Input TRS:
    1: app(perfectp(),|0|()) -> false()
    2: app(perfectp(),app(s(),x)) -> app(app(app(app(f(),x),app(s(),|0|())),app(s(),x)),app(s(),x))
    3: app(app(app(app(f(),|0|()),y),|0|()),u) -> true()
    4: app(app(app(app(f(),|0|()),y),app(s(),z)),u) -> false()
    5: 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)
    6: 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))
    7: app(app(map(),|fun|),nil()) -> nil()
    8: app(app(map(),|fun|),app(app(cons(),x),xs)) -> app(app(cons(),app(|fun|,x)),app(app(map(),|fun|),xs))
    9: app(app(filter(),|fun|),nil()) -> nil()
    10: app(app(filter(),|fun|),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(|fun|,x)),|fun|),x),xs)
    11: app(app(app(app(filter2(),true()),|fun|),x),xs) -> app(app(cons(),x),app(app(filter(),|fun|),xs))
    12: app(app(app(app(filter2(),false()),|fun|),x),xs) -> app(app(filter(),|fun|),xs)
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... failed.
Freezing app
1: app❆1_perfectp(|0|()) -> false()
2: app❆1_perfectp(app❆1_s(x)) -> app❆4_f(x,app❆1_s(|0|()),app❆1_s(x),app❆1_s(x))
3: app❆4_f(|0|(),y,|0|(),u) -> true()
4: app❆4_f(|0|(),y,app❆1_s(z),u) -> false()
5: app❆4_f(app❆1_s(x),|0|(),z,u) -> app❆4_f(x,u,app❆2_minus(z,app❆1_s(x)),u)
6: app❆4_f(app❆1_s(x),app❆1_s(y),z,u) -> app❆3_if(app❆2_le(x,y),app❆4_f(app❆1_s(x),app❆2_minus(y,x),z,u),app❆4_f(x,u,z,u))
7: app❆2_map(|fun|,nil()) -> nil()
8: app❆2_map(|fun|,app❆2_cons(x,xs)) -> app❆2_cons(app(|fun|,x),app❆2_map(|fun|,xs))
9: app❆2_filter(|fun|,nil()) -> nil()
10: app❆2_filter(|fun|,app❆2_cons(x,xs)) -> app❆4_filter2(app(|fun|,x),|fun|,x,xs)
11: app❆4_filter2(true(),|fun|,x,xs) -> app❆2_cons(x,app❆2_filter(|fun|,xs))
12: app❆4_filter2(false(),|fun|,x,xs) -> app❆2_filter(|fun|,xs)
13: app(le(),_1) ->= app❆1_le(_1)
14: app(app❆1_le(_1),_2) ->= app❆2_le(_1,_2)
15: app(if(),_1) ->= app❆1_if(_1)
16: app(app❆1_if(_1),_2) ->= app❆2_if(_1,_2)
17: app(app❆2_if(_1,_2),_3) ->= app❆3_if(_1,_2,_3)
18: app(cons(),_1) ->= app❆1_cons(_1)
19: app(app❆1_cons(_1),_2) ->= app❆2_cons(_1,_2)
20: app(s(),_1) ->= app❆1_s(_1)
21: app(filter(),_1) ->= app❆1_filter(_1)
22: app(app❆1_filter(_1),_2) ->= app❆2_filter(_1,_2)
23: app(perfectp(),_1) ->= app❆1_perfectp(_1)
24: app(map(),_1) ->= app❆1_map(_1)
25: app(app❆1_map(_1),_2) ->= app❆2_map(_1,_2)
26: app(minus(),_1) ->= app❆1_minus(_1)
27: app(app❆1_minus(_1),_2) ->= app❆2_minus(_1,_2)
28: app(f(),_1) ->= app❆1_f(_1)
29: app(app❆1_f(_1),_2) ->= app❆2_f(_1,_2)
