[Add] Initial files for Domain theory - Continuation of #2721 #2809
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Thanks for picking up from @jmougeot 's initial contribution.
I don't think this design is quite ready yet, and needs more thought wrt:
- placement of concepts
- naming of them
- parametrisation of them
At a first cut, I'm not quite clear what should go where, so will need to think about this more, but this is a prompt to other reviewers also to think about these questions. It seems that I might be in conflict with @JacquesCarette over what the right answers might be!?
| record IsDirectedFamily {b : Level} {B : Set b} (_≤_ : Rel A ℓ₂) (f : B → A) : | ||
| Set (a ⊔ b ⊔ ℓ₂) where | ||
| field | ||
| elt : B |
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We might want to revisit this name before committing to it? eg inhabitant as an alternative?
| Set (a ⊔ b ⊔ ℓ₂) where | ||
| field | ||
| elt : B | ||
| isSemidirected : SemiDirected _≤_ f |
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This doesn't seem to be consistent... wrt our naming conventions camelCase : CamelCase, nor wrt use of isX to refer to an IsX *structure...
| isSemidirected : SemiDirected _≤_ f | |
| semiDirected : SemiDirected _≤_ f |
| module _ (P : Poset c ℓ₁ ℓ₂) where | ||
| open Poset P |
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I think that this has come up before in this context; we tend to define this kind of parametrised record on the underlying Raw structure (because none of the axioms for a Poset are involved in the definition, only the underlying relations).
Not least because domain theory ought better to be built on Preorder-based foundations? Antisymmetry plays almost no direct role in the axiomatisation of the concepts you define?
Anyway: meta design principle is/ought to be 'structures don't depend on bundles'
I think (but happy to be contradicted) that UpperBound, Lub, DirectedFamily all belong in Relation.Binary.Domain.Definitions
| module _ (P : Poset c ℓ₁ ℓ₂) where | ||
| open Poset P | ||
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| record IsLub {b : Level} {B : Set b} (f : B → Carrier) |
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b is already implicitly bound by a variable declaration, so shouldn't need explicit mention here.
| record IsLub {b : Level} {B : Set b} (f : B → Carrier) | ||
| (lub : Carrier) : Set (b ⊔ c ⊔ ℓ₁ ⊔ ℓ₂) where | ||
| field | ||
| isUpperBound : UpperBound _≤_ f lub |
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camelCase : CamelCase is the usual convention in stdlib, accordingly:
| isUpperBound : UpperBound _≤_ f lub | |
| upperBound : UpperBound _≤_ f lub |
| (lub : Carrier) : Set (b ⊔ c ⊔ ℓ₁ ⊔ ℓ₂) where | ||
| field | ||
| isUpperBound : UpperBound _≤_ f lub | ||
| isLeast : ∀ y → UpperBound _≤_ f y → lub ≤ y |
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| isLeast : ∀ y → UpperBound _≤_ f y → lub ≤ y | |
| minimal : ∀ y → UpperBound _≤_ f y → lub ≤ y |
because being least only follows from the Poset axioms, whereas in their absence (as above), we only have/need a minimal upper bound?
| field | ||
| ⋁ : ∀ {B : Set c} → | ||
| (f : B → Carrier) → | ||
| IsDirectedFamily _≈_ _≤_ f → |
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The definition you give of IsDirectedFamily is only accidentally dependent on the underlying Setoid equality relation _≈_, so either you need to involve it by making f a suitable congruence wrt it, or else reconsider whether IsDirectedFamily really belongs in Relation.Binary.Structures?
As this stands, I think the design doesn't quite fit properly with the existing stdlib hierarchy.
| -- DirectedFamily | ||
| ------------------------------------------------------------------------ | ||
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| record IsDirectedFamily {b : Level} {B : Set b} (_≤_ : Rel A ℓ₂) (f : B → A) : |
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I'm not sure that this definition belongs in Relation.Binary.Structures, because it's not actually (an extension of) any of the existing relational structures, rather it's a property that any one of them might additionally possess?
This pull request aims to continue and complete the work started in #2721
The first version already includes changes requested by @JacquesCarette, such as moving the definition of
DirectedFamilytoRelation.Binary.Definitions.agdaandIsDirectedFamilytoRelation.Binary.Structures.agda, as well as the proper bundling ofLub.