MAYBE

Problem:
 cond1(true(),x) -> cond2(even(x),x)
 cond2(true(),x) -> cond1(neq(x,0()),div2(x))
 cond2(false(),x) -> cond1(neq(x,0()),p(x))
 neq(0(),0()) -> false()
 neq(0(),s(x)) -> true()
 neq(s(x),0()) -> true()
 neq(s(x),s(y())) -> neq(x,y())
 even(0()) -> true()
 even(s(0())) -> false()
 even(s(s(x))) -> even(x)
 div2(0()) -> 0()
 div2(s(0())) -> 0()
 div2(s(s(x))) -> s(div2(x))
 p(0()) -> 0()
 p(s(x)) -> x

Proof:
 DP Processor:
  DPs:
   cond1#(true(),x) -> even#(x)
   cond1#(true(),x) -> cond2#(even(x),x)
   cond2#(true(),x) -> div2#(x)
   cond2#(true(),x) -> neq#(x,0())
   cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
   cond2#(false(),x) -> p#(x)
   cond2#(false(),x) -> neq#(x,0())
   cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
   neq#(s(x),s(y())) -> neq#(x,y())
   even#(s(s(x))) -> even#(x)
   div2#(s(s(x))) -> div2#(x)
  TRS:
   cond1(true(),x) -> cond2(even(x),x)
   cond2(true(),x) -> cond1(neq(x,0()),div2(x))
   cond2(false(),x) -> cond1(neq(x,0()),p(x))
   neq(0(),0()) -> false()
   neq(0(),s(x)) -> true()
   neq(s(x),0()) -> true()
   neq(s(x),s(y())) -> neq(x,y())
   even(0()) -> true()
   even(s(0())) -> false()
   even(s(s(x))) -> even(x)
   div2(0()) -> 0()
   div2(s(0())) -> 0()
   div2(s(s(x))) -> s(div2(x))
   p(0()) -> 0()
   p(s(x)) -> x
  Usable Rule Processor:
   DPs:
    cond1#(true(),x) -> even#(x)
    cond1#(true(),x) -> cond2#(even(x),x)
    cond2#(true(),x) -> div2#(x)
    cond2#(true(),x) -> neq#(x,0())
    cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
    cond2#(false(),x) -> p#(x)
    cond2#(false(),x) -> neq#(x,0())
    cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
    neq#(s(x),s(y())) -> neq#(x,y())
    even#(s(s(x))) -> even#(x)
    div2#(s(s(x))) -> div2#(x)
   TRS:
    f17(x,y) -> x
    f17(x,y) -> y
    even(0()) -> true()
    even(s(0())) -> false()
    even(s(s(x))) -> even(x)
    div2(0()) -> 0()
    div2(s(0())) -> 0()
    div2(s(s(x))) -> s(div2(x))
    neq(0(),0()) -> false()
    neq(s(x),0()) -> true()
    p(0()) -> 0()
    p(s(x)) -> x
   CDG Processor:
    DPs:
     cond1#(true(),x) -> even#(x)
     cond1#(true(),x) -> cond2#(even(x),x)
     cond2#(true(),x) -> div2#(x)
     cond2#(true(),x) -> neq#(x,0())
     cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
     cond2#(false(),x) -> p#(x)
     cond2#(false(),x) -> neq#(x,0())
     cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
     neq#(s(x),s(y())) -> neq#(x,y())
     even#(s(s(x))) -> even#(x)
     div2#(s(s(x))) -> div2#(x)
    TRS:
     f17(x,y) -> x
     f17(x,y) -> y
     even(0()) -> true()
     even(s(0())) -> false()
     even(s(s(x))) -> even(x)
     div2(0()) -> 0()
     div2(s(0())) -> 0()
     div2(s(s(x))) -> s(div2(x))
     neq(0(),0()) -> false()
     neq(s(x),0()) -> true()
