[ new ] symmetric core of a binary relation

CHANGELOG.md

@@ -51,6 +51,13 @@ New modules
   Relation.Binary.Construct.Interior.Symmetric
   ```
 
+* `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
+
+* Nagata's construction of the "idealization of a module":
+  ```agda
+  Algebra.Module.Construct.Idealization
+  ```
+
 Additions to existing modules
 -----------------------------
 
@@ -166,7 +173,42 @@ Additions to existing modules
   map : (Char → Char) → String → String
   ```
 
-* Added new definitions in `Relation.Binary.Definitions`:
+* In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
+  be used infix.
+
+* Added new definitions in `Relation.Binary.Definitions`
+  ```
+  Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
+  Empty  _∼_ = ∀ {x y} → x ∼ y → ⊥
+  ```
+
+* Added new definitions in `Relation.Nullary`
+  ```
+  Recomputable    : Set _
+  WeaklyDecidable : Set _
+  ```
+
+* Added new definitions in `Relation.Unary`
+  ```
+  Stable          : Pred A ℓ → Set _
+  WeaklyDecidable : Pred A ℓ → Set _
+  ```
+
+* Added new proof in `Relation.Nullary.Decidable`:
+  ```agda
+  ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
+  ```
+
+* Added module `Data.Vec.Functional.Relation.Binary.Permutation`:
+  ```agda
+  _↭_ : IRel (Vector A) _
+  ```
+
+* Added new file `Data.Vec.Functional.Relation.Binary.Permutation.Properties`:
   ```agda
-  Empty _∼_ = ∀ {x y} → x ∼ y → ⊥
+  ↭-refl      : Reflexive (Vector A) _↭_
+  ↭-reflexive : xs ≡ ys → xs ↭ ys
+  ↭-sym       : Symmetric (Vector A) _↭_
+  ↭-trans     : Transitive (Vector A) _↭_
+>>>>>>> master
   ```

src/Relation/Binary/Bundles.agda

@@ -76,15 +76,6 @@ record DecSetoid c ℓ : Set (suc (c ⊔ ℓ)) where
 -- Preorders
 ------------------------------------------------------------------------
 
-record Proset c ℓ : Set (suc (c ⊔ ℓ)) where
-  infix 4 _≤_
-  field
-    Carrier : Set c
-    _≤_ : Rel Carrier ℓ
-    refl : Reflexive _≤_
-    trans : Transitive _≤_
-
-
 record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   infix 4 _≈_ _≲_
   field

src/Relation/Binary/Construct/Interior/Symmetric.agda

@@ -16,9 +16,7 @@ private
   variable
     a b c ℓ r s t : Level
     A : Set a
-    R : Rel A r
-    S : Rel A s
-    T : Rel A t
+    R S T : Rel A r
 
 ------------------------------------------------------------------------
 -- Definition
@@ -28,6 +26,7 @@ record SymInterior (R : Rel A ℓ) (x y : A) : Set ℓ where
   field
     lhs≤rhs : R x y
     rhs≤lhs : R y x
+
 open SymInterior public
 
 ------------------------------------------------------------------------
@@ -60,25 +59,25 @@ asymmetric⇒empty asym (r , r′) = asym r r′
 -- A reflexive transitive relation _≤_ gives rise to a poset in which the
 -- equivalence relation is SymInterior _≤_.
 
-isEquivalence : ∀ {a ℓ} {A : Set a} {≤ : Rel A ℓ} →
-  Reflexive ≤ → Transitive ≤ → IsEquivalence (SymInterior ≤)
-isEquivalence ≤-refl ≤-trans = record
-  { refl = reflexive ≤-refl
+isEquivalence : Reflexive R → Transitive R → IsEquivalence (SymInterior R)
+isEquivalence refl trans = record
+  { refl = reflexive refl
   ; sym = symmetric
-  ; trans = transitive ≤-trans
+  ; trans = transitive trans
   }
 
-isPartialOrder : ∀ {a ℓ} {A : Set a} {≤ : Rel A ℓ} →
-  Reflexive ≤ → Transitive ≤ → IsPartialOrder (SymInterior ≤) ≤
-isPartialOrder ≤-refl ≤-trans = record
+isPartialOrder : Reflexive R → Transitive R → IsPartialOrder (SymInterior R) R
+isPartialOrder refl trans = record
   { isPreorder = record
-    { isEquivalence = isEquivalence ≤-refl ≤-trans
+    { isEquivalence = isEquivalence refl trans
     ; reflexive = lhs≤rhs
-    ; trans = ≤-trans
+    ; trans = trans
     }
   ; antisym = _,_
   }
 
-poset : Proset c ℓ → Poset c ℓ ℓ
-poset record { _≤_ = ≤; refl = refl; trans = trans } =
-  record { _≤_ = ≤; isPartialOrder = isPartialOrder refl trans }
+poset : ∀ {a} {A : Set a} {R : Rel A ℓ} → Reflexive R → Transitive R → Poset a ℓ ℓ
+poset {R = R} refl trans = record
+  { _≤_ = R
+  ; isPartialOrder = isPartialOrder refl trans
+  }