The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))

(2) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
top(ok(X)) → top(active(X))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
tail(ok(X)) → ok(tail(X))
snd(ok(X)) → ok(snd(X))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
U64(mark(X1), X2) → mark(U64(X1, X2))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
head(mark(X)) → mark(head(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
natsFrom(ok(X)) → ok(natsFrom(X))
fst(mark(X)) → mark(fst(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
tail(mark(X)) → mark(tail(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
s(ok(X)) → ok(s(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
proper(0) → ok(0)
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
snd(mark(X)) → mark(snd(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
head(ok(X)) → ok(head(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
fst(ok(X)) → ok(fst(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

Rewrite Strategy: FULL

(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
top(ok(X)) → top(active(X))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
tail(ok(X)) → ok(tail(X))
snd(ok(X)) → ok(snd(X))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
U64(mark(X1), X2) → mark(U64(X1, X2))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
head(mark(X)) → mark(head(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
natsFrom(ok(X)) → ok(natsFrom(X))
fst(mark(X)) → mark(fst(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
tail(mark(X)) → mark(tail(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
s(ok(X)) → ok(s(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
proper(0) → ok(0)
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
snd(mark(X)) → mark(snd(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
head(ok(X)) → ok(head(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
fst(ok(X)) → ok(fst(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

Rewrite Strategy: INNERMOST

(5) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
nil0() → 0
00() → 0
U620(0, 0, 0, 0) → 1
top0(0) → 2
U810(0, 0, 0) → 3
U320(0, 0) → 4
cons0(0, 0) → 5
tail0(0) → 6
snd0(0) → 7
U520(0, 0) → 8
pair0(0, 0) → 9
sel0(0, 0) → 10
splitAt0(0, 0) → 11
U420(0, 0, 0) → 12
proper0(0) → 13
U640(0, 0) → 14
U820(0, 0, 0) → 15
U110(0, 0, 0) → 16
U310(0, 0) → 17
head0(0) → 18
U610(0, 0, 0, 0) → 19
U630(0, 0, 0, 0) → 20
natsFrom0(0) → 21
fst0(0) → 22
afterNth0(0, 0) → 23
U210(0, 0) → 24
s0(0) → 25
U710(0, 0) → 26
take0(0, 0) → 27
U510(0, 0) → 28
U720(0, 0) → 29
U120(0, 0, 0) → 30
U220(0, 0) → 31
U410(0, 0, 0) → 32
U621(0, 0, 0, 0) → 33
ok1(33) → 1
active1(0) → 34
top1(34) → 2
U621(0, 0, 0, 0) → 35
mark1(35) → 1
U811(0, 0, 0) → 36
mark1(36) → 3
U321(0, 0) → 37
mark1(37) → 4
cons1(0, 0) → 38
ok1(38) → 5
U811(0, 0, 0) → 39
ok1(39) → 3
tail1(0) → 40
ok1(40) → 6
snd1(0) → 41
ok1(41) → 7
U521(0, 0) → 42
ok1(42) → 8
pair1(0, 0) → 43
mark1(43) → 9
U321(0, 0) → 44
ok1(44) → 4
sel1(0, 0) → 45
ok1(45) → 10
splitAt1(0, 0) → 46
mark1(46) → 11
sel1(0, 0) → 47
mark1(47) → 10
U421(0, 0, 0) → 48
mark1(48) → 12
tt1() → 49
ok1(49) → 13
nil1() → 50
ok1(50) → 13
U641(0, 0) → 51
