WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            *'(z0,+(z1,z2)) -> c(*'(z0,z1))
            *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
        - Weak TRS:
            *(z0,+(z1,z2)) -> +(*(z0,z1),*(z0,z2))
        - Signature:
            {*/2,*'/2} / {+/2,c/1,c1/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,*'} and constructors {+,c,c1}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            *'(z0,+(z1,z2)) -> c(*'(z0,z1))
            *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
        - Weak TRS:
            *(z0,+(z1,z2)) -> +(*(z0,z1),*(z0,z2))
        - Signature:
            {*/2,*'/2} / {+/2,c/1,c1/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,*'} and constructors {+,c,c1}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            *'(z0,+(z1,z2)) -> c(*'(z0,z1))
            *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
        - Weak TRS:
            *(z0,+(z1,z2)) -> +(*(z0,z1),*(z0,z2))
        - Signature:
            {*/2,*'/2} / {+/2,c/1,c1/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,*'} and constructors {+,c,c1}
    + 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, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "*'") :: ["A"(1, 1, 1) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "+") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "c") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "c1") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(1, 1, 1) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          *'(z0,+(z1,z2)) -> c(*'(z0,z1))
          *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
          main(x1,x2) -> *'(x1,x2)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            *'(z0,+(z1,z2)) -> c(*'(z0,z1))
            *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
        - Weak TRS:
            *(z0,+(z1,z2)) -> +(*(z0,z1),*(z0,z2))
        - Signature:
            {*/2,*'/2} / {+/2,c/1,c1/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,*'} and constructors {+,c,c1}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          *'(x,y){y -> +(y,z)} =
            *'(x,+(y,z)) ->^+ c(*'(x,y))
              = C[*'(x,y) = *'(x,y){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            *'(z0,+(z1,z2)) -> c(*'(z0,z1))
            *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
        - Weak TRS:
            *(z0,+(z1,z2)) -> +(*(z0,z1),*(z0,z2))
        - Signature:
            {*/2,*'/2} / {+/2,c/1,c1/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,*'} and constructors {+,c,c1}
    + 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(c) = {1},
          uargs(c1) = {1}
        
        Following symbols are considered usable:
          {*,*'}
        TcT has computed the following interpretation:
           p(*) = [8] x2 + [0]         
          p(*') = [8] x1 + [1] x2 + [0]
           p(+) = [1] x1 + [1] x2 + [1]
           p(c) = [1] x1 + [0]         
          p(c1) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        *'(z0,+(z1,z2)) = [8] z0 + [1] z1 + [1] z2 + [1]
                        > [8] z0 + [1] z1 + [0]         
                        = c(*'(z0,z1))                  
        
        *'(z0,+(z1,z2)) = [8] z0 + [1] z1 + [1] z2 + [1]
                        > [8] z0 + [1] z2 + [0]         
                        = c1(*'(z0,z2))                 
        
        
        Following rules are (at-least) weakly oriented:
        *(z0,+(z1,z2)) =  [8] z1 + [8] z2 + [8]
                       >= [8] z1 + [8] z2 + [1]
                       =  +(*(z0,z1),*(z0,z2)) 
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(z0,+(z1,z2)) -> +(*(z0,z1),*(z0,z2))
            *'(z0,+(z1,z2)) -> c(*'(z0,z1))
            *'(z0,+(z1,z2)) -> c1(*'(z0,z2))
        - Signature:
            {*/2,*'/2} / {+/2,c/1,c1/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,*'} and constructors {+,c,c1}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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