MAYBE

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

Strict Trs:
  { and(false(), false()) -> false()
  , and(false(), true()) -> false()
  , and(true(), false()) -> false()
  , and(true(), true()) -> true()
  , eq(nil(), nil()) -> true()
  , eq(nil(), cons(T, L)) -> false()
  , eq(cons(T, L), nil()) -> false()
  , eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
  , eq(var(L), var(Lp)) -> eq(L, Lp)
  , eq(var(L), apply(T, S)) -> false()
  , eq(var(L), lambda(X, T)) -> false()
  , eq(apply(T, S), var(L)) -> false()
  , eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
  , eq(apply(T, S), lambda(X, Tp)) -> false()
  , eq(lambda(X, T), var(L)) -> false()
  , eq(lambda(X, T), apply(Tp, Sp)) -> false()
  , eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
  , if(false(), var(K), var(L)) -> var(L)
  , if(true(), var(K), var(L)) -> var(K)
  , ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
  , ren(X, Y, lambda(Z, T)) ->
    lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
           ren(X,
               Y,
               ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
  , ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) }
Obligation:
  runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
   following reason:
   
   Computation stopped due to timeout after 60.0 seconds.

2) 'Best' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)'
      failed due to the following reason:
      
      Computation stopped due to timeout after 30.0 seconds.
   
   2) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed
      due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Bounds with minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   
   3) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due
         to the following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
      2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the
         following reason:
         
         The processor is inapplicable, reason:
           Processor only applicable for innermost runtime complexity analysis
      
   

3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { and^#(false(), false()) -> c_1()
     , and^#(false(), true()) -> c_2()
     , and^#(true(), false()) -> c_3()
     , and^#(true(), true()) -> c_4()
     , eq^#(nil(), nil()) -> c_5()
     , eq^#(nil(), cons(T, L)) -> c_6()
     , eq^#(cons(T, L), nil()) -> c_7()
     , eq^#(cons(T, L), cons(Tp, Lp)) ->
       c_8(and^#(eq(T, Tp), eq(L, Lp)))
     , eq^#(var(L), var(Lp)) -> c_9(eq^#(L, Lp))
     , eq^#(var(L), apply(T, S)) -> c_10()
     , eq^#(var(L), lambda(X, T)) -> c_11()
     , eq^#(apply(T, S), var(L)) -> c_12()
     , eq^#(apply(T, S), apply(Tp, Sp)) ->
       c_13(and^#(eq(T, Tp), eq(S, Sp)))
     , eq^#(apply(T, S), lambda(X, Tp)) -> c_14()
     , eq^#(lambda(X, T), var(L)) -> c_15()
     , eq^#(lambda(X, T), apply(Tp, Sp)) -> c_16()
     , eq^#(lambda(X, T), lambda(Xp, Tp)) ->
       c_17(and^#(eq(T, Tp), eq(X, Xp)))
