MAYBE

Problem:
 times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
 times(x,1()) -> x
 times(x,0()) -> 0()
 plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
 plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
 plus(zero(),y) -> y
 plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
 id(x) -> x
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 not(x) -> if(x,false(),true())
 gt(s(x),zero()) -> true()
 gt(zero(),y) -> false()
 gt(s(x),s(y)) -> gt(x,y)

Proof:
 DP Processor:
  DPs:
   times#(x,plus(y,1())) -> times#(1(),0())
   times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
   times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
   times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
   plus#(s(x),s(y)) -> id#(y)
   plus#(s(x),s(y)) -> id#(x)
   plus#(s(x),s(y)) -> not#(gt(x,y))
   plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
   plus#(s(x),s(y)) -> gt#(x,y)
   plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
   plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
   plus#(s(x),x) -> id#(x)
   plus#(s(x),x) -> gt#(x,x)
   plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
   plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
   plus#(id(x),s(y)) -> gt#(s(y),y)
   plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
   plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
   not#(x) -> if#(x,false(),true())
   gt#(s(x),s(y)) -> gt#(x,y)
  TRS:
   times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
   times(x,1()) -> x
   times(x,0()) -> 0()
   plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
   plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
   plus(zero(),y) -> y
   plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
   id(x) -> x
   if(true(),x,y) -> x
   if(false(),x,y) -> y
   not(x) -> if(x,false(),true())
   gt(s(x),zero()) -> true()
   gt(zero(),y) -> false()
   gt(s(x),s(y)) -> gt(x,y)
  TDG Processor:
   DPs:
    times#(x,plus(y,1())) -> times#(1(),0())
    times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),x) -> id#(x)
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    not#(x) -> if#(x,false(),true())
    gt#(s(x),s(y)) -> gt#(x,y)
   TRS:
    times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
    times(x,1()) -> x
    times(x,0()) -> 0()
    plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
    plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
    plus(zero(),y) -> y
    plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
    id(x) -> x
    if(true(),x,y) -> x
    if(false(),x,y) -> y
    not(x) -> if(x,false(),true())
    gt(s(x),zero()) -> true()
    gt(zero(),y) -> false()
    gt(s(x),s(y)) -> gt(x,y)
   graph:
    gt#(s(x),s(y)) -> gt#(x,y) -> gt#(s(x),s(y)) -> gt#(x,y)
    plus#(id(x),s(y)) -> gt#(s(y),y) ->
    gt#(s(x),s(y)) -> gt#(x,y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> gt#(x,x)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> id#(x)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> id#(x)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),s(y)) -> gt#(x,y) -> gt#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> not#(gt(x,y)) ->
    not#(x) -> if#(x,false(),true())
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> id#(x)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),x) -> gt#(x,x) ->
    gt#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> id#(x)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> id#(y)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),x) -> gt#(x,x)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),x) -> id#(x)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> id#(x)
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x) ->
    plus#(s(x),s(y)) -> id#(y)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),x) -> gt#(x,x)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),x) -> id#(x)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> id#(x)
    times#(x,plus(y,1())) -> plus#(y,times(1(),0())) ->
    plus#(s(x),s(y)) -> id#(y)
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
    times#(x,plus(y,1())) -> times#(1(),0()) ->
    times#(x,plus(y,1())) -> times#(1(),0())
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
    times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0()))) ->
    times#(x,plus(y,1())) -> times#(1(),0())
   Restore Modifier:
    DPs:
     times#(x,plus(y,1())) -> times#(1(),0())
     times#(x,plus(y,1())) -> plus#(y,times(1(),0()))
     times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
     times#(x,plus(y,1())) -> plus#(times(x,plus(y,times(1(),0()))),x)
     plus#(s(x),s(y)) -> id#(y)
     plus#(s(x),s(y)) -> id#(x)
     plus#(s(x),s(y)) -> not#(gt(x,y))
     plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
     plus#(s(x),s(y)) -> gt#(x,y)
     plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
     plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
     plus#(s(x),x) -> id#(x)
     plus#(s(x),x) -> gt#(x,x)
     plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
     plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
     plus#(id(x),s(y)) -> gt#(s(y),y)
     plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
     plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
     not#(x) -> if#(x,false(),true())
     gt#(s(x),s(y)) -> gt#(x,y)
    TRS:
     times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
     times(x,1()) -> x
     times(x,0()) -> 0()
     plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
     plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
     plus(zero(),y) -> y
     plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
     id(x) -> x
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     not(x) -> if(x,false(),true())
     gt(s(x),zero()) -> true()
     gt(zero(),y) -> false()
     gt(s(x),s(y)) -> gt(x,y)
    SCC Processor:
     #sccs: 3
     #rules: 6
     #arcs: 83/400
     DPs:
      times#(x,plus(y,1())) -> times#(1(),0())
      times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
     TRS:
      times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
      times(x,1()) -> x
      times(x,0()) -> 0()
      plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
      plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
      plus(zero(),y) -> y
      plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
      id(x) -> x
      if(true(),x,y) -> x
      if(false(),x,y) -> y
      not(x) -> if(x,false(),true())
      gt(s(x),zero()) -> true()
      gt(zero(),y) -> false()
      gt(s(x),s(y)) -> gt(x,y)
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [times#](x0, x1) = x1,
       
