MAYBE

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

Strict Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , or(true(), y) -> true()
  , or(false(), y) -> y
  , union(empty(), h) -> h
  , union(edge(x, y, i), h) -> edge(x, y, union(i, h))
  , reach(x, y, empty(), h) -> false()
  , reach(x, y, edge(u, v, i), h) ->
    if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
  , if_reach_1(true(), x, y, edge(u, v, i), h) ->
    if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
  , if_reach_1(false(), x, y, edge(u, v, i), h) ->
    reach(x, y, i, edge(u, v, h))
  , if_reach_2(true(), x, y, edge(u, v, i), h) -> true()
  , if_reach_2(false(), x, y, edge(u, v, i), h) ->
    or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(x)) -> c_2()
  , eq^#(s(x), 0()) -> c_3()
  , eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , or^#(true(), y) -> c_5()
  , or^#(false(), y) -> c_6()
  , union^#(empty(), h) -> c_7()
  , union^#(edge(x, y, i), h) -> c_8(union^#(i, h))
  , reach^#(x, y, empty(), h) -> c_9()
  , reach^#(x, y, edge(u, v, i), h) ->
    c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
  , if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
    c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
  , if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
    c_12(reach^#(x, y, i, edge(u, v, h)))
  , if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13()
  , if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty())),
         reach^#(x, y, i, h),
         reach^#(v, y, union(i, h), empty()),
         union^#(i, h)) }

and mark the set of starting terms.

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

Strict DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(x)) -> c_2()
  , eq^#(s(x), 0()) -> c_3()
  , eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , or^#(true(), y) -> c_5()
  , or^#(false(), y) -> c_6()
  , union^#(empty(), h) -> c_7()
  , union^#(edge(x, y, i), h) -> c_8(union^#(i, h))
  , reach^#(x, y, empty(), h) -> c_9()
  , reach^#(x, y, edge(u, v, i), h) ->
    c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
  , if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
    c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
  , if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
    c_12(reach^#(x, y, i, edge(u, v, h)))
  , if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13()
  , if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty())),
         reach^#(x, y, i, h),
         reach^#(v, y, union(i, h), empty()),
         union^#(i, h)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , or(true(), y) -> true()
  , or(false(), y) -> y
  , union(empty(), h) -> h
  , union(edge(x, y, i), h) -> edge(x, y, union(i, h))
  , reach(x, y, empty(), h) -> false()
  , reach(x, y, edge(u, v, i), h) ->
    if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
  , if_reach_1(true(), x, y, edge(u, v, i), h) ->
    if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
  , if_reach_1(false(), x, y, edge(u, v, i), h) ->
    reach(x, y, i, edge(u, v, h))
  , if_reach_2(true(), x, y, edge(u, v, i), h) -> true()
  , if_reach_2(false(), x, y, edge(u, v, i), h) ->
    or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,5,6,7,9,13} by
applications of Pre({1,2,3,5,6,7,9,13}) = {4,8,10,11,12,14}. Here
rules are labeled as follows:

  DPs:
    { 1: eq^#(0(), 0()) -> c_1()
    , 2: eq^#(0(), s(x)) -> c_2()
    , 3: eq^#(s(x), 0()) -> c_3()
    , 4: eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
    , 5: or^#(true(), y) -> c_5()
    , 6: or^#(false(), y) -> c_6()
    , 7: union^#(empty(), h) -> c_7()
    , 8: union^#(edge(x, y, i), h) -> c_8(union^#(i, h))
    , 9: reach^#(x, y, empty(), h) -> c_9()
    , 10: reach^#(x, y, edge(u, v, i), h) ->
          c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
    , 11: if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
          c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
    , 12: if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
          c_12(reach^#(x, y, i, edge(u, v, h)))
    , 13: if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13()
    , 14: if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
          c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty())),
               reach^#(x, y, i, h),
               reach^#(v, y, union(i, h), empty()),
               union^#(i, h)) }

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

Strict DPs:
  { eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , union^#(edge(x, y, i), h) -> c_8(union^#(i, h))
  , reach^#(x, y, edge(u, v, i), h) ->
    c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
  , if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
    c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
  , if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
    c_12(reach^#(x, y, i, edge(u, v, h)))
  , if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty())),
         reach^#(x, y, i, h),
         reach^#(v, y, union(i, h), empty()),
         union^#(i, h)) }
Weak DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(x)) -> c_2()
  , eq^#(s(x), 0()) -> c_3()
  , or^#(true(), y) -> c_5()
  , or^#(false(), y) -> c_6()
  , union^#(empty(), h) -> c_7()
  , reach^#(x, y, empty(), h) -> c_9()
  , if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13() }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , or(true(), y) -> true()
  , or(false(), y) -> y
  , union(empty(), h) -> h
  , union(edge(x, y, i), h) -> edge(x, y, union(i, h))
  , reach(x, y, empty(), h) -> false()
  , reach(x, y, edge(u, v, i), h) ->
    if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
  , if_reach_1(true(), x, y, edge(u, v, i), h) ->
    if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
  , if_reach_1(false(), x, y, edge(u, v, i), h) ->
    reach(x, y, i, edge(u, v, h))
  , if_reach_2(true(), x, y, edge(u, v, i), h) -> true()
  , if_reach_2(false(), x, y, edge(u, v, i), h) ->
    or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) }
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.

