MAYBE

Problem:
 cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
 cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
 cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
 gr(0(),x) -> false()
 gr(s(x),0()) -> true()
 gr(s(x),s(y)) -> gr(x,y)
 p(0()) -> 0()
 p(s(x)) -> x
 eq(0(),0()) -> true()
 eq(s(x),0()) -> false()
 eq(0(),s(x)) -> false()
 eq(s(x),s(y)) -> eq(x,y)
 and(true(),true()) -> true()
 and(false(),x) -> false()
 and(x,false()) -> false()

Proof:
 DP Processor:
  DPs:
   cond1#(true(),x,y) -> gr#(y,0())
   cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
   cond2#(true(),x,y) -> p#(y)
   cond2#(true(),x,y) -> p#(x)
   cond2#(true(),x,y) -> gr#(y,0())
   cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
   cond2#(false(),x,y) -> gr#(x,0())
   cond2#(false(),x,y) -> eq#(x,y)
   cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
   cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
   gr#(s(x),s(y)) -> gr#(x,y)
   eq#(s(x),s(y)) -> eq#(x,y)
  TRS:
   cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
   cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
   cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
   gr(0(),x) -> false()
   gr(s(x),0()) -> true()
   gr(s(x),s(y)) -> gr(x,y)
   p(0()) -> 0()
   p(s(x)) -> x
   eq(0(),0()) -> true()
   eq(s(x),0()) -> false()
   eq(0(),s(x)) -> false()
   eq(s(x),s(y)) -> eq(x,y)
   and(true(),true()) -> true()
   and(false(),x) -> false()
   and(x,false()) -> false()
  TDG Processor:
   DPs:
    cond1#(true(),x,y) -> gr#(y,0())
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
    cond2#(true(),x,y) -> p#(y)
    cond2#(true(),x,y) -> p#(x)
    cond2#(true(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
    cond2#(false(),x,y) -> gr#(x,0())
    cond2#(false(),x,y) -> eq#(x,y)
    cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
    cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
    gr#(s(x),s(y)) -> gr#(x,y)
    eq#(s(x),s(y)) -> eq#(x,y)
   TRS:
    cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
    cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
    cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
    gr(0(),x) -> false()
    gr(s(x),0()) -> true()
    gr(s(x),s(y)) -> gr(x,y)
    p(0()) -> 0()
    p(s(x)) -> x
    eq(0(),0()) -> true()
    eq(s(x),0()) -> false()
    eq(0(),s(x)) -> false()
    eq(s(x),s(y)) -> eq(x,y)
    and(true(),true()) -> true()
    and(false(),x) -> false()
    and(x,false()) -> false()
   graph:
    eq#(s(x),s(y)) -> eq#(x,y) -> eq#(s(x),s(y)) -> eq#(x,y)
    cond2#(false(),x,y) -> eq#(x,y) -> eq#(s(x),s(y)) -> eq#(x,y)
    cond2#(false(),x,y) -> gr#(x,0()) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y) ->
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
    cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y) ->
    cond1#(true(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(false(),x,y) -> eq#(x,y)
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(false(),x,y) -> gr#(x,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(true(),x,y) -> gr#(y,0())
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(true(),x,y) -> p#(x)
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
    cond2#(true(),x,y) -> p#(y)
    cond2#(true(),x,y) -> gr#(y,0()) -> gr#(s(x),s(y)) -> gr#(x,y)
    gr#(s(x),s(y)) -> gr#(x,y) ->
    gr#(s(x),s(y)) -> gr#(x,y)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> eq#(x,y)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(false(),x,y) -> gr#(x,0())
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> gr#(y,0())
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> p#(x)
    cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
    cond2#(true(),x,y) -> p#(y)
    cond1#(true(),x,y) -> gr#(y,0()) -> gr#(s(x),s(y)) -> gr#(x,y)
   EDG Processor:
    DPs:
     cond1#(true(),x,y) -> gr#(y,0())
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
     cond2#(true(),x,y) -> p#(y)
     cond2#(true(),x,y) -> p#(x)
     cond2#(true(),x,y) -> gr#(y,0())
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
     cond2#(false(),x,y) -> gr#(x,0())
     cond2#(false(),x,y) -> eq#(x,y)
     cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
     cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
     gr#(s(x),s(y)) -> gr#(x,y)
     eq#(s(x),s(y)) -> eq#(x,y)
    TRS:
     cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
     cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
     cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
     gr(0(),x) -> false()
     gr(s(x),0()) -> true()
     gr(s(x),s(y)) -> gr(x,y)
     p(0()) -> 0()
     p(s(x)) -> x
     eq(0(),0()) -> true()
     eq(s(x),0()) -> false()
     eq(0(),s(x)) -> false()
     eq(s(x),s(y)) -> eq(x,y)
     and(true(),true()) -> true()
     and(false(),x) -> false()
     and(x,false()) -> false()
    graph:
     eq#(s(x),s(y)) -> eq#(x,y) -> eq#(s(x),s(y)) -> eq#(x,y)
     cond2#(false(),x,y) -> eq#(x,y) ->
     eq#(s(x),s(y)) -> eq#(x,y)
     cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y) ->
     cond1#(true(),x,y) -> gr#(y,0())
     cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y) ->
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(true(),x,y) -> p#(y)
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(true(),x,y) -> p#(x)
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(true(),x,y) -> gr#(y,0())
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(false(),x,y) -> gr#(x,0())
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(false(),x,y) -> eq#(x,y)
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y)) ->
     cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
     gr#(s(x),s(y)) -> gr#(x,y) ->
     gr#(s(x),s(y)) -> gr#(x,y)
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(true(),x,y) -> p#(y)
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(true(),x,y) -> p#(x)
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(true(),x,y) -> gr#(y,0())
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(false(),x,y) -> gr#(x,0())
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(false(),x,y) -> eq#(x,y)
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(false(),x,y) -> and#(eq(x,y),gr(x,0()))
     cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y) ->
     cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
    SCC Processor:
     #sccs: 3
     #rules: 5
     #arcs: 21/144
     DPs:
      gr#(s(x),s(y)) -> gr#(x,y)
     TRS:
      cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
      cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
      cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      p(0()) -> 0()
      p(s(x)) -> x
      eq(0(),0()) -> true()
      eq(s(x),0()) -> false()
      eq(0(),s(x)) -> false()
      eq(s(x),s(y)) -> eq(x,y)
      and(true(),true()) -> true()
      and(false(),x) -> false()
      and(x,false()) -> false()
     KBO Processor:
      argument filtering:
       pi(true) = []
       pi(cond1) = 2
       pi(0) = []
       pi(gr) = 0
       pi(cond2) = 2
       pi(p) = 0
       pi(false) = []
       pi(eq) = []
       pi(and) = [1]
       pi(s) = [0]
       pi(gr#) = 1
      weight function:
       w0 = 1
       w(gr#) = w(s) = w(eq) = w(false) = w(p) = w(cond2) = w(gr) = w(
       0) = w(cond1) = w(true) = 1
       w(and) = 0
      precedence:
       gr# ~ and ~ eq ~ p ~ 0 > s ~ false ~ cond2 ~ gr ~ cond1 ~ true
      problem:
       DPs:
        