30: app(app❆2_f(_1,_2),_3) ->= app❆3_f(_1,_2,_3)
31: app(app❆3_f(_1,_2,_3),_4) ->= app❆4_f(_1,_2,_3,_4)
32: app(filter2(),_1) ->= app❆1_filter2(_1)
33: app(app❆1_filter2(_1),_2) ->= app❆2_filter2(_1,_2)
34: app(app❆2_filter2(_1,_2),_3) ->= app❆3_filter2(_1,_2,_3)
35: app(app❆3_filter2(_1,_2,_3),_4) ->= app❆4_filter2(_1,_2,_3,_4)
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #app❆1_perfectp(app❆1_s(x)) -> #app❆4_f(x,app❆1_s(|0|()),app❆1_s(x),app❆1_s(x))
   #2: #app(app❆3_filter2(_1,_2,_3),_4) ->? #app❆4_filter2(_1,_2,_3,_4)
   #3: #app❆4_f(app❆1_s(x),app❆1_s(y),z,u) -> #app❆4_f(app❆1_s(x),app❆2_minus(y,x),z,u)
   #4: #app❆4_f(app❆1_s(x),app❆1_s(y),z,u) -> #app❆4_f(x,u,z,u)
   #5: #app❆4_filter2(true(),|fun|,x,xs) -> #app❆2_filter(|fun|,xs)
   #6: #app(perfectp(),_1) ->? #app❆1_perfectp(_1)
   #7: #app❆4_filter2(false(),|fun|,x,xs) -> #app❆2_filter(|fun|,xs)
   #8: #app(app❆3_f(_1,_2,_3),_4) ->? #app❆4_f(_1,_2,_3,_4)
   #9: #app(app❆1_map(_1),_2) ->? #app❆2_map(_1,_2)
   #10: #app❆2_filter(|fun|,app❆2_cons(x,xs)) -> #app❆4_filter2(app(|fun|,x),|fun|,x,xs)
   #11: #app❆2_filter(|fun|,app❆2_cons(x,xs)) -> #app(|fun|,x)
   #12: #app❆4_f(app❆1_s(x),|0|(),z,u) -> #app❆4_f(x,u,app❆2_minus(z,app❆1_s(x)),u)
   #13: #app(app❆1_filter(_1),_2) ->? #app❆2_filter(_1,_2)
   #14: #app❆2_map(|fun|,app❆2_cons(x,xs)) -> #app(|fun|,x)
   #15: #app❆2_map(|fun|,app❆2_cons(x,xs)) -> #app❆2_map(|fun|,xs)
Number of SCCs: 2, DPs: 11, edges: 24
	SCC { #4 #12 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
app❆2_minus(x1,x2)	weight: (/ 1 2)
    le()	weight: 0
app❆1_le(x1)	weight: 0
#app❆2_map(x1,x2)	weight: 0
app❆4_f(x1,x2,x3,x4)	weight: 0
     s()	weight: 0
app❆1_map(x1)	weight: 0
app❆1_minus(x1)	weight: 0
app❆1_if(x1)	weight: 0
app❆2_f(x1,x2)	weight: 0
 minus()	weight: 0
app❆4_filter2(x1,x2,x3,x4)	weight: 0
app❆2_map(x1,x2)	weight: 0
 false()	weight: 0
app❆2_if(x1,x2)	weight: 0
  true()	weight: 0
     f()	weight: 0
filter2()	weight: 0
app❆2_filter(x1,x2)	weight: 0
#app❆1_perfectp(x1)	weight: 0
app❆2_le(x1,x2)	weight: 0
    if()	weight: 0
app❆1_f(x1)	weight: 0
app❆3_f(x1,x2,x3)	weight: 0
   nil()	weight: 0
app❆1_filter(x1)	weight: 0
app❆3_if(x1,x2,x3)	weight: 0
 #app(x1,x2)	weight: 0
   map()	weight: 0
#app❆4_filter2(x1,x2,x3,x4)	weight: 0
app❆2_filter2(x1,x2)	weight: 0
#app❆2_filter(x1,x2)	weight: 0
app❆1_perfectp(x1)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: 0
filter()	weight: 0
app❆1_s(x1)	weight: (/ 1 2) + x1
