MAYBE

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

Strict Trs:
  { active(terms(X)) -> terms(active(X))
  , active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
  , active(cons(X1, X2)) -> cons(active(X1), X2)
  , active(recip(X)) -> recip(active(X))
  , active(sqr(X)) -> sqr(active(X))
  , active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
  , active(sqr(0())) -> mark(0())
  , active(s(X)) -> s(active(X))
  , active(add(X1, X2)) -> add(X1, active(X2))
  , active(add(X1, X2)) -> add(active(X1), X2)
  , active(add(s(X), Y)) -> mark(s(add(X, Y)))
  , active(add(0(), X)) -> mark(X)
  , active(dbl(X)) -> dbl(active(X))
  , active(dbl(s(X))) -> mark(s(s(dbl(X))))
  , active(dbl(0())) -> mark(0())
  , active(first(X1, X2)) -> first(X1, active(X2))
  , active(first(X1, X2)) -> first(active(X1), X2)
  , active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
  , active(first(0(), X)) -> mark(nil())
  , terms(mark(X)) -> mark(terms(X))
  , terms(ok(X)) -> ok(terms(X))
  , cons(mark(X1), X2) -> mark(cons(X1, X2))
  , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
  , recip(mark(X)) -> mark(recip(X))
  , recip(ok(X)) -> ok(recip(X))
  , sqr(mark(X)) -> mark(sqr(X))
  , sqr(ok(X)) -> ok(sqr(X))
  , s(mark(X)) -> mark(s(X))
  , s(ok(X)) -> ok(s(X))
  , add(X1, mark(X2)) -> mark(add(X1, X2))
  , add(mark(X1), X2) -> mark(add(X1, X2))
  , add(ok(X1), ok(X2)) -> ok(add(X1, X2))
  , dbl(mark(X)) -> mark(dbl(X))
  , dbl(ok(X)) -> ok(dbl(X))
  , first(X1, mark(X2)) -> mark(first(X1, X2))
  , first(mark(X1), X2) -> mark(first(X1, X2))
  , first(ok(X1), ok(X2)) -> ok(first(X1, X2))
  , proper(terms(X)) -> terms(proper(X))
  , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
  , proper(recip(X)) -> recip(proper(X))
  , proper(sqr(X)) -> sqr(proper(X))
  , proper(s(X)) -> s(proper(X))
  , proper(0()) -> ok(0())
  , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
  , proper(dbl(X)) -> dbl(proper(X))
  , proper(first(X1, X2)) -> first(proper(X1), proper(X2))
  , proper(nil()) -> ok(nil())
  , top(mark(X)) -> top(proper(X))
  , top(ok(X)) -> top(active(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) '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:
     { active^#(terms(X)) -> c_1(terms^#(active(X)))
     , active^#(terms(N)) -> c_2(cons^#(recip(sqr(N)), terms(s(N))))
