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

(0) 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:

active(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X1), X2) → mark(U61(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
U82(mark(X1), X2) → mark(U82(X1, X2))
U91(mark(X1), X2) → mark(U91(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
isLNat(ok(X)) → ok(isLNat(X))
isPLNat(ok(X)) → ok(isPLNat(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

(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(U101(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(X)
active(U31(tt, N)) → mark(N)
active(U41(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U51(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U61(tt, Y)) → mark(Y)
active(U71(tt, XS)) → mark(pair(nil, XS))
active(U81(tt, N, X, XS)) → mark(U82(splitAt(N, XS), X))
active(U82(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U91(tt, XS)) → mark(XS)
active(afterNth(N, XS)) → mark(U11(and(isNatural(N), isLNat(XS)), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(isLNat(X), isLNat(Y)), X))
active(head(cons(N, XS))) → mark(U31(and(isNatural(N), isLNat(XS)), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(cons(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isLNat(fst(V1))) → mark(isPLNat(V1))
active(isLNat(natsFrom(V1))) → mark(isNatural(V1))
active(isLNat(snd(V1))) → mark(isPLNat(V1))
active(isLNat(tail(V1))) → mark(isLNat(V1))
active(isLNat(take(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(isLNat(V1))
active(isNatural(s(V1))) → mark(isNatural(V1))
active(isNatural(sel(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(isPLNat(pair(V1, V2))) → mark(and(isLNat(V1), isLNat(V2)))
active(isPLNat(splitAt(V1, V2))) → mark(and(isNatural(V1), isLNat(V2)))
active(natsFrom(N)) → mark(U41(isNatural(N), N))
active(sel(N, XS)) → mark(U51(and(isNatural(N), isLNat(XS)), N, XS))
active(snd(pair(X, Y))) → mark(U61(and(isLNat(X), isLNat(Y)), Y))
active(splitAt(0, XS)) → mark(U71(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS))
active(tail(cons(N, XS))) → mark(U91(and(isNatural(N), isLNat(XS)), XS))
active(take(N, XS)) → mark(U101(and(isNatural(N), isLNat(XS)), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U51(X1, X2, X3)) → U51(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(U61(X1, X2)) → U61(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U81(X1, X2, X3, X4)) → U81(active(X1), X2, X3, X4)
active(U82(X1, X2)) → U82(active(X1), X2)
active(U91(X1, X2)) → U91(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(U81(X1, X2, X3, X4)) → U81(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U82(X1, X2)) → U82(proper(X1), proper(X2))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

(2) 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:

top(ok(X)) → top(active(X))
isNatural(ok(X)) → ok(isNatural(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U82(mark(X1), X2) → mark(U82(X1, X2))
tail(ok(X)) → ok(tail(X))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
snd(ok(X)) → ok(snd(X))
U82(ok(X1), ok(X2)) → ok(U82(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
pair(mark(X1), X2) → mark(pair(X1, X2))
U41(mark(X1), X2) → mark(U41(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))
isPLNat(ok(X)) → ok(isPLNat(X))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
U11(mark(X1), X2, X3) → mark(U11(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))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U61(mark(X1), X2) → mark(U61(X1, X2))
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))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
proper(0) → ok(0)
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U81(mark(X1), X2, X3, X4) → mark(U81(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U81(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U81(X1, X2, X3, X4))
snd(mark(X)) → mark(snd(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
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))
and(mark(X1), X2) → mark(and(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
isLNat(ok(X)) → ok(isLNat(X))
natsFrom(mark(X)) → mark(natsFrom(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
fst(ok(X)) → ok(fst(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))