     , if^#(false(), var(K), var(L)) -> c_18(L)
     , if^#(true(), var(K), var(L)) -> c_19(K)
     , ren^#(X, Y, apply(T, S)) -> c_20(ren^#(X, Y, T), ren^#(X, Y, S))
     , ren^#(X, Y, lambda(Z, T)) ->
       c_21(X,
            Y,
            Z,
            T,
            ren^#(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren^#(var(L), var(K), var(Lp)) ->
       c_22(if^#(eq(L, Lp), var(K), var(Lp))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { and^#(false(), false()) -> c_1()
     , and^#(false(), true()) -> c_2()
     , and^#(true(), false()) -> c_3()
     , and^#(true(), true()) -> c_4()
     , eq^#(nil(), nil()) -> c_5()
     , eq^#(nil(), cons(T, L)) -> c_6()
     , eq^#(cons(T, L), nil()) -> c_7()
     , eq^#(cons(T, L), cons(Tp, Lp)) ->
       c_8(and^#(eq(T, Tp), eq(L, Lp)))
     , eq^#(var(L), var(Lp)) -> c_9(eq^#(L, Lp))
     , eq^#(var(L), apply(T, S)) -> c_10()
     , eq^#(var(L), lambda(X, T)) -> c_11()
     , eq^#(apply(T, S), var(L)) -> c_12()
     , eq^#(apply(T, S), apply(Tp, Sp)) ->
       c_13(and^#(eq(T, Tp), eq(S, Sp)))
     , eq^#(apply(T, S), lambda(X, Tp)) -> c_14()
     , eq^#(lambda(X, T), var(L)) -> c_15()
     , eq^#(lambda(X, T), apply(Tp, Sp)) -> c_16()
     , eq^#(lambda(X, T), lambda(Xp, Tp)) ->
       c_17(and^#(eq(T, Tp), eq(X, Xp)))
     , if^#(false(), var(K), var(L)) -> c_18(L)
     , if^#(true(), var(K), var(L)) -> c_19(K)
     , ren^#(X, Y, apply(T, S)) -> c_20(ren^#(X, Y, T), ren^#(X, Y, S))
     , ren^#(X, Y, lambda(Z, T)) ->
       c_21(X,
            Y,
            Z,
            T,
            ren^#(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren^#(var(L), var(K), var(Lp)) ->
       c_22(if^#(eq(L, Lp), var(K), var(Lp))) }
   Strict Trs:
     { and(false(), false()) -> false()
     , and(false(), true()) -> false()
     , and(true(), false()) -> false()
     , and(true(), true()) -> true()
     , eq(nil(), nil()) -> true()
     , eq(nil(), cons(T, L)) -> false()
     , eq(cons(T, L), nil()) -> false()
     , eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
     , eq(var(L), var(Lp)) -> eq(L, Lp)
     , eq(var(L), apply(T, S)) -> false()
     , eq(var(L), lambda(X, T)) -> false()
     , eq(apply(T, S), var(L)) -> false()
     , eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
     , eq(apply(T, S), lambda(X, Tp)) -> false()
     , eq(lambda(X, T), var(L)) -> false()
     , eq(lambda(X, T), apply(Tp, Sp)) -> false()
     , eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
     , if(false(), var(K), var(L)) -> var(L)
     , if(true(), var(K), var(L)) -> var(K)
     , ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
     , ren(X, Y, lambda(Z, T)) ->
       lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
              ren(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of
   {1,2,3,4,5,6,7,10,11,12,14,15,16} by applications of
   Pre({1,2,3,4,5,6,7,10,11,12,14,15,16}) = {8,9,13,17,18,19,21}. Here
   rules are labeled as follows:
   
