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

Strict DPs
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
plus#(x,0()) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(0(),s(y),z) -> c_7()
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
times#(0(),y) -> c_9()
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
times#(s(0()),y) -> c_11()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
plus#(x,0()) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(0(),s(y),z) -> c_7()
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
times#(0(),y) -> c_9()
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
times#(s(0()),y) -> c_11()
- Weak TRS:
div(x,y) -> quot(x,y,y)
div(0(),y) -> 0()
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
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)
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
plus#(x,0()) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(0(),s(y),z) -> c_7()
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
times#(0(),y) -> c_9()
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
times#(s(0()),y) -> c_11()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
div#(0(),y) -> c_2()
plus#(x,0()) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(0(),s(y),z) -> c_7()
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
times#(0(),y) -> c_9()
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
times#(s(0()),y) -> c_11()
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,3,4,7,9,11}
by application of
Pre({2,3,4,7,9,11}) = {1,5,6,8,10}.
Here rules are labelled as follows:
1: div#(x,y) -> c_1(quot#(x,y,y))
2: div#(0(),y) -> c_2()
3: plus#(x,0()) -> c_3()
4: plus#(0(),y) -> c_4()
5: plus#(s(x),y) -> c_5(plus#(x,y))
6: quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
7: quot#(0(),s(y),z) -> c_7()
8: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
9: times#(0(),y) -> c_9()
10: times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
11: times#(s(0()),y) -> c_11()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak DPs:
div#(0(),y) -> c_2()
plus#(x,0()) -> c_3()
plus#(0(),y) -> c_4()
quot#(0(),s(y),z) -> c_7()
times#(0(),y) -> c_9()
times#(s(0()),y) -> c_11()
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_8(quot#(x,y,z)):4
-->_1 quot#(0(),s(y),z) -> c_7():9

2:S:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(0(),y) -> c_4():8
-->_1 plus#(x,0()) -> c_3():7
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

3:S:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
-->_1 div#(0(),y) -> c_2():6
-->_1 div#(x,y) -> c_1(quot#(x,y,y)):1

4:S:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
-->_1 quot#(0(),s(y),z) -> c_7():9
-->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):4
-->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):3

5:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
-->_2 times#(s(0()),y) -> c_11():11
-->_2 times#(0(),y) -> c_9():10
-->_1 plus#(0(),y) -> c_4():8
-->_1 plus#(x,0()) -> c_3():7
-->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):5
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

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

7:W:plus#(x,0()) -> c_3()

8:W:plus#(0(),y) -> c_4()

9:W:quot#(0(),s(y),z) -> c_7()

10:W:times#(0(),y) -> c_9()

11:W:times#(s(0()),y) -> c_11()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: times#(0(),y) -> c_9()
11: times#(s(0()),y) -> c_11()
7: plus#(x,0()) -> c_3()
8: plus#(0(),y) -> c_4()
6: div#(0(),y) -> c_2()
9: quot#(0(),s(y),z) -> c_7()
* Step 5: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
plus#(s(x),y) -> c_5(plus#(x,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}

Problem (S)
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
** Step 5.a:1: 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_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_8(quot#(x,y,z)):4

2:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

3:S:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
-->_1 div#(x,y) -> c_1(quot#(x,y,y)):1

4:S:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
-->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):3
-->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):4

5:W:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
-->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):5

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
2: plus#(s(x),y) -> c_5(plus#(x,y))
** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
** Step 5.a:3: 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_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_8(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_6(div#(x,s(z)))
3: quot#(s(x),s(y),z) -> c_8(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:3.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_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_6) = {1},
uargs(c_8) = {1}

Following symbols are considered usable:
{div#,plus#,quot#,times#}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [2] x1 + [8] x2 + [0]
p(plus) = [2] x2 + [2]
p(quot) = [8] x1 + [1] x2 + [1]
p(s) = [1] x1 + [3]
p(times) = [1]
p(div#) = [4] x1 + [10] x2 + [0]
p(plus#) = [1] x1 + [0]
p(quot#) = [4] x1 + [10] x3 + [0]
p(times#) = [1] x2 + [1]
p(c_1) = [1] x1 + [0]
p(c_2) = [4]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [10]
p(c_9) = [8]
p(c_10) = [4] x1 + [0]
p(c_11) = [0]

Following rules are strictly oriented:
quot#(s(x),s(y),z) = [4] x + [10] z + [12]
> [4] x + [10] z + [10]
= c_8(quot#(x,y,z))

Following rules are (at-least) weakly oriented:
div#(x,y) =  [4] x + [10] y + [0]
>= [4] x + [10] y + [0]
=  c_1(quot#(x,y,y))

quot#(x,0(),s(z)) =  [4] x + [10] z + [30]
>= [4] x + [10] z + [30]
=  c_6(div#(x,s(z)))

