On Intuitionistic Linear Logic - Microsoft Research

§2. Natural Deduction 37
commutes to
Γ1 ···
A1 . . .
t
Γn ···
An
(tI)
Γ1 . . . Γn
(tI).
t
In fact, we could have been more refined for these two cases. If the minor premises were the
conclusion of an elimination rule, then it would be the case that it was a subformula of an open
assumption. We shall keep these more global rules as we shall see later that they correspond more
closely with the steps in the cut elimination process.
Now consider the Promotion rule (the introduction rule for the exponential)
·
!A1 . . .
·
!An
!B
[!A1 · · · !An ]
·
B
Promotion.
We have a problem here as we do not know immediately that the !Ai are subformulae of an assumption
or of !B. Consider a deduction in β-normal form ending with an application of the Promotion
rule. If !Ai is the conclusion of a bad elimination then a similar argument to that used earlier gives
that the path need not be a subformula path. 9 For example, consider the β-normal deduction
!C⊗!(C−◦A−◦B)
[!C] [!(C−◦A−◦B)]
[!C] Der
C
[!(C−◦A−◦B)] Der
C−◦A−◦B (−◦E)
A−◦B Prom
!(A−◦B) (⊗E)
!(A−◦B) !A
!B
[!(A−◦B)] Der
A−◦B
B Prom.
[!A] Der
Clearly the formula !(A−◦B) is not a subformula of either an open assumption nor of of the conclusion.
As before we shall introduce further commuting conversions to eliminate the problematic
occurrence when the conclusion of a bad elimination rule is a minor premise of an application of the
Promotion rule. Thus a deduction of the form
commutes to
·
!A1 . . .
·
C
·
C
!Ai
·
!A1 . . .
!B
·
!Ai
(badE)
!B
·
!Ai . . .
. . .
·
!An
·
!An
!B (badE).
[!A1 · · ·!An ]
·
B
Promotion
[!A1 · · ·!An ]
·
B
Promotion
9 The behaviour of the Promotion rule with respect to the subformula property was omitted from an earlier
paper [16].
A (−◦E)