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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

active'(__'(__'(X, Y), Z)) → mark'(__'(X, __'(Y, Z)))
active'(__'(X, nil')) → mark'(X)
active'(__'(nil', X)) → mark'(X)
active'(U11'(tt')) → mark'(U12'(tt'))
active'(U12'(tt')) → mark'(tt')
active'(isNePal'(__'(I, __'(P, I)))) → mark'(U11'(tt'))
active'(__'(X1, X2)) → __'(active'(X1), X2)
active'(__'(X1, X2)) → __'(X1, active'(X2))
active'(U11'(X)) → U11'(active'(X))
active'(U12'(X)) → U12'(active'(X))
active'(isNePal'(X)) → isNePal'(active'(X))
__'(mark'(X1), X2) → mark'(__'(X1, X2))
__'(X1, mark'(X2)) → mark'(__'(X1, X2))
U11'(mark'(X)) → mark'(U11'(X))
U12'(mark'(X)) → mark'(U12'(X))
isNePal'(mark'(X)) → mark'(isNePal'(X))
proper'(__'(X1, X2)) → __'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U11'(X)) → U11'(proper'(X))
proper'(tt') → ok'(tt')
proper'(U12'(X)) → U12'(proper'(X))
proper'(isNePal'(X)) → isNePal'(proper'(X))
__'(ok'(X1), ok'(X2)) → ok'(__'(X1, X2))
U11'(ok'(X)) → ok'(U11'(X))
U12'(ok'(X)) → ok'(U12'(X))
isNePal'(ok'(X)) → ok'(isNePal'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(__'(__'(X, Y), Z)) → mark'(__'(X, __'(Y, Z)))
active'(__'(X, nil')) → mark'(X)
active'(__'(nil', X)) → mark'(X)
active'(U11'(tt')) → mark'(U12'(tt'))
active'(U12'(tt')) → mark'(tt')
active'(isNePal'(__'(I, __'(P, I)))) → mark'(U11'(tt'))
active'(__'(X1, X2)) → __'(active'(X1), X2)
active'(__'(X1, X2)) → __'(X1, active'(X2))
active'(U11'(X)) → U11'(active'(X))
active'(U12'(X)) → U12'(active'(X))
active'(isNePal'(X)) → isNePal'(active'(X))
__'(mark'(X1), X2) → mark'(__'(X1, X2))
__'(X1, mark'(X2)) → mark'(__'(X1, X2))
U11'(mark'(X)) → mark'(U11'(X))
U12'(mark'(X)) → mark'(U12'(X))
isNePal'(mark'(X)) → mark'(isNePal'(X))
proper'(__'(X1, X2)) → __'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(U11'(X)) → U11'(proper'(X))
proper'(tt') → ok'(tt')
proper'(U12'(X)) → U12'(proper'(X))
proper'(isNePal'(X)) → isNePal'(proper'(X))
__'(ok'(X1), ok'(X2)) → ok'(__'(X1, X2))
U11'(ok'(X)) → ok'(U11'(X))
U12'(ok'(X)) → ok'(U12'(X))
isNePal'(ok'(X)) → ok'(isNePal'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':nil':tt':ok' → mark':nil':tt':ok'
__' :: mark':nil':tt':ok' → mark':nil':tt':ok' → mark':nil':tt':ok'
mark' :: mark':nil':tt':ok' → mark':nil':tt':ok'
nil' :: mark':nil':tt':ok'
U11' :: mark':nil':tt':ok' → mark':nil':tt':ok'
tt' :: mark':nil':tt':ok'
U12' :: mark':nil':tt':ok' → mark':nil':tt':ok'
isNePal' :: mark':nil':tt':ok' → mark':nil':tt':ok'
proper' :: mark':nil':tt':ok' → mark':nil':tt':ok'
ok' :: mark':nil':tt':ok' → mark':nil':tt':ok'
top' :: mark':nil':tt':ok' → top'
___hole_mark':nil':tt':ok'1 :: mark':nil':tt':ok'
___hole_top'2 :: top'
___gen_mark':nil':tt':ok'3 :: Nat → mark':nil':tt':ok'

