ceta_eq: termination proof not accepted 1: error below switch to dependency pairs 1.1: error below the dependency graph processor 1.1.1: error when applying the reduction pair processor to remove from the DP problem pairs: a__U101#(tt, M, N) -> a__U102#(a__isNatKind(M), M, N) a__U102#(tt, M, N) -> a__U103#(a__isNat(N), M, N) a__U103#(tt, M, N) -> a__U104#(a__isNatKind(N), M, N) a__U104#(tt, M, N) -> a__plus#(a__x(mark(N), mark(M)), mark(N)) a__U104#(tt, M, N) -> a__x#(mark(N), mark(M)) a__U104#(tt, M, N) -> mark#(N) a__U104#(tt, M, N) -> mark#(M) a__U71#(tt, N) -> a__U72#(a__isNatKind(N), N) a__U72#(tt, N) -> mark#(N) a__U81#(tt, M, N) -> a__U82#(a__isNatKind(M), M, N) a__U82#(tt, M, N) -> a__U83#(a__isNat(N), M, N) a__U83#(tt, M, N) -> a__U84#(a__isNatKind(N), M, N) a__U84#(tt, M, N) -> a__plus#(mark(N), mark(M)) a__U84#(tt, M, N) -> mark#(N) a__U84#(tt, M, N) -> mark#(M) a__plus#(N, 0) -> a__U71#(a__isNat(N), N) a__plus#(N, s(M)) -> a__U81#(a__isNat(M), M, N) a__x#(N, s(M)) -> a__U101#(a__isNat(M), M, N) mark#(U101(X1, X2, X3)) -> a__U101#(mark(X1), X2, X3) mark#(U101(X1, X2, X3)) -> mark#(X1) mark#(U102(X1, X2, X3)) -> a__U102#(mark(X1), X2, X3) mark#(U102(X1, X2, X3)) -> mark#(X1) mark#(U103(X1, X2, X3)) -> a__U103#(mark(X1), X2, X3) mark#(U103(X1, X2, X3)) -> mark#(X1) mark#(U104(X1, X2, X3)) -> a__U104#(mark(X1), X2, X3) mark#(U104(X1, X2, X3)) -> mark#(X1) mark#(plus(X1, X2)) -> a__plus#(mark(X1), mark(X2)) mark#(plus(X1, X2)) -> mark#(X1) mark#(plus(X1, X2)) -> mark#(X2) mark#(x(X1, X2)) -> a__x#(mark(X1), mark(X2)) mark#(x(X1, X2)) -> mark#(X1) mark#(x(X1, X2)) -> mark#(X2) mark#(U11(X1, X2, X3)) -> mark#(X1) mark#(U12(X1, X2, X3)) -> mark#(X1) mark#(U13(X1, X2, X3)) -> mark#(X1) mark#(U14(X1, X2, X3)) -> mark#(X1) mark#(U15(X1, X2)) -> mark#(X1) mark#(U16(X)) -> mark#(X) mark#(U21(X1, X2)) -> mark#(X1) mark#(U22(X1, X2)) -> mark#(X1) mark#(U23(X)) -> mark#(X) mark#(U31(X1, X2, X3)) -> mark#(X1) mark#(U32(X1, X2, X3)) -> mark#(X1) mark#(U33(X1, X2, X3)) -> mark#(X1) mark#(U34(X1, X2, X3)) -> mark#(X1) mark#(U35(X1, X2)) -> mark#(X1) mark#(U36(X)) -> mark#(X) mark#(U41(X1, X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U51(X)) -> mark#(X) mark#(U61(X1, X2)) -> mark#(X1) mark#(U62(X)) -> mark#(X) mark#(U71(X1, X2)) -> a__U71#(mark(X1), X2) mark#(U71(X1, X2)) -> mark#(X1) mark#(U72(X1, X2)) -> a__U72#(mark(X1), X2) mark#(U72(X1, X2)) -> mark#(X1) mark#(U81(X1, X2, X3)) -> a__U81#(mark(X1), X2, X3) mark#(U81(X1, X2, X3)) -> mark#(X1) mark#(U82(X1, X2, X3)) -> a__U82#(mark(X1), X2, X3) mark#(U82(X1, X2, X3)) -> mark#(X1) mark#(U83(X1, X2, X3)) -> a__U83#(mark(X1), X2, X3) mark#(U83(X1, X2, X3)) -> mark#(X1) mark#(U84(X1, X2, X3)) -> a__U84#(mark(X1), X2, X3) mark#(U84(X1, X2, X3)) -> mark#(X1) mark#(U91(X1, X2)) -> mark#(X1) mark#(U92(X)) -> mark#(X) mark#(s(X)) -> mark#(X) rules: