From 3bdbdd9421ab57835c9a23bb68a778c63ab302df Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 07:42:49 +0000
Subject: [PATCH 01/20] =?UTF-8?q?[=20breaking=20]=20change=20fieldname=20`?=
 =?UTF-8?q?=5F=E2=89=9F=5F`=20to=20`=5F=E2=89=88=3F=5F`=20in=20`IsDecEquiv?=
 =?UTF-8?q?alence`?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 src/Relation/Binary/Bundles.agda              | 12 ++--
 .../Binary/Construct/Flip/EqAndOrd.agda       |  4 +-
 src/Relation/Binary/Properties/DecSetoid.agda |  6 +-
 src/Relation/Binary/Properties/Setoid.agda    |  1 -
 .../PropositionalEquality/Properties.agda     |  4 +-
 src/Relation/Binary/Structures.agda           | 59 ++++++++++++-------
 6 files changed, 50 insertions(+), 36 deletions(-)

diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index cd66369dec..22be21b028 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -64,7 +64,7 @@ record DecSetoid c ℓ : Set (suc (c ⊔ ℓ)) where
     isDecEquivalence : IsDecEquivalence _≈_
 
   open IsDecEquivalence isDecEquivalence public
-    using (_≟_; isEquivalence)
+    using (_≈?_; _≟_; isEquivalence)
 
   setoid : Setoid c ℓ
   setoid = record
@@ -140,7 +140,7 @@ record DecPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   private module DPO = IsDecPreorder isDecPreorder
 
   open DPO public
-    using (_≟_; _≲?_; isPreorder)
+    using (_≈?_; _≲?_; isPreorder)
 
   preorder : Preorder c ℓ₁ ℓ₂
   preorder = record
@@ -203,7 +203,7 @@ record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   private module DPO = IsDecPartialOrder isDecPartialOrder
 
   open DPO public
-    using (_≟_; _≤?_; isPartialOrder; isDecPreorder)
+    using (_≈?_; _≤?_; isPartialOrder; isDecPreorder)
 
   poset : Poset c ℓ₁ ℓ₂
   poset = record
@@ -258,7 +258,7 @@ record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂
   private module DSPO = IsDecStrictPartialOrder isDecStrictPartialOrder
 
   open DSPO public
-    using (_<?_; _≟_; isStrictPartialOrder)
+    using (_<?_; _≈?_; isStrictPartialOrder)
 
   strictPartialOrder : StrictPartialOrder c ℓ₁ ℓ₂
   strictPartialOrder = record
@@ -318,7 +318,7 @@ record DecTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   private module DTO = IsDecTotalOrder isDecTotalOrder
 
   open DTO public
-    using (_≟_; _≤?_; isTotalOrder; isDecPartialOrder)
+    using (_≈?_; _≤?_; isTotalOrder; isDecPartialOrder)
 
   totalOrder : TotalOrder c ℓ₁ ℓ₂
   totalOrder = record
@@ -352,7 +352,7 @@ record StrictTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) wh
 
   open IsStrictTotalOrder isStrictTotalOrder public
     using
-    ( _≟_; _<?_; compare; isStrictPartialOrder
+    ( _≈?_; _<?_; compare; isStrictPartialOrder
     ; isDecStrictPartialOrder; isDecEquivalence
     )
 
diff --git a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
index 402804f1b6..b3234d289e 100644
--- a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
+++ b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
@@ -102,7 +102,7 @@ isEquivalence {≈ = ≈} eq = record
 isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)
 isDecEquivalence {≈ = ≈} eq = record
   { isEquivalence = isEquivalence Dec.isEquivalence
-  ; _≟_           = dec ≈ Dec._≟_
+  ; _≈?_          = dec ≈ Dec._≈?_
   } where module Dec = IsDecEquivalence eq
 
 isPreorder : IsPreorder ≈ ∼ → IsPreorder ≈ (flip ∼)
@@ -133,7 +133,7 @@ isTotalOrder O = record
 isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder ≈ (flip ≤)
 isDecTotalOrder O = record
   { isTotalOrder = isTotalOrder O.isTotalOrder
-  ; _≟_          = O._≟_
+  ; _≈?_         = O._≈?_
   ; _≤?_         = dec _ O._≤?_
   } where module O = IsDecTotalOrder O
 
diff --git a/src/Relation/Binary/Properties/DecSetoid.agda b/src/Relation/Binary/Properties/DecSetoid.agda
index 7f9b323d85..f241e4053e 100644
--- a/src/Relation/Binary/Properties/DecSetoid.agda
+++ b/src/Relation/Binary/Properties/DecSetoid.agda
@@ -20,11 +20,11 @@ open import Relation.Binary.Structures
 open import Relation.Nullary.Decidable.Core
   using (yes; no; decidable-stable)
 
-open DecSetoid S using (_≈_; _≉_; _≟_; setoid; trans)
+open DecSetoid S using (_≈_; _≉_; _≈?_; setoid; trans)
 open SetoidProperties setoid
 
 ≉-cotrans : Cotransitive _≉_
-≉-cotrans {x} {y} x≉y z with x ≟ z | z ≟ y
+≉-cotrans {x} {y} x≉y z with x ≈? z | z ≈? y
 ... | _ | no z≉y = inj₂ z≉y
 ... | no x≉z | _ = inj₁ x≉z
 ... | yes x≈z | yes z≈y = inj₁ λ _ → x≉y (trans x≈z z≈y)
@@ -40,4 +40,4 @@ open SetoidProperties setoid
 ≉-apartnessRelation = record { isApartnessRelation = ≉-isApartnessRelation }
 
 ≉-tight : Tight _≈_ _≉_
-≉-tight x y = decidable-stable (x ≟ y) , ≉-irrefl
+≉-tight x y = decidable-stable (x ≈? y) , ≉-irrefl
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index a2f277b754..3277314b6f 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -19,7 +19,6 @@ open import Relation.Binary.Definitions
   using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_; Irreflexive)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; ¬[x≢x])
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
-open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 
 open Setoid S renaming (Carrier to A)
 
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index 5addbfd132..e9af17a238 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -159,9 +159,9 @@ isEquivalence = record
   }
 
 isDecEquivalence : DecidableEquality A → IsDecEquivalence _≡_
-isDecEquivalence _≟_ = record
+isDecEquivalence _≈?_ = record
   { isEquivalence = isEquivalence
-  ; _≟_           = _≟_
+  ; _≈?_          = _≈?_
   }
 
 setoid : Set a → Setoid _ _
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index 4987eb9d6b..72c3617a3b 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -62,13 +62,19 @@ record IsEquivalence : Set (a ⊔ ℓ) where
 
 
 record IsDecEquivalence : Set (a ⊔ ℓ) where
-  infix 4 _≟_
+  infix 4 _≈?_ _≟_
   field
     isEquivalence : IsEquivalence
-    _≟_           : Decidable _≈_
+    _≈?_          : Decidable _≈_
 
   open IsEquivalence isEquivalence public
 
+  _≟_ = _≈?_
+  {-# WARNING_ON_USAGE _≟_
+  "Warning: _≟_ was deprecated in v2.4.
+  Please use _≈?_ instead. "
+  #-}
+
 
 ------------------------------------------------------------------------
 -- Preorders
@@ -123,7 +129,7 @@ record IsTotalPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
 record IsDecPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   field
     isPreorder : IsPreorder _≲_
-    _≟_        : Decidable _≈_
+    _≈?_       : Decidable _≈_
     _≲?_       : Decidable _≲_
 
   open IsPreorder isPreorder public
@@ -134,7 +140,7 @@ record IsDecPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
     isDecEquivalence : IsDecEquivalence
     isDecEquivalence = record
       { isEquivalence = isEquivalence
-      ; _≟_           = _≟_
+      ; _≈?_          = _≈?_
       }
 
     open IsDecEquivalence isDecEquivalence public
@@ -151,17 +157,17 @@ record IsPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
 
   open IsPreorder isPreorder public
     renaming
-    ( ∼-respˡ-≈ to ≤-respˡ-≈
-    ; ∼-respʳ-≈ to ≤-respʳ-≈
-    ; ∼-resp-≈  to ≤-resp-≈
+    ( ≲-respˡ-≈ to ≤-respˡ-≈
+    ; ≲-respʳ-≈ to ≤-respʳ-≈
+    ; ≲-resp-≈  to ≤-resp-≈
     )
 
 
 record IsDecPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
-  infix 4 _≟_ _≤?_
+  infix 4 _≈?_ _≤?_
   field
     isPartialOrder : IsPartialOrder _≤_
-    _≟_            : Decidable _≈_
+    _≈?_           : Decidable _≈_
     _≤?_           : Decidable _≤_
 
   open IsPartialOrder isPartialOrder public
@@ -170,13 +176,15 @@ record IsDecPartialOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) whe
   isDecPreorder : IsDecPreorder _≤_
   isDecPreorder = record
     { isPreorder = isPreorder
-    ; _≟_ = _≟_
+    ; _≈?_ = _≈?_
     ; _≲?_ = _≤?_
     }
 
   open IsDecPreorder isDecPreorder public
     using (module Eq)
 
+  open Eq public using (_≟_)
+
 
 record IsStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   field
@@ -198,10 +206,10 @@ record IsStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wh
 
 
 record IsDecStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
-  infix 4 _≟_ _<?_
+  infix 4 _≈?_ _<?_
   field
     isStrictPartialOrder : IsStrictPartialOrder _<_
-    _≟_                  : Decidable _≈_
+    _≈?_                 : Decidable _≈_
     _<?_                 : Decidable _<_
 
   private
@@ -214,11 +222,13 @@ record IsDecStrictPartialOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂)
     isDecEquivalence : IsDecEquivalence
     isDecEquivalence = record
       { isEquivalence = SPO.isEquivalence
-      ; _≟_           = _≟_
+      ; _≈?_          = _≈?_
       }
 
     open IsDecEquivalence isDecEquivalence public
 
+  open Eq public using (_≟_)
+
 
 ------------------------------------------------------------------------
 -- Total orders
@@ -239,10 +249,10 @@ record IsTotalOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
 
 
 record IsDecTotalOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
-  infix 4 _≟_ _≤?_
+  infix 4 _≈?_ _≤?_
   field
     isTotalOrder : IsTotalOrder _≤_
-    _≟_          : Decidable _≈_
+    _≈?_         : Decidable _≈_
     _≤?_         : Decidable _≤_
 
   open IsTotalOrder isTotalOrder public
@@ -251,13 +261,16 @@ record IsDecTotalOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   isDecPartialOrder : IsDecPartialOrder _≤_
   isDecPartialOrder = record
     { isPartialOrder = isPartialOrder
-    ; _≟_            = _≟_
+    ; _≈?_           = _≈?_
     ; _≤?_           = _≤?_
     }
 
   open IsDecPartialOrder isDecPartialOrder public
     using (isDecPreorder; module Eq)
 
+  open Eq public using (_≟_)
+
+
 -- Note that these orders are decidable. The current implementation
 -- of `Trichotomous` subsumes irreflexivity and asymmetry. See
 -- `Relation.Binary.Structures.Biased` for ways of constructing this
@@ -273,10 +286,10 @@ record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wher
 
   -- `Trichotomous` necessarily separates out the equality case so
   --  it implies decidability.
-  infix 4 _≟_ _<?_
+  infix 4 _≈?_ _<?_
 
-  _≟_ : Decidable _≈_
-  _≟_ = tri⇒dec≈ compare
+  _≈?_ : Decidable _≈_
+  _≈?_ = tri⇒dec≈ compare
 
   _<?_ : Decidable _<_
   _<?_ = tri⇒dec< compare
@@ -284,7 +297,7 @@ record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wher
   isDecStrictPartialOrder : IsDecStrictPartialOrder _<_
   isDecStrictPartialOrder = record
     { isStrictPartialOrder = isStrictPartialOrder
-    ; _≟_                  = _≟_
+    ; _≈?_                  = _≈?_
     ; _<?_                 = _<?_
     }
 
@@ -294,15 +307,17 @@ record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wher
     isDecEquivalence : IsDecEquivalence
     isDecEquivalence = record
       { isEquivalence = isEquivalence
-      ; _≟_           = _≟_
+      ; _≈?_          = _≈?_
       }
 
     open IsDecEquivalence isDecEquivalence public
 
+  open Eq public using (_≟_)
+
   isDecEquivalence : IsDecEquivalence
   isDecEquivalence = record
     { isEquivalence = isEquivalence
-    ; _≟_           = _≟_
+    ; _≈?_          = _≈?_
     }
   {-# WARNING_ON_USAGE isDecEquivalence
   "Warning: isDecEquivalence was deprecated in v2.0.

From 1dd260a185061325e12ee95427182e01faeab08f Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 08:11:07 +0000
Subject: [PATCH 02/20] refactor: knock-ons, plus deprecations

---
 src/Data/Bool/Properties.agda                 | 25 +++++++---
 src/Data/Nat/Properties.agda                  | 49 +++++++++++++------
 .../Binary/Construct/NaturalOrder/Left.agda   | 14 +++---
 .../Binary/Construct/NonStrictToStrict.agda   | 16 +++---
 src/Relation/Binary/Construct/On.agda         |  6 +--
 src/Relation/Binary/Properties/Poset.agda     |  2 +-
 .../Binary/Properties/TotalOrder.agda         |  6 +--
 .../Binary/PropositionalEquality.agda         | 23 +++++++--
 8 files changed, 92 insertions(+), 49 deletions(-)

diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index 53b4a1f4a7..e874cfe85e 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -59,19 +59,19 @@ private
 ------------------------------------------------------------------------
 -- Properties of _≡_
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality Bool
-true  ≟ true  = yes refl
-false ≟ false = yes refl
-true  ≟ false = no λ()
-false ≟ true  = no λ()
+_≡?_ : DecidableEquality Bool
+true  ≡? true  = yes refl
+false ≡? false = yes refl
+true  ≡? false = no λ()
+false ≡? true  = no λ()
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid Bool
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Properties of _≤_
@@ -139,7 +139,7 @@ true  ≤? true  = yes b≤b
 ≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≟_
+  ; _≈?_         = _≡?_
   ; _≤?_         = _≤?_
   }
 
@@ -884,3 +884,12 @@ push-function-into-if = if-float
 "Warning: push-function-into-if was deprecated in v2.0.
 Please use if-float instead."
 #-}
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 5976ba29de..1ddc89e2d6 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -104,23 +104,23 @@ suc-injective = cong pred
 -- We expect the main benefit to be visible in compiled code as the
 -- backend erases proofs.
 
-infix 4 _≟_
-_≟_ : DecidableEquality ℕ
-m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
+infix 4 _≡?_
+_≡?_ : DecidableEquality ℕ
+m ≡? n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 
 ≡-irrelevant : Irrelevant {A = ℕ} _≡_
-≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≟_
+≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≡?_
 
-≟-diag : (eq : m ≡ n) → (m ≟ n) ≡ yes eq
-≟-diag = ≡-≟-identity _≟_
+≡?-diag : (eq : m ≡ n) → (m ≡? n) ≡ yes eq
+≡?-diag = ≡-≡?-identity _≡?_
 
-≟-≡ : (m≢n : m ≢ n) → (m ≟ n) ≡ no m≢n
-≟-≡ = ≢-≟-identity _≟_
+≡?-≡ : (m≢n : m ≢ n) → (m ≡? n) ≡ no m≢n
+≡?-≡ = ≢-≡?-identity _≡?_
 
 ≡-isDecEquivalence : IsDecEquivalence (_≡_ {A = ℕ})
 ≡-isDecEquivalence = record
   { isEquivalence = isEquivalence
-  ; _≟_           = _≟_
+  ; _≈?_           = _≡?_
   }
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
@@ -250,7 +250,7 @@ _≥?_ = flip _≤?_
 ≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≟_
+  ; _≈?_         = _≡?_
   ; _≤?_         = _≤?_
   }
 
@@ -400,7 +400,7 @@ n≤1⇒n≡0∨n≡1 (s≤s z≤n) = inj₂ refl
 -- backend erases proofs.
 
