/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(f(),0()) -> true()
 app(f(),1()) -> false()
 app(f(),app(s(),x)) -> app(f(),x)
 app(app(app(if(),true()),app(s(),x)),app(s(),y)) -> app(s(),x)
 app(app(app(if(),false()),app(s(),x)),app(s(),y)) -> app(s(),y)
 app(app(g(),x),app(c(),y)) -> app(c(),app(app(g(),x),y))
 app(app(g(),x),app(c(),y)) ->
 app(app(g(),x),app(app(app(if(),app(f(),x)),app(c(),app(app(g(),app(s(),x)),y))),app(c(),y)))
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   filter2 ==> filter20/0 filter21/1 filter22/2 filter23/3 filter24/4
   filter ==> filter0/0 filter1/1 filter2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   c ==> c0/0 c1/1
   g ==> g0/0 g1/1 g2/2
   if ==> if0/0 if1/1 if2/2 if3/3
   s ==> s0/0 s1/1
   false ==> false0/0
   1 ==> 10/0
   true ==> true0/0
   0 ==> 00/0
   f ==> f0/0 f1/1
   
  uncurry-rules:
   app(f0(),x4) -> f1(x4)
   app(s0(),x10) -> s1(x10)
   app(if2(x12,x13),x14) -> if3(x12,x13,x14)
   app(if1(x12),x13) -> if2(x12,x13)
   app(if0(),x12) -> if1(x12)
   app(g1(x16),x17) -> g2(x16,x17)
   app(g0(),x16) -> g1(x16)
   app(c0(),x19) -> c1(x19)
   app(map1(x21),x22) -> map2(x21,x22)
   app(map0(),x21) -> map1(x21)
   app(cons1(x25),x26) -> cons2(x25,x26)
   app(cons0(),x25) -> cons1(x25)
   app(filter1(x28),x29) -> filter2(x28,x29)
   app(filter0(),x28) -> filter1(x28)
   app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
   app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
   app(filter21(x31),x32) -> filter22(x31,x32)
   app(filter20(),x31) -> filter21(x31)
  eta-rules:
   
  problem:
   f1(00()) -> true0()
   f1(10()) -> false0()
   f1(s1(x)) -> f1(x)
   if3(true0(),s1(x),s1(y)) -> s1(x)
   if3(false0(),s1(x),s1(y)) -> s1(y)
   g2(x,c1(y)) -> c1(g2(x,y))
   g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
   map2(fun,nil0()) -> nil0()
   map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
   filter2(fun,nil0()) -> nil0()
   filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
   filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
   filter24(false0(),fun,x,xs) -> filter2(fun,xs)
   app(f0(),x4) -> f1(x4)
   app(s0(),x10) -> s1(x10)
   app(if2(x12,x13),x14) -> if3(x12,x13,x14)
   app(if1(x12),x13) -> if2(x12,x13)
   app(if0(),x12) -> if1(x12)
   app(g1(x16),x17) -> g2(x16,x17)
   app(g0(),x16) -> g1(x16)
   app(c0(),x19) -> c1(x19)
   app(map1(x21),x22) -> map2(x21,x22)
   app(map0(),x21) -> map1(x21)
   app(cons1(x25),x26) -> cons2(x25,x26)
   app(cons0(),x25) -> cons1(x25)
   app(filter1(x28),x29) -> filter2(x28,x29)
   app(filter0(),x28) -> filter1(x28)
   app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
   app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
   app(filter21(x31),x32) -> filter22(x31,x32)
   app(filter20(),x31) -> filter21(x31)
  DP Processor:
   DPs:
    f{1,#}(s1(x)) -> f{1,#}(x)
    g{2,#}(x,c1(y)) -> g{2,#}(x,y)
    g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
    g{2,#}(x,c1(y)) -> f{1,#}(x)
    g{2,#}(x,c1(y)) -> if{3,#}(f1(x),c1(g2(s1(x),y)),c1(y))
    g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
    map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
    map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
    filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
    filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
    app#(f0(),x4) -> f{1,#}(x4)
    app#(if2(x12,x13),x14) -> if{3,#}(x12,x13,x14)
    app#(g1(x16),x17) -> g{2,#}(x16,x17)
    app#(map1(x21),x22) -> map{2,#}(x21,x22)
    app#(filter1(x28),x29) -> filter{2,#}(x28,x29)
    app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34)
   TRS:
    f1(00()) -> true0()
    f1(10()) -> false0()
    f1(s1(x)) -> f1(x)
    if3(true0(),s1(x),s1(y)) -> s1(x)
    if3(false0(),s1(x),s1(y)) -> s1(y)
    g2(x,c1(y)) -> c1(g2(x,y))