     , active^#(cons(X1, X2)) -> c_3(cons^#(active(X1), X2))
     , active^#(recip(X)) -> c_4(recip^#(active(X)))
     , active^#(sqr(X)) -> c_5(sqr^#(active(X)))
     , active^#(sqr(s(X))) -> c_6(s^#(add(sqr(X), dbl(X))))
     , active^#(sqr(0())) -> c_7()
     , active^#(s(X)) -> c_8(s^#(active(X)))
     , active^#(add(X1, X2)) -> c_9(add^#(X1, active(X2)))
     , active^#(add(X1, X2)) -> c_10(add^#(active(X1), X2))
     , active^#(add(s(X), Y)) -> c_11(s^#(add(X, Y)))
     , active^#(add(0(), X)) -> c_12(X)
     , active^#(dbl(X)) -> c_13(dbl^#(active(X)))
     , active^#(dbl(s(X))) -> c_14(s^#(s(dbl(X))))
     , active^#(dbl(0())) -> c_15()
     , active^#(first(X1, X2)) -> c_16(first^#(X1, active(X2)))
     , active^#(first(X1, X2)) -> c_17(first^#(active(X1), X2))
     , active^#(first(s(X), cons(Y, Z))) -> c_18(cons^#(Y, first(X, Z)))
     , active^#(first(0(), X)) -> c_19()
     , terms^#(mark(X)) -> c_20(terms^#(X))
     , terms^#(ok(X)) -> c_21(terms^#(X))
     , cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2))
     , recip^#(mark(X)) -> c_24(recip^#(X))
     , recip^#(ok(X)) -> c_25(recip^#(X))
     , sqr^#(mark(X)) -> c_26(sqr^#(X))
     , sqr^#(ok(X)) -> c_27(sqr^#(X))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , add^#(X1, mark(X2)) -> c_30(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_31(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2))
     , dbl^#(mark(X)) -> c_33(dbl^#(X))
     , dbl^#(ok(X)) -> c_34(dbl^#(X))
     , first^#(X1, mark(X2)) -> c_35(first^#(X1, X2))
     , first^#(mark(X1), X2) -> c_36(first^#(X1, X2))
     , first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2))
     , proper^#(terms(X)) -> c_38(terms^#(proper(X)))
     , proper^#(cons(X1, X2)) -> c_39(cons^#(proper(X1), proper(X2)))
     , proper^#(recip(X)) -> c_40(recip^#(proper(X)))
     , proper^#(sqr(X)) -> c_41(sqr^#(proper(X)))
     , proper^#(s(X)) -> c_42(s^#(proper(X)))
     , proper^#(0()) -> c_43()
     , proper^#(add(X1, X2)) -> c_44(add^#(proper(X1), proper(X2)))
     , proper^#(dbl(X)) -> c_45(dbl^#(proper(X)))
     , proper^#(first(X1, X2)) -> c_46(first^#(proper(X1), proper(X2)))
     , proper^#(nil()) -> c_47()
     , top^#(mark(X)) -> c_48(top^#(proper(X)))
     , top^#(ok(X)) -> c_49(top^#(active(X))) }
   
