### (0) Obligation:

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

exp(x, 0) → s(0)
exp(x, s(y)) → *(x, exp(x, y))
*(0, y) → 0
*(s(x), y) → +(y, *(x, y))
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(x, y)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
exp(s(x35_1), s(y)) →+ +(exp(s(x35_1), y), *(x35_1, exp(s(x35_1), y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / s(y)].
The result substitution is [ ].

The rewrite sequence
exp(s(x35_1), s(y)) →+ +(exp(s(x35_1), y), *(x35_1, exp(s(x35_1), y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [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:

exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
+'/0

### (6) Obligation:

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

exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

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

Heuristically decided to analyse the following defined symbols:
exp, *', -

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

### (10) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
*', exp, -

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

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

Proved the following rewrite lemma:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)

Induction Base:
*'(gen_0':s:+'2_0(+(1, 0)), gen_0':s:+'2_0(b))

Induction Step:
*'(gen_0':s:+'2_0(+(1, +(n4_0, 1))), gen_0':s:+'2_0(b)) →RΩ(1)
+'(*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b))) →IH
+'(*3_0)

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

### (13) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
exp, -

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

Proved the following rewrite lemma:
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1057_0)) → *3_0, rt ∈ Ω(n10570)

Induction Base:
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(0))

Induction Step:
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(+(n1057_0, 1))) →RΩ(1)
*'(gen_0':s:+'2_0(0), exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1057_0))) →IH
*'(gen_0':s:+'2_0(0), *3_0)

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

### (16) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1057_0)) → *3_0, rt ∈ Ω(n10570)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
-

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

Proved the following rewrite lemma:
-(gen_0':s:+'2_0(n5653_0), gen_0':s:+'2_0(n5653_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n56530)

Induction Base:
-(gen_0':s:+'2_0(0), gen_0':s:+'2_0(0)) →RΩ(1)
0'

Induction Step:
-(gen_0':s:+'2_0(+(n5653_0, 1)), gen_0':s:+'2_0(+(n5653_0, 1))) →RΩ(1)
-(gen_0':s:+'2_0(n5653_0), gen_0':s:+'2_0(n5653_0)) →IH
gen_0':s:+'2_0(0)

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

### (19) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1057_0)) → *3_0, rt ∈ Ω(n10570)
-(gen_0':s:+'2_0(n5653_0), gen_0':s:+'2_0(n5653_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n56530)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)

### (22) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1057_0)) → *3_0, rt ∈ Ω(n10570)
-(gen_0':s:+'2_0(n5653_0), gen_0':s:+'2_0(n5653_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n56530)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)

### (25) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)
exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1057_0)) → *3_0, rt ∈ Ω(n10570)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)

### (28) Obligation:

TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) → *3_0, rt ∈ Ω(n40)