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:

ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ackin(s(X), s(Y)) →⁺ u21(ackin(s(X), Y), X)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [Y / s(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:

ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Types:
ackin :: s → s → ackout:u22
s :: s → s
u21 :: ackout:u22 → s → ackout:u22
ackout :: s → ackout:u22
u22 :: ackout:u22 → ackout:u22
hole_ackout:u221_0 :: ackout:u22
hole_s2_0 :: s
gen_ackout:u223_0 :: Nat → ackout:u22
gen_s4_0 :: Nat → s

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

Heuristically decided to analyse the following defined symbols:
ackin, u21

They will be analysed ascendingly in the following order:
ackin = u21

(8) Obligation:

TRS:
Rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Types:
ackin :: s → s → ackout:u22
s :: s → s
u21 :: ackout:u22 → s → ackout:u22
ackout :: s → ackout:u22
u22 :: ackout:u22 → ackout:u22
hole_ackout:u221_0 :: ackout:u22
hole_s2_0 :: s
gen_ackout:u223_0 :: Nat → ackout:u22
gen_s4_0 :: Nat → s

Generator Equations:
gen_ackout:u223_0(0) ⇔ ackout(hole_s2_0)
gen_ackout:u223_0(+(x, 1)) ⇔ u22(gen_ackout:u223_0(x))
gen_s4_0(0) ⇔ hole_s2_0
gen_s4_0(+(x, 1)) ⇔ s(gen_s4_0(x))

The following defined symbols remain to be analysed:
u21, ackin

They will be analysed ascendingly in the following order:
ackin = u21

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

Could not prove a rewrite lemma for the defined symbol u21.

(10) Obligation:

TRS:
Rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Types:
ackin :: s → s → ackout:u22
s :: s → s
u21 :: ackout:u22 → s → ackout:u22
ackout :: s → ackout:u22
u22 :: ackout:u22 → ackout:u22
hole_ackout:u221_0 :: ackout:u22
hole_s2_0 :: s
gen_ackout:u223_0 :: Nat → ackout:u22
gen_s4_0 :: Nat → s

Generator Equations:
gen_ackout:u223_0(0) ⇔ ackout(hole_s2_0)
gen_ackout:u223_0(+(x, 1)) ⇔ u22(gen_ackout:u223_0(x))
gen_s4_0(0) ⇔ hole_s2_0
gen_s4_0(+(x, 1)) ⇔ s(gen_s4_0(x))

The following defined symbols remain to be analysed:
ackin

They will be analysed ascendingly in the following order:
ackin = u21

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ackin(gen_s4_0(1), gen_s4_0(+(1, n116_0))) → *5_0, rt ∈ Ω(n116₀)

Induction Base:
ackin(gen_s4_0(1), gen_s4_0(+(1, 0)))

Induction Step:
ackin(gen_s4_0(1), gen_s4_0(+(1, +(n116_0, 1)))) →_R^Ω(1)
u21(ackin(s(gen_s4_0(0)), gen_s4_0(+(1, n116_0))), gen_s4_0(0)) →_IH
u21(*5_0, gen_s4_0(0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Types:
ackin :: s → s → ackout:u22
s :: s → s
u21 :: ackout:u22 → s → ackout:u22
ackout :: s → ackout:u22
u22 :: ackout:u22 → ackout:u22
hole_ackout:u221_0 :: ackout:u22
hole_s2_0 :: s
gen_ackout:u223_0 :: Nat → ackout:u22
gen_s4_0 :: Nat → s

Lemmas:
ackin(gen_s4_0(1), gen_s4_0(+(1, n116_0))) → *5_0, rt ∈ Ω(n116₀)

Generator Equations:
gen_ackout:u223_0(0) ⇔ ackout(hole_s2_0)
gen_ackout:u223_0(+(x, 1)) ⇔ u22(gen_ackout:u223_0(x))
gen_s4_0(0) ⇔ hole_s2_0
gen_s4_0(+(x, 1)) ⇔ s(gen_s4_0(x))

The following defined symbols remain to be analysed:
u21

They will be analysed ascendingly in the following order:
ackin = u21

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol u21.

(15) Obligation:

TRS:
Rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Types:
ackin :: s → s → ackout:u22
s :: s → s
u21 :: ackout:u22 → s → ackout:u22
ackout :: s → ackout:u22
u22 :: ackout:u22 → ackout:u22
hole_ackout:u221_0 :: ackout:u22
hole_s2_0 :: s
gen_ackout:u223_0 :: Nat → ackout:u22
gen_s4_0 :: Nat → s

Lemmas:
ackin(gen_s4_0(1), gen_s4_0(+(1, n116_0))) → *5_0, rt ∈ Ω(n116₀)

Generator Equations:
gen_ackout:u223_0(0) ⇔ ackout(hole_s2_0)
gen_ackout:u223_0(+(x, 1)) ⇔ u22(gen_ackout:u223_0(x))
gen_s4_0(0) ⇔ hole_s2_0
gen_s4_0(+(x, 1)) ⇔ s(gen_s4_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ackin(gen_s4_0(1), gen_s4_0(+(1, n116_0))) → *5_0, rt ∈ Ω(n116₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Types:
ackin :: s → s → ackout:u22
s :: s → s
u21 :: ackout:u22 → s → ackout:u22
ackout :: s → ackout:u22
u22 :: ackout:u22 → ackout:u22
hole_ackout:u221_0 :: ackout:u22
hole_s2_0 :: s
gen_ackout:u223_0 :: Nat → ackout:u22
gen_s4_0 :: Nat → s

Lemmas:
ackin(gen_s4_0(1), gen_s4_0(+(1, n116_0))) → *5_0, rt ∈ Ω(n116₀)

Generator Equations:
gen_ackout:u223_0(0) ⇔ ackout(hole_s2_0)
gen_ackout:u223_0(+(x, 1)) ⇔ u22(gen_ackout:u223_0(x))
gen_s4_0(0) ⇔ hole_s2_0
gen_s4_0(+(x, 1)) ⇔ s(gen_s4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ackin(gen_s4_0(1), gen_s4_0(+(1, n116_0))) → *5_0, rt ∈ Ω(n116₀)

(20) BOUNDS(n^1, INF)