MAYBE

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

Strict Trs:
  { not(tt()) -> ff()
  , not(ff()) -> tt()
  , or(tt(), x) -> tt()
  , or(ff(), x) -> x
  , eq(0(), 0()) -> tt()
  , eq(0(), s(y)) -> ff()
  , eq(s(x), 0()) -> ff()
  , eq(s(x), s(y)) -> eq(x, y)
  , main(phi) -> ver(phi, nil())
  , ver(Var(x), t()) -> in(x, t())
  , ver(Or(phi, psi), t()) -> or(ver(phi, t()), ver(psi, t()))
  , ver(Not(phi), t()) -> not(ver(phi, t()))
  , ver(Exists(n, phi), t()) ->
    or(ver(phi, cons(n, t())), ver(phi, t()))
  , in(x, nil()) -> ff()
  , in(x, cons(a, l)) -> or(eq(x, a), in(x, l)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { not^#(tt()) -> c_1()
  , not^#(ff()) -> c_2()
  , or^#(tt(), x) -> c_3()
  , or^#(ff(), x) -> c_4()
  , eq^#(0(), 0()) -> c_5()
  , eq^#(0(), s(y)) -> c_6()
  , eq^#(s(x), 0()) -> c_7()
  , eq^#(s(x), s(y)) -> c_8(eq^#(x, y))
  , main^#(phi) -> c_9(ver^#(phi, nil()))
  , ver^#(Var(x), t()) -> c_10(in^#(x, t()))
  , ver^#(Or(phi, psi), t()) ->
    c_11(or^#(ver(phi, t()), ver(psi, t())),
         ver^#(phi, t()),
         ver^#(psi, t()))
  , ver^#(Not(phi), t()) ->
    c_12(not^#(ver(phi, t())), ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) ->
    c_13(or^#(ver(phi, cons(n, t())), ver(phi, t())),
         ver^#(phi, cons(n, t())),
         ver^#(phi, t()))
  , in^#(x, nil()) -> c_14()
  , in^#(x, cons(a, l)) ->
    c_15(or^#(eq(x, a), in(x, l)), eq^#(x, a), in^#(x, l)) }

and mark the set of starting terms.

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

Strict DPs:
  { not^#(tt()) -> c_1()
  , not^#(ff()) -> c_2()
  , or^#(tt(), x) -> c_3()
  , or^#(ff(), x) -> c_4()
  , eq^#(0(), 0()) -> c_5()
  , eq^#(0(), s(y)) -> c_6()
  , eq^#(s(x), 0()) -> c_7()
  , eq^#(s(x), s(y)) -> c_8(eq^#(x, y))
  , main^#(phi) -> c_9(ver^#(phi, nil()))
  , ver^#(Var(x), t()) -> c_10(in^#(x, t()))
  , ver^#(Or(phi, psi), t()) ->
    c_11(or^#(ver(phi, t()), ver(psi, t())),
         ver^#(phi, t()),
         ver^#(psi, t()))
  , ver^#(Not(phi), t()) ->
    c_12(not^#(ver(phi, t())), ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) ->
    c_13(or^#(ver(phi, cons(n, t())), ver(phi, t())),
         ver^#(phi, cons(n, t())),
         ver^#(phi, t()))
  , in^#(x, nil()) -> c_14()
  , in^#(x, cons(a, l)) ->
    c_15(or^#(eq(x, a), in(x, l)), eq^#(x, a), in^#(x, l)) }
Weak Trs:
  { not(tt()) -> ff()
  , not(ff()) -> tt()
  , or(tt(), x) -> tt()
  , or(ff(), x) -> x
  , eq(0(), 0()) -> tt()
  , eq(0(), s(y)) -> ff()
  , eq(s(x), 0()) -> ff()
  , eq(s(x), s(y)) -> eq(x, y)
  , main(phi) -> ver(phi, nil())
  , ver(Var(x), t()) -> in(x, t())
  , ver(Or(phi, psi), t()) -> or(ver(phi, t()), ver(psi, t()))
  , ver(Not(phi), t()) -> not(ver(phi, t()))
  , ver(Exists(n, phi), t()) ->
    or(ver(phi, cons(n, t())), ver(phi, t()))
  , in(x, nil()) -> ff()
  , in(x, cons(a, l)) -> or(eq(x, a), in(x, l)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,4,5,6,7,9,10,14} by
applications of Pre({1,2,3,4,5,6,7,9,10,14}) = {8,11,12,13,15}.
Here rules are labeled as follows:

