MAYBE

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

Strict Trs:
  { times(x, y) -> help(x, y, 0())
  , help(x, y, c) -> if(lt(c, y), x, y, c)
  , if(true(), x, y, c) -> plus(x, help(x, y, s(c)))
  , if(false(), x, y, c) -> 0()
  , lt(0(), s(x)) -> true()
  , lt(s(x), 0()) -> false()
  , lt(s(x), s(y)) -> lt(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { times^#(x, y) -> c_1(help^#(x, y, 0()))
  , help^#(x, y, c) -> c_2(if^#(lt(c, y), x, y, c), lt^#(c, y))
  , if^#(true(), x, y, c) ->
    c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c)))
  , if^#(false(), x, y, c) -> c_4()
  , lt^#(0(), s(x)) -> c_5()
  , lt^#(s(x), 0()) -> c_6()
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , plus^#(x, 0()) -> c_8()
  , plus^#(x, s(y)) -> c_9(plus^#(x, y))
  , plus^#(0(), x) -> c_10()
  , plus^#(s(x), y) -> c_11(plus^#(x, y)) }

and mark the set of starting terms.

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

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

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

  DPs:
    { 1: times^#(x, y) -> c_1(help^#(x, y, 0()))
    , 2: help^#(x, y, c) -> c_2(if^#(lt(c, y), x, y, c), lt^#(c, y))
    , 3: if^#(true(), x, y, c) ->
         c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c)))
    , 4: if^#(false(), x, y, c) -> c_4()
    , 5: lt^#(0(), s(x)) -> c_5()
    , 6: lt^#(s(x), 0()) -> c_6()
    , 7: lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
    , 8: plus^#(x, 0()) -> c_8()
    , 9: plus^#(x, s(y)) -> c_9(plus^#(x, y))
    , 10: plus^#(0(), x) -> c_10()
    , 11: plus^#(s(x), y) -> c_11(plus^#(x, y)) }

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

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

{ if^#(false(), x, y, c) -> c_4()
, lt^#(0(), s(x)) -> c_5()
, lt^#(s(x), 0()) -> c_6()
, plus^#(x, 0()) -> c_8()
, plus^#(0(), x) -> c_10() }

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

Strict DPs:
  { times^#(x, y) -> c_1(help^#(x, y, 0()))
  , help^#(x, y, c) -> c_2(if^#(lt(c, y), x, y, c), lt^#(c, y))
  , if^#(true(), x, y, c) ->
    c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , plus^#(x, s(y)) -> c_9(plus^#(x, y))
  , plus^#(s(x), y) -> c_11(plus^#(x, y)) }
Weak Trs:
  { times(x, y) -> help(x, y, 0())
  , help(x, y, c) -> if(lt(c, y), x, y, c)
  , if(true(), x, y, c) -> plus(x, help(x, y, s(c)))
  , if(false(), x, y, c) -> 0()
  , lt(0(), s(x)) -> true()
  , lt(s(x), 0()) -> false()
  , lt(s(x), s(y)) -> lt(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: times^#(x, y) -> c_1(help^#(x, y, 0()))
     -->_1 help^#(x, y, c) ->
           c_2(if^#(lt(c, y), x, y, c), lt^#(c, y)) :2
  
  2: help^#(x, y, c) -> c_2(if^#(lt(c, y), x, y, c), lt^#(c, y))
     -->_2 lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) :4
     -->_1 if^#(true(), x, y, c) ->
           c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c))) :3
  
  3: if^#(true(), x, y, c) ->
     c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c)))
     -->_1 plus^#(s(x), y) -> c_11(plus^#(x, y)) :6
     -->_1 plus^#(x, s(y)) -> c_9(plus^#(x, y)) :5
     -->_2 help^#(x, y, c) ->
           c_2(if^#(lt(c, y), x, y, c), lt^#(c, y)) :2
  
  4: lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
     -->_1 lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) :4
  
  5: plus^#(x, s(y)) -> c_9(plus^#(x, y))
     -->_1 plus^#(s(x), y) -> c_11(plus^#(x, y)) :6
     -->_1 plus^#(x, s(y)) -> c_9(plus^#(x, y)) :5
  
  6: plus^#(s(x), y) -> c_11(plus^#(x, y))
     -->_1 plus^#(s(x), y) -> c_11(plus^#(x, y)) :6
     -->_1 plus^#(x, s(y)) -> c_9(plus^#(x, y)) :5
  

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).

  { times^#(x, y) -> c_1(help^#(x, y, 0())) }


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

Strict DPs:
  { help^#(x, y, c) -> c_2(if^#(lt(c, y), x, y, c), lt^#(c, y))
  , if^#(true(), x, y, c) ->
    c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , plus^#(x, s(y)) -> c_9(plus^#(x, y))
  , plus^#(s(x), y) -> c_11(plus^#(x, y)) }
Weak Trs:
  { times(x, y) -> help(x, y, 0())
  , help(x, y, c) -> if(lt(c, y), x, y, c)
  , if(true(), x, y, c) -> plus(x, help(x, y, s(c)))
  , if(false(), x, y, c) -> 0()
  , lt(0(), s(x)) -> true()
  , lt(s(x), 0()) -> false()
  , lt(s(x), s(y)) -> lt(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { help(x, y, c) -> if(lt(c, y), x, y, c)
    , if(true(), x, y, c) -> plus(x, help(x, y, s(c)))
    , if(false(), x, y, c) -> 0()
    , lt(0(), s(x)) -> true()
    , lt(s(x), 0()) -> false()
    , lt(s(x), s(y)) -> lt(x, y)
    , plus(x, 0()) -> x
    , plus(x, s(y)) -> s(plus(x, y))
    , plus(0(), x) -> x
    , plus(s(x), y) -> s(plus(x, y)) }

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

Strict DPs:
  { help^#(x, y, c) -> c_2(if^#(lt(c, y), x, y, c), lt^#(c, y))
  , if^#(true(), x, y, c) ->
    c_3(plus^#(x, help(x, y, s(c))), help^#(x, y, s(c)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , plus^#(x, s(y)) -> c_9(plus^#(x, y))
  , plus^#(s(x), y) -> c_11(plus^#(x, y)) }
Weak Trs:
  { help(x, y, c) -> if(lt(c, y), x, y, c)
  , if(true(), x, y, c) -> plus(x, help(x, y, s(c)))
  , if(false(), x, y, c) -> 0()
  , lt(0(), s(x)) -> true()
  , lt(s(x), 0()) -> false()
  , lt(s(x), s(y)) -> lt(x, y)
  , plus(x, 0()) -> x
  , plus(x, s(y)) -> s(plus(x, y))
  , plus(0(), x) -> x
  , plus(s(x), y) -> s(plus(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..