The Runtime Complexity (full) 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:

int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
int(0, s(y)) →⁺ .(0, int_list(int(0, y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
The pumping substitution is [y / s(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:

int(0', 0') → .(0', nil)
int(0', s(y)) → .(0', int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
./0

(6) Obligation:

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

int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

(9) OrderProof (LOWER BOUND(ID) transformation)

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

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

(10) Obligation:

TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)

Induction Base:
int_list(gen_nil:.3_0(0)) →_R^Ω(1)
nil

Induction Step:
int_list(gen_nil:.3_0(+(n6_0, 1))) →_R^Ω(1)
.(int_list(gen_nil:.3_0(n6_0))) →_IH
.(gen_nil:.3_0(c7_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
int

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0)) → gen_nil:.3_0(1), rt ∈ Ω(1 + n184₀)

Induction Base:
int(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
.(nil)

Induction Step:
int(gen_0':s4_0(+(n184_0, 1)), gen_0':s4_0(+(n184_0, 1))) →_R^Ω(1)
int_list(int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0))) →_IH
int_list(gen_nil:.3_0(1)) →_L^Ω(2)
gen_nil:.3_0(1)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)
int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0)) → gen_nil:.3_0(1), rt ∈ Ω(1 + n184₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)
int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0)) → gen_nil:.3_0(1), rt ∈ Ω(1 + n184₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n6₀)

(24) BOUNDS(n^1, INF)