Variables:+ ==, <, >, <=, >=, != Each variable is an expression. For any expressions M and N and variable a, the following are also expressions: +`==` is just what we nonprogrammers normally express by `=`. It's a relation that holds or not between two values. Here we'll treat it as a function that takes two values as arguments and returns a **boolean** value, that is a truthvalue, as a result. The reason for using the doubled `=` symbol is that the single `=` symbol tends to get used in lots of different roles in programming, so we reserve `==` to express this meaning. I will deliberately try to minimize the uses of single `=` in this madeup language (but not eliminate it entirely), to reduce ambiguity and confusion. The `==` relationor as we're treating it here, the `==` *function* that returns a boolean valuecan at least take two numbers as arguments. Probably it makes sense for it to take other kinds of values as arguments, too. For example, it should operate on two truthvalues as well. Maybe we'd want it to operate on a number and a truthvalue, too? and always return false in that case? What about operating on two functions? Here we encounter the difficulty that the computer can't in general *decide* when two functions are equivalent. Let's not try to sort this all out just yet. We'll suppose that `==` can at least take two numbers as arguments, or two truthvalues. x
,y
,z
... 
Abstract: (λa M)

+As mentioned in class, we represent the truthvalues like this:
We'll tend to write (λa M)
as just `(\a M)`, so we don't have to write out the markup code for the λ
. You can yourself write (λa M)
or `(\a M)` or `(lambda a M)`.
+ 'true, 'false

Application: (M N)