 <-cmp : Trichotomous _≡_ _<_
-<-cmp m n with m ≟ n | T? (m <ᵇ n)
+<-cmp m n with m ≡? n | T? (m <ᵇ n)
 ... | yes m≡n | _       = tri≈ (<-irrefl m≡n) m≡n (<-irrefl (sym m≡n))
 ... | no  m≢n | yes m<n = tri< (<ᵇ⇒< m n m<n) m≢n (<⇒≯ (<ᵇ⇒< m n m<n))
 ... | no  m≢n | no  m≮n = tri> (m≮n ∘ <⇒<ᵇ)   m≢n (≤∧≢⇒< (≮⇒≥ (m≮n ∘ <⇒<ᵇ)) (m≢n ∘ sym))
@@ -2284,7 +2284,7 @@ _>‴?_ = flip _<‴?_
 -- decidable equality.
 
 eq? : ∀ {a} {A : Set a} → A ↣ ℕ → DecidableEquality A
-eq? inj = via-injection inj _≟_
+eq? inj = via-injection inj _≡?_
 
 -- It's possible to decide existential and universal predicates up to
 -- a limit.
@@ -2299,7 +2299,7 @@ module _ {p} {P : Pred ℕ p} (P? : U.Decidable P) where
   ... | no ¬Pv | no ¬Pn<v           = no ¬Pn<1+v
     where
     ¬Pn<1+v : ¬ (∃ λ n → n < suc v × P n)
-    ¬Pn<1+v (n , s≤s n≤v , Pn) with n ≟ v
+    ¬Pn<1+v (n , s≤s n≤v , Pn) with n ≡? v
     ... | yes refl = ¬Pv Pn
     ... | no  n≢v  = ¬Pn<v (n , ≤∧≢⇒< n≤v n≢v , Pn)
 
@@ -2311,7 +2311,7 @@ module _ {p} {P : Pred ℕ p} (P? : U.Decidable P) where
   ... | yes Pn | yes Pn<v = yes Pn<1+v
     where
       Pn<1+v : ∀ {n} → n < suc v → P n
-      Pn<1+v {n} (s≤s n≤v) with n ≟ v
+      Pn<1+v {n} (s≤s n≤v) with n ≡? v
       ... | yes refl = Pn
       ... | no  n≢v  = Pn<v (≤∧≢⇒< n≤v n≢v)
 
@@ -2460,3 +2460,24 @@ open Data.Nat.Base public
 {-# WARNING_ON_USAGE <-transˡ
 "Warning: <-transˡ was deprecated in v2.0. Please use <-≤-trans instead. "
 #-}
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+
+≟-diag = ≡?-diag
+{-# WARNING_ON_USAGE ≟-diag
+"Warning: ≟-diag was deprecated in v2.4.
+Please use ≡?-diag instead."
+#-}
+
+≟-≡ = ≡?-≡
+{-# WARNING_ON_USAGE ≟-≡
+"Warning: ≟-≡ was deprecated in v2.4.
+Please use ≡?-≡ instead."
+#-}
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Left.agda b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
index 0c5f1cad15..4300a208ac 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Left.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
@@ -91,7 +91,7 @@ resp₂ : IsMagma _∙_ →  _≤_ Respects₂ _≈_
 resp₂ magma = respʳ magma , respˡ magma
 
 dec : Decidable _≈_ → Decidable _≤_
-dec _≟_ x y = x ≟ (x ∙ y)
+dec _≈?_ x y = x ≈? (x ∙ y)
 
 module _ (semi : IsSemilattice _∙_) where
 
@@ -141,10 +141,10 @@ isPartialOrder semilattice = record
 
 isDecPartialOrder : IsSemilattice _∙_ → Decidable _≈_ →
                     IsDecPartialOrder _≈_ _≤_
-isDecPartialOrder semilattice _≟_ = record
+isDecPartialOrder semilattice _≈?_ = record
   { isPartialOrder = isPartialOrder semilattice
-  ; _≟_            = _≟_
-  ; _≤?_           = dec _≟_
+  ; _≈?_           = _≈?_
+  ; _≤?_           = dec _≈?_
   }
 
 isTotalOrder : IsSemilattice _∙_ → Selective _∙_ → IsTotalOrder _≈_ _≤_
@@ -156,10 +156,10 @@ isTotalOrder latt sel  = record
 
 isDecTotalOrder : IsSemilattice _∙_ → Selective _∙_ →
                   Decidable _≈_ → IsDecTotalOrder _≈_ _≤_
-isDecTotalOrder latt sel _≟_ = record
+isDecTotalOrder latt sel _≈?_ = record
   { isTotalOrder = isTotalOrder latt sel
-  ; _≟_          = _≟_
-  ; _≤?_         = dec _≟_
+  ; _≈?_         = _≈?_
+  ; _≤?_         = dec _≈?_
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Construct/NonStrictToStrict.agda b/src/Relation/Binary/Construct/NonStrictToStrict.agda
index a4744a2fd9..559e3f534c 100644
--- a/src/Relation/Binary/Construct/NonStrictToStrict.agda
+++ b/src/Relation/Binary/Construct/NonStrictToStrict.agda
@@ -64,7 +64,7 @@ x < y = x ≤ y × x ≉ y
 
 ≮⇒≥ : Symmetric _≈_ → Decidable _≈_ → _≈_ ⇒ _≤_ → Total _≤_ →
       ∀ {x y} → ¬ (x < y) → y ≤ x
-≮⇒≥ sym _≟_ ≤-refl _≤?_ {x} {y} x≮y with x ≟ y | y ≤? x
+≮⇒≥ sym _≈?_ ≤-refl _≤?_ {x} {y} x≮y with x ≈? y | y ≤? x
 ... | yes x≈y  | _        = ≤-refl (sym x≈y)
 ... | _        | inj₁ y≤x = y≤x
 ... | no  x≉y  | inj₂ x≤y = contradiction (x≤y , x≉y) x≮y
@@ -110,14 +110,14 @@ x < y = x ≤ y × x ≉ y
 <-trichotomous : Symmetric _≈_ → Decidable _≈_ →
                  Antisymmetric _≈_ _≤_ → Total _≤_ →
                  Trichotomous _≈_ _<_
-<-trichotomous ≈-sym _≟_ antisym total x y with x ≟ y
+<-trichotomous ≈-sym _≈?_ antisym total x y with x ≈? y
 ... | yes x≈y = tri≈ (<-irrefl x≈y) x≈y (<-irrefl (≈-sym x≈y))
 ... | no  x≉y with total x y
 ...   | inj₁ x≤y = tri< (x≤y , x≉y) x≉y (x≉y ∘ antisym x≤y ∘ proj₁)
 ...   | inj₂ y≤x = tri> (x≉y ∘ flip antisym y≤x ∘ proj₁) x≉y (y≤x , x≉y ∘ ≈-sym)
 
 <-decidable : Decidable _≈_ → Decidable _≤_ → Decidable _<_
-<-decidable _≟_ _≤?_ x y = x ≤? y ×? ¬? (x ≟ y)
+<-decidable _≈?_ _≤?_ x y = x ≤? y ×? ¬? (x ≈? y)
 
 ------------------------------------------------------------------------
 -- Structures
@@ -135,18 +135,18 @@ x < y = x ≤ y × x ≉ y
                             IsDecStrictPartialOrder _≈_ _<_
 <-isDecStrictPartialOrder dpo = record
   { isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder
-  ; _≟_ = _≟_
-  ; _<?_ = <-decidable _≟_ _≤?_
+  ; _≈?_ = _≈?_
+  ; _<?_ = <-decidable _≈?_ _≤?_
   } where open IsDecPartialOrder dpo
 
 <-isStrictTotalOrder₁ : Decidable _≈_ → IsTotalOrder _≈_ _≤_ →
                         IsStrictTotalOrder _≈_ _<_
-<-isStrictTotalOrder₁ ≟ tot = record
+<-isStrictTotalOrder₁ ≈? tot = record
   { isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder
-  ; compare              = <-trichotomous Eq.sym ≟ antisym total
+  ; compare              = <-trichotomous Eq.sym ≈? antisym total
   } where open IsTotalOrder tot
 
 <-isStrictTotalOrder₂ : IsDecTotalOrder _≈_ _≤_ →
                         IsStrictTotalOrder _≈_ _<_
-<-isStrictTotalOrder₂ dtot = <-isStrictTotalOrder₁ _≟_ isTotalOrder
+<-isStrictTotalOrder₂ dtot = <-isStrictTotalOrder₁ _≈?_ isTotalOrder
   where open IsDecTotalOrder dtot
diff --git a/src/Relation/Binary/Construct/On.agda b/src/Relation/Binary/Construct/On.agda
index 08fb8f80dd..e60c418467 100644
--- a/src/Relation/Binary/Construct/On.agda
+++ b/src/Relation/Binary/Construct/On.agda
@@ -100,7 +100,7 @@ module _ (f : B → A) {≈ : Rel A ℓ₁} where
                      IsDecEquivalence (≈ on f)
   isDecEquivalence dec = record
     { isEquivalence = isEquivalence Dec.isEquivalence
-    ; _≟_           = decidable f ≈ Dec._≟_
+    ; _≈?_          = decidable f ≈ Dec._≈?_
     } where module Dec = IsDecEquivalence dec
 
 module _ (f : B → A) {≈ : Rel A ℓ₁} {∼ : Rel A ℓ₂} where
@@ -123,7 +123,7 @@ module _ (f : B → A) {≈ : Rel A ℓ₁} {∼ : Rel A ℓ₂} where
                       IsDecPartialOrder (≈ on f) (∼ on f)
   isDecPartialOrder dpo = record
     { isPartialOrder = isPartialOrder DPO.isPartialOrder
-    ; _≟_            = decidable f _ DPO._≟_
+    ; _≈?_           = decidable f _ DPO._≈?_
     ; _≤?_           = decidable f _ DPO._≤?_
     } where module DPO = IsDecPartialOrder dpo
 
@@ -147,7 +147,7 @@ module _ (f : B → A) {≈ : Rel A ℓ₁} {∼ : Rel A ℓ₂} where
                     IsDecTotalOrder (≈ on f) (∼ on f)
   isDecTotalOrder dec = record
     { isTotalOrder = isTotalOrder Dec.isTotalOrder
-    ; _≟_          = decidable f ≈ Dec._≟_
+    ; _≈?_         = decidable f ≈ Dec._≈?_
     ; _≤?_         = decidable f ∼ Dec._≤?_
     } where module Dec = IsDecTotalOrder dec
 
diff --git a/src/Relation/Binary/Properties/Poset.agda b/src/Relation/Binary/Properties/Poset.agda
index 9dd3a3047b..198ea3393d 100644
--- a/src/Relation/Binary/Properties/Poset.agda
+++ b/src/Relation/Binary/Properties/Poset.agda
@@ -114,7 +114,7 @@ open StrictPartialOrder <-strictPartialOrder public
 ≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_
 ≤-dec⇒isDecPartialOrder _≤?_ = record
   { isPartialOrder = isPartialOrder
-  ; _≟_            = ≤-dec⇒≈-dec _≤?_
+  ; _≈?_           = ≤-dec⇒≈-dec _≤?_
   ; _≤?_           = _≤?_
   }
 
diff --git a/src/Relation/Binary/Properties/TotalOrder.agda b/src/Relation/Binary/Properties/TotalOrder.agda
index 33a8500b35..7d10058d51 100644
--- a/src/Relation/Binary/Properties/TotalOrder.agda
+++ b/src/Relation/Binary/Properties/TotalOrder.agda
@@ -27,11 +27,11 @@ import Relation.Binary.Properties.Poset poset as PosetProperties
 -- Total orders are almost decidable total orders
 
 decTotalOrder : Decidable _≈_ → DecTotalOrder _ _ _
-decTotalOrder ≟ = record
+decTotalOrder ≈? = record
   { isDecTotalOrder = record
     { isTotalOrder = isTotalOrder
-    ; _≟_          = ≟
-    ; _≤?_         = total∧dec⇒dec reflexive antisym total ≟
+    ; _≈?_         = ≈?
+    ; _≤?_         = total∧dec⇒dec reflexive antisym total ≈?
     }
   }
 
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 283623cc96..2e75071c25 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -82,13 +82,13 @@ cong-≡id {f = f} {x} f≡id = begin
   f≡id (f x)                                     ∎
   where open ≡-Reasoning; fx≡x = f≡id x; f²x≡x = f≡id (f x)
 
-module _ (_≟_ : DecidableEquality A) {x y : A} where
+module _ (_≡?_ : DecidableEquality A) {x y : A} where
 
-  ≡-≟-identity : (eq : x ≡ y) → x ≟ y ≡ yes eq
-  ≡-≟-identity eq = dec-yes-irr (x ≟ y) (Decidable⇒UIP.≡-irrelevant _≟_) eq
+  ≡-≡?-identity : (eq : x ≡ y) → x ≡? y ≡ yes eq
+  ≡-≡?-identity eq = dec-yes-irr (x ≡? y) (Decidable⇒UIP.≡-irrelevant _≡?_) eq
 
-  ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
-  ≢-≟-identity = dec-no (x ≟ y)
+  ≢-≡?-identity : (x≢y : x ≢ y) → x ≡? y ≡ no x≢y
+  ≢-≡?-identity = dec-no (x ≡? y)
 
 
 ------------------------------------------------------------------------
@@ -130,3 +130,16 @@ isPropositional = Irrelevant
 Please use Relation.Nullary.Irrelevant instead. "
 #-}
 
+-- Version 2.4
+
+≡-≟-identity = ≡-≡?-identity
+{-# WARNING_ON_USAGE ≡-≟-identity
+"Warning: ≡-≟-identity was deprecated in v2.4.
+Please use ≡-≡?-identity instead."
+#-}
+
+≢-≟-identity = ≢-≡?-identity
+{-# WARNING_ON_USAGE ≢-≟-identity
+"Warning: ≢-≟-identity was deprecated in v2.4.
+Please use ≢-≡?-identity instead."
+#-}

From d35166dade26c69bd5433e52e5f3b266715a9d8d Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 09:37:49 +0000
Subject: [PATCH 03/20] refactor: MANY MORE knock-ons, plus deprecations

