MAYBE

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

Strict Trs:
  { gt(s(x), s(y)) -> gt(x, y)
  , gt(s(x), 0()) -> true()
  , gt(0(), y) -> false()
  , divides(x, y) -> div(x, y, y)
  , div(s(x), s(y), z) -> div(x, y, z)
  , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
  , div(0(), s(x), z) -> false()
  , div(0(), 0(), z) -> true()
  , prime(x) -> test(x, s(s(0())))
  , test(x, y) -> if1(gt(x, y), x, y)
  , if1(true(), x, y) -> if2(divides(x, y), x, y)
  , if1(false(), x, y) -> true()
  , if2(true(), x, y) -> false()
  , if2(false(), x, y) -> test(x, s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
  , gt^#(s(x), 0()) -> c_2()
  , gt^#(0(), y) -> c_3()
  , divides^#(x, y) -> c_4(div^#(x, y, y))
  , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
  , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
  , div^#(0(), s(x), z) -> c_7()
  , div^#(0(), 0(), z) -> c_8()
  , prime^#(x) -> c_9(test^#(x, s(s(0()))))
  , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
  , if1^#(true(), x, y) ->
    c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
  , if1^#(false(), x, y) -> c_12()
  , if2^#(true(), x, y) -> c_13()
  , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }

and mark the set of starting terms.

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

Strict DPs:
  { gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
  , gt^#(s(x), 0()) -> c_2()
  , gt^#(0(), y) -> c_3()
  , divides^#(x, y) -> c_4(div^#(x, y, y))
  , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
  , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
  , div^#(0(), s(x), z) -> c_7()
  , div^#(0(), 0(), z) -> c_8()
  , prime^#(x) -> c_9(test^#(x, s(s(0()))))
  , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
  , if1^#(true(), x, y) ->
    c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
  , if1^#(false(), x, y) -> c_12()
  , if2^#(true(), x, y) -> c_13()
  , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }
Weak Trs:
  { gt(s(x), s(y)) -> gt(x, y)
  , gt(s(x), 0()) -> true()
  , gt(0(), y) -> false()
  , divides(x, y) -> div(x, y, y)
  , div(s(x), s(y), z) -> div(x, y, z)
  , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
  , div(0(), s(x), z) -> false()
  , div(0(), 0(), z) -> true()
  , prime(x) -> test(x, s(s(0())))
  , test(x, y) -> if1(gt(x, y), x, y)
  , if1(true(), x, y) -> if2(divides(x, y), x, y)
  , if1(false(), x, y) -> true()
  , if2(true(), x, y) -> false()
  , if2(false(), x, y) -> test(x, s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2,3,7,8,12,13} by
applications of Pre({2,3,7,8,12,13}) = {1,4,5,10,11}. Here rules
are labeled as follows:

  DPs:
    { 1: gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
    , 2: gt^#(s(x), 0()) -> c_2()
    , 3: gt^#(0(), y) -> c_3()
    , 4: divides^#(x, y) -> c_4(div^#(x, y, y))
    , 5: div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
    , 6: div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
    , 7: div^#(0(), s(x), z) -> c_7()
    , 8: div^#(0(), 0(), z) -> c_8()
    , 9: prime^#(x) -> c_9(test^#(x, s(s(0()))))
    , 10: test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
    , 11: if1^#(true(), x, y) ->
          c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
    , 12: if1^#(false(), x, y) -> c_12()
    , 13: if2^#(true(), x, y) -> c_13()
    , 14: if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }

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

Strict DPs:
  { gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
  , divides^#(x, y) -> c_4(div^#(x, y, y))
  , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
  , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
  , prime^#(x) -> c_9(test^#(x, s(s(0()))))
  , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
  , if1^#(true(), x, y) ->
    c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
  , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }
Weak DPs:
  { gt^#(s(x), 0()) -> c_2()
  , gt^#(0(), y) -> c_3()
  , div^#(0(), s(x), z) -> c_7()
  , div^#(0(), 0(), z) -> c_8()
  , if1^#(false(), x, y) -> c_12()
  , if2^#(true(), x, y) -> c_13() }
Weak Trs:
  { gt(s(x), s(y)) -> gt(x, y)
  , gt(s(x), 0()) -> true()
  , gt(0(), y) -> false()
  , divides(x, y) -> div(x, y, y)
  , div(s(x), s(y), z) -> div(x, y, z)
  , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
  , div(0(), s(x), z) -> false()
  , div(0(), 0(), z) -> true()
  , prime(x) -> test(x, s(s(0())))
  , test(x, y) -> if1(gt(x, y), x, y)
  , if1(true(), x, y) -> if2(divides(x, y), x, y)
  , if1(false(), x, y) -> true()
  , if2(true(), x, y) -> false()
  , if2(false(), x, y) -> test(x, s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ gt^#(s(x), 0()) -> c_2()
, gt^#(0(), y) -> c_3()
, div^#(0(), s(x), z) -> c_7()
, div^#(0(), 0(), z) -> c_8()
, if1^#(false(), x, y) -> c_12()
, if2^#(true(), x, y) -> c_13() }

