(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: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f

(6) Obligation:

Innermost 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

(7) 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))) →RΩ(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)))) →RΩ(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 ircR ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost 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.

(10) 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)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost 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.

(13) 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)

(14) BOUNDS(n^1, INF)