WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(0()) -> c()
            F(s(z0)) -> c1(G(f(z0)),F(z0))
            F(s(z0)) -> c2(F(z0))
            F(s(z0)) -> c3(F(z0))
            G(z0) -> c4()
        - Weak TRS:
            f(0()) -> 1()
            f(s(z0)) -> +(f(z0),s(f(z0)))
            f(s(z0)) -> g(f(z0))
            g(z0) -> +(z0,s(z0))
        - Signature:
            {F/1,G/1,f/1,g/1} / {+/2,0/0,1/0,c/0,c1/2,c2/1,c3/1,c4/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,G,f,g} and constructors {+,0,1,c,c1,c2,c3,c4,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F(0()) -> c()
            F(s(z0)) -> c1(G(f(z0)),F(z0))
            F(s(z0)) -> c2(F(z0))
            F(s(z0)) -> c3(F(z0))
            G(z0) -> c4()
        - Weak TRS:
            f(0()) -> 1()
            f(s(z0)) -> +(f(z0),s(f(z0)))
            f(s(z0)) -> g(f(z0))
            g(z0) -> +(z0,s(z0))
        - Signature:
            {F/1,G/1,f/1,g/1} / {+/2,0/0,1/0,c/0,c1/2,c2/1,c3/1,c4/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,G,f,g} and constructors {+,0,1,c,c1,c2,c3,c4,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F(0()) -> c()
            F(s(z0)) -> c1(G(f(z0)),F(z0))
            F(s(z0)) -> c2(F(z0))
            F(s(z0)) -> c3(F(z0))
            G(z0) -> c4()
        - Weak TRS:
            f(0()) -> 1()
            f(s(z0)) -> +(f(z0),s(f(z0)))
            f(s(z0)) -> g(f(z0))
            g(z0) -> +(z0,s(z0))
        - Signature:
            {F/1,G/1,f/1,g/1} / {+/2,0/0,1/0,c/0,c1/2,c2/1,c3/1,c4/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,G,f,g} and constructors {+,0,1,c,c1,c2,c3,c4,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          F(x){x -> s(x)} =
            F(s(x)) ->^+ c1(G(f(x)),F(x))
              = C[F(x) = F(x){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(0()) -> c()
            F(s(z0)) -> c1(G(f(z0)),F(z0))
            F(s(z0)) -> c2(F(z0))
            F(s(z0)) -> c3(F(z0))
            G(z0) -> c4()
        - Weak TRS:
            f(0()) -> 1()
            f(s(z0)) -> +(f(z0),s(f(z0)))
            f(s(z0)) -> g(f(z0))
            g(z0) -> +(z0,s(z0))
        - Signature:
            {F/1,G/1,f/1,g/1} / {+/2,0/0,1/0,c/0,c1/2,c2/1,c3/1,c4/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,G,f,g} and constructors {+,0,1,c,c1,c2,c3,c4,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(+) = {1,2},
          uargs(G) = {1},
          uargs(c1) = {1,2},
          uargs(c2) = {1},
          uargs(c3) = {1},
          uargs(g) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {F,G,f,g}
        TcT has computed the following interpretation:
           p(+) = [1] x1 + [1] x2 + [0]
           p(0) = [1]                  
           p(1) = [0]                  
           p(F) = [8] x1 + [2]         
           p(G) = [1] x1 + [0]         
           p(c) = [1]                  
          p(c1) = [1] x1 + [1] x2 + [0]
          p(c2) = [1] x1 + [0]         
          p(c3) = [1] x1 + [0]         
          p(c4) = [0]                  
           p(f) = [0]                  
           p(g) = [8] x1 + [0]         
           p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        F(0()) = [10]
               > [1] 
               = c() 
        
        
        Following rules are (at-least) weakly oriented:
        F(s(z0)) =  [8] z0 + [2]      
                 >= [8] z0 + [2]      
                 =  c1(G(f(z0)),F(z0))
        
        F(s(z0)) =  [8] z0 + [2]      
                 >= [8] z0 + [2]      
                 =  c2(F(z0))         
        
        F(s(z0)) =  [8] z0 + [2]      
                 >= [8] z0 + [2]      
                 =  c3(F(z0))         
        
           G(z0) =  [1] z0 + [0]      
                 >= [0]               
                 =  c4()              
        
          f(0()) =  [0]               
                 >= [0]               
                 =  1()               
        
        f(s(z0)) =  [0]               
                 >= [0]               
                 =  +(f(z0),s(f(z0))) 
        
        f(s(z0)) =  [0]               
                 >= [0]               
                 =  g(f(z0))          
        
           g(z0) =  [8] z0 + [0]      
                 >= [2] z0 + [0]      
                 =  +(z0,s(z0))       
        
** Step 1.b:2: Ara.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(s(z0)) -> c1(G(f(z0)),F(z0))
            F(s(z0)) -> c2(F(z0))
            F(s(z0)) -> c3(F(z0))
            G(z0) -> c4()
        - Weak TRS:
            F(0()) -> c()
            f(0()) -> 1()
            f(s(z0)) -> +(f(z0),s(f(z0)))
            f(s(z0)) -> g(f(z0))
            g(z0) -> +(z0,s(z0))
        - Signature:
            {F/1,G/1,f/1,g/1} / {+/2,0/0,1/0,c/0,c1/2,c2/1,c3/1,c4/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,G,f,g} and constructors {+,0,1,c,c1,c2,c3,c4,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 "0") :: [] -(0)-> "A"(2)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "1") :: [] -(0)-> "A"(0)
          F (TrsFun "F") :: ["A"(2)] -(0)-> "A"(0)
          F (TrsFun "G") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "c") :: [] -(0)-> "A"(0)
          F (TrsFun "c1") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "c2") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "c3") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "c4") :: [] -(0)-> "A"(0)
          F (TrsFun "f") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "g") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "s") :: ["A"(2)] -(2)-> "A"(2)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          F(s(z0)) -> c1(G(f(z0)),F(z0))
          F(s(z0)) -> c2(F(z0))
          F(s(z0)) -> c3(F(z0))
          G(z0) -> c4()
        2. Weak:
          
        

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