MAYBE

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

Strict Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , eq(0(), 0()) -> true()
  , eq(0(), s(y)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , minsort(nil()) -> nil()
  , minsort(cons(x, y)) ->
    cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
  , min(x, nil()) -> x
  , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
  , del(x, nil()) -> nil()
  , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , eq^#(0(), 0()) -> c_4()
  , eq^#(0(), s(y)) -> c_5()
  , eq^#(s(x), 0()) -> c_6()
  , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
  , if^#(true(), x, y) -> c_8()
  , if^#(false(), x, y) -> c_9()
  , minsort^#(nil()) -> c_10()
  , minsort^#(cons(x, y)) ->
    c_11(min^#(x, y),
         minsort^#(del(min(x, y), cons(x, y))),
         del^#(min(x, y), cons(x, y)),
         min^#(x, y))
  , min^#(x, nil()) -> c_12()
  , min^#(x, cons(y, z)) ->
    c_13(if^#(le(x, y), min(x, z), min(y, z)),
         le^#(x, y),
         min^#(x, z),
         min^#(y, z))
  , del^#(x, nil()) -> c_14()
  , del^#(x, cons(y, z)) ->
    c_15(if^#(eq(x, y), z, cons(y, del(x, z))),
         eq^#(x, y),
         del^#(x, z)) }

and mark the set of starting terms.

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

Strict DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , eq^#(0(), 0()) -> c_4()
  , eq^#(0(), s(y)) -> c_5()
  , eq^#(s(x), 0()) -> c_6()
  , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
  , if^#(true(), x, y) -> c_8()
  , if^#(false(), x, y) -> c_9()
  , minsort^#(nil()) -> c_10()
  , minsort^#(cons(x, y)) ->
    c_11(min^#(x, y),
         minsort^#(del(min(x, y), cons(x, y))),
         del^#(min(x, y), cons(x, y)),
         min^#(x, y))
  , min^#(x, nil()) -> c_12()
  , min^#(x, cons(y, z)) ->
    c_13(if^#(le(x, y), min(x, z), min(y, z)),
         le^#(x, y),
         min^#(x, z),
         min^#(y, z))
  , del^#(x, nil()) -> c_14()
  , del^#(x, cons(y, z)) ->
    c_15(if^#(eq(x, y), z, cons(y, del(x, z))),
         eq^#(x, y),
         del^#(x, z)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , eq(0(), 0()) -> true()
  , eq(0(), s(y)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , minsort(nil()) -> nil()
  , minsort(cons(x, y)) ->
    cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
  , min(x, nil()) -> x
  , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
  , del(x, nil()) -> nil()
  , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,5,6,8,9,10,12,14}
by applications of Pre({1,2,4,5,6,8,9,10,12,14}) = {3,7,11,13,15}.
Here rules are labeled as follows:

