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)
  , rm(N, nil()) -> nil()
  , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
  , ifrm(true(), N, add(M, X)) -> rm(N, X)
  , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
  , purge(nil()) -> nil()
  , purge(add(N, X)) -> add(N, purge(rm(N, X))) }
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))
  , rm^#(N, nil()) -> c_5()
  , rm^#(N, add(M, X)) ->
    c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
  , ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
  , ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
  , purge^#(nil()) -> c_9()
  , purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }

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))
  , rm^#(N, nil()) -> c_5()
  , rm^#(N, add(M, X)) ->
    c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
  , ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
  , ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
  , purge^#(nil()) -> c_9()
  , purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(X)) -> false()
  , eq(s(X), 0()) -> false()
  , eq(s(X), s(Y)) -> eq(X, Y)
  , rm(N, nil()) -> nil()
  , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
  , ifrm(true(), N, add(M, X)) -> rm(N, X)
  , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
  , purge(nil()) -> nil()
  , purge(add(N, X)) -> add(N, purge(rm(N, X))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,5,9} by
applications of Pre({1,2,3,5,9}) = {4,6,7,8,10}. 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: rm^#(N, nil()) -> c_5()
    , 6: rm^#(N, add(M, X)) ->
         c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
    , 7: ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
    , 8: ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
    , 9: purge^#(nil()) -> c_9()
    , 10: purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }

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))
  , rm^#(N, add(M, X)) ->
    c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
  , ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
  , ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
  , purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(X)) -> c_2()
  , eq^#(s(X), 0()) -> c_3()
  , rm^#(N, nil()) -> c_5()
  , purge^#(nil()) -> c_9() }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(X)) -> false()
  , eq(s(X), 0()) -> false()
  , eq(s(X), s(Y)) -> eq(X, Y)
  , rm(N, nil()) -> nil()
  , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
  , ifrm(true(), N, add(M, X)) -> rm(N, X)
  , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
  , purge(nil()) -> nil()
  , purge(add(N, X)) -> add(N, purge(rm(N, 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.

{ eq^#(0(), 0()) -> c_1()
, eq^#(0(), s(X)) -> c_2()
, eq^#(s(X), 0()) -> c_3()
, rm^#(N, nil()) -> c_5()
, purge^#(nil()) -> c_9() }

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))
  , rm^#(N, add(M, X)) ->
    c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
  , ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
  , ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
  , purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(X)) -> false()
  , eq(s(X), 0()) -> false()
  , eq(s(X), s(Y)) -> eq(X, Y)
  , rm(N, nil()) -> nil()
  , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
  , ifrm(true(), N, add(M, X)) -> rm(N, X)
  , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X))
  , purge(nil()) -> nil()
  , purge(add(N, X)) -> add(N, purge(rm(N, X))) }
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)
    , rm(N, nil()) -> nil()
    , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
    , ifrm(true(), N, add(M, X)) -> rm(N, X)
    , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }

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))
  , rm^#(N, add(M, X)) ->
    c_6(ifrm^#(eq(N, M), N, add(M, X)), eq^#(N, M))
  , ifrm^#(true(), N, add(M, X)) -> c_7(rm^#(N, X))
  , ifrm^#(false(), N, add(M, X)) -> c_8(rm^#(N, X))
  , purge^#(add(N, X)) -> c_10(purge^#(rm(N, X)), rm^#(N, X)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(X)) -> false()
  , eq(s(X), 0()) -> false()
  , eq(s(X), s(Y)) -> eq(X, Y)
  , rm(N, nil()) -> nil()
  , rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
  , ifrm(true(), N, add(M, X)) -> rm(N, X)
  , ifrm(false(), N, add(M, X)) -> add(M, rm(N, X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..