(1) Complete the proof of Proposition 3.1.7 and show the (slightly) more general result that a poset which has all suprema also has all infima, and vice versa. (2) Which of the following sets are complete lattices: (a) The set of flat natural numbers N_{bot} (b) The set of flat booleans with a top element added: B_{bot}^{top} (c) The set Omega (= N + {omega}, with the ordering we have seen before) (d) The set of monotone functions from B_{bot}^{top} to B_{bot}^{top} (3) Check the following property (that we have seen for cpo's) If D is a finite complete lattice, E a complete lattice, then f : D -> E is monotone iff f : D -> E is continuous (4) If we take A = IN (the natural numbers) and App = * (multiplication), this cannot be made into a model of the untyped lambda calculus. Prove this. Hint: assume that there is an interpretation [[-]] satisfying the definition that we have given (a) Show that [[\x.\y. xy]] = [[\x.\y. yx]] (b) Conclude that d = e for any element in IN So: contradicion. (5) Prove that the theory that equates all terms that don't have a normal form is inconsistent by showing that the following equation is inconsistent in untyped lambda calculus: \x y. x y Omega = \x y. y x Omega