MAYBE

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

Strict Trs:
  { add(true(), x, xs) ->
    add(and(isNat(x), isList(xs)), x, Cons(x, xs))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> true()
  , isList(Cons(x, xs)) -> isList(xs)
  , isList(nil()) -> true()
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        and^#(isNat(x), isList(xs)),
        isNat^#(x),
        isList^#(xs))
  , and^#(x, false()) -> c_2()
  , and^#(true(), true()) -> c_3()
  , and^#(false(), x) -> c_4()
  , isNat^#(s(x)) -> c_5(isNat^#(x))
  , isNat^#(0()) -> c_6()
  , isList^#(Cons(x, xs)) -> c_7(isList^#(xs))
  , isList^#(nil()) -> c_8()
  , if^#(true(), x, y) -> c_9()
  , if^#(false(), x, y) -> c_10() }

and mark the set of starting terms.

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

Strict DPs:
  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        and^#(isNat(x), isList(xs)),
        isNat^#(x),
        isList^#(xs))
  , and^#(x, false()) -> c_2()
  , and^#(true(), true()) -> c_3()
  , and^#(false(), x) -> c_4()
  , isNat^#(s(x)) -> c_5(isNat^#(x))
  , isNat^#(0()) -> c_6()
  , isList^#(Cons(x, xs)) -> c_7(isList^#(xs))
  , isList^#(nil()) -> c_8()
  , if^#(true(), x, y) -> c_9()
  , if^#(false(), x, y) -> c_10() }
Weak Trs:
  { add(true(), x, xs) ->
    add(and(isNat(x), isList(xs)), x, Cons(x, xs))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> true()
  , isList(Cons(x, xs)) -> isList(xs)
  , isList(nil()) -> true()
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2,3,4,6,8,9,10} by
applications of Pre({2,3,4,6,8,9,10}) = {1,5,7}. Here rules are
labeled as follows:

  DPs:
    { 1: add^#(true(), x, xs) ->
         c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
             and^#(isNat(x), isList(xs)),
             isNat^#(x),
             isList^#(xs))
    , 2: and^#(x, false()) -> c_2()
    , 3: and^#(true(), true()) -> c_3()
    , 4: and^#(false(), x) -> c_4()
    , 5: isNat^#(s(x)) -> c_5(isNat^#(x))
    , 6: isNat^#(0()) -> c_6()
    , 7: isList^#(Cons(x, xs)) -> c_7(isList^#(xs))
    , 8: isList^#(nil()) -> c_8()
    , 9: if^#(true(), x, y) -> c_9()
    , 10: if^#(false(), x, y) -> c_10() }

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

Strict DPs:
  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        and^#(isNat(x), isList(xs)),
        isNat^#(x),
        isList^#(xs))
  , isNat^#(s(x)) -> c_5(isNat^#(x))
  , isList^#(Cons(x, xs)) -> c_7(isList^#(xs)) }
Weak DPs:
  { and^#(x, false()) -> c_2()
  , and^#(true(), true()) -> c_3()
  , and^#(false(), x) -> c_4()
  , isNat^#(0()) -> c_6()
  , isList^#(nil()) -> c_8()
  , if^#(true(), x, y) -> c_9()
  , if^#(false(), x, y) -> c_10() }
Weak Trs:
  { add(true(), x, xs) ->
    add(and(isNat(x), isList(xs)), x, Cons(x, xs))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> true()
  , isList(Cons(x, xs)) -> isList(xs)
  , isList(nil()) -> true()
  , if(true(), x, y) -> x
  , if(false(), x, y) -> 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.

{ and^#(x, false()) -> c_2()
, and^#(true(), true()) -> c_3()
, and^#(false(), x) -> c_4()
, isNat^#(0()) -> c_6()
, isList^#(nil()) -> c_8()
, if^#(true(), x, y) -> c_9()
, if^#(false(), x, y) -> c_10() }

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

Strict DPs:
  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        and^#(isNat(x), isList(xs)),
        isNat^#(x),
        isList^#(xs))
  , isNat^#(s(x)) -> c_5(isNat^#(x))
  , isList^#(Cons(x, xs)) -> c_7(isList^#(xs)) }
Weak Trs:
  { add(true(), x, xs) ->
    add(and(isNat(x), isList(xs)), x, Cons(x, xs))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> true()
  , isList(Cons(x, xs)) -> isList(xs)
  , isList(nil()) -> true()
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        and^#(isNat(x), isList(xs)),
        isNat^#(x),
        isList^#(xs)) }

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

Strict DPs:
  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        isNat^#(x),
        isList^#(xs))
  , isNat^#(s(x)) -> c_2(isNat^#(x))
  , isList^#(Cons(x, xs)) -> c_3(isList^#(xs)) }
Weak Trs:
  { add(true(), x, xs) ->
    add(and(isNat(x), isList(xs)), x, Cons(x, xs))
  , and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> true()
  , isList(Cons(x, xs)) -> isList(xs)
  , isList(nil()) -> true()
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { and(x, false()) -> false()
    , and(true(), true()) -> true()
    , and(false(), x) -> false()
    , isNat(s(x)) -> isNat(x)
    , isNat(0()) -> true()
    , isList(Cons(x, xs)) -> isList(xs)
    , isList(nil()) -> true() }

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

Strict DPs:
  { add^#(true(), x, xs) ->
    c_1(add^#(and(isNat(x), isList(xs)), x, Cons(x, xs)),
        isNat^#(x),
        isList^#(xs))
  , isNat^#(s(x)) -> c_2(isNat^#(x))
  , isList^#(Cons(x, xs)) -> c_3(isList^#(xs)) }
Weak Trs:
  { and(x, false()) -> false()
  , and(true(), true()) -> true()
  , and(false(), x) -> false()
  , isNat(s(x)) -> isNat(x)
  , isNat(0()) -> true()
  , isList(Cons(x, xs)) -> isList(xs)
  , isList(nil()) -> true() }
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..