### (0) Obligation:

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

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat → t:1':constant:0':+':*':-:minus:div:2':pow:ln

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

Heuristically decided to analyse the following defined symbols:
D

### (8) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat → t:1':constant:0':+':*':-:minus:div:2':pow:ln

Generator Equations:
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) ⇔ t
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) ⇔ +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x))

The following defined symbols remain to be analysed:
D

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

Proved the following rewrite lemma:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Induction Base:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0))

Induction Step:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(n4_0, 1))) →RΩ(1)
+'(D(t), D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0))) →RΩ(1)
+'(1', D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0))) →IH
+'(1', *3_0)

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

### (11) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat → t:1':constant:0':+':*':-:minus:div:2':pow:ln

Lemmas:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) ⇔ t
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) ⇔ +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x))

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

### (14) Obligation:

TRS:
Rules:
D(t) → 1'
D(constant) → 0'
D(+'(x, y)) → +'(D(x), D(y))
D(*'(x, y)) → +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln → t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat → t:1':constant:0':+':*':-:minus:div:2':pow:ln

Lemmas:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) ⇔ t
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) ⇔ +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) → *3_0, rt ∈ Ω(n40)