YES
We show the termination of the TRS R:

  is_empty(nil()) -> true()
  is_empty(cons(x,l)) -> false()
  hd(cons(x,l)) -> x
  tl(cons(x,l)) -> l
  append(l1,l2) -> ifappend(l1,l2,l1)
  ifappend(l1,l2,nil()) -> l2
  ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))


-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: append#(l1,l2) -> ifappend#(l1,l2,l1)
p2: ifappend#(l1,l2,cons(x,l)) -> append#(l,l2)

and R consists of:

r1: is_empty(nil()) -> true()
r2: is_empty(cons(x,l)) -> false()
r3: hd(cons(x,l)) -> x
r4: tl(cons(x,l)) -> l
r5: append(l1,l2) -> ifappend(l1,l2,l1)
r6: ifappend(l1,l2,nil()) -> l2
r7: ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: append#(l1,l2) -> ifappend#(l1,l2,l1)
p2: ifappend#(l1,l2,cons(x,l)) -> append#(l,l2)

and R consists of:

r1: is_empty(nil()) -> true()
r2: is_empty(cons(x,l)) -> false()
r3: hd(cons(x,l)) -> x
r4: tl(cons(x,l)) -> l
r5: append(l1,l2) -> ifappend(l1,l2,l1)
r6: ifappend(l1,l2,nil()) -> l2
r7: ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))

The set of usable rules consists of

  r1, r2, r3, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        append#_A(x1,x2) = max{3, x1 - 2}
        ifappend#_A(x1,x2,x3) = max{1, x3 - 3}
        cons_A(x1,x2) = max{17, x1 + 7, x2 + 2}
        is_empty_A(x1) = max{1, x1 - 8}
        nil_A = 1
        true_A = 0
        false_A = 8
        hd_A(x1) = x1
        tl_A(x1) = x1
  
    base order 2:
  
      precedence:
      
        append# = cons = is_empty = nil = hd = tl > ifappend# = true = false
      
  
    partial status:
  
      pi(append#) = []
      pi(ifappend#) = []
      pi(cons) = [2]
      pi(is_empty) = []
      pi(nil) = []
      pi(true) = []
      pi(false) = []
      pi(hd) = [1]
      pi(tl) = [1]

The following rules are removed:

  p1



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: ifappend#(l1,l2,cons(x,l)) -> append#(l,l2)

and R consists of:

r1: is_empty(nil()) -> true()
r2: is_empty(cons(x,l)) -> false()
r3: hd(cons(x,l)) -> x
r4: tl(cons(x,l)) -> l
r5: append(l1,l2) -> ifappend(l1,l2,l1)
r6: ifappend(l1,l2,nil()) -> l2
r7: ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2))

The estimated dependency graph contains the following SCCs:

  (no SCCs)