* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          le(x,y){x -> s(x),y -> s(y)} =
            le(s(x),s(y)) ->^+ le(x,y)
              = C[le(x,y) = le(x,y){}]

** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true}
    + 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(if_mod) = {1},
            uargs(mod) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                           
             p(false) = [0]                           
            p(if_mod) = [1] x1 + [1] x2 + [1] x3 + [0]
                p(le) = [11]                          
             p(minus) = [1] x1 + [1]                  
               p(mod) = [1] x1 + [1] x2 + [1]         
                 p(s) = [1] x1 + [8]                  
              p(true) = [0]                           
          
          Following rules are strictly oriented:
          if_mod(false(),s(x),s(y)) = [1] x + [1] y + [16]
                                    > [1] x + [8]         
                                    = s(x)                
          
           if_mod(true(),s(x),s(y)) = [1] x + [1] y + [16]
                                    > [1] x + [1] y + [10]
                                    = mod(minus(x,y),s(y))
          
                          le(0(),y) = [11]                
                                    > [0]                 
                                    = true()              
          
                       le(s(x),0()) = [11]                
                                    > [0]                 
                                    = false()             
          
                       minus(x,0()) = [1] x + [1]         
                                    > [1] x + [0]         
                                    = x                   
          
                   minus(s(x),s(y)) = [1] x + [9]         
                                    > [1] x + [1]         
                                    = minus(x,y)          
          
                         mod(0(),y) = [1] y + [1]         
                                    > [0]                 
                                    = 0()                 
          
                      mod(s(x),0()) = [1] x + [9]         
                                    > [0]                 
                                    = 0()                 
          
          
          Following rules are (at-least) weakly oriented:
           le(s(x),s(y)) =  [11]                     
                         >= [11]                     
                         =  le(x,y)                  
          
          mod(s(x),s(y)) =  [1] x + [1] y + [17]     
                         >= [1] x + [1] y + [27]     
                         =  if_mod(le(y,x),s(x),s(y))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            le(s(x),s(y)) -> le(x,y)
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
        - Weak TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true}
    + 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(if_mod) = {1},
            uargs(mod) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [4]                  
             p(false) = [0]                  
            p(if_mod) = [1] x1 + [1] x2 + [0]
                p(le) = [0]                  
             p(minus) = [1] x1 + [1]         
               p(mod) = [1] x1 + [2]         
                 p(s) = [1] x1 + [4]         
              p(true) = [0]                  
          
          Following rules are strictly oriented:
          mod(s(x),s(y)) = [1] x + [6]              
                         > [1] x + [4]              
                         = if_mod(le(y,x),s(x),s(y))
          
          
          Following rules are (at-least) weakly oriented:
          if_mod(false(),s(x),s(y)) =  [1] x + [4]         
                                    >= [1] x + [4]         
                                    =  s(x)                
          
           if_mod(true(),s(x),s(y)) =  [1] x + [4]         
                                    >= [1] x + [3]         
                                    =  mod(minus(x,y),s(y))
          
                          le(0(),y) =  [0]                 
                                    >= [0]                 
                                    =  true()              
          
                       le(s(x),0()) =  [0]                 
                                    >= [0]                 
                                    =  false()             
          
                      le(s(x),s(y)) =  [0]                 
                                    >= [0]                 
                                    =  le(x,y)             
          
                       minus(x,0()) =  [1] x + [1]         
                                    >= [1] x + [0]         
                                    =  x                   
          
                   minus(s(x),s(y)) =  [1] x + [5]         
                                    >= [1] x + [1]         
                                    =  minus(x,y)          
          
                         mod(0(),y) =  [6]                 
                                    >= [4]                 
                                    =  0()                 
          
                      mod(s(x),0()) =  [1] x + [6]         
                                    >= [4]                 
                                    =  0()                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            le(s(x),s(y)) -> le(x,y)
        - Weak TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true}
    + 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(if_mod) = {1},
          uargs(mod) = {1}
        
        Following symbols are considered usable:
          {if_mod,le,minus,mod}
        TcT has computed the following interpretation:
               p(0) = 0                               
           p(false) = 0                               
          p(if_mod) = 2 + 2*x1 + 2*x2 + 6*x2*x3 + x3^2
              p(le) = 1 + 2*x1                        
           p(minus) = x1                              
             p(mod) = 2*x1 + 6*x1*x2 + 6*x2 + x2^2    
               p(s) = 1 + x1                          
            p(true) = 0                               
        
        Following rules are strictly oriented:
        le(s(x),s(y)) = 3 + 2*x
                      > 1 + 2*x
                      = le(x,y)
        
        
        Following rules are (at-least) weakly oriented:
        if_mod(false(),s(x),s(y)) =  11 + 8*x + 6*x*y + 8*y + y^2 
                                  >= 1 + x                        
                                  =  s(x)                         
        
         if_mod(true(),s(x),s(y)) =  11 + 8*x + 6*x*y + 8*y + y^2 
                                  >= 7 + 8*x + 6*x*y + 8*y + y^2  
                                  =  mod(minus(x,y),s(y))         
        
                        le(0(),y) =  1                            
                                  >= 0                            
                                  =  true()                       
        
                     le(s(x),0()) =  3 + 2*x                      
                                  >= 0                            
                                  =  false()                      
        
                     minus(x,0()) =  x                            
                                  >= x                            
                                  =  x                            
        
                 minus(s(x),s(y)) =  1 + x                        
                                  >= x                            
                                  =  minus(x,y)                   
        
                       mod(0(),y) =  6*y + y^2                    
                                  >= 0                            
                                  =  0()                          
        
                    mod(s(x),0()) =  2 + 2*x                      
                                  >= 0                            
                                  =  0()                          
        
                   mod(s(x),s(y)) =  15 + 8*x + 6*x*y + 14*y + y^2
                                  >= 13 + 8*x + 6*x*y + 12*y + y^2
                                  =  if_mod(le(y,x),s(x),s(y))    
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} and constructors {0,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))