MAYBE

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

Strict Trs:
  { 1024() -> 1024_1(0())
  , 1024_1(x) -> if(lt(x, 10()), x)
  , if(true(), x) -> double(1024_1(s(x)))
  , if(false(), x) -> s(0())
  , lt(x, 0()) -> false()
  , lt(0(), s(y)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , 10() -> double(s(double(s(s(0())))))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { 1024^#() -> c_1(1024_1^#(0()))
  , 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
  , if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
  , if^#(false(), x) -> c_4()
  , lt^#(x, 0()) -> c_5()
  , lt^#(0(), s(y)) -> c_6()
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , 10^#() ->
    c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
  , double^#(0()) -> c_9()
  , double^#(s(x)) -> c_10(double^#(x)) }

and mark the set of starting terms.

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

Strict DPs:
  { 1024^#() -> c_1(1024_1^#(0()))
  , 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
  , if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
  , if^#(false(), x) -> c_4()
  , lt^#(x, 0()) -> c_5()
  , lt^#(0(), s(y)) -> c_6()
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , 10^#() ->
    c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
  , double^#(0()) -> c_9()
  , double^#(s(x)) -> c_10(double^#(x)) }
Weak Trs:
  { 1024() -> 1024_1(0())
  , 1024_1(x) -> if(lt(x, 10()), x)
  , if(true(), x) -> double(1024_1(s(x)))
  , if(false(), x) -> s(0())
  , lt(x, 0()) -> false()
  , lt(0(), s(y)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , 10() -> double(s(double(s(s(0())))))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: 1024^#() -> c_1(1024_1^#(0()))
    , 2: 1024_1^#(x) ->
         c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
    , 3: if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
    , 4: if^#(false(), x) -> c_4()
    , 5: lt^#(x, 0()) -> c_5()
    , 6: lt^#(0(), s(y)) -> c_6()
    , 7: lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
    , 8: 10^#() ->
         c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
    , 9: double^#(0()) -> c_9()
    , 10: double^#(s(x)) -> c_10(double^#(x)) }

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

Strict DPs:
  { 1024^#() -> c_1(1024_1^#(0()))
  , 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
  , if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , 10^#() ->
    c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
  , double^#(s(x)) -> c_10(double^#(x)) }
Weak DPs:
  { if^#(false(), x) -> c_4()
  , lt^#(x, 0()) -> c_5()
  , lt^#(0(), s(y)) -> c_6()
  , double^#(0()) -> c_9() }
Weak Trs:
  { 1024() -> 1024_1(0())
  , 1024_1(x) -> if(lt(x, 10()), x)
  , if(true(), x) -> double(1024_1(s(x)))
  , if(false(), x) -> s(0())
  , lt(x, 0()) -> false()
  , lt(0(), s(y)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , 10() -> double(s(double(s(s(0())))))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(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.

{ if^#(false(), x) -> c_4()
, lt^#(x, 0()) -> c_5()
, lt^#(0(), s(y)) -> c_6()
, double^#(0()) -> c_9() }

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

Strict DPs:
  { 1024^#() -> c_1(1024_1^#(0()))
  , 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
  , if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , 10^#() ->
    c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
  , double^#(s(x)) -> c_10(double^#(x)) }
Weak Trs:
  { 1024() -> 1024_1(0())
  , 1024_1(x) -> if(lt(x, 10()), x)
  , if(true(), x) -> double(1024_1(s(x)))
  , if(false(), x) -> s(0())
  , lt(x, 0()) -> false()
  , lt(0(), s(y)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , 10() -> double(s(double(s(s(0())))))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: 1024^#() -> c_1(1024_1^#(0()))
     -->_1 1024_1^#(x) ->
           c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#()) :2
  
  2: 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
     -->_3 10^#() ->
           c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0())))) :5
     -->_2 lt^#(s(x), s(y)) -> c_7(lt^#(x, y)) :4
     -->_1 if^#(true(), x) ->
           c_3(double^#(1024_1(s(x))), 1024_1^#(s(x))) :3
  
  3: if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
     -->_1 double^#(s(x)) -> c_10(double^#(x)) :6
     -->_2 1024_1^#(x) ->
           c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#()) :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: 10^#() ->
     c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
     -->_2 double^#(s(x)) -> c_10(double^#(x)) :6
     -->_1 double^#(s(x)) -> c_10(double^#(x)) :6
  
  6: double^#(s(x)) -> c_10(double^#(x))
     -->_1 double^#(s(x)) -> c_10(double^#(x)) :6
  

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).

  { 1024^#() -> c_1(1024_1^#(0())) }


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

Strict DPs:
  { 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
  , if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , 10^#() ->
    c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
  , double^#(s(x)) -> c_10(double^#(x)) }
Weak Trs:
  { 1024() -> 1024_1(0())
  , 1024_1(x) -> if(lt(x, 10()), x)
  , if(true(), x) -> double(1024_1(s(x)))
  , if(false(), x) -> s(0())
  , lt(x, 0()) -> false()
  , lt(0(), s(y)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , 10() -> double(s(double(s(s(0())))))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { 1024_1(x) -> if(lt(x, 10()), x)
    , if(true(), x) -> double(1024_1(s(x)))
    , if(false(), x) -> s(0())
    , lt(x, 0()) -> false()
    , lt(0(), s(y)) -> true()
    , lt(s(x), s(y)) -> lt(x, y)
    , 10() -> double(s(double(s(s(0())))))
    , double(0()) -> 0()
    , double(s(x)) -> s(s(double(x))) }

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

Strict DPs:
  { 1024_1^#(x) -> c_2(if^#(lt(x, 10()), x), lt^#(x, 10()), 10^#())
  , if^#(true(), x) -> c_3(double^#(1024_1(s(x))), 1024_1^#(s(x)))
  , lt^#(s(x), s(y)) -> c_7(lt^#(x, y))
  , 10^#() ->
    c_8(double^#(s(double(s(s(0()))))), double^#(s(s(0()))))
  , double^#(s(x)) -> c_10(double^#(x)) }
Weak Trs:
  { 1024_1(x) -> if(lt(x, 10()), x)
  , if(true(), x) -> double(1024_1(s(x)))
  , if(false(), x) -> s(0())
  , lt(x, 0()) -> false()
  , lt(0(), s(y)) -> true()
  , lt(s(x), s(y)) -> lt(x, y)
  , 10() -> double(s(double(s(s(0())))))
  , double(0()) -> 0()
  , double(s(x)) -> s(s(double(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..