On Intuitionistic Linear Logic - Microsoft Research

§2. Natural Deduction 33
(We have introduced a shorthand notation in the last two cases, where Weakening ∗ and Contraction ∗
represent multiple applications of the Weakening and Contraction rule respectively.) The process
above gives rise to a relation which we shall denote by ❀β. We shall describe one of the steps as
a β-reduction rule and say that a deduction is β-reducible if we can apply one of the β-reduction
rules.
Definition 6. A deduction D is said to be in β-normal form if it is not β-reducible.
2.2 Subformula Property and Commuting Conversions
We have already considered the subformula property in §1.3, in the context of the sequent calculus
formulation. Whereas in that case the property held simply by inspection of the derivation rules,
things are more delicate in the natural deduction formulation. For example, consider a deduction
which ends with an application of the −◦E rule
·
A−◦B
· D1
B
· D2
A
(−◦E).
Even if the deduction is in β-normal form, it is not simply the case by inspection that the subformula
property holds, i.e. that A−◦B is a subformula of Γ or ∆ (it is certainly not a subformula of B!).
We also note that for most of the elimination rules the conclusion is not a subformula of its major
premise. 8 These rules are (⊗E), (IE), (⊕E), (fE), Weakening and Contraction; which we shall refer
to collectively as bad eliminations, with those remaining being known as good eliminations.
We shall introduce the notion of a path through a deduction. The idea is that we trace downwards
through a deduction from an assumption, with the hope that each trace yields a path with the
property that the every formula is a subformula of either an open assumption or of the final formula
in the path. Our notion of a path is based on Prawitz’s treatment of IL [62], although he attributes
the idea to Martin-Löf.
Definition 7. A path in a β-normal deduction, D, is a sequence of formulae, A0, . . .,An, such that
1. A0 is an open assumption or axiom; and
2. Ai+1 follows Ai if
(a) Ai+1 is the conclusion of an introduction rule (excluding (tI)) and Ai is a premise of the
rule (if the rule is Promotion, then Ai must not be a minor premise); or
(b) Ai+1 is the conclusion of an elimination rule and Ai is either the major premise of a good
elimination rule, or the minor premise of a bad elimination rule (excluding (fE)); or
(c) Ai is the major premise of a bad elimination rule (excluding (fE)) and Ai+1 is an assumption
discharged by that application; or
(d) Ai is the minor premise of an application of the Promotion rule and Ai+1 is the corresponding
assumption discharged by that assumption; and
3. An is either the conclusion of D, or the major premise of (IE) or Weakening, or a premise of
(fE).
We shall identify a particular path as mentioned earlier.
Definition 8. A subformula path is a path in a deduction, D, such that every formula in it is either
a subformula of an assumption or of the final formula in the path.
Consider a β-normal deduction, D, of the form
8 Girard [34] calls such conclusions parasitic formulae.