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:

intlist(nil) → nil
int(s(x), 0) → nil
int(x, x) → cons(x, nil)
intlist(cons(x, y)) → cons(s(x), intlist(y))
int(s(x), s(y)) → intlist(int(x, y))
int(0, s(y)) → cons(0, int(s(0), s(y)))
intlist(cons(x, nil)) → cons(s(x), nil)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

intlist(nil) → nil
int(s(x), 0') → nil
int(x, x) → cons(x, nil)
intlist(cons(x, y)) → cons(s(x), intlist(y))
int(s(x), s(y)) → intlist(int(x, y))
int(0', s(y)) → cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) → cons(s(x), nil)

S is empty.
Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
intlist(nil) → nil
int(s(x), 0') → nil
int(x, x) → cons(x, nil)
intlist(cons(x, y)) → cons(s(x), intlist(y))
int(s(x), s(y)) → intlist(int(x, y))
int(0', s(y)) → cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) → cons(s(x), nil)

Types:
intlist :: nil:cons → nil:cons
nil :: nil:cons
int :: s:0' → s:0' → nil:cons
s :: s:0' → s:0'
0' :: s:0'
cons :: s:0' → nil:cons → nil:cons
hole_nil:cons1_0 :: nil:cons
hole_s:0'2_0 :: s:0'
gen_nil:cons3_0 :: Nat → nil:cons
gen_s:0'4_0 :: Nat → s:0'

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
intlist, int

They will be analysed ascendingly in the following order:
intlist < int

(8) Obligation:

Innermost TRS:
Rules:
intlist(nil) → nil
int(s(x), 0') → nil
int(x, x) → cons(x, nil)
intlist(cons(x, y)) → cons(s(x), intlist(y))
int(s(x), s(y)) → intlist(int(x, y))
int(0', s(y)) → cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) → cons(s(x), nil)

Types:
intlist :: nil:cons → nil:cons
nil :: nil:cons
int :: s:0' → s:0' → nil:cons
s :: s:0' → s:0'
0' :: s:0'
cons :: s:0' → nil:cons → nil:cons
hole_nil:cons1_0 :: nil:cons
hole_s:0'2_0 :: s:0'
gen_nil:cons3_0 :: Nat → nil:cons
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_nil:cons3_0(0) ⇔ nil
gen_nil:cons3_0(+(x, 1)) ⇔ cons(0', gen_nil:cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
intlist, int

They will be analysed ascendingly in the following order:
intlist < int