MAYBE

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

Strict Trs:
  { from(X) -> cons(X, from(s(X)))
  , sel(s(N), cons(X, XS)) -> sel(N, XS)
  , sel(0(), cons(X, XS)) -> X
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), zWquot(XS, YS))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following weak dependency pairs:

Strict DPs:
  { from^#(X) -> c_1(from^#(s(X)))
  , sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
  , sel^#(0(), cons(X, XS)) -> c_3()
  , minus^#(X, 0()) -> c_4()
  , minus^#(s(X), s(Y)) -> c_5(minus^#(X, Y))
  , quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
  , quot^#(0(), s(Y)) -> c_7()
  , zWquot^#(XS, nil()) -> c_8()
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_9(quot^#(X, Y), zWquot^#(XS, YS))
  , zWquot^#(nil(), XS) -> c_10() }

and mark the set of starting terms.

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

Strict DPs:
  { from^#(X) -> c_1(from^#(s(X)))
  , sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
  , sel^#(0(), cons(X, XS)) -> c_3()
  , minus^#(X, 0()) -> c_4()
  , minus^#(s(X), s(Y)) -> c_5(minus^#(X, Y))
  , quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
  , quot^#(0(), s(Y)) -> c_7()
  , zWquot^#(XS, nil()) -> c_8()
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_9(quot^#(X, Y), zWquot^#(XS, YS))
  , zWquot^#(nil(), XS) -> c_10() }
Strict Trs:
  { from(X) -> cons(X, from(s(X)))
  , sel(s(N), cons(X, XS)) -> sel(N, XS)
  , sel(0(), cons(X, XS)) -> X
  , minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y)
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
  , quot(0(), s(Y)) -> 0()
  , zWquot(XS, nil()) -> nil()
  , zWquot(cons(X, XS), cons(Y, YS)) ->
    cons(quot(X, Y), zWquot(XS, YS))
  , zWquot(nil(), XS) -> nil() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { minus(X, 0()) -> 0()
    , minus(s(X), s(Y)) -> minus(X, Y) }

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

Strict DPs:
  { from^#(X) -> c_1(from^#(s(X)))
  , sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
  , sel^#(0(), cons(X, XS)) -> c_3()
  , minus^#(X, 0()) -> c_4()
  , minus^#(s(X), s(Y)) -> c_5(minus^#(X, Y))
  , quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
  , quot^#(0(), s(Y)) -> c_7()
  , zWquot^#(XS, nil()) -> c_8()
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_9(quot^#(X, Y), zWquot^#(XS, YS))
  , zWquot^#(nil(), XS) -> c_10() }
Strict Trs:
  { minus(X, 0()) -> 0()
  , 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_1) = {1}, Uargs(c_2) = {1}, Uargs(c_5) = {1},
  Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(c_9) = {1, 2}

TcT has computed following constructor-restricted matrix
interpretation.

      [cons](x1, x2) = [1] x1 + [1] x2 + [1]
                                            
             [s](x1) = [1] x1 + [2]         
                                            
                 [0] = [0]                  
                                            
     [minus](x1, x2) = [1] x1 + [2]         
                                            
               [nil] = [2]                  
                                            
        [from^#](x1) = [1] x1 + [1]         
                                            
           [c_1](x1) = [1] x1 + [1]         
                                            
     [sel^#](x1, x2) = [2] x1 + [2] x2 + [1]
                                            
           [c_2](x1) = [1] x1 + [1]         
                                            
               [c_3] = [1]                  
                                            
   [minus^#](x1, x2) = [1] x1 + [1] x2 + [2]
                                            
               [c_4] = [1]                  
                                            
           [c_5](x1) = [1] x1 + [1]         
                                            
    [quot^#](x1, x2) = [2] x1 + [0]         
                                            
           [c_6](x1) = [1] x1 + [2]         
                                            
               [c_7] = [1]                  
                                            
  [zWquot^#](x1, x2) = [2] x1 + [2] x2 + [2]
                                            
               [c_8] = [1]                  
                                            
       [c_9](x1, x2) = [1] x1 + [1] x2 + [1]
                                            
              [c_10] = [1]                  

This order satisfies following ordering constraints:

      [minus(X, 0())] = [1] X + [2]  
                      > [0]          
                      = [0()]        
                                     
  [minus(s(X), s(Y))] = [1] X + [4]  
                      > [1] X + [2]  
                      = [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:
  { from^#(X) -> c_1(from^#(s(X)))
  , quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
  , quot^#(0(), s(Y)) -> c_7() }
Weak DPs:
  { sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
  , sel^#(0(), cons(X, XS)) -> c_3()
  , minus^#(X, 0()) -> c_4()
  , minus^#(s(X), s(Y)) -> c_5(minus^#(X, Y))
  , zWquot^#(XS, nil()) -> c_8()
  , zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_9(quot^#(X, Y), zWquot^#(XS, YS))
  , zWquot^#(nil(), XS) -> c_10() }
Weak Trs:
  { minus(X, 0()) -> 0()
  , 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.

{ sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
, sel^#(0(), cons(X, XS)) -> c_3()
, minus^#(X, 0()) -> c_4()
, minus^#(s(X), s(Y)) -> c_5(minus^#(X, Y))
, zWquot^#(XS, nil()) -> c_8()
, zWquot^#(nil(), XS) -> c_10() }

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

Strict DPs:
  { from^#(X) -> c_1(from^#(s(X)))
  , quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
  , quot^#(0(), s(Y)) -> c_7() }
Weak DPs:
  { zWquot^#(cons(X, XS), cons(Y, YS)) ->
    c_9(quot^#(X, Y), zWquot^#(XS, YS)) }
Weak Trs:
  { minus(X, 0()) -> 0()
  , minus(s(X), s(Y)) -> minus(X, Y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..