MAYBE

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

Strict Trs:
  { active(fact(X)) -> fact(active(X))
  , active(fact(X)) -> mark(if(zero(X), s(0()), prod(X, fact(p(X)))))
  , active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
  , active(if(true(), X, Y)) -> mark(X)
  , active(if(false(), X, Y)) -> mark(Y)
  , active(zero(X)) -> zero(active(X))
  , active(zero(s(X))) -> mark(false())
  , active(zero(0())) -> mark(true())
  , active(s(X)) -> s(active(X))
  , active(prod(X1, X2)) -> prod(X1, active(X2))
  , active(prod(X1, X2)) -> prod(active(X1), X2)
  , active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
  , active(prod(0(), X)) -> mark(0())
  , active(p(X)) -> p(active(X))
  , active(p(s(X))) -> mark(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)
  , fact(mark(X)) -> mark(fact(X))
  , fact(ok(X)) -> ok(fact(X))
  , if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
  , if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
  , zero(mark(X)) -> mark(zero(X))
  , zero(ok(X)) -> ok(zero(X))
  , s(mark(X)) -> mark(s(X))
  , s(ok(X)) -> ok(s(X))
  , prod(X1, mark(X2)) -> mark(prod(X1, X2))
  , prod(mark(X1), X2) -> mark(prod(X1, X2))
  , prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
  , p(mark(X)) -> mark(p(X))
  , p(ok(X)) -> ok(p(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))
  , proper(fact(X)) -> fact(proper(X))
  , proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
  , proper(zero(X)) -> zero(proper(X))
  , proper(s(X)) -> s(proper(X))
  , proper(0()) -> ok(0())
  , proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
  , proper(p(X)) -> p(proper(X))
  , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
  , proper(true()) -> ok(true())
  , proper(false()) -> ok(false())
  , 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) '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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
   due to the following reason:
   
   We add the following weak dependency pairs:
   
