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:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

Trs:
  { ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
  , u21(ackout(X), Y) -> u22(ackin(Y, X)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(u21) = {1}, Uargs(u22) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
    [ackin](x1, x2) = [1] x2 + [1]
                                  
            [s](x1) = [1] x1 + [4]
                                  
      [u21](x1, x2) = [1] x1 + [1]
                                  
       [ackout](x1) = [1] x1 + [7]
                                  
          [u22](x1) = [1] x1 + [4]
  
  The order satisfies the following ordering constraints:
  
    [ackin(s(X), s(Y))] = [1] Y + [5]             
                        > [1] Y + [2]             
                        = [u21(ackin(s(X), Y), X)]
                                                  
    [u21(ackout(X), Y)] = [1] X + [8]             
                        > [1] X + [5]             
                        = [u22(ackin(Y, X))]      
                                                  

We return to the main proof.

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:
  runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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