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a27982f7
Commit
a27982f7
authored
Dec 08, 2017
by
Ralf Jung
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theories/base_logic/upred.v
theories/base_logic/upred.v
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theories/base_logic/upred.v
View file @
a27982f7
...
...
@@ -6,6 +6,40 @@ Set Default Proof Using "Type".
base_logic.base_logic; that will also give you all the primitive
and many derived laws for the logic. *)
(* A good way of understanding this definition 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. 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, obtaining this quotient requires
definig both a custom Equiv and Dist.
It is worth noting that this equivalence relation admits canonical
representatives. More precisely, one can show that every
equivalence class contains 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)
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.
*)
Record
uPred
(
M
:
ucmraT
)
:
Type
:
=
IProp
{
uPred_holds
:
>
nat
→
M
→
Prop
;
...
...
@@ -23,40 +57,6 @@ Arguments uPred_holds {_} _%I _ _.
Section
cofe
.
Context
{
M
:
ucmraT
}.
(* A good way of understanding this definition 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. 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, obtaining this quotient requires
definig both a custom Equiv and Dist.
It is worth noting that this equivalence relation admits canonical
representatives. More precisely, one can show that every
equivalence class contains 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)
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.
*)
Inductive
uPred_equiv'
(
P
Q
:
uPred
M
)
:
Prop
:
=
{
uPred_in_equiv
:
∀
n
x
,
✓
{
n
}
x
→
P
n
x
↔
Q
n
x
}.
Instance
uPred_equiv
:
Equiv
(
uPred
M
)
:
=
uPred_equiv'
.
...
...
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