*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        cons(x,y) -> x
        cons(x,y) -> y
        f(s(a()),s(b()),x) -> f(x,x,x)
        g(f(s(x),s(y),z)) -> g(f(x,y,z))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {cons/2,f/3,g/1} / {a/0,b/0,s/1}
      Obligation:
        Full
        basic terms: {cons,f,g}/{a,b,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        cons#(x,y) -> c_1(x)
        cons#(x,y) -> c_2(y)
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        cons#(x,y) -> c_1(x)
        cons#(x,y) -> c_2(y)
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
      Strict TRS Rules:
        cons(x,y) -> x
        cons(x,y) -> y
        f(s(a()),s(b()),x) -> f(x,x,x)
        g(f(s(x),s(y),z)) -> g(f(x,y,z))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
      Obligation:
        Full
        basic terms: {cons#,f#,g#}/{a,b,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        cons#(x,y) -> c_1(x)
        cons#(x,y) -> c_2(y)
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        cons#(x,y) -> c_1(x)
        cons#(x,y) -> c_2(y)
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
      Obligation:
        Full
        basic terms: {cons#,f#,g#}/{a,b,s}
    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(a) = [0]         
            p(b) = [0]         
         p(cons) = [0]         
            p(f) = [0]         
            p(g) = [0]         
            p(s) = [11]        
        p(cons#) = [9]         
           p(f#) = [0]         
           p(g#) = [0]         
          p(c_1) = [1]         
          p(c_2) = [9]         
          p(c_3) = [0]         
          p(c_4) = [1] x1 + [0]
      
      Following rules are strictly oriented:
      cons#(x,y) = [9]   
                 > [1]   
                 = c_1(x)
      
      
      Following rules are (at-least) weakly oriented:
               cons#(x,y) =  [9]           
                          >= [9]           
                          =  c_2(y)        
      
      f#(s(a()),s(b()),x) =  [0]           
                          >= [0]           
                          =  c_3(f#(x,x,x))
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        cons#(x,y) -> c_2(y)
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
      Strict TRS Rules:
        
      Weak DP Rules:
        cons#(x,y) -> c_1(x)
      Weak TRS Rules:
        
      Signature:
        {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
      Obligation:
        Full
        basic terms: {cons#,f#,g#}/{a,b,s}
    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(a) = [0]
            p(b) = [0]
         p(cons) = [0]
            p(f) = [0]
            p(g) = [0]
            p(s) = [0]
        p(cons#) = [4]
           p(f#) = [0]
           p(g#) = [0]
          p(c_1) = [4]
          p(c_2) = [0]
          p(c_3) = [0]
          p(c_4) = [0]
      
      Following rules are strictly oriented:
      cons#(x,y) = [4]   
                 > [0]   
                 = c_2(y)
      
      
      Following rules are (at-least) weakly oriented:
               cons#(x,y) =  [4]           
                          >= [4]           
                          =  c_1(x)        
      
      f#(s(a()),s(b()),x) =  [0]           
                          >= [0]           
                          =  c_3(f#(x,x,x))
      
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
      Strict TRS Rules:
        
      Weak DP Rules:
        cons#(x,y) -> c_1(x)
        cons#(x,y) -> c_2(y)
      Weak TRS Rules:
        
      Signature:
        {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
      Obligation:
        Full
        basic terms: {cons#,f#,g#}/{a,b,s}
    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(a) = [0]          
              p(b) = [0]          
           p(cons) = [0]          
              p(f) = [0]          
              p(g) = [0]          
              p(s) = [0]          
          p(cons#) = [0]          
             p(f#) = [8] x3 + [11]
             p(g#) = [1] x1 + [0] 
            p(c_1) = [0]          
            p(c_2) = [0]          
            p(c_3) = [4]          
            p(c_4) = [0]          
        
        Following rules are strictly oriented:
        f#(s(a()),s(b()),x) = [8] x + [11]  
                            > [4]           
                            = c_3(f#(x,x,x))
        
        
        Following rules are (at-least) weakly oriented:
        cons#(x,y) =  [0]   
                   >= [0]   
                   =  c_1(x)
        
        cons#(x,y) =  [0]   
                   >= [0]   
                   =  c_2(y)
        
      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:
        cons#(x,y) -> c_1(x)
        cons#(x,y) -> c_2(y)
        f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
      Weak TRS Rules:
        
      Signature:
        {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
      Obligation:
        Full
        basic terms: {cons#,f#,g#}/{a,b,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).