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:

h(x, c(y, z)) → h(c(s(y), x), z)
h(c(s(x), c(s(0), y)), z) → h(y, c(s(0), c(x, z)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

h(x, c(y, z)) → h(c(s(y), x), z)
h(c(s(x), c(s(0'), y)), z) → h(y, c(s(0'), c(x, z)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
h(x, c(y, z)) → h(c(s(y), x), z)
h(c(s(x), c(s(0'), y)), z) → h(y, c(s(0'), c(x, z)))

Types:
h :: c → c → h
c :: s:0' → c → c
s :: s:0' → s:0'
0' :: s:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_s:0'3_0 :: s:0'
gen_c4_0 :: Nat → c
gen_s:0'5_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
h

(8) Obligation:

TRS:
Rules:
h(x, c(y, z)) → h(c(s(y), x), z)
h(c(s(x), c(s(0'), y)), z) → h(y, c(s(0'), c(x, z)))

Types:
h :: c → c → h
c :: s:0' → c → c
s :: s:0' → s:0'
0' :: s:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_s:0'3_0 :: s:0'
gen_c4_0 :: Nat → c
gen_s:0'5_0 :: Nat → s:0'

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(0', gen_c4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

The following defined symbols remain to be analysed:
h

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

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

(10) Obligation:

TRS:
Rules:
h(x, c(y, z)) → h(c(s(y), x), z)
h(c(s(x), c(s(0'), y)), z) → h(y, c(s(0'), c(x, z)))

Types:
h :: c → c → h
c :: s:0' → c → c
s :: s:0' → s:0'
0' :: s:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_s:0'3_0 :: s:0'
gen_c4_0 :: Nat → c
gen_s:0'5_0 :: Nat → s:0'

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(0', gen_c4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

No more defined symbols left to analyse.