Our next topic, a large one, is type systems — static program analyses that classify expressions according to the "shapes" of their results. We'll begin with a typed version of a very simple language with just booleans and numbers, to introduce the basic ideas of types, typing rules, and the fundamental theorems about type systems: type preservation and progress. Then we'll move on to the simply typed lambda-calculus, which lives at the core of every modern functional programming language (including Coq).

More Automation

Before we start, let's spend a little time learning to use some of Coq's more powerful automation features...

The auto and eauto Tactics

The auto tactic solves goals that are solvable by any combination of

• intros,
• apply (with a local hypothesis, by default), and
• reflexivity.

The eauto tactic works just like auto, except that it uses eapply instead of apply. Using auto is always "safe" in the sense that it will never fail and will never change the proof state: either it completely solves the current goal, or it does nothing. Here is a contrived example:

When searching for potential proofs of the current goal, auto and eauto consider the hypotheses in the current context together with a hint database of other lemmas and constructors. Some of the lemmas and constructors we've already seen — e.g., conj, or_introl, and or_intror — are installed in this hint database by default.

We can extend the hint database just for the purposes of one application of auto or eauto by writing auto using .... E.g., if conj, or_introl, and or_intror had not already been in the hint database, we could have done this instead:

Of course, in any given development there will also be some of our own specific constructors and lemmas that are used very often in proofs. We can add these to the global hint database by writing at the top level, where l is a top-level lemma theorem or a constructor of an inductively defined proposition (i.e., anything whose type is an implication). As a shorthand, we can write to tell Coq to do a Hint Resolve for all of the constructors from the inductive definition of c. It is also sometimes necessary to add where d is a defined symbol, so that auto knows to expand uses of d and enable further possibilities for applying lemmas that it knows about. Here are some Hints we will find useful.

Warning: Just as with Coq's other automation facilities, it is easy to overuse auto and eauto and wind up with proofs that are impossible to understand later! Also, overuse of eauto can make proof scripts very slow. Get in the habit of using auto most of the time and eauto only when necessary. For much more detailed information about using auto and eauto, see the chapter UseAuto.v.

The Proof with Tactic

If you start a proof by saying Proof with (tactic) instead of just Proof, then writing ... instead of . after a tactic in the body of the proof will try to solve all generated subgoals with tactic (and fail if this doesn't work). One common use of this facility is "Proof with auto" (or eauto). We'll see many examples of this later in the file.

The solve by inversion Tactic

Here's another nice automation feature: it often arises that the context contains a contradictory assumption and we want to use inversion on it to solve the goal. We'd like to be able to say to Coq, "find a contradictory assumption and invert it" without giving its name explicitly. Doing solve by inversion will find a hypothesis that can be inverted to solve the goal, if there is one. The tactics solve by inversion 2 and solve by inversion 3 are slightly fancier versions which will perform two or three inversions in a row, if necessary, to solve the goal. (These tactics are not actually built into Coq — their definitions are in Sflib.v.) Caution: Overuse of solve by inversion can lead to slow proof scripts.

The try solve Tactic

If t is a tactic, then try solve [t] is a tactic that

• if t solves the goal, behaves just like t, or
• if t cannot completely solve the goal, does nothing.

More generally, try solve [t1 | t2 | ...] will try to solve the goal by using t1, t2, etc. If none of them succeeds in completely solving the goal, then try solve [t1 | t2 | ...] does nothing. The fact that it does nothing when it doesn't completely succeed in solving the goal means that there is no harm in using try solve T in situations where T might actually make no sense. In particular, if we put it after a ; it will solve as many subgoals as it can and leave the rest for us to solve by other methods. It will not leave any of the subgoals in a partially solved state.

The f_equal Tactic

This handy tactic replaces a goal of the form f x1 x2 ... xn = f y1 y2 ... yn, where f is some function, with the subgoals x1 = y1, x2 = y2,...,xn = yn. It is useful for avoiding explicit rewriting steps, and often the generated subgoals can be quickly cleared by auto.

The normalize Tactic

When experimenting with definitions of programming languages in Coq, we often want to see what a particular concrete term steps to — i.e., we want to find proofs for goals of the form t ⇒* t', where t is a completely concrete term and t' is unknown. These proofs are simple but repetitive to do by hand. Consider for example reducing an arithmetic expression using the small-step relation astep defined in the previous chapter:

We repeatedly applied rsc_step until we got to a normal form. The proofs that the intermediate steps are possible are simple enough that auto, with appropriate hints, can solve them.

The following custom Tactic Notation definition captures this pattern. In addition, before each rsc_step we print out the current goal, so that the user can follow how the term is being evaluated.

Finally, this is also useful as a simple way to calculate what the normal form of a term is, by proving a goal with an existential variable in it.

