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:

ack_in(0, n) → ack_out(s(n))
ack_in(s(m), 0) → u11(ack_in(m, s(0)))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
ack_in(s(s(m36_3)), 0) →⁺ u11(u21(ack_in(s(m36_3), 0), m36_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [m36_3 / s(m36_3)].
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:

ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

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

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

(8) Obligation:

TRS:
Rules:
ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

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

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

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

(10) Obligation:

TRS:
Rules:
ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
ack_in

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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.