    g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
    map2(fun,nil0()) -> nil0()
    map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
    filter2(fun,nil0()) -> nil0()
    filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
    filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
    filter24(false0(),fun,x,xs) -> filter2(fun,xs)
    app(f0(),x4) -> f1(x4)
    app(s0(),x10) -> s1(x10)
    app(if2(x12,x13),x14) -> if3(x12,x13,x14)
    app(if1(x12),x13) -> if2(x12,x13)
    app(if0(),x12) -> if1(x12)
    app(g1(x16),x17) -> g2(x16,x17)
    app(g0(),x16) -> g1(x16)
    app(c0(),x19) -> c1(x19)
    app(map1(x21),x22) -> map2(x21,x22)
    app(map0(),x21) -> map1(x21)
    app(cons1(x25),x26) -> cons2(x25,x26)
    app(cons0(),x25) -> cons1(x25)
    app(filter1(x28),x29) -> filter2(x28,x29)
    app(filter0(),x28) -> filter1(x28)
    app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
    app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
    app(filter21(x31),x32) -> filter22(x31,x32)
    app(filter20(),x31) -> filter21(x31)
   TDG Processor:
    DPs:
     f{1,#}(s1(x)) -> f{1,#}(x)
     g{2,#}(x,c1(y)) -> g{2,#}(x,y)
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
     g{2,#}(x,c1(y)) -> f{1,#}(x)
     g{2,#}(x,c1(y)) -> if{3,#}(f1(x),c1(g2(s1(x),y)),c1(y))
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(f0(),x4) -> f{1,#}(x4)
     app#(if2(x12,x13),x14) -> if{3,#}(x12,x13,x14)
     app#(g1(x16),x17) -> g{2,#}(x16,x17)
     app#(map1(x21),x22) -> map{2,#}(x21,x22)
     app#(filter1(x28),x29) -> filter{2,#}(x28,x29)
     app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34)
    TRS:
     f1(00()) -> true0()
     f1(10()) -> false0()
     f1(s1(x)) -> f1(x)
     if3(true0(),s1(x),s1(y)) -> s1(x)
     if3(false0(),s1(x),s1(y)) -> s1(y)
     g2(x,c1(y)) -> c1(g2(x,y))
     g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     map2(fun,nil0()) -> nil0()
     map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
     filter2(fun,nil0()) -> nil0()
     filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
     filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
     filter24(false0(),fun,x,xs) -> filter2(fun,xs)
     app(f0(),x4) -> f1(x4)
     app(s0(),x10) -> s1(x10)
     app(if2(x12,x13),x14) -> if3(x12,x13,x14)
     app(if1(x12),x13) -> if2(x12,x13)
     app(if0(),x12) -> if1(x12)
     app(g1(x16),x17) -> g2(x16,x17)
     app(g0(),x16) -> g1(x16)
     app(c0(),x19) -> c1(x19)
     app(map1(x21),x22) -> map2(x21,x22)
     app(map0(),x21) -> map1(x21)
     app(cons1(x25),x26) -> cons2(x25,x26)
     app(cons0(),x25) -> cons1(x25)
     app(filter1(x28),x29) -> filter2(x28,x29)
     app(filter0(),x28) -> filter1(x28)
     app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
     app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
     app(filter21(x31),x32) -> filter22(x31,x32)
     app(filter20(),x31) -> filter21(x31)
    graph:
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x28),x29) -> filter{2,#}(x28,x29)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x21),x22) -> map{2,#}(x21,x22)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(g1(x16),x17) -> g{2,#}(x16,x17)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(if2(x12,x13),x14) -> if{3,#}(x12,x13,x14)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f0(),x4) -> f{1,#}(x4)
     app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter1(x28),x29) -> filter{2,#}(x28,x29) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     app#(filter1(x28),x29) -> filter{2,#}(x28,x29) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x21),x22) -> map{2,#}(x21,x22) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x21),x22) -> map{2,#}(x21,x22) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     app#(g1(x16),x17) -> g{2,#}(x16,x17) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     app#(g1(x16),x17) -> g{2,#}(x16,x17) ->
