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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, n____(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

__'(__'(X, Y), Z) → __'(X, __'(Y, Z))
__'(X, nil') → X
__'(nil', X) → X
U11'(tt') → tt'
U21'(tt', V2) → U22'(isList'(activate'(V2)))
U22'(tt') → tt'
U31'(tt') → tt'
U41'(tt', V2) → U42'(isNeList'(activate'(V2)))
U42'(tt') → tt'
U51'(tt', V2) → U52'(isList'(activate'(V2)))
U52'(tt') → tt'
U61'(tt') → tt'
U71'(tt', P) → U72'(isPal'(activate'(P)))
U72'(tt') → tt'
U81'(tt') → tt'
isList'(V) → U11'(isNeList'(activate'(V)))
isList'(n__nil') → tt'
isList'(n____'(V1, V2)) → U21'(isList'(activate'(V1)), activate'(V2))
isNeList'(V) → U31'(isQid'(activate'(V)))
isNeList'(n____'(V1, V2)) → U41'(isList'(activate'(V1)), activate'(V2))
isNeList'(n____'(V1, V2)) → U51'(isNeList'(activate'(V1)), activate'(V2))
isNePal'(V) → U61'(isQid'(activate'(V)))
isNePal'(n____'(I, n____'(P, I))) → U71'(isQid'(activate'(I)), activate'(P))
isPal'(V) → U81'(isNePal'(activate'(V)))
isPal'(n__nil') → tt'
isQid'(n__a') → tt'
isQid'(n__e') → tt'
isQid'(n__i') → tt'
isQid'(n__o') → tt'
isQid'(n__u') → tt'
nil' → n__nil'
__'(X1, X2) → n____'(X1, X2)
a' → n__a'
e' → n__e'
i' → n__i'
o' → n__o'
u' → n__u'
activate'(n__nil') → nil'
activate'(n____'(X1, X2)) → __'(activate'(X1), activate'(X2))
activate'(n__a') → a'
activate'(n__e') → e'
activate'(n__i') → i'
activate'(n__o') → o'
activate'(n__u') → u'
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
__'(__'(X, Y), Z) → __'(X, __'(Y, Z))
__'(X, nil') → X
__'(nil', X) → X
U11'(tt') → tt'
U21'(tt', V2) → U22'(isList'(activate'(V2)))
U22'(tt') → tt'
U31'(tt') → tt'
U41'(tt', V2) → U42'(isNeList'(activate'(V2)))
U42'(tt') → tt'
U51'(tt', V2) → U52'(isList'(activate'(V2)))
U52'(tt') → tt'
U61'(tt') → tt'
U71'(tt', P) → U72'(isPal'(activate'(P)))
U72'(tt') → tt'
U81'(tt') → tt'
isList'(V) → U11'(isNeList'(activate'(V)))
isList'(n__nil') → tt'
isList'(n____'(V1, V2)) → U21'(isList'(activate'(V1)), activate'(V2))
isNeList'(V) → U31'(isQid'(activate'(V)))
isNeList'(n____'(V1, V2)) → U41'(isList'(activate'(V1)), activate'(V2))
isNeList'(n____'(V1, V2)) → U51'(isNeList'(activate'(V1)), activate'(V2))
isNePal'(V) → U61'(isQid'(activate'(V)))
isNePal'(n____'(I, n____'(P, I))) → U71'(isQid'(activate'(I)), activate'(P))
isPal'(V) → U81'(isNePal'(activate'(V)))
isPal'(n__nil') → tt'
isQid'(n__a') → tt'
isQid'(n__e') → tt'
isQid'(n__i') → tt'
isQid'(n__o') → tt'
isQid'(n__u') → tt'
nil' → n__nil'
__'(X1, X2) → n____'(X1, X2)
a' → n__a'
e' → n__e'
i' → n__i'
o' → n__o'
u' → n__u'
activate'(n__nil') → nil'
activate'(n____'(X1, X2)) → __'(activate'(X1), activate'(X2))
activate'(n__a') → a'
activate'(n__e') → e'
activate'(n__i') → i'
activate'(n__o') → o'
activate'(n__u') → u'
activate'(X) → X

