%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : CAT005-4 : TPTP v8.1.0. Released v1.0.0. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n018.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 : Fri Jul 15 00:04:49 EDT 2022 % Result : Unsatisfiable 11.14s 11.18s % Output : Proof 11.14s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.12/0.12 % Problem : CAT005-4 : TPTP v8.1.0. Released v1.0.0. % 0.12/0.12 % Command : moca.sh %s % 0.12/0.33 % Computer : n018.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 600 % 0.12/0.33 % DateTime : Sun May 29 17:54:11 EDT 2022 % 0.12/0.33 % CPUTime : % 11.14/11.18 % SZS status Unsatisfiable % 11.14/11.18 % SZS output start Proof % 11.14/11.18 The input problem is unsatisfiable because % 11.14/11.18 % 11.14/11.18 [1] the following set of Horn clauses is unsatisfiable: % 11.14/11.18 % 11.14/11.18 equivalent(X, Y) ==> there_exists(X) % 11.14/11.18 equivalent(X, Y) ==> X = Y % 11.14/11.18 there_exists(X) & X = Y ==> equivalent(X, Y) % 11.14/11.18 there_exists(domain(X)) ==> there_exists(X) % 11.14/11.18 there_exists(codomain(X)) ==> there_exists(X) % 11.14/11.18 there_exists(compose(X, Y)) ==> there_exists(domain(X)) % 11.14/11.18 there_exists(compose(X, Y)) ==> domain(X) = codomain(Y) % 11.14/11.18 there_exists(domain(X)) & domain(X) = codomain(Y) ==> there_exists(compose(X, Y)) % 11.14/11.18 compose(X, compose(Y, Z)) = compose(compose(X, Y), Z) % 11.14/11.18 compose(X, domain(X)) = X % 11.14/11.18 compose(codomain(X), X) = X % 11.14/11.18 there_exists(compose(a, d)) % 11.14/11.18 there_exists(compose(X, d)) ==> compose(X, d) = X % 11.14/11.18 there_exists(compose(d, Y)) ==> compose(d, Y) = Y % 11.14/11.18 domain(a) = d ==> \bottom % 11.14/11.18 % 11.14/11.18 This holds because % 11.14/11.18 % 11.14/11.18 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 11.14/11.18 % 11.14/11.18 E: % 11.14/11.18 compose(X, compose(Y, Z)) = compose(compose(X, Y), Z) % 11.14/11.18 compose(X, domain(X)) = X % 11.14/11.18 compose(codomain(X), X) = X % 11.14/11.18 f1(equivalent(X, Y), X) = true__ % 11.14/11.18 f1(true__, X) = there_exists(X) % 11.14/11.18 f10(codomain(Y), Y, X) = f9(there_exists(domain(X)), X, Y) % 11.14/11.18 f10(domain(X), Y, X) = true__ % 11.14/11.18 f11(there_exists(compose(X, d)), X) = X % 11.14/11.18 f11(true__, X) = compose(X, d) % 11.14/11.18 f12(there_exists(compose(d, Y)), Y) = Y % 11.14/11.18 f12(true__, Y) = compose(d, Y) % 11.14/11.18 f13(d) = false__ % 11.14/11.18 f13(domain(a)) = true__ % 11.14/11.18 f2(equivalent(X, Y), X, Y) = Y % 11.14/11.18 f2(true__, X, Y) = X % 11.14/11.18 f3(true__, X, Y) = equivalent(X, Y) % 11.14/11.18 f4(X, Y, X) = true__ % 11.14/11.18 f4(Y, Y, X) = f3(there_exists(X), X, Y) % 11.14/11.18 f5(there_exists(domain(X)), X) = true__ % 11.14/11.18 f5(true__, X) = there_exists(X) % 11.14/11.18 f6(there_exists(codomain(X)), X) = true__ % 11.14/11.18 f6(true__, X) = there_exists(X) % 11.14/11.18 f7(there_exists(compose(X, Y)), X) = true__ % 11.14/11.18 f7(true__, X) = there_exists(domain(X)) % 11.14/11.18 f8(there_exists(compose(X, Y)), X, Y) = codomain(Y) % 11.14/11.18 f8(true__, X, Y) = domain(X) % 11.14/11.18 f9(true__, X, Y) = there_exists(compose(X, Y)) % 11.14/11.18 there_exists(compose(a, d)) = true__ % 11.14/11.18 G: % 11.14/11.18 true__ = false__ % 