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)
  , minus(0(), y) -> 0()
  , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
  , if_minus(true(), s(x), y) -> 0()
  , if_minus(false(), s(x), y) -> s(minus(x, y))
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
  , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y))
  , if_mod(false(), s(x), s(y)) -> s(x) }
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))
  , minus^#(0(), y) -> c_4()
  , minus^#(s(x), y) ->
    c_5(if_minus^#(le(s(x), y), s(x), y), le^#(s(x), y))
  , if_minus^#(true(), s(x), y) -> c_6()
  , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y))
  , mod^#(0(), y) -> c_8()
  , mod^#(s(x), 0()) -> c_9()
  , mod^#(s(x), s(y)) ->
    c_10(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_mod^#(true(), s(x), s(y)) ->
    c_11(mod^#(minus(x, y), s(y)), minus^#(x, y))
  , if_mod^#(false(), s(x), s(y)) -> c_12() }

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))
  , minus^#(0(), y) -> c_4()
  , minus^#(s(x), y) ->
    c_5(if_minus^#(le(s(x), y), s(x), y), le^#(s(x), y))
  , if_minus^#(true(), s(x), y) -> c_6()
  , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y))
  , mod^#(0(), y) -> c_8()
  , mod^#(s(x), 0()) -> c_9()
  , mod^#(s(x), s(y)) ->
    c_10(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_mod^#(true(), s(x), s(y)) ->
    c_11(mod^#(minus(x, y), s(y)), minus^#(x, y))
  , if_mod^#(false(), s(x), s(y)) -> c_12() }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(0(), y) -> 0()
  , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
  , if_minus(true(), s(x), y) -> 0()
  , if_minus(false(), s(x), y) -> s(minus(x, y))
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
  , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y))
  , if_mod(false(), s(x), s(y)) -> s(x) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,6,8,9,12} by
applications of Pre({1,2,4,6,8,9,12}) = {3,5,7,10,11}. 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: minus^#(0(), y) -> c_4()
    , 5: minus^#(s(x), y) ->
         c_5(if_minus^#(le(s(x), y), s(x), y), le^#(s(x), y))
    , 6: if_minus^#(true(), s(x), y) -> c_6()
    , 7: if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y))
    , 8: mod^#(0(), y) -> c_8()
    , 9: mod^#(s(x), 0()) -> c_9()
    , 10: mod^#(s(x), s(y)) ->
          c_10(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x))
    , 11: if_mod^#(true(), s(x), s(y)) ->
          c_11(mod^#(minus(x, y), s(y)), minus^#(x, y))
    , 12: if_mod^#(false(), s(x), s(y)) -> c_12() }

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))
  , minus^#(s(x), y) ->
    c_5(if_minus^#(le(s(x), y), s(x), y), le^#(s(x), y))
  , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_10(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_mod^#(true(), s(x), s(y)) ->
    c_11(mod^#(minus(x, y), s(y)), minus^#(x, y)) }
Weak DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , minus^#(0(), y) -> c_4()
  , if_minus^#(true(), s(x), y) -> c_6()
  , mod^#(0(), y) -> c_8()
  , mod^#(s(x), 0()) -> c_9()
  , if_mod^#(false(), s(x), s(y)) -> c_12() }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(0(), y) -> 0()
  , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
  , if_minus(true(), s(x), y) -> 0()
  , if_minus(false(), s(x), y) -> s(minus(x, y))
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
  , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y))
  , if_mod(false(), s(x), s(y)) -> s(x) }
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()
, minus^#(0(), y) -> c_4()
, if_minus^#(true(), s(x), y) -> c_6()
, mod^#(0(), y) -> c_8()
, mod^#(s(x), 0()) -> c_9()
, if_mod^#(false(), s(x), s(y)) -> c_12() }

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))
  , minus^#(s(x), y) ->
    c_5(if_minus^#(le(s(x), y), s(x), y), le^#(s(x), y))
  , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_10(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_mod^#(true(), s(x), s(y)) ->
    c_11(mod^#(minus(x, y), s(y)), minus^#(x, y)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(0(), y) -> 0()
  , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
  , if_minus(true(), s(x), y) -> 0()
  , if_minus(false(), s(x), y) -> s(minus(x, y))
  , mod(0(), y) -> 0()
  , mod(s(x), 0()) -> 0()
  , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
  , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y))
  , if_mod(false(), s(x), s(y)) -> s(x) }
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)
    , minus(0(), y) -> 0()
    , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
    , if_minus(true(), s(x), y) -> 0()
    , if_minus(false(), s(x), y) -> s(minus(x, y)) }

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))
  , minus^#(s(x), y) ->
    c_5(if_minus^#(le(s(x), y), s(x), y), le^#(s(x), y))
  , if_minus^#(false(), s(x), y) -> c_7(minus^#(x, y))
  , mod^#(s(x), s(y)) ->
    c_10(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x))
  , if_mod^#(true(), s(x), s(y)) ->
    c_11(mod^#(minus(x, y), s(y)), minus^#(x, y)) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(0(), y) -> 0()
  , minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
  , if_minus(true(), s(x), y) -> 0()
  , if_minus(false(), s(x), y) -> s(minus(x, y)) }
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:
      
      The input cannot be shown compatible
   
   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..