Functional programming in μScheme

COMP 105 Assignment

Due Tuesday, September 25, 2018 at 11:59PM

This assignment is all individual work. There is no pair programming.

Overview

This assignment has two purposes:

The assignment is based primarily on material from sections 2.1 to 2.6 of Programming Languages: Build, Prove, and Compare. You will also need to know the syntax in section 2.11, which starts on page 147, and the initial basis (also in section 2.11). The table on page 159 lists all the functions found in the basis—it is your lifeline. Finally, although it is not necessary, you may find some problems easier to solve if you read ahead from section 2.7 to section 2.9.

You will define many functions and write a few proofs. The functions are small; most are in the range of 4 to 8 lines, and none of my solutions is more than a dozen lines. If you don’t read ahead, a couple of your functions will be a bit longer, which is OK.

Setup

The executable μScheme interpreter is in /comp/105/bin/uscheme; once you have run use comp105 (or set it up to run automatically on login), you should be able to run uscheme as a command. The interpreter accepts a -q (“quiet”) option, which turns off prompting. When using the interpreter interactively, you may find it helpful to use ledit, as in the command

  ledit uscheme

Please also download our template for solution.scm. It contains a skeleton version of each function you must define, but the body of the function calls error. Each call to error should be replaced with a correct implementation.

Diagnostic tracing

μScheme does not ship with a debugger. But in addition to the println and printu functions, it does ship with a tracing facility. The tracing facility can show you the argument and results to every function call, or you can dial it back to show just a limited number.

The tracing facility is described in exercise 74 on page 237 of Build, Prove, and Compare. Our facility takes the approach sketched in part (b). Here are a couple of example calls for you to try:

-> (val &trace 5)
-> (append '(a b c) '(1 2 3))
-> (set &trace 500)
-> (append '(a b c) '(1 2 3))

Used carefully, &trace can save you a lot of time and effort. But do not leave even an unexecuted reference to &trace in your submission.

Dire Warnings

Since we are studying functional programming, the μScheme programs you submit must not use any imperative features. Banish set, while, println, print, printu, and begin from your vocabulary! If you break this rule for any problem, you will get No Credit for that problem. You may find it useful to use begin and println while debugging, but they must not appear in any code you submit. As a substitute for assignment, use let or let*.

Helper functions may be defined at top level only if they meet these criteria:

  • Each helper function has a meaningful name.

  • Each helper function is given an explicit contract—or, as described in the general coding rubric, we can infer the contract by looking at the names of the function and its formal parameters.

  • Each helper function is specified by algebraic laws.

  • Each helper function is tested by check-expect or check-assert, and possibly check-error.

As an alternative to helper functions, you may read ahead and define local functions using lambda along with let, letrec, or let*. If you do define local functions, avoid passing them redundant parameters—a local function already has access to the parameters and let-bound variables of its enclosing function.

Except as specified, functions without algebraic laws will earn failing grades.

Your solutions must be valid μScheme; in particular, they must pass the following test:

   /comp/105/bin/uscheme -q < myfilename

without any error messages or unit-test failures. If your file produces error messages, we won’t test your solution and you will earn No Credit for functional correctness. (You can still earn credit for structure and organization). If your file includes failing unit tests, you might possibly get some credit for functional correctness, but we cannot guarantee it.

Case analysis involving lists and S-expressions must be structural. That is, your case analysis must involve the results of functions like null?, atom?, pair?, and so on, all of which are found in the initial basis. Please note that the length function is not in the initial basis, and length may not be used in case analysis. In fact, you may use length for at most two problems on this assignment:

  • You may use length to test arg-max, but not to implement it.
  • You may pick any one other problem for which you are free to use length in both implementation and in testing.

It is not necessary to use length at all, but there is a problem for which it is convenient.

Code you submit must not even mention &trace.

We will evaluate functional correctness by automated testing. Because testing is automated, each function must be named be exactly as described in each question. Misnamed functions earn No Credit. You may wish to use the template provided above, which has the correct function names.

Reading Comprehension (10 percent)

These problems will help guide you through the reading. Complete them before starting the other problems below. You can download the questions.

  1. Read Section 2 of the handout “Programming With Scheme Values and Algebraic Laws”.

    You are tasked with writing a function that consumes a list of numbers:

    1. How many cases must you consider?

    2. To tell the cases apart, what condition or conditions will you use in if expressions? (List one fewer condition than cases.)