Rewrite Strategy: INNERMOST

(3) 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]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
nil0() → 0
00() → 0
top0(0) → 1
isNatural0(0) → 2
U910(0, 0) → 3
cons0(0, 0) → 4
U820(0, 0) → 5
tail0(0) → 6
U1010(0, 0, 0) → 7
U610(0, 0) → 8
snd0(0) → 9
and0(0, 0) → 10
U510(0, 0, 0) → 11
pair0(0, 0) → 12
U410(0, 0) → 13
sel0(0, 0) → 14
splitAt0(0, 0) → 15
isPLNat0(0) → 16
proper0(0) → 17
U110(0, 0, 0) → 18
U310(0, 0) → 19
head0(0) → 20
natsFrom0(0) → 21
fst0(0) → 22
afterNth0(0, 0) → 23
U210(0, 0) → 24
s0(0) → 25
U710(0, 0) → 26
U810(0, 0, 0, 0) → 27
take0(0, 0) → 28
isLNat0(0) → 29
active1(0) → 30
top1(30) → 1
isNatural1(0) → 31
ok1(31) → 2
U911(0, 0) → 32
mark1(32) → 3
cons1(0, 0) → 33
ok1(33) → 4
U821(0, 0) → 34
mark1(34) → 5
tail1(0) → 35
ok1(35) → 6
U1011(0, 0, 0) → 36
ok1(36) → 7
U611(0, 0) → 37
ok1(37) → 8
snd1(0) → 38
ok1(38) → 9
U821(0, 0) → 39
ok1(39) → 5
and1(0, 0) → 40
ok1(40) → 10
U511(0, 0, 0) → 41
mark1(41) → 11
pair1(0, 0) → 42
mark1(42) → 12
U411(0, 0) → 43
mark1(43) → 13
sel1(0, 0) → 44
ok1(44) → 14
splitAt1(0, 0) → 45
mark1(45) → 15
sel1(0, 0) → 46
mark1(46) → 14
isPLNat1(0) → 47
ok1(47) → 16
tt1() → 48
ok1(48) → 17
nil1() → 49
ok1(49) → 17
U111(0, 0, 0) → 50
mark1(50) → 18
U311(0, 0) → 51
ok1(51) → 19
head1(0) → 52
mark1(52) → 20
U111(0, 0, 0) → 53
ok1(53) → 18
U611(0, 0) → 54
mark1(54) → 8
natsFrom1(0) → 55
ok1(55) → 21
fst1(0) → 56
mark1(56) → 22
afterNth1(0, 0) → 57
ok1(57) → 23
tail1(0) → 58
mark1(58) → 6
U211(0, 0) → 59
ok1(59) → 24
s1(0) → 60
ok1(60) → 25
U1011(0, 0, 0) → 61
mark1(61) → 7
U711(0, 0) → 62
ok1(62) → 26
01() → 63
ok1(63) → 17
afterNth1(0, 0) → 64
mark1(64) → 23
U811(0, 0, 0, 0) → 65
mark1(65) → 27
take1(0, 0) → 66
mark1(66) → 28
U811(0, 0, 0, 0) → 67
ok1(67) → 27
snd1(0) → 68
mark1(68) → 9
take1(0, 0) → 69
ok1(69) → 28
U211(0, 0) → 70
mark1(70) → 24
U511(0, 0, 0) → 71
ok1(71) → 11
head1(0) → 72
ok1(72) → 20
pair1(0, 0) → 73
ok1(73) → 12
and1(0, 0) → 74
mark1(74) → 10
U411(0, 0) → 75
ok1(75) → 13
isLNat1(0) → 76
ok1(76) → 29
natsFrom1(0) → 77
mark1(77) → 21
U711(0, 0) → 78
mark1(78) → 26
s1(0) → 79
mark1(79) → 25
fst1(0) → 80
ok1(80) → 22
splitAt1(0, 0) → 81
ok1(81) → 15
U311(0, 0) → 82
mark1(82) → 19
cons1(0, 0) → 83
mark1(83) → 4
proper1(0) → 84
top1(84) → 1
U911(0, 0) → 85
ok1(85) → 3
ok1(31) → 31
mark1(32) → 32
mark1(32) → 85
ok1(33) → 33
ok1(33) → 83
mark1(34) → 34
mark1(34) → 39
ok1(35) → 35
ok1(35) → 58
ok1(36) → 36
ok1(36) → 61
ok1(37) → 37
ok1(37) → 54
ok1(38) → 38
ok1(38) → 68
ok1(39) → 34
ok1(39) → 39
ok1(40) → 40
ok1(40) → 74
mark1(41) → 41
mark1(41) → 71
mark1(42) → 42
mark1(42) → 73
mark1(43) → 43
mark1(43) → 75
ok1(44) → 44
ok1(44) → 46
mark1(45) → 45
mark1(45) → 81
mark1(46) → 44
mark1(46) → 46
ok1(47) → 47
ok1(48) → 84
ok1(49) → 84
mark1(50) → 50
mark1(50) → 53
ok1(51) → 51
ok1(51) → 82
mark1(52) → 52
mark1(52) → 72
ok1(53) → 50
ok1(53) → 53
mark1(54) → 37
mark1(54) → 54
ok1(55) → 55
ok1(55) → 77
mark1(56) → 56
mark1(56) → 80
ok1(57) → 57
ok1(57) → 64
mark1(58) → 35
mark1(58) → 58
ok1(59) → 59
ok1(59) → 70
ok1(60) → 60
ok1(60) → 79
mark1(61) → 36
mark1(61) → 61
ok1(62) → 62
ok1(62) → 78
ok1(63) → 84
mark1(64) → 57
mark1(64) → 64
mark1(65) → 65
mark1(65) → 67
mark1(66) → 66
mark1(66) → 69
ok1(67) → 65
ok1(67) → 67
mark1(68) → 38
mark1(68) → 68
ok1(69) → 66
ok1(69) → 69
mark1(70) → 59
mark1(70) → 70
ok1(71) → 41
ok1(71) → 71
ok1(72) → 52
ok1(72) → 72
ok1(73) → 42
ok1(73) → 73
mark1(74) → 40
mark1(74) → 74
ok1(75) → 43
ok1(75) → 75
ok1(76) → 76
mark1(77) → 55
mark1(77) → 77
mark1(78) → 62
mark1(78) → 78
mark1(79) → 60
mark1(79) → 79
ok1(80) → 56
ok1(80) → 80
ok1(81) → 45
ok1(81) → 81
mark1(82) → 51
mark1(82) → 82
mark1(83) → 33
mark1(83) → 83
ok1(85) → 32
ok1(85) → 85
active2(48) → 86
top2(86) → 1
active2(49) → 86
active2(63) → 86

