(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: 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:

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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost 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

(5) 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

(6) Obligation:

Innermost 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

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

Innermost 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

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

Proved the following rewrite lemma:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)

Induction Base:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, 0)))

Induction Step:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, +(n4249_0, 1)))) →RΩ(1)
u21(ack_in(s(gen_0':s3_0(0)), gen_0':s3_0(+(1, n4249_0))), gen_0':s3_0(0)) →IH
u21(*4_0, gen_0':s3_0(0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost 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

Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)

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

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

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

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

(13) Obligation:

Innermost 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

Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)

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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost 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

Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)

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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)

(18) BOUNDS(n^1, INF)