/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(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 app(app(app(curry(),g),x),y) -> app(app(g,x),y)
 inc() -> app(map(),app(app(curry(),plus()),app(s(),0())))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   inc ==> inc0/0
   curry ==> curry0/0 curry1/1 curry2/2 curry3/3
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   plus ==> plus0/0 plus1/1 plus2/2
   
  uncurry-rules:
   app(plus1(x5),x6) -> plus2(x5,x6)
   app(plus0(),x5) -> plus1(x5)
   app(s0(),x9) -> s1(x9)
   app(map1(x11),x12) -> map2(x11,x12)
   app(map0(),x11) -> map1(x11)
   app(cons1(x15),x16) -> cons2(x15,x16)
   app(cons0(),x15) -> cons1(x15)
   app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
   app(curry1(x18),x19) -> curry2(x18,x19)
   app(curry0(),x18) -> curry1(x18)
  eta-rules:
   
  problem:
   plus2(00(),y) -> y
   plus2(s1(x),y) -> s1(plus2(x,y))
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   curry3(g,x,y) -> app(app(g,x),y)
   inc0() -> map1(curry2(plus0(),s1(00())))
   app(plus1(x5),x6) -> plus2(x5,x6)
   app(plus0(),x5) -> plus1(x5)
   app(s0(),x9) -> s1(x9)
   app(map1(x11),x12) -> map2(x11,x12)
   app(map0(),x11) -> map1(x11)
   app(cons1(x15),x16) -> cons2(x15,x16)
   app(cons0(),x15) -> cons1(x15)
   app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
   app(curry1(x18),x19) -> curry2(x18,x19)
   app(curry0(),x18) -> curry1(x18)
  DP Processor:
   DPs:
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    curry{3,#}(g,x,y) -> app#(g,x)
    curry{3,#}(g,x,y) -> app#(app(g,x),y)
    app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
    app#(map1(x11),x12) -> map{2,#}(x11,x12)
    app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
   TRS:
    plus2(00(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    curry3(g,x,y) -> app(app(g,x),y)
    inc0() -> map1(curry2(plus0(),s1(00())))
    app(plus1(x5),x6) -> plus2(x5,x6)
    app(plus0(),x5) -> plus1(x5)
    app(s0(),x9) -> s1(x9)
    app(map1(x11),x12) -> map2(x11,x12)
    app(map0(),x11) -> map1(x11)
    app(cons1(x15),x16) -> cons2(x15,x16)
    app(cons0(),x15) -> cons1(x15)
    app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
    app(curry1(x18),x19) -> curry2(x18,x19)
    app(curry0(),x18) -> curry1(x18)
   TDG Processor:
    DPs:
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     curry{3,#}(g,x,y) -> app#(g,x)
     curry{3,#}(g,x,y) -> app#(app(g,x),y)
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     app#(map1(x11),x12) -> map{2,#}(x11,x12)
     app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
    TRS:
     plus2(00(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     curry3(g,x,y) -> app(app(g,x),y)
     inc0() -> map1(curry2(plus0(),s1(00())))
     app(plus1(x5),x6) -> plus2(x5,x6)
     app(plus0(),x5) -> plus1(x5)
     app(s0(),x9) -> s1(x9)
     app(map1(x11),x12) -> map2(x11,x12)
     app(map0(),x11) -> map1(x11)
     app(cons1(x15),x16) -> cons2(x15,x16)
     app(cons0(),x15) -> cons1(x15)
     app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
     app(curry1(x18),x19) -> curry2(x18,x19)
     app(curry0(),x18) -> curry1(x18)
    graph:
     curry{3,#}(g,x,y) -> app#(app(g,x),y) ->
     app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
     curry{3,#}(g,x,y) -> app#(app(g,x),y) ->
     app#(map1(x11),x12) -> map{2,#}(x11,x12)
     curry{3,#}(g,x,y) -> app#(app(g,x),y) ->
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     curry{3,#}(g,x,y) -> app#(g,x) ->
     app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