+These are instances of a broader class of literal values that I called **symbolic atoms**. We'll return to them shortly. The reason we write them with an initial `'` will also be explained shortly. For now, it's enough to note that the expression:
Some authors reserve the term "term" for just variables and abstracts. We'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
+ 1 + 2 == 3
Examples of expressions:
+evaluates to `'true`, and the expression:
 x
 (y x)
 (x x)
 (\x y)
 (\x x)
 (\x (\y x))
 (x (\x x))
 ((\x (x x)) (\x (x x)))
+ 1 + 0 == 3
The lambda calculus has an associated proof theory. For now, we can regard the
proof theory as having just one rule, called the rule of **betareduction** or
"betacontraction". Suppose you have some expression of the form:
+evaluates to `'false`. Something else that evaluates to `'false` is the simple expression:
 ((\a M) N)
+ 'false
that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "betareducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.
+That is, literal values are a limiting case of expression, that evaluate to just themselves. More complex expressions like `1 + 0` don't evaluate to themselves, but rather down to literal values.
The rule of betareduction permits a transition from that expression to the following:
+The functions `succ` and `pred` come before their arguments, like this:
 M [a:=N]
+ succ 1
What this means is just `M`, with any *free occurrences* inside `M` of the variable `a` replaced with the term `N`.
+On the other hand, the functions `+`, `*`, ``, `==`, and so on come in between their arguments, like this:
What is a free occurrence?
+ x < y
> An occurrence of a variable `a` is **bound** in T if T has the form `(\a N)`.
+Functions of this latter sort are said to have an "infix" syntax. This is just a convenience for how we write them. Our language will have to keep rigorous track of which functions have infix syntax and which don't, but we'll just rely on context and our brains to make sense of this for now. Functions with the ordinary, noninfix syntax can take two arguments, as well. If we had defined the lessthan relation (boolean function) in that style, we'd write it like this instead:
> If T has the form `(M N)`, any occurrences of `a` that are bound in `M` are also bound in T, and so too any occurrences of `a` that are bound in `N`.
+ lessthan? (x, y)
> An occurrence of a variable is **free** if it's not bound.
+or perhaps like this:
For instance:
+ lessthan? x y
+We'll get more acquainted with the difference between these next week. For now, I'll just stick to the first form.
> T is defined to be `(x (\x (\y (x (y z)))))`
+Another set of operations we have are:
The first occurrence of `x` in T is free. The `\x` we won't regard as containing an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free.
+ and, or, not
Here's an example of betareduction:
+The first two of these are infix functions that expect two boolean arguments, and gives a boolean result. The third is a function that expects only one boolean argument. Our earlier function `!=` means "doesn't equal", and:
 ((\x (y x)) z)
+ x != y
betareduces to:
+will be just another way to write:
 (y z)
+ not (x == y)
We'll write that like this:
+You see that you can use parentheses in the standard way. By the way, `<=` means ≤ or "less than or equals to", and `>=` means ≥. Just in case you haven't seen them written this way before.
 ((\x (y x)) z) ~~> (y z)
+I've started throwing in some **variables**. We'll say variables are any expression that's written with an initial lowercase letter, then is followed by a sequence of zero or more upper or lowercase letters, or numerals, or underscores (`_`). Then at the end you can optionally have a `?` or `!` or a sequence of `'`s, understood as "primes." Hence, all of these are legal variables:
Different authors use different notations. Some authors use the term "contraction" for a single reduction step, and reserve the term "reduction" for the reflexive transitive closure of that, that is, for zero or more reduction steps. Informally, it seems easiest to us to say "reduction" for one or more reduction steps. So when we write:
+ x
+ x1
+ x_not_y
+ xUBERANT
+ x'
+ x''
+ x?
+ xs
 M ~~> N
+We'll follow a *convention* of using variables with short names and a final `s` to represent collections like sequences (to be discussed below). But this is just a convention to help us remember what we're up to, not a strict rule of the language. We'll also follow a convention of only using variables ending in `?` to represent functions that return a boolean value. Thus, for example, `zero?` will be a function that expects a single number argument and returns a boolean corresponding to whether that number is `0`. `odd?` will be a function that expects a single number argument and returns a boolean corresponding to whether than number is odd. Above, I suggested we might use `lessthan?` to represent a function that expects *two* number arguments, and again returns a boolean result.
We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol →
for onestep contraction, and the symbol ↠
for zeroormore step reduction. Hindley and Seldin use ⊳_{1}
and ⊳
.
+We also conventionally reserve variables ending in `!` for a different special class of functions, that we will explain later in the course.
When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **betaconvertible**. We'll write that like this:
+In fact you can think of `succ` and `pred` and `not` and the rest as also being variables; it's just that these variables have been predefined in our language to be bound to functions we agreed upon in advance. You can even think of `==` and `<` as being variables, too, bound to other functions. But I haven't given you parsing rules yet which would make them legal variables, because they don't start with a lowercase letter. We can make the parsing rules more liberal later.
 M <~~> N
