/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(le(),0()),y) -> true()
 app(app(le(),app(s(),x)),0()) -> false()
 app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
 app(pred(),app(s(),x)) -> x
 app(app(minus(),x),0()) -> x
 app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
 app(app(gcd(),0()),y) -> y
 app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
 app(app(gcd(),app(s(),x)),app(s(),y)) ->
 app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
 app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
 app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
 app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
 app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
 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(filter(),f),nil()) -> nil()
 app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
 app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
 app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),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
   if_gcd ==> if_gcd0/0 if_gcd1/1 if_gcd2/2 if_gcd3/3
   gcd ==> gcd0/0 gcd1/1 gcd2/2
   minus ==> minus0/0 minus1/1 minus2/2
   pred ==> pred0/0 pred1/1
   false ==> false0/0
   s ==> s0/0 s1/1
   true ==> true0/0
   0 ==> 00/0
   le ==> le0/0 le1/1 le2/2
   
  uncurry-rules:
   app(le1(x4),x5) -> le2(x4,x5)
   app(le0(),x4) -> le1(x4)
   app(s0(),x9) -> s1(x9)
   app(pred0(),x12) -> pred1(x12)
   app(minus1(x14),x15) -> minus2(x14,x15)
   app(minus0(),x14) -> minus1(x14)
   app(gcd1(x17),x18) -> gcd2(x17,x18)
   app(gcd0(),x17) -> gcd1(x17)
   app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
   app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
   app(if_gcd0(),x20) -> if_gcd1(x20)
   app(map1(x24),x25) -> map2(x24,x25)
   app(map0(),x24) -> map1(x24)
   app(cons1(x28),x29) -> cons2(x28,x29)
   app(cons0(),x28) -> cons1(x28)
   app(filter1(x31),x32) -> filter2(x31,x32)
   app(filter0(),x31) -> filter1(x31)
   app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
   app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
   app(filter21(x34),x35) -> filter22(x34,x35)
   app(filter20(),x34) -> filter21(x34)
  eta-rules:
   
  problem:
   le2(00(),y) -> true0()
   le2(s1(x),00()) -> false0()
   le2(s1(x),s1(y)) -> le2(x,y)
   pred1(s1(x)) -> x
   minus2(x,00()) -> x
   minus2(x,s1(y)) -> pred1(minus2(x,y))
   gcd2(00(),y) -> y
   gcd2(s1(x),00()) -> s1(x)
   gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
   if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
   if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   filter2(f,nil0()) -> nil0()
   filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
   filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
   filter24(false0(),f,x,xs) -> filter2(f,xs)
   app(le1(x4),x5) -> le2(x4,x5)
   app(le0(),x4) -> le1(x4)
   app(s0(),x9) -> s1(x9)
   app(pred0(),x12) -> pred1(x12)
   app(minus1(x14),x15) -> minus2(x14,x15)
   app(minus0(),x14) -> minus1(x14)
   app(gcd1(x17),x18) -> gcd2(x17,x18)
   app(gcd0(),x17) -> gcd1(x17)
   app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
   app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
   app(if_gcd0(),x20) -> if_gcd1(x20)
   app(map1(x24),x25) -> map2(x24,x25)
   app(map0(),x24) -> map1(x24)
   app(cons1(x28),x29) -> cons2(x28,x29)
   app(cons0(),x28) -> cons1(x28)
   app(filter1(x31),x32) -> filter2(x31,x32)
   app(filter0(),x31) -> filter1(x31)
   app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
   app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
   app(filter21(x34),x35) -> filter22(x34,x35)
   app(filter20(),x34) -> filter21(x34)
  DP Processor:
   DPs:
    le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
    minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
    minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
    gcd{2,#}(s1(x),s1(y)) -> le{2,#}(y,x)
    gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y))
    if_gcd{3,#}(true0(),s1(x),s1(y)) -> minus{2,#}(x,y)
    if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y))
    if_gcd{3,#}(false0(),s1(x),s1(y)) -> minus{2,#}(y,x)
    if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x))
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
    filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
    app#(le1(x4),x5) -> le{2,#}(x4,x5)
