%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP012-2 : TPTP v8.1.0. Released v1.0.0. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n015.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 10:51:48 EDT 2022 % Result : Unsatisfiable 9.30s 9.23s % Output : Proof 9.30s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : GRP012-2 : TPTP v8.1.0. Released v1.0.0. % 0.07/0.13 % Command : moca.sh %s % 0.13/0.34 % Computer : n015.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 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 600 % 0.13/0.34 % DateTime : Tue Jun 14 10:05:54 EDT 2022 % 0.13/0.35 % CPUTime : % 9.30/9.23 % SZS status Unsatisfiable % 9.30/9.23 % SZS output start Proof % 9.30/9.23 The input problem is unsatisfiable because % 9.30/9.23 % 9.30/9.23 [1] the following set of Horn clauses is unsatisfiable: % 9.30/9.23 % 9.30/9.23 product(identity, X, X) % 9.30/9.23 product(X, identity, X) % 9.30/9.23 product(inverse(X), X, identity) % 9.30/9.23 product(X, inverse(X), identity) % 9.30/9.23 product(X, Y, multiply(X, Y)) % 9.30/9.23 product(X, Y, Z) & product(X, Y, W) ==> Z = W % 9.30/9.23 product(X, Y, U) & product(Y, Z, V) & product(U, Z, W) ==> product(X, V, W) % 9.30/9.23 product(X, Y, U) & product(Y, Z, V) & product(X, V, W) ==> product(U, Z, W) % 9.30/9.23 product(a, b, c) % 9.30/9.23 product(inverse(b), inverse(a), d) % 9.30/9.23 inverse(c) = d ==> \bottom % 9.30/9.23 % 9.30/9.23 This holds because % 9.30/9.23 % 9.30/9.23 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 9.30/9.23 % 9.30/9.23 E: % 9.30/9.23 f1(true__, Z, W) = Z % 9.30/9.23 f2(product(X, Y, W), X, Y, Z, W) = W % 9.30/9.23 f2(true__, X, Y, Z, W) = f1(product(X, Y, Z), Z, W) % 9.30/9.23 f3(true__, X, V, W) = product(X, V, W) % 9.30/9.23 f4(true__, X, Y, U, V, W) = f3(product(X, Y, U), X, V, W) % 9.30/9.23 f5(product(U, Z, W), Y, Z, V, X, U, W) = true__ % 9.30/9.23 f5(true__, Y, Z, V, X, U, W) = f4(product(Y, Z, V), X, Y, U, V, W) % 9.30/9.23 f6(true__, U, Z, W) = product(U, Z, W) % 9.30/9.23 f7(true__, X, Y, U, Z, W) = f6(product(X, Y, U), U, Z, W) % 9.30/9.23 f8(product(X, V, W), Y, Z, V, X, U, W) = true__ % 9.30/9.23 f8(true__, Y, Z, V, X, U, W) = f7(product(Y, Z, V), X, Y, U, Z, W) % 9.30/9.23 f9(d) = false__ % 9.30/9.23 f9(inverse(c)) = true__ % 9.30/9.23 product(X, Y, multiply(X, Y)) = true__ % 9.30/9.23 product(X, identity, X) = true__ % 9.30/9.23 product(X, inverse(X), identity) = true__ % 9.30/9.23 product(a, b, c) = true__ % 9.30/9.23 product(identity, X, X) = true__ % 9.30/9.23 product(inverse(X), X, identity) = true__ % 9.30/9.23 product(inverse(b), inverse(a), d) = true__ % 9.30/9.23 G: % 9.30/9.23 true__ = false__ % 9.30/9.23 % 9.30/9.23 This holds because % 9.30/9.23 % 9.30/9.23 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 9.30/9.23 % 9.30/9.23 % 9.30/9.23 c -> multiply(a, b) % 9.30/9.23 d -> multiply(inverse(b), inverse(a)) % 9.30/9.23 