WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "ackin") :: ["A"(1, 1, 1) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "ackout") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(0, 0, 0) x "A"(0, 0, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(2, 2, 1)] -(1)-> "A"(1, 1, 1)
          F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(1, 1, 1)] -(1)-> "A"(1, 0, 1)
          F (TrsFun "u21") :: ["A"(0, 0, 0) x "A"(1, 1, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "u22") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
          u21(ackout(X),Y) -> u22(ackin(Y,X))
          main(x1,x2) -> u21(x1,x2)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          ackin(s(x),y){y -> s(y)} =
            ackin(s(x),s(y)) ->^+ u21(ackin(s(x),y),x)
              = C[ackin(s(x),y) = ackin(s(x),y){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + 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(u21) = {1},
          uargs(u22) = {1}
        
        Following symbols are considered usable:
          {ackin,u21}
        TcT has computed the following interpretation:
           p(ackin) = [1]         
          p(ackout) = [15]        
               p(s) = [1]         
             p(u21) = [1] x1 + [0]
             p(u22) = [1] x1 + [6]
        
        Following rules are strictly oriented:
        u21(ackout(X),Y) = [15]           
                         > [7]            
                         = u22(ackin(Y,X))
        
        
        Following rules are (at-least) weakly oriented:
        ackin(s(X),s(Y)) =  [1]                 
                         >= [1]                 
                         =  u21(ackin(s(X),Y),X)
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
        - Weak TRS:
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(u21) = {1},
          uargs(u22) = {1}
        
        Following symbols are considered usable:
          {ackin,u21}
        TcT has computed the following interpretation:
           p(ackin) = x2     
          p(ackout) = 10 + x1
               p(s) = 2 + x1 
             p(u21) = x1     
             p(u22) = 10 + x1
        
        Following rules are strictly oriented:
        ackin(s(X),s(Y)) = 2 + Y               
                         > Y                   
                         = u21(ackin(s(X),Y),X)
        
        
        Following rules are (at-least) weakly oriented:
        u21(ackout(X),Y) =  10 + X         
                         >= 10 + X         
                         =  u22(ackin(Y,X))
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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