    app#(pred0(),x12) -> pred{1,#}(x12)
    app#(minus1(x14),x15) -> minus{2,#}(x14,x15)
    app#(gcd1(x17),x18) -> gcd{2,#}(x17,x18)
    app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22)
    app#(map1(x24),x25) -> map{2,#}(x24,x25)
    app#(filter1(x31),x32) -> filter{2,#}(x31,x32)
    app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37)
   TRS:
    le2(00(),y) -> true0()
    le2(s1(x),00()) -> false0()
    le2(s1(x),s1(y)) -> le2(x,y)
    pred1(s1(x)) -> x
    minus2(x,00()) -> x
    minus2(x,s1(y)) -> pred1(minus2(x,y))
    gcd2(00(),y) -> y
    gcd2(s1(x),00()) -> s1(x)
    gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
    if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
    if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    filter2(f,nil0()) -> nil0()
    filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
    filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
    filter24(false0(),f,x,xs) -> filter2(f,xs)
    app(le1(x4),x5) -> le2(x4,x5)
    app(le0(),x4) -> le1(x4)
    app(s0(),x9) -> s1(x9)
    app(pred0(),x12) -> pred1(x12)
    app(minus1(x14),x15) -> minus2(x14,x15)
    app(minus0(),x14) -> minus1(x14)
    app(gcd1(x17),x18) -> gcd2(x17,x18)
    app(gcd0(),x17) -> gcd1(x17)
    app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
    app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
    app(if_gcd0(),x20) -> if_gcd1(x20)
    app(map1(x24),x25) -> map2(x24,x25)
    app(map0(),x24) -> map1(x24)
    app(cons1(x28),x29) -> cons2(x28,x29)
    app(cons0(),x28) -> cons1(x28)
    app(filter1(x31),x32) -> filter2(x31,x32)
    app(filter0(),x31) -> filter1(x31)
    app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
    app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
    app(filter21(x34),x35) -> filter22(x34,x35)
    app(filter20(),x34) -> filter21(x34)
   TDG Processor:
    DPs:
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     gcd{2,#}(s1(x),s1(y)) -> le{2,#}(y,x)
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y))
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> minus{2,#}(x,y)
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y))
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> minus{2,#}(y,x)
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x))
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(le1(x4),x5) -> le{2,#}(x4,x5)
     app#(pred0(),x12) -> pred{1,#}(x12)
     app#(minus1(x14),x15) -> minus{2,#}(x14,x15)
     app#(gcd1(x17),x18) -> gcd{2,#}(x17,x18)
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22)
     app#(map1(x24),x25) -> map{2,#}(x24,x25)
     app#(filter1(x31),x32) -> filter{2,#}(x31,x32)
     app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37)
    TRS:
     le2(00(),y) -> true0()
     le2(s1(x),00()) -> false0()
     le2(s1(x),s1(y)) -> le2(x,y)
     pred1(s1(x)) -> x
     minus2(x,00()) -> x
     minus2(x,s1(y)) -> pred1(minus2(x,y))
     gcd2(00(),y) -> y
     gcd2(s1(x),00()) -> s1(x)
     gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
     if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
     if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     filter2(f,nil0()) -> nil0()
     filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
     filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
     filter24(false0(),f,x,xs) -> filter2(f,xs)
     app(le1(x4),x5) -> le2(x4,x5)
     app(le0(),x4) -> le1(x4)
     app(s0(),x9) -> s1(x9)
     app(pred0(),x12) -> pred1(x12)
     app(minus1(x14),x15) -> minus2(x14,x15)
     app(minus0(),x14) -> minus1(x14)
     app(gcd1(x17),x18) -> gcd2(x17,x18)
     app(gcd0(),x17) -> gcd1(x17)
     app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
     app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
     app(if_gcd0(),x20) -> if_gcd1(x20)
     app(map1(x24),x25) -> map2(x24,x25)
     app(map0(),x24) -> map1(x24)
     app(cons1(x28),x29) -> cons2(x28,x29)
     app(cons0(),x28) -> cons1(x28)
     app(filter1(x31),x32) -> filter2(x31,x32)
     app(filter0(),x31) -> filter1(x31)
     app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
     app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
     app(filter21(x34),x35) -> filter22(x34,x35)
     app(filter20(),x34) -> filter21(x34)
    graph:
