MAYBE

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

Strict Trs:
  { isLeaf(leaf()) -> true()
  , isLeaf(cons(u, v)) -> false()
  , left(cons(u, v)) -> u
  , right(cons(u, v)) -> v
  , concat(leaf(), y) -> y
  , concat(cons(u, v), y) -> cons(u, concat(v, y))
  , less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
  , if1(b, true(), u, v) -> false()
  , if1(b, false(), u, v) -> if2(b, u, v)
  , if2(true(), u, v) -> true()
  , if2(false(), u, v) ->
    less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { isLeaf^#(leaf()) -> c_1()
  , isLeaf^#(cons(u, v)) -> c_2()
  , left^#(cons(u, v)) -> c_3()
  , right^#(cons(u, v)) -> c_4()
  , concat^#(leaf(), y) -> c_5()
  , concat^#(cons(u, v), y) -> c_6(concat^#(v, y))
  , less_leaves^#(u, v) ->
    c_7(if1^#(isLeaf(u), isLeaf(v), u, v), isLeaf^#(u), isLeaf^#(v))
  , if1^#(b, true(), u, v) -> c_8()
  , if1^#(b, false(), u, v) -> c_9(if2^#(b, u, v))
  , if2^#(true(), u, v) -> c_10()
  , if2^#(false(), u, v) ->
    c_11(less_leaves^#(concat(left(u), right(u)),
                       concat(left(v), right(v))),
         concat^#(left(u), right(u)),
         left^#(u),
         right^#(u),
         concat^#(left(v), right(v)),
         left^#(v),
         right^#(v)) }

and mark the set of starting terms.

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

Strict DPs:
  { isLeaf^#(leaf()) -> c_1()
  , isLeaf^#(cons(u, v)) -> c_2()
  , left^#(cons(u, v)) -> c_3()
  , right^#(cons(u, v)) -> c_4()
  , concat^#(leaf(), y) -> c_5()
  , concat^#(cons(u, v), y) -> c_6(concat^#(v, y))
  , less_leaves^#(u, v) ->
    c_7(if1^#(isLeaf(u), isLeaf(v), u, v), isLeaf^#(u), isLeaf^#(v))
  , if1^#(b, true(), u, v) -> c_8()
  , if1^#(b, false(), u, v) -> c_9(if2^#(b, u, v))
  , if2^#(true(), u, v) -> c_10()
  , if2^#(false(), u, v) ->
    c_11(less_leaves^#(concat(left(u), right(u)),
                       concat(left(v), right(v))),
         concat^#(left(u), right(u)),
         left^#(u),
         right^#(u),
         concat^#(left(v), right(v)),
         left^#(v),
         right^#(v)) }
Weak Trs:
  { isLeaf(leaf()) -> true()
  , isLeaf(cons(u, v)) -> false()
  , left(cons(u, v)) -> u
  , right(cons(u, v)) -> v
  , concat(leaf(), y) -> y
  , concat(cons(u, v), y) -> cons(u, concat(v, y))
  , less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
  , if1(b, true(), u, v) -> false()
  , if1(b, false(), u, v) -> if2(b, u, v)
  , if2(true(), u, v) -> true()
  , if2(false(), u, v) ->
    less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,4,5,8,10} by
applications of Pre({1,2,3,4,5,8,10}) = {6,7,9,11}. Here rules are
labeled as follows:

  DPs:
    { 1: isLeaf^#(leaf()) -> c_1()
    , 2: isLeaf^#(cons(u, v)) -> c_2()
    , 3: left^#(cons(u, v)) -> c_3()
    , 4: right^#(cons(u, v)) -> c_4()
    , 5: concat^#(leaf(), y) -> c_5()
    , 6: concat^#(cons(u, v), y) -> c_6(concat^#(v, y))
    , 7: less_leaves^#(u, v) ->
         c_7(if1^#(isLeaf(u), isLeaf(v), u, v), isLeaf^#(u), isLeaf^#(v))
    , 8: if1^#(b, true(), u, v) -> c_8()
    , 9: if1^#(b, false(), u, v) -> c_9(if2^#(b, u, v))
    , 10: if2^#(true(), u, v) -> c_10()
    , 11: if2^#(false(), u, v) ->
          c_11(less_leaves^#(concat(left(u), right(u)),
                             concat(left(v), right(v))),
               concat^#(left(u), right(u)),
               left^#(u),
               right^#(u),
               concat^#(left(v), right(v)),
               left^#(v),
               right^#(v)) }

