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:

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0, y, z) → max(y, z)
f(x, 0, z) → max(x, z)
f(x, y, 0) → max(x, y)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
min(s(x), s(y)) →⁺ s(min(x, 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:

min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
min, max, f

They will be analysed ascendingly in the following order:
min < f
max < f

(8) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
min, max, f

They will be analysed ascendingly in the following order:
min < f
max < f

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

Proved the following rewrite lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
min(gen_0':s2_0(0), gen_0':s2_0(0)) →_R^Ω(1)
0'

Induction Step:
min(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
s(min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0))) →_IH
s(gen_0':s2_0(c5_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:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
max, f

They will be analysed ascendingly in the following order:
max < f

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
max(gen_0':s2_0(n312_0), gen_0':s2_0(n312_0)) → gen_0':s2_0(n312_0), rt ∈ Ω(1 + n312₀)

Induction Base:
max(gen_0':s2_0(0), gen_0':s2_0(0)) →_R^Ω(1)
gen_0':s2_0(0)

Induction Step:
max(gen_0':s2_0(+(n312_0, 1)), gen_0':s2_0(+(n312_0, 1))) →_R^Ω(1)
s(max(gen_0':s2_0(n312_0), gen_0':s2_0(n312_0))) →_IH
s(gen_0':s2_0(c313_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
max(gen_0':s2_0(n312_0), gen_0':s2_0(n312_0)) → gen_0':s2_0(n312_0), rt ∈ Ω(1 + n312₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
f

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(16) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
max(gen_0':s2_0(n312_0), gen_0':s2_0(n312_0)) → gen_0':s2_0(n312_0), rt ∈ Ω(1 + n312₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
max(gen_0':s2_0(n312_0), gen_0':s2_0(n312_0)) → gen_0':s2_0(n312_0), rt ∈ Ω(1 + n312₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
p(s(x)) → x
f(s(x), s(y), s(z)) → f(max(s(x), max(s(y), s(z))), p(min(s(x), max(s(y), s(z)))), min(s(x), min(s(y), s(z))))
f(0', y, z) → max(y, z)
f(x, 0', z) → max(x, z)
f(x, y, 0') → max(x, y)

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
f :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

(24) BOUNDS(n^1, INF)