Input TRS:
    1: app(app(app(if(),true()),xs),ys) -> xs
    2: app(app(app(if(),false()),xs),ys) -> ys
    3: app(app(sub(),x),0()) -> x
    4: app(app(sub(),app(s(),x)),app(s(),y)) -> app(app(sub(),x),y)
    5: app(app(gtr(),0()),y) -> false()
    6: app(app(gtr(),app(s(),x)),0()) -> true()
    7: app(app(gtr(),app(s(),x)),app(s(),y)) -> app(app(gtr(),x),y)
    8: app(app(d(),x),0()) -> true()
    9: app(app(d(),app(s(),x)),app(s(),y)) -> app(app(app(if(),app(app(gtr(),x),y)),false()),app(app(d(),app(s(),x)),app(app(sub(),y),x)))
    10: app(len(),nil()) -> 0()
    11: app(len(),app(app(cons(),x),xs)) -> app(s(),app(len(),xs))
    12: app(app(filter(),p),nil()) -> nil()
    13: app(app(filter(),p),app(app(cons(),x),xs)) -> app(app(app(if(),app(p,x)),app(app(cons(),x),app(app(filter(),p),xs))),app(app(filter(),p),xs))
Number of strict rules: 13
Direct Order(PosReal,>,Poly) ... failed.
Freezing app
1: app❆3_if(true(),xs,ys) -> xs
2: app❆3_if(false(),xs,ys) -> ys
3: app❆2_sub(x,0()) -> x
4: app❆2_sub(app❆1_s(x),app❆1_s(y)) -> app❆2_sub(x,y)
5: app❆2_gtr(0(),y) -> false()
6: app❆2_gtr(app❆1_s(x),0()) -> true()
7: app❆2_gtr(app❆1_s(x),app❆1_s(y)) -> app❆2_gtr(x,y)
8: app❆2_d(x,0()) -> true()
9: app❆2_d(app❆1_s(x),app❆1_s(y)) -> app❆3_if(app❆2_gtr(x,y),false(),app❆2_d(app❆1_s(x),app❆2_sub(y,x)))
10: app❆1_len(nil()) -> 0()
11: app❆1_len(app❆2_cons(x,xs)) -> app❆1_s(app❆1_len(xs))
12: app❆2_filter(p,nil()) -> nil()
13: app❆2_filter(p,app❆2_cons(x,xs)) -> app❆3_if(app(p,x),app❆2_cons(x,app❆2_filter(p,xs)),app❆2_filter(p,xs))
14: app(if(),_1) ->= app❆1_if(_1)
15: app(app❆1_if(_1),_2) ->= app❆2_if(_1,_2)
16: app(app❆2_if(_1,_2),_3) ->= app❆3_if(_1,_2,_3)
17: app(cons(),_1) ->= app❆1_cons(_1)
18: app(app❆1_cons(_1),_2) ->= app❆2_cons(_1,_2)
19: app(d(),_1) ->= app❆1_d(_1)
20: app(app❆1_d(_1),_2) ->= app❆2_d(_1,_2)
21: app(s(),_1) ->= app❆1_s(_1)
22: app(filter(),_1) ->= app❆1_filter(_1)
23: app(app❆1_filter(_1),_2) ->= app❆2_filter(_1,_2)
24: app(gtr(),_1) ->= app❆1_gtr(_1)
25: app(app❆1_gtr(_1),_2) ->= app❆2_gtr(_1,_2)
26: app(len(),_1) ->= app❆1_len(_1)
27: app(sub(),_1) ->= app❆1_sub(_1)
28: app(app❆1_sub(_1),_2) ->= app❆2_sub(_1,_2)
Number of strict rules: 13
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #app❆2_filter(p,app❆2_cons(x,xs)) -> #app❆3_if(app(p,x),app❆2_cons(x,app❆2_filter(p,xs)),app❆2_filter(p,xs))
   #2: #app❆2_filter(p,app❆2_cons(x,xs)) -> #app(p,x)
   #3: #app❆2_filter(p,app❆2_cons(x,xs)) -> #app❆2_filter(p,xs)
   #4: #app❆2_filter(p,app❆2_cons(x,xs)) -> #app❆2_filter(p,xs)
   #5: #app❆2_d(app❆1_s(x),app❆1_s(y)) -> #app❆3_if(app❆2_gtr(x,y),false(),app❆2_d(app❆1_s(x),app❆2_sub(y,x)))
   #6: #app❆2_d(app❆1_s(x),app❆1_s(y)) -> #app❆2_gtr(x,y)
   #7: #app❆2_d(app❆1_s(x),app❆1_s(y)) -> #app❆2_d(app❆1_s(x),app❆2_sub(y,x))
