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(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0))) → g(d(1))
g(c(1)) → g(d(h(0)))
g(h(x)) → g(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(h(x)) →⁺ g(x)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / h(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(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0'))) → g(d(1'))
g(c(1')) → g(d(h(0')))
g(h(x)) → g(x)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0'))) → g(d(1'))
g(c(1')) → g(d(h(0')))
g(h(x)) → g(x)

Types:
f :: c:d:0':h:1' → c:d:0':h:1'
c :: c:d:0':h:1' → c:d:0':h:1'
d :: c:d:0':h:1' → c:d:0':h:1'
g :: c:d:0':h:1' → c:d:0':h:1'
h :: c:d:0':h:1' → c:d:0':h:1'
0' :: c:d:0':h:1'
1' :: c:d:0':h:1'
hole_c:d:0':h:1'1_0 :: c:d:0':h:1'
gen_c:d:0':h:1'2_0 :: Nat → c:d:0':h:1'

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

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

(8) Obligation:

TRS:
Rules:
f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0'))) → g(d(1'))
g(c(1')) → g(d(h(0')))
g(h(x)) → g(x)

Types:
f :: c:d:0':h:1' → c:d:0':h:1'
c :: c:d:0':h:1' → c:d:0':h:1'
d :: c:d:0':h:1' → c:d:0':h:1'
g :: c:d:0':h:1' → c:d:0':h:1'
h :: c:d:0':h:1' → c:d:0':h:1'
0' :: c:d:0':h:1'
1' :: c:d:0':h:1'
hole_c:d:0':h:1'1_0 :: c:d:0':h:1'
gen_c:d:0':h:1'2_0 :: Nat → c:d:0':h:1'

Generator Equations:
gen_c:d:0':h:1'2_0(0) ⇔ 1'
gen_c:d:0':h:1'2_0(+(x, 1)) ⇔ c(gen_c:d:0':h:1'2_0(x))

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

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

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

(10) Obligation:

TRS:
Rules:
f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0'))) → g(d(1'))
g(c(1')) → g(d(h(0')))
g(h(x)) → g(x)

Types:
f :: c:d:0':h:1' → c:d:0':h:1'
c :: c:d:0':h:1' → c:d:0':h:1'
d :: c:d:0':h:1' → c:d:0':h:1'
g :: c:d:0':h:1' → c:d:0':h:1'
h :: c:d:0':h:1' → c:d:0':h:1'
0' :: c:d:0':h:1'
1' :: c:d:0':h:1'
hole_c:d:0':h:1'1_0 :: c:d:0':h:1'
gen_c:d:0':h:1'2_0 :: Nat → c:d:0':h:1'

Generator Equations:
gen_c:d:0':h:1'2_0(0) ⇔ 1'
gen_c:d:0':h:1'2_0(+(x, 1)) ⇔ c(gen_c:d:0':h:1'2_0(x))

The following defined symbols remain to be analysed:
g

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

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

(12) Obligation:

TRS:
Rules:
f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0'))) → g(d(1'))
g(c(1')) → g(d(h(0')))
g(h(x)) → g(x)

Types:
f :: c:d:0':h:1' → c:d:0':h:1'
c :: c:d:0':h:1' → c:d:0':h:1'
d :: c:d:0':h:1' → c:d:0':h:1'
g :: c:d:0':h:1' → c:d:0':h:1'
h :: c:d:0':h:1' → c:d:0':h:1'
0' :: c:d:0':h:1'
1' :: c:d:0':h:1'
hole_c:d:0':h:1'1_0 :: c:d:0':h:1'
gen_c:d:0':h:1'2_0 :: Nat → c:d:0':h:1'

Generator Equations:
gen_c:d:0':h:1'2_0(0) ⇔ 1'
gen_c:d:0':h:1'2_0(+(x, 1)) ⇔ c(gen_c:d:0':h:1'2_0(x))

No more defined symbols left to analyse.