MAYBE

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 zero(0()) -> true()
 zero(s(x)) -> false()
 id(0()) -> 0()
 id(s(x)) -> s(id(x))
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
 if_mod(true(),b1,b2,x,y) -> 0()
 if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
 if2(true(),b2,x,y) -> 0()
 if2(false(),b2,x,y) -> if3(b2,x,y)
 if3(true(),x,y) -> mod(minus(x,y),s(y))
 if3(false(),x,y) -> x

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   id#(s(x)) -> id#(x)
   minus#(s(x),s(y)) -> minus#(x,y)
   mod#(x,y) -> id#(y)
   mod#(x,y) -> id#(x)
   mod#(x,y) -> le#(y,x)
   mod#(x,y) -> zero#(y)
   mod#(x,y) -> zero#(x)
   mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
   if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y)
   if2#(false(),b2,x,y) -> if3#(b2,x,y)
   if3#(true(),x,y) -> minus#(x,y)
   if3#(true(),x,y) -> mod#(minus(x,y),s(y))
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   zero(0()) -> true()
   zero(s(x)) -> false()
   id(0()) -> 0()
   id(s(x)) -> s(id(x))
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
   if_mod(true(),b1,b2,x,y) -> 0()
   if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
   if2(true(),b2,x,y) -> 0()
   if2(false(),b2,x,y) -> if3(b2,x,y)
   if3(true(),x,y) -> mod(minus(x,y),s(y))
   if3(false(),x,y) -> x
  Usable Rule Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    id#(s(x)) -> id#(x)
    minus#(s(x),s(y)) -> minus#(x,y)
    mod#(x,y) -> id#(y)
    mod#(x,y) -> id#(x)
    mod#(x,y) -> le#(y,x)
    mod#(x,y) -> zero#(y)
    mod#(x,y) -> zero#(x)
    mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
    if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y)
    if2#(false(),b2,x,y) -> if3#(b2,x,y)
    if3#(true(),x,y) -> minus#(x,y)
    if3#(true(),x,y) -> mod#(minus(x,y),s(y))
   TRS:
    f20(x,y) -> x
    f20(x,y) -> y
    id(0()) -> 0()
    id(s(x)) -> s(id(x))
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    zero(0()) -> true()
    zero(s(x)) -> false()
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
   ADG Processor:
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
     id#(s(x)) -> id#(x)
     minus#(s(x),s(y)) -> minus#(x,y)
     mod#(x,y) -> id#(y)
     mod#(x,y) -> id#(x)
     mod#(x,y) -> le#(y,x)
     mod#(x,y) -> zero#(y)
     mod#(x,y) -> zero#(x)
     mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
     if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y)
     if2#(false(),b2,x,y) -> if3#(b2,x,y)
     if3#(true(),x,y) -> minus#(x,y)
     if3#(true(),x,y) -> mod#(minus(x,y),s(y))
    TRS:
     f20(x,y) -> x
     f20(x,y) -> y
     id(0()) -> 0()
     id(s(x)) -> s(id(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     zero(0()) -> true()
     zero(s(x)) -> false()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
    graph:
     if3#(true(),x,y) -> mod#(minus(x,y),s(y)) ->
     mod#(x,y) -> id#(y)
     if3#(true(),x,y) -> mod#(minus(x,y),s(y)) ->
     mod#(x,y) -> id#(x)
