WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),y) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [0]         
            p(minus) = [1] x1 + [3]
             p(quot) = [1] x1 + [0]
                p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          minus(x,0()) = [1] x + [3]
                       > [1] x + [0]
                       = x          
          
          
          Following rules are (at-least) weakly oriented:
          minus(s(x),s(y)) =  [1] x + [3]             
                           >= [1] x + [3]             
                           =  minus(x,y)              
          
               quot(0(),y) =  [0]                     
                           >= [0]                     
                           =  0()                     
          
           quot(s(x),s(y)) =  [1] x + [0]             
                           >= [1] x + [3]             
                           =  s(quot(minus(x,y),s(y)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),y) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Weak TRS:
            minus(x,0()) -> x
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [1]                  
            p(minus) = [1] x1 + [4]         
             p(quot) = [1] x1 + [1] x2 + [3]
                p(s) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
          minus(s(x),s(y)) = [1] x + [5]
                           > [1] x + [4]
                           = minus(x,y) 
          
               quot(0(),y) = [1] y + [4]
                           > [1]        
                           = 0()        
          
          
          Following rules are (at-least) weakly oriented:
             minus(x,0()) =  [1] x + [4]             
                          >= [1] x + [0]             
                          =  x                       
          
          quot(s(x),s(y)) =  [1] x + [1] y + [5]     
                          >= [1] x + [1] y + [9]     
                          =  s(quot(minus(x,y),s(y)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),y) -> 0()
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(quot) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {minus,quot}
        TcT has computed the following interpretation:
              p(0) = [0]                  
          p(minus) = [1] x1 + [0]         
           p(quot) = [2] x1 + [4] x2 + [5]
              p(s) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
        quot(s(x),s(y)) = [2] x + [4] y + [17]    
                        > [2] x + [4] y + [15]    
                        = s(quot(minus(x,y),s(y)))
        
        
        Following rules are (at-least) weakly oriented:
            minus(x,0()) =  [1] x + [0]
                         >= [1] x + [0]
                         =  x          
        
        minus(s(x),s(y)) =  [1] x + [2]
                         >= [1] x + [0]
                         =  minus(x,y) 
        
             quot(0(),y) =  [4] y + [5]
                         >= [0]        
                         =  0()        
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),y) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))