/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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(curry(),g),x),y) -> app(app(g,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))
 inc() -> app(map(),app(app(curry(),plus()),app(s(),0())))
 double() -> app(map(),app(app(curry(),times()),app(s(),app(s(),0()))))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   double ==> double0/0
   inc ==> inc0/0
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   curry ==> curry0/0 curry1/1 curry2/2 curry3/3
   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(x5),x6) -> plus2(x5,x6)
   app(plus0(),x5) -> plus1(x5)
   app(s0(),x9) -> s1(x9)
   app(times1(x11),x12) -> times2(x11,x12)
   app(times0(),x11) -> times1(x11)
   app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
   app(curry1(x14),x15) -> curry2(x14,x15)
   app(curry0(),x14) -> curry1(x14)
   app(map1(x18),x19) -> map2(x18,x19)
   app(map0(),x18) -> map1(x18)
   app(cons1(x22),x23) -> cons2(x22,x23)
   app(cons0(),x22) -> cons1(x22)
  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)
   curry3(g,x,y) -> app(app(g,x),y)
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   inc0() -> map1(curry2(plus0(),s1(00())))
   double0() -> map1(curry2(times0(),s1(s1(00()))))
   app(plus1(x5),x6) -> plus2(x5,x6)
   app(plus0(),x5) -> plus1(x5)
   app(s0(),x9) -> s1(x9)
   app(times1(x11),x12) -> times2(x11,x12)
   app(times0(),x11) -> times1(x11)
   app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
   app(curry1(x14),x15) -> curry2(x14,x15)
   app(curry0(),x14) -> curry1(x14)
   app(map1(x18),x19) -> map2(x18,x19)
   app(map0(),x18) -> map1(x18)
   app(cons1(x22),x23) -> cons2(x22,x23)
   app(cons0(),x22) -> cons1(x22)
  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)
    curry{3,#}(g,x,y) -> app#(g,x)
    curry{3,#}(g,x,y) -> app#(app(g,x),y)
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
    app#(times1(x11),x12) -> times{2,#}(x11,x12)
    app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
    app#(map1(x18),x19) -> map{2,#}(x18,x19)
   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)
    curry3(g,x,y) -> app(app(g,x),y)
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    inc0() -> map1(curry2(plus0(),s1(00())))
    double0() -> map1(curry2(times0(),s1(s1(00()))))
    app(plus1(x5),x6) -> plus2(x5,x6)
    app(plus0(),x5) -> plus1(x5)
    app(s0(),x9) -> s1(x9)
    app(times1(x11),x12) -> times2(x11,x12)
    app(times0(),x11) -> times1(x11)
    app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
    app(curry1(x14),x15) -> curry2(x14,x15)
    app(curry0(),x14) -> curry1(x14)
    app(map1(x18),x19) -> map2(x18,x19)
    app(map0(),x18) -> map1(x18)
    app(cons1(x22),x23) -> cons2(x22,x23)
    app(cons0(),x22) -> cons1(x22)
   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)
     curry{3,#}(g,x,y) -> app#(g,x)
     curry{3,#}(g,x,y) -> app#(app(g,x),y)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     app#(times1(x11),x12) -> times{2,#}(x11,x12)
     app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
     app#(map1(x18),x19) -> map{2,#}(x18,x19)
    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)
     curry3(g,x,y) -> app(app(g,x),y)
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     inc0() -> map1(curry2(plus0(),s1(00())))
     double0() -> map1(curry2(times0(),s1(s1(00()))))
     app(plus1(x5),x6) -> plus2(x5,x6)
     app(plus0(),x5) -> plus1(x5)
     app(s0(),x9) -> s1(x9)
     app(times1(x11),x12) -> times2(x11,x12)
     app(times0(),x11) -> times1(x11)
     app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