     p(0()) -> 0()
     p(s(x)) -> x
    graph:
     div2#(s(s(x))) -> div2#(x) ->
     div2#(s(s(x))) -> div2#(x)
     cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) ->
     cond1#(true(),x) -> even#(x)
     cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) ->
     cond1#(true(),x) -> cond2#(even(x),x)
     cond2#(true(),x) -> div2#(x) ->
     div2#(s(s(x))) -> div2#(x)
     cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) ->
     cond1#(true(),x) -> even#(x)
     cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) ->
     cond1#(true(),x) -> cond2#(even(x),x)
     even#(s(s(x))) -> even#(x) -> even#(s(s(x))) -> even#(x)
     cond1#(true(),x) -> cond2#(even(x),x) ->
     cond2#(true(),x) -> div2#(x)
     cond1#(true(),x) -> cond2#(even(x),x) ->
     cond2#(true(),x) -> neq#(x,0())
     cond1#(true(),x) -> cond2#(even(x),x) ->
     cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
     cond1#(true(),x) -> cond2#(even(x),x) ->
     cond2#(false(),x) -> p#(x)
     cond1#(true(),x) -> cond2#(even(x),x) ->
     cond2#(false(),x) -> neq#(x,0())
     cond1#(true(),x) -> cond2#(even(x),x) ->
     cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
     cond1#(true(),x) -> even#(x) -> even#(s(s(x))) -> even#(x)
    Restore Modifier:
     DPs:
      cond1#(true(),x) -> even#(x)
      cond1#(true(),x) -> cond2#(even(x),x)
      cond2#(true(),x) -> div2#(x)
      cond2#(true(),x) -> neq#(x,0())
      cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
      cond2#(false(),x) -> p#(x)
      cond2#(false(),x) -> neq#(x,0())
      cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
      neq#(s(x),s(y())) -> neq#(x,y())
      even#(s(s(x))) -> even#(x)
      div2#(s(s(x))) -> div2#(x)
     TRS:
      cond1(true(),x) -> cond2(even(x),x)
      cond2(true(),x) -> cond1(neq(x,0()),div2(x))
      cond2(false(),x) -> cond1(neq(x,0()),p(x))
      neq(0(),0()) -> false()
      neq(0(),s(x)) -> true()
      neq(s(x),0()) -> true()
      neq(s(x),s(y())) -> neq(x,y())
      even(0()) -> true()
      even(s(0())) -> false()
      even(s(s(x))) -> even(x)
      div2(0()) -> 0()
      div2(s(0())) -> 0()
      div2(s(s(x))) -> s(div2(x))
      p(0()) -> 0()
      p(s(x)) -> x
     SCC Processor:
      #sccs: 3
      #rules: 5
      #arcs: 14/121
      DPs:
       cond2#(false(),x) -> cond1#(neq(x,0()),p(x))
       cond1#(true(),x) -> cond2#(even(x),x)
       cond2#(true(),x) -> cond1#(neq(x,0()),div2(x))
      TRS:
       cond1(true(),x) -> cond2(even(x),x)
       cond2(true(),x) -> cond1(neq(x,0()),div2(x))
       cond2(false(),x) -> cond1(neq(x,0()),p(x))
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y())) -> neq(x,y())
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
       p(0()) -> 0()
       p(s(x)) -> x
      Open
      
