/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(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
 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))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   inc ==> inc0/0 inc1/1
   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(inc0(),x10) -> inc1(x10)
   app(map1(x12),x13) -> map2(x12,x13)
   app(map0(),x12) -> map1(x12)
   app(cons1(x16),x17) -> cons2(x16,x17)
   app(cons0(),x16) -> cons1(x16)
  eta-rules:
   
  problem:
   plus2(00(),y) -> y
   plus2(s1(x),y) -> s1(plus2(x,y))
   inc1(xs) -> map2(plus1(s1(00())),xs)
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   app(plus1(x4),x5) -> plus2(x4,x5)
   app(plus0(),x4) -> plus1(x4)
   app(s0(),x8) -> s1(x8)
   app(inc0(),x10) -> inc1(x10)
   app(map1(x12),x13) -> map2(x12,x13)
   app(map0(),x12) -> map1(x12)
   app(cons1(x16),x17) -> cons2(x16,x17)
   app(cons0(),x16) -> cons1(x16)
  DP Processor:
   DPs:
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
    app#(inc0(),x10) -> inc{1,#}(x10)
    app#(map1(x12),x13) -> map{2,#}(x12,x13)
   TRS:
    plus2(00(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    inc1(xs) -> map2(plus1(s1(00())),xs)
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    app(plus1(x4),x5) -> plus2(x4,x5)
    app(plus0(),x4) -> plus1(x4)
    app(s0(),x8) -> s1(x8)
    app(inc0(),x10) -> inc1(x10)
    app(map1(x12),x13) -> map2(x12,x13)
    app(map0(),x12) -> map1(x12)
    app(cons1(x16),x17) -> cons2(x16,x17)
    app(cons0(),x16) -> cons1(x16)
   TDG Processor:
    DPs:
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     app#(inc0(),x10) -> inc{1,#}(x10)
     app#(map1(x12),x13) -> map{2,#}(x12,x13)
    TRS:
     plus2(00(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     inc1(xs) -> map2(plus1(s1(00())),xs)
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     app(plus1(x4),x5) -> plus2(x4,x5)
     app(plus0(),x4) -> plus1(x4)
     app(s0(),x8) -> s1(x8)
     app(inc0(),x10) -> inc1(x10)
     app(map1(x12),x13) -> map2(x12,x13)
     app(map0(),x12) -> map1(x12)
     app(cons1(x16),x17) -> cons2(x16,x17)
     app(cons0(),x16) -> cons1(x16)
    graph:
     app#(map1(x12),x13) -> map{2,#}(x12,x13) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x12),x13) -> map{2,#}(x12,x13) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(inc0(),x10) -> inc{1,#}(x10) ->
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x12),x13) -> map{2,#}(x12,x13)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(inc0(),x10) -> inc{1,#}(x10)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     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)
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),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: 6
     #arcs: 12/49
     DPs:
      app#(map1(x12),x13) -> map{2,#}(x12,x13)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(inc0(),x10) -> inc{1,#}(x10)
      inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      inc1(xs) -> map2(plus1(s1(00())),xs)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(inc0(),x10) -> inc1(x10)
      app(map1(x12),x13) -> map2(x12,x13)
      app(map0(),x12) -> map1(x12)
      app(cons1(x16),x17) -> cons2(x16,x17)
      app(cons0(),x16) -> cons1(x16)
     Subterm Criterion Processor:
      simple projection:
       pi(inc{1,#}) = 0
       pi(map{2,#}) = 1
       pi(app#) = 1
      problem:
       DPs:
        app#(map1(x12),x13) -> map{2,#}(x12,x13)
        app#(inc0(),x10) -> inc{1,#}(x10)
        inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        inc1(xs) -> map2(plus1(s1(00())),xs)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(inc0(),x10) -> inc1(x10)
        app(map1(x12),x13) -> map2(x12,x13)
        app(map0(),x12) -> map1(x12)
        app(cons1(x16),x17) -> cons2(x16,x17)
        app(cons0(),x16) -> cons1(x16)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 9/9
       
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      inc1(xs) -> map2(plus1(s1(00())),xs)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(inc0(),x10) -> inc1(x10)
      app(map1(x12),x13) -> map2(x12,x13)
      app(map0(),x12) -> map1(x12)
      app(cons1(x16),x17) -> cons2(x16,x17)
      app(cons0(),x16) -> cons1(x16)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        inc1(xs) -> map2(plus1(s1(00())),xs)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(inc0(),x10) -> inc1(x10)
        app(map1(x12),x13) -> map2(x12,x13)
        app(map0(),x12) -> map1(x12)
        app(cons1(x16),x17) -> cons2(x16,x17)
        app(cons0(),x16) -> cons1(x16)
      Qed