    You are tasked with writing a function that consumes an ordinary S-expression.

    1. How many cases must you consider?

    2. To tell the cases apart, what condition or conditions will you use in if expressions? (List one fewer condition than cases.)

    You are ready to write algebraic laws using Scheme data.

  2. Read Section 4 of the same handout (“Programming With Scheme Values and Algebraic Laws”). Understand the three faults that are described there, then answer these questions:

    1. Here is an algebraic law:

      (binary (* 2 m)) == (* 10 (binary (/ m 2)))

      Which of the three faults in the section, if any, manifest themselves in this algebraic law? (Answer “first”, “second,” “third”, or any combination thereof, including “none”.)

    2. Here is an algebraic law:

      (double-digit (+ (* 10 m) b) == 
         (+ (* (double-digit (/ n 10)) 100) (double-digit (mod n 10)))

      Which of the three faults in the section, if any, manifest themselves in this algebraic law?

    3. Here is an algebraic law:

      (population-count (* m 2)) == 
         (+ (population-count (mod m 10))
            (population-count (/ m 10))

      Which of the three faults in the section, if any, manifest themselves in this algebraic law?

    You are able to avoid these common faults.

  3. In the main textbook, review section 2.2 on values, S-expressions, and primitives, and say what is the value of each of the expressions below. If a run-time error would occur, please say so.

    (car '(a b 1 2))  
    (cdr '(a b 1 2))  
    (= 'a 'b)         

    Write your answers as S-expression literals, like '(a b c), #t, or 17.

    You are on your way to being ready for exercise F.

  4. In the main textbook, review the first few pages of section 2.3, through the end of section 2.3.2, and also section 2.3.5, which starts on page 103. Which of the following expressions evaluates to #t for every list of ordinary S-expressions xs?

    (=      (reverse (reverse xs)) xs)
    (equal? (reverse (reverse xs)) xs)
    1. Only the first
    2. Only the second
    3. Both the first and the second
    4. None
  5. Read section 2.3.2, then please explain in your own words the difference between simple-reverse and reverse.

    You are now mostly ready for exercise 35.

  6. Read about association lists in section 2.3.8, which starts on page 106. Given the definition

    (val mascots 
       '((Tufts Jumbo) (MIT Beaver) (Northeastern Husky) (BU Terrier)))

    Say what is the value of each of these expressions:

    (find 'Tufts   mascots)
    (find 'MIT     mascots)
    (find 'Harvard mascots)
    (find 'MIT (bind 'MIT 'Engineer mascots))
  7. Read Section 3 of the handout “Programming With Scheme Values and Algebraic Laws” and the first part of section 2.4 of the main textbook, up to and including section 2.4.4.

    Now complete the following law, which should represent a true property of the association-list functions find and bind:

    (find x (bind ...)) = ...

    You may use variables, and you may use forms of data made with '() and with cons. You may not use any atomic literals. Write your property in the style of section 2.4.4.

    You are now prepared for the algebraic laws in exercises A, B, and C.

  8. Read section 2.16.6, which starts on page 194. Imagine that μScheme is given the following definition:

    (record 3point (x y z))

    This definition puts five functions into the environment ρ. What are their names?

    You are now mostly ready for exercise E.

  9. Read section 2.5, which explains let, let*, and letrec. This question asks you to decide if any or all these forms can appropriately express the following function (written in C):

    bool parity(int m) {
      int half_m     = m / 2;
      int other_half = m - half_m;
      return half_m == other_half;
    }      

    Scheme does not have local variables, so to translate this function into μScheme, you must use let, let*, or letrec. For each of these syntactic forms, we ask you if a translation sensibly and faithfully captures the intent and behavior of the original C function.

    ;; Translation A
    (define parity (m)
       (let ([half_m     (/ m 2)]
             [other_half (- m half_m)])
         (= half_m other_half)))

    Is translation A sensible and faithful (yes or no)?

    ;; Translation B
    (define parity (m)
       (let* ([half_m     (/ m 2)]
              [other_half (- m half_m)])
         (= half_m other_half)))

    Is translation B sensible and faithful (yes or no)?

    ;; Translation C
    (define parity (m)
       (letrec ([half_m     (/ m 2)]
                [other_half (- m half_m)])
         (= half_m other_half)))

    Is translation C sensible and faithful (yes or no)?

    You are now ready for exercise 30.

