YES(O(1),O(1))

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

Strict Trs:
  { gcd(x, 0()) -> x
  , gcd(0(), y) -> y
  , gcd(s(x), s(y)) ->
    if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { gcd^#(x, 0()) -> c_1()
  , gcd^#(0(), y) -> c_2()
  , gcd^#(s(x), s(y)) ->
    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }

and mark the set of starting terms.

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

Strict DPs:
  { gcd^#(x, 0()) -> c_1()
  , gcd^#(0(), y) -> c_2()
  , gcd^#(s(x), s(y)) ->
    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
Strict Trs:
  { gcd(x, 0()) -> x
  , gcd(0(), y) -> y
  , gcd(s(x), s(y)) ->
    if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),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),O(1)).

Strict DPs:
  { gcd^#(x, 0()) -> c_1()
  , gcd^#(0(), y) -> c_2()
  , gcd^#(s(x), s(y)) ->
    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

The following argument positions are usable:
  none

TcT has computed following constructor-restricted matrix
interpretation.

              [0] = [0]         
                                
          [s](x1) = [2]         
                                
      [-](x1, x2) = [0]         
                                
  [gcd^#](x1, x2) = [2] x2 + [0]
                                
            [c_1] = [1]         
                                
            [c_2] = [1]         
                                
    [c_3](x1, x2) = [1]         

This order satisfies following ordering constraints:


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)).

Strict DPs:
  { gcd^#(x, 0()) -> c_1()
  , gcd^#(0(), y) -> c_2() }
Weak DPs:
  { gcd^#(s(x), s(y)) ->
    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {1,2} by applications of
Pre({1,2}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: gcd^#(x, 0()) -> c_1()
    , 2: gcd^#(0(), y) -> c_2()
    , 3: gcd^#(s(x), s(y)) ->
         c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }

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

Weak DPs:
  { gcd^#(x, 0()) -> c_1()
  , gcd^#(0(), y) -> c_2()
  , gcd^#(s(x), s(y)) ->
    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ gcd^#(x, 0()) -> c_1()
, gcd^#(0(), y) -> c_2()
, gcd^#(s(x), s(y)) ->
  c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }

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

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

Empty rules are trivially bounded

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