---
 doc/README/Data/Trie/NonDependent.agda        |  4 +-
 src/Data/Char/Properties.agda                 | 31 +++++---
 src/Data/Fin/Permutation.agda                 | 28 ++++----
 src/Data/Fin/Permutation/Components.agda      | 24 +++----
 src/Data/Fin/Properties.agda                  | 70 +++++++++++++------
 src/Data/Integer.agda                         |  2 +-
 src/Data/Integer/Properties.agda              | 26 ++++---
 src/Data/List/Relation/Binary/Lex/Strict.agda |  4 +-
 src/Data/List/Relation/Binary/Pointwise.agda  |  2 +-
 .../List/Relation/Unary/Any/Properties.agda   |  4 +-
 src/Data/Nat/DivMod/Core.agda                 |  2 +-
 src/Data/Nat/Divisibility.agda                |  2 +-
 .../Binary/Pointwise/NonDependent.agda        |  2 +-
 src/Data/String.agda                          |  6 +-
 src/Data/String/Properties.agda               | 33 ++++++---
 src/Data/Sum/Relation/Binary/Pointwise.agda   |  2 +-
 src/Data/Unit.agda                            |  2 +-
 src/Data/Unit/Polymorphic/Properties.agda     | 28 ++++++--
 src/Data/Unit/Properties.agda                 | 28 ++++++--
 .../Construct/Add/Infimum/NonStrict.agda      |  8 +--
 .../Binary/Construct/Add/Infimum/Strict.agda  |  4 +-
 .../Binary/Construct/Add/Point/Equality.agda  |  2 +-
 .../Construct/Add/Supremum/NonStrict.agda     |  8 +--
 .../Binary/Construct/Add/Supremum/Strict.agda |  4 +-
 .../Closure/Reflexive/Properties.agda         |  4 +-
 .../Binary/Construct/Intersection.agda        |  2 +-
 .../Binary/Properties/DecTotalOrder.agda      |  2 +-
 27 files changed, 214 insertions(+), 120 deletions(-)

diff --git a/doc/README/Data/Trie/NonDependent.agda b/doc/README/Data/Trie/NonDependent.agda
index 385e981880..f367b78587 100644
--- a/doc/README/Data/Trie/NonDependent.agda
+++ b/doc/README/Data/Trie/NonDependent.agda
@@ -83,7 +83,7 @@ record Lexer t : Set (suc t) where
 
   -- Two keywords are considered distinct if the strings are not equal
   Distinct : Rel Keyword 0ℓ
-  Distinct a b = ⌊ ¬? ((proj₁ a) String.≟ (proj₁ b)) ⌋
+  Distinct a b = ⌊ ¬? ((proj₁ a) String.≡? (proj₁ b)) ⌋
 
   field
 
@@ -175,7 +175,7 @@ module LetIn where
     LPAR RPAR : TOK
     ID : String → TOK
 
-  keywords : List# (String × TOK) (λ a b → ⌊ ¬? ((proj₁ a) String.≟ (proj₁ b)) ⌋)
+  keywords : List# (String × TOK) (λ a b → ⌊ ¬? ((proj₁ a) String.≡? (proj₁ b)) ⌋)
   keywords =  ("let" , LET)
            ∷# ("="   , EQ)
            ∷# ("in"  , IN)
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index 9e515f7e3a..5d158e6ae7 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -14,7 +14,7 @@ import Data.Nat.Base as ℕ using (ℕ; _<_; _≤_)
 import Data.Nat.Properties as ℕ
   using (_<?_; <-cmp; <-isStrictPartialOrder; <-isStrictTotalOrder
         ; <-strictPartialOrder; <-strictTotalOrder; <-irrefl; <-trans; <-asym
-        ; _≟_)
+        ; _≡?_)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (const; _∘′_)
 open import Relation.Nullary using (¬_; yes; no)
@@ -64,18 +64,18 @@ open import Agda.Builtin.Char.Properties
 ------------------------------------------------------------------------
 -- Properties of _≡_
 
-infix 4 _≟_
-_≟_ : DecidableEquality Char
-x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≟ toℕ y)
+infix 4 _≡?_
+_≡?_ : DecidableEquality Char
+x ≡? y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≡? toℕ y)
 
 setoid : Setoid _ _
 setoid = ≡.setoid Char
 
 decSetoid : DecSetoid _ _
-decSetoid = ≡.decSetoid _≟_
+decSetoid = ≡.decSetoid _≡?_
 
 isDecEquivalence : IsDecEquivalence _≡_
-isDecEquivalence = ≡.isDecEquivalence _≟_
+isDecEquivalence = ≡.isDecEquivalence _≡?_
 
 ------------------------------------------------------------------------
 -- Boolean equality test.
@@ -87,7 +87,7 @@ isDecEquivalence = ≡.isDecEquivalence _≟_
 
 infix 4 _==_
 _==_ : Char → Char → Bool
-c₁ == c₂ = isYes (c₁ ≟ c₂)
+c₁ == c₂ = isYes (c₁ ≡? c₂)
 
 private
 
@@ -179,7 +179,7 @@ _≤?_ = Refl.decidable <-cmp
 ≤-isDecPartialOrder : IsDecPartialOrder _≡_ _≤_
 ≤-isDecPartialOrder = record
   { isPartialOrder = ≤-isPartialOrder
-  ; _≟_            = _≟_
+  ; _≈?_           = _≡?_
   ; _≤?_           = _≤?_
   }
 
@@ -230,7 +230,7 @@ Please use Propositional Equality's subst instead."
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-x ≈? y = toℕ x ℕ.≟ toℕ y
+x ≈? y = toℕ x ℕ.≡? toℕ y
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -245,7 +245,7 @@ x ≈? y = toℕ x ℕ.≟ toℕ y
 ≈-isDecEquivalence : IsDecEquivalence _≈_
 ≈-isDecEquivalence = record
   { isEquivalence = ≈-isEquivalence
-  ; _≟_           = _≈?_
+  ; _≈?_           = _≈?_
   }
 ≈-decSetoid : DecSetoid _ _
 ≈-decSetoid = record
@@ -253,7 +253,7 @@ x ≈? y = toℕ x ℕ.≟ toℕ y
   }
 {-# WARNING_ON_USAGE _≈?_
 "Warning: _≈?_ was deprecated in v1.5.
-Please use _≟_ instead."
+Please use _≡?_ instead."
 #-}
 {-# WARNING_ON_USAGE ≈-isEquivalence
 "Warning: ≈-isEquivalence was deprecated in v1.5.
@@ -314,3 +314,12 @@ Please use <-strictPartialOrder instead."
 Please use <-strictTotalOrder instead."
 #-}
 
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 078e53c3bb..ef3dd75cf2 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base using (Fin; suc; cast; opposite; punchIn; punchOut)
 open import Data.Fin.Patterns using (0F; 1F)
 open import Data.Fin.Properties
-  using (¬Fin0; _≟_; ≟-≡-refl; ≟-≢
+  using (¬Fin0; _≡?_; ≡?-≡-refl; ≡?-≢
         ; cast-involutive; opposite-involutive
         ; punchInᵢ≢i; punchOut-punchIn; punchIn-punchOut
         ; punchOut-cong; punchOut-cong′)
@@ -218,21 +218,21 @@ insert : ∀ {m n} → Fin (suc m) → Fin (suc n) → Permutation m n → Permu
 insert {m} {n} i j π = permutation to from inverseˡ′ inverseʳ′
   where
   to : Fin (suc m) → Fin (suc n)
-  to k with i ≟ k
+  to k with i ≡? k
   ... | yes i≡k = j
   ... | no  i≢k = punchIn j (π ⟨$⟩ʳ punchOut i≢k)
 
   from : Fin (suc n) → Fin (suc m)
-  from k with j ≟ k
+  from k with j ≡? k
   ... | yes j≡k = i
   ... | no  j≢k = punchIn i (π ⟨$⟩ˡ punchOut j≢k)
 
   inverseʳ′ : StrictlyInverseʳ _≡_ to from
-  inverseʳ′ k with i ≟ k
-  ... | yes i≡k rewrite ≟-≡-refl j = i≡k
+  inverseʳ′ k with i ≡? k
+  ... | yes i≡k rewrite ≡?-≡-refl j = i≡k
   ... | no  i≢k
     with j≢punchInⱼπʳpunchOuti≢k ← punchInᵢ≢i j (π ⟨$⟩ʳ punchOut i≢k) ∘ sym
-    rewrite ≟-≢ j≢punchInⱼπʳpunchOuti≢k
+    rewrite ≡?-≢ j≢punchInⱼπʳpunchOuti≢k
     = begin
     punchIn i (π ⟨$⟩ˡ punchOut j≢punchInⱼπʳpunchOuti≢k)                    ≡⟨ cong (λ l → punchIn i (π ⟨$⟩ˡ l)) (punchOut-cong j refl) ⟩
     punchIn i (π ⟨$⟩ˡ punchOut (punchInᵢ≢i j (π ⟨$⟩ʳ punchOut i≢k) ∘ sym)) ≡⟨ cong (λ l → punchIn i (π ⟨$⟩ˡ l)) (punchOut-punchIn j) ⟩
@@ -241,11 +241,11 @@ insert {m} {n} i j π = permutation to from inverseˡ′ inverseʳ′
     k                                                                      ∎
 
   inverseˡ′ : StrictlyInverseˡ _≡_ to from
-  inverseˡ′ k with j ≟ k
-  ... | yes j≡k rewrite ≟-≡-refl i = j≡k
+  inverseˡ′ k with j ≡? k
+  ... | yes j≡k rewrite ≡?-≡-refl i = j≡k
   ... | no  j≢k
     with i≢punchInᵢπˡpunchOutj≢k ← punchInᵢ≢i i (π ⟨$⟩ˡ punchOut j≢k) ∘ sym
-    rewrite ≟-≢ i≢punchInᵢπˡpunchOutj≢k
+    rewrite ≡?-≢ i≢punchInᵢπˡpunchOutj≢k
     = begin
     punchIn j (π ⟨$⟩ʳ punchOut i≢punchInᵢπˡpunchOutj≢k)                    ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-cong i refl) ⟩
     punchIn j (π ⟨$⟩ʳ punchOut (punchInᵢ≢i i (π ⟨$⟩ˡ punchOut j≢k) ∘ sym)) ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-punchIn i) ⟩
@@ -317,14 +317,14 @@ lift₀-cong π ρ f (suc i) = cong suc (f i)
 
 lift₀-transpose : ∀ (i j : Fin n) → transpose (suc i) (suc j) ≈ lift₀ (transpose i j)
 lift₀-transpose i j 0F      = refl
-lift₀-transpose i j (suc k) with does (k ≟ i)
+lift₀-transpose i j (suc k) with does (k ≡? i)
 ... | true = refl
-... | false with does (k ≟ j)
+... | false with does (k ≡? j)
 ...   | false = refl
 ...   | true = refl
 
 insert-punchIn : ∀ i j (π : Permutation m n) k → insert i j π ⟨$⟩ʳ punchIn i k ≡ punchIn j (π ⟨$⟩ʳ k)
-insert-punchIn i j π k with i ≟ punchIn i k
+insert-punchIn i j π k with i ≡? punchIn i k
 ... | yes i≡punchInᵢk = contradiction (sym i≡punchInᵢk) (punchInᵢ≢i i k)
 ... | no  i≢punchInᵢk = begin
   punchIn j (π ⟨$⟩ʳ punchOut i≢punchInᵢk)            ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-cong i refl) ⟩
@@ -332,7 +332,7 @@ insert-punchIn i j π k with i ≟ punchIn i k
   punchIn j (π ⟨$⟩ʳ k)                               ∎
 
 insert-remove : ∀ i (π : Permutation (suc m) (suc n)) → insert i (π ⟨$⟩ʳ i) (remove i π) ≈ π
-insert-remove {m = m} {n = n} i π j with i ≟ j
+insert-remove {m = m} {n = n} i π j with i ≡? j
 ... | yes i≡j = cong (π ⟨$⟩ʳ_) i≡j
 ... | no  i≢j = begin
   punchIn (π ⟨$⟩ʳ i) (punchOut (punchInᵢ≢i i (punchOut i≢j) ∘ sym ∘ Injection.injective (↔⇒↣ π))) ≡⟨ punchIn-punchOut _ ⟩
@@ -340,7 +340,7 @@ insert-remove {m = m} {n = n} i π j with i ≟ j
   π ⟨$⟩ʳ j ∎
 
 remove-insert : ∀ i j (π : Permutation m n) → remove i (insert i j π) ≈ π
-remove-insert i j π k rewrite ≟-≡-refl i = begin
+remove-insert i j π k rewrite ≡?-≡-refl i = begin
   punchOut {i = j} _                               ≡⟨ punchOut-cong j (insert-punchIn i j π k) ⟩
   punchOut {i = j} (punchInᵢ≢i j (π ⟨$⟩ʳ k) ∘ sym) ≡⟨ punchOut-punchIn j ⟩
   π ⟨$⟩ʳ k                                         ∎
diff --git a/src/Data/Fin/Permutation/Components.agda b/src/Data/Fin/Permutation/Components.agda
index b8597674f8..f536a37519 100644
--- a/src/Data/Fin/Permutation/Components.agda
+++ b/src/Data/Fin/Permutation/Components.agda
@@ -11,7 +11,7 @@ module Data.Fin.Permutation.Components where
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.Fin.Base using (Fin; suc; opposite; toℕ)
 open import Data.Fin.Properties
-  using (_≟_; ≟-≡; ≟-≡-refl
+  using (_≡?_; ≡?-≡; ≡?-≡-refl
         ; opposite-prop; opposite-involutive; opposite-suc)
 open import Relation.Nullary.Decidable.Core using (does; yes; no)
 open import Relation.Binary.PropositionalEquality.Core
@@ -24,9 +24,9 @@ open import Relation.Binary.PropositionalEquality.Core
 -- 'transpose i j' swaps the places of 'i' and 'j'.
 
 transpose : ∀ {n} → Fin n → Fin n → Fin n → Fin n
-transpose i j k with does (k ≟ i)
+transpose i j k with does (k ≡? i)
 ... | true  = j
-... | false with does (k ≟ j)
+... | false with does (k ≡? j)
 ...   | true  = i
 ...   | false = k
 
@@ -35,25 +35,25 @@ transpose i j k with does (k ≟ i)
 ------------------------------------------------------------------------
 
 transpose[i,i,j]≡j : ∀ {n} (i j : Fin n) → transpose i i j ≡ j
-transpose[i,i,j]≡j i j with j ≟ i in j≟i
+transpose[i,i,j]≡j i j with j ≡? i in j≡?i
 ... | yes j≡i           = sym j≡i
-... | no  _ rewrite j≟i = refl
+... | no  _ rewrite j≡?i = refl
 
 transpose[i,j,j]≡i : ∀ {n} (i j : Fin n) → transpose i j j ≡ i
-transpose[i,j,j]≡i i j with j ≟ i
+transpose[i,j,j]≡i i j with j ≡? i
 ... | yes j≡i                     = j≡i
-... | no  _ rewrite ≟-≡-refl j = refl
+... | no  _ rewrite ≡?-≡-refl j = refl
 
 transpose[i,j,i]≡j : ∀ {n} (i j : Fin n) → transpose i j i ≡ j
-transpose[i,j,i]≡j i j rewrite ≟-≡-refl i = refl
+transpose[i,j,i]≡j i j rewrite ≡?-≡-refl i = refl
 
 transpose-transpose : ∀ {n} {i j k l : Fin n} →
                       transpose i j k ≡ l → transpose j i l ≡ k
-transpose-transpose {n} {i} {j} {k} {l} eq with k ≟ i in k≟i
-... | yes k≡i rewrite ≟-≡ (sym eq) = sym k≡i
-... | no k≢i with k ≟ j in k≟j
+transpose-transpose {n} {i} {j} {k} {l} eq with k ≡? i in k≡?i
+... | yes k≡i rewrite ≡?-≡ (sym eq) = sym k≡i
+... | no k≢i with k ≡? j in k≡?j
 ...   | yes k≡j rewrite eq | transpose[i,j,j]≡i j l = sym k≡j
-...   | no  k≢j rewrite eq | k≟j | k≟i = refl
+...   | no  k≢j rewrite eq | k≡?j | k≡?i = refl
 
 transpose-inverse : ∀ {n} (i j : Fin n) {k} →
                     transpose i j (transpose j i k) ≡ k
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index ec5767c34c..f3ea2a31b9 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -46,7 +46,7 @@ open import Relation.Binary.PropositionalEquality.Core as ≡
 open import Relation.Binary.PropositionalEquality.Properties as ≡
   using (module ≡-Reasoning)
 open import Relation.Binary.PropositionalEquality as ≡
-  using (≡-≟-identity; ≢-≟-identity)
+  using (≡-≡?-identity; ≢-≡?-identity)
 open import Relation.Nullary.Decidable as Dec
   using (Dec; _because_; yes; no; _×?_; _⊎?_; map′; decidable-stable)
 open import Relation.Nullary.Negation.Core
@@ -97,25 +97,25 @@ nonZeroIndex {n = suc _} _ = _
 suc-injective : Fin.suc i ≡ suc j → i ≡ j
 suc-injective refl = refl
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality (Fin n)
-zero  ≟ zero  = yes refl
-zero  ≟ suc y = no λ()
-suc x ≟ zero  = no λ()
-suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)
+_≡?_ : DecidableEquality (Fin n)
+zero  ≡? zero  = yes refl
+zero  ≡? suc y = no λ()
+suc x ≡? zero  = no λ()
+suc x ≡? suc y = map′ (cong suc) suc-injective (x ≡? y)
 