     DPs:
       { 1: and^#(false(), false()) -> c_1()
       , 2: and^#(false(), true()) -> c_2()
       , 3: and^#(true(), false()) -> c_3()
       , 4: and^#(true(), true()) -> c_4()
       , 5: eq^#(nil(), nil()) -> c_5()
       , 6: eq^#(nil(), cons(T, L)) -> c_6()
       , 7: eq^#(cons(T, L), nil()) -> c_7()
       , 8: eq^#(cons(T, L), cons(Tp, Lp)) ->
            c_8(and^#(eq(T, Tp), eq(L, Lp)))
       , 9: eq^#(var(L), var(Lp)) -> c_9(eq^#(L, Lp))
       , 10: eq^#(var(L), apply(T, S)) -> c_10()
       , 11: eq^#(var(L), lambda(X, T)) -> c_11()
       , 12: eq^#(apply(T, S), var(L)) -> c_12()
       , 13: eq^#(apply(T, S), apply(Tp, Sp)) ->
             c_13(and^#(eq(T, Tp), eq(S, Sp)))
       , 14: eq^#(apply(T, S), lambda(X, Tp)) -> c_14()
       , 15: eq^#(lambda(X, T), var(L)) -> c_15()
       , 16: eq^#(lambda(X, T), apply(Tp, Sp)) -> c_16()
       , 17: eq^#(lambda(X, T), lambda(Xp, Tp)) ->
             c_17(and^#(eq(T, Tp), eq(X, Xp)))
       , 18: if^#(false(), var(K), var(L)) -> c_18(L)
       , 19: if^#(true(), var(K), var(L)) -> c_19(K)
       , 20: ren^#(X, Y, apply(T, S)) ->
             c_20(ren^#(X, Y, T), ren^#(X, Y, S))
       , 21: ren^#(X, Y, lambda(Z, T)) ->
             c_21(X,
                  Y,
                  Z,
                  T,
                  ren^#(X,
                        Y,
                        ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
       , 22: ren^#(var(L), var(K), var(Lp)) ->
             c_22(if^#(eq(L, Lp), var(K), var(Lp))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { eq^#(cons(T, L), cons(Tp, Lp)) ->
       c_8(and^#(eq(T, Tp), eq(L, Lp)))
     , eq^#(var(L), var(Lp)) -> c_9(eq^#(L, Lp))
     , eq^#(apply(T, S), apply(Tp, Sp)) ->
       c_13(and^#(eq(T, Tp), eq(S, Sp)))
     , eq^#(lambda(X, T), lambda(Xp, Tp)) ->
       c_17(and^#(eq(T, Tp), eq(X, Xp)))
     , if^#(false(), var(K), var(L)) -> c_18(L)
     , if^#(true(), var(K), var(L)) -> c_19(K)
     , ren^#(X, Y, apply(T, S)) -> c_20(ren^#(X, Y, T), ren^#(X, Y, S))
     , ren^#(X, Y, lambda(Z, T)) ->
       c_21(X,
            Y,
            Z,
            T,
            ren^#(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren^#(var(L), var(K), var(Lp)) ->
       c_22(if^#(eq(L, Lp), var(K), var(Lp))) }
   Strict Trs:
     { and(false(), false()) -> false()
     , and(false(), true()) -> false()
     , and(true(), false()) -> false()
     , and(true(), true()) -> true()
     , eq(nil(), nil()) -> true()
     , eq(nil(), cons(T, L)) -> false()
     , eq(cons(T, L), nil()) -> false()
     , eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
     , eq(var(L), var(Lp)) -> eq(L, Lp)
     , eq(var(L), apply(T, S)) -> false()
     , eq(var(L), lambda(X, T)) -> false()
     , eq(apply(T, S), var(L)) -> false()
     , eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
     , eq(apply(T, S), lambda(X, Tp)) -> false()
     , eq(lambda(X, T), var(L)) -> false()
     , eq(lambda(X, T), apply(Tp, Sp)) -> false()
     , eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
     , if(false(), var(K), var(L)) -> var(L)
     , if(true(), var(K), var(L)) -> var(K)
     , ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
     , ren(X, Y, lambda(Z, T)) ->
       lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
              ren(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) }
   Weak DPs:
     { and^#(false(), false()) -> c_1()
     , and^#(false(), true()) -> c_2()
     , and^#(true(), false()) -> c_3()
     , and^#(true(), true()) -> c_4()
     , eq^#(nil(), nil()) -> c_5()
     , eq^#(nil(), cons(T, L)) -> c_6()
     , eq^#(cons(T, L), nil()) -> c_7()
     , eq^#(var(L), apply(T, S)) -> c_10()
     , eq^#(var(L), lambda(X, T)) -> c_11()
     , eq^#(apply(T, S), var(L)) -> c_12()
     , eq^#(apply(T, S), lambda(X, Tp)) -> c_14()
     , eq^#(lambda(X, T), var(L)) -> c_15()
     , eq^#(lambda(X, T), apply(Tp, Sp)) -> c_16() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,3,4} by applications of
   Pre({1,3,4}) = {2,5,6,8}. Here rules are labeled as follows:
   
