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:

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

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: c:s → c:s → c:s → c:s
c :: c:s → c:s
s :: c:s → c:s
g :: g → g → g
hole_c:s1_0 :: c:s
hole_g2_0 :: g
gen_c:s3_0 :: Nat → c:s

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

Heuristically decided to analyse the following defined symbols:
f

(8) Obligation:

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

Types:
f :: c:s → c:s → c:s → c:s
c :: c:s → c:s
s :: c:s → c:s
g :: g → g → g
hole_c:s1_0 :: c:s
hole_g2_0 :: g
gen_c:s3_0 :: Nat → c:s

Generator Equations:
gen_c:s3_0(0) ⇔ hole_c:s1_0
gen_c:s3_0(+(x, 1)) ⇔ c(gen_c:s3_0(x))

The following defined symbols remain to be analysed:
f

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

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

(10) Obligation:

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

Types:
f :: c:s → c:s → c:s → c:s
c :: c:s → c:s
s :: c:s → c:s
g :: g → g → g
hole_c:s1_0 :: c:s
hole_g2_0 :: g
gen_c:s3_0 :: Nat → c:s

Generator Equations:
gen_c:s3_0(0) ⇔ hole_c:s1_0
gen_c:s3_0(+(x, 1)) ⇔ c(gen_c:s3_0(x))

No more defined symbols left to analyse.