*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        div(0(),s(Y)) -> 0()
        div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
        minus(X,0()) -> X
        minus(s(X),s(Y)) -> p(minus(X,Y))
        p(s(X)) -> X
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {div/2,minus/2,p/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {div,minus,p}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(minus) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
              p(0) = [0]         
            p(div) = [1] x1 + [0]
          p(minus) = [1] x1 + [0]
              p(p) = [1] x1 + [9]
              p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        p(s(X)) = [1] X + [9]
                > [1] X + [0]
                = X          
        
        
        Following rules are (at-least) weakly oriented:
           div(0(),s(Y)) =  [0]                    
                         >= [0]                    
                         =  0()                    
        
          div(s(X),s(Y)) =  [1] X + [0]            
                         >= [1] X + [0]            
                         =  s(div(minus(X,Y),s(Y)))
        
            minus(X,0()) =  [1] X + [0]            
                         >= [1] X + [0]            
                         =  X                      
        
        minus(s(X),s(Y)) =  [1] X + [0]            
                         >= [1] X + [9]            
                         =  p(minus(X,Y))          
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        div(0(),s(Y)) -> 0()
        div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
        minus(X,0()) -> X
        minus(s(X),s(Y)) -> p(minus(X,Y))
      Weak DP Rules:
        
      Weak TRS Rules:
        p(s(X)) -> X
      Signature:
        {div/2,minus/2,p/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {div,minus,p}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(minus) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
              p(0) = [0]                  
            p(div) = [1] x1 + [8] x2 + [0]
          p(minus) = [1] x1 + [6]         
              p(p) = [1] x1 + [9]         
              p(s) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
        div(0(),s(Y)) = [8] Y + [16]
                      > [0]         
                      = 0()         
        
         minus(X,0()) = [1] X + [6] 
                      > [1] X + [0] 
                      = X           
        
        
        Following rules are (at-least) weakly oriented:
          div(s(X),s(Y)) =  [1] X + [8] Y + [18]   
                         >= [1] X + [8] Y + [24]   
                         =  s(div(minus(X,Y),s(Y)))
        
        minus(s(X),s(Y)) =  [1] X + [8]            
                         >= [1] X + [15]           
                         =  p(minus(X,Y))          
        
                 p(s(X)) =  [1] X + [11]           
                         >= [1] X + [0]            
                         =  X                      
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
        minus(s(X),s(Y)) -> p(minus(X,Y))
      Weak DP Rules:
        
      Weak TRS Rules:
        div(0(),s(Y)) -> 0()
        minus(X,0()) -> X
        p(s(X)) -> X
      Signature:
        {div/2,minus/2,p/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {div,minus,p}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(minus) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
              p(0) = [12]                 
            p(div) = [1] x1 + [5] x2 + [9]
          p(minus) = [1] x1 + [4]         
              p(p) = [1] x1 + [0]         
              p(s) = [1] x1 + [1]         
        
        Following rules are strictly oriented:
        minus(s(X),s(Y)) = [1] X + [5]  
                         > [1] X + [4]  
                         = p(minus(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
         div(0(),s(Y)) =  [5] Y + [26]           
                       >= [12]                   
                       =  0()                    
        
        div(s(X),s(Y)) =  [1] X + [5] Y + [15]   
                       >= [1] X + [5] Y + [19]   
                       =  s(div(minus(X,Y),s(Y)))
        
          minus(X,0()) =  [1] X + [4]            
                       >= [1] X + [0]            
                       =  X                      
        
               p(s(X)) =  [1] X + [1]            
                       >= [1] X + [0]            
                       =  X                      
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
      Weak DP Rules:
        
      Weak TRS Rules:
        div(0(),s(Y)) -> 0()
        minus(X,0()) -> X
        minus(s(X),s(Y)) -> p(minus(X,Y))
        p(s(X)) -> X
      Signature:
        {div/2,minus/2,p/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {div,minus,p}/{0,s}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 1, 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:
      The following argument positions are considered usable:
        uargs(div) = {1},
        uargs(minus) = {1},
        uargs(p) = {1},
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {}
      TcT has computed the following interpretation:
            p(0) = [0]         
          p(div) = [2] x1 + [0]
        p(minus) = [1] x1 + [0]
            p(p) = [1] x1 + [0]
            p(s) = [1] x1 + [1]
      
      Following rules are strictly oriented:
      div(s(X),s(Y)) = [2] X + [2]            
                     > [2] X + [1]            
                     = s(div(minus(X,Y),s(Y)))
      
      
      Following rules are (at-least) weakly oriented:
         div(0(),s(Y)) =  [0]          
                       >= [0]          
                       =  0()          
      
          minus(X,0()) =  [1] X + [0]  
                       >= [1] X + [0]  
                       =  X            
      
      minus(s(X),s(Y)) =  [1] X + [1]  
                       >= [1] X + [0]  
                       =  p(minus(X,Y))
      
               p(s(X)) =  [1] X + [1]  
                       >= [1] X + [0]  
                       =  X            
      
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        div(0(),s(Y)) -> 0()
        div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
        minus(X,0()) -> X
        minus(s(X),s(Y)) -> p(minus(X,Y))
        p(s(X)) -> X
      Signature:
        {div/2,minus/2,p/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {div,minus,p}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).