*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        *(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))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2,+/2} / {0/0,i/1}
      Obligation:
        Innermost
        basic terms: {*,+}/{0,i}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(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))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        *(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
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        +#(x,0()) -> c_3()
        +#(x,i(x)) -> c_4()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        +#(x,0()) -> c_3()
        +#(x,i(x)) -> c_4()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      Trivial
    Proof:
      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.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).