Input TRS:
    1: a(a(append(),nil()),ys) -> ys
    2: a(a(append(),a(a(cons(),x),xs)),ys) -> a(a(cons(),x),a(a(append(),xs),ys))
    3: a(a(filter(),f),nil()) -> nil()
    4: a(a(filter(),f),a(a(cons(),x),xs)) -> a(a(a(if(),a(f,x)),x),a(a(filter(),f),xs))
    5: a(a(le(),0()),y) -> true()
    6: a(a(le(),a(s(),x)),0()) -> false()
    7: a(a(le(),a(s(),x)),a(s(),y)) -> a(a(le(),x),y)
    8: a(a(a(if(),true()),x),xs) -> a(a(cons(),x),xs)
    9: a(a(a(if(),false()),x),xs) -> xs
    10: a(a(not(),f),b) -> a(not2(),a(f,b))
    11: a(not2(),true()) -> false()
    12: a(not2(),false()) -> true()
    13: a(qs(),nil()) -> nil()
    14: a(qs(),a(a(cons(),x),xs)) -> a(a(append(),a(qs(),a(a(filter(),a(le(),x)),xs))),a(a(cons(),x),a(qs(),a(a(filter(),a(not(),a(le(),x))),xs))))
Number of strict rules: 14
Direct Order(PosReal,>,Poly) ... failed.
Freezing a
1: a❆2_append(nil(),ys) -> ys
2: a❆2_append(a❆2_cons(x,xs),ys) -> a❆2_cons(x,a❆2_append(xs,ys))
3: a❆2_filter(f,nil()) -> nil()
4: a❆2_filter(f,a❆2_cons(x,xs)) -> a❆3_if(a(f,x),x,a❆2_filter(f,xs))
5: a❆2_le(0(),y) -> true()
6: a❆2_le(a❆1_s(x),0()) -> false()
7: a❆2_le(a❆1_s(x),a❆1_s(y)) -> a❆2_le(x,y)
8: a❆3_if(true(),x,xs) -> a❆2_cons(x,xs)
9: a❆3_if(false(),x,xs) -> xs
10: a❆2_not(f,b) -> a❆1_not2(a(f,b))
11: a❆1_not2(true()) -> false()
12: a❆1_not2(false()) -> true()
13: a❆1_qs(nil()) -> nil()
14: a❆1_qs(a❆2_cons(x,xs)) -> a❆2_append(a❆1_qs(a❆2_filter(a❆1_le(x),xs)),a❆2_cons(x,a❆1_qs(a❆2_filter(a❆1_not(a❆1_le(x)),xs))))
15: a(qs(),_1) ->= a❆1_qs(_1)
16: a(le(),_1) ->= a❆1_le(_1)
17: a(a❆1_le(_1),_2) ->= a❆2_le(_1,_2)
18: a(append(),_1) ->= a❆1_append(_1)
19: a(a❆1_append(_1),_2) ->= a❆2_append(_1,_2)
20: a(if(),_1) ->= a❆1_if(_1)
21: a(a❆1_if(_1),_2) ->= a❆2_if(_1,_2)
22: a(a❆2_if(_1,_2),_3) ->= a❆3_if(_1,_2,_3)
23: a(cons(),_1) ->= a❆1_cons(_1)
24: a(a❆1_cons(_1),_2) ->= a❆2_cons(_1,_2)
25: a(s(),_1) ->= a❆1_s(_1)
26: a(filter(),_1) ->= a❆1_filter(_1)
27: a(a❆1_filter(_1),_2) ->= a❆2_filter(_1,_2)
28: a(not2(),_1) ->= a❆1_not2(_1)
29: a(not(),_1) ->= a❆1_not(_1)
30: a(a❆1_not(_1),_2) ->= a❆2_not(_1,_2)
Number of strict rules: 14
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #a❆2_append(a❆2_cons(x,xs),ys) -> #a❆2_append(xs,ys)
   #2: #a❆1_qs(a❆2_cons(x,xs)) -> #a❆2_append(a❆1_qs(a❆2_filter(a❆1_le(x),xs)),a❆2_cons(x,a❆1_qs(a❆2_filter(a❆1_not(a❆1_le(x)),xs))))
   #3: #a❆1_qs(a❆2_cons(x,xs)) -> #a❆1_qs(a❆2_filter(a❆1_le(x),xs))
   #4: #a❆1_qs(a❆2_cons(x,xs)) -> #a❆2_filter(a❆1_le(x),xs)
   #5: #a❆1_qs(a❆2_cons(x,xs)) -> #a❆1_qs(a❆2_filter(a❆1_not(a❆1_le(x)),xs))