*** Step 5.a:3.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_6(div#(x,s(z)))
- Weak DPs:
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_8(quot#(x,y,z)):3

2:W:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
-->_1 div#(x,y) -> c_1(quot#(x,y,y)):1

3:W:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
-->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):3
-->_1 quot#(x,0(),s(z)) -> c_6(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_6(div#(x,s(z)))
3: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
*** Step 5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak DPs:
div#(x,y) -> c_1(quot#(x,y,y))
quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1

2:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
-->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):2
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1

3:W:div#(x,y) -> c_1(quot#(x,y,y))
-->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):5

4:W:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
-->_1 div#(x,y) -> c_1(quot#(x,y,y)):3

5:W:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
-->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):5
-->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):4

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: div#(x,y) -> c_1(quot#(x,y,y))
4: quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
5: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
** Step 5.b:2: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak DPs:
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}

Problem (S)
- Strict DPs:
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
*** Step 5.b:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak DPs:
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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: plus#(s(x),y) -> c_5(plus#(x,y))

The strictly oriented rules are moved into the weak component.
**** Step 5.b:2.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak DPs:
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_5) = {1},
uargs(c_10) = {1,2}

Following symbols are considered usable:
{div#,plus#,quot#,times#}
TcT has computed the following interpretation:
p(0) = 0
p(div) = 2 + 4*x1 + x1*x2 + x1^2 + x2
p(plus) = 3 + x1*x2 + 2*x1^2
p(quot) = 2 + x1 + x1*x3 + 4*x1^2
p(s) = 1 + x1
p(times) = 2*x1
p(div#) = x1*x2 + x1^2
p(plus#) = x1
p(quot#) = 4*x1 + x1*x2 + 2*x1^2 + 4*x2 + x2^2
p(times#) = x1 + 5*x1*x2 + 6*x2^2
p(c_1) = 1
p(c_2) = 1
p(c_3) = 0
p(c_4) = 1
p(c_5) = x1
p(c_6) = 0
p(c_7) = 0
p(c_8) = 0
p(c_9) = 0
p(c_10) = 1 + x1 + x2
p(c_11) = 1

Following rules are strictly oriented:
plus#(s(x),y) = 1 + x
> x
= c_5(plus#(x,y))

Following rules are (at-least) weakly oriented:
times#(s(x),y) =  1 + x + 5*x*y + 5*y + 6*y^2
>= 1 + x + 5*x*y + y + 6*y^2
=  c_10(plus#(y,times(x,y)),times#(x,y))

**** Step 5.b:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 5.b:2.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1

2:W:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
-->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):2
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
1: plus#(s(x),y) -> c_5(plus#(x,y))
**** Step 5.b:2.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

*** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
-->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):1

2:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: plus#(s(x),y) -> c_5(plus#(x,y))
*** Step 5.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
-->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
times#(s(x),y) -> c_10(times#(x,y))
*** Step 5.b:2.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(x),y) -> c_10(times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
times(0(),y) -> 0()
times(s(x),y) -> plus(y,times(x,y))
times(s(0()),y) -> y
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
times#(s(x),y) -> c_10(times#(x,y))
*** Step 5.b:2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(x),y) -> c_10(times#(x,y))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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: times#(s(x),y) -> c_10(times#(x,y))

The strictly oriented rules are moved into the weak component.
**** Step 5.b:2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(s(x),y) -> c_10(times#(x,y))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_10) = {1}

Following symbols are considered usable:
{div#,plus#,quot#,times#}
TcT has computed the following interpretation:
p(0) = [1]
p(div) = [2]
p(plus) = [2] x2 + [0]
p(quot) = [1] x1 + [4] x2 + [8] x3 + [1]
p(s) = [1] x1 + [3]
p(times) = [1]
p(div#) = [1] x1 + [1]
p(plus#) = [1] x2 + [0]
p(quot#) = [1] x1 + [4] x2 + [1]
p(times#) = [2] x1 + [2]
p(c_1) = [1] x1 + [0]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [2]
p(c_6) = [8] x1 + [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [4]
p(c_10) = [1] x1 + [3]
p(c_11) = [0]

Following rules are strictly oriented:
times#(s(x),y) = [2] x + [8]
> [2] x + [5]
= c_10(times#(x,y))

Following rules are (at-least) weakly oriented:

**** Step 5.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(s(x),y) -> c_10(times#(x,y))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(s(x),y) -> c_10(times#(x,y))
- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:times#(s(x),y) -> c_10(times#(x,y))
-->_1 times#(s(x),y) -> c_10(times#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: times#(s(x),y) -> c_10(times#(x,y))
**** Step 5.b:2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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