The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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

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

Infered types.

(4) 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

(5) 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

(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

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

(7) 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)) →_R^Ω(1)
0'

Induction Step:
min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) 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

(10) 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)) →_R^Ω(1)
gen_0':s3_0(0)

Induction Step:
max(gen_0':s3_0(+(n327_0, 1)), gen_0':s3_0(+(n327_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) 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

(13) 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)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n733_0, 1)), gen_0':s3_0(b)) →_R^Ω(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 irc_R ∈ Ω(n).

(14) Complex Obligation (BEST)

(15) 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

(16) 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)) →_R^Ω(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n1330_0, 1)), gen_0':s3_0(+(n1330_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(17) Complex Obligation (BEST)

(18) 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

(19) 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)) →_R^Ω(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n1634_0, 1))) →_R^Ω(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))) →_L^{Ω(1 + a)}
gen_0':s3_0(+(a, *(n1634_0, a)))

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

(20) Complex Obligation (BEST)

(21) 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

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(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))

No more defined symbols left to analyse.

(24) 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)

(25) BOUNDS(n^2, INF)

(26) 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.

(27) 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) BOUNDS(n^2, INF)

(29) 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.

(30) 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)

(31) BOUNDS(n^1, INF)

(32) 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.

(33) 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) BOUNDS(n^1, INF)

(35) 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.

(36) 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) BOUNDS(n^1, INF)

(38) 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.

(39) 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) BOUNDS(n^1, INF)