a__U101(tt, M, N) -> a__U102(a__isNatKind(M), M, N) a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(tt, M, N) -> a__U103(a__isNat(N), M, N) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__U103(tt, M, N) -> a__U104(a__isNatKind(N), M, N) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__U104(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U11(tt, V1, V2) -> a__U12(a__isNatKind(V1), V1, V2) a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(tt, V1, V2) -> a__U13(a__isNatKind(V2), V1, V2) a__U12(X1, X2, X3) -> U12(X1, X2, X3) a__U13(tt, V1, V2) -> a__U14(a__isNatKind(V2), V1, V2) a__U13(X1, X2, X3) -> U13(X1, X2, X3) a__U14(tt, V1, V2) -> a__U15(a__isNat(V1), V2) a__U14(X1, X2, X3) -> U14(X1, X2, X3) a__U15(tt, V2) -> a__U16(a__isNat(V2)) a__U15(X1, X2) -> U15(X1, X2) a__U16(tt) -> tt a__U16(X) -> U16(X) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V1, V2) -> a__U32(a__isNatKind(V1), V1, V2) a__U31(X1, X2, X3) -> U31(X1, X2, X3) a__U32(tt, V1, V2) -> a__U33(a__isNatKind(V2), V1, V2) a__U32(X1, X2, X3) -> U32(X1, X2, X3) a__U33(tt, V1, V2) -> a__U34(a__isNatKind(V2), V1, V2) a__U33(X1, X2, X3) -> U33(X1, X2, X3) a__U34(tt, V1, V2) -> a__U35(a__isNat(V1), V2) a__U34(X1, X2, X3) -> U34(X1, X2, X3) a__U35(tt, V2) -> a__U36(a__isNat(V2)) a__U35(X1, X2) -> U35(X1, X2) a__U36(tt) -> tt a__U36(X) -> U36(X) a__U41(tt, V2) -> a__U42(a__isNatKind(V2)) a__U41(X1, X2) -> U41(X1, X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt) -> tt a__U51(X) -> U51(X) a__U61(tt, V2) -> a__U62(a__isNatKind(V2)) a__U61(X1, X2) -> U61(X1, X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt, N) -> a__U72(a__isNatKind(N), N) a__U71(X1, X2) -> U71(X1, X2) a__U72(tt, N) -> mark(N) a__U72(X1, X2) -> U72(X1, X2) a__U81(tt, M, N) -> a__U82(a__isNatKind(M), M, N) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(tt, M, N) -> a__U83(a__isNat(N), M, N) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__U83(tt, M, N) -> a__U84(a__isNatKind(N), M, N) a__U83(X1, X2, X3) -> U83(X1, X2, X3) a__U84(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U84(X1, X2, X3) -> U84(X1, X2, X3) a__U91(tt, N) -> a__U92(a__isNatKind(N)) a__U91(X1, X2) -> U91(X1, X2) a__U92(tt) -> 0 a__U92(X) -> U92(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__U11(a__isNatKind(V1), V1, V2) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(x(V1, V2)) -> a__U31(a__isNatKind(V1), V1, V2) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(plus(V1, V2)) -> a__U41(a__isNatKind(V1), V2) a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1)) a__isNatKind(x(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(X) -> isNatKind(X) a__plus(N, 0) -> a__U71(a__isNat(N), N) a__plus(N, s(M)) -> a__U81(a__isNat(M), M, N) a__plus(X1, X2) -> plus(X1, X2) a__x(N, 0) -> a__U91(a__isNat(N), N) a__x(N, s(M)) -> a__U101(a__isNat(M), M, N) a__x(X1, X2) -> x(X1, X2) mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNat(X)) -> a__isNat(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(U13(X1, X2, X3)) -> a__U13(mark(X1), X2, X3) mark(U14(X1, X2, X3)) -> a__U14(mark(X1), X2, X3) mark(U15(X1, X2)) -> a__U15(mark(X1), X2) mark(U16(X)) -> a__U16(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3) mark(U32(X1, X2, X3)) -> a__U32(mark(X1), X2, X3) mark