i.e any sequent of the form
Γ
1
, A,
Γ
2
-→
A,
(2)
for any formula
A
∈ F
and any sequences Γ
1
,
Γ
2
∈ F
*
.
Inference rules of LI
The set inference rules is divided into two groups: the structural rules and the
logical rules. They are defined as follows.
Structural Rules of LI
Weakening
(
→
weak
)
Γ
-→
Γ
-→
A
.
A
is called the weakening formula.
Contraction
(
contr
→
)
A, A,
Γ
-→
Δ
A,
Γ
-→
Δ
,
A
is called the contraction formula , Δ contains at most one formula.
Exchange
(
exchange
→
)
Γ
1
, A, B,
Γ
2
-→
Δ
Γ
1
, B, A,
Γ
2
-→
Δ
,
Δ contains at most one formula.
Logical Rules of LI
Conjunction rules
(
∩ →
)
A, B,
Γ
-→
Δ
(
A
∩
B
)
,
Γ
-→
Δ
,
(
→ ∩
)
Γ
-→
A
; Γ
-→
B
Γ
-→
(
A
∩
B
)
,
Δ contains at most one formula.
Disjunction rules
(
→ ∪
)
1
Γ
-→
A
Γ
-→
(
A
∪
B
)
,
(
→ ∪
)
2
Γ
-→
B
Γ
-→
(
A
∪
B
)
,
2