### Fix comment.

parent e14e9ec2
 ... @@ -23,37 +23,38 @@ Arguments uPred_holds {_} _%I _ _. ... @@ -23,37 +23,38 @@ Arguments uPred_holds {_} _%I _ _. Section cofe. Section cofe. Context {M : ucmraT}. Context {M : ucmraT}. (* A good way of understanding this defintion of the uPred OFE is to (* A good way of understanding this definition of the uPred OFE is to consider the OFE uPred0 of monotonous SProp predicates. That is, consider the OFE uPred0 of monotonous SProp predicates. That is, uPred0 is the OFE of non-expansive functions from M to SProp that uPred0 is the OFE of non-expansive functions from M to SProp that are monotonous with respect to CMRA inclusion. This notion of are monotonous with respect to CMRA inclusion. This notion of monotonicity has to be stated in the SProp logic. It is exactly monotonicity has to be stated in the SProp logic. Together with the uPred_mono. usual closedness property of SProp, this gives exactly uPred_mono. Then, we quotient uPred0 *in the sProp logic* with respect to Then, we quotient uPred0 *in the sProp logic* with respect to equivalence on valid elements of M. That is, we quotient with equivalence on valid elements of M. That is, we quotient with respect to the following *sProp* equivalence relation: respect to the following *sProp* equivalence relation: P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x)) (1) P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x)) (1) When seen from the ambiant logic, computing this logic require When seen from the ambiant logic, obtaining this quotient requires redefinig both a custom Equiv and Dist. 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 admits canonical representatives. More precisely, one can show that every representatives. More precisely, one can show that every equivalence class contain exactly one element P0 such that: equivalence class contains exactly one element P0 such that: ∀ x, (✓ x → P(x)) → P(x) (2) ∀ x, (✓ x → P(x)) → P(x) (2) (Again, this assertion has to be understood in sProp). Starting (Again, this assertion has to be understood in sProp). Starting from an element P of a given class, one can build this canonical from an element P of a given class, one can build this canonical representative by chosing: 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 Hence, as an alternative definition of uPred, we could use the set of canonical representatives (i.e., the subtype of monotonous of canonical representatives (i.e., the subtype of monotonous sProp predicates that verify (2)). This alternative definition would sProp predicates that verify (2)). This alternative definition would save us from using a quotient. However, the definitions of the various save us from using a quotient. However, the definitions of the various connectives would get more complicated, because we have to make sure connectives would get more complicated, because we have to make sure they all verify (2), which sometimes requires some adjustments. We would they all verify (2), which sometimes requires some adjustments. We moreover need to prove one more property for every logical connective. would moreover need to prove one more property for every logical connective. *) *) Inductive uPred_equiv' (P Q : uPred M) : Prop := Inductive uPred_equiv' (P Q : uPred M) : Prop := ... ...
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