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

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

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,0()) -> c_1()
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
minus#(x,0()) -> c_1()
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,0()) -> c_1()
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4}
by application of
Pre({1,3,4}) = {2,5}.
Here rules are labelled as follows:
1: minus#(x,0()) -> c_1()
2: minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
3: pred#(s(x)) -> c_3()
4: quot#(0(),s(y)) -> c_4()
5: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
minus#(x,0()) -> c_1()
pred#(s(x)) -> c_3()
quot#(0(),s(y)) -> c_4()
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
-->_1 pred#(s(x)) -> c_3():4
-->_2 minus#(x,0()) -> c_1():3
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1

2:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_4():5
-->_2 minus#(x,0()) -> c_1():3
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1

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

4:W:pred#(s(x)) -> c_3()

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

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: quot#(0(),s(y)) -> c_4()
3: minus#(x,0()) -> c_1()
4: pred#(s(x)) -> c_3()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1

2:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(x,s(y)) -> c_2(pred#(minus(x,y)),minus#(x,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(x,s(y)) -> c_2(minus#(x,y))
* Step 6: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,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#(x,s(y)) -> c_2(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}

Problem (S)
- Strict DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
minus#(x,s(y)) -> c_2(minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,s(y)) -> c_2(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: minus#(x,s(y)) -> c_2(minus#(x,y))

The strictly oriented rules are moved into the weak component.
*** Step 6.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(x,s(y)) -> c_2(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_5) = {1,2}

Following symbols are considered usable:
{minus,pred,minus#,pred#,quot#}
TcT has computed the following interpretation:
p(0) = 0
p(minus) = x1
p(pred) = x1
p(quot) = 1 + 2*x1 + 2*x1*x2 + x1^2 + x2 + x2^2
p(s) = 1 + x1
p(minus#) = 7 + x2
p(pred#) = 1
p(quot#) = x1*x2 + 7*x1^2 + x2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 0
p(c_4) = 0
p(c_5) = 1 + x1 + x2

Following rules are strictly oriented:
minus#(x,s(y)) = 8 + y
> 7 + y
= c_2(minus#(x,y))

Following rules are (at-least) weakly oriented:
quot#(s(x),s(y)) =  9 + 15*x + x*y + 7*x^2 + 2*y
>= 9 + x + x*y + 7*x^2 + 2*y
=  c_5(quot#(minus(x,y),s(y)),minus#(x,y))

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

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

pred(s(x)) =  1 + x
>= x
=  x

*** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(x,s(y)) -> c_2(minus#(x,y))
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

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

2:W:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(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.
2: quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
1: minus#(x,s(y)) -> c_2(minus#(x,y))
*** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
minus#(x,s(y)) -> c_2(minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_2(minus#(x,y)):2
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):1

2:W:minus#(x,s(y)) -> c_2(minus#(x,y))
-->_1 minus#(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#(x,s(y)) -> c_2(minus#(x,y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)),minus#(x,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,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_5(quot#(minus(x,y),s(y)))

The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,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_5) = {1}

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

Following rules are strictly oriented:
quot#(s(x),s(y)) =  x + 
>  x + 
= c_5(quot#(minus(x,y),s(y)))

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

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

pred(s(x)) =   x + 
>=  x + 
=  x

*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:quot#(s(x),s(y)) -> c_5(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_5(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_5(quot#(minus(x,y),s(y)))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{minus/2,pred/1,quot/2,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))
```