*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        *(x,0()) -> 0()
        *(*(x,y),z) -> *(x,*(y,z))
        *(1(),y) -> y
        *(i(x),x) -> 1()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2} / {0/0,1/0,i/1}
      Obligation:
        Full
        basic terms: {*}/{0,1,i}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        *#(x,0()) -> c_1()
        *#(*(x,y),z) -> c_2(*#(x,*(y,z)))
        *#(1(),y) -> c_3(y)
        *#(i(x),x) -> c_4()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(x,0()) -> c_1()
        *#(*(x,y),z) -> c_2(*#(x,*(y,z)))
        *#(1(),y) -> c_3(y)
        *#(i(x),x) -> c_4()
      Strict TRS Rules:
        *(x,0()) -> 0()
        *(*(x,y),z) -> *(x,*(y,z))
        *(1(),y) -> y
        *(i(x),x) -> 1()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2,*#/2} / {0/0,1/0,i/1,c_1/0,c_2/1,c_3/1,c_4/0}
      Obligation:
        Full
        basic terms: {*#}/{0,1,i}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        *#(x,0()) -> c_1()
        *#(1(),y) -> c_3(y)
        *#(i(x),x) -> c_4()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(x,0()) -> c_1()
        *#(1(),y) -> c_3(y)
        *#(i(x),x) -> c_4()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2,*#/2} / {0/0,1/0,i/1,c_1/0,c_2/1,c_3/1,c_4/0}
      Obligation:
        Full
        basic terms: {*#}/{0,1,i}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3}
      by application of
        Pre({1,3}) = {2}.
      Here rules are labelled as follows:
        1: *#(x,0()) -> c_1() 
        2: *#(1(),y) -> c_3(y)
        3: *#(i(x),x) -> c_4()
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(1(),y) -> c_3(y)
      Strict TRS Rules:
        
      Weak DP Rules:
        *#(x,0()) -> c_1()
        *#(i(x),x) -> c_4()
      Weak TRS Rules:
        
      Signature:
        {*/2,*#/2} / {0/0,1/0,i/1,c_1/0,c_2/1,c_3/1,c_4/0}
      Obligation:
        Full
        basic terms: {*#}/{0,1,i}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:*#(1(),y) -> c_3(y)
           -->_1 *#(i(x),x) -> c_4():3
           -->_1 *#(x,0()) -> c_1():2
           -->_1 *#(1(),y) -> c_3(y):1
        
        2:W:*#(x,0()) -> c_1()
           
        
        3:W:*#(i(x),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,0()) -> c_1() 
        3: *#(i(x),x) -> c_4()
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(1(),y) -> c_3(y)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2,*#/2} / {0/0,1/0,i/1,c_1/0,c_2/1,c_3/1,c_4/0}
      Obligation:
        Full
        basic terms: {*#}/{0,1,i}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
    Proof:
      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) = [0]         
          p(1) = [0]         
          p(i) = [0]         
         p(*#) = [1] x2 + [8]
        p(c_1) = [0]         
        p(c_2) = [0]         
        p(c_3) = [0]         
        p(c_4) = [0]         
      
      Following rules are strictly oriented:
      *#(1(),y) = [1] y + [8]
                > [0]        
                = c_3(y)     
      
      
      Following rules are (at-least) weakly oriented:
      
*** 1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        *#(1(),y) -> c_3(y)
      Weak TRS Rules:
        
      Signature:
        {*/2,*#/2} / {0/0,1/0,i/1,c_1/0,c_2/1,c_3/1,c_4/0}
      Obligation:
        Full
        basic terms: {*#}/{0,1,i}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).