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:

max(L(x)) → x
max(N(L(0), L(y))) → y
max(N(L(s(x)), L(s(y)))) → s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) → max(N(L(x), L(max(N(y, z)))))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

max(L(x)) → x
max(N(L(0'), L(y))) → y
max(N(L(s(x)), L(s(y)))) → s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) → max(N(L(x), L(max(N(y, z)))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
max(L(x)) → x
max(N(L(0'), L(y))) → y
max(N(L(s(x)), L(s(y)))) → s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) → max(N(L(x), L(max(N(y, z)))))

Types:
max :: L:N → 0':s
L :: 0':s → L:N
N :: L:N → L:N → L:N
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_L:N2_0 :: L:N
gen_0':s3_0 :: Nat → 0':s
gen_L:N4_0 :: Nat → L:N

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

Heuristically decided to analyse the following defined symbols:
max

(8) Obligation:

TRS:
Rules:
max(L(x)) → x
max(N(L(0'), L(y))) → y
max(N(L(s(x)), L(s(y)))) → s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) → max(N(L(x), L(max(N(y, z)))))

Types:
max :: L:N → 0':s
L :: 0':s → L:N
N :: L:N → L:N → L:N
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_L:N2_0 :: L:N
gen_0':s3_0 :: Nat → 0':s
gen_L:N4_0 :: Nat → L:N

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_L:N4_0(0) ⇔ L(0')
gen_L:N4_0(+(x, 1)) ⇔ N(L(0'), gen_L:N4_0(x))

The following defined symbols remain to be analysed:
max

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
max(gen_L:N4_0(+(1, n6_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Induction Base:
max(gen_L:N4_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
max(gen_L:N4_0(+(1, +(n6_0, 1)))) →_R^Ω(1)
max(N(L(0'), L(max(N(L(0'), gen_L:N4_0(n6_0)))))) →_IH
max(N(L(0'), L(gen_0':s3_0(0)))) →_R^Ω(1)
gen_0':s3_0(0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
max(L(x)) → x
max(N(L(0'), L(y))) → y
max(N(L(s(x)), L(s(y)))) → s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) → max(N(L(x), L(max(N(y, z)))))

Types:
max :: L:N → 0':s
L :: 0':s → L:N
N :: L:N → L:N → L:N
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_L:N2_0 :: L:N
gen_0':s3_0 :: Nat → 0':s
gen_L:N4_0 :: Nat → L:N

Lemmas:
max(gen_L:N4_0(+(1, n6_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_L:N4_0(0) ⇔ L(0')
gen_L:N4_0(+(x, 1)) ⇔ N(L(0'), gen_L:N4_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
max(gen_L:N4_0(+(1, n6_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
max(L(x)) → x
max(N(L(0'), L(y))) → y
max(N(L(s(x)), L(s(y)))) → s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) → max(N(L(x), L(max(N(y, z)))))

Types:
max :: L:N → 0':s
L :: 0':s → L:N
N :: L:N → L:N → L:N
0' :: 0':s
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_L:N2_0 :: L:N
gen_0':s3_0 :: Nat → 0':s
gen_L:N4_0 :: Nat → L:N

Lemmas:
max(gen_L:N4_0(+(1, n6_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_L:N4_0(0) ⇔ L(0')
gen_L:N4_0(+(x, 1)) ⇔ N(L(0'), gen_L:N4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
max(gen_L:N4_0(+(1, n6_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)