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

Strict DPs:
  { gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
  , divides^#(x, y) -> c_4(div^#(x, y, y))
  , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
  , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
  , prime^#(x) -> c_9(test^#(x, s(s(0()))))
  , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
  , if1^#(true(), x, y) ->
    c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
  , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }
Weak Trs:
  { gt(s(x), s(y)) -> gt(x, y)
  , gt(s(x), 0()) -> true()
  , gt(0(), y) -> false()
  , divides(x, y) -> div(x, y, y)
  , div(s(x), s(y), z) -> div(x, y, z)
  , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
  , div(0(), s(x), z) -> false()
  , div(0(), 0(), z) -> true()
  , prime(x) -> test(x, s(s(0())))
  , test(x, y) -> if1(gt(x, y), x, y)
  , if1(true(), x, y) -> if2(divides(x, y), x, y)
  , if1(false(), x, y) -> true()
  , if2(true(), x, y) -> false()
  , if2(false(), x, y) -> test(x, s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
     -->_1 gt^#(s(x), s(y)) -> c_1(gt^#(x, y)) :1
  
  2: divides^#(x, y) -> c_4(div^#(x, y, y))
     -->_1 div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) :3
  
  3: div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
     -->_1 div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z))) :4
     -->_1 div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) :3
  
  4: div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
     -->_1 div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) :3
  
  5: prime^#(x) -> c_9(test^#(x, s(s(0()))))
     -->_1 test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y)) :6
  
  6: test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
     -->_1 if1^#(true(), x, y) ->
           c_11(if2^#(divides(x, y), x, y), divides^#(x, y)) :7
     -->_2 gt^#(s(x), s(y)) -> c_1(gt^#(x, y)) :1
  
  7: if1^#(true(), x, y) ->
     c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
     -->_1 if2^#(false(), x, y) -> c_14(test^#(x, s(y))) :8
     -->_2 divides^#(x, y) -> c_4(div^#(x, y, y)) :2
  
  8: if2^#(false(), x, y) -> c_14(test^#(x, s(y)))
     -->_1 test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y)) :6
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { prime^#(x) -> c_9(test^#(x, s(s(0())))) }


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

Strict DPs:
  { gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
  , divides^#(x, y) -> c_4(div^#(x, y, y))
  , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
  , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
  , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
  , if1^#(true(), x, y) ->
    c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
  , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }
Weak Trs:
  { gt(s(x), s(y)) -> gt(x, y)
  , gt(s(x), 0()) -> true()
  , gt(0(), y) -> false()
  , divides(x, y) -> div(x, y, y)
  , div(s(x), s(y), z) -> div(x, y, z)
  , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
  , div(0(), s(x), z) -> false()
  , div(0(), 0(), z) -> true()
  , prime(x) -> test(x, s(s(0())))
  , test(x, y) -> if1(gt(x, y), x, y)
  , if1(true(), x, y) -> if2(divides(x, y), x, y)
  , if1(false(), x, y) -> true()
  , if2(true(), x, y) -> false()
  , if2(false(), x, y) -> test(x, s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { gt(s(x), s(y)) -> gt(x, y)
    , gt(s(x), 0()) -> true()
    , gt(0(), y) -> false()
    , divides(x, y) -> div(x, y, y)
    , div(s(x), s(y), z) -> div(x, y, z)
    , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
    , div(0(), s(x), z) -> false()
    , div(0(), 0(), z) -> true() }

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

Strict DPs:
  { gt^#(s(x), s(y)) -> c_1(gt^#(x, y))
  , divides^#(x, y) -> c_4(div^#(x, y, y))
  , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z))
  , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z)))
  , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y), gt^#(x, y))
  , if1^#(true(), x, y) ->
    c_11(if2^#(divides(x, y), x, y), divides^#(x, y))
  , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) }
Weak Trs:
  { gt(s(x), s(y)) -> gt(x, y)
  , gt(s(x), 0()) -> true()
  , gt(0(), y) -> false()
  , divides(x, y) -> div(x, y, y)
  , div(s(x), s(y), z) -> div(x, y, z)
  , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z))
  , div(0(), s(x), z) -> false()
  , div(0(), 0(), z) -> true() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..