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:

a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(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(b(x), b(y)) →⁺ b(f(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / b(x), y / b(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:

a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))

Types:
a :: b → b
f :: b → b → b
b :: b → b
hole_b1_0 :: b
gen_b2_0 :: Nat → b

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

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

They will be analysed ascendingly in the following order:
a = f

(8) Obligation:

TRS:
Rules:
a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))

Types:
a :: b → b
f :: b → b → b
b :: b → b
hole_b1_0 :: b
gen_b2_0 :: Nat → b

Generator Equations:
gen_b2_0(0) ⇔ hole_b1_0
gen_b2_0(+(x, 1)) ⇔ b(gen_b2_0(x))

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

They will be analysed ascendingly in the following order:
a = f

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

Proved the following rewrite lemma:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
f(gen_b2_0(+(1, 0)), gen_b2_0(+(1, 0)))

Induction Step:
f(gen_b2_0(+(1, +(n4_0, 1))), gen_b2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
b(f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0)))) →_IH
b(*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:
a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))

Types:
a :: b → b
f :: b → b → b
b :: b → b
hole_b1_0 :: b
gen_b2_0 :: Nat → b

Lemmas:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_b2_0(0) ⇔ hole_b1_0
gen_b2_0(+(x, 1)) ⇔ b(gen_b2_0(x))

The following defined symbols remain to be analysed:
a

They will be analysed ascendingly in the following order:
a = f

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))

Types:
a :: b → b
f :: b → b → b
b :: b → b
hole_b1_0 :: b
gen_b2_0 :: Nat → b

Lemmas:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_b2_0(0) ⇔ hole_b1_0
gen_b2_0(+(x, 1)) ⇔ b(gen_b2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) → a(f(x, y))
f(b(x), b(y)) → b(f(x, y))

Types:
a :: b → b
f :: b → b → b
b :: b → b
hole_b1_0 :: b
gen_b2_0 :: Nat → b

Lemmas:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_b2_0(0) ⇔ hole_b1_0
gen_b2_0(+(x, 1)) ⇔ b(gen_b2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(18) BOUNDS(n^1, INF)