The Runtime Complexity (innermost) 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:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__and(tt, X) → mark(X)
a__isList(V) → a__isNeList(V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__and(a__isList(V1), isList(V2))
a__isNeList(V) → a__isQid(V)
a__isNeList(__(V1, V2)) → a__and(a__isList(V1), isNeList(V2))
a__isNeList(__(V1, V2)) → a__and(a__isNeList(V1), isList(V2))
a__isNePal(V) → a__isQid(V)
a__isNePal(__(I, __(P, I))) → a__and(a__isQid(I), isPal(P))
a__isPal(V) → a__isNePal(V)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(isNeList(X)) → a__isNeList(X)
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isList(X) → isList(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)
a__isPal(X) → isPal(X)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__and(tt, and(tt, X213414_0)) →⁺ a__and(tt, X213414_0)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X213414_0 / and(tt, X213414_0)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)