WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            +Full(0(),y) -> y
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
            goal(xs) -> map(xs)
            map(Cons(x,xs)) -> Cons(f(x),map(xs))
            map(Nil()) -> Nil()
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
        - Signature:
            {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +Full(0(),y) -> y
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
            goal(xs) -> map(xs)
            map(Cons(x,xs)) -> Cons(f(x),map(xs))
            map(Nil()) -> Nil()
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
        - Signature:
            {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +Full(x,y){x -> S(x)} =
            +Full(S(x),y) ->^+ +Full(x,S(y))
              = C[+Full(x,S(y)) = +Full(x,y){y -> S(y)}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +Full(0(),y) -> y
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
            goal(xs) -> map(xs)
            map(Cons(x,xs)) -> Cons(f(x),map(xs))
            map(Nil()) -> Nil()
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
        - Signature:
            {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,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(+) = {2},
          uargs(Cons) = {1,2}
        
        Following symbols are considered usable:
          {*,+Full,f,goal,map}
        TcT has computed the following interpretation:
              p(*) = 2*x1 + 5*x2
              p(+) = x2         
          p(+Full) = 4 + 4*x2   
              p(0) = 4          
           p(Cons) = 1 + x1 + x2
            p(Nil) = 1          
              p(S) = x1         
              p(f) = 7*x1       
           p(goal) = 10*x1      
            p(map) = 10*x1      
        
        Following rules are strictly oriented:
           +Full(0(),y) = 4 + 4*y           
                        > y                 
                        = y                 
        
        map(Cons(x,xs)) = 10 + 10*x + 10*xs 
                        > 1 + 7*x + 10*xs   
                        = Cons(f(x),map(xs))
        
             map(Nil()) = 10                
                        > 1                 
                        = Nil()             
        
        
        Following rules are (at-least) weakly oriented:
             *(x,0()) =  20 + 2*x      
                      >= 4             
                      =  0()           
        
          *(x,S(0())) =  20 + 2*x      
                      >= x             
                      =  x             
        
         *(x,S(S(y))) =  2*x + 5*y     
                      >= 2*x + 5*y     
                      =  +(x,*(x,S(y)))
        
             *(0(),y) =  8 + 5*y       
                      >= 4             
                      =  0()           
        
        +Full(S(x),y) =  4 + 4*y       
                      >= 4 + 4*y       
                      =  +Full(x,S(y)) 
        
                 f(x) =  7*x           
                      >= 7*x           
                      =  *(x,x)        
        
             goal(xs) =  10*xs         
                      >= 10*xs         
                      =  map(xs)       
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
            goal(xs) -> map(xs)
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
            +Full(0(),y) -> y
            map(Cons(x,xs)) -> Cons(f(x),map(xs))
            map(Nil()) -> Nil()
        - Signature:
            {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,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(+) = {2},
          uargs(Cons) = {1,2}
        
        Following symbols are considered usable:
          {*,+Full,f,goal,map}
        TcT has computed the following interpretation:
              p(*) = 4 + 5*x1 + x2
              p(+) = x2           
          p(+Full) = 7 + 4*x1 + x2
              p(0) = 4            
           p(Cons) = 2 + x1 + x2  
            p(Nil) = 2            
              p(S) = 4 + x1       
              p(f) = 12 + 6*x1    
           p(goal) = 6 + 10*x1    
            p(map) = 6 + 9*x1     
        
        Following rules are strictly oriented:
        +Full(S(x),y) = 23 + 4*x + y 
                      > 11 + 4*x + y 
                      = +Full(x,S(y))
        
                 f(x) = 12 + 6*x     
                      > 4 + 6*x      
                      = *(x,x)       
        
        
        Following rules are (at-least) weakly oriented:
               *(x,0()) =  8 + 5*x           
                        >= 4                 
                        =  0()               
        
            *(x,S(0())) =  12 + 5*x          
                        >= x                 
                        =  x                 
        
           *(x,S(S(y))) =  12 + 5*x + y      
                        >= 8 + 5*x + y       
                        =  +(x,*(x,S(y)))    
        
               *(0(),y) =  24 + y            
                        >= 4                 
                        =  0()               
        
           +Full(0(),y) =  23 + y            
                        >= y                 
                        =  y                 
        
               goal(xs) =  6 + 10*xs         
                        >= 6 + 9*xs          
                        =  map(xs)           
        
        map(Cons(x,xs)) =  24 + 9*x + 9*xs   
                        >= 20 + 6*x + 9*xs   
                        =  Cons(f(x),map(xs))
        
             map(Nil()) =  24                
                        >= 2                 
                        =  Nil()             
        
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> map(xs)
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
            +Full(0(),y) -> y
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
            map(Cons(x,xs)) -> Cons(f(x),map(xs))
            map(Nil()) -> Nil()
        - Signature:
            {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,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(+) = {2},
          uargs(Cons) = {1,2}
        
        Following symbols are considered usable:
          {*,+Full,f,goal,map}
        TcT has computed the following interpretation:
              p(*) = 2*x1       
              p(+) = x2         
          p(+Full) = 2*x2       
              p(0) = 0          
           p(Cons) = 2 + x1 + x2
            p(Nil) = 1          
              p(S) = x1         
              p(f) = 1 + 2*x1   
           p(goal) = 9 + 14*x1  
            p(map) = 8*x1       
        
        Following rules are strictly oriented:
        goal(xs) = 9 + 14*xs
                 > 8*xs     
                 = map(xs)  
        
        
        Following rules are (at-least) weakly oriented:
               *(x,0()) =  2*x               
                        >= 0                 
                        =  0()               
        
            *(x,S(0())) =  2*x               
                        >= x                 
                        =  x                 
        
           *(x,S(S(y))) =  2*x               
                        >= 2*x               
                        =  +(x,*(x,S(y)))    
        
               *(0(),y) =  0                 
                        >= 0                 
                        =  0()               
        
           +Full(0(),y) =  2*y               
                        >= y                 
                        =  y                 
        
          +Full(S(x),y) =  2*y               
                        >= 2*y               
                        =  +Full(x,S(y))     
        
                   f(x) =  1 + 2*x           
                        >= 2*x               
                        =  *(x,x)            
        
        map(Cons(x,xs)) =  16 + 8*x + 8*xs   
                        >= 3 + 2*x + 8*xs    
                        =  Cons(f(x),map(xs))
        
             map(Nil()) =  8                 
                        >= 1                 
                        =  Nil()             
        
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
            +Full(0(),y) -> y
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
            goal(xs) -> map(xs)
            map(Cons(x,xs)) -> Cons(f(x),map(xs))
            map(Nil()) -> Nil()
        - Signature:
            {*/2,+Full/2,f/1,goal/1,map/1} / {+/2,0/0,Cons/2,Nil/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+Full,f,goal,map} and constructors {+,0,Cons,Nil,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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