```* 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(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
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:
runtime complexity wrt. defined symbols {minus,plus,quot} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:

Strict DPs
minus#(x,0()) -> c_1(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(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(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(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(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
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(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(x,0()) -> c_1(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
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_5) = {1},
uargs(c_7) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [10]
p(minus) = [1] x1 + [1]
p(plus) = [0]
p(quot) = [0]
p(s) = [1] x1 + [1]
p(minus#) = [1] x2 + [0]
p(plus#) = [0]
p(quot#) = [1] x1 + [2] x2 + [15]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [0]
p(c_7) = [1] x1 + [0]

Following rules are strictly oriented:
minus#(x,0()) = [10]
> [0]
= c_1(x)

minus#(s(x),s(y)) = [1] y + [1]
> [1] y + [0]
= c_2(minus#(x,y))

quot#(0(),s(y)) = [2] y + [27]
> [0]
= c_6()

minus(x,0()) = [1] x + [1]
> [1] x + [0]
= x

minus(s(x),s(y)) = [1] x + [2]
> [1] x + [1]
= minus(x,y)

Following rules are (at-least) weakly oriented:
plus#(0(),y) =  [0]
>= [0]
=  c_3(y)

plus#(minus(x,s(0())),minus(y,s(s(z)))) =  [0]
>= [0]
=  c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))

plus#(s(x),y) =  [0]
>= [0]
=  c_5(plus#(x,y))

quot#(s(x),s(y)) =  [1] x + [2] y + [18]
>= [1] x + [2] y + [18]
=  c_7(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:
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
- Weak DPs:
minus#(x,0()) -> c_1(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
quot#(0(),s(y)) -> c_6()
- 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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:plus#(0(),y) -> c_3(y)
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):6
-->_1 minus#(x,0()) -> c_1(x):5
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):4
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
-->_1 quot#(0(),s(y)) -> c_6():7
-->_1 plus#(0(),y) -> c_3(y):1

2:S:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
-->_1 plus#(0(),y) -> c_3(y):1

3:S:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
-->_1 plus#(0(),y) -> c_3(y):1

4:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
-->_1 quot#(0(),s(y)) -> c_6():7
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):4

5:W:minus#(x,0()) -> c_1(x)
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):6
-->_1 quot#(0(),s(y)) -> c_6():7
-->_1 minus#(x,0()) -> c_1(x):5
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):4
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):3
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):2
-->_1 plus#(0(),y) -> c_3(y):1

6:W:minus#(s(x),s(y)) -> c_2(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):6
-->_1 minus#(x,0()) -> c_1(x):5

7:W:quot#(0(),s(y)) -> c_6()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: quot#(0(),s(y)) -> c_6()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
- Weak DPs:
minus#(x,0()) -> c_1(x)
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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
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: plus#(0(),y) -> c_3(y)
2: plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
3: plus#(s(x),y) -> c_5(plus#(x,y))
4: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))

The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
- Weak DPs:
minus#(x,0()) -> c_1(x)
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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
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},
uargs(c_5) = {1},
uargs(c_7) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [5]
p(minus) = [1] x1 + [0]
p(plus) = [1]
p(quot) = [0]
p(s) = [1] x1 + [4]
p(minus#) = [0]
p(plus#) = [4] x1 + [1]
p(quot#) = [2] x1 + [4]
p(c_1) = [0]
p(c_2) = [8] x1 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [1] x1 + [4]

Following rules are strictly oriented:
plus#(0(),y) = [21]
> [0]
= c_3(y)

plus#(minus(x,s(0())),minus(y,s(s(z)))) = [4] x + [1]
> [0]
= c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))

plus#(s(x),y) = [4] x + [17]
> [4] x + [1]
= c_5(plus#(x,y))

quot#(s(x),s(y)) = [2] x + [12]
> [2] x + [8]
= c_7(quot#(minus(x,y),s(y)))

Following rules are (at-least) weakly oriented:
minus#(x,0()) =  [0]
>= [0]
=  c_1(x)

minus#(s(x),s(y)) =  [0]
>= [0]
=  c_2(minus#(x,y))

minus(x,0()) =  [1] x + [0]
>= [1] x + [0]
=  x

minus(s(x),s(y)) =  [1] x + [4]
>= [1] x + [0]
=  minus(x,y)

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(x,0()) -> c_1(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(s(x),s(y)) -> c_7(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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
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:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(x,0()) -> c_1(x)
minus#(s(x),s(y)) -> c_2(minus#(x,y))
plus#(0(),y) -> c_3(y)
plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(s(x),s(y)) -> c_7(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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
runtime complexity wrt. defined symbols {minus#,plus#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:minus#(x,0()) -> c_1(x)
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):6
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):5
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):4
-->_1 plus#(0(),y) -> c_3(y):3
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):2
-->_1 minus#(x,0()) -> c_1(x):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
-->_1 minus#(x,0()) -> c_1(x):1

3:W:plus#(0(),y) -> c_3(y)
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):6
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):5
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):4
-->_1 plus#(0(),y) -> c_3(y):3
-->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):2
-->_1 minus#(x,0()) -> c_1(x):1

4:W:plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):5
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):4
-->_1 plus#(0(),y) -> c_3(y):3

5:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):5
-->_1 plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0())))):4
-->_1 plus#(0(),y) -> c_3(y):3

6:W:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):6

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: minus#(x,0()) -> c_1(x)
3: plus#(0(),y) -> c_3(y)
5: plus#(s(x),y) -> c_5(plus#(x,y))
4: plus#(minus(x,s(0())),minus(y,s(s(z)))) -> c_4(plus#(minus(y,s(s(z))),minus(x,s(0()))))
2: minus#(s(x),s(y)) -> c_2(minus#(x,y))
6: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)))
** Step 5.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_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1}
- Obligation:
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))
```