(0) Obligation:

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

D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
D(c(x, y)) →+ b(c(y, D(x)), c(x, D(y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [x / c(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) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1'))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

S is empty.
Rewrite Strategy: FULL

Infered types.

(6) Obligation:

TRS:
Rules:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1'))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat → t:h:s:constant:c:m:opp:div:2':pow:ln:1'

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

Heuristically decided to analyse the following defined symbols:
D, b

They will be analysed ascendingly in the following order:
b < D

(8) Obligation:

TRS:
Rules:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1'))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat → t:h:s:constant:c:m:opp:div:2':pow:ln:1'

Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) ⇔ t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) ⇔ s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))

The following defined symbols remain to be analysed:
b, D

They will be analysed ascendingly in the following order:
b < D

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

Proved the following rewrite lemma:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Induction Base:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, 0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, 0)))

Induction Step:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, +(n4_0, 1))), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, +(n4_0, 1)))) →RΩ(1)
s(s(b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))))) →IH
s(s(*3_0))

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

(11) Obligation:

TRS:
Rules:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1'))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat → t:h:s:constant:c:m:opp:div:2':pow:ln:1'

Lemmas:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) ⇔ t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) ⇔ s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))

The following defined symbols remain to be analysed:
D

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1'))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat → t:h:s:constant:c:m:opp:div:2':pow:ln:1'

Lemmas:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) ⇔ t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) ⇔ s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

(16) Obligation:

TRS:
Rules:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2')))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1'))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' → t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat → t:h:s:constant:c:m:opp:div:2':pow:ln:1'

Lemmas:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) ⇔ t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) ⇔ s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)