   Strict DPs:
     { active^#(fact(X)) -> c_1(fact^#(active(X)))
     , active^#(fact(X)) ->
       c_2(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
     , active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3))
     , active^#(if(true(), X, Y)) -> c_4(X)
     , active^#(if(false(), X, Y)) -> c_5(Y)
     , active^#(zero(X)) -> c_6(zero^#(active(X)))
     , active^#(zero(s(X))) -> c_7()
     , active^#(zero(0())) -> c_8()
     , active^#(s(X)) -> c_9(s^#(active(X)))
     , active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2)))
     , active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2))
     , active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y)))
     , active^#(prod(0(), X)) -> c_13()
     , active^#(p(X)) -> c_14(p^#(active(X)))
     , active^#(p(s(X))) -> c_15(X)
     , active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2)))
     , active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2))
     , active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y)))
     , active^#(add(0(), X)) -> c_19(X)
     , fact^#(mark(X)) -> c_20(fact^#(X))
     , fact^#(ok(X)) -> c_21(fact^#(X))
     , if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3))
     , if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3))
     , zero^#(mark(X)) -> c_24(zero^#(X))
     , zero^#(ok(X)) -> c_25(zero^#(X))
     , s^#(mark(X)) -> c_26(s^#(X))
     , s^#(ok(X)) -> c_27(s^#(X))
     , prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2))
     , prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2))
     , prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2))
     , add^#(X1, mark(X2)) -> c_33(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_34(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2))
     , p^#(mark(X)) -> c_31(p^#(X))
     , p^#(ok(X)) -> c_32(p^#(X))
     , proper^#(fact(X)) -> c_36(fact^#(proper(X)))
     , proper^#(if(X1, X2, X3)) ->
       c_37(if^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(zero(X)) -> c_38(zero^#(proper(X)))
     , proper^#(s(X)) -> c_39(s^#(proper(X)))
     , proper^#(0()) -> c_40()
     , proper^#(prod(X1, X2)) -> c_41(prod^#(proper(X1), proper(X2)))
     , proper^#(p(X)) -> c_42(p^#(proper(X)))
     , proper^#(add(X1, X2)) -> c_43(add^#(proper(X1), proper(X2)))
     , proper^#(true()) -> c_44()
     , proper^#(false()) -> c_45()
     , top^#(mark(X)) -> c_46(top^#(proper(X)))
     , top^#(ok(X)) -> c_47(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^#(fact(X)) -> c_1(fact^#(active(X)))
     , active^#(fact(X)) ->
       c_2(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
     , active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3))
     , active^#(if(true(), X, Y)) -> c_4(X)
     , active^#(if(false(), X, Y)) -> c_5(Y)
     , active^#(zero(X)) -> c_6(zero^#(active(X)))
     , active^#(zero(s(X))) -> c_7()
     , active^#(zero(0())) -> c_8()
     , active^#(s(X)) -> c_9(s^#(active(X)))
     , active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2)))
     , active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2))
     , active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y)))
     , active^#(prod(0(), X)) -> c_13()
     , active^#(p(X)) -> c_14(p^#(active(X)))
     , active^#(p(s(X))) -> c_15(X)
     , active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2)))
     , active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2))
     , active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y)))
     , active^#(add(0(), X)) -> c_19(X)
     , fact^#(mark(X)) -> c_20(fact^#(X))
     , fact^#(ok(X)) -> c_21(fact^#(X))
     , if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3))
     , if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3))
     , zero^#(mark(X)) -> c_24(zero^#(X))
     , zero^#(ok(X)) -> c_25(zero^#(X))
     , s^#(mark(X)) -> c_26(s^#(X))
     , s^#(ok(X)) -> c_27(s^#(X))
     , prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2))
     , prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2))
     , prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2))
     , add^#(X1, mark(X2)) -> c_33(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_34(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2))
     , p^#(mark(X)) -> c_31(p^#(X))
     , p^#(ok(X)) -> c_32(p^#(X))
     , proper^#(fact(X)) -> c_36(fact^#(proper(X)))
     , proper^#(if(X1, X2, X3)) ->
       c_37(if^#(proper(X1), proper(X2), proper(X3)))
     , proper^#(zero(X)) -> c_38(zero^#(proper(X)))
     , proper^#(s(X)) -> c_39(s^#(proper(X)))
     , proper^#(0()) -> c_40()
     , proper^#(prod(X1, X2)) -> c_41(prod^#(proper(X1), proper(X2)))
     , proper^#(p(X)) -> c_42(p^#(proper(X)))
     , proper^#(add(X1, X2)) -> c_43(add^#(proper(X1), proper(X2)))
     , proper^#(true()) -> c_44()
     , proper^#(false()) -> c_45()
     , top^#(mark(X)) -> c_46(top^#(proper(X)))
     , top^#(ok(X)) -> c_47(top^#(active(X))) }
   Strict Trs:
     { active(fact(X)) -> fact(active(X))
     , active(fact(X)) -> mark(if(zero(X), s(0()), prod(X, fact(p(X)))))
     , active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
     , active(if(true(), X, Y)) -> mark(X)
     , active(if(false(), X, Y)) -> mark(Y)
     , active(zero(X)) -> zero(active(X))
     , active(zero(s(X))) -> mark(false())
     , active(zero(0())) -> mark(true())
     , active(s(X)) -> s(active(X))
     , active(prod(X1, X2)) -> prod(X1, active(X2))
     , active(prod(X1, X2)) -> prod(active(X1), X2)
     , active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
     , active(prod(0(), X)) -> mark(0())
     , active(p(X)) -> p(active(X))
     , active(p(s(X))) -> mark(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)
     , fact(mark(X)) -> mark(fact(X))
     , fact(ok(X)) -> ok(fact(X))
     , if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
     , if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
     , zero(mark(X)) -> mark(zero(X))
     , zero(ok(X)) -> ok(zero(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , prod(X1, mark(X2)) -> mark(prod(X1, X2))
     , prod(mark(X1), X2) -> mark(prod(X1, X2))
     , prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
     , p(mark(X)) -> mark(p(X))
     , p(ok(X)) -> ok(p(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))
     , proper(fact(X)) -> fact(proper(X))
     , proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
     , proper(zero(X)) -> zero(proper(X))
     , proper(s(X)) -> s(proper(X))
     , proper(0()) -> ok(0())
     , proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
     , proper(p(X)) -> p(proper(X))
     , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
     , proper(true()) -> ok(true())
     , proper(false()) -> ok(false())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Consider the dependency graph:
   