   and mark the set of starting terms.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { active^#(terms(X)) -> c_1(terms^#(active(X)))
     , active^#(terms(N)) -> c_2(cons^#(recip(sqr(N)), terms(s(N))))
     , active^#(cons(X1, X2)) -> c_3(cons^#(active(X1), X2))
     , active^#(recip(X)) -> c_4(recip^#(active(X)))
     , active^#(sqr(X)) -> c_5(sqr^#(active(X)))
     , active^#(sqr(s(X))) -> c_6(s^#(add(sqr(X), dbl(X))))
     , active^#(sqr(0())) -> c_7()
     , active^#(s(X)) -> c_8(s^#(active(X)))
     , active^#(add(X1, X2)) -> c_9(add^#(X1, active(X2)))
     , active^#(add(X1, X2)) -> c_10(add^#(active(X1), X2))
     , active^#(add(s(X), Y)) -> c_11(s^#(add(X, Y)))
     , active^#(add(0(), X)) -> c_12(X)
     , active^#(dbl(X)) -> c_13(dbl^#(active(X)))
     , active^#(dbl(s(X))) -> c_14(s^#(s(dbl(X))))
     , active^#(dbl(0())) -> c_15()
     , active^#(first(X1, X2)) -> c_16(first^#(X1, active(X2)))
     , active^#(first(X1, X2)) -> c_17(first^#(active(X1), X2))
     , active^#(first(s(X), cons(Y, Z))) -> c_18(cons^#(Y, first(X, Z)))
     , active^#(first(0(), X)) -> c_19()
     , terms^#(mark(X)) -> c_20(terms^#(X))
     , terms^#(ok(X)) -> c_21(terms^#(X))
     , cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2))
     , recip^#(mark(X)) -> c_24(recip^#(X))
     , recip^#(ok(X)) -> c_25(recip^#(X))
     , sqr^#(mark(X)) -> c_26(sqr^#(X))
     , sqr^#(ok(X)) -> c_27(sqr^#(X))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , add^#(X1, mark(X2)) -> c_30(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_31(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2))
     , dbl^#(mark(X)) -> c_33(dbl^#(X))
     , dbl^#(ok(X)) -> c_34(dbl^#(X))
     , first^#(X1, mark(X2)) -> c_35(first^#(X1, X2))
     , first^#(mark(X1), X2) -> c_36(first^#(X1, X2))
     , first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2))
     , proper^#(terms(X)) -> c_38(terms^#(proper(X)))
     , proper^#(cons(X1, X2)) -> c_39(cons^#(proper(X1), proper(X2)))
     , proper^#(recip(X)) -> c_40(recip^#(proper(X)))
     , proper^#(sqr(X)) -> c_41(sqr^#(proper(X)))
     , proper^#(s(X)) -> c_42(s^#(proper(X)))
     , proper^#(0()) -> c_43()
     , proper^#(add(X1, X2)) -> c_44(add^#(proper(X1), proper(X2)))
     , proper^#(dbl(X)) -> c_45(dbl^#(proper(X)))
     , proper^#(first(X1, X2)) -> c_46(first^#(proper(X1), proper(X2)))
     , proper^#(nil()) -> c_47()
     , top^#(mark(X)) -> c_48(top^#(proper(X)))
     , top^#(ok(X)) -> c_49(top^#(active(X))) }
   Strict Trs:
     { active(terms(X)) -> terms(active(X))
     , active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
     , active(cons(X1, X2)) -> cons(active(X1), X2)
     , active(recip(X)) -> recip(active(X))
     , active(sqr(X)) -> sqr(active(X))
     , active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
     , active(sqr(0())) -> mark(0())
     , active(s(X)) -> s(active(X))
     , active(add(X1, X2)) -> add(X1, active(X2))
     , active(add(X1, X2)) -> add(active(X1), X2)
     , active(add(s(X), Y)) -> mark(s(add(X, Y)))
     , active(add(0(), X)) -> mark(X)
     , active(dbl(X)) -> dbl(active(X))
     , active(dbl(s(X))) -> mark(s(s(dbl(X))))
     , active(dbl(0())) -> mark(0())
     , active(first(X1, X2)) -> first(X1, active(X2))
     , active(first(X1, X2)) -> first(active(X1), X2)
     , active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
     , active(first(0(), X)) -> mark(nil())
     , terms(mark(X)) -> mark(terms(X))
     , terms(ok(X)) -> ok(terms(X))
     , cons(mark(X1), X2) -> mark(cons(X1, X2))
     , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
     , recip(mark(X)) -> mark(recip(X))
     , recip(ok(X)) -> ok(recip(X))
     , sqr(mark(X)) -> mark(sqr(X))
     , sqr(ok(X)) -> ok(sqr(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , add(X1, mark(X2)) -> mark(add(X1, X2))
     , add(mark(X1), X2) -> mark(add(X1, X2))
     , add(ok(X1), ok(X2)) -> ok(add(X1, X2))
     , dbl(mark(X)) -> mark(dbl(X))
     , dbl(ok(X)) -> ok(dbl(X))
     , first(X1, mark(X2)) -> mark(first(X1, X2))
     , first(mark(X1), X2) -> mark(first(X1, X2))
     , first(ok(X1), ok(X2)) -> ok(first(X1, X2))
     , proper(terms(X)) -> terms(proper(X))
     , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
     , proper(recip(X)) -> recip(proper(X))
     , proper(sqr(X)) -> sqr(proper(X))
     , proper(s(X)) -> s(proper(X))
     , proper(0()) -> ok(0())
     , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
     , proper(dbl(X)) -> dbl(proper(X))
     , proper(first(X1, X2)) -> first(proper(X1), proper(X2))
     , proper(nil()) -> ok(nil())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Consider the dependency graph:
   
