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:

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(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
g(c(x, s(y))) →⁺ g(c(s(x), y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [y / s(y)].
The result substitution is [x / s(x)].

(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:

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Types:
g :: c:d → g
c :: s → s → c:d
s :: s → s
f :: c:d → c:d
d :: c:d → c:d
hole_g1_0 :: g
hole_c:d2_0 :: c:d
hole_s3_0 :: s
gen_c:d4_0 :: Nat → c:d
gen_s5_0 :: Nat → s

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

Heuristically decided to analyse the following defined symbols:
g, f

(8) Obligation:

TRS:
Rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Types:
g :: c:d → g
c :: s → s → c:d
s :: s → s
f :: c:d → c:d
d :: c:d → c:d
hole_g1_0 :: g
hole_c:d2_0 :: c:d
hole_s3_0 :: s
gen_c:d4_0 :: Nat → c:d
gen_s5_0 :: Nat → s

Generator Equations:
gen_c:d4_0(0) ⇔ c(hole_s3_0, hole_s3_0)
gen_c:d4_0(+(x, 1)) ⇔ d(gen_c:d4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Types:
g :: c:d → g
c :: s → s → c:d
s :: s → s
f :: c:d → c:d
d :: c:d → c:d
hole_g1_0 :: g
hole_c:d2_0 :: c:d
hole_s3_0 :: s
gen_c:d4_0 :: Nat → c:d
gen_s5_0 :: Nat → s

Generator Equations:
gen_c:d4_0(0) ⇔ c(hole_s3_0, hole_s3_0)
gen_c:d4_0(+(x, 1)) ⇔ d(gen_c:d4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))

The following defined symbols remain to be analysed:
f

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Types:
g :: c:d → g
c :: s → s → c:d
s :: s → s
f :: c:d → c:d
d :: c:d → c:d
hole_g1_0 :: g
hole_c:d2_0 :: c:d
hole_s3_0 :: s
gen_c:d4_0 :: Nat → c:d
gen_s5_0 :: Nat → s

Generator Equations:
gen_c:d4_0(0) ⇔ c(hole_s3_0, hole_s3_0)
gen_c:d4_0(+(x, 1)) ⇔ d(gen_c:d4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))

No more defined symbols left to analyse.