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