### (0) Obligation:

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

f(s(x), x) → f(s(x), round(s(x)))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(x)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

f(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

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

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

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

### (8) Obligation:

TRS:
Rules:
f(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
round, f

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

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

Proved the following rewrite lemma:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n50)

Induction Base:
round(gen_s:0'3_0(*(2, 0))) →RΩ(1)
0'

Induction Step:
round(gen_s:0'3_0(*(2, +(n5_0, 1)))) →RΩ(1)
s(s(round(gen_s:0'3_0(*(2, n5_0))))) →IH
s(s(gen_s:0'3_0(*(2, c6_0))))

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

### (11) Obligation:

TRS:
Rules:
f(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_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:
f(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n50)

### (16) Obligation:

TRS:
Rules:
f(s(x), x) → f(s(x), round(s(x)))
round(0') → 0'
round(0') → s(0')
round(s(0')) → s(0')
round(s(s(x))) → s(s(round(x)))

Types:
f :: s:0' → s:0' → f
s :: s:0' → s:0'
round :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
round(gen_s:0'3_0(*(2, n5_0))) → gen_s:0'3_0(*(2, n5_0)), rt ∈ Ω(1 + n50)