*** 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:
        Full
        basic terms: {*,+}/{0,i}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(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 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,*#/2,+#/2} / {0/0,i/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1}
      Obligation:
        Full
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        +#(x,0()) -> c_3(x)
        +#(x,i(x)) -> c_4()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        +#(x,0()) -> c_3(x)
        +#(x,i(x)) -> c_4()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2,+/2,*#/2,+#/2} / {0/0,i/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1}
      Obligation:
        Full
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      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()
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        +#(x,0()) -> c_3(x)
      Strict TRS Rules:
        
      Weak DP Rules:
        +#(x,i(x)) -> c_4()
      Weak TRS Rules:
        
      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:
        Full
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      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()
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        +#(x,0()) -> c_3(x)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      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:
        Full
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(*) = [0]
            p(+) = [0]
            p(0) = [0]
            p(i) = [0]
           p(*#) = [0]
           p(+#) = [1]
          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]   
                  > [0]   
                  = c_3(x)
        
        
        Following rules are (at-least) weakly oriented:
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        +#(x,0()) -> c_3(x)
      Weak TRS Rules:
        
      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:
        Full
        basic terms: {*#,+#}/{0,i}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).