/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem:
 times(x,0()) -> 0()
 times(x,s(y)) -> plus(times(x,y),x)
 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,s(y)) -> times#(x,y)
   times#(x,s(y)) -> plus#(times(x,y),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,0()) -> 0()
   times(x,s(y)) -> plus(times(x,y),x)
   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,s(y)) -> times#(x,y)
    times#(x,s(y)) -> plus#(times(x,y),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,0()) -> 0()
    times(x,s(y)) -> plus(times(x,y),x)
    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,s(y)) -> plus#(times(x,y),x) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),x) -> gt#(x,x)
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),x) -> id#(x)
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> id#(x)
    times#(x,s(y)) -> plus#(times(x,y),x) ->
    plus#(s(x),s(y)) -> id#(y)
    times#(x,s(y)) -> times#(x,y) ->
    times#(x,s(y)) -> plus#(times(x,y),x)
    times#(x,s(y)) -> times#(x,y) -> times#(x,s(y)) -> times#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 5
    #arcs: 63/324
    DPs:
     times#(x,s(y)) -> times#(x,y)
    TRS:
     times(x,0()) -> 0()
     times(x,s(y)) -> plus(times(x,y),x)
     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)
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 1
     problem:
      DPs:
       
      TRS:
       times(x,0()) -> 0()
       times(x,s(y)) -> plus(times(x,y),x)
       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)
     Qed
    
    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,0()) -> 0()
     times(x,s(y)) -> plus(times(x,y),x)
     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)
    Usable Rule Processor:
     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:
      gt(s(x),zero()) -> true()
      gt(s(x),s(y)) -> gt(x,y)
      gt(zero(),y) -> false()
      if(true(),x,y) -> x
      if(false(),x,y) -> y
      id(x) -> x
      not(x) -> if(x,false(),true())
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       id(x) -> x
      interpretation:
       [plus#](x0, x1) = x0 + x1,
       
       [false] = 0,
       
       [true] = 0,
       
       [zero] = 0,
       
       [id](x0) = x0,
       
       [not](x0) = -2x0 + 0,
       
       [if](x0, x1, x2) = x1 + x2,
       
       [gt](x0, x1) = 2x0 + 1x1 + 2,
       
       [s](x0) = 1x0 + 2
      orientation:
       plus#(id(x),s(y)) = x + 1y + 2 >= x + 1y + 2 = plus#(x,if(gt(s(y),y),y,s(y)))
       
       plus#(s(x),s(y)) = 1x + 1y + 2 >= x + y = plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
       
       plus#(s(x),x) = 1x + 2 >= 1x + 2 = plus#(if(gt(x,x),id(x),id(x)),s(x))
       
       gt(s(x),zero()) = 3x + 4 >= 0 = true()
       
       gt(s(x),s(y)) = 3x + 2y + 4 >= 2x + 1y + 2 = gt(x,y)
       
       gt(zero(),y) = 1y + 2 >= 0 = false()
       
       if(true(),x,y) = x + y >= x = x
       
       if(false(),x,y) = x + y >= y = y
       
       id(x) = x >= x = x
       
       not(x) = -2x + 0 >= 0 = if(x,false(),true())
      problem:
       DPs:
        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))
       TRS:
        gt(s(x),zero()) -> true()
        gt(s(x),s(y)) -> gt(x,y)
        gt(zero(),y) -> false()
        if(true(),x,y) -> x
        if(false(),x,y) -> y
        id(x) -> x
        not(x) -> if(x,false(),true())
      Restore Modifier:
       DPs:
        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))
       TRS:
        times(x,0()) -> 0()
        times(x,s(y)) -> plus(times(x,y),x)
        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)
       Subterm Criterion Processor:
        simple projection:
         pi(if) = [1,1,1,2]
         pi(id) = 0
         pi(plus#) = 0
        problem:
         DPs:
          plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
         TRS:
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
          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)
        Size-Change Termination Processor:
         DPs:
          
         TRS:
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
          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)
         The DP: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) has the edges:
           0 >   0
         Qed
    
    DPs:
     gt#(s(x),s(y)) -> gt#(x,y)
    TRS:
     times(x,0()) -> 0()
     times(x,s(y)) -> plus(times(x,y),x)
     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)
    Subterm Criterion Processor:
     simple projection:
      pi(gt#) = 0
     problem:
      DPs:
       
      TRS:
       times(x,0()) -> 0()
       times(x,s(y)) -> plus(times(x,y),x)
       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)
     Qed