In a sense, CIC is built from just two relatively straightforward features: function types and inductive types. From this modest foundation, we can prove effectively all of the theorems of math and carry out effectively all program verifications, with enough effort expended. This chapter introduces induction and recursion for functional programming in Coq.

Enumerations

Coq inductive types generalize the algebraic datatypes found in Haskell and ML. Confusingly enough, inductive types also generalize generalized algebraic datatypes (GADTs), by adding the possibility for type dependency. Even so, it is worth backing up from the examples of the last chapter and going over basic, algebraic datatype uses of inductive datatypes, because the chance to prove things about the values of these types adds new wrinkles beyond usual practice in Haskell and ML.

The singleton type unit is an inductive type:

This vernacular command defines a new inductive type unit whose only value is tt, as we can see by checking the types of the two identifiers:

unit : Set

tt : unit

We can prove that unit is a genuine singleton type.

The important thing about an inductive type is, unsurprisingly, that you can do induction over its values, and induction is the key to proving this theorem. We ask to proceed by induction on the variable x.

The goal changes to:
 tt = tt

...which we can discharge trivially.

It seems kind of odd to write a proof by induction with no inductive hypotheses. We could have arrived at the same result by beginning the proof with:

  destruct x.


...which corresponds to "proof by case analysis" in classical math. For non-recursive inductive types, the two tactics will always have identical behavior. Often case analysis is sufficient, even in proofs about recursive types, and it is nice to avoid introducing unneeded induction hypotheses.

What exactly is the induction principle for unit? We can ask Coq:

unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u

Every Inductive command defining a type T also defines an induction principle named T_ind. Coq follows the Curry-Howard correspondence and includes the ingredients of programming and proving in the same single syntactic class. Thus, our type, operations over it, and principles for reasoning about it all live in the same language and are described by the same type system. The key to telling what is a program and what is a proof lies in the distinction between the type Prop, which appears in our induction principle; and the type Set, which we have seen a few times already.

The convention goes like this: Set is the type of normal types, and the values of such types are programs. Prop is the type of logical propositions, and the values of such types are proofs. Thus, an induction principle has a type that shows us that it is a function for building proofs.

Specifically, unit_ind quantifies over a predicate P over unit values. If we can present a proof that P holds of tt, then we are rewarded with a proof that P holds for any value u of type unit. In our last proof, the predicate was (fun u : unit => u = tt).



We can define an inductive type even simpler than unit:

Empty_set has no elements. We can prove fun theorems about it:

Because Empty_set has no elements, the fact of having an element of this type implies anything. We use destruct 1 instead of destruct x in the proof because unused quantified variables are relegated to being referred to by number. (There is a good reason for this, related to the unity of quantifiers and implication. An implication is just a quantification over a proof, where the quantified variable is never used. It generally makes more sense to refer to implication hypotheses by number than by name, and Coq treats our quantifier over an unused variable as an implication in determining the proper behavior.)

We can see the induction principle that made this proof so easy:

Empty_set_ind : forall (P : Empty_set -> Prop) (e : Empty_set), P e

In other words, any predicate over values from the empty set holds vacuously of every such element. In the last proof, we chose the predicate (fun _ : Empty_set => 2 + 2 = 5).

We can also apply this get-out-of-jail-free card programmatically. Here is a lazy way of converting values of Empty_set to values of unit:

We employ match pattern matching as in the last chapter. Since we match on a value whose type has no constructors, there is no need to provide any branches.



Moving up the ladder of complexity, we can define the booleans:

We can use less vacuous pattern matching to define boolean negation.

An alternative definition desugars to the above:

We might want to prove that not is its own inverse operation.

After we case-analyze on b, we are left with one subgoal for each constructor of bool.


2 subgoals
  
  ============================
   not (not true) = true


subgoal 2 is:
 not (not false) = false
 

The first subgoal follows by Coq's rules of computation, so we can dispatch it easily:

Likewise for the second subgoal, so we can restart the proof and give a very compact justification.

Another theorem about booleans illustrates another useful tactic.

discriminate is used to prove that two values of an inductive type are not equal, whenever the values are formed with different constructors. In this case, the different constructors are true and false.

