%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : HEN004-3 : TPTP v8.1.0. Released v1.0.0. % 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 12:58:57 EDT 2022 % Result : Unsatisfiable 5.68s 5.66s % Output : Proof 5.68s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.13 % Problem : HEN004-3 : TPTP v8.1.0. Released v1.0.0. % 0.07/0.13 % Command : moca.sh %s % 0.13/0.34 % Computer : n021.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.35 % CPULimit : 300 % 0.13/0.35 % WCLimit : 600 % 0.13/0.35 % DateTime : Fri Jul 1 13:12:44 EDT 2022 % 0.13/0.35 % CPUTime : % 5.68/5.66 % SZS status Unsatisfiable % 5.68/5.66 % SZS output start Proof % 5.68/5.66 The input problem is unsatisfiable because % 5.68/5.66 % 5.68/5.66 [1] the following set of Horn clauses is unsatisfiable: % 5.68/5.66 % 5.68/5.66 less_equal(X, Y) ==> divide(X, Y) = zero % 5.68/5.66 divide(X, Y) = zero ==> less_equal(X, Y) % 5.68/5.66 less_equal(divide(X, Y), X) % 5.68/5.66 less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)) % 5.68/5.66 less_equal(zero, X) % 5.68/5.66 less_equal(X, Y) & less_equal(Y, X) ==> X = Y % 5.68/5.66 less_equal(X, identity) % 5.68/5.66 divide(a, zero) = a ==> \bottom % 5.68/5.66 % 5.68/5.66 This holds because % 5.68/5.66 % 5.68/5.66 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 5.68/5.66 % 5.68/5.66 E: % 5.68/5.66 f1(less_equal(X, Y), X, Y) = zero % 5.68/5.66 f1(true__, X, Y) = divide(X, Y) % 5.68/5.66 f2(divide(X, Y), X, Y) = true__ % 5.68/5.66 f2(zero, X, Y) = less_equal(X, Y) % 5.68/5.66 f3(true__, X, Y) = X % 5.68/5.66 f4(less_equal(Y, X), X, Y) = Y % 5.68/5.66 f4(true__, X, Y) = f3(less_equal(X, Y), X, Y) % 5.68/5.66 f5(a) = false__ % 5.68/5.66 f5(divide(a, zero)) = true__ % 5.68/5.66 less_equal(X, identity) = true__ % 5.68/5.66 less_equal(divide(X, Y), X) = true__ % 5.68/5.66 less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)) = true__ % 5.68/5.66 less_equal(zero, X) = true__ % 5.68/5.66 G: % 5.68/5.66 true__ = false__ % 5.68/5.66 % 5.68/5.66 This holds because % 5.68/5.66 % 5.68/5.66 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 5.68/5.66 % 5.68/5.66 % 5.68/5.66 divide(X, Y) -> f1(true__, X, Y) % 5.68/5.66 f1(f2(zero, Y0, Y1), Y0, Y1) -> zero % 5.68/5.66 f1(less_equal(X, Y), X, Y) -> zero % 5.68/5.66 f1(true__, X0, X0) -> zero % 5.68/5.66 f1(true__, X0, f1(true__, X0, zero)) -> zero % 5.68/5.66 f1(true__, X0, f1(true__, identity, zero)) -> zero % 5.68/5.66 f1(true__, Y0, identity) -> zero % 5.68/5.66 f1(true__, Y0, zero) -> Y0 % 5.68/5.66 f1(true__, f1(true__, X0, X0), f1(true__, X1, X0)) -> zero % 5.68/5.66 f1(true__, f1(true__, X0, X1), f1(true__, X0, X1)) -> zero % 5.68/5.66 f1(true__, f1(true__, X0, X1), f1(true__, X0, zero)) -> zero % 5.68/5.66 f1(true__, f1(true__, X0, f1(true__, X0, zero)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, X0, f1(true__, identity, zero)), X1) -> zero % 5.68/5.66 f1(true__, f1(true__, X0, f1(true__, identity, zero)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, Y0, Y0), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, Y0, Y1), f1(true__, f1(true__, Y0, Y1), Y1)) -> zero % 5.68/5.66 f1(true__, f1(true__, Y0, Y2), f1(true__, identity, Y2)) -> zero % 5.68/5.66 f1(true__, f1(true__, Y0, f1(true__, Y0, Y1)), f1(true__, Y1, f1(true__, Y0, Y1))) -> zero % 5.68/5.66 