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


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YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(0(),y) -> 0()
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
 if_minus(true(),s(x),y) -> 0()
 if_minus(false(),s(x),y) -> s(minus(x,y))
 gcd(0(),y) -> y
 gcd(s(x),0()) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(s(x),y) -> le#(s(x),y)
   minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
   if_minus#(false(),s(x),y) -> minus#(x,y)
   gcd#(s(x),s(y)) -> le#(y,x)
   gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
   if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
   if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
   if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
   if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(0(),y) -> 0()
   minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
   if_minus(true(),s(x),y) -> 0()
   if_minus(false(),s(x),y) -> s(minus(x,y))
   gcd(0(),y) -> y
   gcd(s(x),0()) -> s(x)
   gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
   if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
   if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y)
    gcd#(s(x),s(y)) -> le#(y,x)
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    minus(0(),y) -> 0()
    minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
    if_minus(true(),s(x),y) -> 0()
    if_minus(false(),s(x),y) -> s(minus(x,y))
    gcd(0(),y) -> y
    gcd(s(x),0()) -> s(x)
    gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
    if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
    if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
   graph:
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) ->
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) ->
    gcd#(s(x),s(y)) -> le#(y,x)
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x) ->
    minus#(s(x),y) -> le#(s(x),y)
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) ->
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) ->
    gcd#(s(x),s(y)) -> le#(y,x)
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
    gcd#(s(x),s(y)) -> le#(y,x) ->
    le#(s(x),s(y)) -> le#(x,y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) ->
    if_minus#(false(),s(x),y) -> minus#(x,y)
    minus#(s(x),y) -> le#(s(x),y) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 6
    #arcs: 18/100
    DPs:
     if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
     gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
     if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    Usable Rule Processor:
     DPs:
      if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
      gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
      if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
     TRS:
      minus(0(),y) -> 0()
      minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
      if_minus(true(),s(x),y) -> 0()
      if_minus(false(),s(x),y) -> s(minus(x,y))
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      le(0(),y) -> true()
     KBO Processor:
      argument filtering:
       pi(0) = []
       pi(le) = []
       pi(true) = []
       pi(s) = [0]
       pi(false) = []
       pi(minus) = 0
       pi(if_minus) = 1
       pi(gcd#) = [0,1]
       pi(if_gcd#) = [1,2]
      usable rules:
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
      weight function:
       w0 = 1
       w(false) = w(s) = w(true) = w(le) = w(0) = 1
       w(if_gcd#) = w(gcd#) = w(if_minus) = w(minus) = 0
      precedence:
       gcd# > if_gcd# ~ if_minus ~ minus ~ false ~ s ~ true ~ le ~ 0
      problem:
       DPs:
        
       TRS:
        minus(0(),y) -> 0()
        minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        if_minus(true(),s(x),y) -> 0()
        if_minus(false(),s(x),y) -> s(minus(x,y))
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
        le(0(),y) -> true()
      Qed
    
    DPs:
     minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
     if_minus#(false(),s(x),y) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(if_minus#) = 1
     problem:
      DPs:
       minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
       gcd(0(),y) -> y
       gcd(s(x),0()) -> s(x)
       gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
       if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
       if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    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)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 0
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
       gcd(0(),y) -> y
       gcd(s(x),0()) -> s(x)
       gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
       if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
       if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
     Qed