Polymorphism

Polymorphic Lists

Up to this point, we've been working with lists of numbers. Of course, interesting programs also need to be able to manipulate lists whose elements are drawn from other types — lists of strings, lists of booleans, lists of lists, etc. We could just define a new inductive datatype for each of these, for example...

... but this would quickly become tedious, partly because we have to make up different constructor names for each datatype, but mostly because we would also need to define new versions of all our list manipulating functions (length, rev, etc.) for each new datatype definition. To avoid all this repetition, Coq supports polymorphic inductive type definitions. For example, here is a polymorphic list datatype.

This is exactly like the definition of natlist from the previous chapter, except that the nat argument to the cons constructor has been replaced by an arbitrary type X, a binding for X has been added to the header, and the occurrences of natlist in the types of the constructors have been replaced by list X. (We can re-use the constructor names nil and cons because the earlier definition of natlist was inside of a Module definition that is now out of scope.) So what, exactly, is list? One good way to think about it is that list is a function from Types to Inductive definitions; or, to put it another way, list is a function from Types to Types. For any particular type X, the type list X is an Inductively defined set of lists whose elements are things of type X. With this definition, when we use the constructors nil and cons to build lists, we need to specify what type of lists we are building — that is, nil and cons are now polymorphic constructors. Observe the types of these constructors:

The "∀ X" in these types should be read as an additional argument to the constructors that determines the expected types of the arguments that follow. When nil and cons are used, these arguments are supplied in the same way as the others. For example, the list containing 2 and 1 is written like this:

(We are writing nil and cons explicitly here because we haven't yet defined the [] and :: notations. We'll do that in a bit.) We can now go back and make polymorphic (or "generic") versions of all the list-processing functions that we wrote before. Here is length, for example:

Note that the uses of nil and cons in match patterns do not require any type annotations: we already know that the list l contains elements of type X, so there's no reason to include X in the pattern. More formally, the type X is a parameter of the whole definition of list, not of the individual constructors. As with nil and cons, we can use length by applying it first to a type and then to its list argument:

To use our length with other kinds of lists, we simply instantiate it with an appropriate type parameter:

Let's close this subsection by re-implementing a few other standard list functions on our new polymorphic lists:

Type Inference

Let's write the definition of app again, but this time we won't specify the types of any of the arguments. Will Coq still accept it?

Indeed it will. Let's see what type Coq has assigned to app':

It has exactly the same type type as app. Coq was able to use a procedure called type inference to deduce what the types of X, l1, and l2 must be, based on how they are used. For example, since X is used as an argument to cons, it must be a Type, since cons expects a Type as its first argument; matching l1 with nil and cons means it must be a list; and so on. This powerful facility means we don't always have to write explicit type annotations everywhere, although explicit type annotations are still quite useful as documentation and sanity checks. You should try to strike a balance in your own code between too many type annotations (which serve only to clutter and distract) and too few (which force the reader to perform type inference in their head in order to understand your code).

Argument Synthesis

Whenever we use a polymorphic function, we need to pass it one or more types in addition to its other arguments. For example, the recursive call in the body of the length function above must pass along the type X. But just like providing explicit type annotations everywhere, this is heavy and verbose. Since the second argument to length is a list of Xs, it seems entirely obvious that the first argument can only be X — why should we have to write it explicitly? Fortunately, Coq permits us to avoid this kind of redundancy. In place of any type argument we can write the "implicit argument" _, which can be read as "Please figure out for yourself what type belongs here." More precisely, when Coq encounters a _, it will attempt to unify all locally available information — the type of the function being applied, the types of the other arguments, and the type expected by the context in which the application appears — to determine what concrete type should replace the _. This may sound similar to type inference — in fact, the two procedures rely on the same underlying mechanisms. Instead of simply omitting the types of some arguments to a function, like we can also replace the types with _, like which tells Coq to attempt to infer the missing information, just as with argument synthesis. Using implicit arguments, the length function can be written like this:

In this instance, we don't save much by writing _ instead of X. But in many cases the difference can be significant. For example, suppose we want to write down a list containing the numbers 1, 2, and 3. Instead of writing this...