     1: active^#(fact(X)) -> c_1(fact^#(active(X)))
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
     
     2: active^#(fact(X)) ->
        c_2(if^#(zero(X), s(0()), prod(X, fact(p(X)))))
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
     
     3: active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3))
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
     
     4: active^#(if(true(), X, Y)) -> c_4(X)
        -->_1 top^#(ok(X)) -> c_47(top^#(active(X))) :47
        -->_1 top^#(mark(X)) -> c_46(top^#(proper(X))) :46
        -->_1 proper^#(add(X1, X2)) ->
              c_43(add^#(proper(X1), proper(X2))) :43
        -->_1 proper^#(p(X)) -> c_42(p^#(proper(X))) :42
        -->_1 proper^#(prod(X1, X2)) ->
              c_41(prod^#(proper(X1), proper(X2))) :41
        -->_1 proper^#(s(X)) -> c_39(s^#(proper(X))) :39
        -->_1 proper^#(zero(X)) -> c_38(zero^#(proper(X))) :38
        -->_1 proper^#(if(X1, X2, X3)) ->
              c_37(if^#(proper(X1), proper(X2), proper(X3))) :37
        -->_1 proper^#(fact(X)) -> c_36(fact^#(proper(X))) :36
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
        -->_1 active^#(add(0(), X)) -> c_19(X) :19
        -->_1 active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y))) :18
        -->_1 active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2)) :17
        -->_1 active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2))) :16
        -->_1 active^#(p(s(X))) -> c_15(X) :15
        -->_1 active^#(p(X)) -> c_14(p^#(active(X))) :14
        -->_1 active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y))) :12
        -->_1 active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2)) :11
        -->_1 active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2))) :10
        -->_1 active^#(s(X)) -> c_9(s^#(active(X))) :9
        -->_1 active^#(zero(X)) -> c_6(zero^#(active(X))) :6
        -->_1 active^#(if(false(), X, Y)) -> c_5(Y) :5
        -->_1 proper^#(false()) -> c_45() :45
        -->_1 proper^#(true()) -> c_44() :44
        -->_1 proper^#(0()) -> c_40() :40
        -->_1 active^#(prod(0(), X)) -> c_13() :13
        -->_1 active^#(zero(0())) -> c_8() :8
        -->_1 active^#(zero(s(X))) -> c_7() :7
        -->_1 active^#(if(true(), X, Y)) -> c_4(X) :4
        -->_1 active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3)) :3
        -->_1 active^#(fact(X)) ->
              c_2(if^#(zero(X), s(0()), prod(X, fact(p(X))))) :2
        -->_1 active^#(fact(X)) -> c_1(fact^#(active(X))) :1
     
