YES

The TRS could be proven terminating. The proof took 1139 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (525ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4iUR (510ms).
 |    | – Problem 3 was processed with processor DependencyGraph (1ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

splitAt#(s(N), cons(X, XS))activate#(XS)U81#(tt, N, XS)activate#(N)
U61#(tt, N, X, XS)activate#(XS)U41#(tt, N, XS)activate#(N)
U31#(tt, N)U32#(tt, activate(N))U62#(tt, N, X, XS)activate#(XS)
U82#(tt, N, XS)activate#(N)take#(N, XS)U81#(tt, N, XS)
U51#(tt, Y)U52#(tt, activate(Y))U12#(tt, N, XS)snd#(splitAt(activate(N), activate(XS)))
U11#(tt, N, XS)U12#(tt, activate(N), activate(XS))U82#(tt, N, XS)fst#(splitAt(activate(N), activate(XS)))
U42#(tt, N, XS)activate#(N)U42#(tt, N, XS)head#(afterNth(activate(N), activate(XS)))
U42#(tt, N, XS)activate#(XS)U21#(tt, X)U22#(tt, activate(X))
U81#(tt, N, XS)U82#(tt, activate(N), activate(XS))tail#(cons(N, XS))U71#(tt, activate(XS))
U63#(tt, N, X, XS)activate#(XS)snd#(pair(X, Y))U51#(tt, Y)
U12#(tt, N, XS)activate#(N)U62#(tt, N, X, XS)activate#(X)
U52#(tt, Y)activate#(Y)U61#(tt, N, X, XS)U62#(tt, activate(N), activate(X), activate(XS))
fst#(pair(X, Y))U21#(tt, X)U72#(tt, XS)activate#(XS)
U62#(tt, N, X, XS)activate#(N)U61#(tt, N, X, XS)activate#(N)
splitAt#(s(N), cons(X, XS))U61#(tt, N, X, activate(XS))tail#(cons(N, XS))activate#(XS)
U21#(tt, X)activate#(X)U71#(tt, XS)U72#(tt, activate(XS))
afterNth#(N, XS)U11#(tt, N, XS)U51#(tt, Y)activate#(Y)
U41#(tt, N, XS)U42#(tt, activate(N), activate(XS))U42#(tt, N, XS)afterNth#(activate(N), activate(XS))
sel#(N, XS)U41#(tt, N, XS)U82#(tt, N, XS)splitAt#(activate(N), activate(XS))
U11#(tt, N, XS)activate#(N)head#(cons(N, XS))U31#(tt, N)
U63#(tt, N, X, XS)U64#(splitAt(activate(N), activate(XS)), activate(X))U82#(tt, N, XS)activate#(XS)
activate#(n__natsFrom(X))natsFrom#(X)U32#(tt, N)activate#(N)
U61#(tt, N, X, XS)activate#(X)U63#(tt, N, X, XS)activate#(X)
U81#(tt, N, XS)activate#(XS)U31#(tt, N)activate#(N)
U64#(pair(YS, ZS), X)activate#(X)U12#(tt, N, XS)activate#(XS)
U12#(tt, N, XS)splitAt#(activate(N), activate(XS))U71#(tt, XS)activate#(XS)
U11#(tt, N, XS)activate#(XS)U63#(tt, N, X, XS)splitAt#(activate(N), activate(XS))
U62#(tt, N, X, XS)U63#(tt, activate(N), activate(X), activate(XS))U22#(tt, X)activate#(X)
U63#(tt, N, X, XS)activate#(N)U41#(tt, N, XS)activate#(XS)

Rewrite Rules

U11(tt, N, XS)U12(tt, activate(N), activate(XS))U12(tt, N, XS)snd(splitAt(activate(N), activate(XS)))
U21(tt, X)U22(tt, activate(X))U22(tt, X)activate(X)
U31(tt, N)U32(tt, activate(N))U32(tt, N)activate(N)
U41(tt, N, XS)U42(tt, activate(N), activate(XS))U42(tt, N, XS)head(afterNth(activate(N), activate(XS)))
U51(tt, Y)U52(tt, activate(Y))U52(tt, Y)activate(Y)
U61(tt, N, X, XS)U62(tt, activate(N), activate(X), activate(XS))U62(tt, N, X, XS)U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS)U64(splitAt(activate(N), activate(XS)), activate(X))U64(pair(YS, ZS), X)pair(cons(activate(X), YS), ZS)
U71(tt, XS)U72(tt, activate(XS))U72(tt, XS)activate(XS)
U81(tt, N, XS)U82(tt, activate(N), activate(XS))U82(tt, N, XS)fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS)U11(tt, N, XS)fst(pair(X, Y))U21(tt, X)
head(cons(N, XS))U31(tt, N)natsFrom(N)cons(N, n__natsFrom(s(N)))
sel(N, XS)U41(tt, N, XS)snd(pair(X, Y))U51(tt, Y)
splitAt(0, XS)pair(nil, XS)splitAt(s(N), cons(X, XS))U61(tt, N, X, activate(XS))
tail(cons(N, XS))U71(tt, activate(XS))take(N, XS)U81(tt, N, XS)
natsFrom(X)n__natsFrom(X)activate(n__natsFrom(X))natsFrom(X)
activate(X)X

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, U64, activate, fst, U63, U62, U61, U42, U41, head, U21, U22, snd, cons, tail, n__natsFrom, splitAt, U71, U72, 0, s, U51, tt, U82, take, U52, U81, U11, U12, U31, afterNth, U32, sel, nil

