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: Sum.  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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + 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:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "*") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "+") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "+Full") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "Cons") :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "Nil") :: [] -(0)-> "A"(1)
          F (TrsFun "Nil") :: [] -(0)-> "A"(0)
          F (TrsFun "S") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "f") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "goal") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "map") :: ["A"(1)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          +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()
          main(x1) -> map(x1)
        2. Weak:
          
        

*** Step 1.a:1.b: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: Ara.  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:
        Ara {minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "*") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "+") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "+Full") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "Nil") :: [] -(0)-> "A"(0)
          F (TrsFun "S") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "S") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "f") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "goal") :: ["A"(0)] -(2)-> "A"(0)
          F (TrsFun "map") :: ["A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          +Full(0(),y) -> y
          +Full(S(x),y) -> +Full(x,S(y))
          goal(xs) -> map(xs)
          map(Nil()) -> Nil()
        2. Weak:
          f(x) -> *(x,x)
          map(Cons(x,xs)) -> Cons(f(x),map(xs))
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x) -> *(x,x)
            map(Cons(x,xs)) -> Cons(f(x),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))
            goal(xs) -> 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(*) = [6] x1 + [2] x2 + [2]
              p(+) = [1] x2 + [0]         
          p(+Full) = [1] x2 + [0]         
              p(0) = [4]                  
           p(Cons) = [1] x1 + [1] x2 + [2]
            p(Nil) = [0]                  
              p(S) = [0]                  
              p(f) = [8] x1 + [4]         
           p(goal) = [8] x1 + [8]         
            p(map) = [8] x1 + [2]         
        
        Following rules are strictly oriented:
                   f(x) = [8] x + [4]          
                        > [8] x + [2]          
                        = *(x,x)               
        
        map(Cons(x,xs)) = [8] x + [8] xs + [18]
                        > [8] x + [8] xs + [8] 
                        = Cons(f(x),map(xs))   
        
        
        Following rules are (at-least) weakly oriented:
             *(x,0()) =  [6] x + [10]  
                      >= [4]           
                      =  0()           
        
          *(x,S(0())) =  [6] x + [2]   
                      >= [1] x + [0]   
                      =  x             
        
         *(x,S(S(y))) =  [6] x + [2]   
                      >= [6] x + [2]   
                      =  +(x,*(x,S(y)))
        
             *(0(),y) =  [2] y + [26]  
                      >= [4]           
                      =  0()           
        
         +Full(0(),y) =  [1] y + [0]   
                      >= [1] y + [0]   
                      =  y             
        
        +Full(S(x),y) =  [1] y + [0]   
                      >= [0]           
                      =  +Full(x,S(y)) 
        
             goal(xs) =  [8] xs + [8]  
                      >= [8] xs + [2]  
                      =  map(xs)       
        
           map(Nil()) =  [2]           
                      >= [0]           
                      =  Nil()         
        
** Step 1.b:3: 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))