Types:
__' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → n__nil':n____':n__a':n__e':n__i':n__o':n__u' → n__nil':n____':n__a':n__e':n__i':n__o':n__u'
nil' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
U11' :: tt' → tt'
tt' :: tt'
U21' :: tt' → n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
U22' :: tt' → tt'
isList' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
activate' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → n__nil':n____':n__a':n__e':n__i':n__o':n__u'
U31' :: tt' → tt'
U41' :: tt' → n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
U42' :: tt' → tt'
isNeList' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
U51' :: tt' → n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
U52' :: tt' → tt'
U61' :: tt' → tt'
U71' :: tt' → n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
U72' :: tt' → tt'
isPal' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
U81' :: tt' → tt'
n__nil' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
n____' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → n__nil':n____':n__a':n__e':n__i':n__o':n__u' → n__nil':n____':n__a':n__e':n__i':n__o':n__u'
isQid' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
isNePal' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u' → tt'
n__a' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
n__e' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
n__i' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
n__o' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
n__u' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
a' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
e' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
i' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
o' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
u' :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
___hole_n__nil':n____':n__a':n__e':n__i':n__o':n__u'1 :: n__nil':n____':n__a':n__e':n__i':n__o':n__u'
___hole_tt'2 :: tt'
___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3 :: Nat → n__nil':n____':n__a':n__e':n__i':n__o':n__u'

Heuristically decided to analyse the following defined symbols:
__', isList', activate', isNeList', isPal', isNePal'

They will be analysed ascendingly in the following order:
__' < activate'
activate' < isList'
isList' = isNeList'
activate' < isNeList'
activate' < isPal'
activate' < isNePal'
isPal' = isNePal'

Generator Equations:
___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0) ⇔ n__nil'
___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(+(x, 1)) ⇔ n____'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(x), n__nil')

The following defined symbols remain to be analysed:
__', isList', activate', isNeList', isPal', isNePal'

They will be analysed ascendingly in the following order:
__' < activate'
activate' < isList'
isList' = isNeList'
activate' < isNeList'
activate' < isPal'
activate' < isNePal'
isPal' = isNePal'

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

The following defined symbols remain to be analysed:
activate', isList', isNeList', isPal', isNePal'

They will be analysed ascendingly in the following order:
activate' < isList'
isList' = isNeList'
activate' < isNeList'
activate' < isPal'
activate' < isNePal'
isPal' = isNePal'

Proved the following rewrite lemma:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49)) → ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49), rt ∈ Ω(1 + _{__n49})

Induction Base:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0)) →_R^Ω(1)
___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0)

Induction Step:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(+(___$n50, 1))) →_R^Ω(1)
__'(activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___$n50)), activate'(n__nil')) →_IH
__'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___$n50), activate'(n__nil')) →_R^Ω(1)
__'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___$n50), n__nil') →_R^Ω(1)
n____'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___$n50), n__nil')

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

Lemmas:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49)) → ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49), rt ∈ Ω(1 + _{__n49})

The following defined symbols remain to be analysed:
isNePal', isList', isNeList', isPal'

They will be analysed ascendingly in the following order:
isList' = isNeList'
isPal' = isNePal'

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

Lemmas:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49)) → ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49), rt ∈ Ω(1 + _{__n49})

The following defined symbols remain to be analysed:
isPal', isList', isNeList'

They will be analysed ascendingly in the following order:
isList' = isNeList'
isPal' = isNePal'

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

Lemmas:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49)) → ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49), rt ∈ Ω(1 + _{__n49})

The following defined symbols remain to be analysed:
isNeList', isList'

They will be analysed ascendingly in the following order:
isList' = isNeList'

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

Lemmas:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49)) → ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49), rt ∈ Ω(1 + _{__n49})

The following defined symbols remain to be analysed:
isList'

They will be analysed ascendingly in the following order:
isList' = isNeList'

Proved the following rewrite lemma:
isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n12150)) → tt', rt ∈ Ω(1 + _{__n12150} + _{__n12150}²)

Induction Base:
isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0)) →_R^Ω(1)
tt'

Induction Step:
isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(+(___$n12151, 1))) →_R^Ω(1)
U21'(isList'(activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___$n12151))), activate'(n__nil')) →_L^{Ω(1 + _{__$n12151})}
U21'(isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___$n12151)), activate'(n__nil')) →_IH
U21'(tt', activate'(n__nil')) →_L^Ω(1)
U21'(tt', ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0)) →_R^Ω(1)
U22'(isList'(activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0)))) →_L^Ω(1)
U22'(isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(0))) →_R^Ω(1)
U22'(tt') →_R^Ω(1)
tt'

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

Lemmas:
activate'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49)) → ___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n49), rt ∈ Ω(1 + _{__n49})
isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n12150)) → tt', rt ∈ Ω(1 + _{__n12150} + _{__n12150}²)

The following defined symbols remain to be analysed:
isNeList'

They will be analysed ascendingly in the following order:
isList' = isNeList'

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

No more defined symbols left to analyse.

The lowerbound Ω(n²) was proven with the following lemma:
isList'(___gen_n__nil':n____':n__a':n__e':n__i':n__o':n__u'3(___n12150)) → tt', rt ∈ Ω(1 + _{__n12150} + _{__n12150}²)