/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(minus(),0()),y) -> 0()
 app(app(minus(),app(s(),x)),0()) -> app(s(),x)
 app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),x),y)
 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(app(app(if(),true()),x),y) -> x
 app(app(app(if(),false()),x),y) -> y
 app(perfectp(),0()) -> false()
 app(perfectp(),app(s(),x)) -> app(app(app(app(f(),x),app(s(),0())),app(s(),x)),app(s(),x))
 app(app(app(app(f(),0()),y),0()),u) -> true()
 app(app(app(app(f(),0()),y),app(s(),z)),u) -> false()
 app(app(app(app(f(),app(s(),x)),0()),z),u) ->
 app(app(app(app(f(),x),u),app(app(minus(),z),app(s(),x))),u)
 app(app(app(app(f(),app(s(),x)),app(s(),y)),z),u) ->
 app(app(app(if(),app(app(le(),x),y)),app(app(app(app(f(),app(s(),x)),app(app(minus(),y),x)),z),u)),
     app(app(app(app(f(),x),u),z),u))
 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
   f ==> f0/0 f1/1 f2/2 f3/3 f4/4
   perfectp ==> perfectp0/0 perfectp1/1
   if ==> if0/0 if1/1 if2/2 if3/3
   false ==> false0/0
   true ==> true0/0
   le ==> le0/0 le1/1 le2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   minus ==> minus0/0 minus1/1 minus2/2
   
  uncurry-rules:
   app(minus1(x6),x7) -> minus2(x6,x7)
   app(minus0(),x6) -> minus1(x6)
   app(s0(),x10) -> s1(x10)
   app(le1(x12),x13) -> le2(x12,x13)
   app(le0(),x12) -> le1(x12)
   app(if2(x17,x18),x19) -> if3(x17,x18,x19)
   app(if1(x17),x18) -> if2(x17,x18)
   app(if0(),x17) -> if1(x17)
   app(perfectp0(),x21) -> perfectp1(x21)
   app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
   app(f2(x23,x24),x25) -> f3(x23,x24,x25)
   app(f1(x23),x24) -> f2(x23,x24)
   app(f0(),x23) -> f1(x23)
   app(map1(x28),x29) -> map2(x28,x29)
   app(map0(),x28) -> map1(x28)
   app(cons1(x32),x33) -> cons2(x32,x33)
   app(cons0(),x32) -> cons1(x32)
   app(filter1(x35),x36) -> filter2(x35,x36)
   app(filter0(),x35) -> filter1(x35)
   app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
   app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
   app(filter21(x38),x39) -> filter22(x38,x39)
   app(filter20(),x38) -> filter21(x38)
  eta-rules:
   
  problem:
   minus2(00(),y) -> 00()
   minus2(s1(x),00()) -> s1(x)
   minus2(s1(x),s1(y)) -> minus2(x,y)
   le2(00(),y) -> true0()
   le2(s1(x),00()) -> false0()
   le2(s1(x),s1(y)) -> le2(x,y)
   if3(true0(),x,y) -> x
   if3(false0(),x,y) -> y
   perfectp1(00()) -> false0()
   perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
   f4(00(),y,00(),u) -> true0()
   f4(00(),y,s1(z),u) -> false0()
   f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
   f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
   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(minus1(x6),x7) -> minus2(x6,x7)
   app(minus0(),x6) -> minus1(x6)
   app(s0(),x10) -> s1(x10)
   app(le1(x12),x13) -> le2(x12,x13)
   app(le0(),x12) -> le1(x12)
   app(if2(x17,x18),x19) -> if3(x17,x18,x19)
   app(if1(x17),x18) -> if2(x17,x18)
   app(if0(),x17) -> if1(x17)
   app(perfectp0(),x21) -> perfectp1(x21)
   app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
   app(f2(x23,x24),x25) -> f3(x23,x24,x25)
   app(f1(x23),x24) -> f2(x23,x24)
   app(f0(),x23) -> f1(x23)
   app(map1(x28),x29) -> map2(x28,x29)
   app(map0(),x28) -> map1(x28)
   app(cons1(x32),x33) -> cons2(x32,x33)
   app(cons0(),x32) -> cons1(x32)
   app(filter1(x35),x36) -> filter2(x35,x36)
   app(filter0(),x35) -> filter1(x35)
   app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
   app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
   app(filter21(x38),x39) -> filter22(x38,x39)
   app(filter20(),x38) -> filter21(x38)
  DP Processor:
   DPs:
    minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
    perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x))
    f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
    f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
    f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
    f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
    f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
    f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
    f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
