WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(0()) -> c()
            F(s(0())) -> c1()
            F(s(0())) -> c2(F(0()))
        - Weak TRS:
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(0()) -> c()
            F(s(0())) -> c1()
            F(s(0())) -> c2(F(0()))
        - Weak TRS:
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(0()) -> c()
            F(s(0())) -> c1()
            F(s(0())) -> c2(F(0()))
        - Weak TRS:
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          F(x){x -> +(x,y)} =
            F(+(x,y)) ->^+ c4(F(x))
              = C[F(x) = F(x){}]

** Step 1.b:1: Ara.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(0()) -> c()
            F(s(0())) -> c1()
            F(s(0())) -> c2(F(0()))
        - Weak TRS:
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,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 "0") :: [] -(0)-> "A"(0)
          F (TrsFun "F") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "c") :: [] -(0)-> "A"(0)
          F (TrsFun "c1") :: [] -(0)-> "A"(0)
          F (TrsFun "c2") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "c3") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "c4") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "c5") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "f") :: ["A"(0)] -(0)-> "A"(0)
          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(0()) -> c()
          F(s(0())) -> c1()
        2. Weak:
          F(+(z0,z1)) -> c4(F(z0))
          F(+(z0,z1)) -> c5(F(z1))
          F(+(z0,s(0()))) -> c3(F(z0))
          F(s(0())) -> c2(F(0()))
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(s(0())) -> c2(F(0()))
        - Weak TRS:
            F(0()) -> c()
            F(s(0())) -> c1()
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,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(+) = {2},
          uargs(c2) = {1},
          uargs(c3) = {1},
          uargs(c4) = {1},
          uargs(c5) = {1}
        
        Following symbols are considered usable:
          {F,f}
        TcT has computed the following interpretation:
           p(*) = [1] x1 + [1] x2 + [0]
           p(+) = [1] x1 + [1] x2 + [6]
           p(0) = [0]                  
           p(F) = [4] x1 + [0]         
           p(c) = [0]                  
          p(c1) = [0]                  
          p(c2) = [1] x1 + [0]         
          p(c3) = [1] x1 + [11]        
          p(c4) = [1] x1 + [5]         
          p(c5) = [1] x1 + [6]         
           p(f) = [2] x1 + [8]         
           p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
            F(+(z0,z1)) = [4] z0 + [4] z1 + [24]
                        > [4] z0 + [5]          
                        = c4(F(z0))             
        
            F(+(z0,z1)) = [4] z0 + [4] z1 + [24]
                        > [4] z1 + [6]          
                        = c5(F(z1))             
        
        F(+(z0,s(0()))) = [4] z0 + [24]         
                        > [4] z0 + [11]         
                        = c3(F(z0))             
        
        
        Following rules are (at-least) weakly oriented:
                 F(0()) =  [0]                   
                        >= [0]                   
                        =  c()                   
        
              F(s(0())) =  [0]                   
                        >= [0]                   
                        =  c1()                  
        
              F(s(0())) =  [0]                   
                        >= [0]                   
                        =  c2(F(0()))            
        
            f(+(z0,z1)) =  [2] z0 + [2] z1 + [20]
                        >= [2] z0 + [2] z1 + [16]
                        =  *(f(z0),f(z1))        
        
        f(+(z0,s(0()))) =  [2] z0 + [20]         
                        >= [2] z0 + [14]         
                        =  +(s(s(0())),f(z0))    
        
                 f(0()) =  [8]                   
                        >= [0]                   
                        =  s(0())                
        
              f(s(0())) =  [8]                   
                        >= [8]                   
                        =  *(s(s(0())),f(0()))   
        
              f(s(0())) =  [8]                   
                        >= [0]                   
                        =  s(s(0()))             
        
** Step 1.b:3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            F(s(0())) -> c2(F(0()))
        - Weak TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(0()) -> c()
            F(s(0())) -> c1()
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,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(+) = {2},
          uargs(c2) = {1},
          uargs(c3) = {1},
          uargs(c4) = {1},
          uargs(c5) = {1}
        
        Following symbols are considered usable:
          {F,f}
        TcT has computed the following interpretation:
           p(*) = [1] x1 + [1] x2 + [2]
           p(+) = [1] x1 + [1] x2 + [6]
           p(0) = [0]                  
           p(F) = [2] x1 + [1]         
           p(c) = [0]                  
          p(c1) = [9]                  
          p(c2) = [1] x1 + [6]         
          p(c3) = [1] x1 + [3]         
          p(c4) = [1] x1 + [8]         
          p(c5) = [1] x1 + [4]         
           p(f) = [2] x1 + [4]         
           p(s) = [4]                  
        
        Following rules are strictly oriented:
        F(s(0())) = [9]       
                  > [7]       
                  = c2(F(0()))
        
        
        Following rules are (at-least) weakly oriented:
            F(+(z0,z1)) =  [2] z0 + [2] z1 + [13]
                        >= [2] z0 + [9]          
                        =  c4(F(z0))             
        
            F(+(z0,z1)) =  [2] z0 + [2] z1 + [13]
                        >= [2] z1 + [5]          
                        =  c5(F(z1))             
        
        F(+(z0,s(0()))) =  [2] z0 + [21]         
                        >= [2] z0 + [4]          
                        =  c3(F(z0))             
        
                 F(0()) =  [1]                   
                        >= [0]                   
                        =  c()                   
        
              F(s(0())) =  [9]                   
                        >= [9]                   
                        =  c1()                  
        
            f(+(z0,z1)) =  [2] z0 + [2] z1 + [16]
                        >= [2] z0 + [2] z1 + [10]
                        =  *(f(z0),f(z1))        
        
        f(+(z0,s(0()))) =  [2] z0 + [24]         
                        >= [2] z0 + [14]         
                        =  +(s(s(0())),f(z0))    
        
                 f(0()) =  [4]                   
                        >= [4]                   
                        =  s(0())                
        
              f(s(0())) =  [12]                  
                        >= [10]                  
                        =  *(s(s(0())),f(0()))   
        
              f(s(0())) =  [12]                  
                        >= [4]                   
                        =  s(s(0()))             
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            F(+(z0,z1)) -> c4(F(z0))
            F(+(z0,z1)) -> c5(F(z1))
            F(+(z0,s(0()))) -> c3(F(z0))
            F(0()) -> c()
            F(s(0())) -> c1()
            F(s(0())) -> c2(F(0()))
            f(+(z0,z1)) -> *(f(z0),f(z1))
            f(+(z0,s(0()))) -> +(s(s(0())),f(z0))
            f(0()) -> s(0())
            f(s(0())) -> *(s(s(0())),f(0()))
            f(s(0())) -> s(s(0()))
        - Signature:
            {F/1,f/1} / {*/2,+/2,0/0,c/0,c1/0,c2/1,c3/1,c4/1,c5/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,f} and constructors {*,+,0,c,c1,c2,c3,c4,c5,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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