Programming and Proof Problems (90 percent)

For the “programming and proof” part of this assignment, you will do exercises 1, 2, 10, 30, and 35 in the book, plus the problems A through H and N below—but not in that order. There is also one extra-credit problem: problem M.

Problem Details (Theory)

1. A list of S-expressions is an S-expression. Do exercise 1 on page 207 of Build, Prove, and Compare. Do this proof before tackling exercise 2; the proof should give you ideas about how to implement the code.

Related Reading: The definitions of LIST (A) and SEXPFG are on page 116.

35. Calculational proof. Do exercise 35 on page 221 of Build, Prove, and Compare, proving that reversing a list of values does not change its length.

Hint: structural induction.

Related Reading: section 2.4.5, which starts on page 110

A. From operational semantics to algebraic laws. This problem has two parts:

  1. The operational semantics for μScheme includes rules for cons, car, and cdr. Assuming that x and xs are variables and are defined in ρ (rho), use the operational semantics to prove that

    (cdr (cons x xs)) == xs
  2. The preceding law applies only to variables x and xs. In this part, you determine if a similar law applies to expressions.

    Use the operational semantics to prove or disprove the following conjecture: if e1 and e2 are arbitrary expressions, in any context where the evaluation of e1 terminates and the evaluation of e2 terminates, then both of the following are true:

    • The evaluation of (cdr (cons e1 e2)) terminates, and

    • (cdr (cons e1 e2)) == e2

    The conjecture says that two independent evaluations, starting from the same initial state, produce the same value as a result.

    If you believe the conjecture, you can establish it by proving that it’s true for every choice of e1, e2, in any context in which both e1 and e2 terminate. If you disbelieve the conjecture, you need only to find one choice of e1, e2, and context such that both e1 and e2 terminate but at least one of the desired conclusions does not hold.

Related Reading: The operational semantics for cons, car, and cdr can be found on page 157.

Our expectations for your code: Algebraic laws and unit tests

For each function you define, you must specify not only a contract but also algebraic laws and unit tests. Even helper functions! For some problems, algebraic laws are not needed or are already given to you. Those problems are noted below.

Laws and tests make it easy to write code and easy for readers to be confident that code is correct. To get your laws, code, and tests right, use the checklists below.

A checklist for your laws

A good set of algorithmic laws satisfies all these requirements:

A checklist for your code

Your laws will be evaluated not just in isolation but in the context of your code. (The whole purpose of laws is to help write code.) In particular, your laws must be consistent with your code.

A checklist for your tests

While it is often useful to write additional tests for corner cases, here is a checklist for our minimum expectations.

Problem Details (Code)

Related Reading: Many of the following problems ask you to write recursive functions on lists. You can sometimes emulate examples from section 2.3, which starts on page 98. And you will definitely want to take advantage of μScheme’s predefined and primitive functions (the initial basis). These functions are listed in section 2.13, which starts on page 158.

30. Let-binding. Do parts (a) and (b) of exercise 30 on page 221 of Build, Prove, and Compare. Answer each question in at most a few sentences. Put your answers in comments in file solution.scm.

In this problem, you don’t define any functions, so you don’t write any algebraic laws or any unit tests.

Related Reading: Information on let can be found in section 2.5 (pages 114–115).

2. Recursive functions on lists. Do parts a, b, e, and f of exercise 2 on page 207 of Build, Prove, and Compare (everything except mirror and flatten). Expect to write some recursive functions, but you may also read ahead and use the higher-order functions in sections 2.7 through 2.9.

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert, with this exception:

Hints:

10. Taking and dropping a prefix of a list. Do exercise 10 on page 212 of Build, Prove, and Compare.

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert.

B. Take and drop. Function (take n xs) expects a natural number and a list of values. It returns the longest prefix of xs that contains at most n elements.

Function (drop n xs) expects a natural number and a list of values. Roughly, it removes n elements from the front of the list. When acting together, take and drop have this property: for any list xs and natural number n,

  (append (take n xs) (drop n xs)) == xs

Implement take and drop.

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert. Be aware that the property above (the “append/take/drop” law) is not algorithmic. Therefore, it cannot be used as the sole guide to implementations of take and drop. Before defining take, you must write laws that define only what take does. And before defining drop, you must write more laws that define only what drop does.

C. Zip and unzip. Function zip converts a pair of lists to an association list; unzip converts an association list to a pair of lists. If zip is given lists of unequal length, its behavior is not specified.