(U33(X1, X2, X3)) -> a__U33(mark(X1), X2, X3) mark(U34(X1, X2, X3)) -> a__U34(mark(X1), X2, X3) mark(U35(X1, X2)) -> a__U35(mark(X1), X2) mark(U36(X)) -> a__U36(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(U51(X)) -> a__U51(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X1, X2)) -> a__U72(mark(X1), X2) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) mark(U91(X1, X2)) -> a__U91(mark(X1), X2) mark(U92(X)) -> a__U92(mark(X)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 the pairs a__U104#(tt, M, N) -> a__plus#(a__x(mark(N), mark(M)), mark(N)) a__U104#(tt, M, N) -> a__x#(mark(N), mark(M)) a__U104#(tt, M, N) -> mark#(N) a__U104#(tt, M, N) -> mark#(M) a__U84#(tt, M, N) -> mark#(M) a__x#(N, s(M)) -> a__U101#(a__isNat(M), M, N) mark#(U101(X1, X2, X3)) -> mark#(X1) mark#(U102(X1, X2, X3)) -> mark#(X1) mark#(U103(X1, X2, X3)) -> mark#(X1) mark#(U104(X1, X2, X3)) -> mark#(X1) mark#(plus(X1, X2)) -> a__plus#(mark(X1), mark(X2)) mark#(plus(X1, X2)) -> mark#(X1) mark#(plus(X1, X2)) -> mark#(X2) mark#(x(X1, X2)) -> mark#(X1) mark#(x(X1, X2)) -> mark#(X2) mark#(U71(X1, X2)) -> a__U71#(mark(X1), X2) mark#(U71(X1, X2)) -> mark#(X1) mark#(U72(X1, X2)) -> a__U72#(mark(X1), X2) mark#(U72(X1, X2)) -> mark#(X1) mark#(U81(X1, X2, X3)) -> a__U81#(mark(X1), X2, X3) mark#(U81(X1, X2, X3)) -> mark#(X1) mark#(U82(X1, X2, X3)) -> a__U82#(mark(X1), X2, X3) mark#(U82(X1, X2, X3)) -> mark#(X1) mark#(U83(X1, X2, X3)) -> a__U83#(mark(X1), X2, X3) mark#(U83(X1, X2, X3)) -> mark#(X1) mark#(U84(X1, X2, X3)) -> a__U84#(mark(X1), X2, X3) mark#(U84(X1, X2, X3)) -> mark#(X1) mark#(U91(X1, X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) could not apply the generic root reduction pair processor with the following SCNP-version with mu = MAX and the level mapping defined by pi(a__U102#) = [(epsilon,0),(3,2)] pi(a__U103#) = [(epsilon,0),(1,0)] pi(a__U104#) = [(epsilon,0)] pi(a__plus#) = [(epsilon,0),(1,0),(2,0)] pi(a__U71#) = [(epsilon,0),(1,0)] pi(a__U72#) = [(epsilon,0),(1,0)] pi(mark#) = [(1,0)] pi(a__U101#) = [(epsilon,0),(1,0)] pi(a__x#) = [(epsilon,0),(1,59),(2,63)] pi(a__U81#) = [(epsilon,0),(2,58)] pi(a__U82#) = [(epsilon,0),(2,58)] pi(a__U83#) = [(epsilon,0),(2,4),(3,0)] pi(a__U84#) = [(epsilon,0),(2,3)] Argument Filter: pi(a__U102#/3) = [1,2,3] pi(tt/0) = [] pi(a__U103#/3) = [1,2,3] pi(a__isNat/1) = [] pi(a__U104#/3) = [1,2,3] pi(a__isNatKind/1) = [] pi(a__plus#/2) = 2 pi(a__x/2) = [1,2] pi(mark/1) = 1 pi(0/0) = [] pi(a__U71#/2) = 2 pi(a__U72#/2) = 2 pi(mark#/1) = [] pi(U101/3) = [1,2,3] pi(a__U101#/3) = [1,2,3] pi(U102/3) = [1,2,3] pi(U103/3) = [1,2,3] pi(U104/3) = [1,2,3] pi(a__x#/2) = [1,2] pi(s/1) = [1] pi(plus/2) = [1,2] pi(a__U81#/3) = 3 pi(a__U82#/3) = 3 pi(a__U83#/3) = 2 pi(a__U84#/3) = 3 pi(x/2) = [1,2] pi(U11/3) = 1 pi(U12/3) = 1 pi(U13/3) = 1 pi(U14/3) = 1 pi(U15/2) = 1 pi(U16/1) = 1 pi(U21/2) = 1 pi(U22/2) = 1 pi(U23/1) = 1 pi(U31/3) = 1 pi(U32/3) = 1 pi(U33/3) = 1 pi(U34/3) = 1 pi(U35/2) = 1 pi(U36/1) = 1 pi(U41/2) = 1 pi(U42/1) = 1 pi(U51/1) = 1 pi(U61/2) = 1 pi(U62/1) = 1 