     if3#(true(),x,y) -> mod#(minus(x,y),s(y)) ->
     mod#(x,y) -> le#(y,x)
     if3#(true(),x,y) -> mod#(minus(x,y),s(y)) ->
     mod#(x,y) -> zero#(y)
     if3#(true(),x,y) -> mod#(minus(x,y),s(y)) ->
     mod#(x,y) -> zero#(x)
     if3#(true(),x,y) -> mod#(minus(x,y),s(y)) ->
     mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
     if3#(true(),x,y) -> minus#(x,y) ->
     minus#(s(x),s(y)) -> minus#(x,y)
     if2#(false(),b2,x,y) -> if3#(b2,x,y) ->
     if3#(true(),x,y) -> minus#(x,y)
     if2#(false(),b2,x,y) -> if3#(b2,x,y) ->
     if3#(true(),x,y) -> mod#(minus(x,y),s(y))
     if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y) ->
     if2#(false(),b2,x,y) -> if3#(b2,x,y)
     mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y)) ->
     if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y)
     mod#(x,y) -> id#(x) -> id#(s(x)) -> id#(x)
     mod#(x,y) -> id#(y) -> id#(s(x)) -> id#(x)
     mod#(x,y) -> le#(y,x) -> le#(s(x),s(y)) -> le#(x,y)
     minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
     id#(s(x)) -> id#(x) -> id#(s(x)) -> id#(x)
     le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    Restore Modifier:
     DPs:
      le#(s(x),s(y)) -> le#(x,y)
      id#(s(x)) -> id#(x)
      minus#(s(x),s(y)) -> minus#(x,y)
      mod#(x,y) -> id#(y)
      mod#(x,y) -> id#(x)
      mod#(x,y) -> le#(y,x)
      mod#(x,y) -> zero#(y)
      mod#(x,y) -> zero#(x)
      mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
      if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y)
      if2#(false(),b2,x,y) -> if3#(b2,x,y)
      if3#(true(),x,y) -> minus#(x,y)
      if3#(true(),x,y) -> mod#(minus(x,y),s(y))
     TRS:
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      zero(0()) -> true()
      zero(s(x)) -> false()
      id(0()) -> 0()
      id(s(x)) -> s(id(x))
      minus(x,0()) -> x
      minus(s(x),s(y)) -> minus(x,y)
      mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
      if_mod(true(),b1,b2,x,y) -> 0()
      if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
      if2(true(),b2,x,y) -> 0()
      if2(false(),b2,x,y) -> if3(b2,x,y)
      if3(true(),x,y) -> mod(minus(x,y),s(y))
      if3(false(),x,y) -> x
     SCC Processor:
      #sccs: 4
      #rules: 7
      #arcs: 17/169
      DPs:
       if3#(true(),x,y) -> mod#(minus(x,y),s(y))
       mod#(x,y) -> if_mod#(zero(x),zero(y),le(y,x),id(x),id(y))
       if_mod#(false(),b1,b2,x,y) -> if2#(b1,b2,x,y)
       if2#(false(),b2,x,y) -> if3#(b2,x,y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       zero(0()) -> true()
       zero(s(x)) -> false()
       id(0()) -> 0()
       id(s(x)) -> s(id(x))
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
       if_mod(true(),b1,b2,x,y) -> 0()
       if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
       if2(true(),b2,x,y) -> 0()
       if2(false(),b2,x,y) -> if3(b2,x,y)
       if3(true(),x,y) -> mod(minus(x,y),s(y))
       if3(false(),x,y) -> x
      Open
      