     5: active^#(if(false(), X, Y)) -> c_5(Y)
        -->_1 top^#(ok(X)) -> c_47(top^#(active(X))) :47
        -->_1 top^#(mark(X)) -> c_46(top^#(proper(X))) :46
        -->_1 proper^#(add(X1, X2)) ->
              c_43(add^#(proper(X1), proper(X2))) :43
        -->_1 proper^#(p(X)) -> c_42(p^#(proper(X))) :42
        -->_1 proper^#(prod(X1, X2)) ->
              c_41(prod^#(proper(X1), proper(X2))) :41
        -->_1 proper^#(s(X)) -> c_39(s^#(proper(X))) :39
        -->_1 proper^#(zero(X)) -> c_38(zero^#(proper(X))) :38
        -->_1 proper^#(if(X1, X2, X3)) ->
              c_37(if^#(proper(X1), proper(X2), proper(X3))) :37
        -->_1 proper^#(fact(X)) -> c_36(fact^#(proper(X))) :36
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
        -->_1 active^#(add(0(), X)) -> c_19(X) :19
        -->_1 active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y))) :18
        -->_1 active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2)) :17
        -->_1 active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2))) :16
        -->_1 active^#(p(s(X))) -> c_15(X) :15
        -->_1 active^#(p(X)) -> c_14(p^#(active(X))) :14
        -->_1 active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y))) :12
        -->_1 active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2)) :11
        -->_1 active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2))) :10
        -->_1 active^#(s(X)) -> c_9(s^#(active(X))) :9
        -->_1 active^#(zero(X)) -> c_6(zero^#(active(X))) :6
        -->_1 proper^#(false()) -> c_45() :45
        -->_1 proper^#(true()) -> c_44() :44
        -->_1 proper^#(0()) -> c_40() :40
        -->_1 active^#(prod(0(), X)) -> c_13() :13
        -->_1 active^#(zero(0())) -> c_8() :8
        -->_1 active^#(zero(s(X))) -> c_7() :7
        -->_1 active^#(if(false(), X, Y)) -> c_5(Y) :5
        -->_1 active^#(if(true(), X, Y)) -> c_4(X) :4
        -->_1 active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3)) :3
        -->_1 active^#(fact(X)) ->
              c_2(if^#(zero(X), s(0()), prod(X, fact(p(X))))) :2
        -->_1 active^#(fact(X)) -> c_1(fact^#(active(X))) :1
     
     6: active^#(zero(X)) -> c_6(zero^#(active(X)))
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
     
     7: active^#(zero(s(X))) -> c_7()
     
     8: active^#(zero(0())) -> c_8()
     
     9: active^#(s(X)) -> c_9(s^#(active(X)))
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
     
     10: active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2)))
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
     
     11: active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2))
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
     
     12: active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y)))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     13: active^#(prod(0(), X)) -> c_13()
     
     14: active^#(p(X)) -> c_14(p^#(active(X)))
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
     
     15: active^#(p(s(X))) -> c_15(X)
        -->_1 top^#(ok(X)) -> c_47(top^#(active(X))) :47
        -->_1 top^#(mark(X)) -> c_46(top^#(proper(X))) :46
        -->_1 proper^#(add(X1, X2)) ->
              c_43(add^#(proper(X1), proper(X2))) :43
        -->_1 proper^#(p(X)) -> c_42(p^#(proper(X))) :42
        -->_1 proper^#(prod(X1, X2)) ->
              c_41(prod^#(proper(X1), proper(X2))) :41
        -->_1 proper^#(s(X)) -> c_39(s^#(proper(X))) :39
        -->_1 proper^#(zero(X)) -> c_38(zero^#(proper(X))) :38
        -->_1 proper^#(if(X1, X2, X3)) ->
              c_37(if^#(proper(X1), proper(X2), proper(X3))) :37
        -->_1 proper^#(fact(X)) -> c_36(fact^#(proper(X))) :36
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
        -->_1 active^#(add(0(), X)) -> c_19(X) :19
        -->_1 active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y))) :18
        -->_1 active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2)) :17
        -->_1 active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2))) :16
        -->_1 proper^#(false()) -> c_45() :45
        -->_1 proper^#(true()) -> c_44() :44
        -->_1 proper^#(0()) -> c_40() :40
        -->_1 active^#(p(s(X))) -> c_15(X) :15
        -->_1 active^#(p(X)) -> c_14(p^#(active(X))) :14
        -->_1 active^#(prod(0(), X)) -> c_13() :13
        -->_1 active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y))) :12
        -->_1 active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2)) :11
        -->_1 active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2))) :10
        -->_1 active^#(s(X)) -> c_9(s^#(active(X))) :9
        -->_1 active^#(zero(0())) -> c_8() :8
        -->_1 active^#(zero(s(X))) -> c_7() :7
        -->_1 active^#(zero(X)) -> c_6(zero^#(active(X))) :6
        -->_1 active^#(if(false(), X, Y)) -> c_5(Y) :5
        -->_1 active^#(if(true(), X, Y)) -> c_4(X) :4
        -->_1 active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3)) :3
        -->_1 active^#(fact(X)) ->
              c_2(if^#(zero(X), s(0()), prod(X, fact(p(X))))) :2
        -->_1 active^#(fact(X)) -> c_1(fact^#(active(X))) :1
     
