*** 1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        -(x,0()) -> x
        -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
        -(0(),y) -> 0()
        p(0()) -> 0()
        p(s(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {-/2,p/1} / {0/0,greater/2,if/3,s/1}
      Obligation:
        Innermost
        basic terms: {-,p}/{0,greater,if,s}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak innermost dependency pairs:
      
      Strict DPs
        -#(x,0()) -> c_1()
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        -#(0(),y) -> c_3()
        p#(0()) -> c_4()
        p#(s(x)) -> c_5()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        -#(x,0()) -> c_1()
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        -#(0(),y) -> c_3()
        p#(0()) -> c_4()
        p#(s(x)) -> c_5()
      Strict TRS Rules:
        -(x,0()) -> x
        -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
        -(0(),y) -> 0()
        p(0()) -> 0()
        p(s(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
      Obligation:
        Innermost
        basic terms: {-#,p#}/{0,greater,if,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        p(s(x)) -> x
        -#(x,0()) -> c_1()
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        -#(0(),y) -> c_3()
        p#(0()) -> c_4()
        p#(s(x)) -> c_5()
*** 1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        -#(x,0()) -> c_1()
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        -#(0(),y) -> c_3()
        p#(0()) -> c_4()
        p#(s(x)) -> c_5()
      Strict TRS Rules:
        p(s(x)) -> x
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
      Obligation:
        Innermost
        basic terms: {-#,p#}/{0,greater,if,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(-#) = {2},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                p(-) = [0]                           
                p(0) = [0]                           
          p(greater) = [1] x1 + [1] x2 + [0]         
               p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
                p(p) = [1] x1 + [0]                  
                p(s) = [1] x1 + [11]                 
               p(-#) = [1] x2 + [0]                  
               p(p#) = [0]                           
              p(c_1) = [0]                           
              p(c_2) = [1] x1 + [0]                  
              p(c_3) = [0]                           
              p(c_4) = [0]                           
              p(c_5) = [0]                           
        
        Following rules are strictly oriented:
        p(s(x)) = [1] x + [11]
                > [1] x + [0] 
                = x           
        
        
        Following rules are (at-least) weakly oriented:
         -#(x,0()) =  [0]               
                   >= [0]               
                   =  c_1()             
        
        -#(x,s(y)) =  [1] y + [11]      
                   >= [1] y + [11]      
                   =  c_2(-#(x,p(s(y))))
        
         -#(0(),y) =  [1] y + [0]       
                   >= [0]               
                   =  c_3()             
        
           p#(0()) =  [0]               
                   >= [0]               
                   =  c_4()             
        
          p#(s(x)) =  [0]               
                   >= [0]               
                   =  c_5()             
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        -#(x,0()) -> c_1()
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        -#(0(),y) -> c_3()
        p#(0()) -> c_4()
        p#(s(x)) -> c_5()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(x)) -> x
      Signature:
        {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
      Obligation:
        Innermost
        basic terms: {-#,p#}/{0,greater,if,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3,4,5}
      by application of
        Pre({1,3,4,5}) = {2}.
      Here rules are labelled as follows:
        1: -#(x,0()) -> c_1()              
        2: -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        3: -#(0(),y) -> c_3()              
        4: p#(0()) -> c_4()                
        5: p#(s(x)) -> c_5()               
*** 1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
      Strict TRS Rules:
        
      Weak DP Rules:
        -#(x,0()) -> c_1()
        -#(0(),y) -> c_3()
        p#(0()) -> c_4()
        p#(s(x)) -> c_5()
      Weak TRS Rules:
        p(s(x)) -> x
      Signature:
        {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
      Obligation:
        Innermost
        basic terms: {-#,p#}/{0,greater,if,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:-#(x,s(y)) -> c_2(-#(x,p(s(y))))
           -->_1 -#(0(),y) -> c_3():3
           -->_1 -#(x,0()) -> c_1():2
           -->_1 -#(x,s(y)) -> c_2(-#(x,p(s(y)))):1
        
        2:W:-#(x,0()) -> c_1()
           
        
        3:W:-#(0(),y) -> c_3()
           