    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#(minus1(x6),x7) -> minus{2,#}(x6,x7)
    app#(le1(x12),x13) -> le{2,#}(x12,x13)
    app#(if2(x17,x18),x19) -> if{3,#}(x17,x18,x19)
    app#(perfectp0(),x21) -> perfectp{1,#}(x21)
    app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26)
    app#(map1(x28),x29) -> map{2,#}(x28,x29)
    app#(filter1(x35),x36) -> filter{2,#}(x35,x36)
    app#(filter23(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41)
   TRS:
    minus2(00(),y) -> 00()
    minus2(s1(x),00()) -> s1(x)
    minus2(s1(x),s1(y)) -> minus2(x,y)
    le2(00(),y) -> true0()
    le2(s1(x),00()) -> false0()
    le2(s1(x),s1(y)) -> le2(x,y)
    if3(true0(),x,y) -> x
    if3(false0(),x,y) -> y
    perfectp1(00()) -> false0()
    perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
    f4(00(),y,00(),u) -> true0()
    f4(00(),y,s1(z),u) -> false0()
    f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
    f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
    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(minus1(x6),x7) -> minus2(x6,x7)
    app(minus0(),x6) -> minus1(x6)
    app(s0(),x10) -> s1(x10)
    app(le1(x12),x13) -> le2(x12,x13)
    app(le0(),x12) -> le1(x12)
    app(if2(x17,x18),x19) -> if3(x17,x18,x19)
    app(if1(x17),x18) -> if2(x17,x18)
    app(if0(),x17) -> if1(x17)
    app(perfectp0(),x21) -> perfectp1(x21)
    app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
    app(f2(x23,x24),x25) -> f3(x23,x24,x25)
    app(f1(x23),x24) -> f2(x23,x24)
    app(f0(),x23) -> f1(x23)
    app(map1(x28),x29) -> map2(x28,x29)
    app(map0(),x28) -> map1(x28)
    app(cons1(x32),x33) -> cons2(x32,x33)
    app(cons0(),x32) -> cons1(x32)
    app(filter1(x35),x36) -> filter2(x35,x36)
    app(filter0(),x35) -> filter1(x35)
    app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
    app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
    app(filter21(x38),x39) -> filter22(x38,x39)
    app(filter20(),x38) -> filter21(x38)
   TDG Processor:
    DPs:
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x))
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     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#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     app#(le1(x12),x13) -> le{2,#}(x12,x13)
     app#(if2(x17,x18),x19) -> if{3,#}(x17,x18,x19)
     app#(perfectp0(),x21) -> perfectp{1,#}(x21)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26)
     app#(map1(x28),x29) -> map{2,#}(x28,x29)
     app#(filter1(x35),x36) -> filter{2,#}(x35,x36)
     app#(filter23(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41)
    TRS:
     minus2(00(),y) -> 00()
     minus2(s1(x),00()) -> s1(x)
     minus2(s1(x),s1(y)) -> minus2(x,y)
     le2(00(),y) -> true0()
     le2(s1(x),00()) -> false0()
     le2(s1(x),s1(y)) -> le2(x,y)
     if3(true0(),x,y) -> x
     if3(false0(),x,y) -> y
     perfectp1(00()) -> false0()
     perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
     f4(00(),y,00(),u) -> true0()
     f4(00(),y,s1(z),u) -> false0()
     f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
     f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     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(minus1(x6),x7) -> minus2(x6,x7)
     app(minus0(),x6) -> minus1(x6)
     app(s0(),x10) -> s1(x10)
     app(le1(x12),x13) -> le2(x12,x13)
     app(le0(),x12) -> le1(x12)
     app(if2(x17,x18),x19) -> if3(x17,x18,x19)
     app(if1(x17),x18) -> if2(x17,x18)
     app(if0(),x17) -> if1(x17)
     app(perfectp0(),x21) -> perfectp1(x21)
     app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
     app(f2(x23,x24),x25) -> f3(x23,x24,x25)
     app(f1(x23),x24) -> f2(x23,x24)
     app(f0(),x23) -> f1(x23)
     app(map1(x28),x29) -> map2(x28,x29)
     app(map0(),x28) -> map1(x28)
     app(cons1(x32),x33) -> cons2(x32,x33)
     app(cons0(),x32) -> cons1(x32)
     app(filter1(x35),x36) -> filter2(x35,x36)
     app(filter0(),x35) -> filter1(x35)
     app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
     app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
     app(filter21(x38),x39) -> filter22(x38,x39)
     app(filter20(),x38) -> filter21(x38)