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x31),x32) -> filter{2,#}(x31,x32)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x24),x25) -> map{2,#}(x24,x25)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(gcd1(x17),x18) -> gcd{2,#}(x17,x18)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x14),x15) -> minus{2,#}(x14,x15)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(pred0(),x12) -> pred{1,#}(x12)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(le1(x4),x5) -> le{2,#}(x4,x5)
     app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37) ->
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x31),x32) -> filter{2,#}(x31,x32) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x31),x32) -> filter{2,#}(x31,x32) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x24),x25) -> map{2,#}(x24,x25) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x24),x25) -> map{2,#}(x24,x25) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22) ->
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x))
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22) ->
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> minus{2,#}(y,x)
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22) ->
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y))
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22) ->
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(gcd1(x17),x18) -> gcd{2,#}(x17,x18) ->
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y))
     app#(gcd1(x17),x18) -> gcd{2,#}(x17,x18) ->
     gcd{2,#}(s1(x),s1(y)) -> le{2,#}(y,x)
     app#(minus1(x14),x15) -> minus{2,#}(x14,x15) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     app#(minus1(x14),x15) -> minus{2,#}(x14,x15) ->
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     app#(le1(x4),x5) -> le{2,#}(x4,x5) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x31),x32) -> filter{2,#}(x31,x32)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x24),x25) -> map{2,#}(x24,x25)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(if_gcd2(x20,x21),x22) -> if_gcd{3,#}(x20,x21,x22)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(gcd1(x17),x18) -> gcd{2,#}(x17,x18)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x14),x15) -> minus{2,#}(x14,x15)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(pred0(),x12) -> pred{1,#}(x12)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(le1(x4),x5) -> le{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)
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x)) ->
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y))
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x)) ->
     gcd{2,#}(s1(x),s1(y)) -> le{2,#}(y,x)
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> minus{2,#}(y,x) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> minus{2,#}(y,x) ->
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y)) ->
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y))
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y)) ->
     gcd{2,#}(s1(x),s1(y)) -> le{2,#}(y,x)
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y)) ->
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x))
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y)) ->
     if_gcd{3,#}(false0(),s1(x),s1(y)) -> minus{2,#}(y,x)
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y)) ->
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y))
     gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y)) ->
     if_gcd{3,#}(true0(),s1(x),s1(y)) -> minus{2,#}(x,y)
     gcd{2,#}(s1(x),s1(y)) -> le{2,#}(y,x) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> pred{1,#}(minus2(x,y))
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y) -> le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
    SCC Processor:
     #sccs: 4
     #rules: 14
     #arcs: 55/529
     DPs:
      filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x24),x25) -> map{2,#}(x24,x25)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x31),x32) -> filter{2,#}(x31,x32)
      filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
      filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
      app#(filter23(x34,x35,x36),x37) -> filter2{4,#}(x34,x35,x36,x37)
     TRS:
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      gcd2(00(),y) -> y
      gcd2(s1(x),00()) -> s1(x)
      gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
      if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
