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(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isList(ok(X)) → ok(isList(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
isPal(ok(X)) → ok(isPal(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'(and'(tt', X)) → mark'(X)
active'(isList'(V)) → mark'(isNeList'(V))
active'(isList'(nil')) → mark'(tt')
active'(isList'(__'(V1, V2))) → mark'(and'(isList'(V1), isList'(V2)))
active'(isNeList'(V)) → mark'(isQid'(V))
active'(isNeList'(__'(V1, V2))) → mark'(and'(isList'(V1), isNeList'(V2)))
active'(isNeList'(__'(V1, V2))) → mark'(and'(isNeList'(V1), isList'(V2)))
active'(isNePal'(V)) → mark'(isQid'(V))
active'(isNePal'(__'(I, __'(P, I)))) → mark'(and'(isQid'(I), isPal'(P)))
active'(isPal'(V)) → mark'(isNePal'(V))
active'(isPal'(nil')) → mark'(tt')
active'(isQid'(a')) → mark'(tt')
active'(isQid'(e')) → mark'(tt')
active'(isQid'(i')) → mark'(tt')
active'(isQid'(o')) → mark'(tt')
active'(isQid'(u')) → mark'(tt')
active'(__'(X1, X2)) → __'(active'(X1), X2)
active'(__'(X1, X2)) → __'(X1, active'(X2))
active'(and'(X1, X2)) → and'(active'(X1), X2)
__'(mark'(X1), X2) → mark'(__'(X1, X2))
__'(X1, mark'(X2)) → mark'(__'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(__'(X1, X2)) → __'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(isList'(X)) → isList'(proper'(X))
proper'(isNeList'(X)) → isNeList'(proper'(X))
proper'(isQid'(X)) → isQid'(proper'(X))
proper'(isNePal'(X)) → isNePal'(proper'(X))
proper'(isPal'(X)) → isPal'(proper'(X))
proper'(a') → ok'(a')
proper'(e') → ok'(e')
proper'(i') → ok'(i')
proper'(o') → ok'(o')
proper'(u') → ok'(u')
__'(ok'(X1), ok'(X2)) → ok'(__'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isList'(ok'(X)) → ok'(isList'(X))
isNeList'(ok'(X)) → ok'(isNeList'(X))
isQid'(ok'(X)) → ok'(isQid'(X))
isNePal'(ok'(X)) → ok'(isNePal'(X))
isPal'(ok'(X)) → ok'(isPal'(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'(and'(tt', X)) → mark'(X)
active'(isList'(V)) → mark'(isNeList'(V))
active'(isList'(nil')) → mark'(tt')
active'(isList'(__'(V1, V2))) → mark'(and'(isList'(V1), isList'(V2)))
active'(isNeList'(V)) → mark'(isQid'(V))
active'(isNeList'(__'(V1, V2))) → mark'(and'(isList'(V1), isNeList'(V2)))
active'(isNeList'(__'(V1, V2))) → mark'(and'(isNeList'(V1), isList'(V2)))
active'(isNePal'(V)) → mark'(isQid'(V))
active'(isNePal'(__'(I, __'(P, I)))) → mark'(and'(isQid'(I), isPal'(P)))
active'(isPal'(V)) → mark'(isNePal'(V))
active'(isPal'(nil')) → mark'(tt')
active'(isQid'(a')) → mark'(tt')
active'(isQid'(e')) → mark'(tt')
active'(isQid'(i')) → mark'(tt')
active'(isQid'(o')) → mark'(tt')
active'(isQid'(u')) → mark'(tt')
active'(__'(X1, X2)) → __'(active'(X1), X2)
active'(__'(X1, X2)) → __'(X1, active'(X2))
active'(and'(X1, X2)) → and'(active'(X1), X2)
__'(mark'(X1), X2) → mark'(__'(X1, X2))
__'(X1, mark'(X2)) → mark'(__'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(__'(X1, X2)) → __'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(isList'(X)) → isList'(proper'(X))
proper'(isNeList'(X)) → isNeList'(proper'(X))
proper'(isQid'(X)) → isQid'(proper'(X))
proper'(isNePal'(X)) → isNePal'(proper'(X))
proper'(isPal'(X)) → isPal'(proper'(X))
proper'(a') → ok'(a')
proper'(e') → ok'(e')
proper'(i') → ok'(i')
proper'(o') → ok'(o')
proper'(u') → ok'(u')
__'(ok'(X1), ok'(X2)) → ok'(__'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isList'(ok'(X)) → ok'(isList'(X))
isNeList'(ok'(X)) → ok'(isNeList'(X))
isQid'(ok'(X)) → ok'(isQid'(X))
isNePal'(ok'(X)) → ok'(isNePal'(X))
isPal'(ok'(X)) → ok'(isPal'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
__' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
mark' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
nil' :: mark':nil':tt':a':e':i':o':u':ok'
and' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
tt' :: mark':nil':tt':a':e':i':o':u':ok'
isList' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
isNeList' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
isQid' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
isNePal' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
isPal' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
a' :: mark':nil':tt':a':e':i':o':u':ok'
e' :: mark':nil':tt':a':e':i':o':u':ok'
i' :: mark':nil':tt':a':e':i':o':u':ok'
o' :: mark':nil':tt':a':e':i':o':u':ok'
u' :: mark':nil':tt':a':e':i':o':u':ok'
proper' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
ok' :: mark':nil':tt':a':e':i':o':u':ok' → mark':nil':tt':a':e':i':o':u':ok'
top' :: mark':nil':tt':a':e':i':o':u':ok' → top'
___hole_mark':nil':tt':a':e':i':o':u':ok'1 :: mark':nil':tt':a':e':i':o':u':ok'
___hole_top'2 :: top'
___gen_mark':nil':tt':a':e':i':o':u':ok'3 :: Nat → mark':nil':tt':a':e':i':o':u':ok'

