MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following weak dependency pairs:

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(0(), s(y)) -> c_6()
  , quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y))) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(0(), s(y)) -> c_6()
  , quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y))) }
Strict Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , quot(0(), s(y)) -> 0()
  , quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { minus(x, 0()) -> x
    , minus(s(x), s(y)) -> minus(x, y) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(0(), s(y)) -> c_6()
  , quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y))) }
Strict Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

The following argument positions are usable:
  Uargs(c_2) = {1}, Uargs(c_5) = {1}, Uargs(quot^#) = {1},
  Uargs(c_7) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

    [minus](x1, x2) = [1] x1 + [1]         
                                           
                [0] = [0]                  
                                           
            [s](x1) = [1] x1 + [1]         
                                           
  [minus^#](x1, x2) = [1] x1 + [2] x2 + [1]
                                           
              [c_1] = [2]                  
                                           
          [c_2](x1) = [1] x1 + [1]         
                                           
     [le^#](x1, x2) = [2] x1 + [1] x2 + [1]
                                           
              [c_3] = [0]                  
                                           
              [c_4] = [2]                  
                                           
          [c_5](x1) = [1] x1 + [1]         
                                           
   [quot^#](x1, x2) = [2] x1 + [0]         
                                           
              [c_6] = [2]                  
                                           
          [c_7](x1) = [1] x1 + [2]         

This order satisfies following ordering constraints:

      [minus(x, 0())] = [1] x + [1]  
                      > [1] x + [0]  
                      = [x]          
                                     
  [minus(s(x), s(y))] = [1] x + [2]  
                      > [1] x + [1]  
                      = [minus(x, 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 MAYBE.

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , quot^#(0(), s(y)) -> c_6()
  , quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y))) }
Weak DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2} by applications of
Pre({2}) = {3}. Here rules are labeled as follows:

  DPs:
    { 1: minus^#(x, 0()) -> c_1()
    , 2: quot^#(0(), s(y)) -> c_6()
    , 3: quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y)))
    , 4: minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
    , 5: le^#(0(), y) -> c_3()
    , 6: le^#(s(x), 0()) -> c_4()
    , 7: le^#(s(x), s(y)) -> c_5(le^#(x, y)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y))) }
Weak DPs:
  { minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
  , le^#(0(), y) -> c_3()
  , le^#(s(x), 0()) -> c_4()
  , le^#(s(x), s(y)) -> c_5(le^#(x, y))
  , quot^#(0(), s(y)) -> c_6() }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

{ le^#(0(), y) -> c_3()
, le^#(s(x), 0()) -> c_4()
, le^#(s(x), s(y)) -> c_5(le^#(x, y))
, quot^#(0(), s(y)) -> c_6() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(x, 0()) -> c_1()
  , quot^#(s(x), s(y)) -> c_7(quot^#(minus(s(x), s(y)), s(y))) }
Weak DPs: { minus^#(s(x), s(y)) -> c_2(minus^#(x, y)) }
Weak Trs:
  { minus(x, 0()) -> x
  , minus(s(x), s(y)) -> minus(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..