     app(curry1(x14),x15) -> curry2(x14,x15)
     app(curry0(),x14) -> curry1(x14)
     app(map1(x18),x19) -> map2(x18,x19)
     app(map0(),x18) -> map1(x18)
     app(cons1(x22),x23) -> cons2(x22,x23)
     app(cons0(),x22) -> cons1(x22)
    graph:
     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)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x18),x19) -> map{2,#}(x18,x19)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(times1(x11),x12) -> times{2,#}(x11,x12)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     app#(map1(x18),x19) -> map{2,#}(x18,x19) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x18),x19) -> map{2,#}(x18,x19) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16) ->
     curry{3,#}(g,x,y) -> app#(app(g,x),y)
     app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16) ->
     curry{3,#}(g,x,y) -> app#(g,x)
     app#(times1(x11),x12) -> times{2,#}(x11,x12) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     app#(times1(x11),x12) -> times{2,#}(x11,x12) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     curry{3,#}(g,x,y) -> app#(app(g,x),y) ->
     app#(map1(x18),x19) -> map{2,#}(x18,x19)
     curry{3,#}(g,x,y) -> app#(app(g,x),y) ->
     app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
     curry{3,#}(g,x,y) -> app#(app(g,x),y) ->
     app#(times1(x11),x12) -> times{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#(map1(x18),x19) -> map{2,#}(x18,x19)
     curry{3,#}(g,x,y) -> app#(g,x) ->
     app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
     curry{3,#}(g,x,y) -> app#(g,x) ->
     app#(times1(x11),x12) -> times{2,#}(x11,x12)
     curry{3,#}(g,x,y) -> app#(g,x) ->
     app#(plus1(x5),x6) -> plus{2,#}(x5,x6)
     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: 8
     #arcs: 25/121
     DPs:
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
      curry{3,#}(g,x,y) -> app#(g,x)
      app#(map1(x18),x19) -> map{2,#}(x18,x19)
      curry{3,#}(g,x,y) -> app#(app(g,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)
      curry3(g,x,y) -> app(app(g,x),y)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc0() -> map1(curry2(plus0(),s1(00())))
      double0() -> map1(curry2(times0(),s1(s1(00()))))
      app(plus1(x5),x6) -> plus2(x5,x6)
      app(plus0(),x5) -> plus1(x5)
      app(s0(),x9) -> s1(x9)
      app(times1(x11),x12) -> times2(x11,x12)
      app(times0(),x11) -> times1(x11)
      app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
      app(curry1(x14),x15) -> curry2(x14,x15)
      app(curry0(),x14) -> curry1(x14)
      app(map1(x18),x19) -> map2(x18,x19)
      app(map0(),x18) -> map1(x18)
      app(cons1(x22),x23) -> cons2(x22,x23)
      app(cons0(),x22) -> cons1(x22)
     Subterm Criterion Processor:
      simple projection:
       pi(app) = [0,1]
       pi(plus1) = 0
       pi(plus2) = 1
       pi(s1) = 0
       pi(times1) = 0
       pi(times2) = [0,1]
       pi(curry1) = 0
       pi(curry2) = [0,1]
       pi(curry3) = [0,1,2]
       pi(map1) = 0
       pi(map2) = 1
       pi(cons1) = 0
       pi(cons2) = 1
       pi(curry{3,#}) = [0,1]
       pi(app#) = 0
       pi(map{2,#}) = 0
      problem:
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
        map{2,#}(f,cons2(x,xs)) -> app#(f,x)
        app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
        app#(map1(x18),x19) -> map{2,#}(x18,x19)
        curry{3,#}(g,x,y) -> app#(app(g,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)
        curry3(g,x,y) -> app(app(g,x),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(curry2(plus0(),s1(00())))
        double0() -> map1(curry2(times0(),s1(s1(00()))))
