MAYBE

Problem:
 cond(true(),x,y,z) -> cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
 and(true(),true()) -> true()
 and(x,false()) -> false()
 and(false(),x) -> false()
 gr(0(),0()) -> false()
 gr(0(),x) -> false()
 gr(s(x),0()) -> true()
 gr(s(x),s(y)) -> gr(x,y)
 p(0()) -> 0()
 p(s(x)) -> x

Proof:
 DP Processor:
  DPs:
   cond#(true(),x,y,z) -> p#(y)
   cond#(true(),x,y,z) -> p#(x)
   cond#(true(),x,y,z) -> gr#(y,z)
   cond#(true(),x,y,z) -> gr#(x,z)
   cond#(true(),x,y,z) -> and#(gr(x,z),gr(y,z))
   cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z)
   gr#(s(x),s(y)) -> gr#(x,y)
  TRS:
   cond(true(),x,y,z) -> cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
   and(true(),true()) -> true()
   and(x,false()) -> false()
   and(false(),x) -> false()
   gr(0(),0()) -> false()
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   p(0()) -> 0()
   p(s(x)) -> x
  Usable Rule Processor:
   DPs:
    cond#(true(),x,y,z) -> p#(y)
    cond#(true(),x,y,z) -> p#(x)
    cond#(true(),x,y,z) -> gr#(y,z)
    cond#(true(),x,y,z) -> gr#(x,z)
    cond#(true(),x,y,z) -> and#(gr(x,z),gr(y,z))
    cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z)
    gr#(s(x),s(y)) -> gr#(x,y)
   TRS:
    f12(x,y) -> x
    f12(x,y) -> y
    gr(0(),0()) -> false()
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
    p(0()) -> 0()
    p(s(x)) -> x
    and(true(),true()) -> true()
    and(x,false()) -> false()
    and(false(),x) -> false()
   CDG Processor:
    DPs:
     cond#(true(),x,y,z) -> p#(y)
     cond#(true(),x,y,z) -> p#(x)
     cond#(true(),x,y,z) -> gr#(y,z)
     cond#(true(),x,y,z) -> gr#(x,z)
     cond#(true(),x,y,z) -> and#(gr(x,z),gr(y,z))
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z)
     gr#(s(x),s(y)) -> gr#(x,y)
    TRS:
     f12(x,y) -> x
     f12(x,y) -> y
     gr(0(),0()) -> false()
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     p(0()) -> 0()
     p(s(x)) -> x
     and(true(),true()) -> true()
     and(x,false()) -> false()
     and(false(),x) -> false()
    graph:
     gr#(s(x),s(y)) -> gr#(x,y) -> gr#(s(x),s(y)) -> gr#(x,y)
     cond#(true(),x,y,z) -> gr#(y,z) -> gr#(s(x),s(y)) -> gr#(x,y)
     cond#(true(),x,y,z) -> gr#(x,z) ->
     gr#(s(x),s(y)) -> gr#(x,y)
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z) ->
     cond#(true(),x,y,z) -> p#(y)
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z) ->
     cond#(true(),x,y,z) -> p#(x)
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z) ->
     cond#(true(),x,y,z) -> gr#(y,z)
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z) ->
     cond#(true(),x,y,z) -> gr#(x,z)
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z) ->
     cond#(true(),x,y,z) -> and#(gr(x,z),gr(y,z))
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z) ->
     cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z)
    Restore Modifier:
     DPs:
      cond#(true(),x,y,z) -> p#(y)
      cond#(true(),x,y,z) -> p#(x)
      cond#(true(),x,y,z) -> gr#(y,z)
      cond#(true(),x,y,z) -> gr#(x,z)
      cond#(true(),x,y,z) -> and#(gr(x,z),gr(y,z))
      cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z)
      gr#(s(x),s(y)) -> gr#(x,y)
     TRS:
      cond(true(),x,y,z) -> cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
      and(true(),true()) -> true()
      and(x,false()) -> false()
      and(false(),x) -> false()
      gr(0(),0()) -> false()
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      p(0()) -> 0()
      p(s(x)) -> x
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 9/49
      DPs:
       cond#(true(),x,y,z) -> cond#(and(gr(x,z),gr(y,z)),p(x),p(y),z)
      TRS:
       cond(true(),x,y,z) -> cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
       and(true(),true()) -> true()
       and(x,false()) -> false()
       and(false(),x) -> false()
       gr(0(),0()) -> false()
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       p(0()) -> 0()
       p(s(x)) -> x
      Open
      
      DPs:
       gr#(s(x),s(y)) -> gr#(x,y)
      TRS:
       cond(true(),x,y,z) -> cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
       and(true(),true()) -> true()
       and(x,false()) -> false()
       and(false(),x) -> false()
       gr(0(),0()) -> false()
       gr(0(),x) -> false()
       gr(s(x),0()) -> true()
       gr(s(x),s(y)) -> gr(x,y)
       p(0()) -> 0()
       p(s(x)) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [gr#](x0, x1) = x1 + 1,
        
        [s](x0) = x0 + 1,
        
        [0] = 0,
        
        [false] = 0,
        
        [p](x0) = x0,
        
        [and](x0, x1) = 0,
        
        [gr](x0, x1) = 0,
        
        [cond](x0, x1, x2, x3) = 1,
        
        [true] = 0
       orientation:
        gr#(s(x),s(y)) = y + 2 >= y + 1 = gr#(x,y)
        
        cond(true(),x,y,z) = 1 >= 1 = cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
        
        and(true(),true()) = 0 >= 0 = true()
        
        and(x,false()) = 0 >= 0 = false()
        
        and(false(),x) = 0 >= 0 = false()
        
        gr(0(),0()) = 0 >= 0 = false()
        
        gr(0(),x) = 0 >= 0 = false()
        
        gr(s(x),0()) = 0 >= 0 = true()
        
        gr(s(x),s(y)) = 0 >= 0 = gr(x,y)
        
        p(0()) = 0 >= 0 = 0()
        
        p(s(x)) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         cond(true(),x,y,z) -> cond(and(gr(x,z),gr(y,z)),p(x),p(y),z)
         and(true(),true()) -> true()
         and(x,false()) -> false()
         and(false(),x) -> false()
         gr(0(),0()) -> false()
         gr(0(),x) -> false()
         gr(s(x),0()) -> true()
         gr(s(x),s(y)) -> gr(x,y)
         p(0()) -> 0()
         p(s(x)) -> x
       Qed