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