        app(plus1(x5),x6) -> plus2(x5,x6)
        app(plus0(),x5) -> plus1(x5)
        app(s0(),x9) -> s1(x9)
        app(times1(x11),x12) -> times2(x11,x12)
        app(times0(),x11) -> times1(x11)
        app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
        app(curry1(x14),x15) -> curry2(x14,x15)
        app(curry0(),x14) -> curry1(x14)
        app(map1(x18),x19) -> map2(x18,x19)
        app(map0(),x18) -> map1(x18)
        app(cons1(x22),x23) -> cons2(x22,x23)
        app(cons0(),x22) -> cons1(x22)
      Subterm Criterion Processor:
       simple projection:
        pi(curry{3,#}) = 2
        pi(app#) = 1
        pi(map{2,#}) = 1
       problem:
        DPs:
         app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
         app#(map1(x18),x19) -> map{2,#}(x18,x19)
         curry{3,#}(g,x,y) -> app#(app(g,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)
         curry3(g,x,y) -> app(app(g,x),y)
         map2(f,nil0()) -> nil0()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         inc0() -> map1(curry2(plus0(),s1(00())))
         double0() -> map1(curry2(times0(),s1(s1(00()))))
         app(plus1(x5),x6) -> plus2(x5,x6)
         app(plus0(),x5) -> plus1(x5)
         app(s0(),x9) -> s1(x9)
         app(times1(x11),x12) -> times2(x11,x12)
         app(times0(),x11) -> times1(x11)
         app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
         app(curry1(x14),x15) -> curry2(x14,x15)
         app(curry0(),x14) -> curry1(x14)
         app(map1(x18),x19) -> map2(x18,x19)
         app(map0(),x18) -> map1(x18)
         app(cons1(x22),x23) -> cons2(x22,x23)
         app(cons0(),x22) -> cons1(x22)
       SCC Processor:
        #sccs: 1
        #rules: 2
        #arcs: 12/9
        DPs:
         app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
         curry{3,#}(g,x,y) -> app#(app(g,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)
         curry3(g,x,y) -> app(app(g,x),y)
         map2(f,nil0()) -> nil0()
         map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
         inc0() -> map1(curry2(plus0(),s1(00())))
         double0() -> map1(curry2(times0(),s1(s1(00()))))
         app(plus1(x5),x6) -> plus2(x5,x6)
         app(plus0(),x5) -> plus1(x5)
         app(s0(),x9) -> s1(x9)
         app(times1(x11),x12) -> times2(x11,x12)
         app(times0(),x11) -> times1(x11)
         app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
         app(curry1(x14),x15) -> curry2(x14,x15)
         app(curry0(),x14) -> curry1(x14)
         app(map1(x18),x19) -> map2(x18,x19)
         app(map0(),x18) -> map1(x18)
         app(cons1(x22),x23) -> cons2(x22,x23)
         app(cons0(),x22) -> cons1(x22)
        Subterm Criterion Processor:
         simple projection:
          pi(app) = [0,1,1,1]
          pi(plus1) = 0
          pi(plus2) = [1,1]
          pi(s1) = 0
          pi(times1) = [0,0]
          pi(times2) = [0,0,1,1]
          pi(curry1) = [0,0,0]
          pi(curry2) = [0,0,1,1,1]
          pi(curry3) = [0,0,1,1,1,2,2,2]
          pi(map1) = [0,0,0]
          pi(map2) = [1,1,1]
          pi(cons1) = 0
          pi(cons2) = [1,1,1]
          pi(curry{3,#}) = [0,0,1,1,1]
          pi(app#) = 0
         problem:
          DPs:
           app#(curry2(x14,x15),x16) -> curry{3,#}(x14,x15,x16)
          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)
           curry3(g,x,y) -> app(app(g,x),y)
           map2(f,nil0()) -> nil0()
           map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
           inc0() -> map1(curry2(plus0(),s1(00())))
           double0() -> map1(curry2(times0(),s1(s1(00()))))