...we can use argument synthesis to write this:

Implicit Arguments

If fact, we can go further. To avoid having to sprinkle _'s throughout our programs, we can tell Coq always to infer the type argument(s) of a given function.

We can also declare an argument to be implicit while defining the function itself, by surrounding the argument in curly braces. For example:

(Note that we didn't even have to provide a type argument to the recursive call to length''.) We will use this style whenever possible, although we will continue to use use explicit Implicit Argument declarations for Inductive constructors. One small problem with declaring arguments Implicit is that, occasionally, Coq does not have enough local information to determine a type argument; in such cases, we need to tell Coq that we want to give the argument explicitly this time, even though we've globally declared it to be Implicit. For example, if we write:

If we uncomment this definition, Coq will give us an error, because it doesn't know what type argument to supply to nil. We can help it by providing an explicit type declaration (so that Coq has more information available when it gets to the "application" of nil):

Alternatively, we can force the implicit arguments to be explicit by prefixing the function name with @.

Using argument synthesis and implicit arguments, we can define convenient notation for lists, as before. Since we have made the constructor type arguments implicit, Coq will know to automatically infer these when we use the notations.

Now lists can be written just the way we'd hope:

Exercises: Polymorphic Lists

Exercise: 2 stars, optional (poly_exercises)

Here are a few simple exercises, just like ones in Lists.v, for practice with polymorphism. Fill in the definitions and complete the proofs below.

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Polymorphic Pairs

Following the same pattern, the type definition we gave in the last chapter for pairs of numbers can be generalized to polymorphic pairs (or products):

As with lists, we make the type arguments implicit and define the familiar concrete notation.

We can also use the Notation mechanism to define the standard notation for pair types:

(The annotation : type_scope tells Coq that this abbreviation should be used when parsing types. This avoids a clash with the multiplication symbol.) A note of caution: it is easy at first to get (x,y) and X*Y confused. Remember that (x,y) is a value consisting of a pair of values; X*Y is a type consisting of a pair of types. If x has type X and y has type Y, then (x,y) has type X*Y. The first and second projection functions now look pretty much as they would in any functional programming language.

The following function takes two lists and combines them into a list of pairs. In many functional programming languages, it is called zip. We call it combine for consistency with Coq's standard library. Note that the pair notation can be used both in expressions and in patterns...

Indeed, when no ambiguity results, we can even drop the enclosing parens:

Exercise: 1 star (combine_checks)

Try answering the following questions on paper and checking your answers in coq:

• What is the type of combine (i.e., what does Check @combine print?)
• What does
    Eval simpl in (combine [1,2] [false,false,true,true]).
  print? ☐

Exercise: 2 stars, recommended (split)

The function split is the right inverse of combine: it takes a list of pairs and returns a pair of lists. In many functional programing languages, this function is called unzip. Uncomment the material below and fill in the definition of split. Make sure it passes the given unit tests.

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Polymorphic Options

One last polymorphic type for now: polymorphic options. The type declaration generalizes the one for natoption in the previous chapter:

We can now rewrite the index function so that it works with any type of lists.

Exercise: 1 star, optional (hd_opt_poly)

Complete the definition of a polymorphic version of the hd_opt function from the last chapter. Be sure that it passes the unit tests below.

Once again, to force the implicit arguments to be explicit, we can use @ before the name of the function.

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Functions as Data

Higher-Order Functions

Like many other modern programming languages — including, of course, all functional languages — Coq treats functions as first-class citizens, allowing functions to be passed as arguments to other functions, returned as results, stored in data structures, etc. Functions that manipulate other functions are often called higher-order functions. Here's a simple one:

The argument f here is itself a function (from X to X); the body of doit3times applies f three times to some value n.

Partial Application

In fact, the multiple-argument functions we have already seen are also examples of passing functions as data. To see why, recall that the type of plus, for instance, is nat → nat → nat.

