On Intuitionistic Linear Logic - Microsoft Research

§2. Natural Deduction 29
It is possible to present natural deduction rules in a ‘sequent-style’, where given a sequent
Γ − A, Γ represents all the undischarged assumptions so far in the deduction, and A represents
conclusion of the deduction. We can still label the undischarged assumptions with a unique natural
number, but we refrain from doing so. This formulation should not be confused with the sequent
calculus formulation given in §1, which differs by having operations which act on the left and right
of the turnstile, rather than rules for the introduction and elimination of logical connectives. The
‘sequent-style’ formulation of natural deduction is given in Figure 2.3.
2.1 β-Reductions
With a natural deduction formulation we can produce so-called ‘detours’ in a deduction, which arise
where we introduce a logical connective and then eliminate it immediately afterwards. We can define
the normalization procedure by considering each case of an introduction rule followed immediately
by a corresponding elimination rule in turn.
• (−◦I) followed by (−◦E)
normalizes to
• (II) followed by (IE)
normalizes to
• (⊗I) followed by (⊗E)
normalizes to
[A]
·
B
(−◦I)
A−◦B
·
A
A⊗B
B
·
A
A
·
[A]
·
B.
(II)
I (IE)
·
A.
·
B
(⊗I)
C
· ·
· ·
· ·
[A] [B]
·
C.
·
A
(−◦E)
[A] [B]
·
C
(⊗E)