           app(plus1(x5),x6) -> plus2(x5,x6)
           app(plus0(),x5) -> plus1(x5)
           app(s0(),x9) -> s1(x9)
           app(times1(x11),x12) -> times2(x11,x12)
           app(times0(),x11) -> times1(x11)
           app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
           app(curry1(x14),x15) -> curry2(x14,x15)
           app(curry0(),x14) -> curry1(x14)
           app(map1(x18),x19) -> map2(x18,x19)
           app(map0(),x18) -> map1(x18)
           app(cons1(x22),x23) -> cons2(x22,x23)
           app(cons0(),x22) -> cons1(x22)
         SCC Processor:
          #sccs: 0
          #rules: 0
          #arcs: 2/1
          
     
     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)
      curry3(g,x,y) -> app(app(g,x),y)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc0() -> map1(curry2(plus0(),s1(00())))
      double0() -> map1(curry2(times0(),s1(s1(00()))))
      app(plus1(x5),x6) -> plus2(x5,x6)
      app(plus0(),x5) -> plus1(x5)
      app(s0(),x9) -> s1(x9)
      app(times1(x11),x12) -> times2(x11,x12)
      app(times0(),x11) -> times1(x11)
      app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
      app(curry1(x14),x15) -> curry2(x14,x15)
      app(curry0(),x14) -> curry1(x14)
      app(map1(x18),x19) -> map2(x18,x19)
      app(map0(),x18) -> map1(x18)
      app(cons1(x22),x23) -> cons2(x22,x23)
      app(cons0(),x22) -> cons1(x22)
     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)
        curry3(g,x,y) -> app(app(g,x),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(curry2(plus0(),s1(00())))
        double0() -> map1(curry2(times0(),s1(s1(00()))))
        app(plus1(x5),x6) -> plus2(x5,x6)
        app(plus0(),x5) -> plus1(x5)
        app(s0(),x9) -> s1(x9)
        app(times1(x11),x12) -> times2(x11,x12)
        app(times0(),x11) -> times1(x11)
        app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
        app(curry1(x14),x15) -> curry2(x14,x15)
        app(curry0(),x14) -> curry1(x14)
        app(map1(x18),x19) -> map2(x18,x19)
        app(map0(),x18) -> map1(x18)
        app(cons1(x22),x23) -> cons2(x22,x23)
        app(cons0(),x22) -> cons1(x22)
      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)
      curry3(g,x,y) -> app(app(g,x),y)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc0() -> map1(curry2(plus0(),s1(00())))
      double0() -> map1(curry2(times0(),s1(s1(00()))))
      app(plus1(x5),x6) -> plus2(x5,x6)
      app(plus0(),x5) -> plus1(x5)
      app(s0(),x9) -> s1(x9)
      app(times1(x11),x12) -> times2(x11,x12)
      app(times0(),x11) -> times1(x11)
      app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
      app(curry1(x14),x15) -> curry2(x14,x15)
      app(curry0(),x14) -> curry1(x14)
      app(map1(x18),x19) -> map2(x18,x19)
      app(map0(),x18) -> map1(x18)
      app(cons1(x22),x23) -> cons2(x22,x23)
      app(cons0(),x22) -> cons1(x22)
     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)
        curry3(g,x,y) -> app(app(g,x),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(curry2(plus0(),s1(00())))
        double0() -> map1(curry2(times0(),s1(s1(00()))))
        app(plus1(x5),x6) -> plus2(x5,x6)
        app(plus0(),x5) -> plus1(x5)
        app(s0(),x9) -> s1(x9)
        app(times1(x11),x12) -> times2(x11,x12)
        app(times0(),x11) -> times1(x11)
        app(curry2(x14,x15),x16) -> curry3(x14,x15,x16)
        app(curry1(x14),x15) -> curry2(x14,x15)
        app(curry0(),x14) -> curry1(x14)
        app(map1(x18),x19) -> map2(x18,x19)
        app(map0(),x18) -> map1(x18)
        app(cons1(x22),x23) -> cons2(x22,x23)
        app(cons0(),x22) -> cons1(x22)
      Qed