 ≡-irrelevant : Irrelevant {A = Fin n} _≡_
-≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≟_
+≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≡?_
 
-≟-≡ : (eq : i ≡ j) → (i ≟ j) ≡ yes eq
-≟-≡ = ≡-≟-identity _≟_
+≡?-≡ : (eq : i ≡ j) → (i ≡? j) ≡ yes eq
+≡?-≡ = ≡-≡?-identity _≡?_
 
-≟-≡-refl : (i : Fin n)  → (i ≟ i) ≡ yes refl
-≟-≡-refl _ = ≟-≡ refl
+≡?-≡-refl : (i : Fin n)  → (i ≡? i) ≡ yes refl
+≡?-≡-refl _ = ≡?-≡ refl
 
-≟-≢ : (i≢j : i ≢ j) → (i ≟ j) ≡ no i≢j
-≟-≢ = ≢-≟-identity _≟_
+≡?-≢ : (i≢j : i ≢ j) → (i ≡? j) ≡ no i≢j
+≡?-≢ = ≢-≡?-identity _≡?_
 
 ------------------------------------------------------------------------
 -- Structures
@@ -123,7 +123,7 @@ suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)
 ≡-isDecEquivalence : IsDecEquivalence {A = Fin n} _≡_
 ≡-isDecEquivalence = record
   { isEquivalence = ≡.isEquivalence
-  ; _≟_           = _≟_
+  ; _≈?_          = _≡?_
   }
 
 ------------------------------------------------------------------------
@@ -356,7 +356,7 @@ m <? n = suc (toℕ m) ℕ.≤? toℕ n
 ≤-isDecTotalOrder : IsDecTotalOrder {A = Fin n} _≡_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≟_
+  ; _≈?_         = _≡?_
   ; _≤?_         = _≤?_
   }
 
@@ -1114,7 +1114,7 @@ decFinSubset {suc _} {P = P} {Q = Q} Q? P? = dec[Q⊆P]
 
 pigeonhole : m ℕ.< n → (f : Fin n → Fin m) → ∃₂ λ i j → i < j × f i ≡ f j
 pigeonhole z<s               f = contradiction (f zero) λ()
-pigeonhole (s<s m<n@(s≤s _)) f with any? (λ k → f zero ≟ f (suc k))
+pigeonhole (s<s m<n@(s≤s _)) f with any? (λ k → f zero ≡? f (suc k))
 ... | yes (j , f₀≡fⱼ) = zero , suc j , z<s , f₀≡fⱼ
 ... | no  f₀≢fₖ =
   let i , j , i<j , fᵢ≡fⱼ = pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
@@ -1191,14 +1191,14 @@ module _ {f} {F : Set f → Set f} (RF : RawFunctor F) where
 module _ {ℓ} {S : Setoid a ℓ} (inj : Injection S (≡-setoid n)) where
   open Setoid S
 
-  inj⇒≟ : B.Decidable _≈_
-  inj⇒≟ = Dec.via-injection inj _≟_
+  inj⇒≡? : B.Decidable _≈_
+  inj⇒≡? = Dec.via-injection inj _≡?_
 
   inj⇒decSetoid : DecSetoid a ℓ
   inj⇒decSetoid = record
     { isDecEquivalence = record
       { isEquivalence = isEquivalence
-      ; _≟_           = inj⇒≟
+      ; _≈?_           = inj⇒≡?
       }
     }
 
@@ -1282,10 +1282,10 @@ Fin0↔⊥ = 0↔⊥
 Please use 0↔⊥ instead."
 #-}
 eq? : A ↣ Fin n → DecidableEquality A
-eq? = inj⇒≟
+eq? = inj⇒≡?
 {-# WARNING_ON_USAGE eq?
 "Warning: eq? was deprecated in v2.0.
-Please use inj⇒≟ instead."
+Please use inj⇒≡? instead."
 #-}
 
 private
@@ -1329,3 +1329,29 @@ Please use ¬∀⇒∃¬-smallest instead."
 "Warning: ¬∀⟶∃¬ was deprecated in v2.4.
 Please use ¬∀⇒∃¬ instead."
 #-}
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+
+
+≟-≡-refl = ≡?-≡-refl
+{-# WARNING_ON_USAGE ≟-≡-refl
+"Warning: ≟-≡-refl was deprecated in v2.4.
+Please use ≡?-≡-refl instead."
+#-}
+
+≟-≡ = ≡?-≡
+{-# WARNING_ON_USAGE ≟-≡
+"Warning: ≟-≡ was deprecated in v2.4.
+Please use ≡?-≡ instead."
+#-}
+
+inj⇒≟ = inj⇒≡?
+{-# WARNING_ON_USAGE inj⇒≟
+"Warning: inj⇒≟ was deprecated in v2.4.
+Please use inj⇒≡? instead."
+#-}
diff --git a/src/Data/Integer.agda b/src/Data/Integer.agda
index 6e4976f3df..69aaa5500c 100644
--- a/src/Data/Integer.agda
+++ b/src/Data/Integer.agda
@@ -16,7 +16,7 @@ module Data.Integer where
 
 open import Data.Integer.Base public
 open import Data.Integer.Properties public
-  using (_≟_; _≤?_; _<?_)
+  using (_≡?_; _≤?_; _<?_)
 
 ------------------------------------------------------------------------
 -- Deprecated
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 78b58585c5..fe2ff5ab52 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -70,18 +70,18 @@ private
 +[1+-injective : +[1+ m ] ≡ +[1+ n ] → m ≡ n
 +[1+-injective refl = refl
 
-infix 4 _≟_
-_≟_ : DecidableEquality ℤ
-+ m      ≟ + n      = Dec.map′ (cong (+_)) +-injective (m ℕ.≟ n)
-+ m      ≟ -[1+ n ] = no λ()
--[1+ m ] ≟ + n      = no λ()
--[1+ m ] ≟ -[1+ n ] = Dec.map′ (cong -[1+_]) -[1+-injective (m ℕ.≟ n)
+infix 4 _≡?_
+_≡?_ : DecidableEquality ℤ
++ m      ≡? + n      = Dec.map′ (cong (+_)) +-injective (m ℕ.≡? n)
++ m      ≡? -[1+ n ] = no λ()
+-[1+ m ] ≡? + n      = no λ()
+-[1+ m ] ≡? -[1+ n ] = Dec.map′ (cong -[1+_]) -[1+-injective (m ℕ.≡? n)
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid ℤ
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Properties of _≤_
@@ -162,7 +162,7 @@ _≤?_ : Decidable _≤_
 ≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≟_
+  ; _≈?_         = _≡?_
   ; _≤?_         = _≤?_
   }
 
@@ -2395,3 +2395,13 @@ Please use +-0-isAbelianGroup instead."
 {- issue1844/issue1755: raw bundles have moved to `Data.X.Base` -}
 open Data.Integer.Base public
   using (*-rawMagma; *-1-rawMonoid)
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index 26f05b6878..f274241522 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -203,7 +203,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
                           IsDecPartialOrder _≋_ _≤_
     ≤-isDecPartialOrder sto = record
       { isPartialOrder = ≤-isPartialOrder isStrictPartialOrder
-      ; _≟_            = Pointwise.decidable _≟_
+      ; _≈?_            = Pointwise.decidable _≈?_
       ; _≤?_           = ≤-decidable _≟_ _<?_
       } where open IsStrictTotalOrder sto
 
@@ -218,7 +218,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
                         IsDecTotalOrder _≋_ _≤_
     ≤-isDecTotalOrder sto = record
       { isTotalOrder = ≤-isTotalOrder sto
-      ; _≟_          = Pointwise.decidable _≟_
+      ; _≈?_          = Pointwise.decidable _≈?_
       ; _≤?_         = ≤-decidable _≟_ _<?_
       }
       where open IsStrictTotalOrder sto
diff --git a/src/Data/List/Relation/Binary/Pointwise.agda b/src/Data/List/Relation/Binary/Pointwise.agda
index c2139dadfc..b175006b01 100644
--- a/src/Data/List/Relation/Binary/Pointwise.agda
+++ b/src/Data/List/Relation/Binary/Pointwise.agda
@@ -65,7 +65,7 @@ isEquivalence eq = record
 isDecEquivalence : IsDecEquivalence R → IsDecEquivalence (Pointwise R)
 isDecEquivalence eq = record
   { isEquivalence = isEquivalence DE.isEquivalence
-  ; _≟_           = decidable     DE._≟_
+  ; _≈?_          = decidable     DE._≈?_
   } where module DE = IsDecEquivalence eq
 
 isPreorder : IsPreorder R S → IsPreorder (Pointwise R) (Pointwise S)
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index f9b139dfdb..353b3e3405 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -22,7 +22,7 @@ open import Data.List.Membership.Propositional.Properties.Core
 open import Data.List.Relation.Binary.Pointwise
   using (Pointwise; []; _∷_)
 open import Data.Nat.Base using (zero; suc; _<_; z<s; s<s; s≤s)
-open import Data.Nat.Properties using (_≟_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1+n)
+open import Data.Nat.Properties using (_≡?_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1+n)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any as MAny using (just)
 open import Data.Product.Base as Product
@@ -522,7 +522,7 @@ applyUpTo⁻ f {suc n} (there p) =
 -- applyDownFrom
 
 applyDownFrom⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyDownFrom f n)
-applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≟ n
+applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≡? n
 ... | yes refl = here p
 ... | no  i≢n  = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))
 
diff --git a/src/Data/Nat/DivMod/Core.agda b/src/Data/Nat/DivMod/Core.agda
index 7e627f4bfa..7a37c8cbdc 100644
--- a/src/Data/Nat/DivMod/Core.agda
+++ b/src/Data/Nat/DivMod/Core.agda
@@ -172,7 +172,7 @@ private
   -- (suc , suc) cases
   divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₁ (eq , s≤s j≤k , 1+k<mod)) =
     divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d 1+k<mod))
-  divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₂′ (eq , mod≤1+j , (s≤s j≤k))) with modₕ 0 d (suc n) d ≟ 0
+  divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₂′ (eq , mod≤1+j , (s≤s j≤k))) with modₕ 0 d (suc n) d ≡? 0
   ... | yes mod≡0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n mod≡0)) k≤d))
   ... | no  mod≢0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₂′ (eq , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (n≢0⇒n>0 mod≢0) mod≤1+j , j≤k))
   divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₃  (eq , k<mod , mod≤1+j)) =
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index cba883efab..aa1126e428 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -122,7 +122,7 @@ infix 4 _∣?_
 _∣?_ : Decidable _∣_
 zero  ∣? zero   = yes (divides-refl 0)
 zero  ∣? suc m  = no ((λ()) ∘′ ∣-antisym (divides-refl 0))
-n@(suc _) ∣? m  = Dec.map (m%n≡0⇔n∣m m n) (m % n ≟ 0)
+n@(suc _) ∣? m  = Dec.map (m%n≡0⇔n∣m m n) (m % n ≡? 0)
 
 ∣-isPreorder : IsPreorder _≡_ _∣_
 ∣-isPreorder = record
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 20a1d5ce01..e2a12a2a9a 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -108,7 +108,7 @@ Pointwise R S (a , c) (b , d) = (R a b) × (S c d)
 ×-isDecEquivalence eq₁ eq₂ = record
   { isEquivalence = ×-isEquivalence
                       (isEquivalence eq₁) (isEquivalence eq₂)
-  ; _≟_           = ×-decidable (_≟_ eq₁) (_≟_ eq₂)
+  ; _≈?_          = ×-decidable (_≈?_ eq₁) (_≈?_ eq₂)
   } where open IsDecEquivalence
 
 ×-isPreorder : IsPreorder ≈₁ R → IsPreorder ≈₂ S →
diff --git a/src/Data/String.agda b/src/Data/String.agda
index 98b01b3b5f..9186c514d1 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -17,17 +17,17 @@ open import Data.List.Base as List using (List)
 open import Data.List.Extrema ℕ.≤-totalOrder
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Char.Base as Char using (Char)
-import Data.Char.Properties as Char using (_≟_)
+import Data.Char.Properties as Char using (_≡?_)
 open import Relation.Nullary.Decidable.Core using (does)
 
-open import Data.List.Membership.DecPropositional Char._≟_
+open import Data.List.Membership.DecPropositional Char._≡?_
 
 
 ------------------------------------------------------------------------
 -- Re-export contents of base, and decidability of equality
 
 open import Data.String.Base public
-open import Data.String.Properties using (_≈?_; _≟_; _<?_; _==_) public
+open import Data.String.Properties using (_≈?_; _≡?_; _<?_; _==_) public
 
 ------------------------------------------------------------------------
 -- Conversion functions
diff --git a/src/Data/String/Properties.agda b/src/Data/String/Properties.agda
index 2728f47934..89013023c2 100644
--- a/src/Data/String/Properties.agda
+++ b/src/Data/String/Properties.agda
@@ -61,7 +61,7 @@ open import Agda.Builtin.String.Properties public
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-x ≈? y = Pointwise.decidable Char._≟_ (toList x) (toList y)
+x ≈? y = Pointwise.decidable Char._≡?_ (toList x) (toList y)
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -78,7 +78,7 @@ x ≈? y = Pointwise.decidable Char._≟_ (toList x) (toList y)
 ≈-isDecEquivalence : IsDecEquivalence _≈_
 ≈-isDecEquivalence = record
   { isEquivalence = ≈-isEquivalence
-  ; _≟_           = _≈?_
+  ; _≈?_          = _≈?_
   }
 
 ≈-decSetoid : DecSetoid _ _
@@ -89,23 +89,23 @@ x ≈? y = Pointwise.decidable Char._≟_ (toList x) (toList y)
 ------------------------------------------------------------------------
 -- Properties of _≡_
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality String
-x ≟ y = map′ ≈⇒≡ ≈-reflexive $ x ≈? y
+_≡?_ : DecidableEquality String
+x ≡? y = map′ ≈⇒≡ ≈-reflexive $ x ≈? y
 
 ≡-setoid : Setoid _ _
 ≡-setoid = PropEq.setoid String
 
 ≡-decSetoid : DecSetoid _ _
-≡-decSetoid = PropEq.decSetoid _≟_
+≡-decSetoid = PropEq.decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Properties of _<_
 
 infix 4 _<?_
 _<?_ : Decidable _<_
-x <? y = StrictLex.<-decidable Char._≟_ Char._<?_ (toList x) (toList y)
+x <? y = StrictLex.<-decidable Char._≡?_ Char._<?_ (toList x) (toList y)
 
