*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
        h(X,Z) -> f(X,s(X),Z)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,h/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {f,g,h}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak innermost dependency pairs:
      
      Strict DPs
        f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
        g#(X,s(Y)) -> c_2(g#(X,Y))
        g#(0(),Y) -> c_3()
        h#(X,Z) -> c_4(f#(X,s(X),Z))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
        g#(X,s(Y)) -> c_2(g#(X,Y))
        g#(0(),Y) -> c_3()
        h#(X,Z) -> c_4(f#(X,s(X),Z))
      Strict TRS Rules:
        f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
        h(X,Z) -> f(X,s(X),Z)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
        f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
        g#(X,s(Y)) -> c_2(g#(X,Y))
        g#(0(),Y) -> c_3()
        h#(X,Z) -> c_4(f#(X,s(X),Z))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
        g#(X,s(Y)) -> c_2(g#(X,Y))
        g#(0(),Y) -> c_3()
        h#(X,Z) -> c_4(f#(X,s(X),Z))
      Strict TRS Rules:
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(0) = [0]                            
            p(f) = [0]                            
            p(g) = [4] x2 + [2]                   
            p(h) = [1] x1 + [2]                   
            p(s) = [1] x1 + [4]                   
           p(f#) = [10] x1 + [2] x2 + [4] x3 + [0]
           p(g#) = [8] x1 + [1] x2 + [1]          
           p(h#) = [13] x1 + [4] x2 + [2]         
          p(c_1) = [1] x1 + [4]                   
          p(c_2) = [1] x1 + [3]                   
          p(c_3) = [2]                            
          p(c_4) = [1] x1 + [8]                   
        
        Following rules are strictly oriented:
        g#(X,s(Y)) = [8] X + [1] Y + [5]
                   > [8] X + [1] Y + [4]
                   = c_2(g#(X,Y))       
        
         g(X,s(Y)) = [4] Y + [18]       
                   > [4] Y + [2]        
                   = g(X,Y)             
        
          g(0(),Y) = [4] Y + [2]        
                   > [0]                
                   = 0()                
        
        
        Following rules are (at-least) weakly oriented:
        f#(X,Y,g(X,Y)) =  [10] X + [18] Y + [8]
                       >= [16] Y + [14]        
                       =  c_1(h#(0(),g(X,Y)))  
        
             g#(0(),Y) =  [1] Y + [1]          
                       >= [2]                  
                       =  c_3()                
        
               h#(X,Z) =  [13] X + [4] Z + [2] 
                       >= [12] X + [4] Z + [16]
                       =  c_4(f#(X,s(X),Z))    
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
        g#(0(),Y) -> c_3()
        h#(X,Z) -> c_4(f#(X,s(X),Z))
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(X,s(Y)) -> c_2(g#(X,Y))
      Weak TRS Rules:
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      RemoveInapplicable
    Proof:
      Only the nodes
        {2,3,4}
      are reachable from nodes
        {2,3,4}
      that start derivation from marked basic terms.
      The nodes not reachable are removed from the problem.
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(0(),Y) -> c_3()
        h#(X,Z) -> c_4(f#(X,s(X),Z))
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(X,s(Y)) -> c_2(g#(X,Y))
      Weak TRS Rules:
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {2}
      by application of
        Pre({2}) = {}.
      Here rules are labelled as follows:
        1: g#(0(),Y) -> c_3()          
        2: h#(X,Z) -> c_4(f#(X,s(X),Z))
        3: g#(X,s(Y)) -> c_2(g#(X,Y))  
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(0(),Y) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(X,s(Y)) -> c_2(g#(X,Y))
        h#(X,Z) -> c_4(f#(X,s(X),Z))
      Weak TRS Rules:
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:g#(0(),Y) -> c_3()
           
        
        2:W:g#(X,s(Y)) -> c_2(g#(X,Y))
           -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
           -->_1 g#(0(),Y) -> c_3():1
        
        3:W:h#(X,Z) -> c_4(f#(X,s(X),Z))
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        3: h#(X,Z) -> c_4(f#(X,s(X),Z))
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(0(),Y) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(X,s(Y)) -> c_2(g#(X,Y))
      Weak TRS Rules:
        g(X,s(Y)) -> g(X,Y)
        g(0(),Y) -> 0()
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        g#(X,s(Y)) -> c_2(g#(X,Y))
        g#(0(),Y) -> c_3()
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        g#(0(),Y) -> c_3()
      Strict TRS Rules:
        
      Weak DP Rules:
        g#(X,s(Y)) -> c_2(g#(X,Y))
      Weak TRS Rules:
        
      Signature:
        {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Innermost
        basic terms: {f#,g#,h#}/{0,s}
    Applied Processor:
      PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        1: g#(0(),Y) -> c_3()
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          g#(0(),Y) -> c_3()
        Strict TRS Rules:
          
        Weak DP Rules:
          g#(X,s(Y)) -> c_2(g#(X,Y))
        Weak TRS Rules:
          
        Signature:
          {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        Obligation:
          Innermost
          basic terms: {f#,g#,h#}/{0,s}
      Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#,h#}
        TcT has computed the following interpretation:
            p(0) = [4]                  
            p(f) = [1]                  
            p(g) = [2] x1 + [1] x2 + [8]
            p(h) = [1] x1 + [1] x2 + [0]
            p(s) = [1] x1 + [4]         
           p(f#) = [1] x2 + [1]         
           p(g#) = [4] x1 + [4] x2 + [0]
           p(h#) = [2] x2 + [0]         
          p(c_1) = [1] x1 + [0]         
          p(c_2) = [1] x1 + [11]        
          p(c_3) = [13]                 
          p(c_4) = [0]                  
        
        Following rules are strictly oriented:
        g#(0(),Y) = [4] Y + [16]
                  > [13]        
                  = c_3()       
        
        
        Following rules are (at-least) weakly oriented:
        g#(X,s(Y)) =  [4] X + [4] Y + [16]
                   >= [4] X + [4] Y + [11]
                   =  c_2(g#(X,Y))        
        
  *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          g#(X,s(Y)) -> c_2(g#(X,Y))
          g#(0(),Y) -> c_3()
        Weak TRS Rules:
          
        Signature:
          {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        Obligation:
          Innermost
          basic terms: {f#,g#,h#}/{0,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          g#(X,s(Y)) -> c_2(g#(X,Y))
          g#(0(),Y) -> c_3()
        Weak TRS Rules:
          
        Signature:
          {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        Obligation:
          Innermost
          basic terms: {f#,g#,h#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(0(),Y) -> c_3():2
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):1
          
          2:W:g#(0(),Y) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(X,s(Y)) -> c_2(g#(X,Y))
          2: g#(0(),Y) -> c_3()        
  *** 1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        Obligation:
          Innermost
          basic terms: {f#,g#,h#}/{0,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).