*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {minus/2,quot/2} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {minus,quot}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Strict TRS Rules:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Full
        basic terms: {minus#,quot#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Strict TRS Rules:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Full
        basic terms: {minus#,quot#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {3}
      by application of
        Pre({3}) = {1,4}.
      Here rules are labelled as follows:
        1: minus#(x,0()) -> c_1(x)      
        2: minus#(s(x),s(y)) ->         
             c_2(minus#(x,y))           
        3: quot#(0(),s(y)) -> c_3()     
        4: quot#(s(x),s(y)) ->          
             c_4(quot#(minus(x,y),s(y)))
*** 1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Strict TRS Rules:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
      Weak DP Rules:
        quot#(0(),s(y)) -> c_3()
      Weak TRS Rules:
        
      Signature:
        {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Full
        basic terms: {minus#,quot#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, 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(quot#) = {1},
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
               p(0) = [0]                  
           p(minus) = [1] x1 + [0]         
            p(quot) = [0]                  
               p(s) = [1] x1 + [1]         
          p(minus#) = [8] x1 + [5] x2 + [0]
           p(quot#) = [1] x1 + [8] x2 + [7]
             p(c_1) = [8] x1 + [8]         
             p(c_2) = [1] x1 + [0]         
             p(c_3) = [15]                 
             p(c_4) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        minus#(s(x),s(y)) = [8] x + [5] y + [13]       
                          > [8] x + [5] y + [0]        
                          = c_2(minus#(x,y))           
        
         quot#(s(x),s(y)) = [1] x + [8] y + [16]       
                          > [1] x + [8] y + [15]       
                          = c_4(quot#(minus(x,y),s(y)))
        
         minus(s(x),s(y)) = [1] x + [1]                
                          > [1] x + [0]                
                          = minus(x,y)                 
        
        
        Following rules are (at-least) weakly oriented:
          minus#(x,0()) =  [8] x + [0] 
                        >= [8] x + [8] 
                        =  c_1(x)      
        
        quot#(0(),s(y)) =  [8] y + [15]
                        >= [15]        
                        =  c_3()       
        
           minus(x,0()) =  [1] x + [0] 
                        >= [1] x + [0] 
                        =  x           
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        minus#(x,0()) -> c_1(x)
      Strict TRS Rules:
        minus(x,0()) -> x
      Weak DP Rules:
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Weak TRS Rules:
        minus(s(x),s(y)) -> minus(x,y)
      Signature:
        {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Full
        basic terms: {minus#,quot#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, 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(quot#) = {1},
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
               p(0) = [14]                 
           p(minus) = [1] x1 + [0]         
            p(quot) = [8] x1 + [1] x2 + [1]
               p(s) = [1] x1 + [6]         
          p(minus#) = [2] x1 + [11]        
           p(quot#) = [1] x1 + [0]         
             p(c_1) = [2] x1 + [2]         
             p(c_2) = [1] x1 + [1]         
             p(c_3) = [8]                  
             p(c_4) = [1] x1 + [6]         
        
        Following rules are strictly oriented:
        minus#(x,0()) = [2] x + [11]
                      > [2] x + [2] 
                      = c_1(x)      
        
        
        Following rules are (at-least) weakly oriented:
        minus#(s(x),s(y)) =  [2] x + [23]               
                          >= [2] x + [12]               
                          =  c_2(minus#(x,y))           
        
          quot#(0(),s(y)) =  [14]                       
                          >= [8]                        
                          =  c_3()                      
        
         quot#(s(x),s(y)) =  [1] x + [6]                
                          >= [1] x + [6]                
                          =  c_4(quot#(minus(x,y),s(y)))
        
             minus(x,0()) =  [1] x + [0]                
                          >= [1] x + [0]                
                          =  x                          
        
         minus(s(x),s(y)) =  [1] x + [6]                
                          >= [1] x + [0]                
                          =  minus(x,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(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        minus(x,0()) -> x
      Weak DP Rules:
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Weak TRS Rules:
        minus(s(x),s(y)) -> minus(x,y)
      Signature:
        {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Full
        basic terms: {minus#,quot#}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, 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(quot#) = {1},
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
               p(0) = [15]        
           p(minus) = [1] x1 + [1]
            p(quot) = [1] x2 + [0]
               p(s) = [1] x1 + [2]
          p(minus#) = [1]         
           p(quot#) = [1] x1 + [0]
             p(c_1) = [1]         
             p(c_2) = [1] x1 + [0]
             p(c_3) = [2]         
             p(c_4) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        minus(x,0()) = [1] x + [1]
                     > [1] x + [0]
                     = x          
        
        
        Following rules are (at-least) weakly oriented:
            minus#(x,0()) =  [1]                        
                          >= [1]                        
                          =  c_1(x)                     
        
        minus#(s(x),s(y)) =  [1]                        
                          >= [1]                        
                          =  c_2(minus#(x,y))           
        
          quot#(0(),s(y)) =  [15]                       
                          >= [2]                        
                          =  c_3()                      
        
         quot#(s(x),s(y)) =  [1] x + [2]                
                          >= [1] x + [2]                
                          =  c_4(quot#(minus(x,y),s(y)))
        
         minus(s(x),s(y)) =  [1] x + [3]                
                          >= [1] x + [1]                
                          =  minus(x,y)                 
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        minus#(x,0()) -> c_1(x)
        minus#(s(x),s(y)) -> c_2(minus#(x,y))
        quot#(0(),s(y)) -> c_3()
        quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
      Weak TRS Rules:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
      Signature:
        {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
      Obligation:
        Full
        basic terms: {minus#,quot#}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).