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# Strong Normalisation Theorem

In this chapter we shall prove the strong normalisation theorem for the simple typed -calculus, but since we have already discussed this topic at length, and in particular proved weak normalisation, the purpose of the chapter is really to introduce the technique which we shall later apply to system F.

For simple typed -calculus, there is proof theoretic techniques which make it possible to express the argument of the proof in arithmetic, and even in a very weak system. However our method extends straightforwardly to G•odel's system T, which includes a type of integers and hence codes Peano Arithmetic. As a result, strong normalisation implies the consistency of PA, which means that it cannot itself be proved in PA (Second Incompleteness Theorem).

Accordingly we have to use a strong induction hypothesis, for which we introduce an abstract notion called reducibility, originally due to [Tait]. Some of the technical improvements, such as neutrality, are due to [Gir72]. Besides proving strong normalisation, we identify the three important properties (CR 1-3) of reducibility which we shall use for system F in chapter 14.

## 6.1Reducibility

We de ne a set REDT (\reducible1 terms of type T ") by induction on the type T .

1.For t of atomic type T , t is reducible if it is strongly normalisable.

2.For t of type U V , t is reducible if 1t and 2t are reducible.

3. For t of type U!V , t is reducible if, for all reducible u of type U, t u is reducible of type V .

1This is an abstract notion which should not be confused with reduction.

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