Notation: I use - \x to denote lambda abstraction - \\d to denote "meta" lambda abstraction (1) Prove the correctness of Definition 3.2.5. To prove this, you have to show that the function \\d. [[P]]_{rho(x:=d)} is continuous for every P and rho. (2) At the lecture, have seen the interpretations in D_A of I (= \x.x) and K (= \x.\y.x). Compute the interpretation of: (a) \x.xx (b) (\x.\y.y)I (without doing a beta-reduction first) (3) Compute in D_A the interpretation of y and of \x.yx and conclude that the eta-rule does not hold in D_A. (The eta rule says that \x. M x = M if x not in FV(M).) (4) (a) Use the norm (Def 3.3.4) to show that (beta,b) is not an element of beta for all beta in D_a (b)(*) Show that the interpretation of Omega (= (\x.xx)(\x.xx) ) in D_A is the emptyset (5) Use (4b) to (c) Compute the interpretation of \y.Omega in D_A (d) Compute the interpretation of \y.y Omega in D_A