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

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
and(mark(X1), X2) →⁺ mark(and(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, isNatList, isNat, and, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
isNatList < active
isNat < active
and < active
isNatIList < active
cons < active
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
isNatList < proper
isNat < proper
and < proper
isNatIList < proper
cons < proper
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNatList, active, isNat, and, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
isNatList < active
isNat < active
and < active
isNatIList < active
cons < active
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
isNatList < proper
isNat < proper
and < proper
isNatIList < proper
cons < proper
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatList.

(10) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNat, active, and, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
isNat < active
and < active
isNatIList < active
cons < active
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
isNat < proper
and < proper
isNatIList < proper
cons < proper
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNat.

(12) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
and, active, isNatIList, cons, uTake1, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
and < active
isNatIList < active
cons < active
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
and < proper
isNatIList < proper
cons < proper
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)

Induction Base:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b))

Induction Step:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n13_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) →_R^Ω(1)
mark(and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(14) Complex Obligation (BEST)

(15) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNatIList, active, cons, uTake1, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
isNatIList < active
cons < active
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
isNatIList < proper
cons < proper
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatIList.

(17) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, uTake1, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
cons < active
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
cons < proper
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)

Induction Base:
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b))

Induction Step:
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n1538_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
uTake1, active, uTake2, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
uTake1 < active
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
uTake1 < proper
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)

Induction Base:
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)))

Induction Step:
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n3159_0, 1)))) →_R^Ω(1)
mark(uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
uTake2, active, take, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
uTake2 < active
take < active
uLength < active
s < active
length < active
active < top
uTake2 < proper
take < proper
uLength < proper
s < proper
length < proper
proper < top

(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)

Induction Base:
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d))

Induction Step:
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n3921_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) →_R^Ω(1)
mark(uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(25) Complex Obligation (BEST)

(26) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
take, active, uLength, s, length, proper, top

They will be analysed ascendingly in the following order:
take < active
uLength < active
s < active
length < active
active < top
take < proper
uLength < proper
s < proper
length < proper
proper < top

(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)

Induction Base:
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b))

Induction Step:
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n9004_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) →_R^Ω(1)
mark(take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(28) Complex Obligation (BEST)

(29) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
uLength, active, s, length, proper, top

They will be analysed ascendingly in the following order:
uLength < active
s < active
length < active
active < top
uLength < proper
s < proper
length < proper
proper < top

(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)

Induction Base:
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)), gen_tt:mark:0':zeros:nil:ok3_0(b))

Induction Step:
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n11834_0, 1))), gen_tt:mark:0':zeros:nil:ok3_0(b)) →_R^Ω(1)
mark(uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(31) Complex Obligation (BEST)

(32) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, length, proper, top

They will be analysed ascendingly in the following order:
s < active
length < active
active < top
s < proper
length < proper
proper < top

(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14769_0))) → *4_0, rt ∈ Ω(n14769₀)

Induction Base:
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n14769_0, 1)))) →_R^Ω(1)
mark(s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14769_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(34) Complex Obligation (BEST)

(35) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14769_0))) → *4_0, rt ∈ Ω(n14769₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
length, active, proper, top

They will be analysed ascendingly in the following order:
length < active
active < top
length < proper
proper < top

(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n16180_0))) → *4_0, rt ∈ Ω(n16180₀)

Induction Base:
length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, 0)))

Induction Step:
length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, +(n16180_0, 1)))) →_R^Ω(1)
mark(length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n16180_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(37) Complex Obligation (BEST)

(38) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14769_0))) → *4_0, rt ∈ Ω(n14769₀)
length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n16180_0))) → *4_0, rt ∈ Ω(n16180₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top

(39) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14769_0))) → *4_0, rt ∈ Ω(n14769₀)
length(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n16180_0))) → *4_0, rt ∈ Ω(n16180₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(40) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)
s(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n14769_0))) → *4_0, rt ∈ Ω(n14769₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(41) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)
uLength(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n11834_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11834₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(42) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)
take(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n9004_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9004₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(43) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)
uTake2(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3921_0)), gen_tt:mark:0':zeros:nil:ok3_0(b), gen_tt:mark:0':zeros:nil:ok3_0(c), gen_tt:mark:0':zeros:nil:ok3_0(d)) → *4_0, rt ∈ Ω(n3921₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(44) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)
uTake1(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n3159_0))) → *4_0, rt ∈ Ω(n3159₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(45) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)
cons(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n1538_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n1538₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.

(46) Obligation:

TRS:
Rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0')) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0', zeros))
active(take(0', IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
and :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
tt :: tt:mark:0':zeros:nil:ok
mark :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatIList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNatList :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
isNat :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
0' :: tt:mark:0':zeros:nil:ok
s :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
length :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
zeros :: tt:mark:0':zeros:nil:ok
cons :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
nil :: tt:mark:0':zeros:nil:ok
take :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake1 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uTake2 :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
uLength :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
proper :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
ok :: tt:mark:0':zeros:nil:ok → tt:mark:0':zeros:nil:ok
top :: tt:mark:0':zeros:nil:ok → top
hole_tt:mark:0':zeros:nil:ok1_0 :: tt:mark:0':zeros:nil:ok
hole_top2_0 :: top
gen_tt:mark:0':zeros:nil:ok3_0 :: Nat → tt:mark:0':zeros:nil:ok

Lemmas:
and(gen_tt:mark:0':zeros:nil:ok3_0(+(1, n13_0)), gen_tt:mark:0':zeros:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n13₀)

Generator Equations:
gen_tt:mark:0':zeros:nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':zeros:nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':zeros:nil:ok3_0(x))

No more defined symbols left to analyse.