On Intuitionistic Linear Logic - Microsoft Research

§1. Sequent Calculus 21
(p) Contraction.
Π1
Γ − A
π2
A n , !B, !B, ∆ − C Contraction
A n , !B, ∆ − C Cutn
Γ n , !B, ∆ − C
Let Π ′ be the proof obtained by applying the induction hypothesis to the proof
Π1
Γ − A
π2
A n , !B, !B, ∆ − C Cutn.
Γ n , !B, !B, ∆ − C
We have by assumption that c(Π1), c(π2) ≤ |A| and hence c(Π ′ ) ≤ |A|. We can form the
proof, Π,
Π ′
By definition c(Π) ≤ |A| and we are done.
(q) Promotion.
Π1
!Γ − !A
Γ n , !B, !B, ∆ − C Contraction.
Γ n , !B, ∆ − C
π2
!A n , !∆ − B Promotion
!A n , !∆ − !B Cutn
!Γ n , !∆ − !B
Let Π ′ be the proof obtained by applying the induction hypothesis to the proof
Π1
!Γ − !A
!Γ n , !∆ − B
π2
!A n , !∆ − B Cutn.
We have by assumption that c(Π1), c(π2) ≤ |!A| and hence c(Π ′ ) ≤ |!A|. We can form
the proof, Π,
Π ′
By definition c(Π) ≤ |!A| and we are done.
!Γ n , !∆ − B Promotion.
!Γ n , !∆ − !B
3. When either Π1 or Π2 is an instance of the Identity rule. Firstly
which is replaced by
And similarly the proof
Π1
Γ − A
Γ − A
Identity
A − A Cut,
Π1
Γ − A.