     g{2,#}(x,c1(y)) -> if{3,#}(f1(x),c1(g2(s1(x),y)),c1(y))
     app#(g1(x16),x17) -> g{2,#}(x16,x17) ->
     g{2,#}(x,c1(y)) -> f{1,#}(x)
     app#(g1(x16),x17) -> g{2,#}(x16,x17) ->
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
     app#(g1(x16),x17) -> g{2,#}(x16,x17) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,y)
     app#(f0(),x4) -> f{1,#}(x4) ->
     f{1,#}(s1(x)) -> f{1,#}(x)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x28),x29) -> filter{2,#}(x28,x29)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x21),x22) -> map{2,#}(x21,x22)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(g1(x16),x17) -> g{2,#}(x16,x17)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(if2(x12,x13),x14) -> if{3,#}(x12,x13,x14)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f0(),x4) -> f{1,#}(x4)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
     g{2,#}(x,c1(y)) -> if{3,#}(f1(x),c1(g2(s1(x),y)),c1(y))
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
     g{2,#}(x,c1(y)) -> f{1,#}(x)
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,y)
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y))) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y))) ->
     g{2,#}(x,c1(y)) -> if{3,#}(f1(x),c1(g2(s1(x),y)),c1(y))
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y))) ->
     g{2,#}(x,c1(y)) -> f{1,#}(x)
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y))) ->
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y))) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,y)
     g{2,#}(x,c1(y)) -> g{2,#}(x,y) ->
     g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     g{2,#}(x,c1(y)) -> g{2,#}(x,y) ->
     g{2,#}(x,c1(y)) -> if{3,#}(f1(x),c1(g2(s1(x),y)),c1(y))
     g{2,#}(x,c1(y)) -> g{2,#}(x,y) -> g{2,#}(x,c1(y)) -> f{1,#}(x)
     g{2,#}(x,c1(y)) -> g{2,#}(x,y) ->
     g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
     g{2,#}(x,c1(y)) -> g{2,#}(x,y) -> g{2,#}(x,c1(y)) -> g{2,#}(x,y)
     g{2,#}(x,c1(y)) -> f{1,#}(x) -> f{1,#}(s1(x)) -> f{1,#}(x)
     f{1,#}(s1(x)) -> f{1,#}(x) -> f{1,#}(s1(x)) -> f{1,#}(x)
    SCC Processor:
     #sccs: 3
     #rules: 13
     #arcs: 49/324
     DPs:
      filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
      filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(map1(x21),x22) -> map{2,#}(x21,x22)
      map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
      map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(filter1(x28),x29) -> filter{2,#}(x28,x29)
      filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
      filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
      app#(filter23(x31,x32,x33),x34) -> filter2{4,#}(x31,x32,x33,x34)
     TRS:
      f1(00()) -> true0()
      f1(10()) -> false0()
      f1(s1(x)) -> f1(x)
      if3(true0(),s1(x),s1(y)) -> s1(x)
      if3(false0(),s1(x),s1(y)) -> s1(y)
      g2(x,c1(y)) -> c1(g2(x,y))
      g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(f0(),x4) -> f1(x4)
      app(s0(),x10) -> s1(x10)
      app(if2(x12,x13),x14) -> if3(x12,x13,x14)
      app(if1(x12),x13) -> if2(x12,x13)
      app(if0(),x12) -> if1(x12)
      app(g1(x16),x17) -> g2(x16,x17)
      app(g0(),x16) -> g1(x16)
      app(c0(),x19) -> c1(x19)
      app(map1(x21),x22) -> map2(x21,x22)
      app(map0(),x21) -> map1(x21)
      app(cons1(x25),x26) -> cons2(x25,x26)
      app(cons0(),x25) -> cons1(x25)
      app(filter1(x28),x29) -> filter2(x28,x29)
      app(filter0(),x28) -> filter1(x28)
      app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
      app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
      app(filter21(x31),x32) -> filter22(x31,x32)
      app(filter20(),x31) -> filter21(x31)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 0
