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:

minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0, y) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0) → true
p(s(x)) → x
p(0) → 0
div(x, y) → quot(x, y, 0)
quot(s(x), s(y), z) → quot(minus(p(ack(0, x)), y), s(y), s(z))
quot(0, s(y), z) → z
ack(0, x) → s(x)
ack(0, x) → plus(x, s(0))
ack(s(x), 0) → ack(x, s(0))
ack(s(x), s(y)) → ack(x, ack(s(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
minus(s(x), s(y)) →⁺ minus(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), 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:

minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
minus, plus, quot, ack

They will be analysed ascendingly in the following order:
plus < minus
minus < quot
plus < ack
ack < quot

(8) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
plus, minus, quot, ack

They will be analysed ascendingly in the following order:
plus < minus
minus < quot
plus < ack
ack < quot

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
plus(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
plus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
plus(gen_0':s3_0(n5_0), s(gen_0':s3_0(b))) →_IH
gen_0':s3_0(+(+(b, 1), c6_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:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

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:
minus, quot, ack

They will be analysed ascendingly in the following order:
minus < quot
ack < quot

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n851₀)

Induction Base:
minus(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
minus(gen_0':s3_0(+(n851_0, 1)), gen_0':s3_0(+(n851_0, 1))) →_R^Ω(1)
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) →_IH
gen_0':s3_0(0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n851₀)

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, quot

They will be analysed ascendingly in the following order:
ack < quot

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ack(gen_0':s3_0(1), gen_0':s3_0(+(1, n1297_0))) → *4_0, rt ∈ Ω(n1297₀)

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

Induction Step:
ack(gen_0':s3_0(1), gen_0':s3_0(+(1, +(n1297_0, 1)))) →_R^Ω(1)
ack(gen_0':s3_0(0), ack(s(gen_0':s3_0(0)), gen_0':s3_0(+(1, n1297_0)))) →_IH
ack(gen_0':s3_0(0), *4_0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n851₀)
ack(gen_0':s3_0(1), gen_0':s3_0(+(1, n1297_0))) → *4_0, rt ∈ Ω(n1297₀)

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:
quot

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n851₀)
ack(gen_0':s3_0(1), gen_0':s3_0(+(1, n1297_0))) → *4_0, rt ∈ Ω(n1297₀)

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.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n851₀)
ack(gen_0':s3_0(1), gen_0':s3_0(+(1, n1297_0))) → *4_0, rt ∈ Ω(n1297₀)

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.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
minus(gen_0':s3_0(n851_0), gen_0':s3_0(n851_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n851₀)

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.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
p(0') → 0'
div(x, y) → quot(x, y, 0')
quot(s(x), s(y), z) → quot(minus(p(ack(0', x)), y), s(y), s(z))
quot(0', s(y), z) → z
ack(0', x) → s(x)
ack(0', x) → plus(x, s(0'))
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Types:
minus :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
ack :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

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.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(30) BOUNDS(n^1, INF)