YES

Problem:
 del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
 f(true(),x,y,z) -> del(.(y,z))
 f(false(),x,y,z) -> .(x,del(.(y,z)))
 =(nil(),nil()) -> true()
 =(.(x,y),nil()) -> false()
 =(nil(),.(y,z)) -> false()
 =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))

Proof:
 DP Processor:
  DPs:
   del#(.(x,.(y,z))) -> =#(x,y)
   del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
   f#(true(),x,y,z) -> del#(.(y,z))
   f#(false(),x,y,z) -> del#(.(y,z))
   =#(.(x,y),.(u(),v())) -> =#(y,v())
   =#(.(x,y),.(u(),v())) -> =#(x,u())
  TRS:
   del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
   f(true(),x,y,z) -> del(.(y,z))
   f(false(),x,y,z) -> .(x,del(.(y,z)))
   =(nil(),nil()) -> true()
   =(.(x,y),nil()) -> false()
   =(nil(),.(y,z)) -> false()
   =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
  TDG Processor:
   DPs:
    del#(.(x,.(y,z))) -> =#(x,y)
    del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
    f#(true(),x,y,z) -> del#(.(y,z))
    f#(false(),x,y,z) -> del#(.(y,z))
    =#(.(x,y),.(u(),v())) -> =#(y,v())
    =#(.(x,y),.(u(),v())) -> =#(x,u())
   TRS:
    del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
    f(true(),x,y,z) -> del(.(y,z))
    f(false(),x,y,z) -> .(x,del(.(y,z)))
    =(nil(),nil()) -> true()
    =(.(x,y),nil()) -> false()
    =(nil(),.(y,z)) -> false()
    =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
   graph:
    f#(false(),x,y,z) -> del#(.(y,z)) ->
    del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
    f#(false(),x,y,z) -> del#(.(y,z)) ->
    del#(.(x,.(y,z))) -> =#(x,y)
    f#(true(),x,y,z) -> del#(.(y,z)) ->
    del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
    f#(true(),x,y,z) -> del#(.(y,z)) ->
    del#(.(x,.(y,z))) -> =#(x,y)
    =#(.(x,y),.(u(),v())) -> =#(y,v()) ->
    =#(.(x,y),.(u(),v())) -> =#(x,u())
    =#(.(x,y),.(u(),v())) -> =#(y,v()) ->
    =#(.(x,y),.(u(),v())) -> =#(y,v())
    =#(.(x,y),.(u(),v())) -> =#(x,u()) ->
    =#(.(x,y),.(u(),v())) -> =#(x,u())
    =#(.(x,y),.(u(),v())) -> =#(x,u()) ->
    =#(.(x,y),.(u(),v())) -> =#(y,v())
    del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z) ->
    f#(false(),x,y,z) -> del#(.(y,z))
    del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z) ->
    f#(true(),x,y,z) -> del#(.(y,z))
    del#(.(x,.(y,z))) -> =#(x,y) -> =#(.(x,y),.(u(),v())) -> =#(x,u())
    del#(.(x,.(y,z))) -> =#(x,y) -> =#(.(x,y),.(u(),v())) -> =#(y,v())
   EDG Processor:
    DPs:
     del#(.(x,.(y,z))) -> =#(x,y)
     del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
     f#(true(),x,y,z) -> del#(.(y,z))
     f#(false(),x,y,z) -> del#(.(y,z))
     =#(.(x,y),.(u(),v())) -> =#(y,v())
     =#(.(x,y),.(u(),v())) -> =#(x,u())
    TRS:
     del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
     f(true(),x,y,z) -> del(.(y,z))
     f(false(),x,y,z) -> .(x,del(.(y,z)))
     =(nil(),nil()) -> true()
     =(.(x,y),nil()) -> false()
     =(nil(),.(y,z)) -> false()
     =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
    graph:
     f#(false(),x,y,z) -> del#(.(y,z)) ->
     del#(.(x,.(y,z))) -> =#(x,y)
     f#(false(),x,y,z) -> del#(.(y,z)) ->
     del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
     f#(true(),x,y,z) -> del#(.(y,z)) ->
     del#(.(x,.(y,z))) -> =#(x,y)
     f#(true(),x,y,z) -> del#(.(y,z)) ->
     del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
     del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z) ->
     f#(true(),x,y,z) -> del#(.(y,z))
     del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z) ->
     f#(false(),x,y,z) -> del#(.(y,z))
     del#(.(x,.(y,z))) -> =#(x,y) -> =#(.(x,y),.(u(),v())) -> =#(y,v())
     del#(.(x,.(y,z))) -> =#(x,y) -> =#(.(x,y),.(u(),v())) -> =#(x,u())
    SCC Processor:
     #sccs: 1
     #rules: 3
     #arcs: 8/36
     DPs:
      f#(false(),x,y,z) -> del#(.(y,z))
      del#(.(x,.(y,z))) -> f#(=(x,y),x,y,z)
      f#(true(),x,y,z) -> del#(.(y,z))
     TRS:
      del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
      f(true(),x,y,z) -> del(.(y,z))
      f(false(),x,y,z) -> .(x,del(.(y,z)))
      =(nil(),nil()) -> true()
      =(.(x,y),nil()) -> false()
      =(nil(),.(y,z)) -> false()
      =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
     KBO Processor:
      argument filtering:
       pi(.) = [0,1]
       pi(del) = [0]
       pi(=) = []
       pi(f) = [0,1,2,3]
       pi(true) = []
       pi(false) = []
       pi(nil) = []
       pi(u) = []
       pi(v) = []
       pi(and) = 1
       pi(del#) = 0
       pi(f#) = [2,3]
      weight function:
       w0 = 1
       w(f#) = w(and) = w(v) = w(u) = w(nil) = w(false) = w(true) = w(
       f) = w(=) = w(.) = 1
       w(del#) = w(del) = 0
      precedence:
       f# ~ v ~ u ~ nil ~ del > f > = > del# ~ and ~ false ~ true ~ .
      problem:
       DPs:
        
       TRS:
        del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
        f(true(),x,y,z) -> del(.(y,z))
        f(false(),x,y,z) -> .(x,del(.(y,z)))
        =(nil(),nil()) -> true()
        =(.(x,y),nil()) -> false()
        =(nil(),.(y,z)) -> false()
        =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
      Qed