The deep reason why reducibility works where combinatorial intuition fails is its logical complexity. Indeed, we have:

We see that in passing to U!V , REDU has been negated, and a universal quanti er has been added. In particular the normalisation argument cannot be directly formalised in arithmetic because t 2 REDT is not expressed as an arithmetic formula in t and T .

6.2Properties of reducibility

First we introduce a notion of neutrality: a term is called neutral if it is not of the form hu; vi or x: v. In other words, neutral terms are those of the form:

x 1t 2t t u

The conditions that interest us are the following:

(CR 1) If t 2 REDT , then t is strongly normalisable.

(CR 2) If t 2 REDT and t t0, then t0 2 REDT .

(CR 3) If t is neutral, and whenever we convert a redex of t we obtain a term t0 2 REDT , then t 2 REDT .

As a special case of the last clause:

(CR 4) If t is neutral and normal, then t 2 REDT .

We shall verify by induction on the type that RED satis es these conditions.

6.2.1Atomic types

A term of atomic type is reducible i it is strongly normalisable. So we must show that the set of strongly normalisable terms of type T satis es the three conditions:

(CR 1) is a tautology.

(CR 2) If t is strongly normalisable then every term t0 to which t reduces is also.

(CR 3) A reduction path leaving t must pass through one of the terms t0, which are strongly normalisable, and so is nite. In fact, it is immediate that (t) (see 4.4) is equal to the greatest of the numbers (t0) + 1, as t0 varies over the (one-step) conversions of t.

6.2.2Product type

A term of product type is reducible i its projections are.

(CR 1) Suppose that t, of type U V , is reducible; then 1t is reducible and by induction hypothesis (CR 1) for U, 1t is strongly normalisable. Moreover(t) ( 1t), since to any reduction sequence t; t1; t2; : : :, one can apply 1 to construct a reduction sequence 1t; 1t1; 1t2; : : : (in which the 1 is not reduced). So (t) is nite, and t is strongly normalisable.

(CR 3) Let t be neutral and suppose all the t0 one step from t are reducible. Applying a conversion inside 1t, the result is a 1t0, since 1t cannot itself be a redex (t is not a pair), and 1t0 is reducible, since t0 is. But as 1t is neutral, and all the terms one step from 1t are reducible, the induction hypothesis (CR 3) for U ensures that 1t is reducible. Likewise 2t, and so t is reducible.

6.2.3Arrow type

A term of arrow type is reducible i all its applications to reducible terms are reducible.

(CR 1) If t is reducible of type U!V , let x be a variable of type U; the induction hypothesis (CR 3) for U says that the term x, which is neutral and normal, is reducible. So t x is reducible. Just as in the case of the product type, we remark that (t) (t x). The induction hypothesis (CR 1) for V guarantees that (t x) is nite, and so (t) is nite, and t is strongly normalisable.

(CR 3) Let t be neutral and suppose all the t0 one step from t are reducible. Let u be a reducible term of type U; we want to show that t u is reducible. By induction hypothesis (CR 1) for U, we know that u is strongly normalisable; so we can reason by induction on (u).

In one step, t u converts to

t0 u with t0 one step from t; but t0 is reducible, so t0 u is.