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908ea111
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908ea111
authored
Dec 07, 2017
by
Jacques-Henri Jourdan
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Fix comment.
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theories/base_logic/upred.v
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theories/base_logic/upred.v
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908ea111
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@@ -23,37 +23,38 @@ Arguments uPred_holds {_} _%I _ _.
Section
cofe
.
Context
{
M
:
ucmraT
}.
(* A good way of understanding this defintion of the uPred OFE is to
(* A good way of understanding this defin
i
tion of the uPred OFE is to
consider the OFE uPred0 of monotonous SProp predicates. That is,
uPred0 is the OFE of non-expansive functions from M to SProp that
are monotonous with respect to CMRA inclusion. This notion of
monotonicity has to be stated in the SProp logic.
It is exactly
uPred_mono.
monotonicity has to be stated in the SProp logic.
Together with the
usual closedness property of SProp, this gives exactly
uPred_mono.
Then, we quotient uPred0 *in the sProp logic* with respect to
equivalence on valid elements of M. That is, we quotient with
respect to the following *sProp* equivalence relation:
P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x)) (1)
When seen from the ambiant logic,
comput
ing this
logic
require
re
definig both a custom Equiv and Dist.
When seen from the ambiant logic,
obtain
ing this
quotient
require
s
definig both a custom Equiv and Dist.
It is worth noting that this equivalence relation admit canonical
It is worth noting that this equivalence relation admit
s
canonical
representatives. More precisely, one can show that every
equivalence class contain exactly one element P0 such that:
equivalence class contain
s
exactly one element P0 such that:
∀ x, (✓ x → P(x)) → P(x) (2)
(Again, this assertion has to be understood in sProp). Starting
from an element P of a given class, one can build this canonical
representative by chosing:
P0(x) = ✓ x → P(x)
(3)
P0(x)
:
= ✓ x → P(x) (3)
Hence, as an alternative definition of uPred, we could use the set
of canonical representatives (i.e., the subtype of monotonous
sProp predicates that verify (2)). This alternative definition would
save us from using a quotient. However, the definitions of the various
connectives would get more complicated, because we have to make sure
they all verify (2), which sometimes requires some adjustments. We would
moreover need to prove one more property for every logical connective.
they all verify (2), which sometimes requires some adjustments. We
would moreover need to prove one more property for every logical
connective.
*)
Inductive
uPred_equiv'
(
P
Q
:
uPred
M
)
:
Prop
:
=
...
...
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