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__U31#(tt, N) -> mark#(N) a__U41#(tt, M, N) -> a__U42#(a__isNat(N), M, N) a__U42#(tt, M, N) -> a__plus#(mark(N), mark(M)) a__U42#(tt, M, N) -> mark#(N) a__U42#(tt, M, N) -> mark#(M) a__plus#(N, 0) -> a__U31#(a__isNat(N), N) a__plus#(N, s(M)) -> a__U41#(a__isNat(M), M, N) mark#(U11(X1, X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X1, X2)) -> a__U31#(mark(X1), X2) mark#(U31(X1, X2)) -> mark#(X1) mark#(U41(X1, X2, X3)) -> a__U41#(mark(X1), X2, X3) mark#(U41(X1, X2, X3)) -> mark#(X1) mark#(U42(X1, X2, X3)) -> a__U42#(mark(X1), X2, X3) mark#(U42(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#(s(X)) -> mark#(X) rules: a__U11(tt, V2) -> a__U12(a__isNat(V2)) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt) -> tt a__U12(X) -> U12(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt, N) -> mark(N) a__U31(X1, X2) -> U31(X1, X2) a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__plus(N, 0) -> a__U31(a__isNat(N), N) a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) a__plus(X1, X2) -> plus(X1, X2) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 the pairs a__U31#(tt, N) -> mark#(N) a__U42#(tt, M, N) -> mark#(N) a__U42#(tt, M, N) -> mark#(M) a__plus#(N, 0) -> a__U31#(a__isNat(N), N) mark#(U31(X1, X2)) -> a__U31#(mark(X1), X2) mark#(U31(X1, X2)) -> mark#(X1) mark#(U41(X1, X2, X3)) -> a__U41#(mark(X1), X2, X3) mark#(U41(X1, X2, X3)) -> mark#(X1) mark#(U42(X1, X2, X3)) -> a__U42#(mark(X1), X2, X3) mark#(U42(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#(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(mark#) = [(1,0)] pi(a__U31#) = [(2,4)] pi(a__U41#) = [(2,15),(3,8)] pi(a__U42#) = [(2,12),(3,8)] pi(a__plus#) = [(1,8),(2,12)] Argument Filter: pi(U11/2) = 1 pi(U12/1) = 1 pi(U21/1) = 1 pi(U31/2) = [1,2] pi(mark/1) = 1 pi(tt/0) = [] pi(U41/3) = [1,2,3] pi(a__isNat/1) = [] pi(0/0) = [] pi(s/1) = [1] pi(U42/3) = [1,2,3] pi(plus/2) = [1,2] pi(a__U11/2) = 1 pi(a__U12/1) = 1 pi(isNat/1) = [] pi(a__U21/1) = 1 pi(a__U31/2) = [1,2] pi(a__plus/2) = [1,2] pi(a__U41/3) = [1,2,3] pi(a__U42/3) = [1,2,3] RPO with the following precedence precedence(U31[2]) = 0 precedence(tt[0]) = 1 precedence(U41[3]) = 3 precedence(a__isNat[1]) = 1 precedence(0[0]) = 4 precedence(s[1]) = 2 precedence(U42[3]) = 3 precedence(plus[2]) = 3 precedence(isNat[1]) = 1 precedence(a__U31[2]) = 0 precedence(a__plus[2]) = 3 precedence(a__U41[3]) = 3 precedence(a__U42[3]) = 3 precedence(_) = 0 and the following status status(U31[2]) = lex status(tt[0]) = mul status(U41[3]) = mul status(a__isNat[1]) = lex status(0[0]) = mul status(s[1]) = mul status(U42[3]) = mul status(plus[2]) = mul status(isNat[1]) = lex status(a__U31[2]) = lex status(a__plus[2]) = mul status(a__U41[3]) = mul status(a__U42[3]) = mul status(_) = lex problem when orienting (usable) rules could not orient a__U41(tt, M, N) >= a__U42(a__isNat(N), M, N) pi( a__U41(tt, M, N) ) = a__U41(tt, M, N) pi( a__U42(a__isNat(N), M, N) ) = a__U42(a__isNat, M, N) could not orient a__U41(tt, M, N) >=RPO a__U42(a__isNat, M, N)