YES(O(1),O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
  { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
  , u21(ackout(X), Y) -> u22(ackin(Y, X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(u21) = {1}, Uargs(u22) = {1}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [ackin](x1, x2) = [1] x2 + [7]
                                
          [s](x1) = [1] x1 + [7]
                                
    [u21](x1, x2) = [1] x1 + [7]
                                
     [ackout](x1) = [1] x1 + [7]
                                
        [u22](x1) = [1] x1 + [3]

The following symbols are considered usable

  {ackin, u21}

The order satisfies the following ordering constraints:

  [ackin(s(X), s(Y))] =  [1] Y + [14]            
                      >= [1] Y + [14]            
                      =  [u21(ackin(s(X), Y), X)]
                                                 
  [u21(ackout(X), Y)] =  [1] X + [14]            
                      >  [1] X + [10]            
                      =  [u22(ackin(Y, X))]      
                                                 

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs: { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) }
Weak Trs: { u21(ackout(X), Y) -> u22(ackin(Y, X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(u21) = {1}, Uargs(u22) = {1}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [ackin](x1, x2) = [1] x2 + [7]
                                
          [s](x1) = [1] x1 + [7]
                                
    [u21](x1, x2) = [1] x1 + [6]
                                
     [ackout](x1) = [1] x1 + [7]
                                
        [u22](x1) = [1] x1 + [3]

The following symbols are considered usable

  {ackin, u21}

The order satisfies the following ordering constraints:

  [ackin(s(X), s(Y))] = [1] Y + [14]            
                      > [1] Y + [13]            
                      = [u21(ackin(s(X), Y), X)]
                                                
  [u21(ackout(X), Y)] = [1] X + [13]            
                      > [1] X + [10]            
                      = [u22(ackin(Y, X))]      
                                                

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak Trs:
  { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
  , u21(ackout(X), Y) -> u22(ackin(Y, X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^1))