WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
            innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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:
            innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          minus(x,y){x -> s(x),y -> s(y)} =
            minus(s(x),s(y)) ->^+ p(minus(x,y))
              = C[minus(x,y) = minus(x,y){}]

** Step 1.b:1: NaturalPI 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:
            innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {div,minus,p}
        TcT has computed the following interpretation:
              p(0) = 2               
            p(div) = 11 + 4*x1 + 8*x2
          p(minus) = x1              
              p(p) = 1 + x1          
              p(s) = 1 + x1          
        
        Following rules are strictly oriented:
         div(0(),s(Y)) = 27 + 8*Y               
                       > 2                      
                       = 0()                    
        
        div(s(X),s(Y)) = 23 + 4*X + 8*Y         
                       > 20 + 4*X + 8*Y         
                       = s(div(minus(X,Y),s(Y)))
        
               p(s(X)) = 2 + X                  
                       > X                      
                       = X                      
        
        
        Following rules are (at-least) weakly oriented:
            minus(X,0()) =  X            
                         >= X            
                         =  X            
        
        minus(s(X),s(Y)) =  1 + X        
                         >= 1 + X        
                         =  p(minus(X,Y))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {div,minus,p}
        TcT has computed the following interpretation:
              p(0) = 0       
            p(div) = 1 + 4*x1
          p(minus) = x1      
              p(p) = x1      
              p(s) = 1 + x1  
        
        Following rules are strictly oriented:
        minus(s(X),s(Y)) = 1 + X        
                         > X            
                         = p(minus(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
         div(0(),s(Y)) =  1                      
                       >= 0                      
                       =  0()                    
        
        div(s(X),s(Y)) =  5 + 4*X                
                       >= 2 + 4*X                
                       =  s(div(minus(X,Y),s(Y)))
        
          minus(X,0()) =  X                      
                       >= X                      
                       =  X                      
        
               p(s(X)) =  1 + X                  
                       >= X                      
                       =  X                      
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(X,0()) -> X
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            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:
            innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(div) = {1},
          uargs(p) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {div,minus,p}
        TcT has computed the following interpretation:
              p(0) = 0       
            p(div) = 5 + 2*x1
          p(minus) = 3 + x1  
              p(p) = 12 + x1 
              p(s) = 13 + x1 
        
        Following rules are strictly oriented:
        minus(X,0()) = 3 + X
                     > X    
                     = X    
        
        
        Following rules are (at-least) weakly oriented:
           div(0(),s(Y)) =  5                      
                         >= 0                      
                         =  0()                    
        
          div(s(X),s(Y)) =  31 + 2*X               
                         >= 24 + 2*X               
                         =  s(div(minus(X,Y),s(Y)))
        
        minus(s(X),s(Y)) =  16 + X                 
                         >= 15 + X                 
                         =  p(minus(X,Y))          
        
                 p(s(X)) =  25 + X                 
                         >= X                      
                         =  X                      
        
** Step 1.b:4: 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:
            innermost 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(Omega(n^1),O(n^1))