Homework Three -- Sample Evaluation

; sub1sqrd: number --> number
; Given x, return x^2 - 2x +1.
;
(define add1-and-square
  (lambda (x)
    (+ (* x x) (* 2 x) 1)))

(+ (+ (add1-and-square (+ 3 4)) 5) (* 6 7))

  (+ (+ (add1-and-square (+ 3 4)) 5) (* 6 7))
                                     ^^^^^^^
= (+ (+ (add1-and-square (+ 3 4)) 5) 42)
                         ^^^^^^^
= (+ (+ (add1-and-square 7) 5) 42)
         ^^^^^^^^^^^^^^^
      ; Placeholder about to evaluate to it's defn:
= (+ (+ ((lambda (x) (+ (* x x) (* 2 x) 1)) 7) 5) 42)
        ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^        
 ; Eval a func-application: re-write the lambda's body, w/ 7 subbing for x:

= (+ (+ (+ (* 7 7) (* 2 7) 1) 5) 42)
                   ^^^^^^^
= (+ (+ (+ (* 7 7) 14 1) 5) 42)
           ^^^^^^^
= (+ (+ (+ 49 14 1) 5) 42)
        ^^^^^^^^^^^
= (+ (+ 64 5) 42)
     ^^^^^^^^
= (+ 69 42)
  ^^^^^^^^^
= 111

• Lots of repetition here -- you're encouraged to use a word processor to minimize re-typing.
• You should underline the part of each expression you're about to evaluate. (It's fine to use bold font, or underline by hand on the hardcopy, or whatever.)
• Underline exactly the part you're about to evaluate: in particular, know exactly which parentheses are part of the thing about to be evaluated, and which are just carried from one line to the next.
• Be careful not to skip steps because you can anticipate the answer. In this example, the computer doesn't add 42 and 5 directly. (The Law of Scheme is simple and general; it doesn't take into account the specific fact that addition is associative, as humans might.)
• The comments are added for your benefit. In your homework, you don't need to comment.
• If I'm trying to say that two things are equal, it's only grammatical to join them with "=". It would be points off, if you just had a bunch of expressions listed, with no claim that they were related:

  (+ 2 (* 4 5) 6)
       ^^^^^^^
  (+ 2 20 6)
  28
  
   Okay okay, if you look closely, even this skips some steps.
       For example, going from the first line of the answer to the second,
       we start with (* 6 7).
       What actually happens is that first 7
       is evaluated (to 7).  Then, 6 is evaluated (to 6);
       next the placeholder * is evaluated to its value
       (the primitive function multiplication;
       writing this abstract entity is
       difficult, but conventionally we write this with "*").
       All this, before we actually apply the primitive.
       For your hw (as in this example), you can skip the evaluation of values
       (like 6 and 7),
       and placeholders for primitives (*).