/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:
 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) -> ifMinus(le(s(X),Y),s(X),Y)
 ifMinus(true(),s(X),Y) -> 0()
 ifMinus(false(),s(X),Y) -> s(minus(X,Y))
 quot(0(),s(Y)) -> 0()
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))

Proof:
 DP Processor:
  DPs:
   le#(s(X),s(Y)) -> le#(X,Y)
   minus#(s(X),Y) -> le#(s(X),Y)
   minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
   ifMinus#(false(),s(X),Y) -> minus#(X,Y)
   quot#(s(X),s(Y)) -> minus#(X,Y)
   quot#(s(X),s(Y)) -> quot#(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) -> ifMinus(le(s(X),Y),s(X),Y)
   ifMinus(true(),s(X),Y) -> 0()
   ifMinus(false(),s(X),Y) -> s(minus(X,Y))
   quot(0(),s(Y)) -> 0()
   quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
  TDG Processor:
   DPs:
    le#(s(X),s(Y)) -> le#(X,Y)
    minus#(s(X),Y) -> le#(s(X),Y)
    minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
    ifMinus#(false(),s(X),Y) -> minus#(X,Y)
    quot#(s(X),s(Y)) -> minus#(X,Y)
    quot#(s(X),s(Y)) -> quot#(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) -> ifMinus(le(s(X),Y),s(X),Y)
    ifMinus(true(),s(X),Y) -> 0()
    ifMinus(false(),s(X),Y) -> s(minus(X,Y))
    quot(0(),s(Y)) -> 0()
    quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
   graph:
    quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) ->
    quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
    quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) ->
    quot#(s(X),s(Y)) -> minus#(X,Y)
    quot#(s(X),s(Y)) -> minus#(X,Y) ->
    minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
    quot#(s(X),s(Y)) -> minus#(X,Y) ->
    minus#(s(X),Y) -> le#(s(X),Y)
    ifMinus#(false(),s(X),Y) -> minus#(X,Y) ->
    minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
    ifMinus#(false(),s(X),Y) -> minus#(X,Y) ->
    minus#(s(X),Y) -> le#(s(X),Y)
    minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y) ->
    ifMinus#(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: 4
    #arcs: 9/36
    DPs:
     quot#(s(X),s(Y)) -> quot#(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) -> ifMinus(le(s(X),Y),s(X),Y)
     ifMinus(true(),s(X),Y) -> 0()
     ifMinus(false(),s(X),Y) -> s(minus(X,Y))
     quot(0(),s(Y)) -> 0()
     quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
    Usable Rule Processor:
     DPs:
      quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
     TRS:
      minus(0(),Y) -> 0()
      minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
      ifMinus(true(),s(X),Y) -> 0()
      ifMinus(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()
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       minus(0(),Y) -> 0()
       minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
       ifMinus(true(),s(X),Y) -> 0()
       ifMinus(false(),s(X),Y) -> s(minus(X,Y))
      interpretation:
       [quot#](x0, x1) = x0,
       
       [ifMinus](x0, x1, x2) = x1,
       
       [minus](x0, x1) = x0 + -8,
       
       [false] = 8,
       
       [s](x0) = 1x0 + 0,
       
       [true] = 13,
       
       [le](x0, x1) = 3x0 + 3x1 + -16,
       
       [0] = 0
      orientation:
       quot#(s(X),s(Y)) = 1X + 0 >= X + -8 = quot#(minus(X,Y),s(Y))
       
       minus(0(),Y) = 0 >= 0 = 0()
       
       minus(s(X),Y) = 1X + 0 >= 1X + 0 = ifMinus(le(s(X),Y),s(X),Y)
       
       ifMinus(true(),s(X),Y) = 1X + 0 >= 0 = 0()
       
       ifMinus(false(),s(X),Y) = 1X + 0 >= 1X + 0 = s(minus(X,Y))
       
       le(s(X),0()) = 4X + 3 >= 8 = false()
       
       le(s(X),s(Y)) = 4X + 4Y + 3 >= 3X + 3Y + -16 = le(X,Y)
       
       le(0(),Y) = 3Y + 3 >= 13 = true()
      problem:
       DPs:
        
       TRS:
        minus(0(),Y) -> 0()
        minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        ifMinus(true(),s(X),Y) -> 0()
        ifMinus(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) -> ifMinus#(le(s(X),Y),s(X),Y)
     ifMinus#(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) -> ifMinus(le(s(X),Y),s(X),Y)
     ifMinus(true(),s(X),Y) -> 0()
     ifMinus(false(),s(X),Y) -> s(minus(X,Y))
     quot(0(),s(Y)) -> 0()
     quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(ifMinus#) = 1
     problem:
      DPs:
       minus#(s(X),Y) -> ifMinus#(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) -> ifMinus(le(s(X),Y),s(X),Y)
       ifMinus(true(),s(X),Y) -> 0()
       ifMinus(false(),s(X),Y) -> s(minus(X,Y))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
     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) -> ifMinus(le(s(X),Y),s(X),Y)
     ifMinus(true(),s(X),Y) -> 0()
     ifMinus(false(),s(X),Y) -> s(minus(X,Y))
     quot(0(),s(Y)) -> 0()
     quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
    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) -> ifMinus(le(s(X),Y),s(X),Y)
       ifMinus(true(),s(X),Y) -> 0()
       ifMinus(false(),s(X),Y) -> s(minus(X,Y))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
     Qed