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

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

and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
plus#(x,0()) -> c_1()
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(0(),x) -> c_3()
plus#(s(x),y) -> c_4(plus#(x,y))
times#(x,0()) -> c_5()
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5}
by application of
Pre({1,3,5}) = {2,4,6}.
Here rules are labelled as follows:
1: plus#(x,0()) -> c_1()
2: plus#(x,s(y)) -> c_2(plus#(x,y))
3: plus#(0(),x) -> c_3()
4: plus#(s(x),y) -> c_4(plus#(x,y))
5: times#(x,0()) -> c_5()
6: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak DPs:
plus#(x,0()) -> c_1()
plus#(0(),x) -> c_3()
times#(x,0()) -> c_5()
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:plus#(x,s(y)) -> c_2(plus#(x,y))
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
-->_1 plus#(0(),x) -> c_3():5
-->_1 plus#(x,0()) -> c_1():4
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1

2:S:plus#(s(x),y) -> c_4(plus#(x,y))
-->_1 plus#(0(),x) -> c_3():5
-->_1 plus#(x,0()) -> c_1():4
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1

3:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
-->_2 times#(x,0()) -> c_5():6
-->_1 plus#(0(),x) -> c_3():5
-->_1 plus#(x,0()) -> c_1():4
-->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):3
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1

4:W:plus#(x,0()) -> c_1()

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

6:W:times#(x,0()) -> c_5()

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

Problem (S)
- Strict DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
** Step 4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,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#(x,s(y)) -> c_2(plus#(x,y))

The strictly oriented rules are moved into the weak component.
*** Step 4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,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_2) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1,2}

Following symbols are considered usable:
{plus#,times#}
TcT has computed the following interpretation:
p(0) = 1
p(plus) = 2 + 6*x2
p(s) = 1 + x1
p(times) = 2*x2
p(plus#) = 1 + 2*x2
p(times#) = 5 + 4*x1 + 5*x1*x2 + x2
p(c_1) = 1
p(c_2) = x1
p(c_3) = 1
p(c_4) = x1
p(c_5) = 0
p(c_6) = x1 + x2

Following rules are strictly oriented:
plus#(x,s(y)) = 3 + 2*y
> 1 + 2*y
= c_2(plus#(x,y))

Following rules are (at-least) weakly oriented:
plus#(s(x),y) =  1 + 2*y
>= 1 + 2*y
=  c_4(plus#(x,y))

times#(x,s(y)) =  6 + 9*x + 5*x*y + y
>= 6 + 6*x + 5*x*y + y
=  c_6(plus#(times(x,y),x),times#(x,y))

*** Step 4.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 4.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
and a lower component
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
Further, following extension rules are added to the lower component.
times#(x,s(y)) -> plus#(times(x,y),x)
times#(x,s(y)) -> times#(x,y)
**** Step 4.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,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#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))

The strictly oriented rules are moved into the weak component.
***** Step 4.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,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_6) = {1,2}

Following symbols are considered usable:
{plus#,times#}
TcT has computed the following interpretation:
p(0) = [0]
p(plus) = [0]
p(s) = [1] x1 + [2]
p(times) = [0]
p(plus#) = [0]
p(times#) = [10] x2 + [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [1] x2 + [0]

Following rules are strictly oriented:
times#(x,s(y)) = [10] y + [20]
> [10] y + [0]
= c_6(plus#(times(x,y),x),times#(x,y))

Following rules are (at-least) weakly oriented:

***** Step 4.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

***** Step 4.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
-->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),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#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
***** Step 4.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

**** Step 4.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
times#(x,s(y)) -> plus#(times(x,y),x)
times#(x,s(y)) -> times#(x,y)
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,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_4(plus#(x,y))

The strictly oriented rules are moved into the weak component.
***** Step 4.a:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
times#(x,s(y)) -> plus#(times(x,y),x)
times#(x,s(y)) -> times#(x,y)
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,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_2) = {1},
uargs(c_4) = {1}

Following symbols are considered usable:
{plus,times,plus#,times#}
TcT has computed the following interpretation:
p(0) = 0
p(plus) = x1 + x2
p(s) = 1 + x1
p(times) = 2*x1*x2
p(plus#) = 2*x1 + 3*x2^2
p(times#) = 4 + 4*x1*x2 + 3*x1^2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 0
p(c_4) = x1
p(c_5) = 0
p(c_6) = 1 + x2

