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Refactor proofs from #2023 #2301

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Feb 27, 2024
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Refactor proofs from #2023 #2301

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35 changes: 20 additions & 15 deletions CHANGELOG.md
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Original file line number Diff line number Diff line change
Expand Up @@ -141,28 +141,28 @@ Additions to existing modules

* In `Data.List.Relation.Ternary.Appending.Setoid.Properties`:
```agda
through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs →
through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs →
∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs
through← : ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs →
through← : ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs →
∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs
assoc→ : ∃[ xs ] Appending as bs xs × Appending xs cs ds →
assoc→ : ∃[ xs ] Appending as bs xs × Appending xs cs ds →
∃[ ys ] Appending bs cs ys × Appending as ys ds
```

* In `Data.List.Relation.Ternary.Appending.Properties`:
```agda
through→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) →
∃[ xs ] Pointwise U as xs × Appending V R xs bs cs →
∃[ ys ] Appending W S as bs ys × Pointwise T ys cs
through← : ((R ; S) ⇒ T) → ((U ; S) ⇒ (V ; W)) →
∃[ ys ] Appending U R as bs ys × Pointwise S ys cs →
∃[ xs ] Pointwise V as xs × Appending W T xs bs cs
assoc→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → ((Y ; V) ⇒ X) →
∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds →
∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds
assoc← : ((S ; T) ⇒ R) → ((W ; T) ⇒ (U ; V)) → (X ⇒ (Y ; V)) →
∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds →
∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
through→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) →
∃[ xs ] Pointwise U as xs × Appending V R xs bs cs →
∃[ ys ] Appending W S as bs ys × Pointwise T ys cs
through← : ((R ; S) ⇒ T) → ((U ; S) ⇒ (V ; W)) →
∃[ ys ] Appending U R as bs ys × Pointwise S ys cs →
∃[ xs ] Pointwise V as xs × Appending W T xs bs cs
assoc→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → ((Y ; V) ⇒ X) →
∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds →
∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds
assoc← : ((S ; T) ⇒ R) → ((W ; T) ⇒ (U ; V)) → (X ⇒ (Y ; V)) →
∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds →
∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
```

* In `Data.List.Relation.Binary.Pointwise.Base`:
Expand Down Expand Up @@ -210,6 +210,11 @@ Additions to existing modules
* In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
be used infix.

* Added new proofs in `Relation.Binary.Construct.Composition`:
```agda
transitive⇒≈;≈⊆≈ : Transitive ≈ → (≈ ; ≈) ⇒ ≈
```

* Added new definitions in `Relation.Binary.Definitions`
```
Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
Expand Down
3 changes: 3 additions & 0 deletions src/Relation/Binary/Construct/Composition.agda
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Expand Up @@ -82,3 +82,6 @@ module _ (L : Rel A ℓ₁) (R : Rel A ℓ₂) (comm : R ; L ⇒ L ; R) where
; trans = transitive Oˡ.trans Oʳ.trans
}
where module Oˡ = IsPreorder Oˡ; module Oʳ = IsPreorder Oʳ

transitive⇒≈;≈⊆≈ : (≈ : Rel A ℓ) → Transitive ≈ → (≈ ; ≈) ⇒ ≈
transitive⇒≈;≈⊆≈ _ trans (_ , l , r) = trans l r
7 changes: 4 additions & 3 deletions src/Relation/Binary/Properties/Setoid.agda
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Expand Up @@ -15,7 +15,8 @@ open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
open import Relation.Binary.Definitions
using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
open import Relation.Binary.Construct.Composition using (_;_)
open import Relation.Binary.Construct.Composition
using (_;_; impliesˡ; transitive⇒≈;≈⊆≈)

module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where

Expand Down Expand Up @@ -83,10 +84,10 @@ preorder = record
-- Equality is closed under composition

≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_
≈;≈⇒≈ (_ , p , q) = trans p q
≈;≈⇒≈ = transitive⇒≈;≈⊆≈ _ trans

≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_
≈⇒≈;≈ q = _ , q , refl
≈⇒≈;≈ = impliesˡ _≈_ _≈_ refl id

------------------------------------------------------------------------
-- Other properties
Expand Down
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