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

f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)

Types:
f :: g:h → g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat → g:h

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

Heuristically decided to analyse the following defined symbols:
f

(8) Obligation:

TRS:
Rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)

Types:
f :: g:h → g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat → g:h

Generator Equations:
gen_g:h2_0(0) ⇔ hole_g:h1_0
gen_g:h2_0(+(x, 1)) ⇔ h(gen_g:h2_0(x))

The following defined symbols remain to be analysed:
f

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, 0)))

Induction Step:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
f(h(gen_g:h2_0(a)), gen_g:h2_0(+(1, n4_0))) →_IH
*3_0

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)

Types:
f :: g:h → g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat → g:h

Lemmas:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_g:h2_0(0) ⇔ hole_g:h1_0
gen_g:h2_0(+(x, 1)) ⇔ h(gen_g:h2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
f(x, g(x)) → x
f(x, h(y)) → f(h(x), y)

Types:
f :: g:h → g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat → g:h

Lemmas:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_g:h2_0(0) ⇔ hole_g:h1_0
gen_g:h2_0(+(x, 1)) ⇔ h(gen_g:h2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)