MAYBE

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

Strict Trs:
  { U11(tt(), V2) -> U12(isNat(activate(V2)))
  , U12(tt()) -> tt()
  , isNat(n__0()) -> tt()
  , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
  , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
  , activate(X) -> X
  , activate(n__0()) -> 0()
  , activate(n__plus(X1, X2)) -> plus(X1, X2)
  , activate(n__s(X)) -> s(X)
  , U21(tt()) -> tt()
  , U31(tt(), N) -> activate(N)
  , U41(tt(), M, N) ->
    U42(isNat(activate(N)), activate(M), activate(N))
  , U42(tt(), M, N) -> s(plus(activate(N), activate(M)))
  , s(X) -> n__s(X)
  , plus(X1, X2) -> n__plus(X1, X2)
  , plus(N, s(M)) -> U41(isNat(M), M, N)
  , plus(N, 0()) -> U31(isNat(N), N)
  , 0() -> n__0() }
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) 'Best' failed due to the following reason:
      
      None of the processors succeeded.
      
      Details of failed attempt(s):
      -----------------------------
      1) '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
      
      2) '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
      
   
   3) '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 perSymbol-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
      2) 'Bounds with minimal-enrichment and initial automaton 'match''
         failed due to the following reason:
         
         match-boundness of the problem could not be verified.
      
   

3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , U21^#(tt()) -> c_10()
     , activate^#(X) -> c_6(X)
     , activate^#(n__0()) -> c_7(0^#())
     , activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_9(s^#(X))
     , 0^#() -> c_18()
     , plus^#(X1, X2) -> c_15(X1, X2)
     , plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
     , s^#(X) -> c_14(X)
     , U31^#(tt(), N) -> c_11(activate^#(N))
     , U41^#(tt(), M, N) ->
       c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
     , U42^#(tt(), M, N) -> c_13(s^#(plus(activate(N), activate(M)))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , U21^#(tt()) -> c_10()
     , activate^#(X) -> c_6(X)
     , activate^#(n__0()) -> c_7(0^#())
     , activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_9(s^#(X))
     , 0^#() -> c_18()
     , plus^#(X1, X2) -> c_15(X1, X2)
     , plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
     , s^#(X) -> c_14(X)
     , U31^#(tt(), N) -> c_11(activate^#(N))
     , U41^#(tt(), M, N) ->
       c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
     , U42^#(tt(), M, N) -> c_13(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , U21(tt()) -> tt()
     , U31(tt(), N) -> activate(N)
     , U41(tt(), M, N) ->
       U42(isNat(activate(N)), activate(M), activate(N))
     , U42(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U41(isNat(M), M, N)
     , plus(N, 0()) -> U31(isNat(N), N)
     , 0() -> n__0() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {2,3,6,11} by applications
   of Pre({2,3,6,11}) = {1,5,7,8,12,15}. Here rules are labeled as
   follows:
   
     DPs:
       { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
       , 2: U12^#(tt()) -> c_2()
       , 3: isNat^#(n__0()) -> c_3()
       , 4: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(isNat(activate(V1)), activate(V2)))
       , 5: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
       , 6: U21^#(tt()) -> c_10()
       , 7: activate^#(X) -> c_6(X)
       , 8: activate^#(n__0()) -> c_7(0^#())
       , 9: activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
       , 10: activate^#(n__s(X)) -> c_9(s^#(X))
       , 11: 0^#() -> c_18()
       , 12: plus^#(X1, X2) -> c_15(X1, X2)
       , 13: plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
       , 14: plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
       , 15: s^#(X) -> c_14(X)
       , 16: U31^#(tt(), N) -> c_11(activate^#(N))
       , 17: U41^#(tt(), M, N) ->
             c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
       , 18: U42^#(tt(), M, N) ->
             c_13(s^#(plus(activate(N), activate(M)))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , activate^#(X) -> c_6(X)
     , activate^#(n__0()) -> c_7(0^#())
     , activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_9(s^#(X))
     , plus^#(X1, X2) -> c_15(X1, X2)
     , plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
     , s^#(X) -> c_14(X)
     , U31^#(tt(), N) -> c_11(activate^#(N))
     , U41^#(tt(), M, N) ->
       c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
     , U42^#(tt(), M, N) -> c_13(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , U21(tt()) -> tt()
     , U31(tt(), N) -> activate(N)
     , U41(tt(), M, N) ->
       U42(isNat(activate(N)), activate(M), activate(N))
     , U42(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U41(isNat(M), M, N)
     , plus(N, 0()) -> U31(isNat(N), N)
     , 0() -> n__0() }
   Weak DPs:
     { U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , U21^#(tt()) -> c_10()
     , 0^#() -> c_18() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1,3,5} by applications of
   Pre({1,3,5}) = {2,4,8,11,12}. Here rules are labeled as follows:
   
