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())
   SCC Processor:
    #sccs: 2
    #rules: 5
    #arcs: 12/36
    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()))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [f#](x0, x1, x2, x3) = x0 + 2x1 + 3x2 + 4x3 + 0,
      
      [del#](x0) = 2x0 + 0,
      
      [and](x0, x1) = 1x1 + 7,
      
      [v] = 5,
      
      [u] = 7,
      
      [nil] = 4,
      
      [false] = 4,
      
      [true] = 2,
      
      [f](x0, x1, x2, x3) = 2x0 + 4x1 + 5x2 + 6x3,
      
      [=](x0, x1) = x0 + x1,
      
      [del](x0) = 4x0 + 0,
      
      [.](x0, x1) = x0 + 1x1 + 0
     orientation:
      f#(false(),x,y,z) = 2x + 3y + 4z + 4 >= 2y + 3z + 2 = del#(.(y,z))
      
      del#(.(x,.(y,z))) = 2x + 3y + 4z + 3 >= 2x + 3y + 4z + 0 = f#(=(x,y),x,y,z)
      
      f#(true(),x,y,z) = 2x + 3y + 4z + 2 >= 2y + 3z + 2 = del#(.(y,z))
      
      del(.(x,.(y,z))) = 4x + 5y + 6z + 5 >= 4x + 5y + 6z = f(=(x,y),x,y,z)
      
      f(true(),x,y,z) = 4x + 5y + 6z + 4 >= 4y + 5z + 4 = del(.(y,z))
      
      f(false(),x,y,z) = 4x + 5y + 6z + 6 >= x + 5y + 6z + 5 = .(x,del(.(y,z)))
      
      =(nil(),nil()) = 4 >= 2 = true()
      
      =(.(x,y),nil()) = x + 1y + 4 >= 4 = false()
      
      =(nil(),.(y,z)) = y + 1z + 4 >= 4 = false()
      
      =(.(x,y),.(u(),v())) = x + 1y + 7 >= 1y + 7 = and(=(x,u()),=(y,v()))
     problem:
      DPs:
       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()))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [f#](x0, x1, x2, x3) = 1x0 + 5x3 + 0,
       
       [del#](x0) = 2x0 + 0,
       
       [and](x0, x1) = x0 + x1 + 0,
       
       [v] = 2,
       
       [u] = 3,
       
       [nil] = 1,
       
       [false] = 5,
       
       [true] = 4,
       
       [f](x0, x1, x2, x3) = x0 + 5x3 + 0,
       
       [=](x0, x1) = 5,
       
       [del](x0) = 1x0 + 0,
       
       [.](x0, x1) = 2x1 + 2
      orientation:
       del#(.(x,.(y,z))) = 6z + 6 >= 5z + 6 = f#(=(x,y),x,y,z)
       
       f#(true(),x,y,z) = 5z + 5 >= 4z + 4 = del#(.(y,z))
       
       del(.(x,.(y,z))) = 5z + 5 >= 5z + 5 = f(=(x,y),x,y,z)
       
       f(true(),x,y,z) = 5z + 4 >= 3z + 3 = del(.(y,z))
       
       f(false(),x,y,z) = 5z + 5 >= 5z + 5 = .(x,del(.(y,z)))
       
       =(nil(),nil()) = 5 >= 4 = true()
       
       =(.(x,y),nil()) = 5 >= 5 = false()
       
       =(nil(),.(y,z)) = 5 >= 5 = false()
       
       =(.(x,y),.(u(),v())) = 5 >= 5 = and(=(x,u()),=(y,v()))
      problem:
       DPs:
        del#(.(x,.(y,z))) -> f#(=(x,y),x,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()))
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 4/1
       
    
    DPs:
     =#(.(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()))
    Subterm Criterion Processor:
     simple projection:
      pi(=#) = 1
     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