f1(f6(true__, Y0, Y1, Y3), Y3, multiply(Y0, Y1)) -> multiply(Y0, Y1) % 9.30/9.23 f1(f6(true__, Y0, identity, Y3), Y3, inverse(inverse(Y0))) -> inverse(inverse(Y0)) % 9.30/9.23 f1(f6(true__, Y0, inverse(Y0), Y3), Y3, identity) -> identity % 9.30/9.23 f1(f6(true__, Y0, inverse(multiply(X1, Y0)), Y3), Y3, inverse(X1)) -> inverse(X1) % 9.30/9.23 f1(f6(true__, Y0, inverse(multiply(inverse(Y2), Y0)), Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, Y0, multiply(inverse(Y0), Y2), Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, Y2, identity, Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, a, b, Y3), Y3, c) -> c % 9.30/9.23 f1(f6(true__, b, inverse(multiply(a, b)), Y3), Y3, inverse(a)) -> inverse(a) % 9.30/9.23 f1(f6(true__, identity, Y2, Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, inverse(X0), multiply(X0, Y2), Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, inverse(Y1), Y1, Y3), Y3, identity) -> identity % 9.30/9.23 f1(f6(true__, inverse(a), multiply(a, b), Y3), Y3, b) -> b % 9.30/9.23 f1(f6(true__, inverse(b), inverse(a), Y3), Y3, d) -> d % 9.30/9.23 f1(f6(true__, inverse(inverse(Y2)), identity, Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, inverse(multiply(Y1, X1)), Y1, Y3), Y3, inverse(X1)) -> inverse(X1) % 9.30/9.23 f1(f6(true__, inverse(multiply(Y1, inverse(Y2))), Y1, Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, inverse(multiply(a, b)), a, Y3), Y3, inverse(b)) -> inverse(b) % 9.30/9.23 f1(f6(true__, multiply(Y0, X0), inverse(X0), Y2), Y2, Y0) -> Y0 % 9.30/9.23 f1(f6(true__, multiply(Y2, inverse(Y1)), Y1, Y3), Y3, Y2) -> Y2 % 9.30/9.23 f1(f6(true__, multiply(a, b), inverse(b), Y3), Y3, a) -> a % 9.30/9.23 f1(true__, Z, W) -> Z % 9.30/9.23 f2(f6(true__, Y0, Y1, Y2), Y0, Y1, Y3, Y2) -> Y2 % 9.30/9.23 f2(product(X, Y, W), X, Y, Z, W) -> W % 9.30/9.23 f2(true__, X, Y, Z, W) -> f1(product(X, Y, Z), Z, W) % 9.30/9.23 f3(f6(true__, Y0, X0, X0), Y0, identity, identity) -> true__ % 9.30/9.23 f3(f6(true__, Y0, b, identity), Y0, inverse(a), inverse(multiply(a, b))) -> true__ % 9.30/9.23 f3(f6(true__, Y0, identity, X0), Y0, inverse(X0), identity) -> true__ % 9.30/9.23 f3(f6(true__, Y2, Y0, a), Y2, multiply(Y0, b), multiply(a, b)) -> true__ % 9.30/9.23 f3(f6(true__, Y2, Y1, Y3), Y2, Y1, Y3) -> true__ % 9.30/9.23 f3(f6(true__, Y2, a, a), Y2, multiply(a, b), multiply(a, b)) -> true__ % 9.30/9.23 f3(f6(true__, Y2, identity, Y3), Y2, identity, Y3) -> true__ % 9.30/9.23 f3(f6(true__, Y2, identity, a), Y2, b, multiply(a, b)) -> true__ % 9.30/9.23 f3(f6(true__, Y2, inverse(b), a), Y2, identity, multiply(a, b)) -> true__ % 9.30/9.23 f3(f6(true__, Y2, multiply(Y1, inverse(b)), a), Y2, Y1, multiply(a, b)) -> true__ % 9.30/9.23 f3(f6(true__, Y3, Y0, identity), Y3, Y2, multiply(inverse(Y0), Y2)) -> true__ % 9.30/9.23 f3(f6(true__, Y3, Y0, identity), Y3, identity, inverse(Y0)) -> true__ % 9.30/9.23 f3(f6(true__, Y3, Y0, identity), Y3, multiply(Y0, Y1), Y1) -> true__ % 9.30/9.23 