 <-isStrictPartialOrder-≈ : IsStrictPartialOrder _≈_ _<_
 <-isStrictPartialOrder-≈ =
@@ -165,7 +165,7 @@ x <? y = StrictLex.<-decidable Char._≟_ Char._<?_ (toList x) (toList y)
 
 infix 4 _==_
 _==_ : String → String → Bool
-s₁ == s₂ = isYes (s₁ ≟ s₂)
+s₁ == s₂ = isYes (s₁ ≡? s₂)
 
 private
 
@@ -177,3 +177,20 @@ private
 
   unit-test : P (_==_ "")
   unit-test = p _
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index f0d0e7c39f..9acf4961f5 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -142,7 +142,7 @@ drop-inj₂ (inj₂ x) = x
 ⊎-isDecEquivalence eq₁ eq₂ = record
   { isEquivalence =
       ⊎-isEquivalence (isEquivalence eq₁) (isEquivalence eq₂)
-  ; _≟_           = ⊎-decidable (_≟_ eq₁) (_≟_ eq₂)
+  ; _≈?_          = ⊎-decidable (_≈?_ eq₁) (_≈?_ eq₂)
   } where open IsDecEquivalence
 
 ⊎-isPreorder : IsPreorder ≈₁ R → IsPreorder ≈₂ S →
diff --git a/src/Data/Unit.agda b/src/Data/Unit.agda
index 73f269d9d6..b622fc7e33 100644
--- a/src/Data/Unit.agda
+++ b/src/Data/Unit.agda
@@ -17,4 +17,4 @@ open import Data.Unit.Base public
 -- Re-export query operations
 
 open import Data.Unit.Properties public
-  using (_≟_)
+  using (_≡?_)
diff --git a/src/Data/Unit/Polymorphic/Properties.agda b/src/Data/Unit/Polymorphic/Properties.agda
index f5e9fb9be0..d7cbf6542e 100644
--- a/src/Data/Unit/Polymorphic/Properties.agda
+++ b/src/Data/Unit/Polymorphic/Properties.agda
@@ -35,16 +35,16 @@ private
 -- Equality
 ------------------------------------------------------------------------
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality (⊤ {ℓ})
-_ ≟ _ = yes refl
+_≡?_ : DecidableEquality (⊤ {ℓ})
+_ ≡? _ = yes refl
 
 ≡-setoid : ∀ ℓ → Setoid ℓ ℓ
 ≡-setoid _ = setoid ⊤
 
 ≡-decSetoid : ∀ ℓ → DecSetoid ℓ ℓ
-≡-decSetoid _ = decSetoid _≟_
+≡-decSetoid _ = decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Ordering
@@ -81,8 +81,8 @@ _ ≟ _ = yes refl
 ≡-isDecTotalOrder : ∀ ℓ → IsDecTotalOrder {ℓ} _≡_ _≡_
 ≡-isDecTotalOrder ℓ = record
   { isTotalOrder = ≡-isTotalOrder ℓ
-  ; _≟_          = _≟_
-  ; _≤?_         = _≟_
+  ; _≈?_         = _≡?_
+  ; _≤?_         = _≡?_
   }
 
 ------------------------------------------------------------------------
@@ -110,3 +110,19 @@ _ ≟ _ = yes refl
 
 ⊤↔⊤* : ⊤ {ℓ} ↔ ⊤*
 ⊤↔⊤* = mk↔ ((λ _ → refl) , (λ _ → refl))
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Unit/Properties.agda b/src/Data/Unit/Properties.agda
index d93055db8d..bf124bc1ea 100644
--- a/src/Data/Unit/Properties.agda
+++ b/src/Data/Unit/Properties.agda
@@ -34,16 +34,16 @@ open import Relation.Nullary.Irrelevant using (Irrelevant)
 ------------------------------------------------------------------------
 -- Equality
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality ⊤
-_ ≟ _ = yes refl
+_≡?_ : DecidableEquality ⊤
+_ ≡? _ = yes refl
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid ⊤
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Relational properties
@@ -79,8 +79,8 @@ _ ≟ _ = yes refl
 ≡-isDecTotalOrder : IsDecTotalOrder _≡_ _≡_
 ≡-isDecTotalOrder = record
   { isTotalOrder = ≡-isTotalOrder
-  ; _≟_          = _≟_
-  ; _≤?_         = _≟_
+  ; _≈?_         = _≡?_
+  ; _≤?_         = _≡?_
   }
 
 ------------------------------------------------------------------------
@@ -96,3 +96,19 @@ _ ≟ _ = yes refl
   { isDecTotalOrder = ≡-isDecTotalOrder
   }
 
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+
diff --git a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
index 1c59d03972..2e248ed6dc 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
@@ -134,7 +134,7 @@ module _ {e} {_≈_ : Rel A e} where
 ≤₋-isDecPartialOrder-≡ : IsDecPartialOrder _≡_ _≤_ → IsDecPartialOrder _≡_ _≤₋_
 ≤₋-isDecPartialOrder-≡ ≤-isDecPartialOrder = record
   { isPartialOrder = ≤₋-isPartialOrder-≡ isPartialOrder
-  ; _≟_            = ≡-dec _≟_
+  ; _≈?_           = ≡-dec _≈?_
   ; _≤?_           = ≤₋-dec _≤?_
   } where open IsDecPartialOrder ≤-isDecPartialOrder
 
@@ -147,7 +147,7 @@ module _ {e} {_≈_ : Rel A e} where
 ≤₋-isDecTotalOrder-≡ : IsDecTotalOrder _≡_ _≤_ → IsDecTotalOrder _≡_ _≤₋_
 ≤₋-isDecTotalOrder-≡ ≤-isDecTotalOrder = record
   { isTotalOrder = ≤₋-isTotalOrder-≡ isTotalOrder
-  ; _≟_          = ≡-dec _≟_
+  ; _≈?_         = ≡-dec _≈?_
   ; _≤?_         = ≤₋-dec _≤?_
   } where open IsDecTotalOrder ≤-isDecTotalOrder
 
@@ -174,7 +174,7 @@ module _ {e} {_≈_ : Rel A e} where
   ≤₋-isDecPartialOrder : IsDecPartialOrder _≈_ _≤_ → IsDecPartialOrder _≈₋_ _≤₋_
   ≤₋-isDecPartialOrder ≤-isDecPartialOrder = record
     { isPartialOrder = ≤₋-isPartialOrder isPartialOrder
-    ; _≟_            = ≈₋-dec _≟_
+    ; _≈?_           = ≈₋-dec _≈?_
     ; _≤?_           = ≤₋-dec _≤?_
     } where open IsDecPartialOrder ≤-isDecPartialOrder
 
@@ -187,6 +187,6 @@ module _ {e} {_≈_ : Rel A e} where
   ≤₋-isDecTotalOrder : IsDecTotalOrder _≈_ _≤_ → IsDecTotalOrder _≈₋_ _≤₋_
   ≤₋-isDecTotalOrder ≤-isDecTotalOrder = record
     { isTotalOrder = ≤₋-isTotalOrder isTotalOrder
-    ; _≟_          = ≈₋-dec _≟_
+    ; _≈?_         = ≈₋-dec _≈?_
     ; _≤?_         = ≤₋-dec _≤?_
     } where open IsDecTotalOrder ≤-isDecTotalOrder
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index 8769ac6e94..ddaaf9286c 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -166,7 +166,7 @@ module _ {e} {_≈_ : Rel A e} where
                                IsDecStrictPartialOrder _≡_ _<₋_
 <₋-isDecStrictPartialOrder-≡ dectot = record
   { isStrictPartialOrder = <₋-isStrictPartialOrder-≡ isStrictPartialOrder
-  ; _≟_                  = ≡-dec _≟_
+  ; _≈?_                 = ≡-dec _≈?_
   ; _<?_                 = <₋-dec _<?_
   } where open IsDecStrictPartialOrder dectot
 
@@ -197,7 +197,7 @@ module _ {e} {_≈_ : Rel A e} where
                                IsDecStrictPartialOrder _≈₋_ _<₋_
   <₋-isDecStrictPartialOrder dectot = record
     { isStrictPartialOrder = <₋-isStrictPartialOrder isStrictPartialOrder
-    ; _≟_                  = ≈₋-dec _≟_
+    ; _≈?_                 = ≈₋-dec _≈?_
     ; _<?_                 = <₋-dec _<?_
     } where open IsDecStrictPartialOrder dectot
 
diff --git a/src/Relation/Binary/Construct/Add/Point/Equality.agda b/src/Relation/Binary/Construct/Add/Point/Equality.agda
index cf272759cf..fdb3d9fee0 100644
--- a/src/Relation/Binary/Construct/Add/Point/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Point/Equality.agda
@@ -79,5 +79,5 @@ data _≈∙_ : Rel (Pointed A) (a ⊔ ℓ) where
 ≈∙-isDecEquivalence : IsDecEquivalence _≈_ → IsDecEquivalence _≈∙_
 ≈∙-isDecEquivalence ≈-isDecEquivalence = record
   { isEquivalence = ≈∙-isEquivalence isEquivalence
-  ; _≟_           = ≈∙-dec _≟_
+  ; _≈?_          = ≈∙-dec _≈?_
   } where open IsDecEquivalence ≈-isDecEquivalence
diff --git a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
index 124068aa80..b74f98c06d 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
@@ -133,7 +133,7 @@ module _ {e} {_≈_ : Rel A e} where
 ≤⁺-isDecPartialOrder-≡ : IsDecPartialOrder _≡_ _≤_ → IsDecPartialOrder _≡_ _≤⁺_
 ≤⁺-isDecPartialOrder-≡ ≤-isDecPartialOrder = record
   { isPartialOrder = ≤⁺-isPartialOrder-≡ isPartialOrder
-  ; _≟_            = ≡-dec _≟_
+  ; _≈?_           = ≡-dec _≈?_
   ; _≤?_           = ≤⁺-dec _≤?_
   } where open IsDecPartialOrder ≤-isDecPartialOrder
 
@@ -146,7 +146,7 @@ module _ {e} {_≈_ : Rel A e} where
 ≤⁺-isDecTotalOrder-≡ : IsDecTotalOrder _≡_ _≤_ → IsDecTotalOrder _≡_ _≤⁺_
 ≤⁺-isDecTotalOrder-≡ ≤-isDecTotalOrder = record
   { isTotalOrder = ≤⁺-isTotalOrder-≡ isTotalOrder
-  ; _≟_          = ≡-dec _≟_
+  ; _≈?_         = ≡-dec _≈?_
   ; _≤?_         = ≤⁺-dec _≤?_
   } where open IsDecTotalOrder ≤-isDecTotalOrder
 
@@ -173,7 +173,7 @@ module _ {e} {_≈_ : Rel A e} where
   ≤⁺-isDecPartialOrder : IsDecPartialOrder _≈_ _≤_ → IsDecPartialOrder _≈⁺_ _≤⁺_
   ≤⁺-isDecPartialOrder ≤-isDecPartialOrder = record
     { isPartialOrder = ≤⁺-isPartialOrder isPartialOrder
-    ; _≟_            = ≈⁺-dec _≟_
+    ; _≈?_           = ≈⁺-dec _≈?_
     ; _≤?_           = ≤⁺-dec _≤?_
     } where open IsDecPartialOrder ≤-isDecPartialOrder
 
@@ -186,6 +186,6 @@ module _ {e} {_≈_ : Rel A e} where
   ≤⁺-isDecTotalOrder : IsDecTotalOrder _≈_ _≤_ → IsDecTotalOrder _≈⁺_ _≤⁺_
   ≤⁺-isDecTotalOrder ≤-isDecTotalOrder = record
     { isTotalOrder = ≤⁺-isTotalOrder isTotalOrder
-    ; _≟_          = ≈⁺-dec _≟_
+    ; _≈?_         = ≈⁺-dec _≈?_
     ; _≤?_         = ≤⁺-dec _≤?_
     } where open IsDecTotalOrder ≤-isDecTotalOrder
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
index fd4a78d92f..56829bfec3 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
@@ -150,7 +150,7 @@ module _ {e} {_≈_ : Rel A e} where
                                IsDecStrictPartialOrder _≡_ _<⁺_
 <⁺-isDecStrictPartialOrder-≡ dectot = record
   { isStrictPartialOrder = <⁺-isStrictPartialOrder-≡ isStrictPartialOrder
-  ; _≟_                  = ≡-dec _≟_
+  ; _≈?_                 = ≡-dec _≈?_
   ; _<?_                 = <⁺-dec _<?_
   } where open IsDecStrictPartialOrder dectot
 
@@ -181,7 +181,7 @@ module _ {e} {_≈_ : Rel A e} where
                                IsDecStrictPartialOrder _≈⁺_ _<⁺_
   <⁺-isDecStrictPartialOrder dectot = record
     { isStrictPartialOrder = <⁺-isStrictPartialOrder isStrictPartialOrder
-    ; _≟_                  = ≈⁺-dec _≟_
+    ; _≈?_                 = ≈⁺-dec _≈?_
     ; _<?_                 = <⁺-dec _<?_
     } where open IsDecStrictPartialOrder dectot
 
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index e6d494889f..dce7099889 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -134,7 +134,7 @@ module _ {_~_ : Rel A ℓ} where
   isDecPartialOrder : IsDecStrictPartialOrder _≡_ _~_ → IsDecPartialOrder _≡_ _~ᵒ_
   isDecPartialOrder O = record
     { isPartialOrder = isPartialOrder O.isStrictPartialOrder
-    ; _≟_            = O._≟_
+    ; _≈?_           = O._≈?_
     ; _≤?_           = dec O._≟_ O._<?_
     } where module O = IsDecStrictPartialOrder O
 
@@ -147,6 +147,6 @@ module _ {_~_ : Rel A ℓ} where
   isDecTotalOrder : IsStrictTotalOrder _≡_ _~_ → IsDecTotalOrder _≡_ _~ᵒ_
   isDecTotalOrder O = record
     { isTotalOrder = isTotalOrder O
-    ; _≟_          = _≟_
+    ; _≈?_         = _≈?_
     ; _≤?_         = dec _≟_ _<?_
     } where open IsStrictTotalOrder O
diff --git a/src/Relation/Binary/Construct/Intersection.agda b/src/Relation/Binary/Construct/Intersection.agda
index b5dda5737e..d1f5587641 100644
--- a/src/Relation/Binary/Construct/Intersection.agda
+++ b/src/Relation/Binary/Construct/Intersection.agda
@@ -102,7 +102,7 @@ isEquivalence {L = L} {R = R} eqₗ eqᵣ = record
 isDecEquivalence : IsDecEquivalence L → IsDecEquivalence R → IsDecEquivalence (L ∩ R)
 isDecEquivalence eqₗ eqᵣ = record
   { isEquivalence = isEquivalence L.isEquivalence R.isEquivalence
-  ; _≟_           = decidable L._≟_ R._≟_
+  ; _≈?_           = decidable L._≈?_ R._≈?_
   } where module L = IsDecEquivalence eqₗ; module R = IsDecEquivalence eqᵣ
 
 isPreorder : IsPreorder ≈ L → IsPreorder ≈ R → IsPreorder ≈ (L ∩ R)
diff --git a/src/Relation/Binary/Properties/DecTotalOrder.agda b/src/Relation/Binary/Properties/DecTotalOrder.agda
index 50579656ff..70f8805f23 100644
--- a/src/Relation/Binary/Properties/DecTotalOrder.agda
+++ b/src/Relation/Binary/Properties/DecTotalOrder.agda
@@ -94,5 +94,5 @@ open TotalOrderProperties public
   )
 
 ≮⇒≥ : ∀ {x y} → x ≮ y → y ≤ x
-≮⇒≥ = ToStrict.≮⇒≥ Eq.sym _≟_ reflexive total
+≮⇒≥ = ToStrict.≮⇒≥ Eq.sym _≈?_ reflexive total
 

