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 ... ... @@ -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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