     curry{3,#}(g,x,y) -> app#(g,x) ->
     app#(map1(x11),x12) -> map{2,#}(x11,x12)
     curry{3,#}(g,x,y) -> app#(g,x) ->
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20) ->
     curry{3,#}(g,x,y) -> app#(app(g,x),y)
     app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20) ->
     curry{3,#}(g,x,y) -> app#(g,x)
     app#(map1(x11),x12) -> map{2,#}(x11,x12) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x11),x12) -> map{2,#}(x11,x12) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x11),x12) -> map{2,#}(x11,x12)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) -> plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    SCC Processor:
     #sccs: 2
     #rules: 7
     #arcs: 17/64
     DPs:
      curry{3,#}(g,x,y) -> app#(app(g,x),y)
      app#(map1(x11),x12) -> map{2,#}(x11,x12)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
      curry{3,#}(g,x,y) -> app#(g,x)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      curry3(g,x,y) -> app(app(g,x),y)
      inc0() -> map1(curry2(plus0(),s1(00())))
      app(plus1(x5),x6) -> plus2(x5,x6)
      app(plus0(),x5) -> plus1(x5)
      app(s0(),x9) -> s1(x9)
      app(map1(x11),x12) -> map2(x11,x12)
      app(map0(),x11) -> map1(x11)
      app(cons1(x15),x16) -> cons2(x15,x16)
      app(cons0(),x15) -> cons1(x15)
      app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
      app(curry1(x18),x19) -> curry2(x18,x19)
      app(curry0(),x18) -> curry1(x18)
     Subterm Criterion Processor:
      simple projection:
       pi(app) = [0,1]
       pi(plus1) = 0
       pi(plus2) = 1
       pi(s1) = 0
       pi(map1) = 0
       pi(map2) = 1
       pi(cons1) = 0
       pi(cons2) = 1
       pi(curry1) = 0
       pi(curry2) = [0,1]
       pi(curry3) = [0,1,2]
       pi(map{2,#}) = [0,0]
       pi(app#) = [0,0]
       pi(curry{3,#}) = [0,0,1,1]
      problem:
       DPs:
        curry{3,#}(g,x,y) -> app#(app(g,x),y)
        app#(map1(x11),x12) -> map{2,#}(x11,x12)
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
        map{2,#}(f,cons2(x,xs)) -> app#(f,x)
        app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        curry3(g,x,y) -> app(app(g,x),y)
        inc0() -> map1(curry2(plus0(),s1(00())))
        app(plus1(x5),x6) -> plus2(x5,x6)
        app(plus0(),x5) -> plus1(x5)
        app(s0(),x9) -> s1(x9)
        app(map1(x11),x12) -> map2(x11,x12)
        app(map0(),x11) -> map1(x11)
        app(cons1(x15),x16) -> cons2(x15,x16)
        app(cons0(),x15) -> cons1(x15)
        app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
        app(curry1(x18),x19) -> curry2(x18,x19)
        app(curry0(),x18) -> curry1(x18)
      Subterm Criterion Processor:
       simple projection:
        pi(map{2,#}) = 1
        pi(app#) = 1
        pi(curry{3,#}) = 2
       problem:
        DPs:
         curry{3,#}(g,x,y) -> app#(app(g,x),y)
         app#(map1(x11),x12) -> map{2,#}(x11,x12)
         app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
        TRS:
         plus2(00(),y) -> y
         plus2(s1(x),y) -> s1(plus2(x,y))
         map2(f,nil0()) -> nil0()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         curry3(g,x,y) -> app(app(g,x),y)
         inc0() -> map1(curry2(plus0(),s1(00())))
         app(plus1(x5),x6) -> plus2(x5,x6)
         app(plus0(),x5) -> plus1(x5)
         app(s0(),x9) -> s1(x9)
         app(map1(x11),x12) -> map2(x11,x12)
         app(map0(),x11) -> map1(x11)
         app(cons1(x15),x16) -> cons2(x15,x16)
         app(cons0(),x15) -> cons1(x15)
