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Logic and CS / Girard. Proofs and Types.pdf
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 74 CHAPTER 10. SUMS IN NATURAL DEDUCTION [A] [B] A converts to A 1 I _ _ A B C C _E C C [A] [B] B converts to B 2 I _ _ A B C C _E C C [A] converts to A[a= ] A[a= ] 9I 9 : A C 9E C C

Note that, since there is no introduction rule for ?, there is no standard conversion for this symbol.

Let us just think for a moment about the structure of redexes: on the one hand there is an introduction, on the other an elimination, and the elimination follows the introduction. But there are some eliminations (), _, 9) with more premises and we only consider as redexes the case where the introduction ends in the principal premise of the elimination, namely the one which carries the

 eliminated symbol. For example [A] B )I A ) B (A ) B) ) C C )E

is not considered as a redex. This is fortunate, as we would have trouble converting it!

# 10.3The need for extra conversions

To understand how we are naturally led to introducing extra conversions, let us examine the proof of the Subformula Property in the case of the (^; ); 8) fragment in such a way as to see the obstacles to generalising it. 10.3. THE NEED FOR EXTRA CONVERSIONS 75

## 10.3.1Subformula Property

Theorem Let be a normal deduction in the (^ ) 8) fragment. Then

i)every formula in is a subformula of a conclusion or a hypothesis of ;

ii)if ends in an elimination, it has a principal branch, i.e. a sequence of formulae A0; A1; : : : ; An such that:

A0 is an (undischarged) hypothesis;

An is the conclusion;

Ai is the principal premise of an elimination of which the conclusion is Ai+1 (for i = 0; : : : ; n 1).

In particular An is a subformula of A0.

Proof We have three cases to consider:

1.If consists of a hypothesis, there is nothing to do.

2.If ends in an introduction, for example

AB

^I

A ^ B

then it su ces to apply the induction hypothesis above A and B.

3. If ends in an elimination, for example

A ) B A

)E

B

it is not possible that the proof above the principal premise ends in an introduction, so it ends in an elimination and has a principal branch, which can be extended to a principal branch of .