At this point, it is probably not hard to guess what the underlying induction principle for bool is.

bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b

Simple Recursive Types

The natural numbers are the simplest common example of an inductive type that actually deserves the name.

O is zero, and S is the successor function, so that 0 is syntactic sugar for O, 1 for S O, 2 for S (S O), and so on.

Pattern matching works as we demonstrated in the last chapter:

We can prove theorems by case analysis:

We can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting.

Recall that Fixpoint is Coq's mechanism for recursive function definitions. Some theorems about plus can be proved without induction.

Coq's computation rules automatically simplify the application of plus, because unfolding the definition of plus gives us a match expression where the branch to be taken is obvious from syntax alone. If we just reverse the order of the arguments, though, this no longer works, and we need induction.

Our first subgoal is plus O O = O, which is trivial by computation.

Our second subgoal is more work and also demonstrates our first inductive hypothesis.


  n : nat
  IHn : plus n O = n
  ============================
   plus (S n) O = S n
 

We can start out by using computation to simplify the goal as far as we can.

Now the conclusion is S (plus n O) = S n. Using our inductive hypothesis:

...we get a trivial conclusion S n = S n.

Not much really went on in this proof, so the crush tactic from the Tactics module can prove this theorem automatically.

We can check out the induction principle at work here:

  nat_ind : forall P : nat -> Prop,
            P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
 

Each of the two cases of our last proof came from the type of one of the arguments to nat_ind. We chose P to be (fun n : nat => plus n O = n). The first proof case corresponded to P O and the second case to (forall n : nat, P n -> P (S n)). The free variable n and inductive hypothesis IHn came from the argument types given here.

Since nat has a constructor that takes an argument, we may sometimes need to know that that constructor is injective.

injection refers to a premise by number, adding new equalities between the corresponding arguments of equated terms that are formed with the same constructor. We end up needing to prove n = m -> n = m, so it is unsurprising that a tactic named trivial is able to finish the proof.

There is also a very useful tactic called congruence that can prove this theorem immediately. congruence generalizes discriminate and injection, and it also adds reasoning about the general properties of equality, such as that a function returns equal results on equal arguments. That is, congruence is a complete decision procedure for the theory of equality and uninterpreted functions, plus some smarts about inductive types.



We can define a type of lists of natural numbers.

Recursive definitions are straightforward extensions of what we have seen before.

Inductive theorem proving can again be automated quite effectively.

  nat_list_ind
     : forall P : nat_list -> Prop,
       P NNil ->
       (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
       forall n : nat_list, P n



In general, we can implement any "tree" types as inductive types. For example, here are binary trees of naturals.

  nat_btree_ind
     : forall P : nat_btree -> Prop,
       P NLeaf ->
       (forall n : nat_btree,
        P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
       forall n : nat_btree, P n

Parameterized Types

We can also define polymorphic inductive types, as with algebraic datatypes in Haskell and ML.

There is a useful shorthand for writing many definitions that share the same parameter, based on Coq's section mechanism. The following block of code is equivalent to the above:

After we end the section, the Variables we used are added as extra function parameters for each defined identifier, as needed. We verify that this has happened using the Print command, a cousin of Check which shows the definition of a symbol, rather than just its type.

  Inductive list (T : Set) : Set :=
    Nil : list T | Cons : T -> list T -> list Tlist
 

The final definition is the same as what we wrote manually before. The other elements of the section are altered similarly, turning out exactly as they were before, though we managed to write their definitions more succinctly.

  length
     : forall T : Set, list T -> nat

The parameter T is treated as a new argument to the induction principle, too.

  list_ind
     : forall (T : Set) (P : list T -> Prop),
       P (Nil T) ->
       (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
       forall l : list T, P l

Thus, even though we just saw that T is added as an extra argument to the constructor Cons, there is no quantifier for T in the type of the inductive case like there is for each of the other arguments.

Mutually Inductive Types

We can define inductive types that refer to each other:

Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before.

