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

Strict DPs
*#(x,+(y,z)) -> c_1(+#(*(x,y),*(x,z)))
*#(+(x,y),z) -> c_2(+#(*(x,z),*(y,z)))
+#(x,0()) -> c_3(x)
+#(x,i(x)) -> c_4()
+#(+(x,y),z) -> c_5(+#(x,+(y,z)))
Weak DPs

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

2:W:+#(x,i(x)) -> c_4()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: +#(x,i(x)) -> c_4()
* Step 5: MI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
+#(x,0()) -> c_3(x)
- Signature:
{*/2,+/2,*#/2,+#/2} / {0/0,i/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {*#,+#} and constructors {0,i}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 0))), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 0))):

The following argument positions are considered usable:
none

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [0]
p(+) = [0]
p(0) = [2]
p(i) = [0]
p(*#) = [8]
p(+#) = [1] x_1 + [2] x_2 + [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]

Following rules are strictly oriented:
+#(x,0()) = [1] x + [4]
> [0]
= c_3(x)

Following rules are (at-least) weakly oriented:

* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
+#(x,0()) -> c_3(x)
- Signature:
{*/2,+/2,*#/2,+#/2} / {0/0,i/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1}
- Obligation:
runtime complexity wrt. defined symbols {*#,+#} and constructors {0,i}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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