     1: active^#(terms(X)) -> c_1(terms^#(active(X)))
        -->_1 terms^#(ok(X)) -> c_21(terms^#(X)) :21
        -->_1 terms^#(mark(X)) -> c_20(terms^#(X)) :20
     
     2: active^#(terms(N)) -> c_2(cons^#(recip(sqr(N)), terms(s(N))))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
     
     3: active^#(cons(X1, X2)) -> c_3(cons^#(active(X1), X2))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
     
     4: active^#(recip(X)) -> c_4(recip^#(active(X)))
        -->_1 recip^#(ok(X)) -> c_25(recip^#(X)) :25
        -->_1 recip^#(mark(X)) -> c_24(recip^#(X)) :24
     
     5: active^#(sqr(X)) -> c_5(sqr^#(active(X)))
        -->_1 sqr^#(ok(X)) -> c_27(sqr^#(X)) :27
        -->_1 sqr^#(mark(X)) -> c_26(sqr^#(X)) :26
     
     6: active^#(sqr(s(X))) -> c_6(s^#(add(sqr(X), dbl(X))))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     7: active^#(sqr(0())) -> c_7()
     
     8: active^#(s(X)) -> c_8(s^#(active(X)))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     9: active^#(add(X1, X2)) -> c_9(add^#(X1, active(X2)))
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
     
     10: active^#(add(X1, X2)) -> c_10(add^#(active(X1), X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
     
     11: active^#(add(s(X), Y)) -> c_11(s^#(add(X, Y)))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     12: active^#(add(0(), X)) -> c_12(X)
        -->_1 top^#(ok(X)) -> c_49(top^#(active(X))) :49
        -->_1 top^#(mark(X)) -> c_48(top^#(proper(X))) :48
        -->_1 proper^#(first(X1, X2)) ->
              c_46(first^#(proper(X1), proper(X2))) :46
        -->_1 proper^#(dbl(X)) -> c_45(dbl^#(proper(X))) :45
        -->_1 proper^#(add(X1, X2)) ->
              c_44(add^#(proper(X1), proper(X2))) :44
        -->_1 proper^#(s(X)) -> c_42(s^#(proper(X))) :42
        -->_1 proper^#(sqr(X)) -> c_41(sqr^#(proper(X))) :41
        -->_1 proper^#(recip(X)) -> c_40(recip^#(proper(X))) :40
        -->_1 proper^#(cons(X1, X2)) ->
              c_39(cons^#(proper(X1), proper(X2))) :39
        -->_1 proper^#(terms(X)) -> c_38(terms^#(proper(X))) :38
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
        -->_1 dbl^#(ok(X)) -> c_34(dbl^#(X)) :34
        -->_1 dbl^#(mark(X)) -> c_33(dbl^#(X)) :33
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
        -->_1 sqr^#(ok(X)) -> c_27(sqr^#(X)) :27
        -->_1 sqr^#(mark(X)) -> c_26(sqr^#(X)) :26
        -->_1 recip^#(ok(X)) -> c_25(recip^#(X)) :25
        -->_1 recip^#(mark(X)) -> c_24(recip^#(X)) :24
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
        -->_1 terms^#(ok(X)) -> c_21(terms^#(X)) :21
        -->_1 terms^#(mark(X)) -> c_20(terms^#(X)) :20
        -->_1 active^#(first(s(X), cons(Y, Z))) ->
              c_18(cons^#(Y, first(X, Z))) :18
        -->_1 active^#(first(X1, X2)) -> c_17(first^#(active(X1), X2)) :17
        -->_1 active^#(first(X1, X2)) -> c_16(first^#(X1, active(X2))) :16
        -->_1 active^#(dbl(s(X))) -> c_14(s^#(s(dbl(X)))) :14
        -->_1 active^#(dbl(X)) -> c_13(dbl^#(active(X))) :13
        -->_1 proper^#(nil()) -> c_47() :47
        -->_1 proper^#(0()) -> c_43() :43
        -->_1 active^#(first(0(), X)) -> c_19() :19
        -->_1 active^#(dbl(0())) -> c_15() :15
        -->_1 active^#(add(0(), X)) -> c_12(X) :12
        -->_1 active^#(add(s(X), Y)) -> c_11(s^#(add(X, Y))) :11
        -->_1 active^#(add(X1, X2)) -> c_10(add^#(active(X1), X2)) :10
        -->_1 active^#(add(X1, X2)) -> c_9(add^#(X1, active(X2))) :9
        -->_1 active^#(s(X)) -> c_8(s^#(active(X))) :8
        -->_1 active^#(sqr(0())) -> c_7() :7
        -->_1 active^#(sqr(s(X))) -> c_6(s^#(add(sqr(X), dbl(X)))) :6
        -->_1 active^#(sqr(X)) -> c_5(sqr^#(active(X))) :5
        -->_1 active^#(recip(X)) -> c_4(recip^#(active(X))) :4
        -->_1 active^#(cons(X1, X2)) -> c_3(cons^#(active(X1), X2)) :3
        -->_1 active^#(terms(N)) ->
              c_2(cons^#(recip(sqr(N)), terms(s(N)))) :2
        -->_1 active^#(terms(X)) -> c_1(terms^#(active(X))) :1
     