  -> (zip '(1 2 3) '(a b c))
  ((1 a) (2 b) (3 c))
  -> (unzip '((I Magnin) (U Thant) (E Coli)))
  ((I U E) (Magnin Thant Coli))

Provided lists xs and ys are the same length, zip and unzip have these properties:

  (zip (car (unzip pairs)) (cadr (unzip pairs))) ==  pairs
  (unzip (zip xs ys))                            ==  (list2 xs ys)

Neither of these properties is algorithmic. You are excused from writing algebraic laws for unzip, but you must write additional algebraic laws for zip.

Implement zip and unzip.

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert, with this exception:

Related Reading: Information on association lists can be found in section 2.3.8, which starts on page 106.

N. Nonempty lists. Many useful functions operate on nonempty lists. A nonempty list of A’s is notated LIST1(A).1 The usual forms of data don’t work here: '() is not a nonempty list. In this problem, you define nonempty lists in two different ways:

  1. Define LIST1(A) in terms of LIST(A). You may use set notation, a proof system, or the style of “An informal alternative” in Section 2 of the handout “Programming With Scheme Values and Algebraic Laws”. This definition should not be inductive.

    If there are multiple cases in your definition, say what code you would write to distinguish the cases of a value xs in LIST1(A).

    Warning: a useful definition says what LIST1(A) is, not what it isn’t. Saying “a LIST1(A) is a LIST(A) that is not empty” is not useful. Your definition should be useful.

  2. Define LIST1(A) inductively, without any reference to LIST(A). You may use set notation, a proof system, or the style of “An informal alternative” in Section 2 of the handout “Programming With Scheme Values and Algebraic Laws”.

    If there are multiple cases in your definition, say what code you would write to distinguish the cases of a value xs in LIST1(A).

Both definitions are useful for writing code.

Place your definitions with your code, in file solution.scm.

D. Arg max. This problem gives you a taste of higher-order functions, which we’ll explore in more detail in the next homework assignment. Function arg-max expects two arguments: a function f that maps a value in set A to a number, and a nonempty list as of values in set A. It returns an element a in as for which (f a) is as large as possible. This function is commonly used in machine learning to predict the most likely outcome from a model

  -> (define square (a) (* a a))
  -> (arg-max square '(5 4 3 2 1))
  5
  -> (define invert (a) (/ 1000 a))
  -> (arg-max invert '(5 4 3 2 1))
  1

Implement arg-max. Be sure your implementation does not take exponential time.

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert.

Hint: Elements of the list as might not be numbers. Find a way to test your code with a list of non-numbers.

E. Rightmost point. Page 194 of the book defines a point record. Copy that definition into your code. Define a function rightmost-point that takes a nonempty list of point records and returns the one with the largest x coordinate. Break ties arbitrarily.

For this problem, you need not write any algebraic laws. Write unit tests as usual.

To earn full credit for this problem, define rightmost-point so that it does not call itself recursively.

F. Copy removal. Function (remove-one-copy sx sxs) expects an S-expression and a list of S-expressions. The list sxs contains one or more copies of sx. The function returns a new list which is like sxs except that one copy of sx is removed.

-> (remove-one-copy 'a '(a b c))
(b c)
-> (remove-one-copy 'a '(a a b b c c))
(a b b c c)
-> (remove-one-copy 'a '(x y z))        
Run-time error: removed-an-absent-item
-> (remove-one-copy '(b c) '((a b) (b c) (c d)))
((a b) (c d))

Implement remove-one-copy.

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert. In addition, you must write at least one unit test which verifies that in response to the contract violation mentioned above, remove-one-copy correctly signals a checked run-time error. For that test, use check-error.

G. Permutations. Lists xs and ys are permutations if and only if they have exactly the same elements—but possibly in different orders. Repeated elements must be accounted for. Write function permutation?, which tells if two lists of atoms are permutations.

-> (permutation? '(a b c) '(c b a))
#t
-> (permutation? '(a b b) '(a a b))     
#f
-> (permutation? '(a b c) '(c b a d))
#f

Each function you define, including helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert.

H. Splitting a list of values in two. Function split-list takes a list of values xs and splits it into two lists of nearly equal length. More precisely (split-list xs) returns a two-element list (cons ys (cons zs '()) such that these properties hold:

You have a lot of freedom to choose how you want to split the xs. Here are a couple of examples:

-> (split-list '())   
(() ())
-> (split-list '(a b))
((b) (a))   ;; ((a) (b)) would be equally good here

Define split-list.

