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

We add the following weak dependency pairs:

Strict DPs:
  { gcd^#(x, 0()) -> c_1(x)
  , gcd^#(0(), y) -> c_2(y)
  , gcd^#(s(x), s(y)) ->
    c_3(x, y, 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(x)
  , gcd^#(0(), y) -> c_2(y)
  , gcd^#(s(x), s(y)) ->
    c_3(x, y, 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:
  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(x)
  , gcd^#(0(), y) -> c_2(y)
  , gcd^#(s(x), s(y)) ->
    c_3(x, y, gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
Obligation:
  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 the following constructor-restricted matrix
interpretation.

                    [0] = [0]           
                          [0]           
                                        
                [s](x1) = [0]           
                          [2]           
                                        
            [-](x1, x2) = [0]           
                          [0]           
                                        
        [gcd^#](x1, x2) = [0 2] x1 + [0]
                          [0 0]      [0]
                                        
              [c_1](x1) = [0 1] x1 + [1]
                          [0 0]      [0]
                                        
              [c_2](x1) = [1]           
                          [0]           
                                        
  [c_3](x1, x2, x3, x4) = [1]           
                          [0]           

The following symbols are considered usable

  {gcd^#}

The order satisfies the following ordering constraints:

      [gcd^#(x, 0())] = [0 2] x + [0]                                          
                        [0 0]     [0]                                          
                      ? [0 1] x + [1]                                          
                        [0 0]     [0]                                          
                      = [c_1(x)]                                               
                                                                               
      [gcd^#(0(), y)] = [0]                                                    
                        [0]                                                    
                      ? [1]                                                    
                        [0]                                                    
                      = [c_2(y)]                                               
                                                                               
  [gcd^#(s(x), s(y))] = [4]                                                    
                        [0]                                                    
                      > [1]                                                    
                        [0]                                                    
                      = [c_3(x, y, gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y)))]
                                                                               

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

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

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

Sub-proof:
----------
  The following argument positions are usable:
    none
  
  TcT has computed the following constructor-restricted matrix
  interpretation. Note that the diagonal of the component-wise maxima
  of interpretation-entries (of constructors) contains no more than 0
  non-zero entries.
  
            [gcd](x1, x2) = [0]
                               
                      [0] = [0]
                               
                  [s](x1) = [0]
                               
         [if](x1, x2, x3) = [0]
                               
           [<](x1, x2) = [0]
                               
              [-](x1, x2) = [0]
                               
          [gcd^#](x1, x2) = [1]
                               
                [c_1](x1) = [0]
                               
                [c_2](x1) = [0]
                               
    [c_3](x1, x2, x3, x4) = [1]
  
  The following symbols are considered usable
  
    {gcd^#}
  
  The order satisfies the following ordering constraints:
  
        [gcd^#(x, 0())] =  [1]                                                    
                        >  [0]                                                    
                        =  [c_1(x)]                                               
                                                                                  
        [gcd^#(0(), y)] =  [1]                                                    
                        >  [0]                                                    
                        =  [c_2(y)]                                               
                                                                                  
    [gcd^#(s(x), s(y))] =  [1]                                                    
                        >= [1]                                                    
                        =  [c_3(x, y, gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y)))]
                                                                                  

The strictly oriented rules are moved into the weak component.

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

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

Empty rules are trivially bounded

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