* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          minus#(x,0()) -> c_1()
          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.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(x,0()) -> c_1()
            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:
            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)))
        - Signature:
            {minus/2,quot/2,minus#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus#,quot#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          minus#(x,0()) -> c_1()
          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)))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(x,0()) -> c_1()
            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:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - 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 = WgOnTrs}
    + Details:
        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(quot#) = {1},
            uargs(c_2) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                   
             p(minus) = [1] x1 + [9]          
              p(quot) = [0]                   
                 p(s) = [1] x1 + [2]          
            p(minus#) = [12] x1 + [2] x2 + [2]
             p(quot#) = [1] x1 + [3] x2 + [1] 
               p(c_1) = [0]                   
               p(c_2) = [1] x1 + [0]          
               p(c_3) = [0]                   
               p(c_4) = [1] x1 + [0]          
          
          Following rules are strictly oriented:
              minus#(x,0()) = [12] x + [2]         
                            > [0]                  
                            = c_1()                
          
          minus#(s(x),s(y)) = [12] x + [2] y + [30]
                            > [12] x + [2] y + [2] 
                            = c_2(minus#(x,y))     
          
            quot#(0(),s(y)) = [3] y + [7]          
                            > [0]                  
                            = c_3()                
          
               minus(x,0()) = [1] x + [9]          
                            > [1] x + [0]          
                            = x                    
          
           minus(s(x),s(y)) = [1] x + [11]         
                            > [1] x + [9]          
                            = minus(x,y)           
          
          
          Following rules are (at-least) weakly oriented:
          quot#(s(x),s(y)) =  [1] x + [3] y + [9]        
                           >= [1] x + [3] y + [16]       
                           =  c_4(quot#(minus(x,y),s(y)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
        - Weak DPs:
            minus#(x,0()) -> c_1()
            minus#(s(x),s(y)) -> c_2(minus#(x,y))
            quot#(0(),s(y)) -> c_3()
        - Weak TRS:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus#,quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
             -->_1 quot#(0(),s(y)) -> c_3():4
             -->_1 quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y))):1
          
          2:W:minus#(x,0()) -> c_1()
             
          
          3:W:minus#(s(x),s(y)) -> c_2(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_2(minus#(x,y)):3
             -->_1 minus#(x,0()) -> c_1():2
          
          4:W:quot#(0(),s(y)) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: minus#(s(x),s(y)) -> c_2(minus#(x,y))
          2: minus#(x,0()) -> c_1()
          4: quot#(0(),s(y)) -> c_3()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
        - Weak TRS:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus#,quot#} and constructors {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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
          
        The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
        - Weak TRS:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - 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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {minus,minus#,quot#}
        TcT has computed the following interpretation:
               p(0) = [0]         
           p(minus) = [1] x1 + [1]
            p(quot) = [1] x2 + [0]
               p(s) = [1] x1 + [2]
          p(minus#) = [2]         
           p(quot#) = [8] x1 + [8]
             p(c_1) = [8]         
             p(c_2) = [1] x1 + [0]
             p(c_3) = [0]         
             p(c_4) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        quot#(s(x),s(y)) = [8] x + [24]               
                         > [8] x + [17]               
                         = c_4(quot#(minus(x,y),s(y)))
        
        
        Following rules are (at-least) weakly oriented:
            minus(x,0()) =  [1] x + [1]
                         >= [1] x + [0]
                         =  x          
        
        minus(s(x),s(y)) =  [1] x + [3]
                         >= [1] x + [1]
                         =  minus(x,y) 
        
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
        - Weak TRS:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus#,quot#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
        - Weak TRS:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus#,quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: quot#(s(x),s(y)) -> c_4(quot#(minus(x,y),s(y)))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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_1/0,c_2/1,c_3/0,c_4/1}
        - 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))