* Step 1: Sum 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:
            innermost runtime complexity wrt. defined symbols {*,+} and constructors {0,i}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: 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:
            innermost runtime complexity wrt. defined symbols {*,+} and constructors {0,i}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          *#(x,+(y,z)) -> c_1(+#(*(x,y),*(x,z)),*#(x,y),*#(x,z))
          *#(+(x,y),z) -> c_2(+#(*(x,z),*(y,z)),*#(x,z),*#(y,z))
          +#(x,0()) -> c_3()
          +#(x,i(x)) -> c_4()
          +#(+(x,y),z) -> c_5(+#(x,+(y,z)),+#(y,z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            *#(x,+(y,z)) -> c_1(+#(*(x,y),*(x,z)),*#(x,y),*#(x,z))
            *#(+(x,y),z) -> c_2(+#(*(x,z),*(y,z)),*#(x,z),*#(y,z))
            +#(x,0()) -> c_3()
            +#(x,i(x)) -> c_4()
            +#(+(x,y),z) -> c_5(+#(x,+(y,z)),+#(y,z))
        - Weak 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_1/3,c_2/3,c_3/0,c_4/0,c_5/2}
        - Obligation:
            innermost 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,i(x)) -> c_4()
* Step 4: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_3()
            +#(x,i(x)) -> c_4()
        - Signature:
            {*/2,+/2,*#/2,+#/2} / {0/0,i/1,c_1/3,c_2/3,c_3/0,c_4/0,c_5/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#} and constructors {0,i}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:+#(x,0()) -> c_3()
             
          
          2:S:+#(x,i(x)) -> c_4()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {*/2,+/2,*#/2,+#/2} / {0/0,i/1,c_1/3,c_2/3,c_3/0,c_4/0,c_5/2}
        - Obligation:
            innermost 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))