(4) BOUNDS(1, n^1)

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNatural(ok(z0)) → ok(isNatural(z0))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
U82(mark(z0), z1) → mark(U82(z0, z1))
U82(ok(z0), ok(z1)) → ok(U82(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
isPLNat(ok(z0)) → ok(isPLNat(z0))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U81(mark(z0), z1, z2, z3) → mark(U81(z0, z1, z2, z3))
U81(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U81(z0, z1, z2, z3))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
PROPER(tt) → c33
PROPER(nil) → c34
PROPER(0) → c35
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
PROPER(tt) → c33
PROPER(nil) → c34
PROPER(0) → c35
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:none
Defined Rule Symbols:

top, isNatural, U91, cons, U82, tail, U101, U61, snd, and, U51, pair, U41, sel, splitAt, isPLNat, proper, U11, U31, head, natsFrom, fst, afterNth, U21, s, U71, U81, take, isLNat

Defined Pair Symbols:

TOP, ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, PROPER, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

PROPER(nil) → c34
PROPER(tt) → c33
PROPER(0) → c35
TOP(ok(z0)) → c(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNatural(ok(z0)) → ok(isNatural(z0))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
U82(mark(z0), z1) → mark(U82(z0, z1))
U82(ok(z0), ok(z1)) → ok(U82(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
isPLNat(ok(z0)) → ok(isPLNat(z0))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U81(mark(z0), z1, z2, z3) → mark(U81(z0, z1, z2, z3))
U81(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U81(z0, z1, z2, z3))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:none
Defined Rule Symbols:

top, isNatural, U91, cons, U82, tail, U101, U61, snd, and, U51, pair, U41, sel, splitAt, isPLNat, proper, U11, U31, head, natsFrom, fst, afterNth, U21, s, U71, U81, take, isLNat

Defined Pair Symbols:

TOP, ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNatural(ok(z0)) → ok(isNatural(z0))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
U82(mark(z0), z1) → mark(U82(z0, z1))
U82(ok(z0), ok(z1)) → ok(U82(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
isPLNat(ok(z0)) → ok(isPLNat(z0))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U81(mark(z0), z1, z2, z3) → mark(U81(z0, z1, z2, z3))
U81(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U81(z0, z1, z2, z3))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

top, isNatural, U91, cons, U82, tail, U101, U61, snd, and, U51, pair, U41, sel, splitAt, isPLNat, proper, U11, U31, head, natsFrom, fst, afterNth, U21, s, U71, U81, take, isLNat