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

Strict DPs:
  { concat^#(cons(u, v), y) -> c_6(concat^#(v, y))
  , less_leaves^#(u, v) ->
    c_7(if1^#(isLeaf(u), isLeaf(v), u, v), isLeaf^#(u), isLeaf^#(v))
  , if1^#(b, false(), u, v) -> c_9(if2^#(b, u, v))
  , if2^#(false(), u, v) ->
    c_11(less_leaves^#(concat(left(u), right(u)),
                       concat(left(v), right(v))),
         concat^#(left(u), right(u)),
         left^#(u),
         right^#(u),
         concat^#(left(v), right(v)),
         left^#(v),
         right^#(v)) }
Weak DPs:
  { isLeaf^#(leaf()) -> c_1()
  , isLeaf^#(cons(u, v)) -> c_2()
  , left^#(cons(u, v)) -> c_3()
  , right^#(cons(u, v)) -> c_4()
  , concat^#(leaf(), y) -> c_5()
  , if1^#(b, true(), u, v) -> c_8()
  , if2^#(true(), u, v) -> c_10() }
Weak Trs:
  { isLeaf(leaf()) -> true()
  , isLeaf(cons(u, v)) -> false()
  , left(cons(u, v)) -> u
  , right(cons(u, v)) -> v
  , concat(leaf(), y) -> y
  , concat(cons(u, v), y) -> cons(u, concat(v, y))
  , less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
  , if1(b, true(), u, v) -> false()
  , if1(b, false(), u, v) -> if2(b, u, v)
  , if2(true(), u, v) -> true()
  , if2(false(), u, v) ->
    less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) }
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.

{ isLeaf^#(leaf()) -> c_1()
, isLeaf^#(cons(u, v)) -> c_2()
, left^#(cons(u, v)) -> c_3()
, right^#(cons(u, v)) -> c_4()
, concat^#(leaf(), y) -> c_5()
, if1^#(b, true(), u, v) -> c_8()
, if2^#(true(), u, v) -> c_10() }

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

Strict DPs:
  { concat^#(cons(u, v), y) -> c_6(concat^#(v, y))
  , less_leaves^#(u, v) ->
    c_7(if1^#(isLeaf(u), isLeaf(v), u, v), isLeaf^#(u), isLeaf^#(v))
  , if1^#(b, false(), u, v) -> c_9(if2^#(b, u, v))
  , if2^#(false(), u, v) ->
    c_11(less_leaves^#(concat(left(u), right(u)),
                       concat(left(v), right(v))),
         concat^#(left(u), right(u)),
         left^#(u),
         right^#(u),
         concat^#(left(v), right(v)),
         left^#(v),
         right^#(v)) }
Weak Trs:
  { isLeaf(leaf()) -> true()
  , isLeaf(cons(u, v)) -> false()
  , left(cons(u, v)) -> u
  , right(cons(u, v)) -> v
  , concat(leaf(), y) -> y
  , concat(cons(u, v), y) -> cons(u, concat(v, y))
  , less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
  , if1(b, true(), u, v) -> false()
  , if1(b, false(), u, v) -> if2(b, u, v)
  , if2(true(), u, v) -> true()
  , if2(false(), u, v) ->
    less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { less_leaves^#(u, v) ->
    c_7(if1^#(isLeaf(u), isLeaf(v), u, v), isLeaf^#(u), isLeaf^#(v))
  , if2^#(false(), u, v) ->
    c_11(less_leaves^#(concat(left(u), right(u)),
                       concat(left(v), right(v))),
         concat^#(left(u), right(u)),
         left^#(u),
         right^#(u),
         concat^#(left(v), right(v)),
         left^#(v),
         right^#(v)) }

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

Strict DPs:
  { concat^#(cons(u, v), y) -> c_1(concat^#(v, y))
  , less_leaves^#(u, v) -> c_2(if1^#(isLeaf(u), isLeaf(v), u, v))
  , if1^#(b, false(), u, v) -> c_3(if2^#(b, u, v))
  , if2^#(false(), u, v) ->
    c_4(less_leaves^#(concat(left(u), right(u)),
                      concat(left(v), right(v))),
        concat^#(left(u), right(u)),
        concat^#(left(v), right(v))) }
Weak Trs:
  { isLeaf(leaf()) -> true()
  , isLeaf(cons(u, v)) -> false()
  , left(cons(u, v)) -> u
  , right(cons(u, v)) -> v
  , concat(leaf(), y) -> y
  , concat(cons(u, v), y) -> cons(u, concat(v, y))
  , less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
  , if1(b, true(), u, v) -> false()
  , if1(b, false(), u, v) -> if2(b, u, v)
  , if2(true(), u, v) -> true()
  , if2(false(), u, v) ->
    less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { isLeaf(leaf()) -> true()
    , isLeaf(cons(u, v)) -> false()
    , left(cons(u, v)) -> u
    , right(cons(u, v)) -> v
    , concat(leaf(), y) -> y
    , concat(cons(u, v), y) -> cons(u, concat(v, y)) }

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

Strict DPs:
  { concat^#(cons(u, v), y) -> c_1(concat^#(v, y))
  , less_leaves^#(u, v) -> c_2(if1^#(isLeaf(u), isLeaf(v), u, v))
  , if1^#(b, false(), u, v) -> c_3(if2^#(b, u, v))
  , if2^#(false(), u, v) ->
    c_4(less_leaves^#(concat(left(u), right(u)),
                      concat(left(v), right(v))),
        concat^#(left(u), right(u)),
        concat^#(left(v), right(v))) }
Weak Trs:
  { isLeaf(leaf()) -> true()
  , isLeaf(cons(u, v)) -> false()
  , left(cons(u, v)) -> u
  , right(cons(u, v)) -> v
  , concat(leaf(), y) -> y
  , concat(cons(u, v), y) -> cons(u, concat(v, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..