   #8: #app❆2_d(app❆1_s(x),app❆1_s(y)) -> #app❆2_sub(y,x)
   #9: #app❆1_len(app❆2_cons(x,xs)) -> #app❆1_len(xs)
   #10: #app(app❆1_filter(_1),_2) ->? #app❆2_filter(_1,_2)
   #11: #app(app❆1_gtr(_1),_2) ->? #app❆2_gtr(_1,_2)
   #12: #app(app❆1_d(_1),_2) ->? #app❆2_d(_1,_2)
   #13: #app❆2_gtr(app❆1_s(x),app❆1_s(y)) -> #app❆2_gtr(x,y)
   #14: #app(app❆1_sub(_1),_2) ->? #app❆2_sub(_1,_2)
   #15: #app(len(),_1) ->? #app❆1_len(_1)
   #16: #app(app❆2_if(_1,_2),_3) ->? #app❆3_if(_1,_2,_3)
   #17: #app❆2_sub(app❆1_s(x),app❆1_s(y)) -> #app❆2_sub(x,y)
Number of SCCs: 5, DPs: 8, edges: 14
	SCC { #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     d()	weight: 0
     s()	weight: 0
app❆1_gtr(x1)	weight: 0
app❆1_if(x1)	weight: 0
app❆1_d(x1)	weight: 0
#app❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
app❆1_sub(x1)	weight: 0
app❆2_if(x1,x2)	weight: 0
   gtr()	weight: 0
   sub()	weight: 0
#app❆1_len(x1)	weight: x1
  true()	weight: 0
app❆2_filter(x1,x2)	weight: 0
#app❆2_d(x1,x2)	weight: 0
    if()	weight: 0
     0()	weight: 0
   nil()	weight: 0
app❆2_gtr(x1,x2)	weight: 0
app❆1_filter(x1)	weight: 0
app❆3_if(x1,x2,x3)	weight: 0
 #app(x1,x2)	weight: 0
#app❆2_gtr(x1,x2)	weight: 0
#app❆2_filter(x1,x2)	weight: 0
app❆1_len(x1)	weight: 0
app❆2_d(x1,x2)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2) + x2
app❆2_sub(x1,x2)	weight: 0
filter()	weight: 0
app❆1_s(x1)	weight: 0
   len()	weight: 0
app❆1_cons(x1)	weight: 0
#app❆2_sub(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #9
Number of SCCs: 4, DPs: 7, edges: 13
	SCC { #17 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     d()	weight: 0
     s()	weight: 0
app❆1_gtr(x1)	weight: 0
app❆1_if(x1)	weight: 0
app❆1_d(x1)	weight: 0
#app❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
app❆1_sub(x1)	weight: 0
app❆2_if(x1,x2)	weight: 0
   gtr()	weight: 0
   sub()	weight: 0
#app❆1_len(x1)	weight: 0
  true()	weight: 0
app❆2_filter(x1,x2)	weight: 0
#app❆2_d(x1,x2)	weight: 0
    if()	weight: 0
     0()	weight: 0
   nil()	weight: 0
app❆2_gtr(x1,x2)	weight: 0
app❆1_filter(x1)	weight: 0
app❆3_if(x1,x2,x3)	weight: 0
 #app(x1,x2)	weight: 0
#app❆2_gtr(x1,x2)	weight: 0
#app❆2_filter(x1,x2)	weight: 0
app❆1_len(x1)	weight: 0
app❆2_d(x1,x2)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2)
app❆2_sub(x1,x2)	weight: 0
filter()	weight: 0
app❆1_s(x1)	weight: (/ 1 2) + x1
   len()	weight: 0
app❆1_cons(x1)	weight: 0
#app❆2_sub(x1,x2)	weight: x2
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #17
Number of SCCs: 3, DPs: 6, edges: 12
	SCC { #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     d()	weight: 0
     s()	weight: 0
app❆1_gtr(x1)	weight: 0
app❆1_if(x1)	weight: 0
app❆1_d(x1)	weight: 0
#app❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
