### (0) Obligation:

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

min(0, y) → 0
min(x, 0) → 0
min(s(x), s(y)) → s(min(x, y))
max(0, y) → y
max(x, 0) → x
max(s(x), s(y)) → s(max(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(x, *(x, y))
f(s(x)) → f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(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
min(s(x), s(y)) →+ s(min(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x), y / s(y)].
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:

min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
min, max, +', -, *', f

They will be analysed ascendingly in the following order:
max < f
+' < *'
+' < f
- < f
*' < f

### (8) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
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:
min, max, +', -, *', f

They will be analysed ascendingly in the following order:
max < f
+' < *'
+' < f
- < f
*' < f

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

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

Induction Base:
min(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
0'

Induction Step:
min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
s(min(gen_0':s3_0(n5_0), gen_0':s3_0(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:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(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:
max, +', -, *', f

They will be analysed ascendingly in the following order:
max < f
+' < *'
+' < f
- < f
*' < f

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

Proved the following rewrite lemma:
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)

Induction Base:
max(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
gen_0':s3_0(0)

Induction Step:
max(gen_0':s3_0(+(n327_0, 1)), gen_0':s3_0(+(n327_0, 1))) →RΩ(1)
s(max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0))) →IH
s(gen_0':s3_0(c328_0))

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

### (14) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)

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

They will be analysed ascendingly in the following order:
+' < *'
+' < f
- < f
*' < f

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(b)) →RΩ(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n733_0, 1)), gen_0':s3_0(b)) →RΩ(1)
s(+'(gen_0':s3_0(n733_0), gen_0':s3_0(b))) →IH
s(gen_0':s3_0(+(b, c734_0)))

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

### (17) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)

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

They will be analysed ascendingly in the following order:
- < f
*' < f

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

Proved the following rewrite lemma:
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13300)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n1330_0, 1)), gen_0':s3_0(+(n1330_0, 1))) →RΩ(1)
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) →IH
gen_0':s3_0(0)

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

### (20) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13300)

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

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

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

Proved the following rewrite lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n1634_0)) → gen_0':s3_0(*(n1634_0, a)), rt ∈ Ω(1 + a·n16340 + n16340)

Induction Base:
*'(gen_0':s3_0(a), gen_0':s3_0(0)) →RΩ(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n1634_0, 1))) →RΩ(1)
+'(gen_0':s3_0(a), *'(gen_0':s3_0(a), gen_0':s3_0(n1634_0))) →IH
+'(gen_0':s3_0(a), gen_0':s3_0(*(c1635_0, a))) →LΩ(1 + a)
gen_0':s3_0(+(a, *(n1634_0, a)))

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

### (23) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13300)
*'(gen_0':s3_0(a), gen_0':s3_0(n1634_0)) → gen_0':s3_0(*(n1634_0, a)), rt ∈ Ω(1 + a·n16340 + n16340)

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

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

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

### (25) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13300)
*'(gen_0':s3_0(a), gen_0':s3_0(n1634_0)) → gen_0':s3_0(*(n1634_0, a)), rt ∈ Ω(1 + a·n16340 + n16340)

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.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n1634_0)) → gen_0':s3_0(*(n1634_0, a)), rt ∈ Ω(1 + a·n16340 + n16340)

### (28) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13300)
*'(gen_0':s3_0(a), gen_0':s3_0(n1634_0)) → gen_0':s3_0(*(n1634_0, a)), rt ∈ Ω(1 + a·n16340 + n16340)

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.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n1634_0)) → gen_0':s3_0(*(n1634_0, a)), rt ∈ Ω(1 + a·n16340 + n16340)

### (31) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)
-(gen_0':s3_0(n1330_0), gen_0':s3_0(n1330_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13300)

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.

### (32) LowerBoundsProof (EQUIVALENT transformation)

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

### (34) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)
+'(gen_0':s3_0(n733_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n733_0, b)), rt ∈ Ω(1 + n7330)

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.

### (35) LowerBoundsProof (EQUIVALENT transformation)

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

### (37) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
max(gen_0':s3_0(n327_0), gen_0':s3_0(n327_0)) → gen_0':s3_0(n327_0), rt ∈ Ω(1 + n3270)

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.

### (38) LowerBoundsProof (EQUIVALENT transformation)

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

### (40) Obligation:

TRS:
Rules:
min(0', y) → 0'
min(x, 0') → 0'
min(s(x), s(y)) → s(min(x, y))
max(0', y) → y
max(x, 0') → x
max(s(x), s(y)) → s(max(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
*'(x, 0') → 0'
*'(x, s(y)) → +'(x, *'(x, y))
f(s(x)) → f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0'))))))))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
max :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
*' :: 0':s → 0':s → 0':s
f :: 0':s → f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat → 0':s

Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(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.

### (41) LowerBoundsProof (EQUIVALENT transformation)

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