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
isNatural(ok(z0)) → ok(isNatural(z0))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
U82(mark(z0), z1) → mark(U82(z0, z1))
U82(ok(z0), ok(z1)) → ok(U82(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
isPLNat(ok(z0)) → ok(isPLNat(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U81(mark(z0), z1, z2, z3) → mark(U81(z0, z1, z2, z3))
U81(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U81(z0, z1, z2, z3))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
isLNat(ok(z0)) → ok(isLNat(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(z0)) → c1(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = x₁   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = [1]   
POL(ok(x₁)) = [1]   
POL(proper(x₁)) = x₁   
POL(tt) = [1]   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = x₁   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = x₁   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = x₁   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = x₁   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]   
POL(AFTERNTH(x₁, x₂)) = [3]x₂   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = [2]x₁   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₁   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = [3]x₂   
POL(U82'(x₁, x₂)) = [3]x₂   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = [3]   
POL(ok(x₁)) = [3] + x₁   
POL(proper(x₁)) = [3]x₁   
POL(tt) = [2]   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = x₁   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = x₂   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = x₁   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = x₁   
POL(U81'(x₁, x₂, x₃, x₄)) = x₄   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₁   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = x₁   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAIL(mark(z0)) → c10(TAIL(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = [2]x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = [2]x₂ + [2]x₄   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SEL(z0, mark(z1)) → c27(SEL(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = x₂   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = x₄   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1]   
POL(tt) = [1]   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₁   
POL(U11'(x₁, x₂, x₃)) = x₂   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = [1]   
POL(tt) = 0   

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = x₁   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = x₁   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = x₂   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₃   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = x₃   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(mark(z0), z1) → c50(U21'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₂   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = x₁   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = x₂   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1]   
POL(tt) = 0   

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SND(mark(z0)) → c16(SND(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = x₁   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = x₁   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₃   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = x₂   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = x₂   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(37) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = x₂   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = x₁   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = x₁   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₂   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = x₁   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = x₂   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = [2]x₁   
POL(ISPLNAT(x₁)) = x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₂   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = [2]x₃   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = [2]x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = x₃   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = x₂   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = x₁   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = x₁   
POL(U82'(x₁, x₂)) = x₁   
POL(U91'(x₁, x₂)) = x₂   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = [1]   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = x₁   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = x₂   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₂   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

HEAD(mark(z0)) → c40(HEAD(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = x₁   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₂ + x₃   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = x₃   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = x₂   
POL(U31'(x₁, x₂)) = [2]x₂   
POL(U41'(x₁, x₂)) = x₂   
POL(U51'(x₁, x₂, x₃)) = [2]x₂   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = [2]x₂   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = [2]x₂   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = [2]x₂   
POL(U61'(x₁, x₂)) = x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = [2]x₂   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(mark(z0), z1) → c6(CONS(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISLNAT(ok(z0)) → c60(ISLNAT(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = [2]x₁   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = [2]x₂   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = [2]x₃   
POL(U61'(x₁, x₂)) = [2]x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = [2]x₃   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = [2]x₂   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

CONS(mark(z0), z1) → c6(CONS(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = x₁   
POL(CONS(x₁, x₂)) = x₁   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = x₁   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = x₂   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = x₂   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = x₂   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = x₂   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₁   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

TAIL(ok(z0)) → c9(TAIL(z0))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U61'(mark(z0), z1) → c14(U61'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = x₂   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₃   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = x₁   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

TAIL(ok(z0)) → c9(TAIL(z0))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

HEAD(ok(z0)) → c41(HEAD(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = [2]x₁   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = [2]x₂ + [2]x₃   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = [2]x₃   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = [2]x₁ + [2]x₂   
POL(U81'(x₁, x₂, x₃, x₄)) = [2]x₁   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

TAIL(ok(z0)) → c9(TAIL(z0))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = [2]x₂   
POL(AND(x₁, x₂)) = [3]x₂   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = x₁   
POL(HEAD(x₁)) = [2]x₁   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = [2]x₁   
POL(ISPLNAT(x₁)) = [2]x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = x₁   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₃   
POL(U21'(x₁, x₂)) = x₂   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = [2]x₁   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = [3]x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = [2]x₂ + [2]x₃   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

TAIL(ok(z0)) → c9(TAIL(z0))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAIL(ok(z0)) → c9(TAIL(z0))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = [2]x₁   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = [2]x₁   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = [2]x₁   
POL(PAIR(x₁, x₂)) = [2]x₁   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = [2]x₂   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = [2]x₃   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = [2]x₂   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = [2]x₁   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = [2]x₁   
POL(ISPLNAT(x₁)) = [2]x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = [2]x₁   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₁ + x₂   
POL(U21'(x₁, x₂)) = x₂   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U31'(mark(z0), z1) → c39(U31'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = 0   
POL(ISPLNAT(x₁)) = x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₁   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = x₂   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))

