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 26 CHAPTER 4. THE NORMALISATION THEOREM

## 4.3.3Conversion of maximal degree

Lemma Let r be a redex of maximal degree n in t, and suppose that all the redexes strictly contained in r have degree less than n. If u is obtained from t by converting r to c then u has strictly fewer redexes of degree n.

Proof When the conversion is made, the following things happen:

The redexes outside r remain.

The redexes strictly inside r are in general conserved, but sometimes

proliferated: for example if one replaces ( x: hx; xi) s by hs; si, the redexes of s are duplicated. The hypothesis made does not exclude duplication, but it is limited to degrees less than n.

 The redex r is destroyed and possibly replaced by some redexes of strictly smaller degree.

## 4.3.4Proof of the theorem

If t is a term, consider (t) = (n; m) with

n = d(t) m = number of redexes of degree n

Lemma 4.3.3 says that it is possible to choose a redex r of t in such a way that, after conversion of r to c, the result t0 satis es (t0) < (t) for the lexicographic

 order, i.e. if (t0) = (n0; m0) then n0 < n or (n0 = n and m0 < m). So the result is established by a double induction.

# 4.4The strong normalisation theorem

The weak normalisation theorem is in fact a bit better than its statement leads us to believe, because we have a simple algorithm for choosing at each step an appropriate redex which leads us to the normal form. Having said this, it is interesting to ask whether all normalisation strategies converge.

A term t is strongly normalisable when there is no in nite reduction sequence beginning with t. 4.4. THE STRONG NORMALISATION THEOREM 27

Lemma t is strongly normalisable i there is a number (t) which bounds the length of every normalisation sequence beginning with t.

Proof From the existence of (t), it follows immediately that t is strongly normalisable.

The converse uses K•onig's lemma1: one can represent a sequence of conversions by specifying a redex r0 of t0, then a redex r1 of t1, and so on. The possible sequences can then be arranged in the form of a tree, and the fact that a term has only a nite number of subterms assures us that the tree is nitely-branching. Now, the strong normalisation hypothesis tells us that the tree has no in nite branch, and by K•onig's lemma, the whole tree must be nite, which gives us the existence of (t).

There are several methods to prove that every term (of the typed -calculus) is strongly normalisable:

internalisation: this consists of a tortuous translation of the calculus into itself in such a way as to prove strong normalisation by means of weak normalisation. Gandy was the rst to use this technique [Gandy].

reducibility: we introduce a property of \hereditary calculability" which allows us to manipulate complex combinatorial information. This is the method we shall follow, since it is the only one which generalises to very complicated situations. This method will be the subject of chapter 6.

1A nitely branching tree with no in nite branch is nite. Unless the branches are labelled (as they usually are), this requires the axiom of Choice.