MAYBE

Problem:
 minus(0(),y) -> 0()
 minus(s(x),y) -> if(gt(s(x),y),x,y)
 if(true(),x,y) -> s(minus(x,y))
 if(false(),x,y) -> 0()
 mod(x,0()) -> 0()
 mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
 if1(true(),x,y) -> x
 if1(false(),x,y) -> mod(minus(x,y),y)
 gt(0(),y) -> false()
 gt(s(x),0()) -> true()
 gt(s(x),s(y)) -> gt(x,y)
 lt(x,0()) -> false()
 lt(0(),s(x)) -> true()
 lt(s(x),s(y)) -> lt(x,y)

Proof:
 DP Processor:
  DPs:
   minus#(s(x),y) -> gt#(s(x),y)
   minus#(s(x),y) -> if#(gt(s(x),y),x,y)
   if#(true(),x,y) -> minus#(x,y)
   mod#(x,s(y)) -> lt#(x,s(y))
   mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y))
   if1#(false(),x,y) -> minus#(x,y)
   if1#(false(),x,y) -> mod#(minus(x,y),y)
   gt#(s(x),s(y)) -> gt#(x,y)
   lt#(s(x),s(y)) -> lt#(x,y)
  TRS:
   minus(0(),y) -> 0()
   minus(s(x),y) -> if(gt(s(x),y),x,y)
   if(true(),x,y) -> s(minus(x,y))
   if(false(),x,y) -> 0()
   mod(x,0()) -> 0()
   mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
   if1(true(),x,y) -> x
   if1(false(),x,y) -> mod(minus(x,y),y)
   gt(0(),y) -> false()
   gt(s(x),0()) -> true()
   gt(s(x),s(y)) -> gt(x,y)
   lt(x,0()) -> false()
   lt(0(),s(x)) -> true()
   lt(s(x),s(y)) -> lt(x,y)
  TDG Processor:
   DPs:
    minus#(s(x),y) -> gt#(s(x),y)
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if#(true(),x,y) -> minus#(x,y)
    mod#(x,s(y)) -> lt#(x,s(y))
    mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y))
    if1#(false(),x,y) -> minus#(x,y)
    if1#(false(),x,y) -> mod#(minus(x,y),y)
    gt#(s(x),s(y)) -> gt#(x,y)
    lt#(s(x),s(y)) -> lt#(x,y)
   TRS:
    minus(0(),y) -> 0()
    minus(s(x),y) -> if(gt(s(x),y),x,y)
    if(true(),x,y) -> s(minus(x,y))
    if(false(),x,y) -> 0()
    mod(x,0()) -> 0()
    mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
    if1(true(),x,y) -> x
    if1(false(),x,y) -> mod(minus(x,y),y)
    gt(0(),y) -> false()
    gt(s(x),0()) -> true()
    gt(s(x),s(y)) -> gt(x,y)
    lt(x,0()) -> false()
    lt(0(),s(x)) -> true()
    lt(s(x),s(y)) -> lt(x,y)
   graph:
    if1#(false(),x,y) -> mod#(minus(x,y),y) ->
    mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y))
    if1#(false(),x,y) -> mod#(minus(x,y),y) ->
    mod#(x,s(y)) -> lt#(x,s(y))
    if1#(false(),x,y) -> minus#(x,y) ->
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if1#(false(),x,y) -> minus#(x,y) -> minus#(s(x),y) -> gt#(s(x),y)
    lt#(s(x),s(y)) -> lt#(x,y) -> lt#(s(x),s(y)) -> lt#(x,y)
    mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) ->
    if1#(false(),x,y) -> mod#(minus(x,y),y)
    mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) ->
    if1#(false(),x,y) -> minus#(x,y)
    mod#(x,s(y)) -> lt#(x,s(y)) -> lt#(s(x),s(y)) -> lt#(x,y)
    if#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if#(true(),x,y) -> minus#(x,y) -> minus#(s(x),y) -> gt#(s(x),y)
    gt#(s(x),s(y)) -> gt#(x,y) -> gt#(s(x),s(y)) -> gt#(x,y)
    minus#(s(x),y) -> if#(gt(s(x),y),x,y) ->
    if#(true(),x,y) -> minus#(x,y)
    minus#(s(x),y) -> gt#(s(x),y) -> gt#(s(x),s(y)) -> gt#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 6
    #arcs: 13/81
    DPs:
     if1#(false(),x,y) -> mod#(minus(x,y),y)
     mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y))
    TRS:
     minus(0(),y) -> 0()
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     mod(x,0()) -> 0()
     mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
     if1(true(),x,y) -> x
     if1(false(),x,y) -> mod(minus(x,y),y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     lt(x,0()) -> false()
     lt(0(),s(x)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
    Open
    
    DPs:
     minus#(s(x),y) -> if#(gt(s(x),y),x,y)
     if#(true(),x,y) -> minus#(x,y)
    TRS:
     minus(0(),y) -> 0()
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     mod(x,0()) -> 0()
     mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
     if1(true(),x,y) -> x
     if1(false(),x,y) -> mod(minus(x,y),y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     lt(x,0()) -> false()
     lt(0(),s(x)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(if#) = 1
     problem:
      DPs:
       if#(true(),x,y) -> minus#(x,y)
      TRS:
       minus(0(),y) -> 0()
       minus(s(x),y) -> if(gt(s(x),y),x,y)
       if(true(),x,y) -> s(minus(x,y))
       if(false(),x,y) -> 0()
       mod(x,0()) -> 0()
       mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
       if1(true(),x,y) -> x
       if1(false(),x,y) -> mod(minus(x,y),y)
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       lt(x,0()) -> false()
       lt(0(),s(x)) -> true()
       lt(s(x),s(y)) -> lt(x,y)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     gt#(s(x),s(y)) -> gt#(x,y)
    TRS:
     minus(0(),y) -> 0()
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     mod(x,0()) -> 0()
     mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
     if1(true(),x,y) -> x
     if1(false(),x,y) -> mod(minus(x,y),y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     lt(x,0()) -> false()
     lt(0(),s(x)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(gt#) = 1
     problem:
      DPs:
       
      TRS:
       minus(0(),y) -> 0()
       minus(s(x),y) -> if(gt(s(x),y),x,y)
       if(true(),x,y) -> s(minus(x,y))
       if(false(),x,y) -> 0()
       mod(x,0()) -> 0()
       mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
       if1(true(),x,y) -> x
       if1(false(),x,y) -> mod(minus(x,y),y)
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       lt(x,0()) -> false()
       lt(0(),s(x)) -> true()
       lt(s(x),s(y)) -> lt(x,y)
     Qed
    
    DPs:
     lt#(s(x),s(y)) -> lt#(x,y)
    TRS:
     minus(0(),y) -> 0()
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     mod(x,0()) -> 0()
     mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
     if1(true(),x,y) -> x
     if1(false(),x,y) -> mod(minus(x,y),y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     lt(x,0()) -> false()
     lt(0(),s(x)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(lt#) = 1
     problem:
      DPs:
       
      TRS:
       minus(0(),y) -> 0()
       minus(s(x),y) -> if(gt(s(x),y),x,y)
       if(true(),x,y) -> s(minus(x,y))
       if(false(),x,y) -> 0()
       mod(x,0()) -> 0()
       mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y))
       if1(true(),x,y) -> x
       if1(false(),x,y) -> mod(minus(x,y),y)
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       lt(x,0()) -> false()
       lt(0(),s(x)) -> true()
       lt(s(x),s(y)) -> lt(x,y)
     Qed