The → is actually a binary operator on types; that is, Coq primitively supports only one-argument functions. Moreover, this operator is right-associative, so the type of plus is really a shorthand for nat → (nat → nat) — i.e., it can be read as saying that "plus is a one-argument function that takes a nat and returns a one-argument function that takes another nat and returns a nat." In the examples above, we have always applied plus to both of its arguments at once, but if we like we can supply just the first. This is called partial application.

Digression: Currying

Exercise: 2 stars, optional (currying)

In Coq, a function f : A → B → C really has the type A → (B → C). That is, if you give f a value of type A, it will give you function f' : B → C. If you then give f' a value of type B, it will return a value of type C. This allows for partial application, as in plus3. Processing a list of arguments with functions that return functions is called currying, in honor of the logician Haskell Curry. Conversely, we can reinterpret the type A → B → C as (A * B) → C. This is called uncurrying. With an uncurried binary function, both arguments must be given at once as a pair; there is no partial application. We can define currying as follows:

As an exercise, define its inverse, prod_uncurry. Then prove the theorems below to show that the two are inverses.

(Thought exercise: before running these commands, can you calculate the types of prod_curry and prod_uncurry?)

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Filter

Here is a useful higher-order function, which takes a list of Xs and a predicate on X (a function from X to bool) and "filters" the list, returning a new list containing just those elements for which the predicate returns true.

For example, if we apply filter to the predicate evenb and a list of numbers l, it returns a list containing just the even members of l.

We can use filter to give a concise version of the countoddmembers function from Lists.v.

Anonymous Functions

It is a little annoying to be forced to define the function length_is_1 and give it a name just to be able to pass it as an argument to filter, since we will probably never use it again. This is not an isolated example. When using higher-order functions, we often want to pass as arguments "one-off" functions that we will never use again; having to give each of these functions a name would be tedious. Fortunately, there is a better way. It is also possible to construct a function "on the fly" without declaring it at the top level or giving it a name; this is analogous to the notation we've been using for writing down constant lists, natural numbers, and so on.

Here is the motivating example from before, rewritten to use an anonymous function.

Exercise: 2 stars, optional (filter_even_gt7)

Use filter to write a Coq function filter_even_gt7 which takes a list of natural numbers as input and keeps only those numbers which are even and greater than 7.

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Exercise: 3 stars, optional (partition)

Use filter to write a Coq function partition: Given a set X, a test function of type X → bool and a list X, partition should return a pair of lists. The first member of the pair is the sublist of the original list containing the elements that satisfy the test, and the second is the sublist containing those that fail the test. The order of elements in the two sublists should be the same as their order in the original list.

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Map

Another handy higher-order function is called map.

It takes a function f and a list l = [n1, n2, n3, ...] and returns the list [f n1, f n2, f n3,...] , where f has been applied to each element of l in turn. For example:

The element types of the input and output lists need not be the same (map takes two type arguments, X and Y). This version of map can thus be applied to a list of numbers and a function from numbers to booleans to yield a list of booleans:

It can even be applied to a list of numbers and a function from numbers to lists of booleans to yield a list of lists of booleans:

Exercise: 3 stars, optional (map_rev)

Show that map and rev commute. You may need to define an auxiliary lemma.

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Exercise: 2 stars, recommended (flat_map)

The function map maps a list X to a list Y using a function of type X → Y. We can define a similar function, flat_map, which maps a list X to a list Y using a function f of type X → list Y. Your definition should work by 'flattening' the results of f, like so:

☐ Lists are not the only inductive type that we can write a map function for. Here is the definition of map for the option type:

Exercise: 2 stars, optional (implicit_args)

The definitions and uses of filter and map use implicit arguments in many places. Replace the curly braces around the implicit arguments with parentheses, and then fill in explicit type parameters where necessary and use Coq to check that you've done so correctly. This exercise is not to be turned in; it is probably easiest to do it on a copy of this file that you can throw away afterwards. ☐

Fold

An even more powerful higher-order function is called fold. It is the inspiration for the "reduce" operation that lies at the heart of Google's map/reduce distributed programming framework.

