/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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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:
       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())
      interpretation:
       [gt](x0, x1) = 2x1 + 1,
       
       [zero] = 2,
       
       [if](x0, x1, x2) = x0 + x1 + x2 + 0,
       
       [false] = 0,
       
       [id](x0) = x0 + 1,
       
       [plus#](x0, x1) = 2x0 + x1 + 0,
       
       [true] = 0,
       
       [not](x0) = 2x0 + 1,
       
       [s](x0) = 4x0 + 2
      orientation:
       plus#(id(x),s(y)) = 2x + 4y + 3 >= 2x + 4y + 2 = plus#(x,if(gt(s(y),y),y,s(y)))
       
       plus#(s(x),s(y)) = 6x + 4y + 4 >= 2x + 4y + 3 = plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
       
       plus#(s(x),x) = 6x + 4 >= 4x + 3 = plus#(if(gt(x,x),id(x),id(x)),s(x))
       
       gt(s(x),zero()) = 4 >= 0 = true()
       
       gt(s(x),s(y)) = 6y + 4 >= 2y + 1 = gt(x,y)
       
       gt(zero(),y) = 2y + 1 >= 0 = false()
       
       if(true(),x,y) = x + y + 0 >= x = x
       
       if(false(),x,y) = x + y + 0 >= y = y
       
       id(x) = x + 1 >= x = x
       
       not(x) = 2x + 1 >= x + 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),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
       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),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(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)
       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)))
        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:
          [gt](x0, x1) = x0 + 4x1 + 3,
          
          [zero] = 1,
          
          [if](x0, x1, x2) = x1 + x2,
          
          [false] = 1,
          
          [id](x0) = x0,
          
          [plus#](x0, x1) = x0 + x1,
          
          [true] = 5,
          
          [not](x0) = 2,
          
          [s](x0) = 1x0
         orientation:
          plus#(id(x),s(y)) = x + 1y >= x + 1y = plus#(x,if(gt(s(y),y),y,s(y)))
          
          plus#(s(x),s(y)) = 1x + 1y >= x + y = plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
          
          gt(s(x),zero()) = 1x + 5 >= 5 = true()
          
          gt(s(x),s(y)) = 1x + 5y + 3 >= x + 4y + 3 = gt(x,y)
          
          gt(zero(),y) = 4y + 3 >= 1 = false()
          
          if(true(),x,y) = x + y >= x = x
          
          if(false(),x,y) = x + y >= y = y
          
          id(x) = x >= x = x
          
          not(x) = 2 >= 5 = if(x,false(),true())
         problem:
          DPs:
           plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
          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)))
          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