f3(f6(true__, Y3, Y2, identity), Y3, Y2, identity) -> true__ % 9.30/9.23 f3(f6(true__, Y3, a, identity), Y3, multiply(a, b), b) -> true__ % 9.30/9.23 f3(f6(true__, Y3, a, inverse(b)), Y3, multiply(a, b), identity) -> true__ % 9.30/9.23 f3(f6(true__, Y3, identity, identity), Y3, Y2, Y2) -> true__ % 9.30/9.23 f3(f6(true__, Y3, identity, inverse(Y2)), Y3, Y2, identity) -> true__ % 9.30/9.23 f3(f6(true__, Y3, inverse(X0), identity), Y3, Y2, multiply(X0, Y2)) -> true__ % 9.30/9.23 f3(f6(true__, Y3, inverse(Y1), identity), Y3, identity, Y1) -> true__ % 9.30/9.23 f3(f6(true__, Y3, inverse(Y1), inverse(Y1)), Y3, identity, identity) -> true__ % 9.30/9.23 f3(f6(true__, Y3, inverse(a), identity), Y3, b, multiply(a, b)) -> true__ % 9.30/9.23 f3(f6(true__, Y3, multiply(Y2, X1), identity), Y3, Y2, inverse(X1)) -> true__ % 9.30/9.23 f3(f6(true__, Y3, multiply(Y2, inverse(Y1)), identity), Y3, Y2, Y1) -> true__ % 9.30/9.23 f3(f6(true__, Y3, multiply(a, b), identity), Y3, a, inverse(b)) -> true__ % 9.30/9.23 f3(true__, X, V, W) -> product(X, V, W) % 9.30/9.23 f4(f6(true__, Y3, Y1, Y4), Y5, Y3, Y0, Y4, multiply(Y0, Y1)) -> true__ % 9.30/9.23 f4(f6(true__, Y3, Y1, Y4), Y5, Y3, inverse(Y1), Y4, identity) -> true__ % 9.30/9.23 f4(f6(true__, Y3, Y2, Y4), Y5, Y3, identity, Y4, Y2) -> true__ % 9.30/9.23 f4(f6(true__, Y3, b, Y4), Y5, Y3, a, Y4, c) -> true__ % 9.30/9.23 f4(f6(true__, Y3, identity, Y4), Y5, Y3, Y2, Y4, Y2) -> true__ % 9.30/9.23 f4(f6(true__, Y3, inverse(Y0), Y4), Y5, Y3, Y0, Y4, identity) -> true__ % 9.30/9.23 f4(f6(true__, Y3, inverse(a), Y4), Y5, Y3, inverse(b), Y4, d) -> true__ % 9.30/9.23 f4(true__, X, Y, U, V, W) -> f3(product(X, Y, U), X, V, W) % 9.30/9.23 f5(f6(true__, Y0, Y1, Y2), Y3, Y1, Y4, Y5, Y0, Y2) -> true__ % 9.30/9.23 f5(product(U, Z, W), Y, Z, V, X, U, W) -> true__ % 9.30/9.23 f5(true__, Y, Z, V, X, U, W) -> f4(product(Y, Z, V), X, Y, U, V, W) % 9.30/9.23 f6(f6(true__, X0, identity, Y1), Y1, inverse(X0), identity) -> true__ % 9.30/9.23 f6(f6(true__, X0, inverse(X0), Y1), Y1, identity, identity) -> true__ % 9.30/9.23 f6(f6(true__, Y2, Y0, Y3), Y3, inverse(Y0), Y2) -> true__ % 9.30/9.23 f6(f6(true__, Y2, identity, Y3), Y3, identity, Y2) -> true__ % 9.30/9.23 f6(f6(true__, Y2, inverse(Y1), Y3), Y3, Y1, Y2) -> true__ % 9.30/9.23 f6(f6(true__, a, b, Y2), Y2, identity, multiply(a, b)) -> true__ % 9.30/9.23 f6(f6(true__, a, identity, Y2), Y2, b, multiply(a, b)) -> true__ % 9.30/9.23 f6(f6(true__, identity, Y0, Y3), Y3, Y1, multiply(Y0, Y1)) -> true__ % 9.30/9.23 f6(f6(true__, identity, Y0, Y3), Y3, inverse(Y0), identity) -> true__ % 9.30/9.23 f6(f6(true__, identity, Y2, Y3), Y3, identity, Y2) -> true__ % 9.30/9.23 f6(f6(true__, identity, a, Y3), Y3, b, multiply(a, b)) -> true__ % 9.30/9.23 f6(f6(true__, identity, b, Y1), Y1, inverse(multiply(a, b)), inverse(a)) -> true__ % 9.30/9.23 f6(f6(true__, identity, identity, Y3), Y3, Y2, Y2) -> true__ % 9.30/9.23 