+Only a few simple expressions in our language aren't variables. These include the literal values, and also **keywords** like `let` and `case` and so on that we'll discuss below. You can't use `let` as a variable, else the syntax of our language would become too hard to mechanically parse. (And probably too hard for our meager brains to parse, too.)
This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. Personally, I keep confusing that with the relation to be described next, so let's use this notation instead. Note that `M <~~> N` doesn't mean that each of `M` and `N` are reducible to each other; that only holds when `M` and `N` are the same expression. (Or, with our convention of only saying "reducible" for one or more reduction steps, it never holds.)
+The rule for symbolic atoms is that a single quote `'` followed by any single word that could be a legal variable expresses such an atom, a different atom for each different expression.
+Thus `'false` is a symbolic atom, but so too are `'x` and `'succ`. For the time being, I'll restrict myself to only talking about the symbolic atoms `'true` and `'false`. These constitute a special subclass of symbolic atoms that we call the **booleans** or truthvalues. Nothing deep hangs on them being a subclass of a larger type in this way; it just seems elegant. Some other languages make booleans their own special type, not a subclass of another type. Others make them a subclass of the numbers (yuck). We will think of them this way.
In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol ≡
for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing the metatheory. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in:
+Note that when writing a symbolic atom there is no closing `'`, just a `'` at the beginning. That's enough to make the whole word, up to the next space (or whatever) count as expressing a symbolic atom. We use the initial `'` to make it easy for us to have a rich set of symbolic atoms, as well as a rich set of variables, without getting them mixed up. Variables never begin with `'`; symbolic atoms always do.
> T is defined to be `(M N)`.
+We call these things symbolic *atoms* because they aren't collections. Thus numbers are also atoms, but not symbolic ones. And functions are also atoms, but again, not symbolic ones.
We'll regard the following two expressions:
+Functions are another class of values we'll have in our language. They aren't "literal" values, though. Numbers and symbolic atoms are simple expressions in the language that evaluate to themselves. That's what we mean by calling them "literals." Functions aren't expressions in the language at all; they have to be generated from the evaluation of more complex expressions.
 (\x (x y))
+(By the way, I really am serious in thinking of *the numbers themselves* as being expressions in this language; rather than some "numerals" that aren't themselves numbers. We'll talk about this down the road. For now, don't worry about it too much.)
 (\z (z y))
+I said we wanted to be starting with a fragment of arithmetic, so we'll keep the function values offstage for the moment, and also all the symbolic atoms except for `'true` and `'false`. So we've got numbers, truthvalues, and some functions and relations (that is, boolean functions) defined on them. We also help ourselves to a notion of bounded quantification, as in ∀`x < M.` φ, where `M` and φ are (simple or complex) expressions that evaluate to a number and a boolean, respectively. We limit ourselves to *bounded* quantification so that the fragment we're dealing with can be "effectively" or mechanically decided. (As we extend the language, we will lose that property, but it will be a topic for later discussion exactly when that happens.)
as syntactically equivalent, since they only involve a typographic change of a bound variable. Read Hankin section 2.3 for discussion of different attitudes one can take about this.
+As I mentioned in class, I will sometimes write ∀ x : ψ . φ in my informal metalanguage, where the ψ clause represents the quantifier's *restrictor*. Other people write this like `[`∀ x : ψ `]` φ, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim & Kratzer) use in writing λ x : ψ . φ, where the ψ clause restricts the range of arguments over which the function designated by the λexpression is defined. Later we will see the colon used in a somewhat similar (but also somewhat different) way in our programming languages. But that's foreshadowing.
Note that neither of those expressions are identical to:
 (\x (x w))
+### Let and lambda ###
because here it's a free variable that's been changed. Nor are they identical to:
+So we have bounded quantification as in ∀ `x < 10.` φ. Obviously we could also make sense of ∀ `x == 5.` φ in just the same way. This would evaluate φ but with the variable `x` now bound to the value `5`, ignoring whatever it may be bound to in broader contexts. I will express this idea in a more perspicuous vocabulary, like this: `let x be 5 in` φ. (I say `be` rather than `=` because, as I mentioned before, it's too easy for the `=` sign to get used for too many subtly different jobs.)
 (\y (y y))
+As one of you was quick to notice in class, when I shift to the `let`vocabulary, I no longer restrict myself to just the case where φ evaluates to a boolean. I also permit myself expressions like this:
because here the second occurrence of `y` is no longer free.
+ let x be 5 in x + 1
There is plenty of discussion of this, and the fine points of how substitution works, in Hankin and in various of the tutorials we've linked to about the lambda calculus. We expect you have a good intuitive understanding of what to do already, though, even if you're not able to articulate it rigorously.
+which evaluates to `6`. That's right. I am moving beyond the ∀ `x==5.` φ idea when I do this. But the rules for how to interpret this are just a straightforward generalization of our existing understanding for how to interpret bound variables. So there's nothing fundamentally novel here.
* MORE
+We can have multiple `let`expressions embedded, as in:
+ let y be (let x be 5 in x + 1) in 2 * y
Shorthand