     DPs:
       { 1: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
       , 2: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(isNat(activate(V1)), activate(V2)))
       , 3: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
       , 4: activate^#(X) -> c_6(X)
       , 5: activate^#(n__0()) -> c_7(0^#())
       , 6: activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
       , 7: activate^#(n__s(X)) -> c_9(s^#(X))
       , 8: plus^#(X1, X2) -> c_15(X1, X2)
       , 9: plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
       , 10: plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
       , 11: s^#(X) -> c_14(X)
       , 12: U31^#(tt(), N) -> c_11(activate^#(N))
       , 13: U41^#(tt(), M, N) ->
             c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
       , 14: U42^#(tt(), M, N) ->
             c_13(s^#(plus(activate(N), activate(M))))
       , 15: U12^#(tt()) -> c_2()
       , 16: isNat^#(n__0()) -> c_3()
       , 17: U21^#(tt()) -> c_10()
       , 18: 0^#() -> c_18() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , activate^#(X) -> c_6(X)
     , activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_9(s^#(X))
     , plus^#(X1, X2) -> c_15(X1, X2)
     , plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
     , s^#(X) -> c_14(X)
     , U31^#(tt(), N) -> c_11(activate^#(N))
     , U41^#(tt(), M, N) ->
       c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
     , U42^#(tt(), M, N) -> c_13(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , U21(tt()) -> tt()
     , U31(tt(), N) -> activate(N)
     , U41(tt(), M, N) ->
       U42(isNat(activate(N)), activate(M), activate(N))
     , U42(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U41(isNat(M), M, N)
     , plus(N, 0()) -> U31(isNat(N), N)
     , 0() -> n__0() }
   Weak DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , U21^#(tt()) -> c_10()
     , activate^#(n__0()) -> c_7(0^#())
     , 0^#() -> c_18() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {1} by applications of
   Pre({1}) = {2,5,8}. Here rules are labeled as follows:
   
     DPs:
       { 1: isNat^#(n__plus(V1, V2)) ->
            c_4(U11^#(isNat(activate(V1)), activate(V2)))
       , 2: activate^#(X) -> c_6(X)
       , 3: activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
       , 4: activate^#(n__s(X)) -> c_9(s^#(X))
       , 5: plus^#(X1, X2) -> c_15(X1, X2)
       , 6: plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
       , 7: plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
       , 8: s^#(X) -> c_14(X)
       , 9: U31^#(tt(), N) -> c_11(activate^#(N))
       , 10: U41^#(tt(), M, N) ->
             c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
       , 11: U42^#(tt(), M, N) ->
             c_13(s^#(plus(activate(N), activate(M))))
       , 12: U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
       , 13: U12^#(tt()) -> c_2()
       , 14: isNat^#(n__0()) -> c_3()
       , 15: isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
       , 16: U21^#(tt()) -> c_10()
       , 17: activate^#(n__0()) -> c_7(0^#())
       , 18: 0^#() -> c_18() }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { activate^#(X) -> c_6(X)
     , activate^#(n__plus(X1, X2)) -> c_8(plus^#(X1, X2))
     , activate^#(n__s(X)) -> c_9(s^#(X))
     , plus^#(X1, X2) -> c_15(X1, X2)
     , plus^#(N, s(M)) -> c_16(U41^#(isNat(M), M, N))
     , plus^#(N, 0()) -> c_17(U31^#(isNat(N), N))
     , s^#(X) -> c_14(X)
     , U31^#(tt(), N) -> c_11(activate^#(N))
     , U41^#(tt(), M, N) ->
       c_12(U42^#(isNat(activate(N)), activate(M), activate(N)))
     , U42^#(tt(), M, N) -> c_13(s^#(plus(activate(N), activate(M)))) }
   Strict Trs:
     { U11(tt(), V2) -> U12(isNat(activate(V2)))
     , U12(tt()) -> tt()
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) -> U11(isNat(activate(V1)), activate(V2))
     , isNat(n__s(V1)) -> U21(isNat(activate(V1)))
     , activate(X) -> X
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(X1, X2)
     , activate(n__s(X)) -> s(X)
     , U21(tt()) -> tt()
     , U31(tt(), N) -> activate(N)
     , U41(tt(), M, N) ->
       U42(isNat(activate(N)), activate(M), activate(N))
     , U42(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , s(X) -> n__s(X)
     , plus(X1, X2) -> n__plus(X1, X2)
     , plus(N, s(M)) -> U41(isNat(M), M, N)
     , plus(N, 0()) -> U31(isNat(N), N)
     , 0() -> n__0() }
   Weak DPs:
     { U11^#(tt(), V2) -> c_1(U12^#(isNat(activate(V2))))
     , U12^#(tt()) -> c_2()
     , isNat^#(n__0()) -> c_3()
     , isNat^#(n__plus(V1, V2)) ->
       c_4(U11^#(isNat(activate(V1)), activate(V2)))
     , isNat^#(n__s(V1)) -> c_5(U21^#(isNat(activate(V1))))
     , U21^#(tt()) -> c_10()
     , activate^#(n__0()) -> c_7(0^#())
     , 0^#() -> c_18() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..