        
        4:W:p#(0()) -> c_4()
           
        
        5:W:p#(s(x)) -> c_5()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        5: p#(s(x)) -> c_5() 
        4: p#(0()) -> c_4()  
        2: -#(x,0()) -> c_1()
        3: -#(0(),y) -> c_3()
*** 1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        -#(x,s(y)) -> c_2(-#(x,p(s(y))))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(x)) -> x
      Signature:
        {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
      Obligation:
        Innermost
        basic terms: {-#,p#}/{0,greater,if,s}
    Applied Processor:
      PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        1: -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          p(s(x)) -> x
        Signature:
          {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        Obligation:
          Innermost
          basic terms: {-#,p#}/{0,greater,if,s}
      Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {p,-#,p#}
        TcT has computed the following interpretation:
                p(-) = [1 0 0]      [1 0 0]      [0] 
                       [0 1 1] x1 + [1 0 1] x2 + [0] 
                       [1 1 0]      [4 4 4]      [0] 
                p(0) = [1]                           
                       [0]                           
                       [1]                           
          p(greater) = [1 1 1]      [0]              
                       [0 0 1] x1 + [2]              
                       [0 0 0]      [1]              
               p(if) = [0 2 2]      [0 0 4]      [0 2
                       1]      [2]                   
                       [0 0 1] x1 + [0 0 0] x2 + [0 0
                       2] x3 + [4]                   
                       [0 0 0]      [0 0 0]      [0 0
                       0]      [0]                   
                p(p) = [2 0 0]      [0]              
                       [4 0 0] x1 + [7]              
                       [0 1 0]      [0]              
                p(s) = [1 1 2]      [0]              
                       [0 0 1] x1 + [0]              
                       [0 0 1]      [1]              
               p(-#) = [6 0 4]      [0 0 2]      [2] 
                       [6 0 4] x1 + [0 1 1] x2 + [7] 
                       [4 1 0]      [0 0 0]      [6] 
               p(p#) = [0 0 1]      [1]              
                       [1 1 4] x1 + [1]              
                       [0 0 1]      [1]              
              p(c_1) = [2]                           
                       [2]                           
                       [1]                           
              p(c_2) = [1 0 0]      [0]              
                       [1 0 0] x1 + [6]              
                       [0 0 0]      [1]              
              p(c_3) = [2]                           
                       [2]                           
                       [1]                           
              p(c_4) = [2]                           
                       [1]                           
                       [0]                           
              p(c_5) = [4]                           
                       [0]                           
                       [4]                           
        
        Following rules are strictly oriented:
        -#(x,s(y)) = [6 0 4]     [0 0 2]     [4]
                     [6 0 4] x + [0 0 2] y + [8]
                     [4 1 0]     [0 0 0]     [6]
                   > [6 0 4]     [0 0 2]     [2]
                     [6 0 4] x + [0 0 2] y + [8]
                     [0 0 0]     [0 0 0]     [1]
                   = c_2(-#(x,p(s(y))))         
        
        
        Following rules are (at-least) weakly oriented:
        p(s(x)) =  [2 2 4]     [0]
                   [4 4 8] x + [7]
                   [0 0 1]     [0]
                >= [1 0 0]     [0]
                   [0 1 0] x + [0]
                   [0 0 1]     [0]
                =  x              
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        Weak TRS Rules:
          p(s(x)) -> x
        Signature:
          {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        Obligation:
          Innermost
          basic terms: {-#,p#}/{0,greater,if,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:
          -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        Weak TRS Rules:
          p(s(x)) -> x
        Signature:
          {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        Obligation:
          Innermost
          basic terms: {-#,p#}/{0,greater,if,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:-#(x,s(y)) -> c_2(-#(x,p(s(y))))
             -->_1 -#(x,s(y)) -> c_2(-#(x,p(s(y)))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: -#(x,s(y)) -> c_2(-#(x,p(s(y))))
  *** 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:
          p(s(x)) -> x
        Signature:
          {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        Obligation:
          Innermost
          basic terms: {-#,p#}/{0,greater,if,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).