+ let x be 5 in let y be x + 1 in 2 * y
The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)
+both of which evaluate to `12`. When we have a stack of `let`expressions as in the second example, I will write it like this:
+ let
+ x be 5;
+ y be x + 1
+ in 2 * y
**Parentheses** Outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:
+It's okay to also write it all inline, like so: `let x be 5; y be x + 1 in 2 * y`. The `;` represents that we have a couple of `let`bindings coming in sequence. The earlier bindings in the sequence are considered to be in effect for the later righthand expressions in the sequence. Thus in:
 (\x (x y))
+ let x be 0 in (let x be 5; y be x + 1 in 2 * y)
as:
+The `x + 1` that is evaluated to give the value that `y` gets bound to uses the (more local) binding of `x` to `5`, not the (previous, less local) binding of `x` to `0`. By the way, the parentheses in that displayed expression were just to focus your attention. It would have parsed and meant the same without them.
 \x (x y)
+Now we can allow ourselves to introduce λexpressions in the following way. If a λexpression is applied to an argument, as in: `(`λ `x.` φ`) M`, for any (simple or complex) expressions φ and `M`, this means the same as: `let x be M in` φ. That is, the argument to the λexpression provides (when evaluated) a value for the variable `x` to be bound to, and then the result of the whole thing is whatever φ evaluates to, under that binding to `x`.
but you should include the parentheses in:
+If we restricted ourselves to only that usage of λexpressions, that is when they were applied to all the arguments they're expecting, then we wouldn't have moved very far from the decidable fragment of arithmetic we began with.
 (\x (x y)) z
+However, it's tempting to help ourselves to the notion of (at least partly) *unapplied* λexpressions, too. If I can make sense of what:
and:
+`(`λ `x. x + 1) 5`
 z (\x (x y))
+means, then I can make sense of what:
+`(`λ `x. x + 1)`
**Dot notation** Dot means "put a left paren here, and put the right
paren as far the right as possible without creating unbalanced
parentheses". So:
+means, too. It's just *the function* that waits for an argument and then returns the result of `x + 1` with `x` bound to that argument.
 \x (\y (x y))
+This does take us beyond our (firstorder) fragment of arithmetic, at least if we allow the bodies and arguments of λexpressions to be any expressible value, including other λexpressions. But we're having too much fun, so why should we hold back?
can be abbreviated as:
+So now we have a new kind of value our language can work with, alongside numbers and booleans. We now have function values, too. We can bind these function values to variables just like other values:
 \x (\y. x y)
+`let id be` λ `x. x; y be id 5 in y`
and that as:
+evaluates to `5`. In reaching that result, the variable `id` was temporarily bound to the identity function, that expects an argument, binds it to the variable `x`, and then returns the result of evaluating `x` under that binding.
 \x. \y. x y
+This is what is going on, behind the scenes, with all the expressions like `succ` and `+` that I said could really be understood as variables. They have just been prebound to certain agreedupon functions rather than others.
This:
 \x. \y. (x y) x
+### Containers ###
abbreviates:
+So far, we've only been talking about *atomic* values. Our language will also have some *container* values, that have other values as members. One example are **ordered sequences**, like:
 \x (\y ((x y) x))
+ [10, 20, 30]
This on the other hand:
+This is a sequence of length 3. It's the result of *cons*ing the value `10` onto the front of the shorter, length2 sequence `[20, 30]`. In this madeup language, we'll represent the sequenceconsing operation like this:
 (\x. \y. (x y)) x
+ 10 & [20, 30]
abbreviates:
+If you want to know why we call it "cons", that's because this is what the operation is called in Scheme, and they call it that as shorthand for "constructing" the longer list (they call it a "list" rather than a "sequence") out of the components `10` and `[20, 30]`. The name is a bit unfortunate, though, because other structured values besides lists also get "constructed", but we don't say "cons" about them. Still, this is the tradition. Let's just take "cons" to be a nonsense label with an interesting backhistory.
 ((\x (\y (x y))) x)
+The sequence `[20, 30]` in turn is the result of:
+ 20 & [30]
**Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
+and the sequence `[30]` is the result of consing `30` onto the empty sequence `[]`. Note that the sequence `[30]` is not the same as the number `30`. The former is a container value, with one element. The latter is an atomic value, and as such won't have any elements. If you try to do this:
 (\x y. M)
+ [30] + 1
Similarly, `(\x (\y (\z M)))` can be abbreviated as:
+it won't work. We haven't discussed what happens with illegal expressions like that, or like `'true + 1`. For the time being, I'll just say these "don't work", or that they "crash". We'll discuss the variety of ways these illegalities might be handled later.
 (\x y z. M)
+Also, if you try to do this:
+ 20 & 30
Lambda terms represent functions