      if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(le1(x4),x5) -> le2(x4,x5)
      app(le0(),x4) -> le1(x4)
      app(s0(),x9) -> s1(x9)
      app(pred0(),x12) -> pred1(x12)
      app(minus1(x14),x15) -> minus2(x14,x15)
      app(minus0(),x14) -> minus1(x14)
      app(gcd1(x17),x18) -> gcd2(x17,x18)
      app(gcd0(),x17) -> gcd1(x17)
      app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
      app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
      app(if_gcd0(),x20) -> if_gcd1(x20)
      app(map1(x24),x25) -> map2(x24,x25)
      app(map0(),x24) -> map1(x24)
      app(cons1(x28),x29) -> cons2(x28,x29)
      app(cons0(),x28) -> cons1(x28)
      app(filter1(x31),x32) -> filter2(x31,x32)
      app(filter0(),x31) -> filter1(x31)
      app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
      app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
      app(filter21(x34),x35) -> filter22(x34,x35)
      app(filter20(),x34) -> filter21(x34)
     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(),f,x,xs) -> filter{2,#}(f,xs)
        filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
        map{2,#}(f,cons2(x,xs)) -> app#(f,x)
        filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
        filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
       TRS:
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        gcd2(00(),y) -> y
        gcd2(s1(x),00()) -> s1(x)
        gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
        if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
        if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(le1(x4),x5) -> le2(x4,x5)
        app(le0(),x4) -> le1(x4)
        app(s0(),x9) -> s1(x9)
        app(pred0(),x12) -> pred1(x12)
        app(minus1(x14),x15) -> minus2(x14,x15)
        app(minus0(),x14) -> minus1(x14)
        app(gcd1(x17),x18) -> gcd2(x17,x18)
        app(gcd0(),x17) -> gcd1(x17)
        app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
        app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
        app(if_gcd0(),x20) -> if_gcd1(x20)
        app(map1(x24),x25) -> map2(x24,x25)
        app(map0(),x24) -> map1(x24)
        app(cons1(x28),x29) -> cons2(x28,x29)
        app(cons0(),x28) -> cons1(x28)
        app(filter1(x31),x32) -> filter2(x31,x32)
        app(filter0(),x31) -> filter1(x31)
        app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
        app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
        app(filter21(x34),x35) -> filter22(x34,x35)
        app(filter20(),x34) -> filter21(x34)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        gcd2(00(),y) -> y
        gcd2(s1(x),00()) -> s1(x)
        gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
        if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
        if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(le1(x4),x5) -> le2(x4,x5)
        app(le0(),x4) -> le1(x4)
        app(s0(),x9) -> s1(x9)
        app(pred0(),x12) -> pred1(x12)
        app(minus1(x14),x15) -> minus2(x14,x15)
        app(minus0(),x14) -> minus1(x14)
        app(gcd1(x17),x18) -> gcd2(x17,x18)
        app(gcd0(),x17) -> gcd1(x17)
        app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
        app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
        app(if_gcd0(),x20) -> if_gcd1(x20)
        app(map1(x24),x25) -> map2(x24,x25)
        app(map0(),x24) -> map1(x24)
        app(cons1(x28),x29) -> cons2(x28,x29)
        app(cons0(),x28) -> cons1(x28)
        app(filter1(x31),x32) -> filter2(x31,x32)
        app(filter0(),x31) -> filter1(x31)
        app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
        app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
        app(filter21(x34),x35) -> filter22(x34,x35)
        app(filter20(),x34) -> filter21(x34)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          le2(00(),y) -> true0()
          le2(s1(x),00()) -> false0()
          le2(s1(x),s1(y)) -> le2(x,y)
          pred1(s1(x)) -> x
          minus2(x,00()) -> x
          minus2(x,s1(y)) -> pred1(minus2(x,y))
          gcd2(00(),y) -> y
          gcd2(s1(x),00()) -> s1(x)
          gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
          if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
          if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          filter2(f,nil0()) -> nil0()
          filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
          filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
          filter24(false0(),f,x,xs) -> filter2(f,xs)
          app(le1(x4),x5) -> le2(x4,x5)
          app(le0(),x4) -> le1(x4)
          app(s0(),x9) -> s1(x9)
          app(pred0(),x12) -> pred1(x12)
          app(minus1(x14),x15) -> minus2(x14,x15)
          app(minus0(),x14) -> minus1(x14)
          app(gcd1(x17),x18) -> gcd2(x17,x18)
          app(gcd0(),x17) -> gcd1(x17)
          app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
          app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
          app(if_gcd0(),x20) -> if_gcd1(x20)
          app(map1(x24),x25) -> map2(x24,x25)
          app(map0(),x24) -> map1(x24)
          app(cons1(x28),x29) -> cons2(x28,x29)
          app(cons0(),x28) -> cons1(x28)
          app(filter1(x31),x32) -> filter2(x31,x32)
          app(filter0(),x31) -> filter1(x31)
          app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
          app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
          app(filter21(x34),x35) -> filter22(x34,x35)
          app(filter20(),x34) -> filter21(x34)
        Qed
       
       DPs:
        filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
        filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
        filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
       TRS:
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        gcd2(00(),y) -> y
        gcd2(s1(x),00()) -> s1(x)
        gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
        if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
        if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(le1(x4),x5) -> le2(x4,x5)
        app(le0(),x4) -> le1(x4)
        app(s0(),x9) -> s1(x9)
        app(pred0(),x12) -> pred1(x12)
        app(minus1(x14),x15) -> minus2(x14,x15)
        app(minus0(),x14) -> minus1(x14)
        app(gcd1(x17),x18) -> gcd2(x17,x18)
        app(gcd0(),x17) -> gcd1(x17)
        app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
        app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
        app(if_gcd0(),x20) -> if_gcd1(x20)
        app(map1(x24),x25) -> map2(x24,x25)
        app(map0(),x24) -> map1(x24)
        app(cons1(x28),x29) -> cons2(x28,x29)
        app(cons0(),x28) -> cons1(x28)
        app(filter1(x31),x32) -> filter2(x31,x32)
        app(filter0(),x31) -> filter1(x31)
        app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
        app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
        app(filter21(x34),x35) -> filter22(x34,x35)
        app(filter20(),x34) -> filter21(x34)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
         pi(filter2{4,#}) = 3
        problem:
         DPs:
          filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
          filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
         TRS:
          le2(00(),y) -> true0()
          le2(s1(x),00()) -> false0()
          le2(s1(x),s1(y)) -> le2(x,y)
          pred1(s1(x)) -> x
          minus2(x,00()) -> x
          minus2(x,s1(y)) -> pred1(minus2(x,y))
          gcd2(00(),y) -> y
          gcd2(s1(x),00()) -> s1(x)
          gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
          if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
          if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          filter2(f,nil0()) -> nil0()
          filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
          filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
          filter24(false0(),f,x,xs) -> filter2(f,xs)
          app(le1(x4),x5) -> le2(x4,x5)
          app(le0(),x4) -> le1(x4)
          app(s0(),x9) -> s1(x9)
          app(pred0(),x12) -> pred1(x12)
          app(minus1(x14),x15) -> minus2(x14,x15)
          app(minus0(),x14) -> minus1(x14)
          app(gcd1(x17),x18) -> gcd2(x17,x18)
          app(gcd0(),x17) -> gcd1(x17)
          app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
          app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
          app(if_gcd0(),x20) -> if_gcd1(x20)
          app(map1(x24),x25) -> map2(x24,x25)
          app(map0(),x24) -> map1(x24)
          app(cons1(x28),x29) -> cons2(x28,x29)
          app(cons0(),x28) -> cons1(x28)