  DPs:
    { 1: not^#(tt()) -> c_1()
    , 2: not^#(ff()) -> c_2()
    , 3: or^#(tt(), x) -> c_3()
    , 4: or^#(ff(), x) -> c_4()
    , 5: eq^#(0(), 0()) -> c_5()
    , 6: eq^#(0(), s(y)) -> c_6()
    , 7: eq^#(s(x), 0()) -> c_7()
    , 8: eq^#(s(x), s(y)) -> c_8(eq^#(x, y))
    , 9: main^#(phi) -> c_9(ver^#(phi, nil()))
    , 10: ver^#(Var(x), t()) -> c_10(in^#(x, t()))
    , 11: ver^#(Or(phi, psi), t()) ->
          c_11(or^#(ver(phi, t()), ver(psi, t())),
               ver^#(phi, t()),
               ver^#(psi, t()))
    , 12: ver^#(Not(phi), t()) ->
          c_12(not^#(ver(phi, t())), ver^#(phi, t()))
    , 13: ver^#(Exists(n, phi), t()) ->
          c_13(or^#(ver(phi, cons(n, t())), ver(phi, t())),
               ver^#(phi, cons(n, t())),
               ver^#(phi, t()))
    , 14: in^#(x, nil()) -> c_14()
    , 15: in^#(x, cons(a, l)) ->
          c_15(or^#(eq(x, a), in(x, l)), eq^#(x, a), in^#(x, l)) }

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

Strict DPs:
  { eq^#(s(x), s(y)) -> c_8(eq^#(x, y))
  , ver^#(Or(phi, psi), t()) ->
    c_11(or^#(ver(phi, t()), ver(psi, t())),
         ver^#(phi, t()),
         ver^#(psi, t()))
  , ver^#(Not(phi), t()) ->
    c_12(not^#(ver(phi, t())), ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) ->
    c_13(or^#(ver(phi, cons(n, t())), ver(phi, t())),
         ver^#(phi, cons(n, t())),
         ver^#(phi, t()))
  , in^#(x, cons(a, l)) ->
    c_15(or^#(eq(x, a), in(x, l)), eq^#(x, a), in^#(x, l)) }
Weak DPs:
  { not^#(tt()) -> c_1()
  , not^#(ff()) -> c_2()
  , or^#(tt(), x) -> c_3()
  , or^#(ff(), x) -> c_4()
  , eq^#(0(), 0()) -> c_5()
  , eq^#(0(), s(y)) -> c_6()
  , eq^#(s(x), 0()) -> c_7()
  , main^#(phi) -> c_9(ver^#(phi, nil()))
  , ver^#(Var(x), t()) -> c_10(in^#(x, t()))
  , in^#(x, nil()) -> c_14() }
Weak Trs:
  { not(tt()) -> ff()
  , not(ff()) -> tt()
  , or(tt(), x) -> tt()
  , or(ff(), x) -> x
  , eq(0(), 0()) -> tt()
  , eq(0(), s(y)) -> ff()
  , eq(s(x), 0()) -> ff()
  , eq(s(x), s(y)) -> eq(x, y)
  , main(phi) -> ver(phi, nil())
  , ver(Var(x), t()) -> in(x, t())
  , ver(Or(phi, psi), t()) -> or(ver(phi, t()), ver(psi, t()))
  , ver(Not(phi), t()) -> not(ver(phi, t()))
  , ver(Exists(n, phi), t()) ->
    or(ver(phi, cons(n, t())), ver(phi, t()))
  , in(x, nil()) -> ff()
  , in(x, cons(a, l)) -> or(eq(x, a), in(x, l)) }
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.