     16: active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2)))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     17: active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     18: active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y)))
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
     
     19: active^#(add(0(), X)) -> c_19(X)
        -->_1 top^#(ok(X)) -> c_47(top^#(active(X))) :47
        -->_1 top^#(mark(X)) -> c_46(top^#(proper(X))) :46
        -->_1 proper^#(add(X1, X2)) ->
              c_43(add^#(proper(X1), proper(X2))) :43
        -->_1 proper^#(p(X)) -> c_42(p^#(proper(X))) :42
        -->_1 proper^#(prod(X1, X2)) ->
              c_41(prod^#(proper(X1), proper(X2))) :41
        -->_1 proper^#(s(X)) -> c_39(s^#(proper(X))) :39
        -->_1 proper^#(zero(X)) -> c_38(zero^#(proper(X))) :38
        -->_1 proper^#(if(X1, X2, X3)) ->
              c_37(if^#(proper(X1), proper(X2), proper(X3))) :37
        -->_1 proper^#(fact(X)) -> c_36(fact^#(proper(X))) :36
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
        -->_1 proper^#(false()) -> c_45() :45
        -->_1 proper^#(true()) -> c_44() :44
        -->_1 proper^#(0()) -> c_40() :40
        -->_1 active^#(add(0(), X)) -> c_19(X) :19
        -->_1 active^#(add(s(X), Y)) -> c_18(s^#(add(X, Y))) :18
        -->_1 active^#(add(X1, X2)) -> c_17(add^#(active(X1), X2)) :17
        -->_1 active^#(add(X1, X2)) -> c_16(add^#(X1, active(X2))) :16
        -->_1 active^#(p(s(X))) -> c_15(X) :15
        -->_1 active^#(p(X)) -> c_14(p^#(active(X))) :14
        -->_1 active^#(prod(0(), X)) -> c_13() :13
        -->_1 active^#(prod(s(X), Y)) -> c_12(add^#(Y, prod(X, Y))) :12
        -->_1 active^#(prod(X1, X2)) -> c_11(prod^#(active(X1), X2)) :11
        -->_1 active^#(prod(X1, X2)) -> c_10(prod^#(X1, active(X2))) :10
        -->_1 active^#(s(X)) -> c_9(s^#(active(X))) :9
        -->_1 active^#(zero(0())) -> c_8() :8
        -->_1 active^#(zero(s(X))) -> c_7() :7
        -->_1 active^#(zero(X)) -> c_6(zero^#(active(X))) :6
        -->_1 active^#(if(false(), X, Y)) -> c_5(Y) :5
        -->_1 active^#(if(true(), X, Y)) -> c_4(X) :4
        -->_1 active^#(if(X1, X2, X3)) -> c_3(if^#(active(X1), X2, X3)) :3
        -->_1 active^#(fact(X)) ->
              c_2(if^#(zero(X), s(0()), prod(X, fact(p(X))))) :2
        -->_1 active^#(fact(X)) -> c_1(fact^#(active(X))) :1
     