app❆1_sub(x1)	weight: 0
app❆2_if(x1,x2)	weight: 0
   gtr()	weight: 0
   sub()	weight: 0
#app❆1_len(x1)	weight: 0
  true()	weight: 0
app❆2_filter(x1,x2)	weight: 0
#app❆2_d(x1,x2)	weight: 0
    if()	weight: 0
     0()	weight: 0
   nil()	weight: 0
app❆2_gtr(x1,x2)	weight: 0
app❆1_filter(x1)	weight: 0
app❆3_if(x1,x2,x3)	weight: 0
 #app(x1,x2)	weight: 0
#app❆2_gtr(x1,x2)	weight: x2
#app❆2_filter(x1,x2)	weight: 0
app❆1_len(x1)	weight: 0
app❆2_d(x1,x2)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2)
app❆2_sub(x1,x2)	weight: 0
filter()	weight: 0
app❆1_s(x1)	weight: (/ 1 2) + x1
   len()	weight: 0
app❆1_cons(x1)	weight: 0
#app❆2_sub(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #13
Number of SCCs: 2, DPs: 5, edges: 11
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     d()	weight: 0
     s()	weight: 0
app❆1_gtr(x1)	weight: 0
app❆1_if(x1)	weight: 0
app❆1_d(x1)	weight: 0
#app❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
app❆1_sub(x1)	weight: 0
app❆2_if(x1,x2)	weight: 0
   gtr()	weight: 0
   sub()	weight: 0
#app❆1_len(x1)	weight: 0
  true()	weight: 0
app❆2_filter(x1,x2)	weight: 0
#app❆2_d(x1,x2)	weight: x2
    if()	weight: 0
     0()	weight: 0
   nil()	weight: 0
app❆2_gtr(x1,x2)	weight: 0
app❆1_filter(x1)	weight: 0
app❆3_if(x1,x2,x3)	weight: 0
 #app(x1,x2)	weight: 0
#app❆2_gtr(x1,x2)	weight: 0
#app❆2_filter(x1,x2)	weight: 0
app❆1_len(x1)	weight: 0
app❆2_d(x1,x2)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 4)
app❆2_sub(x1,x2)	weight: (/ 1 4) + x1
filter()	weight: 0
app❆1_s(x1)	weight: (/ 1 2) + x1
   len()	weight: 0
app❆1_cons(x1)	weight: 0
#app❆2_sub(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { 3 4 }
    Removed DPs: #7
Number of SCCs: 1, DPs: 4, edges: 10
	SCC { #2..4 #10 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     d()	weight: 0
     s()	weight: 0
app❆1_gtr(x1)	weight: 0
app❆1_if(x1)	weight: 0
app❆1_d(x1)	weight: 0
#app❆3_if(x1,x2,x3)	weight: 0
 false()	weight: 0
app❆1_sub(x1)	weight: 0
app❆2_if(x1,x2)	weight: 0
   gtr()	weight: 0
   sub()	weight: 0
#app❆1_len(x1)	weight: 0
  true()	weight: 0
app❆2_filter(x1,x2)	weight: 0
#app❆2_d(x1,x2)	weight: x2
    if()	weight: 0
     0()	weight: 0
   nil()	weight: 0
app❆2_gtr(x1,x2)	weight: 0
app❆1_filter(x1)	weight: (/ 1 4)
app❆3_if(x1,x2,x3)	weight: 0
 #app(x1,x2)	weight: (/ 1 4) + x2
#app❆2_gtr(x1,x2)	weight: 0
#app❆2_filter(x1,x2)	weight: x2
app❆1_len(x1)	weight: 0
app❆2_d(x1,x2)	weight: 0
  cons()	weight: 0
app❆2_cons(x1,x2)	weight: (/ 1 2) + x1 + x2
app❆2_sub(x1,x2)	weight: (/ 1 4) + x1
filter()	weight: 0
app❆1_s(x1)	weight: (/ 1 2) + x1
   len()	weight: 0
app❆1_cons(x1)	weight: 0
#app❆2_sub(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2..4 #10
Number of SCCs: 0, DPs: 0, edges: 0
YES