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:

double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0
permute(y, x, c) → s(s(permute(x, y, a)))
p(0) → 0
p(s(x)) → 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))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0)) → s(x)
plus(x, 0) → x
isZero(0) → true
isZero(s(x)) → false

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
permute(s(x36500_1), y, a) →⁺ s(s(permute(x36500_1, ack(s(x36500_1), s(x36500_1)), a)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x36500_1 / s(x36500_1)].
The result substitution is [y / ack(s(x36500_1), s(x36500_1))].

(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:

double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

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

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

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

(8) Obligation:

TRS:
Rules:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

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

The following defined symbols remain to be analysed:
plus, permute, ack

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

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

Proved the following rewrite lemma:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, 0))))

Induction Step:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, +(n5_0, 1))))) →_R^Ω(1)
s(plus(s(gen_false:true:0':s3_0(a)), gen_false:true:0':s3_0(+(2, *(2, n5_0))))) →_IH
s(*4_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:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)

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

The following defined symbols remain to be analysed:
ack, permute

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

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

Proved the following rewrite lemma:
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, n2476_0))) → *4_0, rt ∈ Ω(n2476₀)

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

Induction Step:
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, +(n2476_0, 1)))) →_R^Ω(1)
ack(gen_false:true:0':s3_0(0), ack(s(gen_false:true:0':s3_0(0)), gen_false:true:0':s3_0(+(1, n2476_0)))) →_IH
ack(gen_false:true:0':s3_0(0), *4_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:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, n2476_0))) → *4_0, rt ∈ Ω(n2476₀)

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

The following defined symbols remain to be analysed:
permute

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, n2476_0))) → *4_0, rt ∈ Ω(n2476₀)

Generator Equations:
gen_false:true:0':s3_0(0) ⇔ true
gen_false:true:0':s3_0(+(x, 1)) ⇔ s(gen_false:true: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:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, n2476_0))) → *4_0, rt ∈ Ω(n2476₀)

Generator Equations:
gen_false:true:0':s3_0(0) ⇔ true
gen_false:true:0':s3_0(+(x, 1)) ⇔ s(gen_false:true: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_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
double(x) → permute(x, x, a)
permute(x, y, a) → permute(isZero(x), x, b)
permute(false, x, b) → permute(ack(x, x), p(x), c)
permute(true, x, b) → 0'
permute(y, x, c) → s(s(permute(x, y, a)))
p(0') → 0'
p(s(x)) → 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))
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(x, s(0')) → s(x)
plus(x, 0') → x
isZero(0') → true
isZero(s(x)) → false

Types:
double :: false:true:0':s → false:true:0':s
permute :: false:true:0':s → false:true:0':s → a:b:c → false:true:0':s
a :: a:b:c
isZero :: false:true:0':s → false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s → false:true:0':s → false:true:0':s
p :: false:true:0':s → false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s → false:true:0':s
plus :: false:true:0':s → false:true:0':s → false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat → false:true:0':s

Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_false:true:0':s3_0(0) ⇔ true
gen_false:true:0':s3_0(+(x, 1)) ⇔ s(gen_false:true: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_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) → *4_0, rt ∈ Ω(n5₀)

(24) BOUNDS(n^1, INF)