[Add] Properties for DCPOs in Relation.Binary.Properties.Domain
#2734
Conversation
jmougeot
changed the title
Relation.Binary.Properties.DomainRelation.Binary.Properties.Domain
| → (scott : IsScottContinuous P Q f) | ||
| → IsOrderHomomorphism (Poset._≈_ P) (Poset._≈_ Q) (Poset._≤_ P) (Poset._≤_ Q) f | ||
| DirectedCompletePartialOrder+scott→monotone P-DirectedCompletePartialOrder f scott = record | ||
| { cong = λ {x} {y} x≈y → IsScottContinuous.preserveEquality scott x≈y |
There was a problem hiding this comment.
Don't name implicits that you do not use. Try to eta-reduce too.
| → IsOrderHomomorphism (Poset._≈_ P) (Poset._≈_ Q) (Poset._≤_ P) (Poset._≤_ Q) f | ||
| DirectedCompletePartialOrder+scott→monotone P-DirectedCompletePartialOrder f scott = record | ||
| { cong = λ {x} {y} x≈y → IsScottContinuous.preserveEquality scott x≈y | ||
| ; mono = λ {x} {y} x≤y → mono-proof x y x≤y |
There was a problem hiding this comment.
See if you can get away with _ for x and y in the rhs. If so, then make them implicit in the proof below. Otherwise leave a comment about not being able to do that.
| record IsScottContinuous (f : P.Carrier → Q.Carrier) | ||
| : Set (suc (c₁ ⊔ ℓ₁₁ ⊔ ℓ₁₂ ⊔ c₂ ⊔ ℓ₂₁ ⊔ ℓ₂₂)) where | ||
| field | ||
| preserveLub : ∀ {B : Set c₁} {g : B → P.Carrier} |
There was a problem hiding this comment.
I don't think this layout is as per the style guide.
| IsLub-cong : ∀ {s : Ix → D.Carrier} → {x y : D.Carrier} → x D.≈ y → | ||
| IsLub D.poset s x → IsLub D.poset s y | ||
| IsLub-cong x≈y x-lub = record | ||
| { isLeastUpperBound = |
There was a problem hiding this comment.
I think this would be clearer if the two parts were defined in a where clause
| isMonotone : (P-DirectedCompletePartialOrder : IsDirectedCompletePartialOrder P) → | ||
| (f : P.Carrier → Q.Carrier) → (scott : IsScottContinuous P Q f) → | ||
| IsOrderHomomorphism P._≈_ Q._≈_ P._≤_ Q._≤_ f | ||
| isMonotone P-DirectedCompletePartialOrder f scott = record |
There was a problem hiding this comment.
scott -> cts. It is the fact that it is continuous that is important, not the fact that this is 'Scott' continuity. isCts would be fine too.
| } | ||
|
|
||
| s-lub : IsLub P s y | ||
| s-lub = record { isLeastUpperBound = sx≤sfalse , (λ z proof → proof (lift false))} |
There was a problem hiding this comment.
elide z as it is not used.
| IsDirectedFamily P s → IsDirectedFamily Q (f ∘ s) | ||
| map-directed f ismonotone dir = record | ||
| { elt = IsDirectedFamily.elt dir | ||
| ; isSemidirected = λ i j → let (k , s[i]≤s[k] , s[j]≤s[k]) = IsDirectedFamily.isSemidirected dir i j |
There was a problem hiding this comment.
put the rhs of isSemiDirected in the where clause to improve readability
| module Q = Poset Q | ||
| module R = Poset R | ||
|
|
||
| ScottId : {P : Poset c ℓ₁ ℓ₂} → IsScottContinuous P P id |
| { preserveLub = λ _ _ → id | ||
| ; cong = id } | ||
|
|
||
| scott-cong : (f : R.Carrier → Q.Carrier) (g : P.Carrier → R.Carrier) → |
| ------------------------------------------------------------------------ | ||
| -- Scott continuity module | ||
|
|
||
| module Scott |
There was a problem hiding this comment.
Scott -> ScottContinuity
| EP = E.poset | ||
|
|
||
| module _ (f : D.Carrier → E.Carrier) | ||
| (isScott : IsScottContinuous DP EP f) |
| (λ i → f.mono (D.⋁-≤ i)) | ||
| ) | ||
|
|
||
| to-scott : (∀ {Ix} (s : Ix → D.Carrier) (dir : IsDirectedFamily DP s) → |
There was a problem hiding this comment.
toScott -> isScottContinuous
This pull request introduces new modules and properties for directed complete partial orders (DCPOs) in the Agda standard library. These additions are adapted from the 1Lab library.
The first part of this pull request is available at [Add] Initial files for Domain theory #2721 .
Please comment only on the
src/Relation/Binary/Properties/Domain.agdafile hereKey Changes:
1. Properties of Least Upper Bounds:
uniqueLub:Proves the uniqueness of least upper bounds.IsLub-cong: Demonstrates congruence of least upper bounds under equivalence.2. Scott Continuity and Monotonicity:
DirectedCompletePartialOrder+scott→monotone: Proves that Scott continuous functions are monotone.3. Scott Continuous Functions:
ScottId: Identity function as a Scott continuous function.scott-∘: Composition of Scott continuous functions.4. Suprema and Pointwise Ordering:
⋃-pointwise: Proves pointwise ordering of suprema in directed families.5. Scott Continuity Module:
pres-⋁: Proves preservation of least upper bounds under Scott continuous functions.6. Conversion to Scott Continuity:
-Added
to-scott: Converts monotone functions with preservation of least upper bounds into Scott continuous functions.Source :
1Lab library