      DPs:
       even#(s(s(x))) -> even#(x)
      TRS:
       cond1(true(),x) -> cond2(even(x),x)
       cond2(true(),x) -> cond1(neq(x,0()),div2(x))
       cond2(false(),x) -> cond1(neq(x,0()),p(x))
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y())) -> neq(x,y())
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [even#](x0) = x0,
        
        [y] = 1,
        
        [s](x0) = x0 + 1,
        
        [p](x0) = x0,
        
        [false] = 1,
        
        [div2](x0) = x0,
        
        [neq](x0, x1) = 1,
        
        [0] = 1,
        
        [cond2](x0, x1) = x0 + 1,
        
        [even](x0) = 1,
        
        [cond1](x0, x1) = x0 + 1,
        
        [true] = 1
       orientation:
        even#(s(s(x))) = x + 2 >= x = even#(x)
        
        cond1(true(),x) = 2 >= 2 = cond2(even(x),x)
        
        cond2(true(),x) = 2 >= 2 = cond1(neq(x,0()),div2(x))
        
        cond2(false(),x) = 2 >= 2 = cond1(neq(x,0()),p(x))
        
        neq(0(),0()) = 1 >= 1 = false()
        
        neq(0(),s(x)) = 1 >= 1 = true()
        
        neq(s(x),0()) = 1 >= 1 = true()
        
        neq(s(x),s(y())) = 1 >= 1 = neq(x,y())
        
        even(0()) = 1 >= 1 = true()
        
        even(s(0())) = 1 >= 1 = false()
        
        even(s(s(x))) = 1 >= 1 = even(x)
        
        div2(0()) = 1 >= 1 = 0()
        
        div2(s(0())) = 2 >= 1 = 0()
        
        div2(s(s(x))) = x + 2 >= x + 1 = s(div2(x))
        
        p(0()) = 1 >= 1 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond1(true(),x) -> cond2(even(x),x)
         cond2(true(),x) -> cond1(neq(x,0()),div2(x))
         cond2(false(),x) -> cond1(neq(x,0()),p(x))
         neq(0(),0()) -> false()
         neq(0(),s(x)) -> true()
         neq(s(x),0()) -> true()
         neq(s(x),s(y())) -> neq(x,y())
         even(0()) -> true()
         even(s(0())) -> false()
         even(s(s(x))) -> even(x)
         div2(0()) -> 0()
         div2(s(0())) -> 0()
         div2(s(s(x))) -> s(div2(x))
         p(0()) -> 0()
         p(s(x)) -> x
       Qed
      
      DPs:
       div2#(s(s(x))) -> div2#(x)
      TRS:
       cond1(true(),x) -> cond2(even(x),x)
       cond2(true(),x) -> cond1(neq(x,0()),div2(x))
       cond2(false(),x) -> cond1(neq(x,0()),p(x))
       neq(0(),0()) -> false()
       neq(0(),s(x)) -> true()
       neq(s(x),0()) -> true()
       neq(s(x),s(y())) -> neq(x,y())
       even(0()) -> true()
       even(s(0())) -> false()
       even(s(s(x))) -> even(x)
       div2(0()) -> 0()
       div2(s(0())) -> 0()
       div2(s(s(x))) -> s(div2(x))
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [div2#](x0) = x0,
        
        [y] = 1,
        
        [s](x0) = x0 + 1,
        
        [p](x0) = x0,
        
        [false] = 1,
        
        [div2](x0) = x0,
        
        [neq](x0, x1) = 1,
        
        [0] = 1,
        
        [cond2](x0, x1) = x0 + 1,
        
        [even](x0) = 1,
        
        [cond1](x0, x1) = x0 + 1,
        
        [true] = 1
       orientation:
        div2#(s(s(x))) = x + 2 >= x = div2#(x)
        
        cond1(true(),x) = 2 >= 2 = cond2(even(x),x)
        
        cond2(true(),x) = 2 >= 2 = cond1(neq(x,0()),div2(x))
        
        cond2(false(),x) = 2 >= 2 = cond1(neq(x,0()),p(x))
        
        neq(0(),0()) = 1 >= 1 = false()
        
        neq(0(),s(x)) = 1 >= 1 = true()
        
        neq(s(x),0()) = 1 >= 1 = true()
        
        neq(s(x),s(y())) = 1 >= 1 = neq(x,y())
        
        even(0()) = 1 >= 1 = true()
        
        even(s(0())) = 1 >= 1 = false()
        
        even(s(s(x))) = 1 >= 1 = even(x)
        
        div2(0()) = 1 >= 1 = 0()
        
        div2(s(0())) = 2 >= 1 = 0()
        
        div2(s(s(x))) = x + 2 >= x + 1 = s(div2(x))
        
        p(0()) = 1 >= 1 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond1(true(),x) -> cond2(even(x),x)
         cond2(true(),x) -> cond1(neq(x,0()),div2(x))
         cond2(false(),x) -> cond1(neq(x,0()),p(x))
         neq(0(),0()) -> false()
         neq(0(),s(x)) -> true()
         neq(s(x),0()) -> true()
         neq(s(x),s(y())) -> neq(x,y())
         even(0()) -> true()
         even(s(0())) -> false()
         even(s(s(x))) -> even(x)
         div2(0()) -> 0()
         div2(s(0())) -> 0()
         div2(s(s(x))) -> s(div2(x))
         p(0()) -> 0()
         p(s(x)) -> x
       Qed