    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(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x35),x36) -> filter{2,#}(x35,x36)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x28),x29) -> map{2,#}(x28,x29)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(perfectp0(),x21) -> perfectp{1,#}(x21)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(if2(x17,x18),x19) -> if{3,#}(x17,x18,x19)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(le1(x12),x13) -> le{2,#}(x12,x13)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     app#(filter23(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter23(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter1(x35),x36) -> filter{2,#}(x35,x36) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     app#(filter1(x35),x36) -> filter{2,#}(x35,x36) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x28),x29) -> map{2,#}(x28,x29) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x28),x29) -> map{2,#}(x28,x29) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26) ->
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
     app#(perfectp0(),x21) -> perfectp{1,#}(x21) ->
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x))
     app#(le1(x12),x13) -> le{2,#}(x12,x13) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x35),x36) -> filter{2,#}(x35,x36)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x28),x29) -> map{2,#}(x28,x29)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f3(x23,x24,x25),x26) -> f{4,#}(x23,x24,x25,x26)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(perfectp0(),x21) -> perfectp{1,#}(x21)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(if2(x17,x18),x19) -> if{3,#}(x17,x18,x19)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(le1(x12),x13) -> le{2,#}(x12,x13)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x6),x7) -> minus{2,#}(x6,x7)
     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)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x)) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(00()),s1(x),s1(x)) ->
     f{4,#}(s1(x),00(),z,u) -> minus{2,#}(z,s1(x))
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) -> minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    SCC Processor:
     #sccs: 4
     #rules: 14
     #arcs: 73/576
     DPs:
      filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
      filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(map1(x28),x29) -> map{2,#}(x28,x29)
      map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
      map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(filter1(x35),x36) -> filter{2,#}(x35,x36)
      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(x38,x39,x40),x41) -> filter2{4,#}(x38,x39,x40,x41)
     TRS:
      minus2(00(),y) -> 00()
      minus2(s1(x),00()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true0(),x,y) -> x
      if3(false0(),x,y) -> y
      perfectp1(00()) -> false0()
      perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
      f4(00(),y,00(),u) -> true0()
      f4(00(),y,s1(z),u) -> false0()
      f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      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(minus1(x6),x7) -> minus2(x6,x7)
      app(minus0(),x6) -> minus1(x6)
      app(s0(),x10) -> s1(x10)
      app(le1(x12),x13) -> le2(x12,x13)
      app(le0(),x12) -> le1(x12)
      app(if2(x17,x18),x19) -> if3(x17,x18,x19)
      app(if1(x17),x18) -> if2(x17,x18)
      app(if0(),x17) -> if1(x17)
      app(perfectp0(),x21) -> perfectp1(x21)
      app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
      app(f2(x23,x24),x25) -> f3(x23,x24,x25)
      app(f1(x23),x24) -> f2(x23,x24)
      app(f0(),x23) -> f1(x23)
      app(map1(x28),x29) -> map2(x28,x29)
      app(map0(),x28) -> map1(x28)
      app(cons1(x32),x33) -> cons2(x32,x33)
      app(cons0(),x32) -> cons1(x32)
      app(filter1(x35),x36) -> filter2(x35,x36)
      app(filter0(),x35) -> filter1(x35)
      app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
      app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
      app(filter21(x38),x39) -> filter22(x38,x39)
      app(filter20(),x38) -> filter21(x38)
     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:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
