Situations from events to proofs 1 Introduction

(i)
every (finite or infinite) subset X of E to
X +
def
= X ∪ {past(e) | e ∈ X and (∃e ′ ) e ′ < e} ∪
{future(e) | e ∈ X and (∃e ′ ) e < e ′ }
(ii) ○ to overlap ○ + on E + , setting for all e, e ′ ∈ E,
e ○ + e ′
def
⇔ e ○ e ′
with past(e) and future(e) given by existential quantification
past(e) ○ + e ′
future(e) ○ + e ′
past(e) ○ + future(e ′ )
def
⇔ (∃e ′′ < e) e ′′ ○ e ′
def
⇔ (∃e ′′ > e) e ′′ ○ e ′
def
⇔ (∃e ′′ < e)(∃e ′′′ > e ′ ) e ′′ ○ e ′′′
and so on, and
(iii)
< to precedence < + on E + , defining < + to be the union
< ∪ {(past(e), e ′ ) ∈ E + × E + | not past(e) ○ + e ′ }
∪ {(e, future(e ′ )) ∈ E + × E + | not e ○ + future(e ′ )}
∪ {(past(e), future(e ′ )) ∈ E + × E + | not past(e) ○ + future(e ′ )}
so that for all x, y ∈ E + ,
x < + y iff neither x ○ + y nor y < + x .
One can show that for every subset X of E, 〈X + , < X + , ○ X + 〉 is an event structure,
where < X + and ○ X + are the restrictions of < + and ○ + to X + , respectively
< X +
def
= < + ∩ (X + × X + ) and ○ X +
def
= ○ + ∩ (X + × X + ) .
We can then form a compaction p of 〈E,