/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /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) -> 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()
     KBO Processor:
      argument filtering:
       pi(0) = []
       pi(le) = []
       pi(true) = []
       pi(s) = [0]
       pi(false) = []
       pi(minus) = 0
       pi(ifMinus) = 1
       pi(quot#) = [0,1]
      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))
      weight function:
       w0 = 1
       w(false) = w(s) = w(true) = w(le) = w(0) = 1
       w(quot#) = w(ifMinus) = w(minus) = 0
      precedence:
       0 > ifMinus > quot# > false > s > true > le > minus
      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