       TRS:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          minus2(00(),y) -> 00()
          minus2(s1(x),00()) -> s1(x)
          minus2(s1(x),s1(y)) -> minus2(x,y)
          le2(00(),y) -> true0()
          le2(s1(x),00()) -> false0()
          le2(s1(x),s1(y)) -> le2(x,y)
          if3(true0(),x,y) -> x
          if3(false0(),x,y) -> y
          perfectp1(00()) -> false0()
          perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
          f4(00(),y,00(),u) -> true0()
          f4(00(),y,s1(z),u) -> false0()
          f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
          f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
          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(minus1(x6),x7) -> minus2(x6,x7)
          app(minus0(),x6) -> minus1(x6)
          app(s0(),x10) -> s1(x10)
          app(le1(x12),x13) -> le2(x12,x13)
          app(le0(),x12) -> le1(x12)
          app(if2(x17,x18),x19) -> if3(x17,x18,x19)
          app(if1(x17),x18) -> if2(x17,x18)
          app(if0(),x17) -> if1(x17)
          app(perfectp0(),x21) -> perfectp1(x21)
          app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
          app(f2(x23,x24),x25) -> f3(x23,x24,x25)
          app(f1(x23),x24) -> f2(x23,x24)
          app(f0(),x23) -> f1(x23)
          app(map1(x28),x29) -> map2(x28,x29)
          app(map0(),x28) -> map1(x28)
          app(cons1(x32),x33) -> cons2(x32,x33)
          app(cons0(),x32) -> cons1(x32)
          app(filter1(x35),x36) -> filter2(x35,x36)
          app(filter0(),x35) -> filter1(x35)
          app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
          app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
          app(filter21(x38),x39) -> filter22(x38,x39)
          app(filter20(),x38) -> filter21(x38)
        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:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
       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:
          minus2(00(),y) -> 00()
          minus2(s1(x),00()) -> s1(x)
          minus2(s1(x),s1(y)) -> minus2(x,y)
          le2(00(),y) -> true0()
          le2(s1(x),00()) -> false0()
          le2(s1(x),s1(y)) -> le2(x,y)
          if3(true0(),x,y) -> x
          if3(false0(),x,y) -> y
          perfectp1(00()) -> false0()
          perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
          f4(00(),y,00(),u) -> true0()
          f4(00(),y,s1(z),u) -> false0()
          f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
          f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
          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(minus1(x6),x7) -> minus2(x6,x7)
          app(minus0(),x6) -> minus1(x6)
          app(s0(),x10) -> s1(x10)
          app(le1(x12),x13) -> le2(x12,x13)
          app(le0(),x12) -> le1(x12)
          app(if2(x17,x18),x19) -> if3(x17,x18,x19)
          app(if1(x17),x18) -> if2(x17,x18)
          app(if0(),x17) -> if1(x17)
          app(perfectp0(),x21) -> perfectp1(x21)
          app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
          app(f2(x23,x24),x25) -> f3(x23,x24,x25)
          app(f1(x23),x24) -> f2(x23,x24)
          app(f0(),x23) -> f1(x23)
          app(map1(x28),x29) -> map2(x28,x29)
          app(map0(),x28) -> map1(x28)
          app(cons1(x32),x33) -> cons2(x32,x33)
          app(cons0(),x32) -> cons1(x32)
          app(filter1(x35),x36) -> filter2(x35,x36)
          app(filter0(),x35) -> filter1(x35)
          app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
          app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
          app(filter21(x38),x39) -> filter22(x38,x39)
          app(filter20(),x38) -> filter21(x38)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      f{4,#}(s1(x),00(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
      f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
      f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     TRS:
      minus2(00(),y) -> 00()
      minus2(s1(x),00()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true0(),x,y) -> x
      if3(false0(),x,y) -> y
      perfectp1(00()) -> false0()
      perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