Typed Arithmetic Expressions

To motivate the discussion of type systems, let's begin as usual with an extremely simple toy language. We want it to have the potential for programs "going wrong" because of runtime type errors, so we need something a tiny bit more complex than the language of constants and addition that we used in Smallstep.v: a single kind of data (just numbers) is too simple, but just two kinds (numbers and booleans) already gives us enough material to tell an interesting story. The language definition is completely routine. The only thing to notice is that we are not using the asnum/aslist trick that we used in ImpList.v to make all the operations total by forcibly coercing the arguments to + (for example) into numbers. Instead, we simply let terms get stuck if they try to use an operator with the wrong kind of operands: the step relation doesn't relate them to anything.

Syntax

Small-Step Reduction

Notice that the step relation doesn't care about whether expressions make global sense — it just checks that the operation in the next reduction step is being applied to the right kinds of operands. For example, the term tm_succ tm_true cannot take a step, but the almost-as-obviously-nonsensical term can.

Normal Forms and Values

The first interesting thing about the step relation in this language is that the strong progress theorem from the Smallstep chapter fails! That is, there are terms that are normal forms (they can't take a step) but not values (because we have not included them in our definition of possible "results of evaluation"). Such terms are stuck.

Exercise: 2 stars, optional (some_tm_is_stuck)

☐ However, although values and normal forms are not the same in this language, the former set is included in the latter. This is important because it shows we did not accidentally define things so that some value could still take a step.

Exercise: 3 stars, optional (value_is_nf)

Hint: You will reach a point in this proof where you need to use an induction to reason about a term that is known to be a numeric value. This induction can be performed either over the term itself or over the evidence that it is a numeric value. The proof goes through in either case, but you will find that one way is quite a bit shorter than the other. For the sake of the exercise, try to complete the proof both ways.

☐

Exercise: 3 stars, optional (step_deterministic)

Using value_is_nf, we can show that the step relation is also deterministic...

☐

Typing

The next critical observation about this language is that, although there are stuck terms, they are all "nonsensical", mixing booleans and numbers in a way that we don't even want to have a meaning. We can easily exclude such ill-typed terms by defining a typing relation that relates terms to the types (either numeric or boolean) of their final results.

The typing relation: In informal notation, the typing relation is often written ⊢ t : T, pronounced "t has type T." The ⊢ symbol is called a "turnstile". (Below, we're going to see richer typing relations where an additional "context" argument is written to the left of the turnstile. Here, the context is always empty.)

	(T_True)
⊢ true : Bool

	(T_False)
⊢ false : Bool

⊢ t1 : Bool    ⊢ t2 : T    ⊢ t3 : T	(T_If)
⊢ if t1 then t2 else t3 : T

	(T_Zero)
⊢ 0 : Nat

⊢ t1 : Nat	(T_Succ)
⊢ succ t1 : Nat

⊢ t1 : Nat	(T_Pred)
⊢ pred t1 : Nat

⊢ t1 : Nat	(T_IsZero)
⊢ iszero t1 : Bool

Examples

It's important to realize that the typing relation is a conservative (or static) approximation: it does not calculate the type of the normal form of a term.

(Since we've included all the constructors of the typing relation in the hint database, the auto tactic can actually find this proof automatically.)

Exercise: 1 star (succ_hastype_nat__hastype_nat)

☐

Progress

The typing relation enjoys two critical properties. The first is that well-typed normal forms are values (i.e., not stuck).

Exercise: 3 stars, recommended (finish_progress_informal)

Complete the following proof: Theorem: If ⊢ t : T, then either t is a value or else t ⇒ t' for some t'. Proof: By induction on a derivation of ⊢ t : T.

• If the last rule in the derivation is T_If, then t = if t1 then t2 else t3, with ⊢ t1 : Bool, ⊢ t2 : T and ⊢ t3 : T. By the IH, either t1 is a value or else t1 can step to some t1'.
  • If t1 is a value, then it is either an nvalue or a bvalue. But it cannot be an nvalue, because we know ⊢ t1 : Bool and there are no rules assigning type Bool to any term that could be an nvalue. So t1 is a bvalue — i.e., it is either true or false. If t1 = true, then t steps to t2 by ST_IfTrue, while if t1 = false, then t steps to t3 by ST_IfFalse. Either way, t can step, which is what we wanted to show.
  • If t1 itself can take a step, then, by ST_If, so can t.

(* FILL IN HERE *)
☐

Exercise: 3 stars (finish_progress)

☐ This is more interesting than the strong progress theorem that we saw in the Smallstep chapter, where all normal forms were values. Here, a term can be stuck, but only if it is ill typed.

Exercise: 1 star (step_review)

Quick review. Answer true or false. (As usual, no need to hand these in.)