       pi(app#) = 0
       pi(filter{2,#}) = 0
       pi(filter2{4,#}) = 1
      problem:
       DPs:
        filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
        filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
        map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
        filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
        filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
       TRS:
        f1(00()) -> true0()
        f1(10()) -> false0()
        f1(s1(x)) -> f1(x)
        if3(true0(),s1(x),s1(y)) -> s1(x)
        if3(false0(),s1(x),s1(y)) -> s1(y)
        g2(x,c1(y)) -> c1(g2(x,y))
        g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x4) -> f1(x4)
        app(s0(),x10) -> s1(x10)
        app(if2(x12,x13),x14) -> if3(x12,x13,x14)
        app(if1(x12),x13) -> if2(x12,x13)
        app(if0(),x12) -> if1(x12)
        app(g1(x16),x17) -> g2(x16,x17)
        app(g0(),x16) -> g1(x16)
        app(c0(),x19) -> c1(x19)
        app(map1(x21),x22) -> map2(x21,x22)
        app(map0(),x21) -> map1(x21)
        app(cons1(x25),x26) -> cons2(x25,x26)
        app(cons0(),x25) -> cons1(x25)
        app(filter1(x28),x29) -> filter2(x28,x29)
        app(filter0(),x28) -> filter1(x28)
        app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
        app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
        app(filter21(x31),x32) -> filter22(x31,x32)
        app(filter20(),x31) -> filter21(x31)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
       TRS:
        f1(00()) -> true0()
        f1(10()) -> false0()
        f1(s1(x)) -> f1(x)
        if3(true0(),s1(x),s1(y)) -> s1(x)
        if3(false0(),s1(x),s1(y)) -> s1(y)
        g2(x,c1(y)) -> c1(g2(x,y))
        g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x4) -> f1(x4)
        app(s0(),x10) -> s1(x10)
        app(if2(x12,x13),x14) -> if3(x12,x13,x14)
        app(if1(x12),x13) -> if2(x12,x13)
        app(if0(),x12) -> if1(x12)
        app(g1(x16),x17) -> g2(x16,x17)
        app(g0(),x16) -> g1(x16)
        app(c0(),x19) -> c1(x19)
        app(map1(x21),x22) -> map2(x21,x22)
        app(map0(),x21) -> map1(x21)
        app(cons1(x25),x26) -> cons2(x25,x26)
        app(cons0(),x25) -> cons1(x25)
        app(filter1(x28),x29) -> filter2(x28,x29)
        app(filter0(),x28) -> filter1(x28)
        app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
        app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
        app(filter21(x31),x32) -> filter22(x31,x32)
        app(filter20(),x31) -> filter21(x31)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          f1(00()) -> true0()
          f1(10()) -> false0()
          f1(s1(x)) -> f1(x)
          if3(true0(),s1(x),s1(y)) -> s1(x)
          if3(false0(),s1(x),s1(y)) -> s1(y)
          g2(x,c1(y)) -> c1(g2(x,y))
          g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
          map2(fun,nil0()) -> nil0()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil0()) -> nil0()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false0(),fun,x,xs) -> filter2(fun,xs)
          app(f0(),x4) -> f1(x4)
          app(s0(),x10) -> s1(x10)
          app(if2(x12,x13),x14) -> if3(x12,x13,x14)
          app(if1(x12),x13) -> if2(x12,x13)
          app(if0(),x12) -> if1(x12)
          app(g1(x16),x17) -> g2(x16,x17)
          app(g0(),x16) -> g1(x16)
          app(c0(),x19) -> c1(x19)
          app(map1(x21),x22) -> map2(x21,x22)
          app(map0(),x21) -> map1(x21)
          app(cons1(x25),x26) -> cons2(x25,x26)
          app(cons0(),x25) -> cons1(x25)
          app(filter1(x28),x29) -> filter2(x28,x29)
          app(filter0(),x28) -> filter1(x28)
          app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
          app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