One goal remains:

  n : nat
  o : odd_list
  el2 : even_list
  ============================
   S (olength (oapp o el2)) = S (plus (olength o) (elength el2))

We have no induction hypothesis, so we cannot prove this goal without starting another induction, which would reach a similar point, sending us into a futile infinite chain of inductions. The problem is that Coq's generation of T_ind principles is incomplete. We only get non-mutual induction principles generated by default.

  even_list_ind
     : forall P : even_list -> Prop,
       P ENil ->
       (forall (n : nat) (o : odd_list), P (ECons n o)) ->
       forall e : even_list, P e
 

We see that no inductive hypotheses are included anywhere in the type. To get them, we must ask for mutual principles as we need them, using the Scheme command.

  even_list_mut
     : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
       P ENil ->
       (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
       (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
       forall e : even_list, P e
 

This is the principle we wanted in the first place. There is one more wrinkle left in using it: the induction tactic will not apply it for us automatically. It will be helpful to look at how to prove one of our past examples without using induction, so that we can then generalize the technique to mutual inductive types.

From this example, we can see that induction is not magic. It only does some bookkeeping for us to make it easy to apply a theorem, which we can do directly with the apply tactic. We apply not just an identifier but a partial application of it, specifying the predicate we mean to prove holds for all naturals.

This technique generalizes to our mutual example:

We simply need to specify two predicates, one for each of the mutually inductive types. In general, it would not be a good idea to assume that a proof assistant could infer extra predicates, so this way of applying mutual induction is about as straightforward as we could hope for.

Reflexive Types

A kind of inductive type called a reflexive type is defined in terms of functions that have the type being defined as their range. One very useful class of examples is in modeling variable binders. For instance, here is a type for encoding the syntax of a subset of first-order logic:

Our kinds of formulas are equalities between naturals, conjunction, and universal quantification over natural numbers. We avoid needing to include a notion of "variables" in our type, by using Coq functions to encode quantification. For instance, here is the encoding of forall x : nat, x = x:

We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions.

We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands.

It is helpful to prove that this transformation does not make true formulas false.

We can take a look at the induction principle behind this proof.

  formula_ind
     : forall P : formula -> Prop,
       (forall n n0 : nat, P (Eq n n0)) ->
       (forall f0 : formula,
        P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
       (forall f1 : nat -> formula,
        (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
       forall f2 : formula, P f2
 

Focusing on the Forall case, which comes third, we see that we are allowed to assume that the theorem holds for any application of the argument function f1. That is, Coq induction principles do not follow a simple rule that the textual representations of induction variables must get shorter in appeals to induction hypotheses. Luckily for us, the people behind the metatheory of Coq have verified that this flexibility does not introduce unsoundness.



Up to this point, we have seen how to encode in Coq more and more of what is possible with algebraic datatypes in Haskell and ML. This may have given the inaccurate impression that inductive types are a strict extension of algebraic datatypes. In fact, Coq must rule out some types allowed by Haskell and ML, for reasons of soundness. Reflexive types provide our first good example of such a case.

Given our last example of an inductive type, many readers are probably eager to try encoding the syntax of lambda calculus. Indeed, the function-based representation technique that we just used, called higher-order abstract syntax (HOAS), is the representation of choice for lambda calculi in Twelf and in many applications implemented in Haskell and ML. Let us try to import that choice to Coq:

Inductive term : Set :=
| App : term -> term -> term
| Abs : (term -> term) -> term.

Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
 

We have run afoul of the strict positivity requirement for inductive definitions, which says that the type being defined may not occur to the left of an arrow in the type of a constructor argument. It is important that the type of a constructor is viewed in terms of a series of arguments and a result, since obviously we need recursive occurrences to the lefts of the outermost arrows if we are to have recursive occurrences at all.

Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:


Definition uhoh (t : term) : term :=
  match t with
    | Abs f => f t
    | _ => t
  end.


Using an informal idea of Coq's semantics, it is easy to verify that the application uhoh (Abs uhoh) will run forever. This would be a mere curiosity in OCaml and Haskell, where non-termination is commonplace, though the fact that we have a non-terminating program without explicit recursive function definitions is unusual.

For Coq, however, this would be a disaster. The possibility of writing such a function would destroy all our confidence that proving a theorem means anything. Since Coq combines programs and proofs in one language, we would be able to prove every theorem with an infinite loop.

Nonetheless, the basic insight of HOAS is a very useful one, and there are ways to realize most benefits of HOAS in Coq. We will study a particular technique of this kind in the later chapters on programming language syntax and semantics.

