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(x, 0()) -> x
  , minus(0(), s(y)) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , mod(x, s(y)) -> help(x, s(y), 0())
  , mod(s(x), 0()) -> 0()
  , help(x, s(y), c) -> if(le(c, x), x, s(y), c)
  , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y)))
  , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) }
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^#(x, 0()) -> c_4()
  , minus^#(0(), s(y)) -> c_5()
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , plus^#(x, 0()) -> c_7()
  , plus^#(x, s(y)) -> c_8(plus^#(x, y))
  , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0()))
  , mod^#(s(x), 0()) -> c_10()
  , help^#(x, s(y), c) ->
    c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
  , if^#(true(), x, s(y), c) ->
    c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
  , if^#(false(), x, s(y), c) ->
    c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) }

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^#(x, 0()) -> c_4()
  , minus^#(0(), s(y)) -> c_5()
  , minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
  , plus^#(x, 0()) -> c_7()
  , plus^#(x, s(y)) -> c_8(plus^#(x, y))
  , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0()))
  , mod^#(s(x), 0()) -> c_10()
  , help^#(x, s(y), c) ->
    c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
  , if^#(true(), x, s(y), c) ->
    c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
  , if^#(false(), x, s(y), c) ->
    c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), s(y)) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , mod(x, s(y)) -> help(x, s(y), 0())
  , mod(s(x), 0()) -> 0()
  , help(x, s(y), c) -> if(le(c, x), x, s(y), c)
  , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y)))
  , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,5,7,10} by
applications of Pre({1,2,4,5,7,10}) = {3,6,8,11,13}. 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^#(x, 0()) -> c_4()
    , 5: minus^#(0(), s(y)) -> c_5()
    , 6: minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
    , 7: plus^#(x, 0()) -> c_7()
    , 8: plus^#(x, s(y)) -> c_8(plus^#(x, y))
    , 9: mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0()))
    , 10: mod^#(s(x), 0()) -> c_10()
    , 11: help^#(x, s(y), c) ->
          c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
    , 12: if^#(true(), x, s(y), c) ->
          c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
    , 13: if^#(false(), x, s(y), c) ->
          c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(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), s(y)) -> c_6(minus^#(x, y))
  , plus^#(x, s(y)) -> c_8(plus^#(x, y))
  , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0()))
  , help^#(x, s(y), c) ->
    c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
  , if^#(true(), x, s(y), c) ->
    c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
  , if^#(false(), x, s(y), c) ->
    c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) }
Weak DPs:
  { le^#(0(), y) -> c_1()
  , le^#(s(x), 0()) -> c_2()
  , minus^#(x, 0()) -> c_4()
  , minus^#(0(), s(y)) -> c_5()
  , plus^#(x, 0()) -> c_7()
  , mod^#(s(x), 0()) -> c_10() }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), s(y)) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , mod(x, s(y)) -> help(x, s(y), 0())
  , mod(s(x), 0()) -> 0()
  , help(x, s(y), c) -> if(le(c, x), x, s(y), c)
  , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y)))
  , if(false(), x, s(y), c) -> minus(x, minus(c, s(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_1()
, le^#(s(x), 0()) -> c_2()
, minus^#(x, 0()) -> c_4()
, minus^#(0(), s(y)) -> c_5()
, plus^#(x, 0()) -> c_7()
, mod^#(s(x), 0()) -> c_10() }

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), s(y)) -> c_6(minus^#(x, y))
  , plus^#(x, s(y)) -> c_8(plus^#(x, y))
  , mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0()))
  , help^#(x, s(y), c) ->
    c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
  , if^#(true(), x, s(y), c) ->
    c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
  , if^#(false(), x, s(y), c) ->
    c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), s(y)) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , mod(x, s(y)) -> help(x, s(y), 0())
  , mod(s(x), 0()) -> 0()
  , help(x, s(y), c) -> if(le(c, x), x, s(y), c)
  , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y)))
  , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: le^#(s(x), s(y)) -> c_3(le^#(x, y))
     -->_1 le^#(s(x), s(y)) -> c_3(le^#(x, y)) :1
  
  2: minus^#(s(x), s(y)) -> c_6(minus^#(x, y))
     -->_1 minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) :2
  
  3: plus^#(x, s(y)) -> c_8(plus^#(x, y))
     -->_1 plus^#(x, s(y)) -> c_8(plus^#(x, y)) :3
  
  4: mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0()))
     -->_1 help^#(x, s(y), c) ->
           c_11(if^#(le(c, x), x, s(y), c), le^#(c, x)) :5
  
  5: help^#(x, s(y), c) ->
     c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
     -->_1 if^#(false(), x, s(y), c) ->
           c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) :7
     -->_1 if^#(true(), x, s(y), c) ->
           c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y))) :6
     -->_2 le^#(s(x), s(y)) -> c_3(le^#(x, y)) :1
  
  6: if^#(true(), x, s(y), c) ->
     c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
     -->_1 help^#(x, s(y), c) ->
           c_11(if^#(le(c, x), x, s(y), c), le^#(c, x)) :5
     -->_2 plus^#(x, s(y)) -> c_8(plus^#(x, y)) :3
  
  7: if^#(false(), x, s(y), c) ->
     c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y)))
     -->_2 minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) :2
     -->_1 minus^#(s(x), s(y)) -> c_6(minus^#(x, y)) :2
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { mod^#(x, s(y)) -> c_9(help^#(x, s(y), 0())) }


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), s(y)) -> c_6(minus^#(x, y))
  , plus^#(x, s(y)) -> c_8(plus^#(x, y))
  , help^#(x, s(y), c) ->
    c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
  , if^#(true(), x, s(y), c) ->
    c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
  , if^#(false(), x, s(y), c) ->
    c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), s(y)) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , mod(x, s(y)) -> help(x, s(y), 0())
  , mod(s(x), 0()) -> 0()
  , help(x, s(y), c) -> if(le(c, x), x, s(y), c)
  , if(true(), x, s(y), c) -> help(x, s(y), plus(c, s(y)))
  , if(false(), x, s(y), c) -> minus(x, minus(c, s(y))) }
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(x, 0()) -> x
    , minus(0(), s(y)) -> 0()
    , minus(s(x), s(y)) -> minus(x, y)
    , plus(x, 0()) -> x
    , plus(x, s(y)) -> s(plus(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), s(y)) -> c_6(minus^#(x, y))
  , plus^#(x, s(y)) -> c_8(plus^#(x, y))
  , help^#(x, s(y), c) ->
    c_11(if^#(le(c, x), x, s(y), c), le^#(c, x))
  , if^#(true(), x, s(y), c) ->
    c_12(help^#(x, s(y), plus(c, s(y))), plus^#(c, s(y)))
  , if^#(false(), x, s(y), c) ->
    c_13(minus^#(x, minus(c, s(y))), minus^#(c, s(y))) }
Weak Trs:
  { le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , minus(x, 0()) -> x
  , minus(0(), s(y)) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(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..