• In this language, every well-typed normal form is a value.
• Every value is a normal form.
• The single-step evaluation relation is a partial function.
• The single-step evaluation relation is a total function.

☐

Type Preservation

The second critical property of typing is that, when a well-typed term takes a step, the result is also a well-typed term. This theorem is often called the subject reduction property, because it tells us what happens when the "subject" of the typing relation is reduced. This terminology comes from thinking of typing statements as sentences, where the term is the subject and the type is the predicate.

Exercise: 3 stars, recommended (finish_preservation_informal)

Complete the following proof: Theorem: If ⊢ t : T and t ⇒ t', then ⊢ t' : T. Proof: By induction on a derivation of ⊢ t : T.

• If the last rule in the derivation is T_If, then t = if t1 then t2 else t3, with ⊢ t1 : Bool, ⊢ t2 : T and ⊢ t3 : T.
  Inspecting the rules for the small-step reduction relation and remembering that t has the form if ..., we see that the only ones that could have been used to prove t ⇒ t' are ST_IfTrue, ST_IfFalse, or ST_If.
  • If the last rule was ST_IfTrue, then t' = t2. But we know that ⊢ t2 : T, so we are done.
  • If the last rule was ST_IfFalse, then t' = t3. But we know that ⊢ t3 : T, so we are done.
  • If the last rule was ST_If, then t' = if t1' then t2 else t3, where t1 ⇒ t1'. We know ⊢ t1 : Bool so, by the IH, ⊢ t1' : Bool. The T_If rule then gives us ⊢ if t1' then t2 else t3 : T, as required.

(* FILL IN HERE *)
☐

Exercise: 2 stars (finish_preservation)

☐

Exercise: 3 stars (preservation_alternate_proof)

Now prove the same property again by induction on the evaluation derivation instead of on the typing derivation. Begin by carefully reading and thinking about the first few lines of the above proof to make sure you understand what each one is doing. The set-up for this proof is similar, but not exactly the same.

☐

Type Soundness

Putting progress and preservation together, we can see that a well-typed term can never reach a stuck state.

Indeed, in the present — extremely simple — language, every well-typed term can be reduced to a normal form: this is the normalization property. (The proof is straightforward, though somewhat tedious.) In richer languages, this property often fails, though there are some interesting languages (such as Coq's Fixpoint language, and the simply typed lambda-calculus, which we'll be looking at next) where all well-typed terms can be reduced to normal forms.

Additional Exercises

Exercise: 2 stars, recommended (subject_expansion)

Having seen the subject reduction property, it is reasonable to wonder whether the opposity property — subject expansion — also holds. That is, is it always the case that, if t ⇒ t' and has_type t' T, then has_type t T? If so, prove it. If not, give a counter-example. (* FILL IN HERE *)
☐

Exercise: 2 stars (variation1)

Suppose we add the following two new rules to the evaluation relation: Which of the following properties remain true in the presence of these rules? For each one, write either "remains true" or else "becomes false." If a property becomes false, give a counterexample.

• Determinacy of step
• Normalization of step for well-typed terms
• Progress
• Preservation

☐

Exercise: 2 stars (variation2)

Suppose, instead, that we add this new rule to the typing relation: Which of the following properties remain true in the presence of this rule? (Answer in the same style as above.)

• Determinacy of step
• Normalization of step for well-typed terms
• Progress
• Preservation

☐

Exercise: 2 stars (variation3)

Suppose, instead, that we add this new rule to the typing relation: Which of the following properties remain true in the presence of this rule? (Answer in the same style as above.)

• Determinacy of step
• Normalization of step for well-typed terms
• Progress
• Preservation

☐

Exercise: 2 stars (variation4)

Suppose, instead, that we add this new rule to the step relation: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. ☐

Exercise: 2 stars (variation5)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. ☐

Exercise: 2 stars (variation6)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. ☐

Exercise: 2 stars (variation7)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. ☐

Exercise: 2 stars (variation8)

Suppose instead that we add this rule: Which of the above properties become false in the presence of this rule? For each one that does, give a counter-example. ☐

Exercise: 3 stars, optional (more_variations)

Make up some exercises of your own along the same lines as the ones above. Try to find ways of selectively breaking properties — i.e., ways of changing the definitions that break just one of the properties and leave the others alone. ☐

Exercise: 1 star (remove_predzero)

The evaluation rule E_PredZero is a bit counter-intuitive: we might feel that it makes more sense for the predecessor of zero to be undefined, rather than being defined to be zero. Can we achieve this simply by removing the rule from the definition of step? (* FILL IN HERE *)
☐

Exercise: 4 stars, optional (prog_pres_bigstep)

Suppose our evaluation relation is defined in the big-step style. What are the appropriate analogs of the progress and preservation properties? (* FILL IN HERE *)
☐