The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → x

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

f(s(x)) → s(f(f(p(s(x)))))
f(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0') → 0'
p(s(x)) → x

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
p :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
f

(8) Obligation:

TRS:
Rules:
f(s(x)) → s(f(f(p(s(x)))))
f(0') → 0'
p(s(x)) → x

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
p :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
f

(9) RewriteLemmaProof (EQUIVALENT transformation)

Proved the following rewrite lemma:
f(gen_s:0'2_0(n4_0)) → gen_s:0'2_0(n4_0), rt ∈ Ω(2ⁿ)

Induction Base:
f(gen_s:0'2_0(0)) →_R^Ω(1)
0'

Induction Step:
f(gen_s:0'2_0(+(n4_0, 1))) →_R^Ω(1)
s(f(f(p(s(gen_s:0'2_0(n4_0)))))) →_R^Ω(1)
s(f(f(gen_s:0'2_0(n4_0)))) →_IH
s(f(gen_s:0'2_0(c5_0))) →_IH
s(gen_s:0'2_0(c5_0))

We have rt ∈ Ω(2ⁿ) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(2ⁿ)

(10) BOUNDS(2^n, INF)