The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
exp(s(x29228_1), s(y)) →⁺ plus(exp(s(x29228_1), y), times(p(s(x29228_1)), exp(s(x29228_1), y)))
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 [ ].

The rewrite sequence
exp(s(x29228_1), s(y)) →⁺ plus(exp(s(x29228_1), y), times(p(s(x29228_1)), exp(s(x29228_1), y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [y / s(y)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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:

plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
plus, p, times, exp, towerIter

They will be analysed ascendingly in the following order:
p < plus
plus < times
p < times
p < towerIter
times < exp
exp < towerIter

(8) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
p, plus, times, exp, towerIter

They will be analysed ascendingly in the following order:
p < plus
plus < times
p < times
p < towerIter
times < exp
exp < towerIter

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

Proved the following rewrite lemma:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
p(gen_0':s2_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
p(gen_0':s2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
s(p(s(gen_0':s2_0(n4_0)))) →_IH
s(gen_0':s2_0(c5_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:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
plus, times, exp, towerIter

They will be analysed ascendingly in the following order:
plus < times
times < exp
exp < towerIter

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

Proved the following rewrite lemma:
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) →_R^Ω(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n201_0, 1)), gen_0':s2_0(b)) →_R^Ω(1)
s(plus(p(s(gen_0':s2_0(n201_0))), gen_0':s2_0(b))) →_L^{Ω(1 + n201₀)}
s(plus(gen_0':s2_0(n201_0), gen_0':s2_0(b))) →_IH
s(gen_0':s2_0(+(b, c202_0)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
times, exp, towerIter

They will be analysed ascendingly in the following order:
times < exp
exp < towerIter

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

Proved the following rewrite lemma:
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s2_0(+(n625_0, 1)), gen_0':s2_0(b)) →_R^Ω(1)
plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n625_0))), gen_0':s2_0(b))) →_L^{Ω(1 + n625₀)}
plus(gen_0':s2_0(b), times(gen_0':s2_0(n625_0), gen_0':s2_0(b))) →_IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c626_0, b))) →_L^{Ω(1 + b + b2)}
gen_0':s2_0(+(b, *(n625_0, b)))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
exp, towerIter

They will be analysed ascendingly in the following order:
exp < towerIter

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

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

(19) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
towerIter

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

(23) BOUNDS(n^3, INF)

(24) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n625₀ + b²·n625₀ + n625₀ + n625₀²)

(26) BOUNDS(n^3, INF)

(27) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n201₀ + n201₀²)

(29) BOUNDS(n^2, INF)

(30) Obligation:

TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

(32) BOUNDS(n^1, INF)