mark1(51) → 14
U821(0, 0, 0) → 52
ok1(52) → 15
U111(0, 0, 0) → 53
mark1(53) → 16
U821(0, 0, 0) → 54
mark1(54) → 15
U311(0, 0) → 55
ok1(55) → 17
head1(0) → 56
mark1(56) → 18
U611(0, 0, 0, 0) → 57
ok1(57) → 19
U111(0, 0, 0) → 58
ok1(58) → 16
U421(0, 0, 0) → 59
ok1(59) → 12
U631(0, 0, 0, 0) → 60
ok1(60) → 20
natsFrom1(0) → 61
ok1(61) → 21
fst1(0) → 62
mark1(62) → 22
afterNth1(0, 0) → 63
ok1(63) → 23
tail1(0) → 64
mark1(64) → 6
U211(0, 0) → 65
ok1(65) → 24
s1(0) → 66
ok1(66) → 25
U711(0, 0) → 67
ok1(67) → 26
01() → 68
ok1(68) → 13
afterNth1(0, 0) → 69
mark1(69) → 23
take1(0, 0) → 70
mark1(70) → 27
U511(0, 0) → 71
mark1(71) → 28
U721(0, 0) → 72
mark1(72) → 29
U121(0, 0, 0) → 73
ok1(73) → 30
U221(0, 0) → 74
mark1(74) → 31
U121(0, 0, 0) → 75
mark1(75) → 30
U511(0, 0) → 76
ok1(76) → 28
snd1(0) → 77
mark1(77) → 7
take1(0, 0) → 78
ok1(78) → 27
U721(0, 0) → 79
ok1(79) → 29
U211(0, 0) → 80
mark1(80) → 24
U521(0, 0) → 81
mark1(81) → 8
head1(0) → 82
ok1(82) → 18
pair1(0, 0) → 83
ok1(83) → 9
U631(0, 0, 0, 0) → 84
mark1(84) → 20
U611(0, 0, 0, 0) → 85
mark1(85) → 19
U641(0, 0) → 86
ok1(86) → 14
U711(0, 0) → 87
mark1(87) → 26
natsFrom1(0) → 88
mark1(88) → 21
s1(0) → 89
mark1(89) → 25
splitAt1(0, 0) → 90
ok1(90) → 11
fst1(0) → 91
ok1(91) → 22
U411(0, 0, 0) → 92
mark1(92) → 32
U221(0, 0) → 93
ok1(93) → 31
U311(0, 0) → 94
mark1(94) → 17
cons1(0, 0) → 95
mark1(95) → 5
proper1(0) → 96
top1(96) → 2
U411(0, 0, 0) → 97
ok1(97) → 32
ok1(33) → 33
ok1(33) → 35
mark1(35) → 33
mark1(35) → 35
mark1(36) → 36
mark1(36) → 39
mark1(37) → 37
mark1(37) → 44
ok1(38) → 38
ok1(38) → 95
ok1(39) → 36
ok1(39) → 39
ok1(40) → 40
ok1(40) → 64
ok1(41) → 41
ok1(41) → 77
ok1(42) → 42
ok1(42) → 81
mark1(43) → 43
mark1(43) → 83
ok1(44) → 37
ok1(44) → 44
ok1(45) → 45
ok1(45) → 47
mark1(46) → 46
mark1(46) → 90
mark1(47) → 45
mark1(47) → 47
mark1(48) → 48
mark1(48) → 59
ok1(49) → 96
ok1(50) → 96
mark1(51) → 51
mark1(51) → 86
ok1(52) → 52
ok1(52) → 54
mark1(53) → 53
mark1(53) → 58
mark1(54) → 52
mark1(54) → 54
ok1(55) → 55
ok1(55) → 94
mark1(56) → 56
mark1(56) → 82
ok1(57) → 57
ok1(57) → 85
ok1(58) → 53
ok1(58) → 58
ok1(59) → 48
ok1(59) → 59
ok1(60) → 60
ok1(60) → 84
ok1(61) → 61
ok1(61) → 88
mark1(62) → 62
mark1(62) → 91
ok1(63) → 63
ok1(63) → 69
mark1(64) → 40
mark1(64) → 64
ok1(65) → 65
ok1(65) → 80
ok1(66) → 66
ok1(66) → 89
ok1(67) → 67
ok1(67) → 87
ok1(68) → 96
mark1(69) → 63
mark1(69) → 69
mark1(70) → 70
mark1(70) → 78
mark1(71) → 71
mark1(71) → 76
mark1(72) → 72
mark1(72) → 79
ok1(73) → 73
ok1(73) → 75
mark1(74) → 74
mark1(74) → 93
mark1(75) → 73
mark1(75) → 75
ok1(76) → 71
ok1(76) → 76
mark1(77) → 41
mark1(77) → 77
ok1(78) → 70
ok1(78) → 78
ok1(79) → 72
ok1(79) → 79
mark1(80) → 65
mark1(80) → 80
mark1(81) → 42
mark1(81) → 81
ok1(82) → 56
ok1(82) → 82
ok1(83) → 43
ok1(83) → 83
mark1(84) → 60
mark1(84) → 84
mark1(85) → 57
mark1(85) → 85
ok1(86) → 51
ok1(86) → 86
mark1(87) → 67
mark1(87) → 87
mark1(88) → 61
mark1(88) → 88
mark1(89) → 66
mark1(89) → 89
ok1(90) → 46
ok1(90) → 90
ok1(91) → 62
ok1(91) → 91
mark1(92) → 92
mark1(92) → 97
ok1(93) → 74
ok1(93) → 93
mark1(94) → 55
mark1(94) → 94
mark1(95) → 38
mark1(95) → 95
ok1(97) → 92
ok1(97) → 97
active2(49) → 98
top2(98) → 2
active2(50) → 98
active2(68) → 98

(6) BOUNDS(1, n^1)