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

Strict DPs
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(s(x),y) -> c_4(plus#(x,y))
quot#(0(),s(y)) -> c_5()
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(s(x),y) -> c_4(plus#(x,y))
quot#(0(),s(y)) -> c_5()
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Strict TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(s(x),y) -> c_4(plus#(x,y))
quot#(0(),s(y)) -> c_5()
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(x,0()) -> c_1()
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3()
plus#(s(x),y) -> c_4(plus#(x,y))
quot#(0(),s(y)) -> c_5()
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Strict TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(quot#) = {1},
uargs(c_2) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(minus) =  x1 + 
p(plus) = 
p(quot) =  x2 + 
p(s) =  x1 + 
p(minus#) = 
p(plus#) =  x1 + 
p(quot#) =  x1 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) =  x1 + 

Following rules are strictly oriented:
minus#(x,0()) = 
> 
= c_1()

plus#(0(),y) = 
> 
= c_3()

plus#(s(x),y) =  x + 
>  x + 
= c_4(plus#(x,y))

quot#(0(),s(y)) = 
> 
= c_5()

minus(x,0()) =  x + 
>  x + 
= x

minus(s(x),s(y)) =  x + 
>  x + 
= minus(x,y)

Following rules are (at-least) weakly oriented:
minus#(s(x),s(y)) =  
>= 
=  c_2(minus#(x,y))

quot#(s(x),s(y)) =   x + 
>=  x + 
=  c_6(quot#(minus(x,y),s(y)))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak DPs:
minus#(x,0()) -> c_1()
plus#(0(),y) -> c_3()
plus#(s(x),y) -> c_4(plus#(x,y))
quot#(0(),s(y)) -> c_5()
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(x,0()) -> c_1():3
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1

2:S:quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
-->_1 quot#(0(),s(y)) -> c_5():6
-->_1 quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y))):2

3:W:minus#(x,0()) -> c_1()

4:W:plus#(0(),y) -> c_3()

5:W:plus#(s(x),y) -> c_4(plus#(x,y))
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):5
-->_1 plus#(0(),y) -> c_3():4

6:W:quot#(0(),s(y)) -> c_5()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: plus#(s(x),y) -> c_4(plus#(x,y))
4: plus#(0(),y) -> c_3()
6: quot#(0(),s(y)) -> c_5()
3: minus#(x,0()) -> c_1()
* Step 5: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.

Problem (R)
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}

Problem (S)
- Strict DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1

2:W:quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y))):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
** Step 5.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,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:
1: minus#(s(x),s(y)) -> c_2(minus#(x,y))

The strictly oriented rules are moved into the weak component.
*** Step 5.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,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_2) = {1}

Following symbols are considered usable:
{minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = 
p(minus) = 
p(plus) = 
p(quot) =  x2 + 
p(s) =  x1 + 
p(minus#) =  x2 + 
p(plus#) = 
p(quot#) =  x1 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) = 
p(c_6) = 

Following rules are strictly oriented:
minus#(s(x),s(y)) =  y + 
>  y + 
= c_2(minus#(x,y))

Following rules are (at-least) weakly oriented:

*** Step 5.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 5.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: minus#(s(x),s(y)) -> c_2(minus#(x,y))
*** Step 5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak DPs:
minus#(s(x),s(y)) -> c_2(minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y))):1

2:W:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: minus#(s(x),s(y)) -> c_2(minus#(x,y))
** Step 5.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,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:
1: quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))

The strictly oriented rules are moved into the weak component.
*** Step 5.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,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_6) = {1}

Following symbols are considered usable:
{minus,minus#,plus#,quot#}
TcT has computed the following interpretation:
p(0) = 
p(minus) =  x1 + 
p(plus) =  x2 + 
p(quot) =  x2 + 
p(s) =  x1 + 
p(minus#) =  x1 +  x2 + 
p(plus#) =  x2 + 
p(quot#) =  x1 +  x2 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) =  x1 + 
p(c_5) = 
p(c_6) =  x1 + 

Following rules are strictly oriented:
quot#(s(x),s(y)) =  x +  y + 
>  x +  y + 
= c_6(quot#(minus(x,y),s(y)))

Following rules are (at-least) weakly oriented:
minus(x,0()) =   x + 
>=  x + 
=  x

minus(s(x),s(y)) =   x + 
>=  x + 
=  minus(x,y)

*** Step 5.b:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: quot#(s(x),s(y)) -> c_6(quot#(minus(x,y),s(y)))
*** Step 5.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{minus/2,plus/2,quot/2,minus#/2,plus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,plus#,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))
```