{ eq^#(0(), 0()) -> c_1()
, eq^#(0(), s(x)) -> c_2()
, eq^#(s(x), 0()) -> c_3()
, or^#(true(), y) -> c_5()
, or^#(false(), y) -> c_6()
, union^#(empty(), h) -> c_7()
, reach^#(x, y, empty(), h) -> c_9()
, if_reach_2^#(true(), x, y, edge(u, v, i), h) -> c_13() }

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

Strict DPs:
  { eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , union^#(edge(x, y, i), h) -> c_8(union^#(i, h))
  , reach^#(x, y, edge(u, v, i), h) ->
    c_10(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
  , if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
    c_11(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
  , if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
    c_12(reach^#(x, y, i, edge(u, v, h)))
  , if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty())),
         reach^#(x, y, i, h),
         reach^#(v, y, union(i, h), empty()),
         union^#(i, h)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , or(true(), y) -> true()
  , or(false(), y) -> y
  , union(empty(), h) -> h
  , union(edge(x, y, i), h) -> edge(x, y, union(i, h))
  , reach(x, y, empty(), h) -> false()
  , reach(x, y, edge(u, v, i), h) ->
    if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
  , if_reach_1(true(), x, y, edge(u, v, i), h) ->
    if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
  , if_reach_1(false(), x, y, edge(u, v, i), h) ->
    reach(x, y, i, edge(u, v, h))
  , if_reach_2(true(), x, y, edge(u, v, i), h) -> true()
  , if_reach_2(false(), x, y, edge(u, v, i), h) ->
    or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_14(or^#(reach(x, y, i, h), reach(v, y, union(i, h), empty())),
         reach^#(x, y, i, h),
         reach^#(v, y, union(i, h), empty()),
         union^#(i, h)) }

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

Strict DPs:
  { eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
  , union^#(edge(x, y, i), h) -> c_2(union^#(i, h))
  , reach^#(x, y, edge(u, v, i), h) ->
    c_3(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
  , if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
    c_4(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
  , if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
    c_5(reach^#(x, y, i, edge(u, v, h)))
  , if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_6(reach^#(x, y, i, h),
        reach^#(v, y, union(i, h), empty()),
        union^#(i, h)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , or(true(), y) -> true()
  , or(false(), y) -> y
  , union(empty(), h) -> h
  , union(edge(x, y, i), h) -> edge(x, y, union(i, h))
  , reach(x, y, empty(), h) -> false()
  , reach(x, y, edge(u, v, i), h) ->
    if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
  , if_reach_1(true(), x, y, edge(u, v, i), h) ->
    if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
  , if_reach_1(false(), x, y, edge(u, v, i), h) ->
    reach(x, y, i, edge(u, v, h))
  , if_reach_2(true(), x, y, edge(u, v, i), h) -> true()
  , if_reach_2(false(), x, y, edge(u, v, i), h) ->
    or(reach(x, y, i, h), reach(v, y, union(i, h), empty())) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { eq(0(), 0()) -> true()
    , eq(0(), s(x)) -> false()
    , eq(s(x), 0()) -> false()
    , eq(s(x), s(y)) -> eq(x, y)
    , union(empty(), h) -> h
    , union(edge(x, y, i), h) -> edge(x, y, union(i, h)) }

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

Strict DPs:
  { eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
  , union^#(edge(x, y, i), h) -> c_2(union^#(i, h))
  , reach^#(x, y, edge(u, v, i), h) ->
    c_3(if_reach_1^#(eq(x, u), x, y, edge(u, v, i), h), eq^#(x, u))
  , if_reach_1^#(true(), x, y, edge(u, v, i), h) ->
    c_4(if_reach_2^#(eq(y, v), x, y, edge(u, v, i), h), eq^#(y, v))
  , if_reach_1^#(false(), x, y, edge(u, v, i), h) ->
    c_5(reach^#(x, y, i, edge(u, v, h)))
  , if_reach_2^#(false(), x, y, edge(u, v, i), h) ->
    c_6(reach^#(x, y, i, h),
        reach^#(v, y, union(i, h), empty()),
        union^#(i, h)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , union(empty(), h) -> h
  , union(edge(x, y, i), h) -> edge(x, y, union(i, h)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..