+it won't work. The consing operator `&` always requires a container (here, a sequence) on its righthand side. And `30` is not a container.
The untyped lambda calculus is Turing complete: all (recursively computable) functions can be represented by lambda terms. For some lambda terms, it is easy to see what function they represent:
+We've said that:
> `(\x x)` represents the identity function: given any argument `M`, this function
simply returns `M`: `((\x x) M) ~~> M`.
+ [10, 20, 30]
> `(\x (x x))` duplicates its argument:
`((\x (x x)) M) ~~> (M M)`
+is the same as;
> `(\x (\y x))` throws away its second argument:
`(((\x (\y x)) M) N) ~~> M`
+ 10 & (20 & (30 & []))
and so on.
+and the latter can also be written without the parentheses. Our language knows that `&` should always be understood as "implicitly associating to the right", that is, that:
It is easy to see that distinct lambda expressions can represent the same
function, considered as a mapping from input to outputs. Obviously:
+ 10 & 20 & 30 & []
 (\x x)
+should be interpreted like the expression displayed before. Other operators like `` should be understood as "implicitly associating to the left." If we write:
and:
+ 30  2  1
 (\z z)
+we presumably want it to be understood as:
both represent the same function, the identity function. However, we said above that we would be regarding these expressions as synactically equivalent, so they aren't yet really examples of *distinct* lambda expressions representing a single function. However, all three of these are distinct lambda expressions:
+ (30  2)  1
 (\y x. y x) (\z z)
+not as:
 (\x. (\z z) x)
+ 30  (2  1)
 (\z z)
+Other operators don't implicitly associate at all. For example, you may understand the expression:
yet when applied to any argument M, all of these will always return M. So they have the same extension. It's also true, though you may not yet be in a position to see, that no other function can differentiate between them when they're supplied as an argument to it. However, these expressions are all syntactically distinct.
+ 10 < x < 20
The first two expressions are *convertible*: in particular the first reduces to the second. So they can be regarded as prooftheoretically equivalent even though they're not syntactically identical. However, the proof theory we've given so far doesn't permit you to reduce the second expression to the third. So these lambda expressions are nonequivalent.
+because we have familiar conventions about what it means. But what it means is not:
There's an extension of the prooftheory we've presented so far which does permit this further move. And in that extended proof theory, all computable functions with the same extension do turn out to be equivalent (convertible). However, at that point, we still won't be working with the traditional mathematical notion of a function as a set of ordered pairs. One reason is that the latter but not the former permits many uncomputable functions. A second reason is that the latter but not the former prohibits functions from applying to themselves. We discussed this some at the end of Monday's meeting (and further discussion is best pursued in person).
+ (10 < x) < 20
+The result of the parenthesized expression is either `'true` or `'false`, assuming `x` evaluates to a number. But `'true < 20` doesn't mean anything, much less what we expect `10 < x < 20` to mean. So `<` doesn't implicitly associate to the left. Neither does it implicitly associate to the right. If you want expressions like `10 < x < 20` to be meaningful, they will need their own special rules.
+Sequences are containers that keep track of the order of their arguments, and also those arguments' multiplicity (how many times each one appears). Other containers might also keep track of these things, and more structural properties too, or they might keep track of less. Let's say we also have **set containers** too, like this:
Booleans and pairs
==================
+ {10, 20, 30}
Our definition of these is reviewed in [[Assignment1]].
+Whereas the sequences `[10, 20, 10]`, `[10, 20]`, and `[20, 10]` are three different sequences, `{10, 20, 10}`, `{10, 20}`, and `{20, 10}` would just be different ways of expressing a single set.
+We can let the `&` operator do extraduty, and express the "consing" relation for sets, too:
It's possible to do the assignment without using a Scheme interpreter, however
you should take this opportunity to [get Scheme installed on your
computer](/how_to_get_the_programming_languages_running_on_your_computer), and
[get started learning Scheme](/learning_scheme). It will help you test out
proposed answers to the assignment.
+ 10 & {20}
+evaluates to `{10, 20}`, and so too does:
There's also a (slow, barebones, but perfectly adequate) version of Scheme available for online use at Scheme (functional part)  OCaml (functional part)  C, Java, Pasval Scheme (imperative part) OCaml (imperative part) 
untyped lambda calculus combinatorial logic  
 Turing complete   
  more advanced type systems, such as polymorphic types    
  simplytyped lambda calculus (what linguists mostly use)    
∀x. (F x or ∀x (not (F x)))


