diff git a/theories/base_logic/upred.v b/theories/base_logic/upred.v
index d51b589e125c07c5b891f08091498c2055e8ca60..6a7aa906bb6587c659406613a70c816319eb0d1b 100644
 a/theories/base_logic/upred.v
+++ b/theories/base_logic/upred.v
@@ 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 nonexpansive 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 nonexpansive 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'.