From 9481e20a617a8361262eece7c09a391dd774cbe0 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 12:16:17 +0000
Subject: [PATCH 04/20] refactor: last round of knock-ons

---
 doc/README/Data/List/Membership.agda          |   4 +-
 .../Data/List/Relation/Binary/Subset.agda     |   4 +-
 doc/README/Data/Record.agda                   |   2 +-
 doc/README/Foreign/Haskell.agda               |   4 +-
 doc/README/Function/Reasoning.agda            |   2 +-
 doc/README/Nary.agda                          |   4 +-
 .../Solver/CommutativeMonoid/Normal.agda      |  15 +-
 .../IdempotentCommutativeMonoid/Normal.agda   |  15 +-
 src/Algebra/Solver/Monoid/Normal.agda         |  25 +-
 .../Ring/NaturalCoefficients/Default.agda     |   4 +-
 src/Data/Bool.agda                            |   2 +-
 src/Data/Bool/Instances.agda                  |   4 +-
 src/Data/Bool/Solver.agda                     |   6 +-
 src/Data/Char.agda                            |   2 +-
 src/Data/Digit/Properties.agda                |   2 +-
 src/Data/Fin.agda                             |   2 +-
 src/Data/Fin/Subset.agda                      |   4 +-
 src/Data/Fin/Subset/Properties.agda           |   4 +-
 src/Data/Float.agda                           |   2 +-
 src/Data/Float/Instances.agda                 |   4 +-
 src/Data/Float/Properties.agda                |  30 +-
 src/Data/Graph/Acyclic.agda                   |   4 +-
 src/Data/Integer/Divisibility/Signed.agda     |   2 +-
 src/Data/Integer/Instances.agda               |   4 +-
 src/Data/Integer/Solver.agda                  |   4 +-
 src/Data/List/Instances.agda                  |   4 +-
 .../Binary/Infix/Homogeneous/Properties.agda  |   2 +-
 .../List/Relation/Binary/Lex/NonStrict.agda   |   6 +-
 .../Binary/Prefix/Homogeneous/Properties.agda |   2 +-
 .../Binary/Sublist/DecPropositional.agda      |   8 +-
 .../Relation/Binary/Sublist/DecSetoid.agda    |   4 +-
 .../Sublist/Heterogeneous/Properties.agda     |   2 +-
 .../Binary/Sublist/Heterogeneous/Solver.agda  |   2 +-
 .../Binary/Suffix/Homogeneous/Properties.agda |   2 +-
 src/Data/Maybe/Relation/Binary/Pointwise.agda |   2 +-
 src/Data/Nat.agda                             |   2 +-
 src/Data/Nat/Binary/Instances.agda            |   4 +-
 src/Data/Nat/Binary/Properties.agda           |  39 +-
 src/Data/Nat/Combinatorics/Specification.agda |   2 +-
 src/Data/Nat/Coprimality.agda                 |   2 +-
 src/Data/Nat/GCD.agda                         |   2 +-
 src/Data/Nat/Instances.agda                   |   4 +-
 src/Data/Nat/Primality.agda                   |   4 +-
 src/Data/Nat/Solver.agda                      |   2 +-
 src/Data/Product/Instances.agda               |   2 +-
 .../Relation/Binary/Lex/NonStrict.agda        |   6 +-
 src/Data/Rational/Instances.agda              |   4 +-
 src/Data/Rational/Properties.agda             |  23 +-
 src/Data/Rational/Solver.agda                 |   4 +-
 .../Rational/Unnormalised/Properties.agda     |   6 +-
 src/Data/String/Instances.agda                |   4 +-
 src/Data/Sum/Instances.agda                   |   4 +-
 src/Data/Sum/Relation/Binary/LeftOrder.agda   |   2 +-
 src/Data/These/Instances.agda                 |   4 +-
 .../Membership/Propositional/Properties.agda  |   2 +-
 .../Tree/AVL/Sets/Membership/Properties.agda  |   2 +-
 src/Data/Unit/Instances.agda                  |   4 +-
 src/Data/Unit/Polymorphic/Instances.agda      |   4 +-
 .../Relation/Binary/Pointwise/Properties.agda |   2 +-
 src/Data/Vec/Instances.agda                   |   4 +-
 src/Data/Vec/Relation/Binary/Lex/Core.agda    |   4 +-
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  12 +-
 .../Binary/Pointwise/Extensional.agda         |   2 +-
 .../Relation/Binary/Pointwise/Inductive.agda  |   2 +-
 src/Data/Word64/Instances.agda                |   4 +-
 src/Data/Word64/Properties.agda               |  29 +-
 src/Reflection.agda                           |  64 +-
 src/Reflection/AST/Abstraction.agda           |   8 +-
 src/Reflection/AST/AlphaEquality.agda         |  12 +-
 src/Reflection/AST/Argument.agda              |   4 +-
 src/Reflection/AST/Argument/Information.agda  |  24 +-
 src/Reflection/AST/Argument/Modality.agda     |  25 +-
 src/Reflection/AST/Argument/Quantity.agda     |  27 +-
 src/Reflection/AST/Argument/Relevance.agda    |  27 +-
 src/Reflection/AST/Argument/Visibility.agda   |  38 +-
 src/Reflection/AST/Definition.agda            |  92 +--
 src/Reflection/AST/Instances.agda             |  26 +-
 src/Reflection/AST/Literal.agda               | 119 ++--
 src/Reflection/AST/Meta.agda                  |  26 +-
 src/Reflection/AST/Name.agda                  |  26 +-
 src/Reflection/AST/Pattern.agda               |   4 +-
 src/Reflection/AST/Term.agda                  | 548 ++++++++++--------
 src/Relation/Binary/Construct/Flip/Ord.agda   |   4 +-
 .../Binary/Construct/Interior/Symmetric.agda  |   2 +-
 .../Binary/Construct/NaturalOrder/Right.agda  |  12 +-
 .../Binary/Construct/StrictToNonStrict.agda   |   2 +-
 .../Binary/HeterogeneousEquality.agda         |   2 +-
 .../Indexed/Homogeneous/Construct/At.agda     |   2 +-
 .../Indexed/Homogeneous/Structures.agda       |  10 +-
 .../Lattice/Properties/JoinSemilattice.agda   |   6 +-
 .../Lattice/Properties/MeetSemilattice.agda   |   8 +-
 .../Binary/Morphism/OrderMonomorphism.agda    |   2 +-
 .../Binary/Morphism/RelMonomorphism.agda      |   2 +-
 src/Relation/Binary/TypeClasses.agda          |   2 +-
 src/Tactic/Cong.agda                          |   2 +-
 src/Test/Golden.agda                          |   8 +-
 src/Text/Regex/Properties.agda                |   2 +-
 97 files changed, 903 insertions(+), 613 deletions(-)

diff --git a/doc/README/Data/List/Membership.agda b/doc/README/Data/List/Membership.agda
index 38732fe631..d29a2cf9d8 100644
--- a/doc/README/Data/List/Membership.agda
+++ b/doc/README/Data/List/Membership.agda
@@ -34,7 +34,7 @@ import Data.List.Membership.DecPropositional as DecPropMembership
 -- propositional equality over `ℕ` and it is also decidable. Therefore
 -- the module `DecPropMembership` should be opened as follows:
 
-open DecPropMembership ℕ._≟_
+open DecPropMembership ℕ._≡?_
 
 -- As membership is just an instance of `Any` we also need to import
 -- the constructors `here` and `there`. (See issue #553 on Github for
@@ -66,7 +66,7 @@ import Data.List.Membership.Propositional.Properties as PropProperties
 -- following the first `∈` refers to lists of type `List ℕ` whereas
 -- the second `∈` refers to lists of type `List Char`.
 
-open DecPropMembership Char._≟_ renaming (_∈_ to _∈ᶜ_)
+open DecPropMembership Char._≡?_ renaming (_∈_ to _∈ᶜ_)
 open SetoidProperties using (∈-map⁺)
 
 lem₂ : {v : ℕ} {xs : List ℕ} → v ∈ xs → fromℕ v ∈ᶜ map fromℕ xs
diff --git a/doc/README/Data/List/Relation/Binary/Subset.agda b/doc/README/Data/List/Relation/Binary/Subset.agda
index ccef725cca..646c4c6e54 100644
--- a/doc/README/Data/List/Relation/Binary/Subset.agda
+++ b/doc/README/Data/List/Relation/Binary/Subset.agda
@@ -22,10 +22,10 @@ module README.Data.List.Relation.Binary.Subset where
 
 -- Decidable equality over Strings
 open import Data.String.Base using (String)
-open import Data.String.Properties using (_≟_)
+open import Data.String.Properties using (_≡?_)
 
 -- Open the decidable membership module using Decidable ≡ over Strings
-open import Data.List.Membership.DecPropositional _≟_
+open import Data.List.Membership.DecPropositional _≡?_
 
 
 -- Simple cases are inductive proofs
diff --git a/doc/README/Data/Record.agda b/doc/README/Data/Record.agda
index 63fea328ae..53b3d50fb2 100644
--- a/doc/README/Data/Record.agda
+++ b/doc/README/Data/Record.agda
@@ -21,7 +21,7 @@ import Data.Record as Record
 
 -- Let us use strings as labels.
 
-open Record String _≟_
+open Record String _≡?_
 
 -- Partial equivalence relations.
 
diff --git a/doc/README/Foreign/Haskell.agda b/doc/README/Foreign/Haskell.agda
index 38c6c78c4e..e608334197 100644
--- a/doc/README/Foreign/Haskell.agda
+++ b/doc/README/Foreign/Haskell.agda
@@ -92,12 +92,12 @@ main = run $ do
   content ← readFiniteFile "README/Foreign/Haskell.agda"
   let chars = toList content
   let cleanup = catMaybes ∘ List.map (λ c → if testChar c then just c else nothing)
-  let cleaned = dropWhile ('\n' ≟_) $ cleanup chars
+  let cleaned = dropWhile ('\n' ≡?_) $ cleanup chars
   case uncons cleaned of λ where
     nothing         → putStrLn "I cannot believe this file is filed with dashes only!"
     (just (c , cs)) → putStrLn $ unlines
                     $ ("First (non dash) character: " ++ Char.show c)
-                    ∷ ("Rest (dash free) of the line: " ++ fromList (takeWhile (¬? ∘ ('\n' ≟_)) cs))
+                    ∷ ("Rest (dash free) of the line: " ++ fromList (takeWhile (¬? ∘ ('\n' ≡?_)) cs))
                     ∷ []
 
 -- You can compile and run this test by writing:
diff --git a/doc/README/Function/Reasoning.agda b/doc/README/Function/Reasoning.agda
index 810f048aa3..7c2d3ed7d0 100644
--- a/doc/README/Function/Reasoning.agda
+++ b/doc/README/Function/Reasoning.agda
@@ -57,7 +57,7 @@ subpalindromes str = let Chars = List Char in
   |> filter (λ cs → 2 ≤? length cs)        ∶ List Chars
   -- only keep the ones that are palindromes
   |> map < fromList , fromList ∘ reverse > ∶ List (String × String)
-  |> filter (uncurry String._≟_)           ∶ List (String × String)
+  |> filter (uncurry String._≡?_)           ∶ List (String × String)
   |> map proj₁                             ∶ List String
 
 -- Test cases
diff --git a/doc/README/Nary.agda b/doc/README/Nary.agda
index b8b337832a..e06c3a8455 100644
--- a/doc/README/Nary.agda
+++ b/doc/README/Nary.agda
@@ -338,14 +338,14 @@ exist₁ = 19 , 793 , 3059 , 10 , refl
 -- ∀[_] P = ∀ {a₁} → ⋯ → ∀ {aₙ} → P a₁ ⋯ aₙ
 
 all₁ : ∀[ (λ (a₁ a₂ : ℕ) → Dec (a₁ ≡ a₂)) ]
-all₁ {a₁} {a₂} = a₁ ≟ a₂
+all₁ {a₁} {a₂} = a₁ ≡? a₂
 
 ------------------------------------------------------------------------
 -- Π : (A₁ → ⋯ → Aₙ → Set r) → Set _
 -- Π P = ∀ a₁ → ⋯ → ∀ aₙ → P a₁ ⋯ aₙ
 
 all₂ : Π[ (λ (a₁ a₂ : ℕ) → Dec (a₁ ≡ a₂)) ]
-all₂ = _≟_
+all₂ = _≡?_
 
 ------------------------------------------------------------------------
 -- _⇒_ : (A₁ → ⋯ → Aₙ → Set r) → (A₁ → ⋯ → Aₙ → Set s) → (A₁ → ⋯ → Aₙ → Set _)
diff --git a/src/Algebra/Solver/CommutativeMonoid/Normal.agda b/src/Algebra/Solver/CommutativeMonoid/Normal.agda
index 7713e497e1..ae564c1f0a 100644
--- a/src/Algebra/Solver/CommutativeMonoid/Normal.agda
+++ b/src/Algebra/Solver/CommutativeMonoid/Normal.agda
@@ -57,10 +57,10 @@ Normal n = Vec ℕ n
 
 -- We can decide if two normal forms are /syntactically/ equal.
 
-infix 5 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality (Normal n)
-nf₁ ≟ nf₂ = Dec.map Pointwise-≡↔≡ (decidable ℕ._≟_ nf₁ nf₂)
+_≡?_ : DecidableEquality (Normal n)
+nf₁ ≡? nf₂ = Dec.map Pointwise-≡↔≡ (decidable ℕ._≡?_ nf₁ nf₂)
   where open Pointwise using (Pointwise-≡↔≡; decidable)
 
 ------------------------------------------------------------------------
@@ -162,3 +162,12 @@ sg-correct = singleton-correct
 "Warning: sg-correct was deprecated in v2.2.
 Please use singleton-correct instead."
 #-}
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid/Normal.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid/Normal.agda
index 0ebdf4351d..4c67728c65 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid/Normal.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid/Normal.agda
@@ -56,10 +56,10 @@ Normal n = Vec Bool n
 
 -- We can decide if two normal forms are /syntactically/ equal.
 
-infix 5 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality (Normal n)
-nf₁ ≟ nf₂ = Dec.map Pointwise-≡↔≡ (decidable Bool._≟_ nf₁ nf₂)
+_≡?_ : DecidableEquality (Normal n)
+nf₁ ≡? nf₂ = Dec.map Pointwise-≡↔≡ (decidable Bool._≡?_ nf₁ nf₂)
   where open Pointwise using (Pointwise-≡↔≡; decidable)
 
 ------------------------------------------------------------------------
@@ -167,3 +167,12 @@ sg-correct = singleton-correct
 "Warning: sg-correct was deprecated in v2.2.
 Please use singleton-correct instead."
 #-}
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Algebra/Solver/Monoid/Normal.agda b/src/Algebra/Solver/Monoid/Normal.agda
index 62a02bfca5..2195eeafac 100644
--- a/src/Algebra/Solver/Monoid/Normal.agda
+++ b/src/Algebra/Solver/Monoid/Normal.agda
@@ -50,11 +50,11 @@ Normal n = List (Fin n)
 
 -- We can decide if two normal forms are /syntactically/ equal.
 
-infix 5 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality (Normal n)
-nf₁ ≟ nf₂ = Dec.map′ ≋⇒≡ ≡⇒≋ (nf₁ ≋? nf₂)
-  where open ≋ Fin._≟_ using (_≋?_; ≋⇒≡; ≡⇒≋)
+_≡?_ : DecidableEquality (Normal n)
+nf₁ ≡? nf₂ = Dec.map′ ≋⇒≡ ≡⇒≋ (nf₁ ≋? nf₂)
+  where open ≋ Fin._≡?_ using (_≋?_; ≋⇒≡; ≡⇒≋)
 
 -- A normaliser.
 
