* 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: 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: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
             p(ackin) = [1] x2 + [0]
            p(ackout) = [1] x1 + [1]
                 p(s) = [1] x1 + [0]
               p(u21) = [1] x1 + [0]
               p(u22) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          u21(ackout(X),Y) = [1] X + [1]    
                           > [1] X + [0]    
                           = u22(ackin(Y,X))
          
          
          Following rules are (at-least) weakly oriented:
          ackin(s(X),s(Y)) =  [1] Y + [0]         
                           >= [1] Y + [0]         
                           =  u21(ackin(s(X),Y),X)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
             p(ackin) = [1] x2 + [0]
            p(ackout) = [1] x1 + [0]
                 p(s) = [1] x1 + [1]
               p(u21) = [1] x1 + [0]
               p(u22) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          ackin(s(X),s(Y)) = [1] Y + [1]         
                           > [1] Y + [0]         
                           = u21(ackin(s(X),Y),X)
          
          
          Following rules are (at-least) weakly oriented:
          u21(ackout(X),Y) =  [1] X + [0]    
                           >= [1] X + [0]    
                           =  u22(ackin(Y,X))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** 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))