       [false] = 0,
       
       [true] = 0,
       
       [zero] = 0,
       
       [id](x0) = x0,
       
       [not](x0) = x0 + 1,
       
       [if](x0, x1, x2) = x0 + x1 + x2,
       
       [gt](x0, x1) = 0,
       
       [s](x0) = 0,
       
       [0] = 0,
       
       [times](x0, x1) = x0 + x1,
       
       [plus](x0, x1) = x1 + 1,
       
       [1] = 1
      orientation:
       times#(x,plus(y,1())) = 2 >= 0 = times#(1(),0())
       
       times#(x,plus(y,1())) = 2 >= 2 = times#(x,plus(y,times(1(),0())))
       
       times(x,plus(y,1())) = x + 2 >= x + 1 = plus(times(x,plus(y,times(1(),0()))),x)
       
       times(x,1()) = x + 1 >= x = x
       
       times(x,0()) = x >= 0 = 0()
       
       plus(s(x),s(y)) = 1 >= 0 = s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
       
       plus(s(x),x) = x + 1 >= 1 = plus(if(gt(x,x),id(x),id(x)),s(x))
       
       plus(zero(),y) = y + 1 >= y = y
       
       plus(id(x),s(y)) = 1 >= 0 = s(plus(x,if(gt(s(y),y),y,s(y))))
       
       id(x) = x >= x = x
       
       if(true(),x,y) = x + y >= x = x
       
       if(false(),x,y) = x + y >= y = y
       
       not(x) = x + 1 >= x = if(x,false(),true())
       
       gt(s(x),zero()) = 0 >= 0 = true()
       
       gt(zero(),y) = 0 >= 0 = false()
       
       gt(s(x),s(y)) = 0 >= 0 = gt(x,y)
      problem:
       DPs:
        times#(x,plus(y,1())) -> times#(x,plus(y,times(1(),0())))
       TRS:
        times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
        times(x,1()) -> x
        times(x,0()) -> 0()
        plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
        plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
        plus(zero(),y) -> y
        plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
        id(x) -> x
        if(true(),x,y) -> x
        if(false(),x,y) -> y
        not(x) -> if(x,false(),true())
        gt(s(x),zero()) -> true()
        gt(zero(),y) -> false()
        gt(s(x),s(y)) -> gt(x,y)
      Open
     
     DPs:
      plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
      plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
      plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
     TRS:
      times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
      times(x,1()) -> x
      times(x,0()) -> 0()
      plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
      plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
      plus(zero(),y) -> y
      plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
      id(x) -> x
      if(true(),x,y) -> x
      if(false(),x,y) -> y
      not(x) -> if(x,false(),true())
      gt(s(x),zero()) -> true()
      gt(zero(),y) -> false()
      gt(s(x),s(y)) -> gt(x,y)
     Open
     
     DPs:
      gt#(s(x),s(y)) -> gt#(x,y)
     TRS:
      times(x,plus(y,1())) -> plus(times(x,plus(y,times(1(),0()))),x)
      times(x,1()) -> x
      times(x,0()) -> 0()
      plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
      plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
      plus(zero(),y) -> y
      plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
      id(x) -> x
      if(true(),x,y) -> x
      if(false(),x,y) -> y
      not(x) -> if(x,false(),true())
      gt(s(x),zero()) -> true()
      gt(zero(),y) -> false()
      gt(s(x),s(y)) -> gt(x,y)
     Open