Normalization for cbv System T

In this document we consider System T with call-by-value reduction, and prove that every (well-typed) term in this system reduces to a value. The proof largely follows the normalization proof given in Chapter 12 of Pierce's "Types and Programming Languages" book, which is recommended reading.

We begin by importing some definitions and facts from the standard library, and define a variant of the inversion tactic (that we seen previously).

Syntax

Identifiers

Identifiers are represented by natural numbers, using the wrapper Id. Two identifiers are equal if and only if they are represented by the same natural number.

Types and terms

The syntax of types and terms are defined as inductive types; values are described by an inductively defined predicate:

Environments

We use a polymorphic type for environments: env X is the type of partial functions from identifiers to X. This has the advantage that we can use env to implement both typing contexts (as env ty) as well as parallel substitutions (as env tm, see below).

The following operation pointwise "lifts" a relation R between X and Y to a relation Pointwise X Y R between env X and env Y: two environments are related if they have the same domain and for each argument the respective images are related by R.

Parallel Substitution

To state and prove the propositions that lead to the normalization proof for System T it is convenient to phrase the operational semantics in terms of a parallel substitution operation. This operation generalizes our earlier substitution [x:=s]t so that we can substitute multiple terms simultaneously, [x1:=s1 ... xn:=sn]t.

We follow our earlier approach and just state the properties of parallel substitution that we need as assumptions, rather than implementing a particular parallel substitution operation. However, to formulate these assumptions in a meaningful way, we restrict substitutions so that s1,...,sn in [x1:=s1 ... xn:=sn]t are closed terms. (Otherwise, we would need to be much more careful to avoid the possible capture of free variables of the terms si by the binders within t.)

Free variables and closed terms

The proposition appears_free_in x t characterizes those variables x that appear free in t:

It is now easy to define when a term t is closed, and when a substitution gamma substitutes only closed terms which we call (somewhat inaccurately) closing:

Assumptions about parallel (simultaneous) substitution

We assume the existence of a parallel (simultaneous) substitution operation simsubst:

and axiomatize its behaviour on terms:

Ordinary substitution is defined as a special case of simultaneous substitution:

We also assume that successive substitutions can be merged, expressed by the following hypothesis:

As a first sanity check for our definitions and the assumptions we made so far, we prove that applying the empty substitution to a term t does not change t.

Reduction Relation

The reduction relation is the one we have seen previously:

We can introduce some nice notation for the single step relation and its reflexive transitive closure. To define the latter, we use the operation clos_refl_trans that is defined in the Coq library.

A useful observation is that values are irreducible (have no successor).

Reductions in (some particular) reduction contexts

In the normalization proof we sometimes need to construct new reduction sequences from given ones. Usually these new reductions will simulate the given ones in a reduction context. To realize these constructions without too much repetition of proofs, it is useful to abstract slightly and consider a notion of compatible relation. (In fact, we have seen this already in the coincidence proof for PCF when we showed the agreement between the small-step reduction semantics and the evaluation relation.)

We then instantiate this definition to three particular cases: reductions of function arguments, reductions below S, and reductions of the first argument of primitive recursions.

Typing

The inductive definition of the typing relation is straightforward:

A simple observation that relates the typing relation and the notion of free variables of a term is given next.

Type preservation

We assume a substitution lemma. This lemma takes a more general form than the one we have seen earlier for PCF - the reason is that we need to adapt it to the parallel substitutions that we are using now.

With the help of this substitution lemma, it is not too difficult to prove the type preservation theorem for System T:

The Logical Relation R

The property that we are interested in eventually is that every typable term reduces to a value:

However, this property turns out to be "too weak" to be established directly by an inductive proof. To get a sufficiently strong induction hypothesis, we consider the following type-indexed family of predicates, the "logical" relation R:

This family is defined recursively on the type T:

• R ty_nat is simply halts, i.e., the property we're interested in, and
• R (ty_arrow T1 T2) additionally expresses that application preserves the property of being in the relation. Accordingly R (ty_arrow T1 T2) it is defined in terms of R T1 and R T2.

The relation can be extended to substitutions straightforwardly, using the pointwise operation, as pointwise R Gamma.

Note that it is not possible to define R using Inductive: the reason is that in the right hand side of the case R (ty_arrow T1 T2) the relation R appears in a "negative" position (to the left of an implication), which conflicts with the "monotonicity" requirement that Coq enforces for inductive definitions.

From the definition it is clear that R T consists of closed terms of type T:

It is also immediate from the definition that R implies normalization:

Closure properties of R

For the normalization proof we need two closure properties of the logical relation:

• closure under reduction
• closure under expansion

To establish these, we must in particular prove that the predicate halts is closed under both reduction and expansion. In turn, to prove the former, we first show that the reduction relation is deterministic:

We prove that each R T is closed under reduction by induction on T.

Similarly, we prove that each R T is closed under expansion by induction on T. Since this requires that both terms are of type T, we must add an additional assumption (why?).

Preservation properties for R

We can now prove that R is preserved by all the typing rules of System T. We begin with the case of abstraction:

This was the most difficult case, because the function body t will in general have an x as an additional free variable. The cases of the remaining preservation lemmas reflect the respective typing rules even more directly. They are also easier to prove, essentially because the free variables of the direct subterms are the same as those of the term itself:

Basic lemma for the logical relation

We prove that typable terms of type T are in the relation R T. For the induction to go through this needs to be generalized to open terms (in context Gamma) and substitutions (that are in pointwise R Gamma). We also need to know that we can modify substitutions, by updating with terms that are in R:

Now we can prove the so-called "basic lemma" for the relation R. Its proof is pretty straightforward - all the hard work has been done in the above preservation lemmas. More precisely, in each of the inductive cases we apply the appropriate preservation lemma. There are only two exceptional cases: First, the case of a variable follows from the assumptions on the substitution gamma. Second, the case of an abstraction which requires some preliminary work before the preservation lemma can be applied.

Since the relation R T is restricted to well-typed terms of type T, this property must be established when proving the basic lemma. To this end the following consequence of the assumed substitution lemma is used.

Normalization

Finally it is a fairly simple matter of connecting all the pieces and conclude normalization. This is left as an exercise. (Hint: Use R_halts and Basic_lemma. The lemma subst_id may also be useful.)