 Solution

Add Fixes [Ident] [Expr] to the abstract syntax tree. For the stipulated nonempty, equally long lists of distinct names and lambda abstractions, define all programming closures against one common extended semantic environment:

eval (Fixes xs es) rho = result (head vs)
  where
    rho' = extend rho (zip xs vs)
    vs = map (\e -> close e rho') es

Expanding the auxiliary function, if $e_i=\mathrm{Lambda}y{s}_i{b}_i$,Every body can therefore call every member of the family, including itself. Constructing the family performs no body computations in the arbitrary computational monad. As in part (c), host lazy evaluation ties the recursive semantic environment; a monadic fixed-point operator is unnecessary.

For example, with two bodies implementing evenness and oddness, respectively, each programming closure sees both names in ${\rho }^{\ast }$. Returning the first programming closure still retains access to the second. Building each programming closure in an independently extended old semantic environment would lose this mutual recursion. For $n=1$, the definition agrees with Fix.