%------------------------------------------------------------------------------ % File : Moca---0.1 % Problem : GRP034-3 : TPTP v8.1.0. Released v1.0.0. % Transfm : none % Format : tptp:raw % Command : moca.sh %s % Computer : n010.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 10:52:03 EDT 2022 % Result : Unsatisfiable 5.80s 5.86s % Output : Proof 5.80s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : GRP034-3 : TPTP v8.1.0. Released v1.0.0. % 0.11/0.13 % Command : moca.sh %s % 0.13/0.34 % Computer : n010.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 01:26:39 EDT 2022 % 0.13/0.34 % CPUTime : % 5.80/5.86 % SZS status Unsatisfiable % 5.80/5.86 % SZS output start Proof % 5.80/5.86 The input problem is unsatisfiable because % 5.80/5.86 % 5.80/5.86 [1] the following set of Horn clauses is unsatisfiable: % 5.80/5.86 % 5.80/5.86 product(identity, X, X) % 5.80/5.86 product(X, identity, X) % 5.80/5.86 product(inverse(X), X, identity) % 5.80/5.86 product(X, inverse(X), identity) % 5.80/5.86 product(X, Y, multiply(X, Y)) % 5.80/5.86 product(X, Y, Z) & product(X, Y, W) ==> Z = W % 5.80/5.86 product(X, Y, U) & product(Y, Z, V) & product(U, Z, W) ==> product(X, V, W) % 5.80/5.86 product(X, Y, U) & product(Y, Z, V) & product(X, V, W) ==> product(U, Z, W) % 5.80/5.86 subgroup_member(A) & subgroup_member(B) & product(A, inverse(B), C) ==> subgroup_member(C) % 5.80/5.86 subgroup_member(a) % 5.80/5.86 subgroup_member(inverse(a)) ==> \bottom % 5.80/5.86 % 5.80/5.86 This holds because % 5.80/5.86 % 5.80/5.86 [2] the following E entails the following G (Claessen-Smallbone's transformation (2018)): % 5.80/5.86 % 5.80/5.86 E: % 5.80/5.86 f1(true__, Z, W) = Z % 5.80/5.86 f10(true__, A, C) = f9(subgroup_member(A), C) % 5.80/5.86 f11(product(A, inverse(B), C), B, A, C) = true__ % 5.80/5.86 f11(true__, B, A, C) = f10(subgroup_member(B), A, C) % 5.80/5.86 f12(subgroup_member(inverse(a))) = true__ % 5.80/5.86 f12(true__) = false__ % 5.80/5.86 f2(product(X, Y, W), X, Y, Z, W) = W % 5.80/5.86 f2(true__, X, Y, Z, W) = f1(product(X, Y, Z), Z, W) % 5.80/5.86 f3(true__, X, V, W) = product(X, V, W) % 5.80/5.86 f4(true__, X, Y, U, V, W) = f3(product(X, Y, U), X, V, W) % 5.80/5.86 f5(product(U, Z, W), Y, Z, V, X, U, W) = true__ % 5.80/5.86 f5(true__, Y, Z, V, X, U, W) = f4(product(Y, Z, V), X, Y, U, V, W) % 5.80/5.86 f6(true__, U, Z, W) = product(U, Z, W) % 5.80/5.86 f7(true__, X, Y, U, Z, W) = f6(product(X, Y, U), U, Z, W) % 5.80/5.86 f8(product(X, V, W), Y, Z, V, X, U, W) = true__ % 5.80/5.86 f8(true__, Y, Z, V, X, U, W) = f7(product(Y, Z, V), X, Y, U, Z, W) % 5.80/5.86 f9(true__, C) = subgroup_member(C) % 5.80/5.86 product(X, Y, multiply(X, Y)) = true__ % 5.80/5.86 product(X, identity, X) = true__ % 5.80/5.86 product(X, inverse(X), identity) = true__ % 5.80/5.86 product(identity, X, X) = true__ % 5.80/5.86 product(inverse(X), X, identity) = true__ % 5.80/5.86 subgroup_member(a) = true__ % 5.80/5.86 G: % 5.80/5.86 true__ = false__ % 5.80/5.86 % 5.80/5.86 This holds because % 5.80/5.86 % 5.80/5.86 [3] E entails the following ordered TRS and the lhs and rhs of G join by the TRS: % 5.80/5.86 % 5.80/5.86 % 5.80/5.86 f1(f3(true__, Y0, Y1, Y3), Y3, multiply(Y0, Y1)) -> multiply(Y0, Y1) % 5.80/5.86 f1(f3(true__, Y0, inverse(Y0), Y3), Y3, identity) -> identity % 5.80/5.86 f1(f3(true__, Y2, identity, Y3), Y3, Y2) -> Y2 % 5.80/5.86 f1(f3(true__, identity, Y2, Y3), Y3, Y2) -> Y2 % 5.80/5.86 f1(f3(true__, inverse(Y1), Y1, Y3), Y3, identity) -> identity % 5.80/5.86 f1(true__, Z, W) -> Z % 5.80/5.86 f10(f9(true__, Y1), Y0, multiply(Y0, inverse(Y1))) -> true__ % 5.80/5.86 f10(f9(true__, Y1), Y1, identity) -> true__ % 5.80/5.86 f10(f9(true__, Y1), identity, inverse(Y1)) -> true__ % 5.80/5.86 f10(f9(true__, Y1), inverse(inverse(Y1)), identity) -> true__ % 5.80/5.86 f10(true__, A, C) -> f9(f9(true__, A), C) % 5.80/5.86 f11(f3(true__, Y0, identity, Y2), identity, Y0, Y2) -> true__ % 5.80/5.86 f11(f6(true__, Y0, identity, Y2), identity, Y0, Y2) -> true__ % 5.80/5.86 f11(product(A, inverse(B), C), B, A, C) -> true__ % 5.80/5.86 f11(true__, B, A, C) -> f10(f9(true__, B), A, C) % 5.80/5.86 f12(f9(true__, inverse(a))) -> true__ % 5.80/5.86 f12(true__) -> false__ % 5.80/5.86 f2(product(X, Y, W), X, Y, Z, W) -> W % 5.80/5.86 f2(true__, X, Y, Z, W) -> f1(f6(true__, X, Y, Z), Z, W) % 5.80/5.86 f3(f3(true__, Y3, identity, Y4), Y3, Y2, multiply(Y4, Y2)) -> true__ % 5.80/5.86 f3(true__, X, V, W) -> f6(true__, X, V, W) % 5.80/5.86 f3(true__, Y0, Y1, multiply(Y0, Y1)) -> true__ % 5.80/5.86 f3(true__, Y0, inverse(Y0), identity) -> true__ % 5.80/5.86 f3(true__, Y2, identity, Y2) -> true__ % 5.80/5.86 f3(true__, identity, Y2, Y2) -> true__ % 5.80/5.86 f3(true__, inverse(Y1), Y1, identity) -> true__ % 5.80/5.86 f4(f3(true__, Y3, Y1, Y4), Y5, Y3, Y0, Y4, multiply(Y0, Y1)) -> true__ % 5.80/5.86 f4(true__, X, Y, U, V, W) -> f3(f6(true__, X, Y, U), X, V, W) % 5.80/5.86 f5(product(U, Z, W), Y, Z, V, X, U, W) -> true__ % 5.80/5.86 f5(true__, Y, Z, V, X, U, W) -> f4(f6(true__, Y, Z, V), X, Y, U, V, W) % 5.80/5.86 f6(true__, Y0, identity, Y0) -> true__ % 5.80/5.86 f6(true__, identity, Y1, Y1) -> true__ % 5.80/5.86 f6(true__, identity, identity, identity) -> true__ % 5.80/5.86 f7(true__, X, Y, U, Z, W) -> f6(f6(true__, X, Y, U), U, Z, W) % 5.80/5.86 f8(product(X, V, W), Y, Z, V, X, U, W) -> true__ % 5.80/5.86 f8(true__, Y, Z, V, X, U, W) -> f7(f6(true__, Y, Z, V), X, Y, U, Z, W) % 5.80/5.86 f9(f9(true__, Y0), Y0) -> true__ % 5.80/5.86 f9(f9(true__, Y1), multiply(Y1, identity)) -> true__ % 5.80/5.86 f9(f9(true__, Y1), multiply(Y1, inverse(a))) -> true__ % 5.80/5.86 f9(f9(true__, inverse(inverse(a))), identity) -> true__ % 5.80/5.86 f9(true__, a) -> true__ % 5.80/5.86 f9(true__, identity) -> true__ % 5.80/5.86 f9(true__, inverse(a)) -> true__ % 5.80/5.86 f9(true__, inverse(inverse(a))) -> true__ % 5.80/5.86 false__ -> true__ % 5.80/5.86 inverse(identity) -> identity % 5.80/5.86 multiply(Y0, identity) -> Y0 % 5.80/5.86 multiply(identity, Y0) -> Y0 % 5.80/5.86 product(U, Z, W) -> f6(true__, U, Z, W) % 5.80/5.86 product(X, Y, multiply(X, Y)) -> true__ % 5.80/5.86 product(X, identity, X) -> true__ % 5.80/5.86 product(X, inverse(X), identity) -> true__ % 5.80/5.86 product(identity, X, X) -> true__ % 5.80/5.86 product(inverse(X), X, identity) -> true__ % 5.80/5.86 subgroup_member(C) -> f9(true__, C) % 5.80/5.86 with the LPO induced by % 5.80/5.86 f2 > f1 > f5 > f4 > f3 > f8 > f7 > product > f6 > a > f11 > f10 > subgroup_member > f9 > multiply > inverse > identity > f12 > false__ > true__ % 5.80/5.86 % 5.80/5.86 % SZS output end Proof % 5.80/5.86 %------------------------------------------------------------------------------