#app❆4_f(x1,x2,x3,x4)	weight: x1
perfectp()	weight: 0
app❆1_filter2(x1)	weight: 0
app❆1_cons(x1)	weight: 0
app❆3_filter2(x1,x2,x3)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #4 #12
Number of SCCs: 1, DPs: 9, edges: 20
	SCC { #2 #5 #7 #9..11 #13..15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
app❆2_minus(x1,x2)	weight: (/ 3 8) + x1 + x2
    le()	weight: 0
app❆1_le(x1)	weight: (/ 3 16)
#app❆2_map(x1,x2)	weight: (/ 1 16) + x1
app❆4_f(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x3 + x4
     s()	weight: 0
app❆1_map(x1)	weight: (/ 3 16) + x1
app❆1_minus(x1)	weight: (/ 3 16)
app❆1_if(x1)	weight: (/ 3 16)
app❆2_f(x1,x2)	weight: (/ 3 8) + x1
 minus()	weight: 0
app❆4_filter2(x1,x2,x3,x4)	weight: (/ 3 4) + x1
app❆2_map(x1,x2)	weight: (/ 3 8)
 false()	weight: 0
app❆2_if(x1,x2)	weight: (/ 3 8) + x1
  true()	weight: 0
     f()	weight: 0
filter2()	weight: 0
app❆2_filter(x1,x2)	weight: (/ 13 16)
#app❆1_perfectp(x1)	weight: 0
app❆2_le(x1,x2)	weight: (/ 3 8) + x1 + x2
    if()	weight: 0
app❆1_f(x1)	weight: (/ 3 16)
app❆3_f(x1,x2,x3)	weight: (/ 9 16) + x2
   nil()	weight: 0
app❆1_filter(x1)	weight: (/ 3 16) + x1
app❆3_if(x1,x2,x3)	weight: (/ 5 8) + x1
 #app(x1,x2)	weight: x1
   map()	weight: 0
#app❆4_filter2(x1,x2,x3,x4)	weight: (/ 1 16) + x2
app❆2_filter2(x1,x2)	weight: (/ 3 8) + x1
#app❆2_filter(x1,x2)	weight: (/ 1 16) + x1
app❆1_perfectp(x1)	weight: (/ 3 16) + x1
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 3 8) + x1 + x2
filter()	weight: 0
app❆1_s(x1)	weight: (/ 3 16) + x1
#app❆4_f(x1,x2,x3,x4)	weight: 0
perfectp()	weight: 0
app❆1_filter2(x1)	weight: (/ 3 16)
app❆1_cons(x1)	weight: (/ 3 16)
app❆3_filter2(x1,x2,x3)	weight: (/ 9 16) + x2 + x3
   app(x1,x2)	weight: (/ 1 8) + x1 + x2
    Usable rules: { }
    Removed DPs: #2 #9 #11 #13 #14
Number of SCCs: 2, DPs: 4, edges: 5
	SCC { #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
app❆2_minus(x1,x2)	weight: (/ 3 8) + x1 + x2
    le()	weight: 0
app❆1_le(x1)	weight: (/ 3 16)
#app❆2_map(x1,x2)	weight: (/ 1 16) + x2
app❆4_f(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x3 + x4
     s()	weight: 0
app❆1_map(x1)	weight: (/ 3 16) + x1
app❆1_minus(x1)	weight: (/ 3 16)
app❆1_if(x1)	weight: (/ 3 16)
app❆2_f(x1,x2)	weight: (/ 3 8) + x1
 minus()	weight: 0
app❆4_filter2(x1,x2,x3,x4)	weight: (/ 3 4) + x1
app❆2_map(x1,x2)	weight: (/ 3 8)
 false()	weight: 0
app❆2_if(x1,x2)	weight: (/ 3 8) + x1
  true()	weight: 0
     f()	weight: 0
filter2()	weight: 0
