WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            equal(Capture(),Capture()) -> True()
            equal(Capture(),Swap()) -> False()
            equal(Swap(),Capture()) -> False()
            equal(Swap(),Swap()) -> True()
            game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
            game(p1,p2,Nil()) -> @(p1,p2)
            game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
            goal(p1,p2,moves) -> game(p1,p2,moves)
        - Signature:
            {@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
            ,Nil,Swap,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            equal(Capture(),Capture()) -> True()
            equal(Capture(),Swap()) -> False()
            equal(Swap(),Capture()) -> False()
            equal(Swap(),Swap()) -> True()
            game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
            game(p1,p2,Nil()) -> @(p1,p2)
            game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
            goal(p1,p2,moves) -> game(p1,p2,moves)
        - Signature:
            {@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
            ,Nil,Swap,True}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          @(y,z){y -> Cons(x,y)} =
            @(Cons(x,y),z) ->^+ Cons(x,@(y,z))
              = C[@(y,z) = @(y,z){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            equal(Capture(),Capture()) -> True()
            equal(Capture(),Swap()) -> False()
            equal(Swap(),Capture()) -> False()
            equal(Swap(),Swap()) -> True()
            game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
            game(p1,p2,Nil()) -> @(p1,p2)
            game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
            goal(p1,p2,moves) -> game(p1,p2,moves)
        - Signature:
            {@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
            ,Nil,Swap,True}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {@,equal,game,goal}
        TcT has computed the following interpretation:
                p(@) = x1 + 2*x2             
          p(Capture) = 4                     
             p(Cons) = x1 + x2               
            p(False) = 15                    
              p(Nil) = 4                     
             p(Swap) = 6                     
             p(True) = 0                     
            p(equal) = 7 + x1 + 3*x2         
             p(game) = 1 + 2*x1 + 2*x2 + 4*x3
             p(goal) = 8 + 8*x1 + 2*x2 + 9*x3
        
        Following rules are strictly oriented:
                                     @(Nil(),ys) = 4 + 2*ys                       
                                                 > ys                             
                                                 = ys                             
        
                      equal(Capture(),Capture()) = 23                             
                                                 > 0                              
                                                 = True()                         
        
                         equal(Capture(),Swap()) = 29                             
                                                 > 15                             
                                                 = False()                        
        
                         equal(Swap(),Capture()) = 25                             
                                                 > 15                             
                                                 = False()                        
        
                            equal(Swap(),Swap()) = 31                             
                                                 > 0                              
                                                 = True()                         
        
                     game(p1,p2,Cons(Swap(),xs)) = 25 + 2*p1 + 2*p2 + 4*xs        
                                                 > 1 + 2*p1 + 2*p2 + 4*xs         
                                                 = game(p2,p1,xs)                 
        
                               game(p1,p2,Nil()) = 17 + 2*p1 + 2*p2               
                                                 > p1 + 2*p2                      
                                                 = @(p1,p2)                       
        
        game(p1,Cons(x',xs'),Cons(Capture(),xs)) = 17 + 2*p1 + 2*x' + 4*xs + 2*xs'
                                                 > 1 + 2*p1 + 2*x' + 4*xs + 2*xs' 
                                                 = game(Cons(x',p1),xs',xs)       
        
                               goal(p1,p2,moves) = 8 + 9*moves + 8*p1 + 2*p2      
                                                 > 1 + 4*moves + 2*p1 + 2*p2      
                                                 = game(p1,p2,moves)              
        
        
        Following rules are (at-least) weakly oriented:
        @(Cons(x,xs),ys) =  x + xs + 2*ys   
                         >= x + xs + 2*ys   
                         =  Cons(x,@(xs,ys))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
        - Weak TRS:
            @(Nil(),ys) -> ys
            equal(Capture(),Capture()) -> True()
            equal(Capture(),Swap()) -> False()
            equal(Swap(),Capture()) -> False()
            equal(Swap(),Swap()) -> True()
            game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
            game(p1,p2,Nil()) -> @(p1,p2)
            game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
            goal(p1,p2,moves) -> game(p1,p2,moves)
        - Signature:
            {@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
            ,Nil,Swap,True}
    + 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(Cons) = {2}
        
        Following symbols are considered usable:
          {@,equal,game,goal}
        TcT has computed the following interpretation:
                p(@) = 8*x1 + 2*x2
          p(Capture) = 0          
             p(Cons) = 2 + x2     
            p(False) = 0          
              p(Nil) = 0          
             p(Swap) = 0          
             p(True) = 0          
            p(equal) = 0          
             p(game) = 8*x1 + 8*x2
             p(goal) = 8*x1 + 8*x2
        
        Following rules are strictly oriented:
        @(Cons(x,xs),ys) = 16 + 8*xs + 2*ys
                         > 2 + 8*xs + 2*ys 
                         = Cons(x,@(xs,ys))
        
        
        Following rules are (at-least) weakly oriented:
                                     @(Nil(),ys) =  2*ys                    
                                                 >= ys                      
                                                 =  ys                      
        
                      equal(Capture(),Capture()) =  0                       
                                                 >= 0                       
                                                 =  True()                  
        
                         equal(Capture(),Swap()) =  0                       
                                                 >= 0                       
                                                 =  False()                 
        
                         equal(Swap(),Capture()) =  0                       
                                                 >= 0                       
                                                 =  False()                 
        
                            equal(Swap(),Swap()) =  0                       
                                                 >= 0                       
                                                 =  True()                  
        
                     game(p1,p2,Cons(Swap(),xs)) =  8*p1 + 8*p2             
                                                 >= 8*p1 + 8*p2             
                                                 =  game(p2,p1,xs)          
        
                               game(p1,p2,Nil()) =  8*p1 + 8*p2             
                                                 >= 8*p1 + 2*p2             
                                                 =  @(p1,p2)                
        
        game(p1,Cons(x',xs'),Cons(Capture(),xs)) =  16 + 8*p1 + 8*xs'       
                                                 >= 16 + 8*p1 + 8*xs'       
                                                 =  game(Cons(x',p1),xs',xs)
        
                               goal(p1,p2,moves) =  8*p1 + 8*p2             
                                                 >= 8*p1 + 8*p2             
                                                 =  game(p1,p2,moves)       
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            equal(Capture(),Capture()) -> True()
            equal(Capture(),Swap()) -> False()
            equal(Swap(),Capture()) -> False()
            equal(Swap(),Swap()) -> True()
            game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
            game(p1,p2,Nil()) -> @(p1,p2)
            game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
            goal(p1,p2,moves) -> game(p1,p2,moves)
        - Signature:
            {@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
            ,Nil,Swap,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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