Each recursive function you define, including helper functions, must be accompanied by algebraic laws. Your algebraic laws for split-list are likely to be more specific than the problem definition—they describe not the problem as a whole, but your particular implementation.

Each top-level function, whether recursive or not, must be accompanied by unit tests written using check-expect or check-assert. (Inner functions defined with letrec can have algebraic laws, but they cannot be unit tested.) If you have a working version of permutation?, it is acceptable for your test cases to call it.

Extra credit: Merge sort

M. Merge and Merge sort. For extra credit, implement merge sort Begin with function merge, which expects two lists of numbers sorted in increasing order and returns a single list sorted in increasing order containing exactly the same elements as the two argument lists together:

  -> (merge '(1 2 3) '(4 5 6))
  (1 2 3 4 5 6)
  -> (merge '(1 3 5) '(2 4 6))
  (1 2 3 4 5 6)

Function merge, plus any helper functions, must be accompanied by algebraic laws and by unit tests written using check-expect or check-assert.

Now use split-list and merge to define a recursive function merge-sort, which is given a list of numbers and returns a sorted version of that list, in increasing order. You need not write any algebraic laws for merge-sort.

What and how to submit

Please submit four files:

As soon as you have the files listed above, run submit105-scheme to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.

How your work will be evaluated

Programming in μScheme

The criteria we will use to assess the structure and organization of your μScheme code, which are described in detail below, are mostly the same as the criteria in the general coding rubric, which we used to assess your Impcore code. But some additional criteria appear below.

Laws must be well formed and algorithmic

Exemplary Satisfactory Must improve
Laws

• When defining function f, each left-hand side applies f to one or more patterns, where a pattern is a form of input (examples: (+ m 1), (cons x xs)).

• When a law applies only to equal inputs, those inputs are notated with the same letter.

• The left-hand side of each algebraic law applies the function being defined.

• On the left-hand side of each algebraic law, the number and types of arguments in the law are the same as the number and types of arguments in the code.

• The only variables used on the right-hand side of each law are those that appear in arguments on the left-hand side.

• When a variable on a left-hand side is part of a form-of-data argument, that variable is used on the right-hand side as a part of the argument.

• For every permissible form of the function’s input or inputs, there is an algebraic law with a matching left-hand side (and a matching side condition, if any).

• The patterns of the left-hand sides of laws defining function f are all mutually exclusive, or

• The patterns of the left-hand sides of laws defining function f are either mutually exclusive or are distinguished with side conditions written on the right-hand side.

• On a left-hand side, f is applied to a form of input, but the form of input is written in a way that is not consistent with code.

• When a law applies only to equal inputs, the equality is written as a side condition.

• Once or twice in an assignment, a variable appears on the right-hand side of a law without also appearing on the left-hand side. The variable appears to name an argument.

• Once or twice, a variable on a left-hand side is part of a form-of-data argument, but on the right-hand side, it is used as if it were the whole argument.

• For every permissible form of the function’s input or inputs, there is an algebraic law with a matching left-hand side, but some inputs might inadvertently be excluded by side conditions that are too restrictive.

• Laws are distinguished by side conditions, but the side conditions appear on the left-hand side.

• There are some inputs that match more than one left-hand side, and these inputs are not distinguished by side conditions, but the laws contain a note that the ambiguity is intentional, and for such inputs, the right-hand sides all specify the same result.

• One or more left-hand sides contain laws that are not applications of f.

• On a left-hand side, f is applied to something that is not a form of input, like an arbitrary sum (+ j k) or an append.

• The left-hand side of an algebraic law applies some function other than the one being defined.

• The left-hand side of an algebraic law the function being defined to the wrong number of arguments, or to arguments of the wrong types.

• The right-hand side of a law refers to a variable that is not part of the left-hand side and which appears not to refer to an argument.

• The assignment shows a pattern of using argument variables on right-hand sides, instead of or in addition to the variables that appear on left-hand sides.

• The assignment shows a pattern of using part-of-data variables as if they were whole arguments.

• There is permissible input whose form is not matched by the left-hand side of any algebraic law.

• There is at least one input to which it is ambiguous which law should apply: the input matches more than one left-hand side, and either there are no side conditions, or the side conditions are insufficient to distinguish the ambiguous laws. And there is no note explaining that the ambiguity is intentional and OK.

Code must be well structured

We’re looking for functional programs that use Boolean and name bindings idiomatically. Case analysis must be kept to a minimum.