f6(f6(true__, identity, inverse(X0), Y3), Y3, multiply(X0, Y2), Y2) -> true__ % 9.30/9.23 f6(f6(true__, identity, inverse(Y1), Y3), Y3, Y1, identity) -> true__ % 9.30/9.23 f6(f6(true__, identity, inverse(a), Y3), Y3, multiply(a, b), b) -> true__ % 9.30/9.23 f6(f6(true__, identity, multiply(Y2, X1), Y3), Y3, inverse(X1), Y2) -> true__ % 9.30/9.23 f6(f6(true__, identity, multiply(Y2, inverse(Y1)), Y3), Y3, Y1, Y2) -> true__ % 9.30/9.23 f6(f6(true__, identity, multiply(a, b), Y3), Y3, inverse(b), a) -> true__ % 9.30/9.23 f6(f6(true__, inverse(Y2), Y2, Y3), Y3, identity, identity) -> true__ % 9.30/9.23 f6(f6(true__, inverse(Y2), identity, Y3), Y3, Y2, identity) -> true__ % 9.30/9.23 f6(f6(true__, inverse(multiply(a, b)), a, Y3), Y3, b, identity) -> true__ % 9.30/9.23 f6(true__, Y0, X0, inverse(multiply(inverse(X0), inverse(Y0)))) -> true__ % 9.30/9.23 f6(true__, Y0, Y1, multiply(Y0, Y1)) -> true__ % 9.30/9.23 f6(true__, Y0, identity, inverse(inverse(Y0))) -> true__ % 9.30/9.23 f6(true__, Y0, inverse(X0), inverse(multiply(X0, inverse(Y0)))) -> true__ % 9.30/9.23 f6(true__, Y0, inverse(Y0), identity) -> true__ % 9.30/9.23 f6(true__, Y0, multiply(inverse(Y0), Y2), Y2) -> true__ % 9.30/9.23 f6(true__, Y2, identity, Y2) -> true__ % 9.30/9.23 f6(true__, Y2, inverse(multiply(X0, Y2)), inverse(X0)) -> true__ % 9.30/9.23 f6(true__, Y2, inverse(multiply(inverse(Y0), Y2)), Y0) -> true__ % 9.30/9.23 f6(true__, a, b, c) -> true__ % 9.30/9.23 f6(true__, b, inverse(multiply(a, b)), inverse(a)) -> true__ % 9.30/9.23 f6(true__, identity, Y1, inverse(inverse(Y1))) -> true__ % 9.30/9.23 f6(true__, identity, Y2, Y2) -> true__ % 9.30/9.23 f6(true__, identity, inverse(inverse(Y0)), Y0) -> true__ % 9.30/9.23 f6(true__, inverse(X0), Y1, inverse(multiply(inverse(Y1), X0))) -> true__ % 9.30/9.23 f6(true__, inverse(X0), inverse(Y1), inverse(multiply(Y1, X0))) -> true__ % 9.30/9.23 f6(true__, inverse(Y1), Y1, identity) -> true__ % 9.30/9.23 f6(true__, inverse(Y1), multiply(Y1, Y2), Y2) -> true__ % 9.30/9.23 f6(true__, inverse(a), multiply(a, b), b) -> true__ % 9.30/9.23 f6(true__, inverse(b), inverse(a), d) -> true__ % 9.30/9.23 f6(true__, inverse(b), inverse(a), inverse(multiply(a, b))) -> true__ % 9.30/9.23 f6(true__, inverse(inverse(Y1)), identity, Y1) -> true__ % 9.30/9.23 f6(true__, inverse(multiply(X0, inverse(Y1))), X0, Y1) -> true__ % 9.30/9.23 f6(true__, inverse(multiply(Y0, X0)), Y0, inverse(X0)) -> true__ % 9.30/9.23 f6(true__, inverse(multiply(a, b)), a, inverse(b)) -> true__ % 9.30/9.23 f6(true__, multiply(Y0, X0), inverse(X0), Y0) -> true__ % 9.30/9.23 f6(true__, multiply(Y0, inverse(Y1)), Y1, Y0) -> true__ % 9.30/9.23 f6(true__, multiply(a, b), inverse(b), a) -> true__ % 9.30/9.23 f6(true__, multiply(a, b), multiply(inverse(b), inverse(a)), identity) -> true__ % 9.30/9.23 f6(true__, multiply(inverse(b), inverse(a)), multiply(a, b), identity) -> true__ % 9.30/9.23 