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(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(h(x, y)) →⁺ g(h(y, f(x)), h(x, f(y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [x / h(x, y)].
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(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e

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

Heuristically decided to analyse the following defined symbols:
f

(8) Obligation:

TRS:
Rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e

Generator Equations:
gen_a:b:c:d:h:e2_0(0) ⇔ a
gen_a:b:c:d:h:e2_0(+(x, 1)) ⇔ h(a, gen_a:b:c:d:h:e2_0(x))

The following defined symbols remain to be analysed:
f

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

Proved the following rewrite lemma:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
f(gen_a:b:c:d:h:e2_0(0))

Induction Step:
f(gen_a:b:c:d:h:e2_0(+(n4_0, 1))) →_R^Ω(1)
g(h(gen_a:b:c:d:h:e2_0(n4_0), f(a)), h(a, f(gen_a:b:c:d:h:e2_0(n4_0)))) →_R^Ω(1)
g(h(gen_a:b:c:d:h:e2_0(n4_0), b), h(a, f(gen_a:b:c:d:h:e2_0(n4_0)))) →_IH
g(h(gen_a:b:c:d:h:e2_0(n4_0), b), h(a, *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(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e

Lemmas:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:b:c:d:h:e2_0(0) ⇔ a
gen_a:b:c:d:h:e2_0(+(x, 1)) ⇔ h(a, gen_a:b:c:d:h:e2_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_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e

Lemmas:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:b:c:d:h:e2_0(0) ⇔ a
gen_a:b:c:d:h:e2_0(+(x, 1)) ⇔ h(a, gen_a:b:c:d:h:e2_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_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)