          app(filter21(x31),x32) -> filter22(x31,x32)
          app(filter20(),x31) -> filter21(x31)
        Qed
       
       DPs:
        filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
        filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
        filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
       TRS:
        f1(00()) -> true0()
        f1(10()) -> false0()
        f1(s1(x)) -> f1(x)
        if3(true0(),s1(x),s1(y)) -> s1(x)
        if3(false0(),s1(x),s1(y)) -> s1(y)
        g2(x,c1(y)) -> c1(g2(x,y))
        g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x4) -> f1(x4)
        app(s0(),x10) -> s1(x10)
        app(if2(x12,x13),x14) -> if3(x12,x13,x14)
        app(if1(x12),x13) -> if2(x12,x13)
        app(if0(),x12) -> if1(x12)
        app(g1(x16),x17) -> g2(x16,x17)
        app(g0(),x16) -> g1(x16)
        app(c0(),x19) -> c1(x19)
        app(map1(x21),x22) -> map2(x21,x22)
        app(map0(),x21) -> map1(x21)
        app(cons1(x25),x26) -> cons2(x25,x26)
        app(cons0(),x25) -> cons1(x25)
        app(filter1(x28),x29) -> filter2(x28,x29)
        app(filter0(),x28) -> filter1(x28)
        app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
        app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
        app(filter21(x31),x32) -> filter22(x31,x32)
        app(filter20(),x31) -> filter21(x31)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
         pi(filter2{4,#}) = 3
        problem:
         DPs:
          filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
          filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
         TRS:
          f1(00()) -> true0()
          f1(10()) -> false0()
          f1(s1(x)) -> f1(x)
          if3(true0(),s1(x),s1(y)) -> s1(x)
          if3(false0(),s1(x),s1(y)) -> s1(y)
          g2(x,c1(y)) -> c1(g2(x,y))
          g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
          map2(fun,nil0()) -> nil0()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil0()) -> nil0()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false0(),fun,x,xs) -> filter2(fun,xs)
          app(f0(),x4) -> f1(x4)
          app(s0(),x10) -> s1(x10)
          app(if2(x12,x13),x14) -> if3(x12,x13,x14)
          app(if1(x12),x13) -> if2(x12,x13)
          app(if0(),x12) -> if1(x12)
          app(g1(x16),x17) -> g2(x16,x17)
          app(g0(),x16) -> g1(x16)
          app(c0(),x19) -> c1(x19)
          app(map1(x21),x22) -> map2(x21,x22)
          app(map0(),x21) -> map1(x21)
          app(cons1(x25),x26) -> cons2(x25,x26)
          app(cons0(),x25) -> cons1(x25)
          app(filter1(x28),x29) -> filter2(x28,x29)
          app(filter0(),x28) -> filter1(x28)
          app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
          app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
          app(filter21(x31),x32) -> filter22(x31,x32)
          app(filter20(),x31) -> filter21(x31)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      g{2,#}(x,c1(y)) -> g{2,#}(x,y)
      g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
      g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
     TRS:
      f1(00()) -> true0()
      f1(10()) -> false0()
      f1(s1(x)) -> f1(x)
      if3(true0(),s1(x),s1(y)) -> s1(x)
      if3(false0(),s1(x),s1(y)) -> s1(y)
      g2(x,c1(y)) -> c1(g2(x,y))
      g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(f0(),x4) -> f1(x4)
      app(s0(),x10) -> s1(x10)
      app(if2(x12,x13),x14) -> if3(x12,x13,x14)
      app(if1(x12),x13) -> if2(x12,x13)
      app(if0(),x12) -> if1(x12)
      app(g1(x16),x17) -> g2(x16,x17)
      app(g0(),x16) -> g1(x16)
      app(c0(),x19) -> c1(x19)
      app(map1(x21),x22) -> map2(x21,x22)
      app(map0(),x21) -> map1(x21)
      app(cons1(x25),x26) -> cons2(x25,x26)
      app(cons0(),x25) -> cons1(x25)
      app(filter1(x28),x29) -> filter2(x28,x29)
