MAYBE

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

Strict Trs:
  { plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(y, times(x, y))
  , exp(x, 0()) -> s(0())
  , exp(x, s(y)) -> times(x, exp(x, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , tower(x, y) -> towerIter(0(), x, y, s(0()))
  , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
  , help(true(), c, x, y, z) -> z
  , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { plus^#(0(), x) -> c_1()
  , plus^#(s(x), y) -> c_2(plus^#(x, y))
  , times^#(0(), y) -> c_3()
  , times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
  , exp^#(x, 0()) -> c_5()
  , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
  , ge^#(x, 0()) -> c_7()
  , ge^#(0(), s(x)) -> c_8()
  , ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
  , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0())))
  , towerIter^#(c, x, y, z) ->
    c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
  , help^#(true(), c, x, y, z) -> c_12()
  , help^#(false(), c, x, y, z) ->
    c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }

and mark the set of starting terms.

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

Strict DPs:
  { plus^#(0(), x) -> c_1()
  , plus^#(s(x), y) -> c_2(plus^#(x, y))
  , times^#(0(), y) -> c_3()
  , times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
  , exp^#(x, 0()) -> c_5()
  , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
  , ge^#(x, 0()) -> c_7()
  , ge^#(0(), s(x)) -> c_8()
  , ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
  , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0())))
  , towerIter^#(c, x, y, z) ->
    c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
  , help^#(true(), c, x, y, z) -> c_12()
  , help^#(false(), c, x, y, z) ->
    c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }
Weak Trs:
  { plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(y, times(x, y))
  , exp(x, 0()) -> s(0())
  , exp(x, s(y)) -> times(x, exp(x, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , tower(x, y) -> towerIter(0(), x, y, s(0()))
  , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
  , help(true(), c, x, y, z) -> z
  , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: plus^#(0(), x) -> c_1()
    , 2: plus^#(s(x), y) -> c_2(plus^#(x, y))
    , 3: times^#(0(), y) -> c_3()
    , 4: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
    , 5: exp^#(x, 0()) -> c_5()
    , 6: exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
    , 7: ge^#(x, 0()) -> c_7()
    , 8: ge^#(0(), s(x)) -> c_8()
    , 9: ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
    , 10: tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0())))
    , 11: towerIter^#(c, x, y, z) ->
          c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
    , 12: help^#(true(), c, x, y, z) -> c_12()
    , 13: help^#(false(), c, x, y, z) ->
          c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }

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

Strict DPs:
  { plus^#(s(x), y) -> c_2(plus^#(x, y))
  , times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
  , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
  , ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
  , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0())))
  , towerIter^#(c, x, y, z) ->
    c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
  , help^#(false(), c, x, y, z) ->
    c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }
Weak DPs:
  { plus^#(0(), x) -> c_1()
  , times^#(0(), y) -> c_3()
  , exp^#(x, 0()) -> c_5()
  , ge^#(x, 0()) -> c_7()
  , ge^#(0(), s(x)) -> c_8()
  , help^#(true(), c, x, y, z) -> c_12() }
Weak Trs:
  { plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(y, times(x, y))
  , exp(x, 0()) -> s(0())
  , exp(x, s(y)) -> times(x, exp(x, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , tower(x, y) -> towerIter(0(), x, y, s(0()))
  , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
  , help(true(), c, x, y, z) -> z
  , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, 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.

{ plus^#(0(), x) -> c_1()
, times^#(0(), y) -> c_3()
, exp^#(x, 0()) -> c_5()
, ge^#(x, 0()) -> c_7()
, ge^#(0(), s(x)) -> c_8()
, help^#(true(), c, x, y, z) -> c_12() }

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

Strict DPs:
  { plus^#(s(x), y) -> c_2(plus^#(x, y))
  , times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
  , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
  , ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
  , tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0())))
  , towerIter^#(c, x, y, z) ->
    c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
  , help^#(false(), c, x, y, z) ->
    c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }
Weak Trs:
  { plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(y, times(x, y))
  , exp(x, 0()) -> s(0())
  , exp(x, s(y)) -> times(x, exp(x, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , tower(x, y) -> towerIter(0(), x, y, s(0()))
  , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
  , help(true(), c, x, y, z) -> z
  , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: plus^#(s(x), y) -> c_2(plus^#(x, y))
     -->_1 plus^#(s(x), y) -> c_2(plus^#(x, y)) :1
  
  2: times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
     -->_2 times^#(s(x), y) ->
           c_4(plus^#(y, times(x, y)), times^#(x, y)) :2
     -->_1 plus^#(s(x), y) -> c_2(plus^#(x, y)) :1
  
  3: exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
     -->_2 exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y)) :3
     -->_1 times^#(s(x), y) ->
           c_4(plus^#(y, times(x, y)), times^#(x, y)) :2
  
  4: ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
     -->_1 ge^#(s(x), s(y)) -> c_9(ge^#(x, y)) :4
  
  5: tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0())))
     -->_1 towerIter^#(c, x, y, z) ->
           c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x)) :6
  
  6: towerIter^#(c, x, y, z) ->
     c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
     -->_1 help^#(false(), c, x, y, z) ->
           c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) :7
     -->_2 ge^#(s(x), s(y)) -> c_9(ge^#(x, y)) :4
  
  7: help^#(false(), c, x, y, z) ->
     c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z))
     -->_1 towerIter^#(c, x, y, z) ->
           c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x)) :6
     -->_2 exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y)) :3
  

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).

  { tower^#(x, y) -> c_10(towerIter^#(0(), x, y, s(0()))) }


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

Strict DPs:
  { plus^#(s(x), y) -> c_2(plus^#(x, y))
  , times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
  , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
  , ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
  , towerIter^#(c, x, y, z) ->
    c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
  , help^#(false(), c, x, y, z) ->
    c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }
Weak Trs:
  { plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(y, times(x, y))
  , exp(x, 0()) -> s(0())
  , exp(x, s(y)) -> times(x, exp(x, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y)
  , tower(x, y) -> towerIter(0(), x, y, s(0()))
  , towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
  , help(true(), c, x, y, z) -> z
  , help(false(), c, x, y, z) -> towerIter(s(c), x, y, exp(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { plus(0(), x) -> x
    , plus(s(x), y) -> s(plus(x, y))
    , times(0(), y) -> 0()
    , times(s(x), y) -> plus(y, times(x, y))
    , exp(x, 0()) -> s(0())
    , exp(x, s(y)) -> times(x, exp(x, y))
    , ge(x, 0()) -> true()
    , ge(0(), s(x)) -> false()
    , ge(s(x), s(y)) -> ge(x, y) }

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

Strict DPs:
  { plus^#(s(x), y) -> c_2(plus^#(x, y))
  , times^#(s(x), y) -> c_4(plus^#(y, times(x, y)), times^#(x, y))
  , exp^#(x, s(y)) -> c_6(times^#(x, exp(x, y)), exp^#(x, y))
  , ge^#(s(x), s(y)) -> c_9(ge^#(x, y))
  , towerIter^#(c, x, y, z) ->
    c_11(help^#(ge(c, x), c, x, y, z), ge^#(c, x))
  , help^#(false(), c, x, y, z) ->
    c_13(towerIter^#(s(c), x, y, exp(y, z)), exp^#(y, z)) }
Weak Trs:
  { plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> plus(y, times(x, y))
  , exp(x, 0()) -> s(0())
  , exp(x, s(y)) -> times(x, exp(x, y))
  , ge(x, 0()) -> true()
  , ge(0(), s(x)) -> false()
  , ge(s(x), s(y)) -> ge(x, y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..