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: $ $ (<M,u>)<M,v> \: \simeq_{\beta} \: <M,u>$ $

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?