WORST_CASE(?,O(n^1))
* Step 1: NaturalMI 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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 + [3] x2 + [0]
              p(+) = [1] x2 + [0]         
          p(+Full) = [8] x2 + [0]         
              p(0) = [1]                  
           p(Cons) = [1] x1 + [1] x2 + [2]
            p(Nil) = [0]                  
              p(S) = [0]                  
              p(f) = [5] x1 + [0]         
           p(goal) = [8] x1 + [2]         
            p(map) = [5] x1 + [2]         
        
        Following rules are strictly oriented:
        map(Cons(x,xs)) = [5] x + [5] xs + [12]
                        > [5] x + [5] xs + [4] 
                        = Cons(f(x),map(xs))   
        
             map(Nil()) = [2]                  
                        > [0]                  
                        = Nil()                
        
        
        Following rules are (at-least) weakly oriented:
             *(x,0()) =  [2] x + [3]   
                      >= [1]           
                      =  0()           
        
          *(x,S(0())) =  [2] x + [0]   
                      >= [1] x + [0]   
                      =  x             
        
         *(x,S(S(y))) =  [2] x + [0]   
                      >= [2] x + [0]   
                      =  +(x,*(x,S(y)))
        
             *(0(),y) =  [3] y + [2]   
                      >= [1]           
                      =  0()           
        
         +Full(0(),y) =  [8] y + [0]   
                      >= [1] y + [0]   
                      =  y             
        
        +Full(S(x),y) =  [8] y + [0]   
                      >= [0]           
                      =  +Full(x,S(y)) 
        
                 f(x) =  [5] x + [0]   
                      >= [5] x + [0]   
                      =  *(x,x)        
        
             goal(xs) =  [8] xs + [2]  
                      >= [5] xs + [2]  
                      =  map(xs)       
        
* Step 2: NaturalMI 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)
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
            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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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 + [8]         
              p(+) = [1] x2 + [0]         
          p(+Full) = [1] x1 + [2] x2 + [3]
              p(0) = [1]                  
           p(Cons) = [1] x1 + [1] x2 + [2]
            p(Nil) = [2]                  
              p(S) = [1] x1 + [0]         
              p(f) = [5] x1 + [8]         
           p(goal) = [8] x1 + [1]         
            p(map) = [5] x1 + [0]         
        
        Following rules are strictly oriented:
        +Full(0(),y) = [2] y + [4] 
                     > [1] y + [0] 
                     = y           
        
            goal(xs) = [8] xs + [1]
                     > [5] xs + [0]
                     = map(xs)     
        
        
        Following rules are (at-least) weakly oriented:
               *(x,0()) =  [2] x + [8]          
                        >= [1]                  
                        =  0()                  
        
            *(x,S(0())) =  [2] x + [8]          
                        >= [1] x + [0]          
                        =  x                    
        
           *(x,S(S(y))) =  [2] x + [8]          
                        >= [2] x + [8]          
                        =  +(x,*(x,S(y)))       
        
               *(0(),y) =  [10]                 
                        >= [1]                  
                        =  0()                  
        
          +Full(S(x),y) =  [1] x + [2] y + [3]  
                        >= [1] x + [2] y + [3]  
                        =  +Full(x,S(y))        
        
                   f(x) =  [5] x + [8]          
                        >= [2] x + [8]          
                        =  *(x,x)               
        
        map(Cons(x,xs)) =  [5] x + [5] xs + [10]
                        >= [5] x + [5] xs + [10]
                        =  Cons(f(x),map(xs))   
        
             map(Nil()) =  [10]                 
                        >= [2]                  
                        =  Nil()                
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +Full(S(x),y) -> +Full(x,S(y))
            f(x) -> *(x,x)
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
            +Full(0(),y) -> y
            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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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 + [4] x2 + [0]
              p(+) = [1] x2 + [0]         
          p(+Full) = [2] x2 + [2]         
              p(0) = [4]                  
           p(Cons) = [1] x1 + [1] x2 + [4]
            p(Nil) = [0]                  
              p(S) = [0]                  
              p(f) = [7] x1 + [1]         
           p(goal) = [7] x1 + [2]         
            p(map) = [7] x1 + [2]         
        
        Following rules are strictly oriented:
        f(x) = [7] x + [1]
             > [6] x + [0]
             = *(x,x)     
        
        
        Following rules are (at-least) weakly oriented:
               *(x,0()) =  [2] x + [16]         
                        >= [4]                  
                        =  0()                  
        
            *(x,S(0())) =  [2] x + [0]          
                        >= [1] x + [0]          
                        =  x                    
        
           *(x,S(S(y))) =  [2] x + [0]          
                        >= [2] x + [0]          
                        =  +(x,*(x,S(y)))       
        
               *(0(),y) =  [4] y + [8]          
                        >= [4]                  
                        =  0()                  
        
           +Full(0(),y) =  [2] y + [2]          
                        >= [1] y + [0]          
                        =  y                    
        
          +Full(S(x),y) =  [2] y + [2]          
                        >= [2]                  
                        =  +Full(x,S(y))        
        
               goal(xs) =  [7] xs + [2]         
                        >= [7] xs + [2]         
                        =  map(xs)              
        
        map(Cons(x,xs)) =  [7] x + [7] xs + [30]
                        >= [7] x + [7] xs + [7] 
                        =  Cons(f(x),map(xs))   
        
             map(Nil()) =  [2]                  
                        >= [0]                  
                        =  Nil()                
        
* Step 4: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +Full(S(x),y) -> +Full(x,S(y))
        - Weak TRS:
            *(x,0()) -> 0()
            *(x,S(0())) -> x
            *(x,S(S(y))) -> +(x,*(x,S(y)))
            *(0(),y) -> 0()
            +Full(0(),y) -> 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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(*) = [1] x1 + [0]         
              p(+) = [1] x2 + [0]         
          p(+Full) = [8] x1 + [4] x2 + [0]
              p(0) = [0]                  
           p(Cons) = [1] x1 + [1] x2 + [0]
            p(Nil) = [1]                  
              p(S) = [1] x1 + [2]         
              p(f) = [2] x1 + [0]         
           p(goal) = [3] x1 + [2]         
            p(map) = [2] x1 + [2]         
        
        Following rules are strictly oriented:
        +Full(S(x),y) = [8] x + [4] y + [16]
                      > [8] x + [4] y + [8] 
                      = +Full(x,S(y))       
        
        
        Following rules are (at-least) weakly oriented:
               *(x,0()) =  [1] x + [0]         
                        >= [0]                 
                        =  0()                 
        
            *(x,S(0())) =  [1] x + [0]         
                        >= [1] x + [0]         
                        =  x                   
        
           *(x,S(S(y))) =  [1] x + [0]         
                        >= [1] x + [0]         
                        =  +(x,*(x,S(y)))      
        
               *(0(),y) =  [0]                 
                        >= [0]                 
                        =  0()                 
        
           +Full(0(),y) =  [4] y + [0]         
                        >= [1] y + [0]         
                        =  y                   
        
                   f(x) =  [2] x + [0]         
                        >= [1] x + [0]         
                        =  *(x,x)              
        
               goal(xs) =  [3] xs + [2]        
                        >= [2] xs + [2]        
                        =  map(xs)             
        
        map(Cons(x,xs)) =  [2] x + [2] xs + [2]
                        >= [2] x + [2] xs + [2]
                        =  Cons(f(x),map(xs))  
        
             map(Nil()) =  [4]                 
                        >= [1]                 
                        =  Nil()               
        
* Step 5: 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(?,O(n^1))