How to find a lambda term to complete a function? I tried to complete this exercise but i stopped...
Defining a $ \lambda $-term M such that:
$$(&lt;M,u&gt;)&lt;M,v&gt; \: \simeq_{\beta} \: &lt;M,u&gt;$$

I chose $M=\lambda m \lambda a \lambda b \lambda p \,((p)m)b \:$ then i have to find a representation T of a function using M that value true if the sequence is empty and false if it's not. A sequence is defined as:
$$[]=\lambda x_0\lambda x_1 \lambda z z \\
[b]=\lambda x_0 \lambda x_1 \lambda z (z) x_b\\ [b_1 b_2]=\lambda x_0 \lambda x_1 \lambda z ((z)x_{b_1})x_{b_2} \\
.\\.
\\ .
\\ [b_1 .. b_n]= \lambda x_0 \lambda x_1 \lambda z (...((z) x_{b_1})x_{b_2}...)x_{b_n} $$
so the sequence of exercise is :
$$[01101]= \lambda x_0 \lambda x_1 \lambda z (((((z)x_0)x_1)x_1)x_0)x_1 $$ 
For example T need to be:
$(T)[01101] \simeq_{\beta}$ false while $ (T) []\simeq_{\beta}$ true. 
I really find that difficult. How i can do that?

edit.

$true=λxλyx$
and $false=λxλyy$
We can remove the dependence on what the values stored in the sequences are by assuming that we pass "the same thing" as the first two arguments. Regardless of what $b_1, b_2, b_3$ are:

$$([b_1b_2b_3]\,x)\,x = \lambda f. ((f\,x)\,x)\,x$$

This gives you an effective way to tell the difference between an empty sequence and a non-empty sequence :

$$(([]\left&lt;M,u\right&gt;)\left&lt;M,u\right&gt;)\,(\lambda x. x) = \lambda x. x$$
$$(([b_1]\left&lt;M,u\right&gt;)\left&lt;M,u\right&gt;)\,(\lambda x. x) = \left&lt;M,u\right&gt;$$
$$(([b_1b_2]\left&lt;M,u\right&gt;)\left&lt;M,u\right&gt;)\,(\lambda x. x) = \left&lt;M,u\right&gt;$$
$$(([b_1b_2b_3]\left&lt;M,u\right&gt;)\left&lt;M,u\right&gt;)\,(\lambda x. x) = \left&lt;M,u\right&gt;$$

So if $s$ is a sequence, consider this:

$$((((s\left&lt;M,u\right&gt;) \left&lt;M,u\right&gt;) (\lambda x. x))\,\hbox{false}$$

What does it reduce to if $s$ is empty? What does it reduce to if it's non-empty?