     DPs:
       { 1: eq^#(cons(T, L), cons(Tp, Lp)) ->
            c_8(and^#(eq(T, Tp), eq(L, Lp)))
       , 2: eq^#(var(L), var(Lp)) -> c_9(eq^#(L, Lp))
       , 3: eq^#(apply(T, S), apply(Tp, Sp)) ->
            c_13(and^#(eq(T, Tp), eq(S, Sp)))
       , 4: eq^#(lambda(X, T), lambda(Xp, Tp)) ->
            c_17(and^#(eq(T, Tp), eq(X, Xp)))
       , 5: if^#(false(), var(K), var(L)) -> c_18(L)
       , 6: if^#(true(), var(K), var(L)) -> c_19(K)
       , 7: ren^#(X, Y, apply(T, S)) ->
            c_20(ren^#(X, Y, T), ren^#(X, Y, S))
       , 8: ren^#(X, Y, lambda(Z, T)) ->
            c_21(X,
                 Y,
                 Z,
                 T,
                 ren^#(X,
                       Y,
                       ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
       , 9: ren^#(var(L), var(K), var(Lp)) ->
            c_22(if^#(eq(L, Lp), var(K), var(Lp)))
       , 10: and^#(false(), false()) -> c_1()
       , 11: and^#(false(), true()) -> c_2()
       , 12: and^#(true(), false()) -> c_3()
       , 13: and^#(true(), true()) -> c_4()
       , 14: eq^#(nil(), nil()) -> c_5()
       , 15: eq^#(nil(), cons(T, L)) -> c_6()
       , 16: eq^#(cons(T, L), nil()) -> c_7()
       , 17: eq^#(var(L), apply(T, S)) -> c_10()
       , 18: eq^#(var(L), lambda(X, T)) -> c_11()
       , 19: eq^#(apply(T, S), var(L)) -> c_12()
       , 20: eq^#(apply(T, S), lambda(X, Tp)) -> c_14()
       , 21: eq^#(lambda(X, T), var(L)) -> c_15()
       , 22: eq^#(lambda(X, T), apply(Tp, Sp)) -> c_16() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { eq^#(var(L), var(Lp)) -> c_9(eq^#(L, Lp))
     , if^#(false(), var(K), var(L)) -> c_18(L)
     , if^#(true(), var(K), var(L)) -> c_19(K)
     , ren^#(X, Y, apply(T, S)) -> c_20(ren^#(X, Y, T), ren^#(X, Y, S))
     , ren^#(X, Y, lambda(Z, T)) ->
       c_21(X,
            Y,
            Z,
            T,
            ren^#(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren^#(var(L), var(K), var(Lp)) ->
       c_22(if^#(eq(L, Lp), var(K), var(Lp))) }
   Strict Trs:
     { and(false(), false()) -> false()
     , and(false(), true()) -> false()
     , and(true(), false()) -> false()
     , and(true(), true()) -> true()
     , eq(nil(), nil()) -> true()
     , eq(nil(), cons(T, L)) -> false()
     , eq(cons(T, L), nil()) -> false()
     , eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
     , eq(var(L), var(Lp)) -> eq(L, Lp)
     , eq(var(L), apply(T, S)) -> false()
     , eq(var(L), lambda(X, T)) -> false()
     , eq(apply(T, S), var(L)) -> false()
     , eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
     , eq(apply(T, S), lambda(X, Tp)) -> false()
     , eq(lambda(X, T), var(L)) -> false()
     , eq(lambda(X, T), apply(Tp, Sp)) -> false()
     , eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
     , if(false(), var(K), var(L)) -> var(L)
     , if(true(), var(K), var(L)) -> var(K)
     , ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
     , ren(X, Y, lambda(Z, T)) ->
       lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil())))),
              ren(X,
                  Y,
                  ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil())))), T)))
     , ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) }
   Weak DPs:
     { and^#(false(), false()) -> c_1()
     , and^#(false(), true()) -> c_2()
     , and^#(true(), false()) -> c_3()
     , and^#(true(), true()) -> c_4()
     , eq^#(nil(), nil()) -> c_5()
     , eq^#(nil(), cons(T, L)) -> c_6()
     , eq^#(cons(T, L), nil()) -> c_7()
     , eq^#(cons(T, L), cons(Tp, Lp)) ->
       c_8(and^#(eq(T, Tp), eq(L, Lp)))
     , eq^#(var(L), apply(T, S)) -> c_10()
     , eq^#(var(L), lambda(X, T)) -> c_11()
     , eq^#(apply(T, S), var(L)) -> c_12()
     , eq^#(apply(T, S), apply(Tp, Sp)) ->
       c_13(and^#(eq(T, Tp), eq(S, Sp)))
     , eq^#(apply(T, S), lambda(X, Tp)) -> c_14()
     , eq^#(lambda(X, T), var(L)) -> c_15()
     , eq^#(lambda(X, T), apply(Tp, Sp)) -> c_16()
     , eq^#(lambda(X, T), lambda(Xp, Tp)) ->
       c_17(and^#(eq(T, Tp), eq(X, Xp))) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..