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:

tower(x) → f(a, x, s(0))
f(a, 0, y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0) → s(0)
exp(s(x)) → double(exp(x))
double(0) → 0
double(s(x)) → s(s(double(x)))
half(0) → double(0)
half(s(0)) → half(0)
half(s(s(x))) → s(half(x))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
f, half, exp, double

They will be analysed ascendingly in the following order:
half < f
exp < f
double < half
double < exp

(8) Obligation:

TRS:
Rules:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
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:
double, f, half, exp

They will be analysed ascendingly in the following order:
half < f
exp < f
double < half
double < exp

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

Proved the following rewrite lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

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

Induction Step:
double(gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
s(s(double(gen_0':s3_0(n5_0)))) →_IH
s(s(gen_0':s3_0(*(2, 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:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), 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:
half, f, exp

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

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

Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n253_0))) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n253₀)

Induction Base:
half(gen_0':s3_0(*(2, 0))) →_R^Ω(1)
double(0') →_L^Ω(1)
gen_0':s3_0(*(2, 0))

Induction Step:
half(gen_0':s3_0(*(2, +(n253_0, 1)))) →_R^Ω(1)
s(half(gen_0':s3_0(*(2, n253_0)))) →_IH
s(gen_0':s3_0(c254_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:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n253_0))) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n253₀)

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:
exp, f

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

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

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

(16) Obligation:

TRS:
Rules:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n253_0))) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n253₀)

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:
f

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n253_0))) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n253₀)

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.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

TRS:
Rules:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)
half(gen_0':s3_0(*(2, n253_0))) → gen_0':s3_0(n253_0), rt ∈ Ω(1 + n253₀)

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.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
tower(x) → f(a, x, s(0'))
f(a, 0', y) → y
f(a, s(x), y) → f(b, y, s(x))
f(b, y, x) → f(a, half(x), exp(y))
exp(0') → s(0')
exp(s(x)) → double(exp(x))
double(0') → 0'
double(s(x)) → s(s(double(x)))
half(0') → double(0')
half(s(0')) → half(0')
half(s(s(x))) → s(half(x))

Types:
tower :: 0':s → 0':s
f :: a:b → 0':s → 0':s → 0':s
a :: a:b
s :: 0':s → 0':s
0' :: 0':s
b :: a:b
half :: 0':s → 0':s
exp :: 0':s → 0':s
double :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat → 0':s

Lemmas:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), 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.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s3_0(n5_0)) → gen_0':s3_0(*(2, n5_0)), rt ∈ Ω(1 + n5₀)

(26) BOUNDS(n^1, INF)