/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:
 app(app(plus(),0()),y) -> y
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 app(app(times(),0()),y) -> 0()
 app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),y)
 app(app(app(comp(),f),g),x) -> app(f,app(g,x))
 app(twice(),f) -> app(app(comp(),f),f)

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   twice ==> twice0/0 twice1/1
   comp ==> comp0/0 comp1/1 comp2/2 comp3/3
   times ==> times0/0 times1/1 times2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   plus ==> plus0/0 plus1/1 plus2/2
   
  uncurry-rules:
   app(plus1(x4),x5) -> plus2(x4,x5)
   app(plus0(),x4) -> plus1(x4)
   app(s0(),x8) -> s1(x8)
   app(times1(x10),x11) -> times2(x10,x11)
   app(times0(),x10) -> times1(x10)
   app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
   app(comp1(x13),x14) -> comp2(x13,x14)
   app(comp0(),x13) -> comp1(x13)
   app(twice0(),x17) -> twice1(x17)
  eta-rules:
   
  problem:
   plus2(00(),y) -> y
   plus2(s1(x),y) -> s1(plus2(x,y))
   times2(00(),y) -> 00()
   times2(s1(x),y) -> plus2(times2(x,y),y)
   comp3(f,g,x) -> app(f,app(g,x))
   twice1(f) -> comp2(f,f)
   app(plus1(x4),x5) -> plus2(x4,x5)
   app(plus0(),x4) -> plus1(x4)
   app(s0(),x8) -> s1(x8)
   app(times1(x10),x11) -> times2(x10,x11)
   app(times0(),x10) -> times1(x10)
   app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
   app(comp1(x13),x14) -> comp2(x13,x14)
   app(comp0(),x13) -> comp1(x13)
   app(twice0(),x17) -> twice1(x17)
  DP Processor:
   DPs:
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    times{2,#}(s1(x),y) -> times{2,#}(x,y)
    times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
    comp{3,#}(f,g,x) -> app#(g,x)
    comp{3,#}(f,g,x) -> app#(f,app(g,x))
    app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
    app#(times1(x10),x11) -> times{2,#}(x10,x11)
    app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15)
    app#(twice0(),x17) -> twice{1,#}(x17)
   TRS:
    plus2(00(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    times2(00(),y) -> 00()
    times2(s1(x),y) -> plus2(times2(x,y),y)
    comp3(f,g,x) -> app(f,app(g,x))
    twice1(f) -> comp2(f,f)
    app(plus1(x4),x5) -> plus2(x4,x5)
    app(plus0(),x4) -> plus1(x4)
    app(s0(),x8) -> s1(x8)
    app(times1(x10),x11) -> times2(x10,x11)
    app(times0(),x10) -> times1(x10)
    app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
    app(comp1(x13),x14) -> comp2(x13,x14)
    app(comp0(),x13) -> comp1(x13)
    app(twice0(),x17) -> twice1(x17)
   TDG Processor:
    DPs:
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     comp{3,#}(f,g,x) -> app#(g,x)
     comp{3,#}(f,g,x) -> app#(f,app(g,x))
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     app#(times1(x10),x11) -> times{2,#}(x10,x11)
     app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15)
     app#(twice0(),x17) -> twice{1,#}(x17)
    TRS:
     plus2(00(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     times2(00(),y) -> 00()
     times2(s1(x),y) -> plus2(times2(x,y),y)
     comp3(f,g,x) -> app(f,app(g,x))
     twice1(f) -> comp2(f,f)
     app(plus1(x4),x5) -> plus2(x4,x5)
     app(plus0(),x4) -> plus1(x4)
     app(s0(),x8) -> s1(x8)
     app(times1(x10),x11) -> times2(x10,x11)
     app(times0(),x10) -> times1(x10)
     app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
     app(comp1(x13),x14) -> comp2(x13,x14)
     app(comp0(),x13) -> comp1(x13)
     app(twice0(),x17) -> twice1(x17)
    graph:
     app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15) ->
     comp{3,#}(f,g,x) -> app#(f,app(g,x))
     app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15) ->
     comp{3,#}(f,g,x) -> app#(g,x)
     app#(times1(x10),x11) -> times{2,#}(x10,x11) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     app#(times1(x10),x11) -> times{2,#}(x10,x11) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     comp{3,#}(f,g,x) -> app#(g,x) ->
     app#(twice0(),x17) -> twice{1,#}(x17)
     comp{3,#}(f,g,x) -> app#(g,x) ->
     app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15)
     comp{3,#}(f,g,x) -> app#(g,x) ->
     app#(times1(x10),x11) -> times{2,#}(x10,x11)
     comp{3,#}(f,g,x) -> app#(g,x) ->
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     comp{3,#}(f,g,x) -> app#(f,app(g,x)) ->
     app#(twice0(),x17) -> twice{1,#}(x17)
     comp{3,#}(f,g,x) -> app#(f,app(g,x)) ->
     app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15)
     comp{3,#}(f,g,x) -> app#(f,app(g,x)) ->
     app#(times1(x10),x11) -> times{2,#}(x10,x11)
     comp{3,#}(f,g,x) -> app#(f,app(g,x)) ->
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     times{2,#}(s1(x),y) -> times{2,#}(x,y) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     times{2,#}(s1(x),y) -> times{2,#}(x,y) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) -> plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    SCC Processor:
     #sccs: 3
     #rules: 5
     #arcs: 17/81
     DPs:
      app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15)
      comp{3,#}(f,g,x) -> app#(g,x)
      comp{3,#}(f,g,x) -> app#(f,app(g,x))
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(00(),y) -> 00()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      comp3(f,g,x) -> app(f,app(g,x))
      twice1(f) -> comp2(f,f)
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(times1(x10),x11) -> times2(x10,x11)
      app(times0(),x10) -> times1(x10)
      app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
      app(comp1(x13),x14) -> comp2(x13,x14)
      app(comp0(),x13) -> comp1(x13)
      app(twice0(),x17) -> twice1(x17)
     Size-Change Termination Processor:
      DPs:
       