Heuristically decided to analyse the following defined symbols:
active', __', U12', U11', isNePal', proper', top'

They will be analysed ascendingly in the following order:
__' < active'
U12' < active'
U11' < active'
isNePal' < active'
active' < top'
__' < proper'
U12' < proper'
U11' < proper'
isNePal' < proper'
proper' < top'

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

The following defined symbols remain to be analysed:
__', active', U12', U11', isNePal', proper', top'

Proved the following rewrite lemma:
__'(___gen_mark':nil':tt':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})

Induction Base:
__'(___gen_mark':nil':tt':ok'3(+(1, 0)), ___gen_mark':nil':tt':ok'3(b))

Induction Step:
__'(___gen_mark':nil':tt':ok'3(+(1, +(___$n6, 1))), ___gen_mark':nil':tt':ok'3(___b826)) →_R^Ω(1)
mark'(__'(___gen_mark':nil':tt':ok'3(+(1, ___$n6)), ___gen_mark':nil':tt':ok'3(___b826))) →_IH
mark'(___*4)

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

Lemmas:
__'(___gen_mark':nil':tt':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

The following defined symbols remain to be analysed:
U12', active', U11', isNePal', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
U11' < active'
isNePal' < active'
active' < top'
U12' < proper'
U11' < proper'
isNePal' < proper'
proper' < top'

Proved the following rewrite lemma:
U12'(___gen_mark':nil':tt':ok'3(+(1, ___n1865))) → ___*4, rt ∈ Ω(_{__n1865})

Induction Base:
U12'(___gen_mark':nil':tt':ok'3(+(1, 0)))

Induction Step:
U12'(___gen_mark':nil':tt':ok'3(+(1, +(___$n1866, 1)))) →_R^Ω(1)
mark'(U12'(___gen_mark':nil':tt':ok'3(+(1, ___$n1866)))) →_IH
mark'(___*4)

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

Lemmas:
__'(___gen_mark':nil':tt':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})
U12'(___gen_mark':nil':tt':ok'3(+(1, ___n1865))) → ___*4, rt ∈ Ω(_{__n1865})

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

The following defined symbols remain to be analysed:
U11', active', isNePal', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
isNePal' < active'
active' < top'
U11' < proper'
isNePal' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(___gen_mark':nil':tt':ok'3(+(1, ___n3027))) → ___*4, rt ∈ Ω(_{__n3027})

Induction Base:
U11'(___gen_mark':nil':tt':ok'3(+(1, 0)))

Induction Step:
U11'(___gen_mark':nil':tt':ok'3(+(1, +(___$n3028, 1)))) →_R^Ω(1)
mark'(U11'(___gen_mark':nil':tt':ok'3(+(1, ___$n3028)))) →_IH
mark'(___*4)

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

Lemmas:
__'(___gen_mark':nil':tt':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})
U12'(___gen_mark':nil':tt':ok'3(+(1, ___n1865))) → ___*4, rt ∈ Ω(_{__n1865})
U11'(___gen_mark':nil':tt':ok'3(+(1, ___n3027))) → ___*4, rt ∈ Ω(_{__n3027})

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

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

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

Proved the following rewrite lemma:
isNePal'(___gen_mark':nil':tt':ok'3(+(1, ___n4313))) → ___*4, rt ∈ Ω(_{__n4313})

Induction Base:
isNePal'(___gen_mark':nil':tt':ok'3(+(1, 0)))

Induction Step:
isNePal'(___gen_mark':nil':tt':ok'3(+(1, +(___$n4314, 1)))) →_R^Ω(1)
mark'(isNePal'(___gen_mark':nil':tt':ok'3(+(1, ___$n4314)))) →_IH
mark'(___*4)

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

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(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'

Could not prove a rewrite lemma for the defined symbol active'.

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

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

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

Could not prove a rewrite lemma for the defined symbol proper'.

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

The following defined symbols remain to be analysed:
top'

Could not prove a rewrite lemma for the defined symbol top'.

Generator Equations:
___gen_mark':nil':tt':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
__'(___gen_mark':nil':tt':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})