K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = x₂   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(ISLNAT(x₁)) = 0   
POL(ISNATURAL(x₁)) = x₁   
POL(ISPLNAT(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = x₁ + x₂   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U101'(x₁, x₂, x₃)) = x₃   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂)) = 0   
POL(U51'(x₁, x₂, x₃)) = 0   
POL(U61'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃, x₄)) = 0   
POL(U82'(x₁, x₂)) = 0   
POL(U91'(x₁, x₂)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c27(x₁)) = x₁   
POL(c28(x₁)) = x₁   
POL(c29(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
TAIL(mark(z0)) → c10(TAIL(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
SND(mark(z0)) → c16(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
HEAD(ok(z0)) → c41(HEAD(z0))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
FST(ok(z0)) → c45(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:none
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
U51'(mark(z0), z1, z2) → c19(U51'(z0, z1, z2))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
NATSFROM(mark(z0)) → c43(NATSFROM(z0))
FST(mark(z0)) → c44(FST(z0))
U82'(ok(z0), ok(z1)) → c8(U82'(z0, z1))
U11'(mark(z0), z1, z2) → c36(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c37(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c46(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c47(AFTERNTH(z0, z1))
S(ok(z0)) → c51(S(z0))
S(mark(z0)) → c52(S(z0))
U81'(ok(z0), ok(z1), ok(z2), ok(z3)) → c56(U81'(z0, z1, z2, z3))
U41'(mark(z0), z1) → c24(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c25(U41'(z0, z1))
NATSFROM(ok(z0)) → c42(NATSFROM(z0))
U71'(ok(z0), ok(z1)) → c53(U71'(z0, z1))
U71'(mark(z0), z1) → c54(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U91'(mark(z0), z1) → c3(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c12(U101'(z0, z1, z2))
TAIL(mark(z0)) → c10(TAIL(z0))
U61'(ok(z0), ok(z1)) → c13(U61'(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
U101'(ok(z0), ok(z1), ok(z2)) → c11(U101'(z0, z1, z2))
ISNATURAL(ok(z0)) → c2(ISNATURAL(z0))
PAIR(mark(z0), z1) → c21(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c23(PAIR(z0, z1))
U21'(mark(z0), z1) → c50(U21'(z0, z1))
SND(mark(z0)) → c16(SND(z0))
U91'(ok(z0), ok(z1)) → c4(U91'(z0, z1))
SND(ok(z0)) → c15(SND(z0))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U31'(ok(z0), ok(z1)) → c38(U31'(z0, z1))
AFTERNTH(mark(z0), z1) → c48(AFTERNTH(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c20(U51'(z0, z1, z2))
ISPLNAT(ok(z0)) → c32(ISPLNAT(z0))
U82'(mark(z0), z1) → c7(U82'(z0, z1))
SPLITAT(mark(z0), z1) → c29(SPLITAT(z0, z1))
U81'(mark(z0), z1, z2, z3) → c55(U81'(z0, z1, z2, z3))
SPLITAT(z0, mark(z1)) → c30(SPLITAT(z0, z1))
HEAD(mark(z0)) → c40(HEAD(z0))
U21'(ok(z0), ok(z1)) → c49(U21'(z0, z1))
CONS(ok(z0), ok(z1)) → c5(CONS(z0, z1))
ISLNAT(ok(z0)) → c60(ISLNAT(z0))
CONS(mark(z0), z1) → c6(CONS(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
PAIR(z0, mark(z1)) → c22(PAIR(z0, z1))
U61'(mark(z0), z1) → c14(U61'(z0, z1))
HEAD(ok(z0)) → c41(HEAD(z0))
FST(ok(z0)) → c45(FST(z0))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAIL(ok(z0)) → c9(TAIL(z0))
SPLITAT(ok(z0), ok(z1)) → c31(SPLITAT(z0, z1))
U31'(mark(z0), z1) → c39(U31'(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

ISNATURAL, U91', CONS, U82', TAIL, U101', U61', SND, AND, U51', PAIR, U41', SEL, SPLITAT, ISPLNAT, U11', U31', HEAD, NATSFROM, FST, AFTERNTH, U21', S, U71', U81', TAKE, ISLNAT, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c1

(73) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(74) BOUNDS(1, 1)