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:

append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Rewrite Strategy: INNERMOST

(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

S is empty.
Rewrite Strategy: INNERMOST

(3) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
append(::(@x3_0, @xs4_0), @l2) →⁺ ::(@x3_0, append(@xs4_0, @l2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [@xs4_0 / ::(@x3_0, @xs4_0)].
The result substitution is [ ].

(4) BOUNDS(n^1, INF)