f1(true__, f1(true__, Y1, X1), Y1) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, X0), zero), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, X1), X2), X0) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, X1), X2), f1(true__, X0, X2)) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, X1), X2), f1(true__, identity, X1)) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, X1), f1(true__, X2, X1)), f1(true__, f1(true__, X0, X2), X1)) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, f1(true__, X0, zero)), X1), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, X0, f1(true__, identity, zero)), zero), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, Y1), f1(true__, Y0, zero)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, Y1), f1(true__, identity, Y1)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, Y1), zero), f1(true__, f1(true__, Y0, Y1), Y1)) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, Y1), zero), f1(true__, f1(true__, Y0, zero), Y1)) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, f1(true__, Y0, Y2)), f1(true__, Y2, f1(true__, Y0, Y2))), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, f1(true__, Y0, zero)), zero), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, Y0, f1(true__, identity, zero)), Y2), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, X0, X0), Y1), f1(true__, X1, X0)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, X0, X0), Y1), zero), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, X0, X1), X2), X3), X0) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, X0, X1), Y1), f1(true__, identity, X1)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, X0, f1(true__, X0, zero)), Y1), zero), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, Y0, Y1), Y2), Y3), f1(true__, Y0, Y2)) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, Y2, X1), Y1), Y2), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, f1(true__, X0, X1), X2), Y1), f1(true__, X0, X2)), zero) -> zero % 5.68/5.66 f1(true__, f1(true__, f1(true__, f1(true__, f1(true__, Y2, X1), X2), Y1), Y2), zero) -> zero % 5.68/5.66 f1(true__, identity, zero) -> identity % 5.68/5.66 f1(true__, zero, Y1) -> zero % 5.68/5.66 f2(Y0, Y0, zero) -> true__ % 5.68/5.66 f2(divide(X, Y), X, Y) -> true__ % 5.68/5.66 f2(f1(true__, Y0, Y1), Y0, Y1) -> true__ % 5.68/5.66 f2(identity, identity, zero) -> true__ % 5.68/5.66 f2(zero, Y0, f1(true__, Y0, zero)) -> true__ % 5.68/5.66 f2(zero, Y0, identity) -> true__ % 5.68/5.66 f2(zero, Y1, Y1) -> true__ % 5.68/5.66 f2(zero, f1(true__, X0, X0), f1(true__, X1, X0)) -> true__ % 5.68/5.66 f2(zero, f1(true__, X0, X1), f1(true__, X0, X1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, X0, X1), f1(true__, identity, X1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, X0, f1(true__, X0, zero)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, Y0, Y0), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, Y0, Y1), Y0) -> true__ % 5.68/5.66 f2(zero, f1(true__, Y0, Y1), f1(true__, f1(true__, Y0, Y1), Y1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, Y0, Y1), f1(true__, f1(true__, Y0, f1(true__, Y1, Y2)), Y1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, Y0, f1(true__, identity, zero)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, X0, X1), X2), f1(true__, X0, X2)) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, X0, X1), X2), f1(true__, identity, X1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, X1, X0), f1(true__, X1, X0)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y0), zero), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y1), f1(true__, Y0, zero)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y1), f1(true__, Y2, Y1)), f1(true__, f1(true__, Y0, Y2), Y1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y1), f1(true__, identity, Y1)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y1), zero), f1(true__, f1(true__, Y0, Y1), Y1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y1), zero), f1(true__, f1(true__, Y0, f1(true__, Y1, X1)), Y1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, Y1), zero), f1(true__, f1(true__, Y0, zero), Y1)) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, f1(true__, Y0, Y2)), f1(true__, Y2, f1(true__, Y0, Y2))), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, f1(true__, Y0, zero)), zero), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, f1(true__, f1(true__, Y0, zero), zero)), zero), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, f1(true__, identity, zero)), Y2), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y0, f1(true__, identity, zero)), zero), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y1, X1), X2), Y1) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, Y1, Y1), f1(true__, Y2, Y1)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, X0, X0), Y1), zero), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, X0, X1), Y1), f1(true__, identity, X1)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, X0, f1(true__, X0, zero)), Y1), zero), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, Y1, X1), X2), X3), Y1) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, Y2, X1), Y1), Y2), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, Y2, X1), Y1), f1(true__, Y2, Y1)), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, f1(true__, Y2, X1), X2), Y1), Y2), zero) -> true__ % 5.68/5.66 f2(zero, f1(true__, f1(true__, f1(true__, f1(true__, Y2, X1), Y1), Y2), zero), zero) -> true__ % 5.68/5.66 f2(zero, zero, Y1) -> true__ % 5.68/5.66 f3(f2(zero, Y1, f1(true__, Y1, X1)), Y1, f1(true__, Y1, X1)) -> f1(true__, Y1, X1) % 5.68/5.66 f3(f2(zero, Y1, f1(true__, f1(true__, Y1, X1), X2)), Y1, f1(true__, f1(true__, Y1, X1), X2)) -> f1(true__, f1(true__, Y1, X1), X2) % 5.68/5.66 f3(f2(zero, Y1, f1(true__, f1(true__, f1(true__, Y1, X1), X2), X3)), Y1, f1(true__, f1(true__, f1(true__, Y1, X1), X2), X3)) -> f1(true__, f1(true__, f1(true__, Y1, X1), X2), X3) % 5.68/5.66 f3(f2(zero, Y1, zero), Y1, zero) -> zero % 5.68/5.66 f3(f2(zero, f1(true__, X0, X2), f1(true__, f1(true__, X0, X1), X2)), f1(true__, X0, X2), f1(true__, f1(true__, X0, X1), X2)) -> f1(true__, f1(true__, X0, X1), X2) % 5.68/5.66 f3(f2(zero, f1(true__, X1, X0), f1(true__, X0, X0)), f1(true__, X1, X0), f1(true__, X0, X0)) -> f1(true__, X0, X0) % 5.68/5.66 f3(f2(zero, f1(true__, f1(true__, X0, X2), X1), f1(true__, f1(true__, X0, X1), f1(true__, X2, X1))), f1(true__, f1(true__, X0, X2), X1), f1(true__, f1(true__, X0, X1), f1(true__, X2, X1))) -> f1(true__, f1(true__, X0, X1), f1(true__, X2, X1)) % 5.68/5.66 f3(f2(zero, f1(true__, f1(true__, Y0, Y2), Y2), f1(true__, f1(true__, Y0, Y2), zero)), f1(true__, f1(true__, Y0, Y2), Y2), f1(true__, f1(true__, Y0, Y2), zero)) -> f1(true__, f1(true__, Y0, Y2), zero) % 5.68/5.66 f3(f2(zero, f1(true__, f1(true__, Y0, zero), Y2), f1(true__, f1(true__, Y0, Y2), zero)), f1(true__, f1(true__, Y0, zero), Y2), f1(true__, f1(true__, Y0, Y2), zero)) -> f1(true__, f1(true__, Y0, Y2), zero) % 5.68/5.66 f3(f2(zero, f1(true__, identity, X1), f1(true__, X0, X1)), f1(true__, identity, X1), f1(true__, X0, X1)) -> f1(true__, X0, X1) % 5.68/5.66 f3(f2(zero, f1(true__, identity, X1), f1(true__, f1(true__, X0, X1), X2)), f1(true__, identity, X1), f1(true__, f1(true__, X0, X1), X2)) -> f1(true__, f1(true__, X0, X1), X2) % 5.68/5.66 f3(f2(zero, identity, Y0), identity, Y0) -> Y0 % 5.68/5.66 f3(true__, X, Y) -> X % 5.68/5.66 f4(f2(zero, Y0, Y1), Y1, Y0) -> Y0 % 5.68/5.66 f4(less_equal(Y, X), X, Y) -> Y % 5.68/5.66 f4(true__, X, Y) -> f3(less_equal(X, Y), X, Y) % 5.68/5.66 f5(a) -> false__ % 5.68/5.66 f5(divide(a, zero)) -> true__ % 5.68/5.66 f5(f1(true__, a, zero)) -> true__ % 5.68/5.66 false__ -> true__ % 5.68/5.66 less_equal(X, Y) -> f2(zero, X, Y) % 5.68/5.66 less_equal(X, identity) -> true__ % 5.68/5.66 less_equal(divide(X, Y), X) -> true__ % 5.68/5.66 less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)) -> true__ % 5.68/5.66 less_equal(zero, X) -> true__ % 5.68/5.66 with the LPO induced by % 5.68/5.66 a > divide > f1 > f4 > f3 > less_equal > f5 > identity > f2 > zero > false__ > true__ % 5.68/5.66 % 5.68/5.66 % SZS output end Proof % 5.68/5.66 %------------------------------------------------------------------------------