pi(U71/2) = [1,2] pi(U72/2) = [1,2] pi(U81/3) = [1,2,3] pi(U82/3) = [1,2,3] pi(U83/3) = [1,2,3] pi(U84/3) = [1,2,3] pi(U91/2) = [2,1] pi(U92/1) = 1 pi(a__U11/3) = 1 pi(a__U21/2) = 1 pi(a__U31/3) = 1 pi(isNat/1) = [] pi(a__U41/2) = 1 pi(a__U51/1) = 1 pi(a__U61/2) = 1 pi(isNatKind/1) = [] pi(a__U102/3) = [1,2,3] pi(a__U103/3) = [1,2,3] pi(a__U104/3) = [1,2,3] pi(a__plus/2) = [1,2] pi(a__U71/2) = [1,2] pi(a__U72/2) = [1,2] pi(a__U101/3) = [1,2,3] pi(a__U12/3) = 1 pi(a__U13/3) = 1 pi(a__U14/3) = 1 pi(a__U15/2) = 1 pi(a__U16/1) = 1 pi(a__U22/2) = 1 pi(a__U23/1) = 1 pi(a__U32/3) = 1 pi(a__U33/3) = 1 pi(a__U34/3) = 1 pi(a__U35/2) = 1 pi(a__U36/1) = 1 pi(a__U42/1) = 1 pi(a__U62/1) = 1 pi(a__U81/3) = [1,2,3] pi(a__U82/3) = [1,2,3] pi(a__U83/3) = [1,2,3] pi(a__U84/3) = [1,2,3] pi(a__U91/2) = [2,1] pi(a__U92/1) = 1 RPO with the following precedence precedence(a__U102#[3]) = 4 precedence(tt[0]) = 2 precedence(a__U103#[3]) = 4 precedence(a__isNat[1]) = 2 precedence(a__U104#[3]) = 4 precedence(a__isNatKind[1]) = 2 precedence(a__x[2]) = 4 precedence(0[0]) = 2 precedence(mark#[1]) = 2 precedence(U101[3]) = 4 precedence(a__U101#[3]) = 4 precedence(U102[3]) = 4 precedence(U103[3]) = 4 precedence(U104[3]) = 4 precedence(a__x#[2]) = 4 precedence(s[1]) = 2 precedence(plus[2]) = 3 precedence(x[2]) = 4 precedence(U71[2]) = 0 precedence(U72[2]) = 0 precedence(U81[3]) = 3 precedence(U82[3]) = 3 precedence(U83[3]) = 3 precedence(U84[3]) = 3 precedence(U91[2]) = 1 precedence(isNat[1]) = 2 precedence(isNatKind[1]) = 2 precedence(a__U102[3]) = 4 precedence(a__U103[3]) = 4 precedence(a__U104[3]) = 4 precedence(a__plus[2]) = 3 precedence(a__U71[2]) = 0 precedence(a__U72[2]) = 0 precedence(a__U101[3]) = 4 precedence(a__U81[3]) = 3 precedence(a__U82[3]) = 3 precedence(a__U83[3]) = 3 precedence(a__U84[3]) = 3 precedence(a__U91[2]) = 1 precedence(_) = 0 and the following status status(a__U102#[3]) = mul status(tt[0]) = mul status(a__U103#[3]) = mul status(a__isNat[1]) = mul status(a__U104#[3]) = mul status(a__isNatKind[1]) = mul status(a__x[2]) = mul status(0[0]) = mul status(mark#[1]) = lex status(U101[3]) = mul status(a__U101#[3]) = mul status(U102[3]) = mul status(U103[3]) = mul status(U104[3]) = mul status(a__x#[2]) = mul status(s[1]) = mul status(plus[2]) = mul status(x[2]) = mul status(U71[2]) = mul status(U72[2]) = mul status(U81[3]) = mul status(U82[3]) = mul status(U83[3]) = mul status(U84[3]) = mul status(U91[2]) = lex status(isNat[1]) = mul status(isNatKind[1]) = mul status(a__U102[3]) = mul status(a__U103[3]) = mul status(a__U104[3]) = mul status(a__plus[2]) = mul status(a__U71[2]) = mul status(a__U72[2]) = mul status(a__U101[3]) = mul status(a__U81[3]) = mul status(a__U82[3]) = mul status(a__U83[3]) = mul status(a__U84[3]) = mul status(a__U91[2]) = lex status(_) = lex problem when orienting (usable) rules could not orient a__U101(tt, M, N) >= a__U102(a__isNatKind(M), M, N) pi( a__U101(tt, M, N) ) = a__U101(tt, M, N) pi( a__U102(a__isNatKind(M), M, N) ) = a__U102(a__isNatKind, M, N) could not orient a__U101(tt, M, N) >=RPO a__U102(a__isNatKind, M, N)