{ not^#(tt()) -> c_1()
, not^#(ff()) -> c_2()
, or^#(tt(), x) -> c_3()
, or^#(ff(), x) -> c_4()
, eq^#(0(), 0()) -> c_5()
, eq^#(0(), s(y)) -> c_6()
, eq^#(s(x), 0()) -> c_7()
, main^#(phi) -> c_9(ver^#(phi, nil()))
, ver^#(Var(x), t()) -> c_10(in^#(x, t()))
, in^#(x, nil()) -> c_14() }

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

Strict DPs:
  { eq^#(s(x), s(y)) -> c_8(eq^#(x, y))
  , ver^#(Or(phi, psi), t()) ->
    c_11(or^#(ver(phi, t()), ver(psi, t())),
         ver^#(phi, t()),
         ver^#(psi, t()))
  , ver^#(Not(phi), t()) ->
    c_12(not^#(ver(phi, t())), ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) ->
    c_13(or^#(ver(phi, cons(n, t())), ver(phi, t())),
         ver^#(phi, cons(n, t())),
         ver^#(phi, t()))
  , in^#(x, cons(a, l)) ->
    c_15(or^#(eq(x, a), in(x, l)), eq^#(x, a), in^#(x, l)) }
Weak Trs:
  { not(tt()) -> ff()
  , not(ff()) -> tt()
  , or(tt(), x) -> tt()
  , or(ff(), x) -> x
  , eq(0(), 0()) -> tt()
  , eq(0(), s(y)) -> ff()
  , eq(s(x), 0()) -> ff()
  , eq(s(x), s(y)) -> eq(x, y)
  , main(phi) -> ver(phi, nil())
  , ver(Var(x), t()) -> in(x, t())
  , ver(Or(phi, psi), t()) -> or(ver(phi, t()), ver(psi, t()))
  , ver(Not(phi), t()) -> not(ver(phi, t()))
  , ver(Exists(n, phi), t()) ->
    or(ver(phi, cons(n, t())), ver(phi, t()))
  , in(x, nil()) -> ff()
  , in(x, cons(a, l)) -> or(eq(x, a), in(x, l)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { ver^#(Or(phi, psi), t()) ->
    c_11(or^#(ver(phi, t()), ver(psi, t())),
         ver^#(phi, t()),
         ver^#(psi, t()))
  , ver^#(Not(phi), t()) ->
    c_12(not^#(ver(phi, t())), ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) ->
    c_13(or^#(ver(phi, cons(n, t())), ver(phi, t())),
         ver^#(phi, cons(n, t())),
         ver^#(phi, t()))
  , in^#(x, cons(a, l)) ->
    c_15(or^#(eq(x, a), in(x, l)), eq^#(x, a), in^#(x, l)) }

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))
  , ver^#(Or(phi, psi), t()) -> c_2(ver^#(phi, t()), ver^#(psi, t()))
  , ver^#(Not(phi), t()) -> c_3(ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) -> c_4(ver^#(phi, t()))
  , in^#(x, cons(a, l)) -> c_5(eq^#(x, a), in^#(x, l)) }
Weak Trs:
  { not(tt()) -> ff()
  , not(ff()) -> tt()
  , or(tt(), x) -> tt()
  , or(ff(), x) -> x
  , eq(0(), 0()) -> tt()
  , eq(0(), s(y)) -> ff()
  , eq(s(x), 0()) -> ff()
  , eq(s(x), s(y)) -> eq(x, y)
  , main(phi) -> ver(phi, nil())
  , ver(Var(x), t()) -> in(x, t())
  , ver(Or(phi, psi), t()) -> or(ver(phi, t()), ver(psi, t()))
  , ver(Not(phi), t()) -> not(ver(phi, t()))
  , ver(Exists(n, phi), t()) ->
    or(ver(phi, cons(n, t())), ver(phi, t()))
  , in(x, nil()) -> ff()
  , in(x, cons(a, l)) -> or(eq(x, a), in(x, l)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

No rule is usable, rules are removed from the input problem.

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))
  , ver^#(Or(phi, psi), t()) -> c_2(ver^#(phi, t()), ver^#(psi, t()))
  , ver^#(Not(phi), t()) -> c_3(ver^#(phi, t()))
  , ver^#(Exists(n, phi), t()) -> c_4(ver^#(phi, t()))
  , in^#(x, cons(a, l)) -> c_5(eq^#(x, a), in^#(x, l)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..