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

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

Strict Trs:
  { main(S(x1)) -> x1
  , main(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { main^#(S(x1)) -> c_1()
  , main^#(0()) -> c_2() }

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:
  { main^#(S(x1)) -> c_1()
  , main^#(0()) -> c_2() }
Strict Trs:
  { main(S(x1)) -> x1
  , main(0()) -> 0() }
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:
  { main^#(S(x1)) -> c_1()
  , main^#(0()) -> c_2() }
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 the following constructor-restricted matrix
interpretation.

       [S](x1) = [2]           
                 [1]           
                               
           [0] = [1]           
                 [1]           
                               
  [main^#](x1) = [1 1] x1 + [2]
                 [1 1]      [1]
                               
         [c_1] = [0]           
                 [0]           
                               
         [c_2] = [1]           
                 [1]           

The following symbols are considered usable

  {main^#}

The order satisfies the following ordering constraints:

  [main^#(S(x1))] = [5]    
                    [4]    
                  > [0]    
                    [0]    
                  = [c_1()]
                           
    [main^#(0())] = [4]    
                    [3]    
                  > [1]    
                    [1]    
                  = [c_2()]
                           

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 DPs:
  { main^#(S(x1)) -> c_1()
  , main^#(0()) -> c_2() }
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.

{ main^#(S(x1)) -> c_1()
, main^#(0()) -> c_2() }

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))