      f4(00(),y,00(),u) -> true0()
      f4(00(),y,s1(z),u) -> false0()
      f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      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(minus1(x6),x7) -> minus2(x6,x7)
      app(minus0(),x6) -> minus1(x6)
      app(s0(),x10) -> s1(x10)
      app(le1(x12),x13) -> le2(x12,x13)
      app(le0(),x12) -> le1(x12)
      app(if2(x17,x18),x19) -> if3(x17,x18,x19)
      app(if1(x17),x18) -> if2(x17,x18)
      app(if0(),x17) -> if1(x17)
      app(perfectp0(),x21) -> perfectp1(x21)
      app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
      app(f2(x23,x24),x25) -> f3(x23,x24,x25)
      app(f1(x23),x24) -> f2(x23,x24)
      app(f0(),x23) -> f1(x23)
      app(map1(x28),x29) -> map2(x28,x29)
      app(map0(),x28) -> map1(x28)
      app(cons1(x32),x33) -> cons2(x32,x33)
      app(cons0(),x32) -> cons1(x32)
      app(filter1(x35),x36) -> filter2(x35,x36)
      app(filter0(),x35) -> filter1(x35)
      app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
      app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
      app(filter21(x38),x39) -> filter22(x38,x39)
      app(filter20(),x38) -> filter21(x38)
     Subterm Criterion Processor:
      simple projection:
       pi(f{4,#}) = 0
      problem:
       DPs:
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
       TRS:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
      EDG Processor:
       DPs:
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
       TRS:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
       graph:
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
       Subterm Criterion Processor:
        simple projection:
         pi(minus2) = 0
         pi(f{4,#}) = 1
        problem:
         DPs:
          
         TRS:
          minus2(00(),y) -> 00()
          minus2(s1(x),00()) -> s1(x)
          minus2(s1(x),s1(y)) -> minus2(x,y)
          le2(00(),y) -> true0()
          le2(s1(x),00()) -> false0()
          le2(s1(x),s1(y)) -> le2(x,y)
          if3(true0(),x,y) -> x
          if3(false0(),x,y) -> y
          perfectp1(00()) -> false0()
          perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
          f4(00(),y,00(),u) -> true0()
          f4(00(),y,s1(z),u) -> false0()
          f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
          f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
          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(minus1(x6),x7) -> minus2(x6,x7)
          app(minus0(),x6) -> minus1(x6)
          app(s0(),x10) -> s1(x10)
          app(le1(x12),x13) -> le2(x12,x13)
          app(le0(),x12) -> le1(x12)
          app(if2(x17,x18),x19) -> if3(x17,x18,x19)
          app(if1(x17),x18) -> if2(x17,x18)
          app(if0(),x17) -> if1(x17)
          app(perfectp0(),x21) -> perfectp1(x21)
          app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
          app(f2(x23,x24),x25) -> f3(x23,x24,x25)
          app(f1(x23),x24) -> f2(x23,x24)
          app(f0(),x23) -> f1(x23)
          app(map1(x28),x29) -> map2(x28,x29)
          app(map0(),x28) -> map1(x28)
          app(cons1(x32),x33) -> cons2(x32,x33)
          app(cons0(),x32) -> cons1(x32)
          app(filter1(x35),x36) -> filter2(x35,x36)
          app(filter0(),x35) -> filter1(x35)
          app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
          app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
          app(filter21(x38),x39) -> filter22(x38,x39)
          app(filter20(),x38) -> filter21(x38)
        Qed
     
     DPs:
      le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     TRS:
      minus2(00(),y) -> 00()
      minus2(s1(x),00()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true0(),x,y) -> x
      if3(false0(),x,y) -> y
      perfectp1(00()) -> false0()
      perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
      f4(00(),y,00(),u) -> true0()
      f4(00(),y,s1(z),u) -> false0()
      f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      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(minus1(x6),x7) -> minus2(x6,x7)
      app(minus0(),x6) -> minus1(x6)
      app(s0(),x10) -> s1(x10)
      app(le1(x12),x13) -> le2(x12,x13)
