MAYBE

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

Strict Trs:
  { terms(N) -> cons(recip(sqr(N)), terms(s(N)))
  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
  , sqr(0()) -> 0()
  , add(s(X), Y) -> s(add(X, Y))
  , add(0(), X) -> X
  , dbl(s(X)) -> s(s(dbl(X)))
  , dbl(0()) -> 0()
  , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
  , first(0(), X) -> nil()
  , half(s(s(X))) -> s(half(X))
  , half(s(0())) -> 0()
  , half(0()) -> 0()
  , half(dbl(X)) -> X }
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) '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) '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:
     { terms^#(N) -> c_1(sqr^#(N), terms^#(s(N)))
     , sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
     , sqr^#(0()) -> c_3()
     , add^#(s(X), Y) -> c_4(add^#(X, Y))
     , add^#(0(), X) -> c_5(X)
     , dbl^#(s(X)) -> c_6(dbl^#(X))
     , dbl^#(0()) -> c_7()
     , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
     , first^#(0(), X) -> c_9()
     , half^#(s(s(X))) -> c_10(half^#(X))
     , half^#(s(0())) -> c_11()
     , half^#(0()) -> c_12()
     , half^#(dbl(X)) -> c_13(X) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { terms^#(N) -> c_1(sqr^#(N), terms^#(s(N)))
     , sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
     , sqr^#(0()) -> c_3()
     , add^#(s(X), Y) -> c_4(add^#(X, Y))
     , add^#(0(), X) -> c_5(X)
     , dbl^#(s(X)) -> c_6(dbl^#(X))
     , dbl^#(0()) -> c_7()
     , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
     , first^#(0(), X) -> c_9()
     , half^#(s(s(X))) -> c_10(half^#(X))
     , half^#(s(0())) -> c_11()
     , half^#(0()) -> c_12()
     , half^#(dbl(X)) -> c_13(X) }
   Strict Trs:
     { terms(N) -> cons(recip(sqr(N)), terms(s(N)))
     , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
     , sqr(0()) -> 0()
     , add(s(X), Y) -> s(add(X, Y))
     , add(0(), X) -> X
     , dbl(s(X)) -> s(s(dbl(X)))
     , dbl(0()) -> 0()
     , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
     , first(0(), X) -> nil()
     , half(s(s(X))) -> s(half(X))
     , half(s(0())) -> 0()
     , half(0()) -> 0()
     , half(dbl(X)) -> X }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {3,7,9,11,12} by
   applications of Pre({3,7,9,11,12}) = {1,5,6,8,10,13}. Here rules
   are labeled as follows:
   
     DPs:
       { 1: terms^#(N) -> c_1(sqr^#(N), terms^#(s(N)))
       , 2: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
       , 3: sqr^#(0()) -> c_3()
       , 4: add^#(s(X), Y) -> c_4(add^#(X, Y))
       , 5: add^#(0(), X) -> c_5(X)
       , 6: dbl^#(s(X)) -> c_6(dbl^#(X))
       , 7: dbl^#(0()) -> c_7()
       , 8: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
       , 9: first^#(0(), X) -> c_9()
       , 10: half^#(s(s(X))) -> c_10(half^#(X))
       , 11: half^#(s(0())) -> c_11()
       , 12: half^#(0()) -> c_12()
       , 13: half^#(dbl(X)) -> c_13(X) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { terms^#(N) -> c_1(sqr^#(N), terms^#(s(N)))
     , sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
     , add^#(s(X), Y) -> c_4(add^#(X, Y))
     , add^#(0(), X) -> c_5(X)
     , dbl^#(s(X)) -> c_6(dbl^#(X))
     , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
     , half^#(s(s(X))) -> c_10(half^#(X))
     , half^#(dbl(X)) -> c_13(X) }
   Strict Trs:
     { terms(N) -> cons(recip(sqr(N)), terms(s(N)))
     , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
     , sqr(0()) -> 0()
     , add(s(X), Y) -> s(add(X, Y))
     , add(0(), X) -> X
     , dbl(s(X)) -> s(s(dbl(X)))
     , dbl(0()) -> 0()
     , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
     , first(0(), X) -> nil()
     , half(s(s(X))) -> s(half(X))
     , half(s(0())) -> 0()
     , half(0()) -> 0()
     , half(dbl(X)) -> X }
   Weak DPs:
     { sqr^#(0()) -> c_3()
     , dbl^#(0()) -> c_7()
     , first^#(0(), X) -> c_9()
     , half^#(s(0())) -> c_11()
     , half^#(0()) -> c_12() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..