An Interlude on Proof Terms

As we have emphasized a few times already, Coq proofs are actually programs, written in the same language we have been using in our examples all along. We can get a first sense of what this means by taking a look at the definitions of some of the induction principles we have used.

  unit_ind =
  fun P : unit -> Prop => unit_rect P
     : forall P : unit -> Prop, P tt -> forall u : unit, P u
 

We see that this induction principle is defined in terms of a more general principle, unit_rect.

  unit_rect
     : forall P : unit -> Type, P tt -> forall u : unit, P u
 

unit_rect gives P type unit -> Type instead of unit -> Prop. Type is another universe, like Set and Prop. In fact, it is a common supertype of both. Later on, we will discuss exactly what the significances of the different universes are. For now, it is just important that we can use Type as a sort of meta-universe that may turn out to be either Set or Prop. We can see the symmetry inherent in the subtyping relationship by printing the definition of another principle that was generated for unit automatically:

  unit_rec =
  fun P : unit -> Set => unit_rect P
     : forall P : unit -> Set, P tt -> forall u : unit, P u
 

This is identical to the definition for unit_ind, except that we have substituted Set for Prop. For most inductive types T, then, we get not just induction principles T_ind, but also recursion principles T_rec. We can use T_rec to write recursive definitions without explicit Fixpoint recursion. For instance, the following two definitions are equivalent:

Going even further down the rabbit hole, unit_rect itself is not even a primitive. It is a functional program that we can write manually.

  unit_rect =
  fun (P : unit -> Type) (f : P tt) (u : unit) =>
  match u as u0 return (P u0) with
  | tt => f
  end
     : forall P : unit -> Type, P tt -> forall u : unit, P u
 

The only new feature we see is an as clause for a match, which is used in concert with the return clause that we saw in the introduction. Since the type of the match is dependent on the value of the object being analyzed, we must give that object a name so that we can refer to it in the return clause.

To prove that unit_rect is nothing special, we can reimplement it manually.

We rely on Coq's heuristics for inferring match annotations.

We can check the implementation of nat_rect as well:

  nat_rect =
  fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
  fix F (n : nat) : P n :=
    match n as n0 return (P n0) with
     | O => f
     | S n0 => f0 n0 (F n0)
     end
      : forall P : nat -> Type,
        P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n

Now we have an actual recursive definition. fix expressions are an anonymous form of Fixpoint, just as fun expressions stand for anonymous non-recursive functions. Beyond that, the syntax of fix mirrors that of Fixpoint. We can understand the definition of nat_rect better by reimplementing nat_ind using sections.

First, we have the property of natural numbers that we aim to prove.

Then we require a proof of the O case.

Next is a proof of the S case, which may assume an inductive hypothesis.

Finally, we define a recursive function to tie the pieces together.

Closing the section adds the Variables and Hypothesises as new fun-bound arguments to nat_ind', and, modulo the use of Prop instead of Type, we end up with the exact same definition that was generated automatically for nat_rect.



We can also examine the definition of even_list_mut, which we generated with Scheme for a mutually-recursive type.

   even_list_mut =
   fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
     (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
     (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
   fix F (e : even_list) : P e :=
     match e as e0 return (P e0) with
     | ENil => f
     | ECons n o => f0 n o (F0 o)
     end
   with F0 (o : odd_list) : P0 o :=
     match o as o0 return (P0 o0) with
     | OCons n e => f1 n e (F e)
     end
   for F
      : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
        P ENil ->
        (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
        (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
        forall e : even_list, P e

We see a mutually-recursive fix, with the different functions separated by with in the same way that they would be separated by and in ML. A final for clause identifies which of the mutually-recursive functions should be the final value of the fix expression. Using this definition as a template, we can reimplement even_list_mut directly.

First, we need the properties that we are proving.

Next, we need proofs of the three cases.

Finally, we define the recursive functions.

Even induction principles for reflexive types are easy to implement directly. For our formula type, we can use a recursive definition much like those we wrote above.

Nested Inductive Types

Suppose we want to extend our earlier type of binary trees to trees with arbitrary finite branching. We can use lists to give a simple definition.

