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