* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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:
             runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1},
            uargs(minus) = {1},
            uargs(p) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          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.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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 TRS:
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1},
            uargs(minus) = {1},
            uargs(p) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          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.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            minus(s(X),s(Y)) -> p(minus(X,Y))
        - Weak TRS:
            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:
             runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1},
            uargs(minus) = {1},
            uargs(p) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          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.
* Step 4: MI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
        - Weak TRS:
            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:
             runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(minus) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(0) = [2]          
            p(div) = [4] x_1 + [0]
          p(minus) = [1] x_1 + [0]
              p(p) = [1] x_1 + [1]
              p(s) = [1] x_1 + [4]
        
        Following rules are strictly oriented:
        div(s(X),s(Y)) = [4] X + [16]           
                       > [4] X + [4]            
                       = s(div(minus(X,Y),s(Y)))
        
        
        Following rules are (at-least) weakly oriented:
           div(0(),s(Y)) =  [8]          
                         >= [2]          
                         =  0()          
        
            minus(X,0()) =  [1] X + [0]  
                         >= [1] X + [0]  
                         =  X            
        
        minus(s(X),s(Y)) =  [1] X + [4]  
                         >= [1] X + [1]  
                         =  p(minus(X,Y))
        
                 p(s(X)) =  [1] X + [5]  
                         >= [1] X + [0]  
                         =  X            
        
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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:
             runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))