      app(le0(),x12) -> le1(x12)
      app(if2(x17,x18),x19) -> if3(x17,x18,x19)
      app(if1(x17),x18) -> if2(x17,x18)
      app(if0(),x17) -> if1(x17)
      app(perfectp0(),x21) -> perfectp1(x21)
      app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
      app(f2(x23,x24),x25) -> f3(x23,x24,x25)
      app(f1(x23),x24) -> f2(x23,x24)
      app(f0(),x23) -> f1(x23)
      app(map1(x28),x29) -> map2(x28,x29)
      app(map0(),x28) -> map1(x28)
      app(cons1(x32),x33) -> cons2(x32,x33)
      app(cons0(),x32) -> cons1(x32)
      app(filter1(x35),x36) -> filter2(x35,x36)
      app(filter0(),x35) -> filter1(x35)
      app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
      app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
      app(filter21(x38),x39) -> filter22(x38,x39)
      app(filter20(),x38) -> filter21(x38)
     Subterm Criterion Processor:
      simple projection:
       pi(le{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
      Qed
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      minus2(00(),y) -> 00()
      minus2(s1(x),00()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(00(),y) -> true0()
      le2(s1(x),00()) -> false0()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true0(),x,y) -> x
      if3(false0(),x,y) -> y
      perfectp1(00()) -> false0()
      perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
      f4(00(),y,00(),u) -> true0()
      f4(00(),y,s1(z),u) -> false0()
      f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      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(minus1(x6),x7) -> minus2(x6,x7)
      app(minus0(),x6) -> minus1(x6)
      app(s0(),x10) -> s1(x10)
      app(le1(x12),x13) -> le2(x12,x13)
      app(le0(),x12) -> le1(x12)
      app(if2(x17,x18),x19) -> if3(x17,x18,x19)
      app(if1(x17),x18) -> if2(x17,x18)
      app(if0(),x17) -> if1(x17)
      app(perfectp0(),x21) -> perfectp1(x21)
      app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
      app(f2(x23,x24),x25) -> f3(x23,x24,x25)
      app(f1(x23),x24) -> f2(x23,x24)
      app(f0(),x23) -> f1(x23)
      app(map1(x28),x29) -> map2(x28,x29)
      app(map0(),x28) -> map1(x28)
      app(cons1(x32),x33) -> cons2(x32,x33)
      app(cons0(),x32) -> cons1(x32)
      app(filter1(x35),x36) -> filter2(x35,x36)
      app(filter0(),x35) -> filter1(x35)
      app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
      app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
      app(filter21(x38),x39) -> filter22(x38,x39)
      app(filter20(),x38) -> filter21(x38)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        minus2(00(),y) -> 00()
        minus2(s1(x),00()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(00(),y) -> true0()
        le2(s1(x),00()) -> false0()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true0(),x,y) -> x
        if3(false0(),x,y) -> y
        perfectp1(00()) -> false0()
        perfectp1(s1(x)) -> f4(x,s1(00()),s1(x),s1(x))
        f4(00(),y,00(),u) -> true0()
        f4(00(),y,s1(z),u) -> false0()
        f4(s1(x),00(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        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(minus1(x6),x7) -> minus2(x6,x7)
        app(minus0(),x6) -> minus1(x6)
        app(s0(),x10) -> s1(x10)
        app(le1(x12),x13) -> le2(x12,x13)
        app(le0(),x12) -> le1(x12)
        app(if2(x17,x18),x19) -> if3(x17,x18,x19)
        app(if1(x17),x18) -> if2(x17,x18)
        app(if0(),x17) -> if1(x17)
        app(perfectp0(),x21) -> perfectp1(x21)
        app(f3(x23,x24,x25),x26) -> f4(x23,x24,x25,x26)
        app(f2(x23,x24),x25) -> f3(x23,x24,x25)
        app(f1(x23),x24) -> f2(x23,x24)
        app(f0(),x23) -> f1(x23)
        app(map1(x28),x29) -> map2(x28,x29)
        app(map0(),x28) -> map1(x28)
        app(cons1(x32),x33) -> cons2(x32,x33)
        app(cons0(),x32) -> cons1(x32)
        app(filter1(x35),x36) -> filter2(x35,x36)
        app(filter0(),x35) -> filter1(x35)
        app(filter23(x38,x39,x40),x41) -> filter24(x38,x39,x40,x41)
        app(filter22(x38,x39),x40) -> filter23(x38,x39,x40)
        app(filter21(x38),x39) -> filter22(x38,x39)
        app(filter20(),x38) -> filter21(x38)
      Qed