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))

We add following weak dependency pairs:

Strict DPs:
  { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X))
  , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) }

and mark the set of starting terms.

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

Strict DPs:
  { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X))
  , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) }
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 constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(u21) = {1}, Uargs(u22) = {1}, Uargs(c_1) = {1},
  Uargs(u21^#) = {1}, Uargs(c_2) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

    [ackin](x1, x2) = [1] x2 + [0]         
                                           
            [s](x1) = [1] x1 + [1]         
                                           
      [u21](x1, x2) = [1] x1 + [0]         
                                           
       [ackout](x1) = [1] x1 + [1]         
                                           
          [u22](x1) = [1] x1 + [0]         
                                           
  [ackin^#](x1, x2) = [2] x1 + [2] x2 + [2]
                                           
          [c_1](x1) = [1] x1 + [2]         
                                           
    [u21^#](x1, x2) = [2] x1 + [2] x2 + [2]
                                           
          [c_2](x1) = [1] x1 + [1]         

This order satisfies following ordering constraints:

  [ackin(s(X), s(Y))] = [1] Y + [1]             
                      > [1] Y + [0]             
                      = [u21(ackin(s(X), Y), X)]
                                                
  [u21(ackout(X), Y)] = [1] X + [1]             
                      > [1] X + [0]             
                      = [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)).

Weak DPs:
  { ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X))
  , u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) }
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))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ ackin^#(s(X), s(Y)) -> c_1(u21^#(ackin(s(X), Y), X))
, u21^#(ackout(X), Y) -> c_2(ackin^#(Y, X)) }

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

No rule is usable, rules are removed from the input problem.

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

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


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