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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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

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

Infered types.

(4) 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

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
max

(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

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

(7) 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).

(8) Complex Obligation (BEST)

(9) 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.

(10) 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)

(11) BOUNDS(n^1, INF)

(12) 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.

(13) 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)

(14) BOUNDS(n^1, INF)