      app(filter0(),x28) -> filter1(x28)
      app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
      app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
      app(filter21(x31),x32) -> filter22(x31,x32)
      app(filter20(),x31) -> filter21(x31)
     EDG Processor:
      DPs:
       g{2,#}(x,c1(y)) -> g{2,#}(x,y)
       g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
       g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
      TRS:
       f1(00()) -> true0()
       f1(10()) -> false0()
       f1(s1(x)) -> f1(x)
       if3(true0(),s1(x),s1(y)) -> s1(x)
       if3(false0(),s1(x),s1(y)) -> s1(y)
       g2(x,c1(y)) -> c1(g2(x,y))
       g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
       map2(fun,nil0()) -> nil0()
       map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
       filter2(fun,nil0()) -> nil0()
       filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
       filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
       filter24(false0(),fun,x,xs) -> filter2(fun,xs)
       app(f0(),x4) -> f1(x4)
       app(s0(),x10) -> s1(x10)
       app(if2(x12,x13),x14) -> if3(x12,x13,x14)
       app(if1(x12),x13) -> if2(x12,x13)
       app(if0(),x12) -> if1(x12)
       app(g1(x16),x17) -> g2(x16,x17)
       app(g0(),x16) -> g1(x16)
       app(c0(),x19) -> c1(x19)
       app(map1(x21),x22) -> map2(x21,x22)
       app(map0(),x21) -> map1(x21)
       app(cons1(x25),x26) -> cons2(x25,x26)
       app(cons0(),x25) -> cons1(x25)
       app(filter1(x28),x29) -> filter2(x28,x29)
       app(filter0(),x28) -> filter1(x28)
       app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
       app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
       app(filter21(x31),x32) -> filter22(x31,x32)
       app(filter20(),x31) -> filter21(x31)
      graph:
       g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
       g{2,#}(x,c1(y)) -> g{2,#}(x,y)
       g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
       g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
       g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) ->
       g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
       g{2,#}(x,c1(y)) -> g{2,#}(x,y) ->
       g{2,#}(x,c1(y)) -> g{2,#}(x,y)
       g{2,#}(x,c1(y)) -> g{2,#}(x,y) ->
       g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
       g{2,#}(x,c1(y)) -> g{2,#}(x,y) ->
       g{2,#}(x,c1(y)) -> g{2,#}(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
      SCC Processor:
       #sccs: 1
       #rules: 2
       #arcs: 6/9
       DPs:
        g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y)
        g{2,#}(x,c1(y)) -> g{2,#}(x,y)
       TRS:
        f1(00()) -> true0()
        f1(10()) -> false0()
        f1(s1(x)) -> f1(x)
        if3(true0(),s1(x),s1(y)) -> s1(x)
        if3(false0(),s1(x),s1(y)) -> s1(y)
        g2(x,c1(y)) -> c1(g2(x,y))
        g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x4) -> f1(x4)
        app(s0(),x10) -> s1(x10)
        app(if2(x12,x13),x14) -> if3(x12,x13,x14)
        app(if1(x12),x13) -> if2(x12,x13)
        app(if0(),x12) -> if1(x12)
        app(g1(x16),x17) -> g2(x16,x17)
        app(g0(),x16) -> g1(x16)
        app(c0(),x19) -> c1(x19)
        app(map1(x21),x22) -> map2(x21,x22)
        app(map0(),x21) -> map1(x21)
        app(cons1(x25),x26) -> cons2(x25,x26)
        app(cons0(),x25) -> cons1(x25)
        app(filter1(x28),x29) -> filter2(x28,x29)
        app(filter0(),x28) -> filter1(x28)
        app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
        app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
        app(filter21(x31),x32) -> filter22(x31,x32)
        app(filter20(),x31) -> filter21(x31)
       Size-Change Termination Processor:
        DPs:
         
        TRS:
         f1(00()) -> true0()