Intuitively, the behavior of the fold operation is to insert a given binary operator f between every pair of elements in a given list. For example, fold plus [1,2,3,4] intuitively means 1+2+3+4. To make this precise, we also need a "starting element" that serves as the initial second input to f. So, for example, yields Here are some more examples:

Exercise: 1 star, optional (fold_types_different)

Observe that the type of fold is parameterized by two type variables, X and Y, and the parameter f is a binary operator that takes an X and a Y and returns a Y. Can you think of a situation where it would be useful for X and Y to be different?

Functions For Constructing Functions

Most of the higher-order functions we have talked about so far take functions as arguments. Now let's look at some examples involving returning functions as the results of other functions. To begin, here is a function that takes a value x (drawn from some type X) and returns a function from nat to X that yields x whenever it is called, ignoring its nat argument.

Similarly, but a bit more interestingly, here is a function that takes a function f from numbers to some type X, a number k, and a value x, and constructs a function that behaves exactly like f except that, when called with the argument k, it returns x.

For example, we can apply override twice to obtain a function from numbers to booleans that returns false on 1 and 3 and returns true on all other arguments.

Exercise: 1 star (override_example)

Before starting to work on the following proof, make sure you understand exactly what the theorem is saying and can paraphrase it in your own words. The proof itself is straightforward.

☐ We'll use function overriding heavily in parts of the rest of the course, and we will end up needing to know quite a bit about its properties. To prove these properties, though, we need to know about a few more of Coq's tactics; developing these is the main topic of the rest of the chapter.

More About Coq

The unfold Tactic

Sometimes, a proof will get stuck because Coq doesn't automatically expand a function call into its definition. (This is a feature, not a bug: if Coq automatically expanded everything possible, our proof goals would quickly become enormous — hard to read and slow for Coq to manipulate!)

The unfold tactic can be used to explicitly replace a defined name by the right-hand side of its definition.

Now we can prove a first property of override: If we override a function at some argument k and then look up k, we get back the overridden value.

This proof was straightforward, but note that it requires unfold to expand the definition of override.

Exercise: 2 stars (override_neq)

☐ As the inverse of unfold, Coq also provides a tactic fold, which can be used to "unexpand" a definition. It is used much less often.

Inversion

Recall the definition of natural numbers: It is clear from this definition that every number has one of two forms: either it is the constructor O or it is built by applying the constructor S to another number. But there is more here than meets the eye: implicit in the definition (and in our informal understanding of how datatype declarations work in other programming languages) are two other facts:

• The constructor S is injective. That is, the only way we can have S n = S m is if n = m.
• The constructors O and S are disjoint. That is, O is not equal to S n for any n.

Similar principles apply to all inductively defined types: all constructors are injective, and the values built from distinct constructors are never equal. For lists, the cons constructor is injective and nil is different from every non-empty list. For booleans, true and false are unequal. (Since neither true nor false take any arguments, their injectivity is not an issue.) Coq provides a tactic, called inversion, that allows us to exploit these principles in making proofs. The inversion tactic is used like this. Suppose H is a hypothesis in the context (or a previously proven lemma) of the form for some constructors c and d and arguments a1 ... an and b1 ... bm. Then inversion H instructs Coq to "invert" this equality to extract the information it contains about these terms:

• If c and d are the same constructor, then we know, by the injectivity of this constructor, that a1 = b1, a2 = b2, etc.; inversion H adds these facts to the context, and tries to use them to rewrite the goal.
• If c and d are different constructors, then the hypothesis H is contradictory. That is, a false assumption has crept into the context, and this means that any goal whatsoever is provable! In this case, inversion H marks the current goal as completed and pops it off the goal stack.

The inversion tactic is probably easier to understand by seeing it in action than from general descriptions like the above. Below you will find example theorems that demonstrate the use of inversion and exercises to test your understanding.

As a convenience, the inversion tactic can also destruct equalities between complex values, binding multiple variables as it goes.

Exercise: 1 star (sillyex1)

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Exercise: 1 star (sillyex2)

☐ While the injectivity of constructors allows us to reason ∀ (n m : nat), S n = S m → n = m, the reverse direction of the implication, provable by standard equational reasoning, is a useful fact to record for cases we will see several times.