   #6: #a❆1_qs(a❆2_cons(x,xs)) -> #a❆2_filter(a❆1_not(a❆1_le(x)),xs)
   #7: #a(a❆1_not(_1),_2) ->? #a❆2_not(_1,_2)
   #8: #a❆2_le(a❆1_s(x),a❆1_s(y)) -> #a❆2_le(x,y)
   #9: #a❆2_not(f,b) -> #a❆1_not2(a(f,b))
   #10: #a❆2_not(f,b) -> #a(f,b)
   #11: #a(not2(),_1) ->? #a❆1_not2(_1)
   #12: #a(a❆2_if(_1,_2),_3) ->? #a❆3_if(_1,_2,_3)
   #13: #a(a❆1_filter(_1),_2) ->? #a❆2_filter(_1,_2)
   #14: #a(a❆1_le(_1),_2) ->? #a❆2_le(_1,_2)
   #15: #a(a❆1_append(_1),_2) ->? #a❆2_append(_1,_2)
   #16: #a(qs(),_1) ->? #a❆1_qs(_1)
   #17: #a❆2_filter(f,a❆2_cons(x,xs)) -> #a❆3_if(a(f,x),x,a❆2_filter(f,xs))
   #18: #a❆2_filter(f,a❆2_cons(x,xs)) -> #a(f,x)
   #19: #a❆2_filter(f,a❆2_cons(x,xs)) -> #a❆2_filter(f,xs)
Number of SCCs: 3, DPs: 12, edges: 29
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1,x2)	weight: 0
    le()	weight: 0
a❆2_le(x1,x2)	weight: 0
a❆1_filter(x1)	weight: 0
a❆2_append(x1,x2)	weight: 0
#a❆3_if(x1,x2,x3)	weight: 0
     s()	weight: 0
#a❆2_le(x1,x2)	weight: 0
a❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
a❆1_cons(x1)	weight: 0
a❆1_qs(x1)	weight: 0
  true()	weight: 0
a❆1_le(x1)	weight: 0
#a❆1_not2(x1)	weight: 0
a❆2_if(x1,x2)	weight: 0
    qs()	weight: 0
a❆1_s(x1)	weight: 0
append()	weight: 0
a❆2_filter(x1,x2)	weight: 0
    if()	weight: 0
     0()	weight: 0
a❆2_cons(x1,x2)	weight: (/ 1 2) + x2
a❆1_if(x1)	weight: 0
   nil()	weight: 0
  not2()	weight: 0
#a❆1_qs(x1)	weight: 0
a❆1_append(x1)	weight: 0
a❆1_not(x1)	weight: 0
#a❆2_not(x1,x2)	weight: 0
  cons()	weight: 0
   #a(x1,x2)	weight: 0
a❆2_not(x1,x2)	weight: 0
filter()	weight: 0
#a❆2_filter(x1,x2)	weight: 0
a❆1_not2(x1)	weight: 0
#a❆2_append(x1,x2)	weight: x1
   not()	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 2, DPs: 11, edges: 28
	SCC { #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1,x2)	weight: 0
    le()	weight: 0
a❆2_le(x1,x2)	weight: 0
a❆1_filter(x1)	weight: 0
a❆2_append(x1,x2)	weight: 0
#a❆3_if(x1,x2,x3)	weight: 0
     s()	weight: 0
#a❆2_le(x1,x2)	weight: x2
a❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
a❆1_cons(x1)	weight: 0
a❆1_qs(x1)	weight: 0
  true()	weight: 0
a❆1_le(x1)	weight: 0
#a❆1_not2(x1)	weight: 0
a❆2_if(x1,x2)	weight: 0
    qs()	weight: 0
a❆1_s(x1)	weight: (/ 1 2) + x1
append()	weight: 0
a❆2_filter(x1,x2)	weight: 0
    if()	weight: 0
     0()	weight: 0
a❆2_cons(x1,x2)	weight: (/ 1 2)
a❆1_if(x1)	weight: 0
   nil()	weight: 0
  not2()	weight: 0
#a❆1_qs(x1)	weight: 0
a❆1_append(x1)	weight: 0
a❆1_not(x1)	weight: 0
#a❆2_not(x1,x2)	weight: 0
  cons()	weight: 0
   #a(x1,x2)	weight: 0
a❆2_not(x1,x2)	weight: 0
filter()	weight: 0