         f1(10()) -> false0()
         f1(s1(x)) -> f1(x)
         if3(true0(),s1(x),s1(y)) -> s1(x)
         if3(false0(),s1(x),s1(y)) -> s1(y)
         g2(x,c1(y)) -> c1(g2(x,y))
         g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
         map2(fun,nil0()) -> nil0()
         map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
         filter2(fun,nil0()) -> nil0()
         filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
         filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
         filter24(false0(),fun,x,xs) -> filter2(fun,xs)
         app(f0(),x4) -> f1(x4)
         app(s0(),x10) -> s1(x10)
         app(if2(x12,x13),x14) -> if3(x12,x13,x14)
         app(if1(x12),x13) -> if2(x12,x13)
         app(if0(),x12) -> if1(x12)
         app(g1(x16),x17) -> g2(x16,x17)
         app(g0(),x16) -> g1(x16)
         app(c0(),x19) -> c1(x19)
         app(map1(x21),x22) -> map2(x21,x22)
         app(map0(),x21) -> map1(x21)
         app(cons1(x25),x26) -> cons2(x25,x26)
         app(cons0(),x25) -> cons1(x25)
         app(filter1(x28),x29) -> filter2(x28,x29)
         app(filter0(),x28) -> filter1(x28)
         app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
         app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
         app(filter21(x31),x32) -> filter22(x31,x32)
         app(filter20(),x31) -> filter21(x31)
        The DP: g{2,#}(x,c1(y)) -> g{2,#}(s1(x),y) has the edges:
          1 >   1
        The DP: g{2,#}(x,c1(y)) -> g{2,#}(x,y) has the edges:
          0 >=  0
          1 >   1
        Qed
     
     DPs:
      f{1,#}(s1(x)) -> f{1,#}(x)
     TRS:
      f1(00()) -> true0()
      f1(10()) -> false0()
      f1(s1(x)) -> f1(x)
      if3(true0(),s1(x),s1(y)) -> s1(x)
      if3(false0(),s1(x),s1(y)) -> s1(y)
      g2(x,c1(y)) -> c1(g2(x,y))
      g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(f0(),x4) -> f1(x4)
      app(s0(),x10) -> s1(x10)
      app(if2(x12,x13),x14) -> if3(x12,x13,x14)
      app(if1(x12),x13) -> if2(x12,x13)
      app(if0(),x12) -> if1(x12)
      app(g1(x16),x17) -> g2(x16,x17)
      app(g0(),x16) -> g1(x16)
      app(c0(),x19) -> c1(x19)
      app(map1(x21),x22) -> map2(x21,x22)
      app(map0(),x21) -> map1(x21)
      app(cons1(x25),x26) -> cons2(x25,x26)
      app(cons0(),x25) -> cons1(x25)
      app(filter1(x28),x29) -> filter2(x28,x29)
      app(filter0(),x28) -> filter1(x28)
      app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
      app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
      app(filter21(x31),x32) -> filter22(x31,x32)
      app(filter20(),x31) -> filter21(x31)
     Subterm Criterion Processor:
      simple projection:
       pi(f{1,#}) = 0
      problem:
       DPs:
        
       TRS:
        f1(00()) -> true0()
        f1(10()) -> false0()
        f1(s1(x)) -> f1(x)
        if3(true0(),s1(x),s1(y)) -> s1(x)
        if3(false0(),s1(x),s1(y)) -> s1(y)
        g2(x,c1(y)) -> c1(g2(x,y))
        g2(x,c1(y)) -> g2(x,if3(f1(x),c1(g2(s1(x),y)),c1(y)))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x4) -> f1(x4)
        app(s0(),x10) -> s1(x10)
        app(if2(x12,x13),x14) -> if3(x12,x13,x14)
        app(if1(x12),x13) -> if2(x12,x13)
        app(if0(),x12) -> if1(x12)
        app(g1(x16),x17) -> g2(x16,x17)
        app(g0(),x16) -> g1(x16)
        app(c0(),x19) -> c1(x19)
        app(map1(x21),x22) -> map2(x21,x22)
        app(map0(),x21) -> map1(x21)
        app(cons1(x25),x26) -> cons2(x25,x26)
        app(cons0(),x25) -> cons1(x25)
        app(filter1(x28),x29) -> filter2(x28,x29)
        app(filter0(),x28) -> filter1(x28)
        app(filter23(x31,x32,x33),x34) -> filter24(x31,x32,x33,x34)
        app(filter22(x31,x32),x33) -> filter23(x31,x32,x33)
        app(filter21(x31),x32) -> filter22(x31,x32)
        app(filter20(),x31) -> filter21(x31)
      Qed