Here is a more realistic use of inversion to prove a property that is useful in many places later on...

Exercise: 2 stars (beq_nat_eq_informal)

Give an informal proof of beq_nat_eq.

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Exercise: 3 stars (beq_nat_eq')

We can also prove beq_nat_eq by induction on m, though we have to be a little careful about which order we introduce the variables, so that we get a general enough induction hypothesis — this is done for you below. Finish the following proof. To get maximum benefit from the exercise, try first to do it without looking back at the one above.

☐ Here's another illustration of inversion. This is a slightly roundabout way of stating a fact that we have already proved above. The extra equalities force us to do a little more equational reasoning and exercise some of the tactics we've seen recently.

Practice Session

Exercise: 2 stars, optional (practice)

Some nontrivial but not-too-complicated proofs to work together in class, and some for you to work as exercises. Some of the exercises may involve applying lemmas from earlier lectures or homeworks.

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Using Tactics on Hypotheses

By default, most tactics work on the goal formula and leave the context unchanged. However, most tactics also have a variant that performs a similar operation on a statement in the context. For example, the tactic simpl in H performs simplification in the hypothesis named H in the context.

Similarly, the tactic apply L in H matches some conditional statement L (of the form L1 → L2, say) against a hypothesis H in the context. However, unlike ordinary apply (which rewrites a goal matching L2 into a subgoal L1), apply L in H matches H against L1 and, if successful, replaces it with L2. In other words, apply L in H gives us a form of "forward reasoning" — from L1 → L2 and a hypothesis matching L1, it gives us a hypothesis matching L2. By contrast, apply L is "backward reasoning" — it says that if we know L1→L2 and we are trying to prove L2, it suffices to prove L1. Here is a variant of a proof from above, using forward reasoning throughout instead of backward reasoning.

Forward reasoning starts from what is given (premises, previously proven theorems) and iteratively draws conclusions from them until the goal is reached. Backward reasoning starts from the goal, and iteratively reasons about what would imply the goal, until premises or previously proven theorems are reached. If you've seen informal proofs before (for example, in a math or computer science class), they probably used forward reasoning. In general, Coq tends to favor backward reasoning, but in some situations the forward style can be easier to use or to think about.

Exercise: 3 stars, recommended (plus_n_n_injective)

You can practice using the "in" variants in this exercise.

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Using destruct on Compound Expressions

We have seen many examples where the destruct tactic is used to perform case analysis of the value of some variable. But sometimes we need to reason by cases on the result of some expression. We can also do this with destruct. Here are some examples:

After unfolding sillyfun in the above proof, we find that we are stuck on if (beq_nat n 3) then ... else .... Well, either n is equal to 3 or it isn't, so we use destruct (beq_nat n 3) to let us reason about the two cases.

Exercise: 1 star (override_shadow)

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Exercise: 3 stars, recommended (combine_split)

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Exercise: 3 stars, optional (split_combine)

Thought exercise: We have just proven that for all lists of pairs, combine is the inverse of split. How would you state the theorem showing that split is the inverse of combine? Hint: what property do you need of l1 and l2 for split combine l1 l2 = (l1,l2) to be true? State this theorem in Coq, and prove it. (Be sure to leave your induction hypothesis general by not doing intros on more things than necessary.)

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The remember Tactic

(Note: the remember tactic is not strictly needed until a bit later, so if necessary this section can be skipped and returned to when needed.) We have seen how the destruct tactic can be used to perform case analysis of the results of arbitrary computations. If e is an expression whose type is some inductively defined type T, then, for each constructor c of T, destruct e generates a subgoal in which all occurrences of e (in the goal and in the context) are replaced by c. Sometimes, however, this substitution process loses information that we need in order to complete the proof. For example, suppose we define a function sillyfun1 like this:

And suppose that we want to convince Coq of the rather obvious observation that sillyfun1 n yields true only when n is odd. By analogy with the proofs we did with sillyfun above, it is natural to start the proof like this:

We get stuck at this point because the context does not contain enough information to prove the goal! The problem is that the substitution peformed by destruct is too brutal — it threw away every occurrence of beq_nat n 3, but we need to keep at least one of these because we need to be able to reason that since, in this branch of the case analysis, beq_nat n 3 = true, it must be that n = 3, from which it follows that n is odd. What we would really like is not to use destruct directly on beq_nat n 3 and substitute away all occurrences of this expression, but rather to use destruct on something else that is equal to beq_nat n 3. For example, if we had a variable that we knew was equal to beq_nat n 3, we could destruct this variable instead. The remember tactic allows us to introduce such a variable.

