### (0) Obligation:

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

g(x, 0) → 0
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0), y) → y
id(x) → f(x, s(0))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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:

g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
g, f

### (8) Obligation:

TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
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:
g, f

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

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

Induction Base:
g(h, gen_0':s3_0(*(2, 0))) →RΩ(1)
0'

Induction Step:
g(h, gen_0':s3_0(*(2, +(n5_0, 1)))) →RΩ(1)
s(g(h, gen_0':s3_0(*(2, n5_0)))) →IH
s(gen_0':s3_0(c6_0))

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

### (11) Obligation:

TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(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:
f

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

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

### (13) Obligation:

TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

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

### (14) LowerBoundsProof (EQUIVALENT transformation)

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

### (16) Obligation:

TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

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

### (17) LowerBoundsProof (EQUIVALENT transformation)

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