Following rules are strictly oriented:
plus#(s(x),y) = 2 + 2*x + 3*y^2
> 2*x + 3*y^2
= c_4(plus#(x,y))

Following rules are (at-least) weakly oriented:
plus#(x,s(y)) =  3 + 2*x + 6*y + 3*y^2
>= 2*x + 3*y^2
=  c_2(plus#(x,y))

times#(x,s(y)) =  4 + 4*x + 4*x*y + 3*x^2
>= 4*x*y + 3*x^2
=  plus#(times(x,y),x)

times#(x,s(y)) =  4 + 4*x + 4*x*y + 3*x^2
>= 4 + 4*x*y + 3*x^2
=  times#(x,y)

plus(x,0()) =  x
>= x
=  x

plus(x,s(y)) =  1 + x + y
>= 1 + x + y
=  s(plus(x,y))

plus(0(),x) =  x
>= x
=  x

plus(s(x),y) =  1 + x + y
>= 1 + x + y
=  s(plus(x,y))

times(x,0()) =  0
>= 0
=  0()

times(x,s(y)) =  2*x + 2*x*y
>= x + 2*x*y
=  plus(times(x,y),x)

***** Step 4.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
times#(x,s(y)) -> plus#(times(x,y),x)
times#(x,s(y)) -> times#(x,y)
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

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

2:W:plus#(s(x),y) -> c_4(plus#(x,y))
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1

3:W:times#(x,s(y)) -> plus#(times(x,y),x)
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1

4:W:times#(x,s(y)) -> times#(x,y)
-->_1 times#(x,s(y)) -> times#(x,y):4
-->_1 times#(x,s(y)) -> plus#(times(x,y),x):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: times#(x,s(y)) -> times#(x,y)
3: times#(x,s(y)) -> plus#(times(x,y),x)
1: plus#(x,s(y)) -> c_2(plus#(x,y))
2: plus#(s(x),y) -> c_4(plus#(x,y))
***** Step 4.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak DPs:
plus#(x,s(y)) -> c_2(plus#(x,y))
plus#(s(x),y) -> c_4(plus#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2
-->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1

2:W:plus#(x,s(y)) -> c_2(plus#(x,y))
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2

3:W:plus#(s(x),y) -> c_4(plus#(x,y))
-->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3
-->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: plus#(s(x),y) -> c_4(plus#(x,y))
2: plus#(x,s(y)) -> c_2(plus#(x,y))
** Step 4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
-->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
times#(x,s(y)) -> c_6(times#(x,y))
** Step 4.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(times#(x,y))
- Weak TRS:
plus(x,0()) -> x
plus(x,s(y)) -> s(plus(x,y))
plus(0(),x) -> x
plus(s(x),y) -> s(plus(x,y))
times(x,0()) -> 0()
times(x,s(y)) -> plus(times(x,y),x)
- Signature:
{plus/2,times/2,plus#/2,times#/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 {plus#,times#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
times#(x,s(y)) -> c_6(times#(x,y))
** Step 4.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(times#(x,y))
- Signature:
{plus/2,times/2,plus#/2,times#/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 {plus#,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#(x,s(y)) -> c_6(times#(x,y))

The strictly oriented rules are moved into the weak component.
*** Step 4.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
times#(x,s(y)) -> c_6(times#(x,y))
- Signature:
{plus/2,times/2,plus#/2,times#/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 {plus#,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_6) = {1}

Following symbols are considered usable:
{plus#,times#}
TcT has computed the following interpretation:
p(0) = [1]
p(plus) = [1] x2 + [1]
p(s) = [1] x1 + [8]
p(times) = [1] x2 + [1]
p(plus#) = [1]
p(times#) = [1] x2 + [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [4]
p(c_6) = [1] x1 + [4]

Following rules are strictly oriented:
times#(x,s(y)) = [1] y + [8]
> [1] y + [4]
= c_6(times#(x,y))

Following rules are (at-least) weakly oriented:

*** Step 4.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(x,s(y)) -> c_6(times#(x,y))
- Signature:
{plus/2,times/2,plus#/2,times#/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 {plus#,times#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 4.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
times#(x,s(y)) -> c_6(times#(x,y))
- Signature:
{plus/2,times/2,plus#/2,times#/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 {plus#,times#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:times#(x,s(y)) -> c_6(times#(x,y))
-->_1 times#(x,s(y)) -> c_6(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#(x,s(y)) -> c_6(times#(x,y))
*** Step 4.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{plus/2,times/2,plus#/2,times#/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 {plus#,times#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))
```