#a❆2_filter(x1,x2)	weight: 0
a❆1_not2(x1)	weight: 0
#a❆2_append(x1,x2)	weight: 0
   not()	weight: 0
    Usable rules: { }
    Removed DPs: #8
Number of SCCs: 1, DPs: 10, edges: 27
	SCC { #3..7 #10 #13 #16 #18 #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1,x2)	weight: (/ 1 8) + x2
    le()	weight: 0
a❆2_le(x1,x2)	weight: (/ 1 4)
a❆1_filter(x1)	weight: (/ 1 4) + x1
a❆2_append(x1,x2)	weight: (/ 1 4)
#a❆3_if(x1,x2,x3)	weight: 0
     s()	weight: 0
#a❆2_le(x1,x2)	weight: 0
a❆3_if(x1,x2,x3)	weight: (/ 1 4) + x1
 false()	weight: 0
a❆1_cons(x1)	weight: (/ 1 4)
a❆1_qs(x1)	weight: (/ 1 4) + x1
  true()	weight: 0
a❆1_le(x1)	weight: 0
#a❆1_not2(x1)	weight: 0
a❆2_if(x1,x2)	weight: (/ 1 4)
    qs()	weight: 0
a❆1_s(x1)	weight: (/ 1 4)
append()	weight: 0
a❆2_filter(x1,x2)	weight: (/ 1 4)
    if()	weight: 0
     0()	weight: 0
a❆2_cons(x1,x2)	weight: (/ 3 8) + x1
a❆1_if(x1)	weight: (/ 1 4)
   nil()	weight: 0
  not2()	weight: 0
#a❆1_qs(x1)	weight: 0
a❆1_append(x1)	weight: (/ 1 4)
a❆1_not(x1)	weight: x1
#a❆2_not(x1,x2)	weight: x1
  cons()	weight: 0
   #a(x1,x2)	weight: x1
a❆2_not(x1,x2)	weight: (/ 1 4) + x1
filter()	weight: 0
#a❆2_filter(x1,x2)	weight: x1
a❆1_not2(x1)	weight: (/ 3 8)
#a❆2_append(x1,x2)	weight: 0
   not()	weight: 0
    Usable rules: { }
    Removed DPs: #13
Number of SCCs: 1, DPs: 9, edges: 23
	SCC { #3..7 #10 #16 #18 #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     a(x1,x2)	weight: (/ 1 4)
    le()	weight: 0
a❆2_le(x1,x2)	weight: (/ 3 8)
a❆1_filter(x1)	weight: (/ 3 8)
a❆2_append(x1,x2)	weight: (/ 3 8) + x2
#a❆3_if(x1,x2,x3)	weight: 0
     s()	weight: 0
#a❆2_le(x1,x2)	weight: 0
a❆3_if(x1,x2,x3)	weight: (/ 1 2) + x2 + x3
 false()	weight: 0
a❆1_cons(x1)	weight: (/ 3 8)
a❆1_qs(x1)	weight: (/ 3 8) + x1
  true()	weight: 0
a❆1_le(x1)	weight: (/ 3 8) + x1
#a❆1_not2(x1)	weight: 0
a❆2_if(x1,x2)	weight: (/ 3 8)
    qs()	weight: 0
a❆1_s(x1)	weight: (/ 3 8)
append()	weight: 0
a❆2_filter(x1,x2)	weight: (/ 3 8) + x2
    if()	weight: 0
     0()	weight: 0
a❆2_cons(x1,x2)	weight: (/ 1 2) + x1 + x2
a❆1_if(x1)	weight: (/ 3 8)
   nil()	weight: 0
  not2()	weight: 0
#a❆1_qs(x1)	weight: (/ 1 4) + x1
a❆1_append(x1)	weight: (/ 3 8)
a❆1_not(x1)	weight: (/ 1 4) + x1
#a❆2_not(x1,x2)	weight: (/ 1 2) + x1 + x2
  cons()	weight: 0
   #a(x1,x2)	weight: (/ 3 8) + x1 + x2
a❆2_not(x1,x2)	weight: (/ 3 8) + x1
filter()	weight: 0
#a❆2_filter(x1,x2)	weight: x1 + x2
a❆1_not2(x1)	weight: (/ 1 2)
#a❆2_append(x1,x2)	weight: 0
   not()	weight: 0
    Usable rules: { 3 4 8 9 }
    Removed DPs: #3..7 #10 #16 #18 #19
Number of SCCs: 0, DPs: 0, edges: 0
YES