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()))
  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()))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [f#](x0, x1, x2, x3) = x0 + x2 + x3,
      
      [del#](x0) = x0,
      
      [and](x0, x1) = x0,
      
      [v] = 0,
      
      [u] = 0,
      
      [nil] = 1,
      
      [false] = 0,
      
      [true] = 1,
      
      [f](x0, x1, x2, x3) = x1 + x2 + x3 + 1,
      
      [=](x0, x1) = x0,
      
      [del](x0) = x0 + 1,
      
      [.](x0, x1) = x0 + x1
     orientation:
      f#(false(),x,y,z) = y + z >= y + z = del#(.(y,z))
      
      del#(.(x,.(y,z))) = x + y + z >= x + y + z = f#(=(x,y),x,y,z)
      
      f#(true(),x,y,z) = y + z + 1 >= y + z = del#(.(y,z))
      
      del(.(x,.(y,z))) = x + y + z + 1 >= x + y + z + 1 = f(=(x,y),x,y,z)
      
      f(true(),x,y,z) = x + y + z + 1 >= y + z + 1 = del(.(y,z))
      
      f(false(),x,y,z) = x + y + z + 1 >= x + y + z + 1 = .(x,del(.(y,z)))
      
      =(nil(),nil()) = 1 >= 1 = true()
      
      =(.(x,y),nil()) = x + y >= 0 = false()
      
      =(nil(),.(y,z)) = 1 >= 0 = false()
      
      =(.(x,y),.(u(),v())) = x + y >= x = and(=(x,u()),=(y,v()))
     problem:
      DPs:
       f#(false(),x,y,z) -> del#(.(y,z))
       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()))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [f#](x0, x1, x2, x3) = x3 + 1,
       
       [del#](x0) = x0,
       
       [and](x0, x1) = x1,
       
       [v] = 0,
       
       [u] = 0,
       
       [nil] = 0,
       
       [false] = 1,
       
       [true] = 0,
       
       [f](x0, x1, x2, x3) = x0 + x3 + 1,
       
       [=](x0, x1) = 1,
       
       [del](x0) = x0,
       
       [.](x0, x1) = x1 + 1
      orientation:
       f#(false(),x,y,z) = z + 1 >= z + 1 = del#(.(y,z))
       
       del#(.(x,.(y,z))) = z + 2 >= z + 1 = f#(=(x,y),x,y,z)
       
       del(.(x,.(y,z))) = z + 2 >= z + 2 = f(=(x,y),x,y,z)
       
       f(true(),x,y,z) = z + 1 >= z + 1 = del(.(y,z))
       
       f(false(),x,y,z) = z + 2 >= z + 2 = .(x,del(.(y,z)))
       
       =(nil(),nil()) = 1 >= 0 = true()
       
       =(.(x,y),nil()) = 1 >= 1 = false()
       
       =(nil(),.(y,z)) = 1 >= 1 = false()
       
       =(.(x,y),.(u(),v())) = 1 >= 1 = and(=(x,u()),=(y,v()))
      problem:
       DPs:
        f#(false(),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()))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1, x2, x3) = 1,
        
        [del#](x0) = 0,
        
        [and](x0, x1) = 0,
        
        [v] = 0,
        
        [u] = 0,
        
        [nil] = 0,
        
        [false] = 0,
        
        [true] = 0,
        
        [f](x0, x1, x2, x3) = 0,
        
        [=](x0, x1) = 0,
        
        [del](x0) = 0,
        
        [.](x0, x1) = 0
       orientation:
        f#(false(),x,y,z) = 1 >= 0 = del#(.(y,z))
        
        del(.(x,.(y,z))) = 0 >= 0 = f(=(x,y),x,y,z)
        
        f(true(),x,y,z) = 0 >= 0 = del(.(y,z))
        
        f(false(),x,y,z) = 0 >= 0 = .(x,del(.(y,z)))
        
        =(nil(),nil()) = 0 >= 0 = true()
        
        =(.(x,y),nil()) = 0 >= 0 = false()
        
        =(nil(),.(y,z)) = 0 >= 0 = false()
        
        =(.(x,y),.(u(),v())) = 0 >= 0 = and(=(x,u()),=(y,v()))
       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