Strategy


The following SCCs where found

splitAt#(s(N), cons(X, XS)) → U61#(tt, N, X, activate(XS))U61#(tt, N, X, XS) → U62#(tt, activate(N), activate(X), activate(XS))
U63#(tt, N, X, XS) → splitAt#(activate(N), activate(XS))U62#(tt, N, X, XS) → U63#(tt, activate(N), activate(X), activate(XS))

Problem 2: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

splitAt#(s(N), cons(X, XS))U61#(tt, N, X, activate(XS))U61#(tt, N, X, XS)U62#(tt, activate(N), activate(X), activate(XS))
U63#(tt, N, X, XS)splitAt#(activate(N), activate(XS))U62#(tt, N, X, XS)U63#(tt, activate(N), activate(X), activate(XS))

Rewrite Rules

U11(tt, N, XS)U12(tt, activate(N), activate(XS))U12(tt, N, XS)snd(splitAt(activate(N), activate(XS)))
U21(tt, X)U22(tt, activate(X))U22(tt, X)activate(X)
U31(tt, N)U32(tt, activate(N))U32(tt, N)activate(N)
U41(tt, N, XS)U42(tt, activate(N), activate(XS))U42(tt, N, XS)head(afterNth(activate(N), activate(XS)))
U51(tt, Y)U52(tt, activate(Y))U52(tt, Y)activate(Y)
U61(tt, N, X, XS)U62(tt, activate(N), activate(X), activate(XS))U62(tt, N, X, XS)U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS)U64(splitAt(activate(N), activate(XS)), activate(X))U64(pair(YS, ZS), X)pair(cons(activate(X), YS), ZS)
U71(tt, XS)U72(tt, activate(XS))U72(tt, XS)activate(XS)
U81(tt, N, XS)U82(tt, activate(N), activate(XS))U82(tt, N, XS)fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS)U11(tt, N, XS)fst(pair(X, Y))U21(tt, X)
head(cons(N, XS))U31(tt, N)natsFrom(N)cons(N, n__natsFrom(s(N)))
sel(N, XS)U41(tt, N, XS)snd(pair(X, Y))U51(tt, Y)
splitAt(0, XS)pair(nil, XS)splitAt(s(N), cons(X, XS))U61(tt, N, X, activate(XS))
tail(cons(N, XS))U71(tt, activate(XS))take(N, XS)U81(tt, N, XS)
natsFrom(X)n__natsFrom(X)activate(n__natsFrom(X))natsFrom(X)
activate(X)X

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, U64, activate, fst, U63, U62, U61, U42, U41, head, U21, U22, snd, cons, tail, n__natsFrom, splitAt, U71, U72, 0, s, U51, tt, U82, take, U52, U81, U11, U12, U31, afterNth, U32, sel, nil

Strategy


Polynomial Interpretation

Improved Usable rules

activate(n__natsFrom(X))natsFrom(X)natsFrom(N)cons(N, n__natsFrom(s(N)))
activate(X)XnatsFrom(X)n__natsFrom(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

splitAt#(s(N), cons(X, XS))U61#(tt, N, X, activate(XS))U62#(tt, N, X, XS)U63#(tt, activate(N), activate(X), activate(XS))

Problem 3: DependencyGraph



Dependency Pair Problem

Dependency Pairs

U61#(tt, N, X, XS)U62#(tt, activate(N), activate(X), activate(XS))U63#(tt, N, X, XS)splitAt#(activate(N), activate(XS))

Rewrite Rules

U11(tt, N, XS)U12(tt, activate(N), activate(XS))U12(tt, N, XS)snd(splitAt(activate(N), activate(XS)))
U21(tt, X)U22(tt, activate(X))U22(tt, X)activate(X)
U31(tt, N)U32(tt, activate(N))U32(tt, N)activate(N)
U41(tt, N, XS)U42(tt, activate(N), activate(XS))U42(tt, N, XS)head(afterNth(activate(N), activate(XS)))
U51(tt, Y)U52(tt, activate(Y))U52(tt, Y)activate(Y)
U61(tt, N, X, XS)U62(tt, activate(N), activate(X), activate(XS))U62(tt, N, X, XS)U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS)U64(splitAt(activate(N), activate(XS)), activate(X))U64(pair(YS, ZS), X)pair(cons(activate(X), YS), ZS)
U71(tt, XS)U72(tt, activate(XS))U72(tt, XS)activate(XS)
U81(tt, N, XS)U82(tt, activate(N), activate(XS))U82(tt, N, XS)fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS)U11(tt, N, XS)fst(pair(X, Y))U21(tt, X)
head(cons(N, XS))U31(tt, N)natsFrom(N)cons(N, n__natsFrom(s(N)))
sel(N, XS)U41(tt, N, XS)snd(pair(X, Y))U51(tt, Y)
splitAt(0, XS)pair(nil, XS)splitAt(s(N), cons(X, XS))U61(tt, N, X, activate(XS))
tail(cons(N, XS))U71(tt, activate(XS))take(N, XS)U81(tt, N, XS)
natsFrom(X)n__natsFrom(X)activate(n__natsFrom(X))natsFrom(X)
activate(X)X

Original Signature

Termination of terms over the following signature is verified: pair, natsFrom, fst, activate, U64, U63, U62, U61, U42, U41, head, U21, U22, snd, cons, tail, n__natsFrom, splitAt, U71, U72, 0, s, U51, tt, take, U82, U81, U52, U11, U12, afterNth, U31, U32, sel, nil

Strategy


There are no SCCs!