          app(filter1(x31),x32) -> filter2(x31,x32)
          app(filter0(),x31) -> filter1(x31)
          app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
          app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
          app(filter21(x34),x35) -> filter22(x34,x35)
          app(filter20(),x34) -> filter21(x34)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      gcd{2,#}(s1(x),s1(y)) -> if_gcd{3,#}(le2(y,x),s1(x),s1(y))
      if_gcd{3,#}(true0(),s1(x),s1(y)) -> gcd{2,#}(minus2(x,y),s1(y))
      if_gcd{3,#}(false0(),s1(x),s1(y)) -> gcd{2,#}(minus2(y,x),s1(x))
     TRS:
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      gcd2(00(),y) -> y
      gcd2(s1(x),00()) -> s1(x)
      gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
      if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
      if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(le1(x4),x5) -> le2(x4,x5)
      app(le0(),x4) -> le1(x4)
      app(s0(),x9) -> s1(x9)
      app(pred0(),x12) -> pred1(x12)
      app(minus1(x14),x15) -> minus2(x14,x15)
      app(minus0(),x14) -> minus1(x14)
      app(gcd1(x17),x18) -> gcd2(x17,x18)
      app(gcd0(),x17) -> gcd1(x17)
      app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
      app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
      app(if_gcd0(),x20) -> if_gcd1(x20)
      app(map1(x24),x25) -> map2(x24,x25)
      app(map0(),x24) -> map1(x24)
      app(cons1(x28),x29) -> cons2(x28,x29)
      app(cons0(),x28) -> cons1(x28)
      app(filter1(x31),x32) -> filter2(x31,x32)
      app(filter0(),x31) -> filter1(x31)
      app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
      app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
      app(filter21(x34),x35) -> filter22(x34,x35)
      app(filter20(),x34) -> filter21(x34)
     Subterm Criterion Processor:
      simple projection:
       pi(pred1) = 0
       pi(minus2) = [0,0]
       pi(filter23) = [2,2]
       pi(gcd{2,#}) = [0,0,1]
       pi(if_gcd{3,#}) = [1,2]
      problem:
       DPs:
        
       TRS:
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        gcd2(00(),y) -> y
        gcd2(s1(x),00()) -> s1(x)
        gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
        if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
        if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(le1(x4),x5) -> le2(x4,x5)
        app(le0(),x4) -> le1(x4)
        app(s0(),x9) -> s1(x9)
        app(pred0(),x12) -> pred1(x12)
        app(minus1(x14),x15) -> minus2(x14,x15)
        app(minus0(),x14) -> minus1(x14)
        app(gcd1(x17),x18) -> gcd2(x17,x18)
        app(gcd0(),x17) -> gcd1(x17)
        app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
        app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
        app(if_gcd0(),x20) -> if_gcd1(x20)
        app(map1(x24),x25) -> map2(x24,x25)
        app(map0(),x24) -> map1(x24)
        app(cons1(x28),x29) -> cons2(x28,x29)
        app(cons0(),x28) -> cons1(x28)
        app(filter1(x31),x32) -> filter2(x31,x32)
        app(filter0(),x31) -> filter1(x31)
        app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
        app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
        app(filter21(x34),x35) -> filter22(x34,x35)
        app(filter20(),x34) -> filter21(x34)
      Qed
     
     DPs:
      minus{2,#}(x,s1(y)) -> minus{2,#}(x,y)
     TRS:
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      gcd2(00(),y) -> y
      gcd2(s1(x),00()) -> s1(x)
      gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
      if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
      if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(le1(x4),x5) -> le2(x4,x5)
      app(le0(),x4) -> le1(x4)
      app(s0(),x9) -> s1(x9)
      app(pred0(),x12) -> pred1(x12)
      app(minus1(x14),x15) -> minus2(x14,x15)
      app(minus0(),x14) -> minus1(x14)
      app(gcd1(x17),x18) -> gcd2(x17,x18)
      app(gcd0(),x17) -> gcd1(x17)
      app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
      app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
      app(if_gcd0(),x20) -> if_gcd1(x20)
      app(map1(x24),x25) -> map2(x24,x25)
      app(map0(),x24) -> map1(x24)
      app(cons1(x28),x29) -> cons2(x28,x29)
      app(cons0(),x28) -> cons1(x28)
      app(filter1(x31),x32) -> filter2(x31,x32)