11.14/11.18 % 11.14/11.18 This holds because % 11.14/11.18 % 11.14/11.18 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 11.14/11.18 % 11.14/11.18 f8(f1(true__, compose(X, Y)), X, Y) = codomain(Y) % 11.14/11.18 codomain(d) -> domain(a) % 11.14/11.18 compose(X, domain(X)) -> X % 11.14/11.18 compose(a, compose(d, Y2)) -> compose(a, Y2) % 11.14/11.18 compose(codomain(X), X) -> X % 11.14/11.18 compose(compose(X, Y), Z) -> compose(X, compose(Y, Z)) % 11.14/11.18 compose(domain(a), d) -> domain(a) % 11.14/11.18 compose(domain(a), domain(a)) -> domain(a) % 11.14/11.18 d -> domain(a) % 11.14/11.18 domain(d) -> domain(a) % 11.14/11.18 domain(domain(a)) -> domain(a) % 11.14/11.18 f1(equivalent(X, Y), X) -> true__ % 11.14/11.18 f1(true__, a) -> true__ % 11.14/11.18 f1(true__, domain(a)) -> true__ % 11.14/11.18 f1(true__, domain(codomain(a))) -> true__ % 11.14/11.18 f1(true__, domain(codomain(domain(a)))) -> true__ % 11.14/11.18 f1(true__, domain(domain(a))) -> true__ % 11.14/11.18 f1(true__, domain(domain(domain(a)))) -> true__ % 11.14/11.18 f1(true__, domain(f8(f1(true__, compose(false__, a)), false__, a))) -> true__ % 11.14/11.18 f10(codomain(Y), Y, X) -> f9(f1(true__, domain(X)), X, Y) % 11.14/11.18 f10(domain(X), Y, X) -> true__ % 11.14/11.18 f10(domain(a), Y1, domain(a)) -> true__ % 11.14/11.18 f11(f1(true__, compose(Y0, d)), Y0) -> Y0 % 11.14/11.18 f11(f1(true__, compose(Y0, domain(a))), Y0) -> Y0 % 11.14/11.18 f11(f1(true__, d), codomain(d)) -> domain(a) % 11.14/11.18 f11(true__, X) -> compose(X, domain(a)) % 11.14/11.18 f11(true__, codomain(d)) -> domain(a) % 11.14/11.18 f11(true__, domain(a)) -> domain(a) % 11.14/11.18 f12(f1(true__, compose(d, Y0)), Y0) -> Y0 % 11.14/11.18 f12(f1(true__, compose(domain(a), Y0)), Y0) -> Y0 % 11.14/11.18 f12(f1(true__, d), domain(d)) -> domain(a) % 11.14/11.18 f12(true__, Y) -> compose(domain(a), Y) % 11.14/11.18 f12(true__, domain(d)) -> domain(a) % 11.14/11.18 f13(d) -> false__ % 11.14/11.18 f13(domain(a)) -> true__ % 11.14/11.18 f2(equivalent(X, Y), X, Y) -> Y % 11.14/11.18 f2(true__, X, Y) -> X % 11.14/11.18 f3(true__, X, Y) -> equivalent(X, Y) % 11.14/11.18 f4(X, Y, X) -> true__ % 11.14/11.18 f4(Y, Y, X) -> f3(f1(true__, X), X, Y) % 11.14/11.18 f5(f1(true__, domain(Y0)), Y0) -> true__ % 11.14/11.18 f5(true__, X) -> f1(true__, X) % 11.14/11.18 f6(f1(true__, codomain(Y0)), Y0) -> true__ % 11.14/11.18 f6(true__, X) -> f1(true__, X) % 11.14/11.18 f7(f1(true__, Y0), Y0) -> true__ % 11.14/11.18 f7(f1(true__, Y1), codomain(Y1)) -> true__ % 11.14/11.18 f7(f1(true__, compose(Y0, Y1)), Y0) -> true__ % 11.14/11.18 f7(true__, X) -> f1(true__, domain(X)) % 11.14/11.18 f8(f1(true__, compose(Y1, d)), Y1, d) -> domain(a) % 11.14/11.18 f8(f1(true__, compose(false__, domain(a))), false__, domain(a)) -> domain(a) % 11.14/11.18 f8(true__, X, Y) -> domain(X) % 11.14/11.18 f9(true__, X, Y) -> f1(true__, compose(X, Y)) % 11.14/11.18 there_exists(X) -> f1(true__, X) % 11.14/11.18 true__ -> false__ % 11.14/11.18 with the LPO induced by % 11.14/11.18 f13 > codomain > f8 > f10 > f9 > f7 > f4 > f2 > f12 > f11 > compose > f3 > f5 > d > domain > equivalent > f6 > there_exists > f1 > a > true__ > false__ % 11.14/11.18 % 11.14/11.18 % SZS output end Proof % 11.14/11.18 %------------------------------------------------------------------------------