     13: active^#(dbl(X)) -> c_13(dbl^#(active(X)))
        -->_1 dbl^#(ok(X)) -> c_34(dbl^#(X)) :34
        -->_1 dbl^#(mark(X)) -> c_33(dbl^#(X)) :33
     
     14: active^#(dbl(s(X))) -> c_14(s^#(s(dbl(X))))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     15: active^#(dbl(0())) -> c_15()
     
     16: active^#(first(X1, X2)) -> c_16(first^#(X1, active(X2)))
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
     
     17: active^#(first(X1, X2)) -> c_17(first^#(active(X1), X2))
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
     
     18: active^#(first(s(X), cons(Y, Z))) ->
         c_18(cons^#(Y, first(X, Z)))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
     
     19: active^#(first(0(), X)) -> c_19()
     
     20: terms^#(mark(X)) -> c_20(terms^#(X))
        -->_1 terms^#(ok(X)) -> c_21(terms^#(X)) :21
        -->_1 terms^#(mark(X)) -> c_20(terms^#(X)) :20
     
     21: terms^#(ok(X)) -> c_21(terms^#(X))
        -->_1 terms^#(ok(X)) -> c_21(terms^#(X)) :21
        -->_1 terms^#(mark(X)) -> c_20(terms^#(X)) :20
     
     22: cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
     
     23: cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
     
     24: recip^#(mark(X)) -> c_24(recip^#(X))
        -->_1 recip^#(ok(X)) -> c_25(recip^#(X)) :25
        -->_1 recip^#(mark(X)) -> c_24(recip^#(X)) :24
     
     25: recip^#(ok(X)) -> c_25(recip^#(X))
        -->_1 recip^#(ok(X)) -> c_25(recip^#(X)) :25
        -->_1 recip^#(mark(X)) -> c_24(recip^#(X)) :24
     
     26: sqr^#(mark(X)) -> c_26(sqr^#(X))
        -->_1 sqr^#(ok(X)) -> c_27(sqr^#(X)) :27
        -->_1 sqr^#(mark(X)) -> c_26(sqr^#(X)) :26
     
     27: sqr^#(ok(X)) -> c_27(sqr^#(X))
        -->_1 sqr^#(ok(X)) -> c_27(sqr^#(X)) :27
        -->_1 sqr^#(mark(X)) -> c_26(sqr^#(X)) :26
     
     28: s^#(mark(X)) -> c_28(s^#(X))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     29: s^#(ok(X)) -> c_29(s^#(X))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     30: add^#(X1, mark(X2)) -> c_30(add^#(X1, X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
     
     31: add^#(mark(X1), X2) -> c_31(add^#(X1, X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
     
     32: add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
     
     33: dbl^#(mark(X)) -> c_33(dbl^#(X))
        -->_1 dbl^#(ok(X)) -> c_34(dbl^#(X)) :34
        -->_1 dbl^#(mark(X)) -> c_33(dbl^#(X)) :33
     
     34: dbl^#(ok(X)) -> c_34(dbl^#(X))
        -->_1 dbl^#(ok(X)) -> c_34(dbl^#(X)) :34
        -->_1 dbl^#(mark(X)) -> c_33(dbl^#(X)) :33
     
     35: first^#(X1, mark(X2)) -> c_35(first^#(X1, X2))
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
     