Exemplary Satisfactory Must improve
Structure

• The assignment does not use set, while, print, or begin.

• Wherever Booleans are called for, code uses Boolean values #t and #f.

• The code has as little case analysis as possible (i.e., the course staff can see no simple way to eliminate any case analysis)

• When possible, inner functions use the parameters and let-bound names of outer functions directly.

• The code contains case analysis that the course staff can see follows from the structure of the data, but that could be simplified away by applying equational reasoning.

• An inner function is passed, as a parameter, the value of a parameter or let-bound variable of an outer function, which it could have accessed directly.

• Some code uses set, while, print, or begin (No Credit).

• Code uses integers, like 0 or 1, where Booleans are called for.

• The code contains superfluous case analysis that is not mandated by the structure of the data.

Code must be well laid out, with attention to vertical space

In addition to following the layout rules in the general coding rubric (80 columns, no offside violations), we expect you to use vertical space wisely.

Exemplary Satisfactory Must improve
Form

• Code is laid out in a way that makes good use of scarce vertical space. Blank lines are used judiciously to break large blocks of code into groups, each of which can be understood as a unit.

• Code has a few too many blank lines.

• Code needs a few more blank lines to break big blocks into smaller chunks that course staff can more easily understand.

• Code wastes scarce vertical space with too many blank lines, block or line comments, or syntactic markers carrying no information.

• Code preserves vertical space too aggressively, using so few blank lines that a reader suffers from a “wall of text” effect.

• Code preserves vertical space too aggressively by crowding multiple expressions onto a line using some kind of greedy algorithm, as opposed to a layout that communicates the syntactic structure of the code.

• In some parts of code, every single line of code is separated form its neighbor by a blank line, throwing away half of the vertical space (serious fault).

Code must load without errors

Ideally you want to pass all of our correctness tests, but at minimum, your own code must load and pass its own unit tests.

Exemplary Satisfactory Must improve
Correctness

• Your μScheme code loads without errors.

• Your code passes all the tests we can devise.

Or, your code passes all tests but one.

• Your code fails a few of our stringent tests.

• Loading your μScheme into uscheme causes an error message (No Credit).

• Your code fails many tests.

Costs of list tests must be appropriate

Be sure you can identify a nonempty list in constant time.

Exemplary Satisfactory Must improve
Cost

• Empty lists are distinguished from non-empty lists in constant time.

• Distinguishing an empty list from a non-empty list might take longer than constant time.

Explaining let

Here is what we expect from your explanation of the strange let in exercise 30.

Exemplary Satisfactory Must improve
Let

• Your explanation of the strange let code is accurate and appeals to the relevant semantic rules by name. The meanings of the rules are explained informally.

• Your explanation of the strange let code is accurate and appeals to the relevant semantic rules by name, but it does not explain the rules.

• Your explanation of the strange let code does not identify which rules of the μScheme semantics must be used to explain the code.

Your proofs

The proofs for this homework are different from the derivations and metatheoretic proofs from the operational-semantics homework, and different criteria apply.

Exemplary Satisfactory Must improve
Proofs

• Course staff find proofs short, clear, and convincing.

• Proofs have exactly as much case analysis as is needed (which could mean no case analysis)

• Proofs by induction explicitly say what data is inducted over and clearly identify the induction hypothesis.

• Each calculational proof is laid out as shown in the textbook, with each term on one line, and every equals sign between two terms has a comment that explains why the two terms are equal.

• Course staff find a proof clear and convincing, but a bit long.

Or, course staff have to work a bit too hard to understand a proof.

• A proof has a case analysis which is complete but could be eliminated.

• A proof by induction doesn’t say explicitly what data is inducted over, but course staff can figure it out.

• A proof by induction is not explicit about what the induction hypothesis is, but course staff can figure it out.

• Each calculational proof is laid out as shown in the textbook, with each term on one line, and most of the the equals signs between terms have comments that explain why the two terms are equal.

• Course staff don’t understand a proof or aren’t convinced by it.

• A proof has an incomplete case analysis: not all cases are covered.

• In a proof by induction, course staff cannot figure out what data is inducted over.

• In a proof by induction, course staff cannot figure out what the induction hypothesis is.

• A calculational proof is laid out correctly, but few of the equalities are explained.

• A calculational proof is called for, but course staff cannot recognize its structure as being the same structure shown in the book.


  1. Because it has at least one element.