*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        div(x,y) -> quot(x,y,y)
        div(0(),y) -> 0()
        quot(x,0(),s(z)) -> s(div(x,s(z)))
        quot(0(),s(y),z) -> 0()
        quot(s(x),s(y),z) -> quot(x,y,z)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {div/2,quot/3} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {div,quot}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak innermost dependency pairs:
      
      Strict DPs
        div#(x,y) -> c_1(quot#(x,y,y))
        div#(0(),y) -> c_2()
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(0(),s(y),z) -> c_4()
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        div#(x,y) -> c_1(quot#(x,y,y))
        div#(0(),y) -> c_2()
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(0(),s(y),z) -> c_4()
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
      Strict TRS Rules:
        div(x,y) -> quot(x,y,y)
        div(0(),y) -> 0()
        quot(x,0(),s(z)) -> s(div(x,s(z)))
        quot(0(),s(y),z) -> 0()
        quot(s(x),s(y),z) -> quot(x,y,z)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {div#,quot#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        div#(x,y) -> c_1(quot#(x,y,y))
        div#(0(),y) -> c_2()
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(0(),s(y),z) -> c_4()
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        div#(x,y) -> c_1(quot#(x,y,y))
        div#(0(),y) -> c_2()
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(0(),s(y),z) -> c_4()
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {div#,quot#}/{0,s}
    Applied Processor:
      Succeeding
    Proof:
      ()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        div#(x,y) -> c_1(quot#(x,y,y))
        div#(0(),y) -> c_2()
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(0(),s(y),z) -> c_4()
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {div#,quot#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {2,4}
      by application of
        Pre({2,4}) = {1,3,5}.
      Here rules are labelled as follows:
        1: div#(x,y) -> c_1(quot#(x,y,y))      
        2: div#(0(),y) -> c_2()                
        3: quot#(x,0(),s(z)) -> c_3(div#(x     
                                        ,s(z)))
        4: quot#(0(),s(y),z) -> c_4()          
        5: quot#(s(x),s(y),z) ->               
             c_5(quot#(x,y,z))                 
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        div#(x,y) -> c_1(quot#(x,y,y))
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
      Strict TRS Rules:
        
      Weak DP Rules:
        div#(0(),y) -> c_2()
        quot#(0(),s(y),z) -> c_4()
      Weak TRS Rules:
        
      Signature:
        {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {div#,quot#}/{0,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:div#(x,y) -> c_1(quot#(x,y,y))
           -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
           -->_1 quot#(0(),s(y),z) -> c_4():5
        
        2:S:quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
           -->_1 div#(0(),y) -> c_2():4
           -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1
        
        3:S:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
           -->_1 quot#(0(),s(y),z) -> c_4():5
           -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
           -->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2
        
        4:W:div#(0(),y) -> c_2()
           
        
        5:W:quot#(0(),s(y),z) -> c_4()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        4: div#(0(),y) -> c_2()      
        5: quot#(0(),s(y),z) -> c_4()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        div#(x,y) -> c_1(quot#(x,y,y))
        quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
      Obligation:
        Innermost
        basic terms: {div#,quot#}/{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:
        3: quot#(s(x),s(y),z) ->
             c_5(quot#(x,y,z))  
        
      Consider the set of all dependency pairs
        1: div#(x,y) -> c_1(quot#(x,y,y))      
        2: quot#(x,0(),s(z)) -> c_3(div#(x     
                                        ,s(z)))
        3: quot#(s(x),s(y),z) ->               
             c_5(quot#(x,y,z))                 
      Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
      SPACE(?,?)on application of the dependency pairs
        {3}
      These cover all (indirect) predecessors of dependency pairs
        {1,2,3}
      their number of applications is equally bounded.
      The dependency pairs are shifted into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          div#(x,y) -> c_1(quot#(x,y,y))
          quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
          quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {div#,quot#}/{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_1) = {1},
          uargs(c_3) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {div#,quot#}
        TcT has computed the following interpretation:
              p(0) = [0]                  
            p(div) = [2] x2 + [8]         
           p(quot) = [1]                  
              p(s) = [1] x1 + [2]         
           p(div#) = [4] x1 + [9] x2 + [5]
          p(quot#) = [4] x1 + [9] x3 + [5]
            p(c_1) = [1] x1 + [0]         
            p(c_2) = [0]                  
            p(c_3) = [1] x1 + [0]         
            p(c_4) = [0]                  
            p(c_5) = [1] x1 + [6]         
        
        Following rules are strictly oriented:
        quot#(s(x),s(y),z) = [4] x + [9] z + [13]
                           > [4] x + [9] z + [11]
                           = c_5(quot#(x,y,z))   
        
        
        Following rules are (at-least) weakly oriented:
                div#(x,y) =  [4] x + [9] y + [5] 
                          >= [4] x + [9] y + [5] 
                          =  c_1(quot#(x,y,y))   
        
        quot#(x,0(),s(z)) =  [4] x + [9] z + [23]
                          >= [4] x + [9] z + [23]
                          =  c_3(div#(x,s(z)))   
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          div#(x,y) -> c_1(quot#(x,y,y))
          quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        Strict TRS Rules:
          
        Weak DP Rules:
          quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        Weak TRS Rules:
          
        Signature:
          {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {div#,quot#}/{0,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          div#(x,y) -> c_1(quot#(x,y,y))
          quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
          quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        Weak TRS Rules:
          
        Signature:
          {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {div#,quot#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:div#(x,y) -> c_1(quot#(x,y,y))
             -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
          
          2:W:quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
             -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1
          
          3:W:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
             -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
             -->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: div#(x,y) -> c_1(quot#(x,y,y))      
          2: quot#(x,0(),s(z)) -> c_3(div#(x     
                                          ,s(z)))
          3: quot#(s(x),s(y),z) ->               
               c_5(quot#(x,y,z))                 
  *** 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:
          {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        Obligation:
          Innermost
          basic terms: {div#,quot#}/{0,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).