Exercise: 2 stars (override_same)

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Exercise: 3 stars, optional (filter_exercise)

This one is a bit challenging. Be sure your initial intros go only up through the parameter on which you want to do induction!

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The apply ... with ... Tactic

The following silly example uses two rewrites in a row to get from [a,b] to [e,f].

Since this is a common pattern, we might abstract it out as a lemma recording once and for all the fact that equality is transitive.

Now, we should be able to use trans_eq to prove the above example. However, to do this we need a slight refinement of the apply tactic.

Actually, we usually don't have to include the name m in the with clause; Coq is often smart enough to figure out which instantiation we're giving. We could instead write: apply trans_eq with c,d.

Exercise: 3 stars, recommended (apply_exercises)

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Review

We've now seen a bunch of Coq's fundamental tactics — enough to do pretty much everything we'll want for a while. We'll introduce one or two more as we go along through the next few lectures, and later in the course we'll introduce some more powerful automation tactics that make Coq do more of the low-level work in many cases. But basically we've got what we need to get work done. Here are the ones we've seen:

• intros: move hypotheses/variables from goal to context
• reflexivity: finish the proof (when the goal looks like e = e)
• apply: prove goal using a hypothesis, lemma, or constructor
• apply... in H: apply a hypothesis, lemma, or constructor to a hypothesis in the context (forward reasoning)
• apply... with...: explicitly specify values for variables that cannot be determined by pattern matching
• simpl: simplify computations in the goal
• simpl in H: ... or a hypothesis
• rewrite: use an equality hypothesis (or lemma) to rewrite the goal
• rewrite ... in H: ... or a hypothesis
• unfold: replace a defined constant by its right-hand side in the goal
• unfold... in H: ... or a hypothesis
• destruct... as...: case analysis on values of inductively defined types
• induction... as...: induction on values of inductively defined types
• inversion: reason by injectivity and distinctness of constructors
• remember (e) as x: give a name (x) to an expression (e) so that we can destruct x without "losing" e
• assert (e) as H: introduce a "local lemma" e and call it H
• generalize dependent e: "re-quantify" e, moving it from the local context back into the goal

Additional Exercises

Exercise: 2 stars, optional (fold_length)

Many common functions on lists can be implemented in terms of fold. For example, here is an alternate definition of length:

Prove the correctness of fold_length.

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Exercise: 3 stars, recommended (fold_map)

We can also define map in terms of fold. Finish fold_map below.

Write down a theorem in Coq stating that fold_map is correct, and prove it.

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Exercise: 2 stars, optional (mumble_grumble)

Consider the following two inductively defined types.

Which of the following are well-typed elements of grumble X for some type X?

• d (b a 5)
• d mumble (b a 5)
• d bool (b a 5)
• e bool true
• e mumble (b c 0)
• e bool (b c 0)
• c

(* FILL IN HERE *)
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Exercise: 2 stars, optional (baz_num_elts)

Consider the following inductive definition:

How many elements does the type baz have? (* FILL IN HERE *)
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Exercise: 4 stars, recommended (forall_exists_challenge)

Challenge problem: Define two recursive Fixpoints, forallb and existsb. The first checks whether every element in a list satisfies a given predicate: The function existsb checks whether there exists an element in the list that satisfies a given predicate: Next, create a nonrecursive Definition, existsb', using forallb and negb. Prove that existsb' and existsb have the same behavior.

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Exercise: 2 stars, optional (index_informal)

Recall the definition of the index function: Write an informal proof of the following theorem: (* FILL IN HERE *)
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