@@ -87,3 +87,20 @@ correct (e₁ ⊕ e₂) ρ = begin
   ⟦ normalise e₁ ++ normalise e₂ ⟧⇓ ρ        ≈⟨ comp-correct (normalise e₁) (normalise e₂) ρ ⟩
   ⟦ normalise e₁ ⟧⇓ ρ ∙ ⟦ normalise e₂ ⟧⇓ ρ  ≈⟨ ∙-cong (correct e₁ ρ) (correct e₂ ρ) ⟩
   ⟦ e₁ ⟧ ρ ∙ ⟦ e₂ ⟧ ρ                        ∎
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+
+_≟_ = _≡?_ 
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
index 324f05eb04..89d4ab537b 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
@@ -18,7 +18,7 @@ module Algebra.Solver.Ring.NaturalCoefficients.Default
 
 import Algebra.Properties.Semiring.Mult as SemiringMultiplication using (_×_)
 open import Data.Maybe.Base using (Maybe; map)
-open import Data.Nat using (_≟_)
+open import Data.Nat using (_≡?_)
 open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 import Relation.Binary.PropositionalEquality.Core as ≡ using (refl; cong)
 
@@ -27,6 +27,6 @@ open SemiringMultiplication semiring
 
 private
   dec : ∀ m n → Maybe (m × 1# ≈ n × 1#)
-  dec m n = map (λ { ≡.refl → refl }) (dec⇒weaklyDec _≟_ m n)
+  dec m n = map (λ { ≡.refl → refl }) (dec⇒weaklyDec _≡?_ m n)
 
 open import Algebra.Solver.Ring.NaturalCoefficients R dec public
diff --git a/src/Data/Bool.agda b/src/Data/Bool.agda
index 55ed146e2c..0ead2d50b2 100644
--- a/src/Data/Bool.agda
+++ b/src/Data/Bool.agda
@@ -17,4 +17,4 @@ open import Data.Bool.Base public
 -- Publicly re-export queries
 
 open import Data.Bool.Properties public
-  using (T?; _≟_; _≤?_; _<?_)
+  using (T?; _≟_; _≡?_; _≤?_; _<?_)
diff --git a/src/Data/Bool/Instances.agda b/src/Data/Bool/Instances.agda
index 3158e2b1b2..e61f94b813 100644
--- a/src/Data/Bool/Instances.agda
+++ b/src/Data/Bool/Instances.agda
@@ -8,10 +8,10 @@
 
 module Data.Bool.Instances where
 
-open import Data.Bool.Properties using (_≟_; ≤-isDecTotalOrder)
+open import Data.Bool.Properties using (_≡?_; ≤-isDecTotalOrder)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  Bool-≡-isDecEquivalence = isDecEquivalence _≟_
+  Bool-≡-isDecEquivalence = isDecEquivalence _≡?_
   Bool-≤-isDecTotalOrder = ≤-isDecTotalOrder
diff --git a/src/Data/Bool/Solver.agda b/src/Data/Bool/Solver.agda
index 901429a095..abc4c4e05d 100644
--- a/src/Data/Bool/Solver.agda
+++ b/src/Data/Bool/Solver.agda
@@ -13,18 +13,18 @@ module Data.Bool.Solver where
 import Algebra.Solver.Ring.Simple as Solver
 import Algebra.Solver.Ring.AlmostCommutativeRing as ACR
 open import Data.Bool.Properties
-  using (_≟_;  ∨-∧-commutativeSemiring; xor-∧-commutativeRing)
+  using (_≡?_;  ∨-∧-commutativeSemiring; xor-∧-commutativeRing)
 
 ------------------------------------------------------------------------
 -- A module for automatically solving propositional equivalences
 -- containing _∨_ and _∧_
 
 module ∨-∧-Solver =
-  Solver (ACR.fromCommutativeSemiring ∨-∧-commutativeSemiring) _≟_
+  Solver (ACR.fromCommutativeSemiring ∨-∧-commutativeSemiring) _≡?_
 
 ------------------------------------------------------------------------
 -- A module for automatically solving propositional equivalences
 -- containing _xor_ and _∧_
 
 module xor-∧-Solver =
-  Solver (ACR.fromCommutativeRing xor-∧-commutativeRing) _≟_
+  Solver (ACR.fromCommutativeRing xor-∧-commutativeRing) _≡?_
diff --git a/src/Data/Char.agda b/src/Data/Char.agda
index fdc4764667..aeb7f6fcee 100644
--- a/src/Data/Char.agda
+++ b/src/Data/Char.agda
@@ -13,4 +13,4 @@ module Data.Char where
 
 open import Data.Char.Base public
 open import Data.Char.Properties
-  using (_≈?_; _≟_; _<?_; _≤?_; _==_) public
+  using (_≈?_; _≟_; _≡?_; _<?_; _≤?_; _==_) public
diff --git a/src/Data/Digit/Properties.agda b/src/Data/Digit/Properties.agda
index 59bc6e0f51..02abe95163 100644
--- a/src/Data/Digit/Properties.agda
+++ b/src/Data/Digit/Properties.agda
@@ -22,7 +22,7 @@ open import Function.Base using (_∘_)
 module Data.Digit.Properties where
 
 digitCharsUnique : Unique digitChars
-digitCharsUnique = from-yes (allPairs? (λ x y → ¬? (x Char.≟ y)) digitChars)
+digitCharsUnique = from-yes (allPairs? (λ x y → ¬? (x Char.≡? y)) digitChars)
 
 module _ (base : ℕ) where
   module _ {base≥2 base≥2′ : True (2 ≤? base)} where
diff --git a/src/Data/Fin.agda b/src/Data/Fin.agda
index d0c0e12c5c..e3ee65bba3 100644
--- a/src/Data/Fin.agda
+++ b/src/Data/Fin.agda
@@ -20,7 +20,7 @@ open import Data.Fin.Base public
 -- Publicly re-export queries
 
 open import Data.Fin.Properties public
-  using (_≟_; _≤?_; _<?_)
+  using (_≟_; _≡?_; _≤?_; _<?_)
 
 -- # m = "m".
 
diff --git a/src/Data/Fin/Subset.agda b/src/Data/Fin/Subset.agda
index 49dd7c84b4..33207061e5 100644
--- a/src/Data/Fin/Subset.agda
+++ b/src/Data/Fin/Subset.agda
@@ -9,7 +9,7 @@
 module Data.Fin.Subset where
 
 open import Algebra.Core using (Op₁; Op₂)
-open import Data.Bool using (not; _∧_; _∨_; _≟_)
+open import Data.Bool using (not; _∧_; _∨_; _≡?_)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base using (List; foldr; foldl)
 open import Data.Nat.Base using (ℕ)
@@ -131,7 +131,7 @@ p - x = p ─ ⁅ x ⁆
 
 -- Size
 ∣_∣ : Subset n → ℕ
-∣ p ∣ = count (_≟ inside) p
+∣ p ∣ = count (_≡? inside) p
 
 ------------------------------------------------------------------------
 -- Properties
diff --git a/src/Data/Fin/Subset/Properties.agda b/src/Data/Fin/Subset/Properties.agda
index 074a4a05b8..321543609f 100644
--- a/src/Data/Fin/Subset/Properties.agda
+++ b/src/Data/Fin/Subset/Properties.agda
@@ -140,7 +140,7 @@ nonempty? p = any? (_∈? p)
 -- ∣_∣
 
 ∣p∣≤n : ∀ (p : Subset n) → ∣ p ∣ ≤ n
-∣p∣≤n = count≤n (_≟ inside)
+∣p∣≤n = count≤n (_≡? inside)
 
 ∣p∣≤∣x∷p∣ : ∀ x (p : Subset n)  → ∣ p ∣ ≤ ∣ x ∷ p ∣
 ∣p∣≤∣x∷p∣ outside p = ℕ.≤-refl
@@ -317,7 +317,7 @@ module _ (n : ℕ) where
   ⊂-isDecStrictPartialOrder : IsDecStrictPartialOrder {A = Subset n} _≡_ _⊂_
   ⊂-isDecStrictPartialOrder = record
     { isStrictPartialOrder = ⊂-isStrictPartialOrder
-    ; _≟_ = ≡-dec _≟_
+    ; _≈?_ = ≡-dec _≡?_
     ; _<?_ = _⊂?_
     }
 
diff --git a/src/Data/Float.agda b/src/Data/Float.agda
index c38f9fddfa..8574c8538e 100644
--- a/src/Data/Float.agda
+++ b/src/Data/Float.agda
@@ -12,4 +12,4 @@ module Data.Float where
 -- Re-export base definitions and decidability of equality
 
 open import Data.Float.Base public
-open import Data.Float.Properties using (_≈?_; _≟_) public
+open import Data.Float.Properties using (_≈?_; _≟_; _≡?_) public
diff --git a/src/Data/Float/Instances.agda b/src/Data/Float/Instances.agda
index 46fc202e08..4bfe137534 100644
--- a/src/Data/Float/Instances.agda
+++ b/src/Data/Float/Instances.agda
@@ -8,9 +8,9 @@
 
 module Data.Float.Instances where
 
-open import Data.Float.Properties using (_≟_)
+open import Data.Float.Properties using (_≡?_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  Float-≡-isDecEquivalence = isDecEquivalence _≟_
+  Float-≡-isDecEquivalence = isDecEquivalence _≡?_
diff --git a/src/Data/Float/Properties.agda b/src/Data/Float/Properties.agda
index 7c6866220f..02b04ad3d9 100644
--- a/src/Data/Float/Properties.agda
+++ b/src/Data/Float/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base as Bool using (Bool)
 open import Data.Float.Base using (Float; _≈_; toWord64)
 import Data.Maybe.Base as Maybe using (Maybe; just; nothing; map)
 import Data.Maybe.Properties as Maybe using (map-injective; ≡-dec)
-import Data.Nat.Properties as ℕ using (_≟_)
+import Data.Nat.Properties as ℕ using (_≡?_)
 import Data.Word64.Base as Word64 using (Word64; toℕ)
 import Data.Word64.Properties as Word64 using (≈⇒≡)
 open import Function.Base using (_∘_)
@@ -59,7 +59,7 @@ open import Agda.Builtin.Float.Properties
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-_≈?_ = On.decidable (Maybe.map Word64.toℕ ∘ toWord64) _≡_ (Maybe.≡-dec ℕ._≟_)
+_≈?_ = On.decidable (Maybe.map Word64.toℕ ∘ toWord64) _≡_ (Maybe.≡-dec ℕ._≡?_)
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -76,7 +76,7 @@ _≈?_ = On.decidable (Maybe.map Word64.toℕ ∘ toWord64) _≡_ (Maybe.≡-dec
 ≈-isDecEquivalence : IsDecEquivalence _≈_
 ≈-isDecEquivalence = record
   { isEquivalence = ≈-isEquivalence
-  ; _≟_           = _≈?_
+  ; _≈?_          = _≈?_
   }
 
 ≈-decSetoid : DecSetoid _ _
@@ -86,22 +86,36 @@ _≈?_ = On.decidable (Maybe.map Word64.toℕ ∘ toWord64) _≡_ (Maybe.≡-dec
 ------------------------------------------------------------------------
 -- Properties of _≡_
 
-infix 4 _≟_
-_≟_ : DecidableEquality Float
-x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
+infix 4 _≡?_
+_≡?_ : DecidableEquality Float
+x ≡? y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
 
 ≡-setoid : Setoid _ _
 ≡-setoid = setoid Float
 
 ≡-decSetoid : DecSetoid _ _
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 
 ------------------------------------------------------------------------
--- DEPRECATIONS
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.1
 
 toWord-injective = toWord64-injective
 {-# WARNING_ON_USAGE toWord-injective
 "Warning: toWord-injective was deprecated in v2.1.
 Please use toWord64-injective instead."
 #-}
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index 418d42446c..5d0c39f1ae 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -17,7 +17,7 @@ open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _<′_)
 open import Data.Nat.Induction using (<′-rec; <′-Rec)
 import Data.Nat.Properties as ℕ using (≤⇒≤′)
 open import Data.Fin as Fin
-  using (Fin; Fin′; zero; suc; #_; toℕ; _≟_; opposite)
+  using (Fin; Fin′; zero; suc; #_; toℕ; _≡?_; opposite)
   renaming (_ℕ-ℕ_ to _-_)
 open import Data.Fin.Properties as Fin using (nℕ-ℕi≤n)
 open import Data.Product.Base as Prod using (∃; _×_; _,_)
@@ -239,7 +239,7 @@ preds (c & g) (suc i) =
             (List.map (Prod.map suc id) $ preds g i)
   where
   p : ∀ {e} {E : Set e} {n} (i : Fin n) → E × Fin n → Maybe (Fin′ (suc i) × E)
-  p i (e , j) = Maybe.map (λ{ refl → zero , e }) (dec⇒weaklyDec _≟_ i j)
+  p i (e , j) = Maybe.map (λ{ refl → zero , e }) (dec⇒weaklyDec _≡?_ i j)
 
 private
 
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index 85e83d468d..47952ca647 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -48,7 +48,7 @@ open _∣_ using (quotient) public
 
 ∣ᵤ⇒∣ : ∀ {k i} → k Unsigned.∣ i → k ∣ i
 ∣ᵤ⇒∣ {k} {i} (Unsigned.divides 0           eq) = divides +0 (∣i∣≡0⇒i≡0 eq)
-∣ᵤ⇒∣ {k} {i} (Unsigned.divides q@(ℕ.suc _) eq) with k ≟ +0
+∣ᵤ⇒∣ {k} {i} (Unsigned.divides q@(ℕ.suc _) eq) with k ≡? +0
 ... | yes refl = divides +0 (∣i∣≡0⇒i≡0 (trans eq (ℕ.*-zeroʳ q)))
 ... | no  neq  = divides s[i*k]◃q (◃-cong sign-eq abs-eq)
   where
diff --git a/src/Data/Integer/Instances.agda b/src/Data/Integer/Instances.agda
index 4bad2a5ec1..cf592e2782 100644
--- a/src/Data/Integer/Instances.agda
+++ b/src/Data/Integer/Instances.agda
@@ -8,10 +8,10 @@
 
 module Data.Integer.Instances where
 
-open import Data.Integer.Properties using (_≟_; ≤-isDecTotalOrder)
+open import Data.Integer.Properties using (_≡?_; ≤-isDecTotalOrder)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  ℤ-≡-isDecEquivalence = isDecEquivalence _≟_
+  ℤ-≡-isDecEquivalence = isDecEquivalence _≡?_
   ℤ-≤-isDecTotalOrder = ≤-isDecTotalOrder
diff --git a/src/Data/Integer/Solver.agda b/src/Data/Integer/Solver.agda
index 46e9398677..daa527bb68 100644
--- a/src/Data/Integer/Solver.agda
+++ b/src/Data/Integer/Solver.agda
@@ -12,11 +12,11 @@ module Data.Integer.Solver where
 
 import Algebra.Solver.Ring.Simple as Solver
 import Algebra.Solver.Ring.AlmostCommutativeRing as ACR
-open import Data.Integer.Properties using (_≟_; +-*-commutativeRing)
+open import Data.Integer.Properties using (_≡?_; +-*-commutativeRing)
 
