// Language 7 examples.
(fun f (lambda x (* x x)))  // f(x) = x^2
(apply f 5)                 // f(5)
(apply f 6)                 // f(6)
(apply f (apply f 4))       // f(f(4))
(var h f)                   // h is an alias for the function f
(print f)                   // x -> x*x
(print h)                   // x -> x*x


// Language 8 examples. 
// We can embed lambda expressions anywhere we want.
(apply (lambda x (* x x)) 5)  // apply a "function literal", "anonymous lambda expression"
(var f (lambda x (* x x)))    // f(x) = x^2 or, f = x -> x^2
(apply f 5)                   // f(5)
(var h (lambda y (+ 2 y)))    // h(y) = 2+y or, h = x -> 2+y
(print (lambda y (+ 2 y)))    // y -> 2+y


// Since we can embed lambda expressions anywhere we want,
// we can put lambda expressions inside of lambda expressions.
// Language 8 (but not Language7) allows nested functions.

(fun f (lambda x
       (+ (* x x) (* x x x)))

// A not too interesting use of nested functions.
(fun f (lambda x
         (begin
           (var g (lambda x (* x x)))    // nested (helper) function
           (var h (lambda x (* x x x)))  // nested (helper) function
           (+ (apply g x) (apply h x))))


// An important example of how we want to use nested functions.
(fun f (lambda x
         (begin
           (var g (lambda y (+ y x)))  // nested function return value (closure)!!!
           g))))

(var f2 (apply f 3))  // f2 increments by 3
(apply f2 5)          // returns 8