```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
sum(0()) -> 0()
sum(s(x)) -> +(sum(x),s(x))
sum1(0()) -> 0()
sum1(s(x)) -> s(+(sum1(x),+(x,x)))
- Signature:
{sum/1,sum1/1} / {+/2,0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:

Strict DPs
sum#(0()) -> c_1()
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(0()) -> c_3()
sum1#(s(x)) -> c_4(sum1#(x),x,x)
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(0()) -> c_1()
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(0()) -> c_3()
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Strict TRS:
sum(0()) -> 0()
sum(s(x)) -> +(sum(x),s(x))
sum1(0()) -> 0()
sum1(s(x)) -> s(+(sum1(x),+(x,x)))
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
sum#(0()) -> c_1()
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(0()) -> c_3()
sum1#(s(x)) -> c_4(sum1#(x),x,x)
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(0()) -> c_1()
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(0()) -> c_3()
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3}
by application of
Pre({1,3}) = {2,4}.
Here rules are labelled as follows:
1: sum#(0()) -> c_1()
2: sum#(s(x)) -> c_2(sum#(x),x)
3: sum1#(0()) -> c_3()
4: sum1#(s(x)) -> c_4(sum1#(x),x,x)
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Weak DPs:
sum#(0()) -> c_1()
sum1#(0()) -> c_3()
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:sum#(s(x)) -> c_2(sum#(x),x)
-->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_2 sum1#(0()) -> c_3():4
-->_2 sum#(0()) -> c_1():3
-->_1 sum#(0()) -> c_1():3
-->_2 sum#(s(x)) -> c_2(sum#(x),x):1
-->_1 sum#(s(x)) -> c_2(sum#(x),x):1

2:S:sum1#(s(x)) -> c_4(sum1#(x),x,x)
-->_3 sum1#(0()) -> c_3():4
-->_2 sum1#(0()) -> c_3():4
-->_1 sum1#(0()) -> c_3():4
-->_3 sum#(0()) -> c_1():3
-->_2 sum#(0()) -> c_1():3
-->_3 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_1 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_3 sum#(s(x)) -> c_2(sum#(x),x):1
-->_2 sum#(s(x)) -> c_2(sum#(x),x):1

3:W:sum#(0()) -> c_1()

4:W:sum1#(0()) -> c_3()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: sum#(0()) -> c_1()
4: sum1#(0()) -> c_3()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} 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: sum#(s(x)) -> c_2(sum#(x),x)

The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} 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_4) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [0]
p(0) = [2]
p(s) = [1] x1 + [2]
p(sum) = [8] x1 + [1]
p(sum1) = [0]
p(sum#) = [1] x1 + [0]
p(sum1#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [8] x1 + [0]

Following rules are strictly oriented:
sum#(s(x)) = [1] x + [2]
> [1] x + [0]
= c_2(sum#(x),x)

Following rules are (at-least) weakly oriented:
sum1#(s(x)) =  [0]
>= [0]
=  c_4(sum1#(x),x,x)

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Weak DPs:
sum#(s(x)) -> c_2(sum#(x),x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Weak DPs:
sum#(s(x)) -> c_2(sum#(x),x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} 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: sum1#(s(x)) -> c_4(sum1#(x),x,x)

The strictly oriented rules are moved into the weak component.
*** Step 5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Weak DPs:
sum#(s(x)) -> c_2(sum#(x),x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} 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_4) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [1]
p(0) = [0]
p(s) = [1] x1 + [2]
p(sum) = [1] x1 + [0]
p(sum1) = [1] x1 + [0]
p(sum#) = [6] x1 + [0]
p(sum1#) = [8] x1 + [4]
p(c_1) = [1]
p(c_2) = [1] x1 + [12]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]

Following rules are strictly oriented:
sum1#(s(x)) = [8] x + [20]
> [8] x + [4]
= c_4(sum1#(x),x,x)

Following rules are (at-least) weakly oriented:
sum#(s(x)) =  [6] x + [12]
>= [6] x + [12]
=  c_2(sum#(x),x)

*** Step 5.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 5.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sum#(s(x)) -> c_2(sum#(x),x)
sum1#(s(x)) -> c_4(sum1#(x),x,x)
- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:sum#(s(x)) -> c_2(sum#(x),x)
-->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_2 sum#(s(x)) -> c_2(sum#(x),x):1
-->_1 sum#(s(x)) -> c_2(sum#(x),x):1

2:W:sum1#(s(x)) -> c_4(sum1#(x),x,x)
-->_3 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_1 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
-->_3 sum#(s(x)) -> c_2(sum#(x),x):1
-->_2 sum#(s(x)) -> c_2(sum#(x),x):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sum#(s(x)) -> c_2(sum#(x),x)
2: sum1#(s(x)) -> c_4(sum1#(x),x,x)
*** Step 5.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
- Obligation:
runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```