      DPs:
       id#(s(x)) -> id#(x)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       zero(0()) -> true()
       zero(s(x)) -> false()
       id(0()) -> 0()
       id(s(x)) -> s(id(x))
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
       if_mod(true(),b1,b2,x,y) -> 0()
       if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
       if2(true(),b2,x,y) -> 0()
       if2(false(),b2,x,y) -> if3(b2,x,y)
       if3(true(),x,y) -> mod(minus(x,y),s(y))
       if3(false(),x,y) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [id#](x0) = x0 + 1,
        
        [if3](x0, x1, x2) = x1 + 1,
        
        [if2](x0, x1, x2, x3) = x2 + 1,
        
        [if_mod](x0, x1, x2, x3, x4) = x3 + 1,
        
        [mod](x0, x1) = x0 + 1,
        
        [minus](x0, x1) = x0,
        
        [id](x0) = x0,
        
        [zero](x0) = 1,
        
        [false] = 1,
        
        [s](x0) = x0 + 1,
        
        [true] = 1,
        
        [le](x0, x1) = 1,
        
        [0] = 1
       orientation:
        id#(s(x)) = x + 2 >= x + 1 = id#(x)
        
        le(0(),y) = 1 >= 1 = true()
        
        le(s(x),0()) = 1 >= 1 = false()
        
        le(s(x),s(y)) = 1 >= 1 = le(x,y)
        
        zero(0()) = 1 >= 1 = true()
        
        zero(s(x)) = 1 >= 1 = false()
        
        id(0()) = 1 >= 1 = 0()
        
        id(s(x)) = x + 1 >= x + 1 = s(id(x))
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        mod(x,y) = x + 1 >= x + 1 = if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
        
        if_mod(true(),b1,b2,x,y) = x + 1 >= 1 = 0()
        
        if_mod(false(),b1,b2,x,y) = x + 1 >= x + 1 = if2(b1,b2,x,y)
        
        if2(true(),b2,x,y) = x + 1 >= 1 = 0()
        
        if2(false(),b2,x,y) = x + 1 >= x + 1 = if3(b2,x,y)
        
        if3(true(),x,y) = x + 1 >= x + 1 = mod(minus(x,y),s(y))
        
        if3(false(),x,y) = x + 1 >= x = x
       problem:
        DPs:
         
        TRS:
         le(0(),y) -> true()
         le(s(x),0()) -> false()
         le(s(x),s(y)) -> le(x,y)
         zero(0()) -> true()
         zero(s(x)) -> false()
         id(0()) -> 0()
         id(s(x)) -> s(id(x))
         minus(x,0()) -> x
         minus(s(x),s(y)) -> minus(x,y)
         mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
         if_mod(true(),b1,b2,x,y) -> 0()
         if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
         if2(true(),b2,x,y) -> 0()
         if2(false(),b2,x,y) -> if3(b2,x,y)
         if3(true(),x,y) -> mod(minus(x,y),s(y))
         if3(false(),x,y) -> x
       Qed
      
      DPs:
       le#(s(x),s(y)) -> le#(x,y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       zero(0()) -> true()
       zero(s(x)) -> false()
       id(0()) -> 0()
       id(s(x)) -> s(id(x))
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
       if_mod(true(),b1,b2,x,y) -> 0()
       if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
       if2(true(),b2,x,y) -> 0()
       if2(false(),b2,x,y) -> if3(b2,x,y)
       if3(true(),x,y) -> mod(minus(x,y),s(y))
       if3(false(),x,y) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [le#](x0, x1) = x0 + 1,
        
        [if3](x0, x1, x2) = x1,
        
        [if2](x0, x1, x2, x3) = x2,
        
        [if_mod](x0, x1, x2, x3, x4) = x3,
        
        [mod](x0, x1) = x0,
        
        [minus](x0, x1) = x0,
        
        [id](x0) = x0,
        
        [zero](x0) = 0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = 0,
        
        [0] = 0
       orientation:
        le#(s(x),s(y)) = x + 2 >= x + 1 = le#(x,y)
        
        le(0(),y) = 0 >= 0 = true()
        
        le(s(x),0()) = 0 >= 0 = false()
        
        le(s(x),s(y)) = 0 >= 0 = le(x,y)
        
        zero(0()) = 0 >= 0 = true()
        
        zero(s(x)) = 0 >= 0 = false()
        
        id(0()) = 0 >= 0 = 0()
        
        id(s(x)) = x + 1 >= x + 1 = s(id(x))
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        mod(x,y) = x >= x = if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
        
        if_mod(true(),b1,b2,x,y) = x >= 0 = 0()
        
        if_mod(false(),b1,b2,x,y) = x >= x = if2(b1,b2,x,y)
        
        if2(true(),b2,x,y) = x >= 0 = 0()
        
        if2(false(),b2,x,y) = x >= x = if3(b2,x,y)
        
        if3(true(),x,y) = x >= x = mod(minus(x,y),s(y))
        
        if3(false(),x,y) = x >= x = x
       problem:
        DPs:
         
