### (0) Obligation:

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

f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

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

Heuristically decided to analyse the following defined symbols:
g, +'

They will be analysed ascendingly in the following order:
+' < g

### (8) Obligation:

TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

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

The following defined symbols remain to be analysed:
+', g

They will be analysed ascendingly in the following order:
+' < g

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

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

Induction Base:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(0)) →RΩ(1)
gen_0':1':s2_0(a)

Induction Step:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(+(n4_0, 1))) →RΩ(1)
s(+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0))) →IH
s(gen_0':1':s2_0(+(a, c5_0)))

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

### (11) Obligation:

TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

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

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

The following defined symbols remain to be analysed:
g

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

Proved the following rewrite lemma:
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n4570 + n45702)

Induction Base:
g(gen_0':1':s2_0(0), gen_0':1':s2_0(b))

Induction Step:
g(gen_0':1':s2_0(+(n457_0, 1)), gen_0':1':s2_0(b)) →RΩ(1)
g(gen_0':1':s2_0(n457_0), +'(gen_0':1':s2_0(b), s(gen_0':1':s2_0(n457_0)))) →LΩ(2 + n4570)
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(+(+(n457_0, 1), b))) →IH
*3_0

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

### (14) Obligation:

TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n4570 + n45702)

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

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n4570 + n45702)

### (17) Obligation:

TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n4570 + n45702)

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

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n4570 + n45702)

### (20) Obligation:

TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

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

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

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

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