     20: fact^#(mark(X)) -> c_20(fact^#(X))
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
     
     21: fact^#(ok(X)) -> c_21(fact^#(X))
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
     
     22: if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3))
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
     
     23: if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3))
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
     
     24: zero^#(mark(X)) -> c_24(zero^#(X))
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
     
     25: zero^#(ok(X)) -> c_25(zero^#(X))
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
     
     26: s^#(mark(X)) -> c_26(s^#(X))
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
     
     27: s^#(ok(X)) -> c_27(s^#(X))
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
     
     28: prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2))
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
     
     29: prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2))
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
     
     30: prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2))
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
     
     31: add^#(X1, mark(X2)) -> c_33(add^#(X1, X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     32: add^#(mark(X1), X2) -> c_34(add^#(X1, X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     33: add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     34: p^#(mark(X)) -> c_31(p^#(X))
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
     
     35: p^#(ok(X)) -> c_32(p^#(X))
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
     
     36: proper^#(fact(X)) -> c_36(fact^#(proper(X)))
        -->_1 fact^#(ok(X)) -> c_21(fact^#(X)) :21
        -->_1 fact^#(mark(X)) -> c_20(fact^#(X)) :20
     
     37: proper^#(if(X1, X2, X3)) ->
         c_37(if^#(proper(X1), proper(X2), proper(X3)))
        -->_1 if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3)) :23
        -->_1 if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3)) :22
     
     38: proper^#(zero(X)) -> c_38(zero^#(proper(X)))
        -->_1 zero^#(ok(X)) -> c_25(zero^#(X)) :25
        -->_1 zero^#(mark(X)) -> c_24(zero^#(X)) :24
     
     39: proper^#(s(X)) -> c_39(s^#(proper(X)))
        -->_1 s^#(ok(X)) -> c_27(s^#(X)) :27
        -->_1 s^#(mark(X)) -> c_26(s^#(X)) :26
     
     40: proper^#(0()) -> c_40()
     
     41: proper^#(prod(X1, X2)) -> c_41(prod^#(proper(X1), proper(X2)))
        -->_1 prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2)) :30
        -->_1 prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2)) :29
        -->_1 prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2)) :28
     
     42: proper^#(p(X)) -> c_42(p^#(proper(X)))
        -->_1 p^#(ok(X)) -> c_32(p^#(X)) :35
        -->_1 p^#(mark(X)) -> c_31(p^#(X)) :34
     
     43: proper^#(add(X1, X2)) -> c_43(add^#(proper(X1), proper(X2)))
        -->_1 add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2)) :33
        -->_1 add^#(mark(X1), X2) -> c_34(add^#(X1, X2)) :32
        -->_1 add^#(X1, mark(X2)) -> c_33(add^#(X1, X2)) :31
     
     44: proper^#(true()) -> c_44()
     
     45: proper^#(false()) -> c_45()
     
     46: top^#(mark(X)) -> c_46(top^#(proper(X)))
        -->_1 top^#(ok(X)) -> c_47(top^#(active(X))) :47
        -->_1 top^#(mark(X)) -> c_46(top^#(proper(X))) :46
     
     47: top^#(ok(X)) -> c_47(top^#(active(X)))
        -->_1 top^#(ok(X)) -> c_47(top^#(active(X))) :47
        -->_1 top^#(mark(X)) -> c_46(top^#(proper(X))) :46
     