      app(filter0(),x31) -> filter1(x31)
      app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
      app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
      app(filter21(x34),x35) -> filter22(x34,x35)
      app(filter20(),x34) -> filter21(x34)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        gcd2(00(),y) -> y
        gcd2(s1(x),00()) -> s1(x)
        gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
        if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
        if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(le1(x4),x5) -> le2(x4,x5)
        app(le0(),x4) -> le1(x4)
        app(s0(),x9) -> s1(x9)
        app(pred0(),x12) -> pred1(x12)
        app(minus1(x14),x15) -> minus2(x14,x15)
        app(minus0(),x14) -> minus1(x14)
        app(gcd1(x17),x18) -> gcd2(x17,x18)
        app(gcd0(),x17) -> gcd1(x17)
        app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
        app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
        app(if_gcd0(),x20) -> if_gcd1(x20)
        app(map1(x24),x25) -> map2(x24,x25)
        app(map0(),x24) -> map1(x24)
        app(cons1(x28),x29) -> cons2(x28,x29)
        app(cons0(),x28) -> cons1(x28)
        app(filter1(x31),x32) -> filter2(x31,x32)
        app(filter0(),x31) -> filter1(x31)
        app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
        app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
        app(filter21(x34),x35) -> filter22(x34,x35)
        app(filter20(),x34) -> filter21(x34)
      Qed
     
     DPs:
      le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     TRS:
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      pred1(s1(x)) -> x
      minus2(x,00()) -> x
      minus2(x,s1(y)) -> pred1(minus2(x,y))
      gcd2(00(),y) -> y
      gcd2(s1(x),00()) -> s1(x)
      gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
      if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
      if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(le1(x4),x5) -> le2(x4,x5)
      app(le0(),x4) -> le1(x4)
      app(s0(),x9) -> s1(x9)
      app(pred0(),x12) -> pred1(x12)
      app(minus1(x14),x15) -> minus2(x14,x15)
      app(minus0(),x14) -> minus1(x14)
      app(gcd1(x17),x18) -> gcd2(x17,x18)
      app(gcd0(),x17) -> gcd1(x17)
      app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
      app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
      app(if_gcd0(),x20) -> if_gcd1(x20)
      app(map1(x24),x25) -> map2(x24,x25)
      app(map0(),x24) -> map1(x24)
      app(cons1(x28),x29) -> cons2(x28,x29)
      app(cons0(),x28) -> cons1(x28)
      app(filter1(x31),x32) -> filter2(x31,x32)
      app(filter0(),x31) -> filter1(x31)
      app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
      app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
      app(filter21(x34),x35) -> filter22(x34,x35)
      app(filter20(),x34) -> filter21(x34)
     Subterm Criterion Processor:
      simple projection:
       pi(le{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        pred1(s1(x)) -> x
        minus2(x,00()) -> x
        minus2(x,s1(y)) -> pred1(minus2(x,y))
        gcd2(00(),y) -> y
        gcd2(s1(x),00()) -> s1(x)
        gcd2(s1(x),s1(y)) -> if_gcd3(le2(y,x),s1(x),s1(y))
        if_gcd3(true0(),s1(x),s1(y)) -> gcd2(minus2(x,y),s1(y))
        if_gcd3(false0(),s1(x),s1(y)) -> gcd2(minus2(y,x),s1(x))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(le1(x4),x5) -> le2(x4,x5)
        app(le0(),x4) -> le1(x4)
        app(s0(),x9) -> s1(x9)
        app(pred0(),x12) -> pred1(x12)
        app(minus1(x14),x15) -> minus2(x14,x15)
        app(minus0(),x14) -> minus1(x14)
        app(gcd1(x17),x18) -> gcd2(x17,x18)
        app(gcd0(),x17) -> gcd1(x17)
        app(if_gcd2(x20,x21),x22) -> if_gcd3(x20,x21,x22)
        app(if_gcd1(x20),x21) -> if_gcd2(x20,x21)
        app(if_gcd0(),x20) -> if_gcd1(x20)
        app(map1(x24),x25) -> map2(x24,x25)
        app(map0(),x24) -> map1(x24)
        app(cons1(x28),x29) -> cons2(x28,x29)
        app(cons0(),x28) -> cons1(x28)
        app(filter1(x31),x32) -> filter2(x31,x32)
        app(filter0(),x31) -> filter1(x31)
        app(filter23(x34,x35,x36),x37) -> filter24(x34,x35,x36,x37)
        app(filter22(x34,x35),x36) -> filter23(x34,x35,x36)
        app(filter21(x34),x35) -> filter22(x34,x35)
        app(filter20(),x34) -> filter21(x34)
      Qed