app❆2_filter(x1,x2)	weight: (/ 13 16)
#app❆1_perfectp(x1)	weight: 0
app❆2_le(x1,x2)	weight: (/ 3 8) + x1 + x2
    if()	weight: 0
app❆1_f(x1)	weight: (/ 3 16)
app❆3_f(x1,x2,x3)	weight: (/ 9 16) + x2
   nil()	weight: 0
app❆1_filter(x1)	weight: (/ 3 16) + x1
app❆3_if(x1,x2,x3)	weight: (/ 5 8) + x1
 #app(x1,x2)	weight: 0
   map()	weight: 0
#app❆4_filter2(x1,x2,x3,x4)	weight: (/ 1 16)
app❆2_filter2(x1,x2)	weight: (/ 3 8) + x1
#app❆2_filter(x1,x2)	weight: (/ 1 16)
app❆1_perfectp(x1)	weight: (/ 3 16) + x1
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 3 8) + x1 + x2
filter()	weight: 0
app❆1_s(x1)	weight: (/ 3 16) + x1
#app❆4_f(x1,x2,x3,x4)	weight: 0
perfectp()	weight: 0
app❆1_filter2(x1)	weight: (/ 3 16)
app❆1_cons(x1)	weight: (/ 3 16)
app❆3_filter2(x1,x2,x3)	weight: (/ 9 16) + x2 + x3
   app(x1,x2)	weight: (/ 1 8) + x1 + x2
    Usable rules: { }
    Removed DPs: #15
Number of SCCs: 1, DPs: 3, edges: 4
	SCC { #5 #7 #10 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
app❆2_minus(x1,x2)	weight: (/ 3 8) + x1 + x2
    le()	weight: 0
app❆1_le(x1)	weight: (/ 3 16)
#app❆2_map(x1,x2)	weight: (/ 1 16)
app❆4_f(x1,x2,x3,x4)	weight: (/ 3 4) + x1 + x3 + x4
     s()	weight: 0
app❆1_map(x1)	weight: (/ 3 16) + x1
app❆1_minus(x1)	weight: (/ 3 16)
app❆1_if(x1)	weight: (/ 3 16)
app❆2_f(x1,x2)	weight: (/ 3 8) + x1
 minus()	weight: 0
app❆4_filter2(x1,x2,x3,x4)	weight: (/ 3 4) + x1
app❆2_map(x1,x2)	weight: (/ 3 8)
 false()	weight: 0
app❆2_if(x1,x2)	weight: (/ 3 8) + x1
  true()	weight: 0
     f()	weight: 0
filter2()	weight: 0
app❆2_filter(x1,x2)	weight: (/ 13 16)
#app❆1_perfectp(x1)	weight: 0
app❆2_le(x1,x2)	weight: (/ 3 8) + x1 + x2
    if()	weight: 0
app❆1_f(x1)	weight: (/ 3 16)
app❆3_f(x1,x2,x3)	weight: (/ 9 16) + x2
   nil()	weight: 0
app❆1_filter(x1)	weight: (/ 3 16) + x1
app❆3_if(x1,x2,x3)	weight: (/ 5 8) + x1
 #app(x1,x2)	weight: 0
   map()	weight: 0
#app❆4_filter2(x1,x2,x3,x4)	weight: (/ 1 8) + x4
app❆2_filter2(x1,x2)	weight: (/ 3 8) + x1
#app❆2_filter(x1,x2)	weight: (/ 1 16) + x2
app❆1_perfectp(x1)	weight: (/ 3 16) + x1
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 3 8) + x1 + x2
filter()	weight: 0
app❆1_s(x1)	weight: (/ 3 16) + x1
#app❆4_f(x1,x2,x3,x4)	weight: 0
perfectp()	weight: 0
app❆1_filter2(x1)	weight: (/ 3 16)
app❆1_cons(x1)	weight: (/ 3 16)
app❆3_filter2(x1,x2,x3)	weight: (/ 9 16) + x2 + x3
   app(x1,x2)	weight: (/ 1 8) + x1 + x2
    Usable rules: { }
    Removed DPs: #5 #7 #10
Number of SCCs: 0, DPs: 0, edges: 0
YES