  DPs:
    { 1: le^#(0(), y) -> c_1()
    , 2: le^#(s(x), 0()) -> c_2()
    , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y))
    , 4: eq^#(0(), 0()) -> c_4()
    , 5: eq^#(0(), s(y)) -> c_5()
    , 6: eq^#(s(x), 0()) -> c_6()
    , 7: eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
    , 8: if^#(true(), x, y) -> c_8()
    , 9: if^#(false(), x, y) -> c_9()
    , 10: minsort^#(nil()) -> c_10()
    , 11: minsort^#(cons(x, y)) ->
          c_11(min^#(x, y),
               minsort^#(del(min(x, y), cons(x, y))),
               del^#(min(x, y), cons(x, y)),
               min^#(x, y))
    , 12: min^#(x, nil()) -> c_12()
    , 13: min^#(x, cons(y, z)) ->
          c_13(if^#(le(x, y), min(x, z), min(y, z)),
               le^#(x, y),
               min^#(x, z),
               min^#(y, z))
    , 14: del^#(x, nil()) -> c_14()
    , 15: del^#(x, cons(y, z)) ->
          c_15(if^#(eq(x, y), z, cons(y, del(x, z))),
               eq^#(x, y),
               del^#(x, z)) }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
  , minsort^#(cons(x, y)) ->
    c_11(min^#(x, y),
         minsort^#(del(min(x, y), cons(x, y))),
         del^#(min(x, y), cons(x, y)),
         min^#(x, y))
  , min^#(x, cons(y, z)) ->
    c_13(if^#(le(x, y), min(x, z), min(y, z)),
         le^#(x, y),
         min^#(x, z),
         min^#(y, z))
  , del^#(x, cons(y, z)) ->
    c_15(if^#(eq(x, y), z, cons(y, del(x, z))),
         eq^#(x, y),
         del^#(x, z)) }
Weak DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , eq^#(0(), 0()) -> c_4()
  , eq^#(0(), s(y)) -> c_5()
  , eq^#(s(x), 0()) -> c_6()
  , if^#(true(), x, y) -> c_8()
  , if^#(false(), x, y) -> c_9()
  , minsort^#(nil()) -> c_10()
  , min^#(x, nil()) -> c_12()
  , del^#(x, nil()) -> c_14() }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , eq(0(), 0()) -> true()
  , eq(0(), s(y)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , minsort(nil()) -> nil()
  , minsort(cons(x, y)) ->
    cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
  , min(x, nil()) -> x
  , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
  , del(x, nil()) -> nil()
  , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }
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_1()
, le^#(s(x), 0()) -> c_2()
, eq^#(0(), 0()) -> c_4()
, eq^#(0(), s(y)) -> c_5()
, eq^#(s(x), 0()) -> c_6()
, if^#(true(), x, y) -> c_8()
, if^#(false(), x, y) -> c_9()
, minsort^#(nil()) -> c_10()
, min^#(x, nil()) -> c_12()
, del^#(x, nil()) -> c_14() }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_3(le^#(x, y))
  , eq^#(s(x), s(y)) -> c_7(eq^#(x, y))
  , minsort^#(cons(x, y)) ->
    c_11(min^#(x, y),
         minsort^#(del(min(x, y), cons(x, y))),
         del^#(min(x, y), cons(x, y)),
         min^#(x, y))
  , min^#(x, cons(y, z)) ->
    c_13(if^#(le(x, y), min(x, z), min(y, z)),
         le^#(x, y),
         min^#(x, z),
         min^#(y, z))
  , del^#(x, cons(y, z)) ->
    c_15(if^#(eq(x, y), z, cons(y, del(x, z))),
         eq^#(x, y),
         del^#(x, z)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , eq(0(), 0()) -> true()
  , eq(0(), s(y)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , minsort(nil()) -> nil()
  , minsort(cons(x, y)) ->
    cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
  , min(x, nil()) -> x
  , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
  , del(x, nil()) -> nil()
  , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { min^#(x, cons(y, z)) ->
    c_13(if^#(le(x, y), min(x, z), min(y, z)),
         le^#(x, y),
         min^#(x, z),
         min^#(y, z))
  , del^#(x, cons(y, z)) ->
    c_15(if^#(eq(x, y), z, cons(y, del(x, z))),
         eq^#(x, y),
         del^#(x, z)) }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_1(le^#(x, y))
  , eq^#(s(x), s(y)) -> c_2(eq^#(x, y))
  , minsort^#(cons(x, y)) ->
    c_3(min^#(x, y),
        minsort^#(del(min(x, y), cons(x, y))),
        del^#(min(x, y), cons(x, y)),
        min^#(x, y))
  , min^#(x, cons(y, z)) -> c_4(le^#(x, y), min^#(x, z), min^#(y, z))
  , del^#(x, cons(y, z)) -> c_5(eq^#(x, y), del^#(x, z)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , eq(0(), 0()) -> true()
  , eq(0(), s(y)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , minsort(nil()) -> nil()
  , minsort(cons(x, y)) ->
    cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
  , min(x, nil()) -> x
  , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
  , del(x, nil()) -> nil()
  , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { le(0(), y) -> true()
    , le(s(x), 0()) -> false()
    , le(s(x), s(y)) -> le(x, y)
    , eq(0(), 0()) -> true()
    , eq(0(), s(y)) -> false()
    , eq(s(x), 0()) -> false()
    , eq(s(x), s(y)) -> eq(x, y)
    , if(true(), x, y) -> x
    , if(false(), x, y) -> y
    , min(x, nil()) -> x
    , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
    , del(x, nil()) -> nil()
    , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }

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

Strict DPs:
  { le^#(s(x), s(y)) -> c_1(le^#(x, y))
  , eq^#(s(x), s(y)) -> c_2(eq^#(x, y))
  , minsort^#(cons(x, y)) ->
    c_3(min^#(x, y),
        minsort^#(del(min(x, y), cons(x, y))),
        del^#(min(x, y), cons(x, y)),
        min^#(x, y))
  , min^#(x, cons(y, z)) -> c_4(le^#(x, y), min^#(x, z), min^#(y, z))
  , del^#(x, cons(y, z)) -> c_5(eq^#(x, y), del^#(x, z)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , eq(0(), 0()) -> true()
  , eq(0(), s(y)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , min(x, nil()) -> x
  , min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
  , del(x, nil()) -> nil()
  , del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      Following exception was raised:
        stack overflow
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..