f7(f6(true__, Y0, Y1, b), a, Y0, Y2, Y1, multiply(a, b)) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, Y1), Y0, Y3, Y5, Y4, multiply(Y0, Y1)) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, Y1), inverse(Y1), Y3, Y5, Y4, identity) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, Y2), identity, Y3, Y5, Y4, Y2) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, b), a, Y3, Y5, Y4, c) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, identity), Y2, Y3, Y5, Y4, Y2) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, inverse(Y0)), Y0, Y3, Y5, Y4, identity) -> true__ % 9.30/9.23 f7(f6(true__, Y3, Y4, inverse(a)), inverse(b), Y3, Y5, Y4, d) -> true__ % 9.30/9.23 f7(true__, X, Y, U, Z, W) -> f6(product(X, Y, U), U, Z, W) % 9.30/9.23 f8(f6(true__, Y0, Y1, Y2), Y3, Y4, Y1, Y0, Y5, Y2) -> true__ % 9.30/9.23 f8(product(X, V, W), Y, Z, V, X, U, W) -> true__ % 9.30/9.23 f8(true__, Y, Z, V, X, U, W) -> f7(product(Y, Z, V), X, Y, U, Z, W) % 9.30/9.23 f9(d) -> false__ % 9.30/9.23 f9(inverse(c)) -> true__ % 9.30/9.23 f9(inverse(multiply(a, b))) -> true__ % 9.30/9.23 f9(multiply(inverse(b), inverse(a))) -> false__ % 9.30/9.23 false__ -> true__ % 9.30/9.23 inverse(identity) -> identity % 9.30/9.23 inverse(inverse(Y1)) -> Y1 % 9.30/9.23 inverse(multiply(inverse(b), inverse(a))) -> multiply(a, b) % 9.30/9.23 multiply(Y0, identity) -> Y0 % 9.30/9.23 multiply(Y0, inverse(Y0)) -> identity % 9.30/9.23 multiply(Y0, multiply(inverse(Y0), Y1)) -> Y1 % 9.30/9.23 multiply(b, inverse(multiply(a, b))) -> inverse(a) % 9.30/9.23 multiply(identity, Y0) -> Y0 % 9.30/9.23 multiply(inverse(Y0), Y0) -> identity % 9.30/9.23 multiply(inverse(Y0), multiply(Y0, Y1)) -> Y1 % 9.30/9.23 multiply(inverse(a), multiply(a, b)) -> b % 9.30/9.23 multiply(inverse(b), inverse(a)) -> inverse(multiply(a, b)) % 9.30/9.23 multiply(inverse(multiply(a, b)), a) -> inverse(b) % 9.30/9.23 multiply(multiply(Y0, X0), inverse(X0)) -> Y0 % 9.30/9.23 multiply(multiply(Y0, inverse(Y1)), Y1) -> Y0 % 9.30/9.23 multiply(multiply(Y0, multiply(a, b)), multiply(inverse(b), inverse(a))) -> Y0 % 9.30/9.23 multiply(multiply(a, b), inverse(b)) -> a % 9.30/9.23 multiply(multiply(a, b), multiply(inverse(b), inverse(a))) -> identity % 9.30/9.23 multiply(multiply(inverse(b), inverse(a)), multiply(a, b)) -> identity % 9.30/9.23 product(U, Z, W) -> f6(true__, U, Z, W) % 9.30/9.23 product(X, Y, multiply(X, Y)) -> true__ % 9.30/9.23 product(X, identity, X) -> true__ % 9.30/9.23 product(X, inverse(X), identity) -> true__ % 9.30/9.23 product(a, b, c) -> true__ % 9.30/9.23 product(identity, X, X) -> true__ % 9.30/9.23 product(inverse(X), X, identity) -> true__ % 9.30/9.23 product(inverse(b), inverse(a), d) -> true__ % 9.30/9.23 with the LPO induced by % 9.30/9.23 f9 > d > c > b > a > f2 > f1 > f5 > f4 > f3 > f8 > f7 > product > f6 > multiply > inverse > identity > false__ > true__ % 9.30/9.23 % 9.30/9.23 % SZS output end Proof % 9.30/9.23 %------------------------------------------------------------------------------