(Y1, multiply(X0, Y1))) -> Y1 % 3.93/3.99 multiply(inverse(identity), Y1) -> Y1 % 3.93/3.99 multiply(inverse(inverse(X0)), X1) -> multiply(X0, X1) % 3.93/3.99 multiply(inverse(inverse(Y1)), identity) -> Y1 % 3.93/3.99 multiply(inverse(least_upper_bound(X0, identity)), least_upper_bound(Y1, multiply(X0, Y1))) -> Y1 % 3.93/3.99 multiply(inverse(least_upper_bound(identity, X0)), least_upper_bound(Y1, multiply(X0, Y1))) -> Y1 % 3.93/3.99 multiply(inverse(multiply(X0, X1)), multiply(X0, multiply(X1, Y1))) -> Y1 % 3.93/3.99 multiply(least_upper_bound(inverse(Y1), Y2), Y1) -> least_upper_bound(identity, multiply(Y2, Y1)) % 3.93/3.99 multiply(multiply(X, Y), Z) -> multiply(X, multiply(Y, Z)) % 3.93/3.99 true__ -> false__ % 3.93/3.99 with the LPO induced by % 3.93/3.99 a > f1 > b > inverse > identity > least_upper_bound > greatest_lower_bound > multiply > true__ > false__ % 3.93/3.99 % 3.93/3.99 % SZS output end Proof % 3.93/3.99 %------------------------------------------------------------------------------