```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(x,y) -> quot(x,y,y)
div(0(),y) -> 0()
quot(x,0(),s(z)) -> s(div(x,s(z)))
quot(0(),s(y),z) -> 0()
quot(s(x),s(y),z) -> quot(x,y,z)
- Signature:
{div/2,quot/3} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(0(),s(y),z) -> c_4()
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(0(),s(y),z) -> c_4()
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Strict TRS:
div(x,y) -> quot(x,y,y)
div(0(),y) -> 0()
quot(x,0(),s(z)) -> s(div(x,s(z)))
quot(0(),s(y),z) -> 0()
quot(s(x),s(y),z) -> quot(x,y,z)
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(0(),s(y),z) -> c_4()
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(0(),s(y),z) -> c_4()
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4}
by application of
Pre({2,4}) = {1,3,5}.
Here rules are labelled as follows:
1: div#(x,y) -> c_1(quot#(x,y,y))
2: div#(0(),y) -> c_2()
3: quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
4: quot#(0(),s(y),z) -> c_4()
5: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Weak DPs:
div#(0(),y) -> c_2()
quot#(0(),s(y),z) -> c_4()
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:div#(x,y) -> c_1(quot#(x,y,y))
-->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
-->_1 quot#(0(),s(y),z) -> c_4():5

2:S:quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
-->_1 div#(0(),y) -> c_2():4
-->_1 div#(x,y) -> c_1(quot#(x,y,y)):1

3:S:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
-->_1 quot#(0(),s(y),z) -> c_4():5
-->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
-->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2

4:W:div#(0(),y) -> c_2()

5:W:quot#(0(),s(y),z) -> c_4()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: div#(0(),y) -> c_2()
5: quot#(0(),s(y),z) -> c_4()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))

Consider the set of all dependency pairs
1: div#(x,y) -> c_1(quot#(x,y,y))
2: quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{3}
These cover all (indirect) predecessors of dependency pairs
{1,2,3}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_3) = {1},
uargs(c_5) = {1}

Following symbols are considered usable:
{div#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [1]
p(quot) = [1] x1 + [1]
p(s) = [1] x1 + [8]
p(div#) = [2] x1 + [0]
p(quot#) = [2] x1 + [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [8]
p(c_3) = [1] x1 + [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [12]

Following rules are strictly oriented:
quot#(s(x),s(y),z) = [2] x + [16]
> [2] x + [12]
= c_5(quot#(x,y,z))

Following rules are (at-least) weakly oriented:
div#(x,y) =  [2] x + [0]
>= [2] x + [0]
=  c_1(quot#(x,y,y))

quot#(x,0(),s(z)) =  [2] x + [0]
>= [2] x + [0]
=  c_3(div#(x,s(z)))

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
- Weak DPs:
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:div#(x,y) -> c_1(quot#(x,y,y))
-->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3

2:W:quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
-->_1 div#(x,y) -> c_1(quot#(x,y,y)):1

3:W:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
-->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
-->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: div#(x,y) -> c_1(quot#(x,y,y))
2: quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```