This is an example of a nested inductive type definition, because we use the type we are defining as an argument to a parametrized type family. Coq will not allow all such definitions; it effectively pretends that we are defining nat_tree mutually with a version of list specialized to nat_tree, checking that the resulting expanded definition satisfies the usual rules. For instance, if we replaced list with a type family that used its parameter as a function argument, then the definition would be rejected as violating the positivity restriction.

Like we encountered for mutual inductive types, we find that the automatically-generated induction principle for nat_tree is too weak.

  nat_tree_ind
     : forall P : nat_tree -> Prop,
       P NLeaf' ->
       (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
       forall n : nat_tree, P n
 

There is no command like Scheme that will implement an improved principle for us. In general, it takes creativity to figure out how to incorporate nested uses to different type families. Now that we know how to implement induction principles manually, we are in a position to apply just such creativity to this problem.

First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list.

It will be useful to look at the definitions of True and /\, since we will want to write manual proofs of them below.

  Inductive True : Prop := I : True
 

That is, True is a proposition with exactly one proof, I, which we may always supply trivially.

Finding the definition of /\ takes a little more work. Coq supports user registration of arbitrary parsing rules, and it is such a rule that is letting us write /\ instead of an application of some inductive type family. We can find the underlying inductive type with the Locate command.

  Notation Scope
  "A /\ B" := and A B : type_scope
                        (default interpretation)

  Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
  For conj: Arguments A, B are implicit
  For and: Argument scopes are [type_scope type_scope]
  For conj: Argument scopes are [type_scope type_scope _ _]
 

In addition to the definition of and itself, we get information on implicit arguments and parsing rules for and and its constructor conj. We will ignore the parsing information for now. The implicit argument information tells us that we build a proof of a conjunction by calling the constructor conj on proofs of the conjuncts, with no need to include the types of those proofs as explicit arguments.



Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter.

A first attempt at writing the induction principle itself follows the intuition that nested inductive type definitions are expanded into mutual inductive definitions.


  Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
    match tr with
      | NLeaf' => NLeaf'_case
      | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
    end

  with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
    match ls with
      | Nil => I
      | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
    end.


Coq rejects this definition, saying "Recursive call to nat_tree_ind' has principal argument equal to "tr" instead of rest." The term "nested inductive type" hints at the solution to the problem. Just like true mutually-inductive types require mutually-recursive induction principles, nested types require nested recursion.

We include an anonymous fix version of list_nat_tree_ind that is literally nested inside the definition of the recursive function corresponding to the inductive definition that had the nested use of list.

We can try our induction principle out by defining some recursive functions on nat_trees and proving a theorem about them. First, we define some helper functions that operate on lists.

Now we can define a size function over our trees.

Notice that Coq was smart enough to expand the definition of map to verify that we are using proper nested recursion, even through a use of a higher-order function.

We have defined another arbitrary notion of tree splicing, similar to before, and we can prove an analogous theorem about its relationship with tree size. We start with a useful lemma about addition.

Now we begin the proof of the theorem, adding the lemma plus_S as a hint.

We know that the standard induction principle is insufficient for the task, so we need to provide a using clause for the induction tactic to specify our alternate principle.

One subgoal remains:
  n : nat
  ls : list nat_tree
  H : All
        (fun tr1 : nat_tree =>
         forall tr2 : nat_tree,
         ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
  tr2 : nat_tree
  ============================
   ntsize
     match ls with
     | Nil => NNode' n (Cons tr2 Nil)
     | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
     end = S (plus (ntsize tr2) (sum (map ntsize ls)))
 

After a few moments of squinting at this goal, it becomes apparent that we need to do a case analysis on the structure of ls. The rest is routine.

We can go further in automating the proof by exploiting the hint mechanism.

We will go into great detail on hints in a later chapter, but the only important thing to note here is that we register a pattern that describes a conclusion we expect to encounter during the proof. The pattern may contain unification variables, whose names are prefixed with question marks, and we may refer to those bound variables in a tactic that we ask to have run whenever the pattern matches.

The advantage of using the hint is not very clear here, because the original proof was so short. However, the hint has fundamentally improved the readability of our proof. Before, the proof referred to the local variable ls, which has an automatically-generated name. To a human reading the proof script without stepping through it interactively, it was not clear where ls came from. The hint explains to the reader the process for choosing which variables to case analyze on, and the hint can continue working even if the rest of the proof structure changes significantly.

