MAYBE

Problem:
 leq(0(),y) -> true()
 leq(s(x),0()) -> false()
 leq(s(x),s(y)) -> leq(x,y)
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 mod(0(),y) -> 0()
 mod(s(x),0()) -> 0()
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))

Proof:
 DP Processor:
  DPs:
   leq#(s(x),s(y)) -> leq#(x,y)
   -#(s(x),s(y)) -> -#(x,y)
   mod#(s(x),s(y)) -> -#(s(x),s(y))
   mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y))
   mod#(s(x),s(y)) -> leq#(y,x)
   mod#(s(x),s(y)) -> if#(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
  TRS:
   leq(0(),y) -> true()
   leq(s(x),0()) -> false()
   leq(s(x),s(y)) -> leq(x,y)
   if(true(),x,y) -> x
   if(false(),x,y) -> y
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   mod(0(),y) -> 0()
   mod(s(x),0()) -> 0()
   mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
  TDG Processor:
   DPs:
    leq#(s(x),s(y)) -> leq#(x,y)
    -#(s(x),s(y)) -> -#(x,y)
    mod#(s(x),s(y)) -> -#(s(x),s(y))
    mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y))
    mod#(s(x),s(y)) -> leq#(y,x)
    mod#(s(x),s(y)) -> if#(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
   TRS:
    leq(0(),y) -> true()
    leq(s(x),0()) -> false()
    leq(s(x),s(y)) -> leq(x,y)
    if(true(),x,y) -> x
    if(false(),x,y) -> y
    -(x,0()) -> x
    -(s(x),s(y)) -> -(x,y)
    mod(0(),y) -> 0()
    mod(s(x),0()) -> 0()
    mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
   graph:
    mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y)) ->
    mod#(s(x),s(y)) -> if#(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
    mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y)) ->
    mod#(s(x),s(y)) -> leq#(y,x)
    mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y)) ->
    mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y))
    mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y)) ->
    mod#(s(x),s(y)) -> -#(s(x),s(y))
    mod#(s(x),s(y)) -> -#(s(x),s(y)) -> -#(s(x),s(y)) -> -#(x,y)
    mod#(s(x),s(y)) -> leq#(y,x) -> leq#(s(x),s(y)) -> leq#(x,y)
    -#(s(x),s(y)) -> -#(x,y) -> -#(s(x),s(y)) -> -#(x,y)
    leq#(s(x),s(y)) -> leq#(x,y) -> leq#(s(x),s(y)) -> leq#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 3
    #arcs: 8/36
    DPs:
     mod#(s(x),s(y)) -> mod#(-(s(x),s(y)),s(y))
    TRS:
     leq(0(),y) -> true()
     leq(s(x),0()) -> false()
     leq(s(x),s(y)) -> leq(x,y)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     mod(0(),y) -> 0()
     mod(s(x),0()) -> 0()
     mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
    Open
    
    DPs:
     leq#(s(x),s(y)) -> leq#(x,y)
    TRS:
     leq(0(),y) -> true()
     leq(s(x),0()) -> false()
     leq(s(x),s(y)) -> leq(x,y)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     mod(0(),y) -> 0()
     mod(s(x),0()) -> 0()
     mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(leq#) = 1
     problem:
      DPs:
       
      TRS:
       leq(0(),y) -> true()
       leq(s(x),0()) -> false()
       leq(s(x),s(y)) -> leq(x,y)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
     Qed
    
    DPs:
     -#(s(x),s(y)) -> -#(x,y)
    TRS:
     leq(0(),y) -> true()
     leq(s(x),0()) -> false()
     leq(s(x),s(y)) -> leq(x,y)
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     mod(0(),y) -> 0()
     mod(s(x),0()) -> 0()
     mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(-#) = 1
     problem:
      DPs:
       
      TRS:
       leq(0(),y) -> true()
       leq(s(x),0()) -> false()
       leq(s(x),s(y)) -> leq(x,y)
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
     Qed