     36: first^#(mark(X1), X2) -> c_36(first^#(X1, X2))
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
     
     37: first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2))
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
     
     38: proper^#(terms(X)) -> c_38(terms^#(proper(X)))
        -->_1 terms^#(ok(X)) -> c_21(terms^#(X)) :21
        -->_1 terms^#(mark(X)) -> c_20(terms^#(X)) :20
     
     39: proper^#(cons(X1, X2)) -> c_39(cons^#(proper(X1), proper(X2)))
        -->_1 cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2)) :23
        -->_1 cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2)) :22
     
     40: proper^#(recip(X)) -> c_40(recip^#(proper(X)))
        -->_1 recip^#(ok(X)) -> c_25(recip^#(X)) :25
        -->_1 recip^#(mark(X)) -> c_24(recip^#(X)) :24
     
     41: proper^#(sqr(X)) -> c_41(sqr^#(proper(X)))
        -->_1 sqr^#(ok(X)) -> c_27(sqr^#(X)) :27
        -->_1 sqr^#(mark(X)) -> c_26(sqr^#(X)) :26
     
     42: proper^#(s(X)) -> c_42(s^#(proper(X)))
        -->_1 s^#(ok(X)) -> c_29(s^#(X)) :29
        -->_1 s^#(mark(X)) -> c_28(s^#(X)) :28
     
     43: proper^#(0()) -> c_43()
     
     44: proper^#(add(X1, X2)) -> c_44(add^#(proper(X1), proper(X2)))
        -->_1 add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2)) :32
        -->_1 add^#(mark(X1), X2) -> c_31(add^#(X1, X2)) :31
        -->_1 add^#(X1, mark(X2)) -> c_30(add^#(X1, X2)) :30
     
     45: proper^#(dbl(X)) -> c_45(dbl^#(proper(X)))
        -->_1 dbl^#(ok(X)) -> c_34(dbl^#(X)) :34
        -->_1 dbl^#(mark(X)) -> c_33(dbl^#(X)) :33
     
     46: proper^#(first(X1, X2)) ->
         c_46(first^#(proper(X1), proper(X2)))
        -->_1 first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2)) :37
        -->_1 first^#(mark(X1), X2) -> c_36(first^#(X1, X2)) :36
        -->_1 first^#(X1, mark(X2)) -> c_35(first^#(X1, X2)) :35
     
     47: proper^#(nil()) -> c_47()
     
     48: top^#(mark(X)) -> c_48(top^#(proper(X)))
        -->_1 top^#(ok(X)) -> c_49(top^#(active(X))) :49
        -->_1 top^#(mark(X)) -> c_48(top^#(proper(X))) :48
     
     49: top^#(ok(X)) -> c_49(top^#(active(X)))
        -->_1 top^#(ok(X)) -> c_49(top^#(active(X))) :49
        -->_1 top^#(mark(X)) -> c_48(top^#(proper(X))) :48
     
