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CHAPTER 7. GODEL'S SYSTEM T

7.2Normalisation theorem

In T, all the reduction sequences are nite and lead to the same normal form.

Proof Part of the result is the extension of Church-Rosser; it is not di cult to extend the proof for the simple system to this more complex case. The other part is a strong normalisation result, for which reducibility is well adapted (it was for T that Tait invented the notion).

First, the notion of neutrality is extended: a term is called neutral if it is not of the form hu; vi, x: v, O, S t, T or F. Then, without changing anything, we show successively:

1. O, T and F are reducible | they are normal terms of atomic type.

2. If t of type Int is reducible (i.e. strongly normalisable), then S t is reducible

| that comes from (S t) = (t).

3.If u, v, t are reducible, then D u v t is reducible | u, v, t are strongly normalisable by (CR 1), and so one can reason by induction on the number(u) + (v) + (t). The neutral term D u v t converts to one of the following terms:

 

D u0 v0 t0

with u, v, t reduced respectively to u0, v0,

t0.

In

this case,

 

 

we have

(u0) + (v0) + (t0) < (u) + (v) + (t),

and

by

induction

 

hypothesis, the term is reducible.

 

 

 

u or v if t is T or F; these two terms are reducible.

 

 

 

We conclude by (CR 3) that D u v t is reducible.

4. If u, v, t are reducible, then R u v t is reducible | here also we reason by induction, but on (u) + (v) + (t) + `(t), where `(t) is the number of symbols of the normal form of t. In one step, R u v t converts to:

R u0 v0 t0 with etc. | reducible by induction.

u (if t = O) | reducible.

v (R u v w) w, where S w = t; since (w) = (t) and `(w) < `(t), the

induction hypothesis tells us that R u v w is reducible.

As v and w are,

v (R u v w) w is reducible by the de nition for U!V .

 

The use of the induction hypothesis in the nal case is really essential: it is the only occasion, in all the uses so far made of reducibility, where we truly use an induction on reducibility. For the other cases, the cognoscenti will see that we really have no need for induction on a complex predicate, by reformulating (CR 3) in an appropriate way.

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