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76

CHAPTER 10. SUMS IN NATURAL DEDUCTION

10.3.2Extension to the full fragment

For the full calculus, we come against an enormous di culty: it is no longer true that the conclusion of an elimination is a subformula of its principal premise: the \C" of the three elimination rules has nothing to do with the eliminated formula. So we are led to restricting the notion of principal branch to those eliminations which are well-behaved (^1E, ^2E, )E and 8E) and we can try to extend our theorem. Of course it will be necessary to restrict part (ii) to the case where ends in a \good" elimination.

The theorem is proved as before in the case of introductions, but the case of eliminations is more complex:

If ends in a good elimination, look at its principal premise A: we shall be embarrassed in the case where A is the conclusion of a bad elimination. Otherwise we conclude the existence of a principal branch.

If ends in a bad elimination, look again at its principal premise A: it is not the conclusion of an introduction. If A is a hypothesis or the conclusion of a good elimination, it is a subformula of a hypothesis, and the result follows easily. There still remains the case where A comes from a bad elimination.

To sum up, it would be necessary to get rid of con gurations formed from a succession of two rules: a bad elimination of which the conclusion C is the principal premise of an elimination, good or bad. Once we have done this, we can recover the Subformula Property. A quick calculation shows that the number of con gurations is 3 7 = 21 and there is no question of considering them one by one. In any case, the removal of these con gurations is certainly necessary, as the

following example shows:

 

 

 

 

 

 

[A] [A]

 

[A] [A]

 

 

 

 

 

^I

 

 

^I

 

A _ A A ^ A

 

 

 

 

A ^ A

 

 

 

A ^ A

 

^1E

 

_E

 

 

 

 

 

 

 

 

 

A

 

 

 

which does not satisfy the Subformula Property.

10.4Commuting conversions

In what follows, C ... r denotes an elimination of the principal premise C, the

D

conclusion is D and the ellipsis represents some possible secondary premises with the corresponding deductions. This symbolic notation covers the seven cases of elimination.

10.4. COMMUTING CONVERSIONS

77

1. commutation of ?E:

 

 

 

 

 

 

 

 

converts to

 

 

 

 

?

 

 

 

 

 

 

?E ..

 

?

?E

C

.

r

 

 

 

D

 

 

 

 

 

 

D

 

 

 

2. commutation of _E:

 

 

 

[A]

[B]

 

 

 

[A]

 

[B]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

converts to

 

 

.

 

 

.

A

 

B

 

 

 

 

 

.

 

 

.

_

C

C

 

 

C .

 

 

C .

 

 

 

 

 

.

 

 

 

 

 

r

 

 

 

r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C

 

 

_E .

 

 

D

 

 

D

 

 

 

 

 

.

 

A _ B

 

 

 

 

 

 

 

D

 

r

 

 

D

 

 

 

_E

3. commutation of 9E:

 

 

[A]

 

 

[A]

 

 

 

 

 

 

 

 

 

converts to

 

 

.

 

 

 

 

 

.

 

: A

C

 

C .

 

9

 

 

.

 

 

 

 

 

r

 

 

 

 

 

 

 

C

 

9E .

 

 

 

D

 

.

 

9 : A

 

 

 

 

D

 

r

D

 

 

9E

Example The most complicated situation is:

 

 

 

 

[A]

 

[B]

 

 

 

 

 

 

 

 

 

 

 

 

 

[C]

[D]

 

 

 

 

 

 

 

 

 

 

 

converts to

 

 

 

 

 

 

 

 

 

 

A

_

B

C

_

D

C

_

D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_E

 

 

 

 

 

 

C _ D

 

 

 

 

 

 

 

 

 

 

 

E

E

_E

 

 

 

 

 

 

 

 

E

 

 

 

78

 

 

 

 

CHAPTER 10.

SUMS IN NATURAL DEDUCTION

 

 

 

[A]

[C]

[D]

 

 

[B]

[C]

[D]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C

_

D

E

E

 

C

_

D

 

E

E

 

 

 

 

 

 

 

 

 

 

 

 

 

_E

 

 

 

 

 

 

 

_E

 

 

 

 

 

 

 

 

 

 

 

 

A _ B

 

 

 

E

 

 

 

 

 

 

E

 

 

 

 

 

 

 

 

E

 

 

 

 

 

_E

 

 

We see in particular that the general case (with an unspeci ed elimination r) is more intelligible than its 21 specialisations.

10.5Properties of conversion

First of all, the normal form, if it exists, is unique: that follows again from a Church-Rosser property. The result remains true in this case, since the con icts of the kind

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[A]

[B]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ I

 

 

 

 

 

 

 

 

 

 

 

 

 

A _ B

 

C

C

_E ..

 

 

 

 

 

 

 

 

 

C

 

 

 

 

 

 

 

 

 

 

 

 

 

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r

 

 

 

 

 

 

 

 

 

 

 

 

D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

which converts in two di erent ways, namely

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[A]

[B]

[A]

 

 

 

 

 

 

 

 

 

 

 

 

 

and

 

 

 

 

 

.

 

.

 

.

 

 

 

 

 

 

.

 

.

 

 

 

 

 

A

 

 

 

C .

C .

 

.

 

 

 

 

_1I

 

 

C .

 

 

 

 

 

 

 

 

 

r

 

 

 

r

 

r

A _ B

 

 

 

 

 

 

 

 

 

 

D

 

D

 

 

 

 

 

D

 

 

 

 

 

 

 

 

 

 

 

_E

 

 

 

 

 

 

 

 

 

D

 

 

are easily resolved, because the second deduction converts to the rst.

It is possible to extend the results obtained for the (^; ); 8) fragment to the full calculus, at the price of boring complications. [Prawitz] gives all the technical details for doing this. The abstract properties of reducibility for this case are in [Gir72], and there are no real problems when we extend this to existential quanti cation over types.

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