   
   Only the nodes
   {20,21,22,23,24,25,26,27,28,29,30,32,31,33,34,35,37,36,43,47,48,49}
   are reachable from nodes
   {20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,43,47,48,49}
   that start derivation from marked basic terms. The nodes not
   reachable are removed from the problem.
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { terms^#(mark(X)) -> c_20(terms^#(X))
     , terms^#(ok(X)) -> c_21(terms^#(X))
     , cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2))
     , recip^#(mark(X)) -> c_24(recip^#(X))
     , recip^#(ok(X)) -> c_25(recip^#(X))
     , sqr^#(mark(X)) -> c_26(sqr^#(X))
     , sqr^#(ok(X)) -> c_27(sqr^#(X))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , add^#(X1, mark(X2)) -> c_30(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_31(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2))
     , dbl^#(mark(X)) -> c_33(dbl^#(X))
     , dbl^#(ok(X)) -> c_34(dbl^#(X))
     , first^#(X1, mark(X2)) -> c_35(first^#(X1, X2))
     , first^#(mark(X1), X2) -> c_36(first^#(X1, X2))
     , first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2))
     , proper^#(0()) -> c_43()
     , proper^#(nil()) -> c_47()
     , top^#(mark(X)) -> c_48(top^#(proper(X)))
     , top^#(ok(X)) -> c_49(top^#(active(X))) }
   Strict Trs:
     { active(terms(X)) -> terms(active(X))
     , active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
     , active(cons(X1, X2)) -> cons(active(X1), X2)
     , active(recip(X)) -> recip(active(X))
     , active(sqr(X)) -> sqr(active(X))
     , active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
     , active(sqr(0())) -> mark(0())
     , active(s(X)) -> s(active(X))
     , active(add(X1, X2)) -> add(X1, active(X2))
     , active(add(X1, X2)) -> add(active(X1), X2)
     , active(add(s(X), Y)) -> mark(s(add(X, Y)))
     , active(add(0(), X)) -> mark(X)
     , active(dbl(X)) -> dbl(active(X))
     , active(dbl(s(X))) -> mark(s(s(dbl(X))))
     , active(dbl(0())) -> mark(0())
     , active(first(X1, X2)) -> first(X1, active(X2))
     , active(first(X1, X2)) -> first(active(X1), X2)
     , active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
     , active(first(0(), X)) -> mark(nil())
     , terms(mark(X)) -> mark(terms(X))
     , terms(ok(X)) -> ok(terms(X))
     , cons(mark(X1), X2) -> mark(cons(X1, X2))
     , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
     , recip(mark(X)) -> mark(recip(X))
     , recip(ok(X)) -> ok(recip(X))
     , sqr(mark(X)) -> mark(sqr(X))
     , sqr(ok(X)) -> ok(sqr(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , add(X1, mark(X2)) -> mark(add(X1, X2))
     , add(mark(X1), X2) -> mark(add(X1, X2))
     , add(ok(X1), ok(X2)) -> ok(add(X1, X2))
     , dbl(mark(X)) -> mark(dbl(X))
     , dbl(ok(X)) -> ok(dbl(X))
     , first(X1, mark(X2)) -> mark(first(X1, X2))
     , first(mark(X1), X2) -> mark(first(X1, X2))
     , first(ok(X1), ok(X2)) -> ok(first(X1, X2))
     , proper(terms(X)) -> terms(proper(X))
     , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
     , proper(recip(X)) -> recip(proper(X))
     , proper(sqr(X)) -> sqr(proper(X))
     , proper(s(X)) -> s(proper(X))
     , proper(0()) -> ok(0())
     , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
     , proper(dbl(X)) -> dbl(proper(X))
     , proper(first(X1, X2)) -> first(proper(X1), proper(X2))
     , proper(nil()) -> ok(nil())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {19,20} by applications of
   Pre({19,20}) = {}. Here rules are labeled as follows:
   