 When a previouslybound variable is rebound in the way we see here, that's called **shadowing**: the outer binding is shadowed during the scope of the inner binding.


Some more comparisons between Scheme and OCaml


11. Simple predefined values

 Numbers in Scheme: `2`, `3`
 In OCaml: `2`, `3`

 Booleans in Scheme: `#t`, `#f`
 In OCaml: `true`, `false`

 The eighth letter in the Latin alphabet, in Scheme: `#\h`
 In OCaml: `'h'`

12. Compound values

 These are values which are built up out of (zero or more) simple values.

 Ordered pairs in Scheme: `'(2 . 3)`
 In OCaml: `(2, 3)`

 Lists in Scheme: `'(2 3)`
 In OCaml: `[2; 3]`
 We'll be explaining the difference between pairs and lists next week.

 The empty list, in Scheme: `'()`
 In OCaml: `[]`

 The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`
 In OCaml: `"h"`

 A longer string, in Scheme: `"horse"`
 In OCaml: `"horse"`

 A shorter string, in Scheme: `""`
 In OCaml: `""`



What "sequencing" is and isn't


We mentioned before the idea that computation is a sequencing of some changes. I said we'd be discussing (fragments of, and in some cases, entire) languages that have no native notion of change.

Neither do they have any useful notion of sequencing. But what this would be takes some care to identify.

First off, the mere concatenation of expressions isn't what we mean by sequencing. Concatenation of expressions is how you build syntactically complex expressions out of simpler ones. The complex expressions often express a computation where a function is applied to one (or more) arguments,

Second, the kind of rebinding we called "shadowing" doesn't involve any changes or sequencing. All the precedence facts about that kind of rebinding are just consequences of the compound syntactic structures in which it occurs.

Third, the kinds of bindings we see in:

 (define foo A)
 (foo 2)

Or even:

 (define foo A)
 (define foo B)
 (foo 2)

don't involve any changes or sequencing in the sense we're trying to identify. As we said, these programs are just syntactic variants of (single) compound syntactic structures involving `let`s and `lambda`s.

Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this:

 (begin A B C)

In OCaml it looks like this:

 begin A; B; C end

Or this:

 (A; B; C)

In the presence of imperatival elements, sequencing order is very relevant. For example, these will behave differently:

 (begin (print "under") (print "water"))

 (begin (print "water") (print "under"))

And so too these:

 begin x := 3; x := 2; x end

 begin x := 2; x := 3; x end

However, if A and B are purely functional, nonimperatival expressions, then:

 begin A; B; C end

just evaluates to C (so long as A and B evaluate to something at all). So:

 begin A; B; C end

contributes no more to a larger context in which it's embedded than C does. This is the sense in which functional languages have no serious notion of sequencing.

We'll discuss this more as the seminar proceeds.
+### That's enough ###
+This was a lot of material, and you may need to read it carefully and think about it, but none of it should seem profoundly different from things you're already accustomed to doing. What we worked our way up to was just the kind of recursive definitions of `factorial` and `length` that you volunteered in class, before learning any programming.
+You have all the materials you need now to do this week's [[assignmentassignment1]]. Some of you may find it easy. Many of you will not. But if you understand what we've done here, and give it your time and attention, we believe you can do it.
+There are also some [[advanced notesweek1 advanced notes]] extending this week's material.