      TRS:
       plus2(00(),y) -> y
       plus2(s1(x),y) -> s1(plus2(x,y))
       times2(00(),y) -> 00()
       times2(s1(x),y) -> plus2(times2(x,y),y)
       comp3(f,g,x) -> app(f,app(g,x))
       twice1(f) -> comp2(f,f)
       app(plus1(x4),x5) -> plus2(x4,x5)
       app(plus0(),x4) -> plus1(x4)
       app(s0(),x8) -> s1(x8)
       app(times1(x10),x11) -> times2(x10,x11)
       app(times0(),x10) -> times1(x10)
       app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
       app(comp1(x13),x14) -> comp2(x13,x14)
       app(comp0(),x13) -> comp1(x13)
       app(twice0(),x17) -> twice1(x17)
      The DP: app#(comp2(x13,x14),x15) -> comp{3,#}(x13,x14,x15) has the edges:
        0 >   1
        0 >   0
        1 >=  2
      The DP: comp{3,#}(f,g,x) -> app#(g,x) has the edges:
        1 >=  0
        2 >=  1
      The DP: comp{3,#}(f,g,x) -> app#(f,app(g,x)) has the edges:
        0 >=  0
      Qed
     
     DPs:
      times{2,#}(s1(x),y) -> times{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(00(),y) -> 00()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      comp3(f,g,x) -> app(f,app(g,x))
      twice1(f) -> comp2(f,f)
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(times1(x10),x11) -> times2(x10,x11)
      app(times0(),x10) -> times1(x10)
      app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
      app(comp1(x13),x14) -> comp2(x13,x14)
      app(comp0(),x13) -> comp1(x13)
      app(twice0(),x17) -> twice1(x17)
     Subterm Criterion Processor:
      simple projection:
       pi(times{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(00(),y) -> 00()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        comp3(f,g,x) -> app(f,app(g,x))
        twice1(f) -> comp2(f,f)
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(times1(x10),x11) -> times2(x10,x11)
        app(times0(),x10) -> times1(x10)
        app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
        app(comp1(x13),x14) -> comp2(x13,x14)
        app(comp0(),x13) -> comp1(x13)
        app(twice0(),x17) -> twice1(x17)
      Qed
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(00(),y) -> 00()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      comp3(f,g,x) -> app(f,app(g,x))
      twice1(f) -> comp2(f,f)
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(times1(x10),x11) -> times2(x10,x11)
      app(times0(),x10) -> times1(x10)
      app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
      app(comp1(x13),x14) -> comp2(x13,x14)
      app(comp0(),x13) -> comp1(x13)
      app(twice0(),x17) -> twice1(x17)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(00(),y) -> 00()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        comp3(f,g,x) -> app(f,app(g,x))
        twice1(f) -> comp2(f,f)
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(times1(x10),x11) -> times2(x10,x11)
        app(times0(),x10) -> times1(x10)
        app(comp2(x13,x14),x15) -> comp3(x13,x14,x15)
        app(comp1(x13),x14) -> comp2(x13,x14)
        app(comp0(),x13) -> comp1(x13)
        app(twice0(),x17) -> twice1(x17)
      Qed