     DPs:
       { 1: terms^#(mark(X)) -> c_20(terms^#(X))
       , 2: terms^#(ok(X)) -> c_21(terms^#(X))
       , 3: cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2))
       , 4: cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2))
       , 5: recip^#(mark(X)) -> c_24(recip^#(X))
       , 6: recip^#(ok(X)) -> c_25(recip^#(X))
       , 7: sqr^#(mark(X)) -> c_26(sqr^#(X))
       , 8: sqr^#(ok(X)) -> c_27(sqr^#(X))
       , 9: s^#(mark(X)) -> c_28(s^#(X))
       , 10: s^#(ok(X)) -> c_29(s^#(X))
       , 11: add^#(X1, mark(X2)) -> c_30(add^#(X1, X2))
       , 12: add^#(mark(X1), X2) -> c_31(add^#(X1, X2))
       , 13: add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2))
       , 14: dbl^#(mark(X)) -> c_33(dbl^#(X))
       , 15: dbl^#(ok(X)) -> c_34(dbl^#(X))
       , 16: first^#(X1, mark(X2)) -> c_35(first^#(X1, X2))
       , 17: first^#(mark(X1), X2) -> c_36(first^#(X1, X2))
       , 18: first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2))
       , 19: proper^#(0()) -> c_43()
       , 20: proper^#(nil()) -> c_47()
       , 21: top^#(mark(X)) -> c_48(top^#(proper(X)))
       , 22: top^#(ok(X)) -> c_49(top^#(active(X))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { terms^#(mark(X)) -> c_20(terms^#(X))
     , terms^#(ok(X)) -> c_21(terms^#(X))
     , cons^#(mark(X1), X2) -> c_22(cons^#(X1, X2))
     , cons^#(ok(X1), ok(X2)) -> c_23(cons^#(X1, X2))
     , recip^#(mark(X)) -> c_24(recip^#(X))
     , recip^#(ok(X)) -> c_25(recip^#(X))
     , sqr^#(mark(X)) -> c_26(sqr^#(X))
     , sqr^#(ok(X)) -> c_27(sqr^#(X))
     , s^#(mark(X)) -> c_28(s^#(X))
     , s^#(ok(X)) -> c_29(s^#(X))
     , add^#(X1, mark(X2)) -> c_30(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_31(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_32(add^#(X1, X2))
     , dbl^#(mark(X)) -> c_33(dbl^#(X))
     , dbl^#(ok(X)) -> c_34(dbl^#(X))
     , first^#(X1, mark(X2)) -> c_35(first^#(X1, X2))
     , first^#(mark(X1), X2) -> c_36(first^#(X1, X2))
     , first^#(ok(X1), ok(X2)) -> c_37(first^#(X1, X2))
     , top^#(mark(X)) -> c_48(top^#(proper(X)))
     , top^#(ok(X)) -> c_49(top^#(active(X))) }
   Strict Trs:
     { active(terms(X)) -> terms(active(X))
     , active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
     , active(cons(X1, X2)) -> cons(active(X1), X2)
     , active(recip(X)) -> recip(active(X))
     , active(sqr(X)) -> sqr(active(X))
     , active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
     , active(sqr(0())) -> mark(0())
     , active(s(X)) -> s(active(X))
     , active(add(X1, X2)) -> add(X1, active(X2))
     , active(add(X1, X2)) -> add(active(X1), X2)
     , active(add(s(X), Y)) -> mark(s(add(X, Y)))
     , active(add(0(), X)) -> mark(X)
     , active(dbl(X)) -> dbl(active(X))
     , active(dbl(s(X))) -> mark(s(s(dbl(X))))
     , active(dbl(0())) -> mark(0())
     , active(first(X1, X2)) -> first(X1, active(X2))
     , active(first(X1, X2)) -> first(active(X1), X2)
     , active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
     , active(first(0(), X)) -> mark(nil())
     , terms(mark(X)) -> mark(terms(X))
     , terms(ok(X)) -> ok(terms(X))
     , cons(mark(X1), X2) -> mark(cons(X1, X2))
     , cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
     , recip(mark(X)) -> mark(recip(X))
     , recip(ok(X)) -> ok(recip(X))
     , sqr(mark(X)) -> mark(sqr(X))
     , sqr(ok(X)) -> ok(sqr(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , add(X1, mark(X2)) -> mark(add(X1, X2))
     , add(mark(X1), X2) -> mark(add(X1, X2))
     , add(ok(X1), ok(X2)) -> ok(add(X1, X2))
     , dbl(mark(X)) -> mark(dbl(X))
     , dbl(ok(X)) -> ok(dbl(X))
     , first(X1, mark(X2)) -> mark(first(X1, X2))
     , first(mark(X1), X2) -> mark(first(X1, X2))
     , first(ok(X1), ok(X2)) -> ok(first(X1, X2))
     , proper(terms(X)) -> terms(proper(X))
     , proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
     , proper(recip(X)) -> recip(proper(X))
     , proper(sqr(X)) -> sqr(proper(X))
     , proper(s(X)) -> s(proper(X))
     , proper(0()) -> ok(0())
     , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
     , proper(dbl(X)) -> dbl(proper(X))
     , proper(first(X1, X2)) -> first(proper(X1), proper(X2))
     , proper(nil()) -> ok(nil())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Weak DPs:
     { proper^#(0()) -> c_43()
     , proper^#(nil()) -> c_47() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..