   
   Only the nodes
   {20,21,22,23,24,25,26,27,28,30,29,31,33,32,34,35,40,44,45,46,47}
   are reachable from nodes
   {20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,40,44,45,46,47}
   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:
     { fact^#(mark(X)) -> c_20(fact^#(X))
     , fact^#(ok(X)) -> c_21(fact^#(X))
     , if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3))
     , if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3))
     , zero^#(mark(X)) -> c_24(zero^#(X))
     , zero^#(ok(X)) -> c_25(zero^#(X))
     , s^#(mark(X)) -> c_26(s^#(X))
     , s^#(ok(X)) -> c_27(s^#(X))
     , prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2))
     , prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2))
     , prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2))
     , add^#(X1, mark(X2)) -> c_33(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_34(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2))
     , p^#(mark(X)) -> c_31(p^#(X))
     , p^#(ok(X)) -> c_32(p^#(X))
     , proper^#(0()) -> c_40()
     , proper^#(true()) -> c_44()
     , proper^#(false()) -> c_45()
     , top^#(mark(X)) -> c_46(top^#(proper(X)))
     , top^#(ok(X)) -> c_47(top^#(active(X))) }
   Strict Trs:
     { active(fact(X)) -> fact(active(X))
     , active(fact(X)) -> mark(if(zero(X), s(0()), prod(X, fact(p(X)))))
     , active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
     , active(if(true(), X, Y)) -> mark(X)
     , active(if(false(), X, Y)) -> mark(Y)
     , active(zero(X)) -> zero(active(X))
     , active(zero(s(X))) -> mark(false())
     , active(zero(0())) -> mark(true())
     , active(s(X)) -> s(active(X))
     , active(prod(X1, X2)) -> prod(X1, active(X2))
     , active(prod(X1, X2)) -> prod(active(X1), X2)
     , active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
     , active(prod(0(), X)) -> mark(0())
     , active(p(X)) -> p(active(X))
     , active(p(s(X))) -> mark(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)
     , fact(mark(X)) -> mark(fact(X))
     , fact(ok(X)) -> ok(fact(X))
     , if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
     , if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
     , zero(mark(X)) -> mark(zero(X))
     , zero(ok(X)) -> ok(zero(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , prod(X1, mark(X2)) -> mark(prod(X1, X2))
     , prod(mark(X1), X2) -> mark(prod(X1, X2))
     , prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
     , p(mark(X)) -> mark(p(X))
     , p(ok(X)) -> ok(p(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))
     , proper(fact(X)) -> fact(proper(X))
     , proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
     , proper(zero(X)) -> zero(proper(X))
     , proper(s(X)) -> s(proper(X))
     , proper(0()) -> ok(0())
     , proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
     , proper(p(X)) -> p(proper(X))
     , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
     , proper(true()) -> ok(true())
     , proper(false()) -> ok(false())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   We estimate the number of application of {17,18,19} by applications
   of Pre({17,18,19}) = {}. Here rules are labeled as follows:
   
     DPs:
       { 1: fact^#(mark(X)) -> c_20(fact^#(X))
       , 2: fact^#(ok(X)) -> c_21(fact^#(X))
       , 3: if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3))
       , 4: if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3))
       , 5: zero^#(mark(X)) -> c_24(zero^#(X))
       , 6: zero^#(ok(X)) -> c_25(zero^#(X))
       , 7: s^#(mark(X)) -> c_26(s^#(X))
       , 8: s^#(ok(X)) -> c_27(s^#(X))
       , 9: prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2))
       , 10: prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2))
       , 11: prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2))
       , 12: add^#(X1, mark(X2)) -> c_33(add^#(X1, X2))
       , 13: add^#(mark(X1), X2) -> c_34(add^#(X1, X2))
       , 14: add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2))
       , 15: p^#(mark(X)) -> c_31(p^#(X))
       , 16: p^#(ok(X)) -> c_32(p^#(X))
       , 17: proper^#(0()) -> c_40()
       , 18: proper^#(true()) -> c_44()
       , 19: proper^#(false()) -> c_45()
       , 20: top^#(mark(X)) -> c_46(top^#(proper(X)))
       , 21: top^#(ok(X)) -> c_47(top^#(active(X))) }
   
   We are left with following problem, upon which TcT provides the
   certificate MAYBE.
   