       TRS:
        cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
        cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
        cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
        gr(0(),x) -> false()
        gr(s(x),0()) -> true()
        gr(s(x),s(y)) -> gr(x,y)
        p(0()) -> 0()
        p(s(x)) -> x
        eq(0(),0()) -> true()
        eq(s(x),0()) -> false()
        eq(0(),s(x)) -> false()
        eq(s(x),s(y)) -> eq(x,y)
        and(true(),true()) -> true()
        and(false(),x) -> false()
        and(x,false()) -> false()
      Qed
     
     DPs:
      cond2#(false(),x,y) -> cond1#(and(eq(x,y),gr(x,0())),x,y)
      cond1#(true(),x,y) -> cond2#(gr(y,0()),x,y)
      cond2#(true(),x,y) -> cond2#(gr(y,0()),p(x),p(y))
     TRS:
      cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
      cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
      cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      p(0()) -> 0()
      p(s(x)) -> x
      eq(0(),0()) -> true()
      eq(s(x),0()) -> false()
      eq(0(),s(x)) -> false()
      eq(s(x),s(y)) -> eq(x,y)
      and(true(),true()) -> true()
      and(false(),x) -> false()
      and(x,false()) -> false()
     Open
     
     DPs:
      eq#(s(x),s(y)) -> eq#(x,y)
     TRS:
      cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
      cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
      cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
      gr(0(),x) -> false()
      gr(s(x),0()) -> true()
      gr(s(x),s(y)) -> gr(x,y)
      p(0()) -> 0()
      p(s(x)) -> x
      eq(0(),0()) -> true()
      eq(s(x),0()) -> false()
      eq(0(),s(x)) -> false()
      eq(s(x),s(y)) -> eq(x,y)
      and(true(),true()) -> true()
      and(false(),x) -> false()
      and(x,false()) -> false()
     KBO Processor:
      argument filtering:
       pi(true) = []
       pi(cond1) = 2
       pi(0) = []
       pi(gr) = 0
       pi(cond2) = 2
       pi(p) = 0
       pi(false) = []
       pi(eq) = []
       pi(and) = [1]
       pi(s) = [0]
       pi(eq#) = 1
      weight function:
       w0 = 1
       w(eq#) = w(s) = w(eq) = w(false) = w(p) = w(cond2) = w(gr) = w(
       0) = w(cond1) = w(true) = 1
       w(and) = 0
      precedence:
       eq# ~ and ~ eq ~ p ~ 0 > s ~ false ~ cond2 ~ gr ~ cond1 ~ true
      problem:
       DPs:
        
       TRS:
        cond1(true(),x,y) -> cond2(gr(y,0()),x,y)
        cond2(true(),x,y) -> cond2(gr(y,0()),p(x),p(y))
        cond2(false(),x,y) -> cond1(and(eq(x,y),gr(x,0())),x,y)
        gr(0(),x) -> false()
        gr(s(x),0()) -> true()
        gr(s(x),s(y)) -> gr(x,y)
        p(0()) -> 0()
        p(s(x)) -> x
        eq(0(),0()) -> true()
        eq(s(x),0()) -> false()
        eq(0(),s(x)) -> false()
        eq(s(x),s(y)) -> eq(x,y)
        and(true(),true()) -> true()
        and(false(),x) -> false()
        and(x,false()) -> false()
      Qed