Heuristically decided to analyse the following defined symbols:
active', __', isNeList', and', isList', isQid', isPal', isNePal', proper', top'

They will be analysed ascendingly in the following order:
__' < active'
isNeList' < active'
and' < active'
isList' < active'
isQid' < active'
isPal' < active'
isNePal' < active'
active' < top'
__' < proper'
isNeList' < proper'
and' < proper'
isList' < proper'
isQid' < proper'
isPal' < proper'
isNePal' < proper'
proper' < top'

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

The following defined symbols remain to be analysed:
__', active', isNeList', and', isList', isQid', isPal', isNePal', proper', top'

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

Induction Base:
__'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, 0)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(b))

Induction Step:
__'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, +(___$n6, 1))), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(___b826)) →_R^Ω(1)
mark'(__'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, ___$n6)), ___gen_mark':nil':tt':a':e':i':o':u':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':a':e':i':o':u':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

The following defined symbols remain to be analysed:
isNeList', active', and', isList', isQid', isPal', isNePal', proper', top'

They will be analysed ascendingly in the following order:
isNeList' < active'
and' < active'
isList' < active'
isQid' < active'
isPal' < active'
isNePal' < active'
active' < top'
isNeList' < proper'
and' < proper'
isList' < proper'
isQid' < proper'
isPal' < proper'
isNePal' < proper'
proper' < top'

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

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

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

The following defined symbols remain to be analysed:
and', active', isList', isQid', isPal', isNePal', proper', top'

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

Proved the following rewrite lemma:
and'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, ___n2478)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(b)) → ___*4, rt ∈ Ω(_{__n2478})

Induction Base:
and'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, 0)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(b))

Induction Step:
and'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, +(___$n2479, 1))), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(___b3407)) →_R^Ω(1)
mark'(and'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, ___$n2479)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(___b3407))) →_IH
mark'(___*4)

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

Lemmas:
__'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, ___n5)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(b)) → ___*4, rt ∈ Ω(_{__n5})
and'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(1, ___n2478)), ___gen_mark':nil':tt':a':e':i':o':u':ok'3(b)) → ___*4, rt ∈ Ω(_{__n2478})

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

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

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

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

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

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

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

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

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

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

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

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

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':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'

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

Generator Equations:
___gen_mark':nil':tt':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':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':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':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':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':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':a':e':i':o':u':ok'3(0) ⇔ nil'
___gen_mark':nil':tt':a':e':i':o':u':ok'3(+(x, 1)) ⇔ mark'(___gen_mark':nil':tt':a':e':i':o':u':ok'3(x))

No more defined symbols left to analyse.

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