   Strict DPs:
     { fact^#(mark(X)) -> c_20(fact^#(X))
     , fact^#(ok(X)) -> c_21(fact^#(X))
     , if^#(mark(X1), X2, X3) -> c_22(if^#(X1, X2, X3))
     , if^#(ok(X1), ok(X2), ok(X3)) -> c_23(if^#(X1, X2, X3))
     , zero^#(mark(X)) -> c_24(zero^#(X))
     , zero^#(ok(X)) -> c_25(zero^#(X))
     , s^#(mark(X)) -> c_26(s^#(X))
     , s^#(ok(X)) -> c_27(s^#(X))
     , prod^#(X1, mark(X2)) -> c_28(prod^#(X1, X2))
     , prod^#(mark(X1), X2) -> c_29(prod^#(X1, X2))
     , prod^#(ok(X1), ok(X2)) -> c_30(prod^#(X1, X2))
     , add^#(X1, mark(X2)) -> c_33(add^#(X1, X2))
     , add^#(mark(X1), X2) -> c_34(add^#(X1, X2))
     , add^#(ok(X1), ok(X2)) -> c_35(add^#(X1, X2))
     , p^#(mark(X)) -> c_31(p^#(X))
     , p^#(ok(X)) -> c_32(p^#(X))
     , top^#(mark(X)) -> c_46(top^#(proper(X)))
     , top^#(ok(X)) -> c_47(top^#(active(X))) }
   Strict Trs:
     { active(fact(X)) -> fact(active(X))
     , active(fact(X)) -> mark(if(zero(X), s(0()), prod(X, fact(p(X)))))
     , active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
     , active(if(true(), X, Y)) -> mark(X)
     , active(if(false(), X, Y)) -> mark(Y)
     , active(zero(X)) -> zero(active(X))
     , active(zero(s(X))) -> mark(false())
     , active(zero(0())) -> mark(true())
     , active(s(X)) -> s(active(X))
     , active(prod(X1, X2)) -> prod(X1, active(X2))
     , active(prod(X1, X2)) -> prod(active(X1), X2)
     , active(prod(s(X), Y)) -> mark(add(Y, prod(X, Y)))
     , active(prod(0(), X)) -> mark(0())
     , active(p(X)) -> p(active(X))
     , active(p(s(X))) -> mark(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)
     , fact(mark(X)) -> mark(fact(X))
     , fact(ok(X)) -> ok(fact(X))
     , if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
     , if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
     , zero(mark(X)) -> mark(zero(X))
     , zero(ok(X)) -> ok(zero(X))
     , s(mark(X)) -> mark(s(X))
     , s(ok(X)) -> ok(s(X))
     , prod(X1, mark(X2)) -> mark(prod(X1, X2))
     , prod(mark(X1), X2) -> mark(prod(X1, X2))
     , prod(ok(X1), ok(X2)) -> ok(prod(X1, X2))
     , p(mark(X)) -> mark(p(X))
     , p(ok(X)) -> ok(p(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))
     , proper(fact(X)) -> fact(proper(X))
     , proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
     , proper(zero(X)) -> zero(proper(X))
     , proper(s(X)) -> s(proper(X))
     , proper(0()) -> ok(0())
     , proper(prod(X1, X2)) -> prod(proper(X1), proper(X2))
     , proper(p(X)) -> p(proper(X))
     , proper(add(X1, X2)) -> add(proper(X1), proper(X2))
     , proper(true()) -> ok(true())
     , proper(false()) -> ok(false())
     , top(mark(X)) -> top(proper(X))
     , top(ok(X)) -> top(active(X)) }
   Weak DPs:
     { proper^#(0()) -> c_40()
     , proper^#(true()) -> c_44()
     , proper^#(false()) -> c_45() }
   Obligation:
     runtime complexity
   Answer:
     MAYBE
   
   Empty strict component of the problem is NOT empty.


Arrrr..