Manual Proofs About Constructors

It can be useful to understand how tactics like discriminate and injection work, so it is worth stepping through a manual proof of each kind. We will start with a proof fit for discriminate.

We begin with the tactic red, which is short for "one step of reduction," to unfold the definition of logical negation.

  ============================
   true = false -> False
 

The negation is replaced with an implication of falsehood. We use the tactic intro H to change the assumption of the implication into a hypothesis named H.

  H : true = false
  ============================
   False
 

This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon.

It is worth recalling the difference between the lowercase and uppercase versions of truth and falsehood: True and False are logical propositions, while true and false are boolean values that we can case-analyze. We have defined f such that our conclusion of False is computationally equivalent to f false. Thus, the change tactic will let us change the conclusion to f false.

  H : true = false
  ============================
   f false
 

Now the righthand side of H's equality appears in the conclusion, so we can rewrite, using the notation <- to request to replace the righthand side the equality with the lefthand side.

  H : true = false
  ============================
   f true
 

We are almost done. Just how close we are to done is revealed by computational simplification.

  H : true = false
  ============================
   True
 

I have no trivial automated version of this proof to suggest, beyond using discriminate or congruence in the first place.



We can perform a similar manual proof of injectivity of the constructor S. I leave a walk-through of the details to curious readers who want to run the proof script interactively.

Exercises

1. Define an inductive type truth with three constructors, Yes, No, and Maybe. Yes stands for certain truth, No for certain falsehood, and Maybe for an unknown situation. Define "not," "and," and "or" for this replacement boolean algebra. Prove that your implementation of "and" is commutative and distributes over your implementation of "or."



2. Modify the first example language of Chapter 2 to include variables, where variables are represented with nat. Extend the syntax and semantics of expressions to accommodate the change. Your new expDenote function should take as a new extra first argument a value of type var -> nat, where var is a synonym for naturals-as-variables, and the function assigns a value to each variable. Define a constant folding function which does a bottom-up pass over an expression, at each stage replacing every binary operation on constants with an equivalent constant. Prove that constant folding preserves the meanings of expressions.



3. Reimplement the second example language of Chapter 2 to use mutually-inductive types instead of dependent types. That is, define two separate (non-dependent) inductive types nat_exp and bool_exp for expressions of the two different types, rather than a single indexed type. To keep things simple, you may consider only the binary operators that take naturals as operands. Add natural number variables to the language, as in the last exercise, and add an "if" expression form taking as arguments one boolean expression and two natural number expressions. Define semantics and constant-folding functions for this new language. Your constant folding should simplify not just binary operations (returning naturals or booleans) with known arguments, but also "if" expressions with known values for their test expressions but possibly undetermined "then" and "else" cases. Prove that constant-folding a natural number expression preserves its meaning.



4. Using a reflexive inductive definition, define a type nat_tree of infinitary trees, with natural numbers at their leaves and a countable infinity of new trees branching out of each internal node. Define a function increment that increments the number in every leaf of a nat_tree. Define a function leapfrog over a natural i and a tree nt. leapfrog should recurse into the ith child of nt, the i+1st child of that node, the i+2nd child of the next node, and so on, until reaching a leaf, in which case leapfrog should return the number at that leaf. Prove that the result of any call to leapfrog is incremented by one by calling increment on the tree.



5. Define a type of trees of trees of trees of (repeat to infinity). That is, define an inductive type trexp, whose members are either base cases containing natural numbers or binary trees of trexps. Base your definition on a parameterized binary tree type btree that you will also define, so that trexp is defined as a nested inductive type. Define a function total that sums all of the naturals at the leaves of a trexp. Define a function increment that increments every leaf of a trexp by one. Prove that, for all tr, total (increment tr) >= total tr. On the way to finishing this proof, you will probably want to prove a lemma and add it as a hint using the syntax Hint Resolve name_of_lemma..



6. Prove discrimination and injectivity theorems for the nat_btree type defined earlier in this chapter. In particular, without using the tactics discriminate, injection, or congruence, prove that no leaf equals any node, and prove that two equal nodes carry the same natural number.