Input TRS:
    1: a(h(),h(),h(),x) -> s(x)
    2: a(l,x,s(y),h()) -> a(l,x,y,s(h()))
    3: a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z))
    4: a(l,s(x),h(),z) -> a(l,x,z,z)
    5: a(s(l),h(),h(),z) -> a(l,z,h(),z)
    6: +(x,h()) -> x
    7: +(h(),x) -> x
    8: +(s(x),s(y)) -> s(s(+(x,y)))
    9: +(+(x,y),z) -> +(x,+(y,z))
    10: s(h()) -> 1()
    11: app(nil(),k) -> k
    12: app(l,nil()) -> l
    13: app(cons(x,l),k) -> cons(x,app(l,k))
    14: sum(cons(x,nil())) -> cons(x,nil())
    15: sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h(),h()),l))
Number of strict rules: 15
Direct Order(PosReal,>,Poly) ... failed.
Freezing sum
1: a(h(),h(),h(),x) -> s(x)
2: a(l,x,s(y),h()) -> a(l,x,y,s(h()))
3: a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z))
4: a(l,s(x),h(),z) -> a(l,x,z,z)
5: a(s(l),h(),h(),z) -> a(l,z,h(),z)
6: +(x,h()) -> x
7: +(h(),x) -> x
8: +(s(x),s(y)) -> s(s(+(x,y)))
9: +(+(x,y),z) -> +(x,+(y,z))
10: s(h()) -> 1()
11: app(nil(),k) -> k
12: app(l,nil()) -> l
13: app(cons(x,l),k) -> cons(x,app(l,k))
14: sum❆1_cons(x,nil()) -> cons(x,nil())
15: sum❆1_cons(x,cons(y,l)) -> sum❆1_cons(a(x,y,h(),h()),l)
16: sum(cons(_1,_2)) ->= sum❆1_cons(_1,_2)
Number of strict rules: 15
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #a(l,x,s(y),h()) -> #a(l,x,y,s(h()))
   #2: #a(l,x,s(y),h()) -> #s(h())
   #3: #app(cons(x,l),k) -> #app(l,k)
   #4: #+(+(x,y),z) -> #+(x,+(y,z))
   #5: #+(+(x,y),z) -> #+(y,z)
   #6: #a(s(l),h(),h(),z) -> #a(l,z,h(),z)
   #7: #sum(cons(_1,_2)) ->? #sum❆1_cons(_1,_2)
   #8: #a(l,x,s(y),s(z)) -> #a(l,x,y,a(l,x,s(y),z))
   #9: #a(l,x,s(y),s(z)) -> #a(l,x,s(y),z)
   #10: #a(h(),h(),h(),x) -> #s(x)
   #11: #+(s(x),s(y)) -> #s(s(+(x,y)))
   #12: #+(s(x),s(y)) -> #s(+(x,y))
   #13: #+(s(x),s(y)) -> #+(x,y)
   #14: #sum❆1_cons(x,cons(y,l)) -> #sum❆1_cons(a(x,y,h(),h()),l)
   #15: #sum❆1_cons(x,cons(y,l)) -> #a(x,y,h(),h())
   #16: #a(l,s(x),h(),z) -> #a(l,x,z,z)
Number of SCCs: 4, DPs: 10, edges: 29
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: 0
     1()	weight: 0
     s(x1)	weight: 0
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: x1
  cons(x1,x2)	weight: (/ 1 2) + x2
   #a(x1,x2,x3,x4)	weight: 0
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #3
Number of SCCs: 3, DPs: 9, edges: 28
	SCC { #14 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: (/ 1 2) + x2 + x3 + x4
     1()	weight: 0
     s(x1)	weight: (/ 1 4) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: x2
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 4) + x1 + x2
   #a(x1,x2,x3,x4)	weight: 0
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #14
Number of SCCs: 2, DPs: 8, edges: 27
	SCC { #4 #5 #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: (/ 1 2) + x2 + x3 + x4
     1()	weight: 0
     s(x1)	weight: (/ 1 4) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 4)
   #a(x1,x2,x3,x4)	weight: 0
     +(x1,x2)	weight: (/ 1 4) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: x1 + x2
   app(x1,x2)	weight: 0
    Usable rules: { 6..10 }
    Removed DPs: #5 #13
Number of SCCs: 2, DPs: 6, edges: 19
	SCC { #4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: (/ 1 2) + x2 + x3 + x4
     1()	weight: 0
     s(x1)	weight: (/ 1 4) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 4)
   #a(x1,x2,x3,x4)	weight: 0
     +(x1,x2)	weight: (/ 1 4) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: x1
   app(x1,x2)	weight: 0
    Usable rules: { 10 }
    Removed DPs: #4
Number of SCCs: 1, DPs: 5, edges: 18
	SCC { #1 #6 #8 #9 #16 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: (/ 1 2) + x2 + x4
     1()	weight: 0
     s(x1)	weight: (/ 1 4) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 4)
   #a(x1,x2,x3,x4)	weight: x1
     +(x1,x2)	weight: (/ 1 4) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { 10 }
    Removed DPs: #6
Number of SCCs: 2, DPs: 4, edges: 13
	SCC { #1 #8 #9 #16 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: (/ 1 2) + x2 + x4
     1()	weight: 0
     s(x1)	weight: (/ 1 4) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 4)
   #a(x1,x2,x3,x4)	weight: x2
     +(x1,x2)	weight: (/ 1 4) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { 10 }
    Removed DPs: #16
Number of SCCs: 2, DPs: 3, edges: 7
	SCC { #1 #8 #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     h()	weight: 0
     a(x1,x2,x3,x4)	weight: (/ 1 4) + x1 + x2
     1()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 4)
   #a(x1,x2,x3,x4)	weight: x3
     +(x1,x2)	weight: (/ 1 4) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { 10 }
    Removed DPs: #1 #8
Number of SCCs: 2, DPs: 1, edges: 1
	SCC { #9 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
     h()	weight: (/ 1 4)
     a(x1,x2,x3,x4)	weight: 0
     1()	weight: (/ 3 4)
     s(x1)	weight: (/ 1 4) + x1
sum❆1_cons(x1,x2)	weight: 0
#sum❆1_cons(x1,x2)	weight: 0
   sum(x1)	weight: 0
   #s(x1)	weight: 0
   nil()	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: 0
   #a(x1,x2,x3,x4)	weight: max{0, (/ 1 4) + x4}
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #9
Number of SCCs: 1, DPs: 0, edges: 0
YES