        TRS:
         le(0(),y) -> true()
         le(s(x),0()) -> false()
         le(s(x),s(y)) -> le(x,y)
         zero(0()) -> true()
         zero(s(x)) -> false()
         id(0()) -> 0()
         id(s(x)) -> s(id(x))
         minus(x,0()) -> x
         minus(s(x),s(y)) -> minus(x,y)
         mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
         if_mod(true(),b1,b2,x,y) -> 0()
         if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
         if2(true(),b2,x,y) -> 0()
         if2(false(),b2,x,y) -> if3(b2,x,y)
         if3(true(),x,y) -> mod(minus(x,y),s(y))
         if3(false(),x,y) -> x
       Qed
      
      DPs:
       minus#(s(x),s(y)) -> minus#(x,y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       zero(0()) -> true()
       zero(s(x)) -> false()
       id(0()) -> 0()
       id(s(x)) -> s(id(x))
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
       if_mod(true(),b1,b2,x,y) -> 0()
       if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
       if2(true(),b2,x,y) -> 0()
       if2(false(),b2,x,y) -> if3(b2,x,y)
       if3(true(),x,y) -> mod(minus(x,y),s(y))
       if3(false(),x,y) -> x
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [minus#](x0, x1) = x0 + 1,
        
        [if3](x0, x1, x2) = x1,
        
        [if2](x0, x1, x2, x3) = x2,
        
        [if_mod](x0, x1, x2, x3, x4) = x3,
        
        [mod](x0, x1) = x0,
        
        [minus](x0, x1) = x0,
        
        [id](x0) = x0,
        
        [zero](x0) = 0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = 0,
        
        [0] = 0
       orientation:
        minus#(s(x),s(y)) = x + 2 >= x + 1 = minus#(x,y)
        
        le(0(),y) = 0 >= 0 = true()
        
        le(s(x),0()) = 0 >= 0 = false()
        
        le(s(x),s(y)) = 0 >= 0 = le(x,y)
        
        zero(0()) = 0 >= 0 = true()
        
        zero(s(x)) = 0 >= 0 = false()
        
        id(0()) = 0 >= 0 = 0()
        
        id(s(x)) = x + 1 >= x + 1 = s(id(x))
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        mod(x,y) = x >= x = if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
        
        if_mod(true(),b1,b2,x,y) = x >= 0 = 0()
        
        if_mod(false(),b1,b2,x,y) = x >= x = if2(b1,b2,x,y)
        
        if2(true(),b2,x,y) = x >= 0 = 0()
        
        if2(false(),b2,x,y) = x >= x = if3(b2,x,y)
        
        if3(true(),x,y) = x >= x = mod(minus(x,y),s(y))
        
        if3(false(),x,y) = x >= x = x
       problem:
        DPs:
         
        TRS:
         le(0(),y) -> true()
         le(s(x),0()) -> false()
         le(s(x),s(y)) -> le(x,y)
         zero(0()) -> true()
         zero(s(x)) -> false()
         id(0()) -> 0()
         id(s(x)) -> s(id(x))
         minus(x,0()) -> x
         minus(s(x),s(y)) -> minus(x,y)
         mod(x,y) -> if_mod(zero(x),zero(y),le(y,x),id(x),id(y))
         if_mod(true(),b1,b2,x,y) -> 0()
         if_mod(false(),b1,b2,x,y) -> if2(b1,b2,x,y)
         if2(true(),b2,x,y) -> 0()
         if2(false(),b2,x,y) -> if3(b2,x,y)
         if3(true(),x,y) -> mod(minus(x,y),s(y))
         if3(false(),x,y) -> x
       Qed