         app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
         app(curry1(x18),x19) -> curry2(x18,x19)
         app(curry0(),x18) -> curry1(x18)
       SCC Processor:
        #sccs: 1
        #rules: 2
        #arcs: 12/9
        DPs:
         curry{3,#}(g,x,y) -> app#(app(g,x),y)
         app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
        TRS:
         plus2(00(),y) -> y
         plus2(s1(x),y) -> s1(plus2(x,y))
         map2(f,nil0()) -> nil0()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         curry3(g,x,y) -> app(app(g,x),y)
         inc0() -> map1(curry2(plus0(),s1(00())))
         app(plus1(x5),x6) -> plus2(x5,x6)
         app(plus0(),x5) -> plus1(x5)
         app(s0(),x9) -> s1(x9)
         app(map1(x11),x12) -> map2(x11,x12)
         app(map0(),x11) -> map1(x11)
         app(cons1(x15),x16) -> cons2(x15,x16)
         app(cons0(),x15) -> cons1(x15)
         app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
         app(curry1(x18),x19) -> curry2(x18,x19)
         app(curry0(),x18) -> curry1(x18)
        Subterm Criterion Processor:
         simple projection:
          pi(app) = [0,1,1,1]
          pi(plus1) = [0,0,0]
          pi(plus2) = [1,1]
          pi(s1) = 0
          pi(map1) = 0
          pi(map2) = 1
          pi(cons1) = [0,0,0]
          pi(cons2) = 1
          pi(curry1) = [0,0,0]
          pi(curry2) = [0,0,1,1,1]
          pi(curry3) = [0,1,1,1,2,2,2]
          pi(app#) = 0
          pi(curry{3,#}) = [0,0,1,1,1]
         problem:
          DPs:
           app#(curry2(x18,x19),x20) -> curry{3,#}(x18,x19,x20)
          TRS:
           plus2(00(),y) -> y
           plus2(s1(x),y) -> s1(plus2(x,y))
           map2(f,nil0()) -> nil0()
           map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
           curry3(g,x,y) -> app(app(g,x),y)
           inc0() -> map1(curry2(plus0(),s1(00())))
           app(plus1(x5),x6) -> plus2(x5,x6)
           app(plus0(),x5) -> plus1(x5)
           app(s0(),x9) -> s1(x9)
           app(map1(x11),x12) -> map2(x11,x12)
           app(map0(),x11) -> map1(x11)
           app(cons1(x15),x16) -> cons2(x15,x16)
           app(cons0(),x15) -> cons1(x15)
           app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
           app(curry1(x18),x19) -> curry2(x18,x19)
           app(curry0(),x18) -> curry1(x18)
         SCC Processor:
          #sccs: 0
          #rules: 0
          #arcs: 2/1
          
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      curry3(g,x,y) -> app(app(g,x),y)
      inc0() -> map1(curry2(plus0(),s1(00())))
      app(plus1(x5),x6) -> plus2(x5,x6)
      app(plus0(),x5) -> plus1(x5)
      app(s0(),x9) -> s1(x9)
      app(map1(x11),x12) -> map2(x11,x12)
      app(map0(),x11) -> map1(x11)
      app(cons1(x15),x16) -> cons2(x15,x16)
      app(cons0(),x15) -> cons1(x15)
      app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
      app(curry1(x18),x19) -> curry2(x18,x19)
      app(curry0(),x18) -> curry1(x18)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        curry3(g,x,y) -> app(app(g,x),y)
        inc0() -> map1(curry2(plus0(),s1(00())))
        app(plus1(x5),x6) -> plus2(x5,x6)
        app(plus0(),x5) -> plus1(x5)
        app(s0(),x9) -> s1(x9)
        app(map1(x11),x12) -> map2(x11,x12)
        app(map0(),x11) -> map1(x11)
        app(cons1(x15),x16) -> cons2(x15,x16)
        app(cons0(),x15) -> cons1(x15)
        app(curry2(x18,x19),x20) -> curry3(x18,x19,x20)
        app(curry1(x18),x19) -> curry2(x18,x19)
        app(curry0(),x18) -> curry1(x18)
      Qed