 ------------------------------------------------------------------------
 -- A module for automatically solving propositional equivalences
 -- containing _+_ and _*_
 
 module +-*-Solver =
-  Solver (ACR.fromCommutativeRing +-*-commutativeRing) _≟_
+  Solver (ACR.fromCommutativeRing +-*-commutativeRing) _≡?_
diff --git a/src/Data/List/Instances.agda b/src/Data/List/Instances.agda
index d239be7eee..c6b3c58ba9 100644
--- a/src/Data/List/Instances.agda
+++ b/src/Data/List/Instances.agda
@@ -28,7 +28,7 @@ open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 open import Relation.Binary.TypeClasses
-  using (IsDecTotalOrder; IsDecEquivalence; _≟_)
+  using (IsDecTotalOrder; IsDecEquivalence; _≈?_)
 
 private
   variable
@@ -52,7 +52,7 @@ instance
   listTMonadT = λ {f} {g} {M} {{inst}} → Trans.monadT {f} {g} {M} inst
 
   List-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → IsDecEquivalence {A = List A} _≡_
-  List-≡-isDecEquivalence = isDecEquivalence (≡-dec _≟_)
+  List-≡-isDecEquivalence = isDecEquivalence (≡-dec _≈?_)
 
   List-Lex-≤-isDecTotalOrder : {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂}
                              → {{IsDecTotalOrder _≈_ _≼_}}
diff --git a/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda
index 08976e70cd..373e49869f 100644
--- a/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda
@@ -42,6 +42,6 @@ isPartialOrder po = record
 isDecPartialOrder : IsDecPartialOrder R S → IsDecPartialOrder (Pointwise R) (Infix S)
 isDecPartialOrder dpo = record
   { isPartialOrder = isPartialOrder DPO.isPartialOrder
-  ; _≟_            = Pointwise.decidable DPO._≟_
+  ; _≈?_           = Pointwise.decidable DPO._≈?_
   ; _≤?_           = infix? DPO._≤?_
   } where module DPO = IsDecPartialOrder dpo
diff --git a/src/Data/List/Relation/Binary/Lex/NonStrict.agda b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
index efd1520f82..893620d3ff 100644
--- a/src/Data/List/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
@@ -111,7 +111,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
 <-strictTotalOrder : ∀ {a ℓ₁ ℓ₂} → DecTotalOrder a ℓ₁ ℓ₂ →
                      StrictTotalOrder _ _ _
 <-strictTotalOrder dtot = record
-  { isStrictTotalOrder = <-isStrictTotalOrder _≟_ isTotalOrder
+  { isStrictTotalOrder = <-isStrictTotalOrder _≈?_ isTotalOrder
   } where open DecTotalOrder dtot
 
 ------------------------------------------------------------------------
@@ -150,8 +150,8 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
     ≤-resp₂ eq resp = Core.respects₂ eq (Conv.<-resp-≈ _ _ eq resp)
 
     ≤-decidable : Decidable _≈_ → Decidable _≼_ → Decidable _≤_
-    ≤-decidable _≟_ _≼?_ =
-      Core.decidable (yes tt) _≟_ (Conv.<-decidable _ _ _≟_ _≼?_)
+    ≤-decidable _≈?_ _≼?_ =
+      Core.decidable (yes tt) _≈?_ (Conv.<-decidable _ _ _≈?_ _≼?_)
 
     ≤-total : Symmetric _≈_ → Decidable _≈_ → Antisymmetric _≈_ _≼_ →
               Total _≼_ → Total _≤_
diff --git a/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda
index 6cb7ebd715..3b8ccd7955 100644
--- a/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda
@@ -43,6 +43,6 @@ isPartialOrder po = record
 isDecPartialOrder : IsDecPartialOrder R S → IsDecPartialOrder (Pointwise R) (Prefix S)
 isDecPartialOrder dpo = record
   { isPartialOrder = isPartialOrder DPO.isPartialOrder
-  ; _≟_            = Pointwise.decidable DPO._≟_
+  ; _≈?_           = Pointwise.decidable DPO._≈?_
   ; _≤?_           = prefix? DPO._≤?_
   } where module DPO = IsDecPartialOrder dpo
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
index a1df8dfd6e..882e93e848 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
@@ -11,10 +11,10 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A)
+  {a} {A : Set a} (_≈?_ : DecidableEquality A)
   where
 
-open import Data.List.Relation.Binary.Equality.DecPropositional _≟_
+open import Data.List.Relation.Binary.Equality.DecPropositional _≈?_
   using (_≡?_)
 import Data.List.Relation.Binary.Sublist.DecSetoid as DecSetoidSublist
 import Data.List.Relation.Binary.Sublist.Propositional as PropositionalSublist
@@ -27,7 +27,7 @@ open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 -- Re-export core definitions and operations
 
 open PropositionalSublist {A = A} public
-open DecSetoidSublist (decSetoid _≟_) using (_⊆?_) public
+open DecSetoidSublist (decSetoid _≈?_) using (_⊆?_) public
 
 ------------------------------------------------------------------------
 -- Additional relational properties
@@ -35,7 +35,7 @@ open DecSetoidSublist (decSetoid _≟_) using (_⊆?_) public
 ⊆-isDecPartialOrder : IsDecPartialOrder _≡_ _⊆_
 ⊆-isDecPartialOrder = record
   { isPartialOrder = ⊆-isPartialOrder
-  ; _≟_            = _≡?_
+  ; _≈?_           = _≡?_
   ; _≤?_           = _⊆?_
   }
 
diff --git a/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda b/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda
index a184386034..8af43b3aa0 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda
@@ -36,12 +36,12 @@ open SetoidSublist setoid public
 -- Additional relational properties
 
 _⊆?_ : Decidable _⊆_
-_⊆?_ = HeterogeneousProperties.sublist? _≟_
+_⊆?_ = HeterogeneousProperties.sublist? _≈?_
 
 ⊆-isDecPartialOrder : IsDecPartialOrder _≋_ _⊆_
 ⊆-isDecPartialOrder = record
   { isPartialOrder = ⊆-isPartialOrder
-  ; _≟_            = _≋?_
+  ; _≈?_           = _≋?_
   ; _≤?_           = _⊆?_
   }
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index af3675fb9c..929f17c0b1 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -448,7 +448,7 @@ module _ {E : Rel A e} {R : Rel A r} where
                       IsDecPartialOrder (Pointwise E) (Sublist R)
   isDecPartialOrder ER-isDecPartialOrder = record
     { isPartialOrder = isPartialOrder ER.isPartialOrder
-    ; _≟_            = Pw.decidable   ER._≟_
+    ; _≈?_           = Pw.decidable   ER._≈?_
     ; _≤?_           = sublist?       ER._≤?_
     } where module ER = IsDecPartialOrder ER-isDecPartialOrder
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index 11db87fa52..3b6d0c6b2f 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -125,7 +125,7 @@ private
 
 -- Solver for items
 solveI : ∀ {n} (a b : Item n) → Maybe (a ⊆I b)
-solveI (var k) (var l) = Maybe.map var $ dec⇒weaklyDec Fin._≟_ k  l
+solveI (var k) (var l) = Maybe.map var $ dec⇒weaklyDec Fin._≡?_ k  l
 solveI (val a) (val b) = Maybe.map val $ dec⇒weaklyDec R? a b
 solveI _ _ = nothing
 
diff --git a/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda
index 0a5298f431..4c7037b84d 100644
--- a/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda
@@ -42,6 +42,6 @@ isPartialOrder po = record
 isDecPartialOrder : IsDecPartialOrder R S → IsDecPartialOrder (Pointwise R) (Suffix S)
 isDecPartialOrder dpo = record
   { isPartialOrder = isPartialOrder DPO.isPartialOrder
-  ; _≟_            = Pointwise.decidable DPO._≟_
+  ; _≈?_            = Pointwise.decidable DPO._≈?_
   ; _≤?_           = suffix? DPO._≤?_
   } where module DPO = IsDecPartialOrder dpo
diff --git a/src/Data/Maybe/Relation/Binary/Pointwise.agda b/src/Data/Maybe/Relation/Binary/Pointwise.agda
index cb42bf5114..18a849ece6 100644
--- a/src/Data/Maybe/Relation/Binary/Pointwise.agda
+++ b/src/Data/Maybe/Relation/Binary/Pointwise.agda
@@ -92,7 +92,7 @@ module _ {a r} {A : Set a} {R : Rel A r} where
   isDecEquivalence : IsDecEquivalence R → IsDecEquivalence (Pointwise R)
   isDecEquivalence R-isDecEquivalence = record
     { isEquivalence = isEquivalence R.isEquivalence
-    ; _≟_           = dec R._≟_
+    ; _≈?_          = dec R._≈?_
     } where module R = IsDecEquivalence R-isDecEquivalence
 
   pointwise⊆any : ∀ {x} → Pointwise R (just x) ⊆ Any (R x)
diff --git a/src/Data/Nat.agda b/src/Data/Nat.agda
index 8811565c11..5bcad86a0d 100644
--- a/src/Data/Nat.agda
+++ b/src/Data/Nat.agda
@@ -24,7 +24,7 @@ open import Data.Nat.Properties public
   -- key values
   ( nonZero?
   -- equalities
-  ; _≟_ ; eq?
+  ; _≟_ ;_≡?_ ; eq?
   -- standard orders & their relationship
   ; _≤?_ ; _≥?_ ; _<?_ ; _>?_
   ; ≤-<-connex ; ≥->-connex ; <-≤-connex ; >-≥-connex
diff --git a/src/Data/Nat/Binary/Instances.agda b/src/Data/Nat/Binary/Instances.agda
index 9ef0ef6373..f8c56ebc38 100644
--- a/src/Data/Nat/Binary/Instances.agda
+++ b/src/Data/Nat/Binary/Instances.agda
@@ -8,9 +8,9 @@
 
 module Data.Nat.Binary.Instances where
 
-open import Data.Nat.Binary.Properties
+open import Data.Nat.Binary.Properties using (_≡?_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  ℕᵇ-≡-isDecEquivalence = isDecEquivalence _≟_
+  ℕᵇ-≡-isDecEquivalence = isDecEquivalence _≡?_
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 6c2298d88a..79ca4687d2 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -61,7 +61,7 @@ private
   variable
     x : ℕᵇ
 
-infix 4  _<?_ _≟_ _≤?_
+infix 4  _<?_ _≡?_ _≤?_
 
 ------------------------------------------------------------------------
 -- Properties of _≡_
@@ -79,25 +79,25 @@ infix 4  _<?_ _≟_ _≤?_
 1+[2_]-injective : Injective _≡_ _≡_ 1+[2_]
 1+[2_]-injective refl = refl
 
-_≟_ : DecidableEquality ℕᵇ
-zero     ≟ zero     =  yes refl
-zero     ≟ 2[1+ _ ] =  no λ()
-zero     ≟ 1+[2 _ ] =  no λ()
-2[1+ _ ] ≟ zero     =  no λ()
-2[1+ x ] ≟ 2[1+ y ] =  Dec.map′ (cong 2[1+_]) 2[1+_]-injective (x ≟ y)
-2[1+ _ ] ≟ 1+[2 _ ] =  no λ()
-1+[2 _ ] ≟ zero     =  no λ()
-1+[2 _ ] ≟ 2[1+ _ ] =  no λ()
-1+[2 x ] ≟ 1+[2 y ] =  Dec.map′ (cong 1+[2_]) 1+[2_]-injective (x ≟ y)
+_≡?_ : DecidableEquality ℕᵇ
+zero     ≡? zero     =  yes refl
+zero     ≡? 2[1+ _ ] =  no λ()
+zero     ≡? 1+[2 _ ] =  no λ()
+2[1+ _ ] ≡? zero     =  no λ()
+2[1+ x ] ≡? 2[1+ y ] =  Dec.map′ (cong 2[1+_]) 2[1+_]-injective (x ≡? y)
+2[1+ _ ] ≡? 1+[2 _ ] =  no λ()
+1+[2 _ ] ≡? zero     =  no λ()
+1+[2 _ ] ≡? 2[1+ _ ] =  no λ()
+1+[2 x ] ≡? 1+[2 y ] =  Dec.map′ (cong 1+[2_]) 1+[2_]-injective (x ≡? y)
 
 ≡-isDecEquivalence : IsDecEquivalence {A = ℕᵇ} _≡_
-≡-isDecEquivalence = isDecEquivalence _≟_
+≡-isDecEquivalence = isDecEquivalence _≡?_
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid ℕᵇ
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Properties of toℕ & fromℕ
@@ -327,7 +327,7 @@ toℕ-cancel-< {2[1+ x ]} {2[1+ y ]} x<y =
 toℕ-cancel-< {2[1+ x ]} {1+[2 y ]} x<y
   rewrite ℕ.*-distribˡ-+ 2 1 (toℕ x) =
   even<odd (toℕ-cancel-< (ℕ.*-cancelˡ-< 2 _ _ (ℕ.≤-trans (s≤s (ℕ.n≤1+n _)) (s<s⁻¹ x<y))))
-toℕ-cancel-< {1+[2 x ]} {2[1+ y ]} x<y with toℕ x ℕ.≟ toℕ y
+toℕ-cancel-< {1+[2 x ]} {2[1+ y ]} x<y with toℕ x ℕ.≡? toℕ y
 ... | yes x≡y = odd<even (inj₂ (toℕ-injective x≡y))
 ... | no  x≢y
   rewrite ℕ.+-suc (toℕ y) (toℕ y ℕ.+ 0) =
@@ -584,7 +584,7 @@ x ≤? y with <-cmp x y
 ≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≟_
+  ; _≈?_         = _≡?_
   ; _≤?_         = _≤?_
   }
 
@@ -1540,3 +1540,12 @@ Please use +-*-commutativeSemiring instead."
 {- issue1858/issue1755: raw bundles have moved to `Data.X.Base` -}
 open Data.Nat.Binary.Base public
   using (+-rawMagma; +-0-rawMonoid; *-rawMagma; *-1-rawMonoid)
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index 9f61a5e452..6bb9e4cd1d 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -85,7 +85,7 @@ nP′n≡n! n = begin-equality
 
 k!∣nP′k : ∀ {n k} → k ≤ n → k ! ∣ n P′ k
 k!∣nP′k {n}         {zero}      k≤n = ∣-refl
-k!∣nP′k n@{suc n-1} k@{suc k-1} k≤n@(s≤s k-1≤n-1) with k-1 ≟ n-1
+k!∣nP′k n@{suc n-1} k@{suc k-1} k≤n@(s≤s k-1≤n-1) with k-1 ≡? n-1
 ... | yes refl = ∣-reflexive (sym (nP′n≡n! n))
 ... | no  k≢n  = begin
   k !                           ≡⟨⟩
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 52946ffeff..5728f7e7cc 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -67,7 +67,7 @@ sym : Symmetric Coprime
 sym c = c ∘ swap
 
 coprime? : Decidable Coprime
-coprime? m n = map′ gcd≡1⇒coprime coprime⇒gcd≡1 (gcd m n ≟ 1)
+coprime? m n = map′ gcd≡1⇒coprime coprime⇒gcd≡1 (gcd m n ≡? 1)
 
 ------------------------------------------------------------------------
 -- Other basic properties
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index 66df3cc615..8943c2d0b6 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -283,7 +283,7 @@ mkGCD m n = gcd m n , gcd-GCD m n
 gcd? : (m n d : ℕ) → Dec (GCD m n d)
 gcd? m n d =
   Dec.map′ (λ { ≡.refl → gcd-GCD m n }) (GCD.unique (gcd-GCD m n))
-           (gcd m n ≟ d)
+           (gcd m n ≡? d)
 
 GCD-* : ∀ {m n d c} .{{_ : NonZero c}} → GCD (m * c) (n * c) (d * c) → GCD m n d
 GCD-* {c = suc _} (GCD.is (dc∣nc , dc∣mc) dc-greatest) =
diff --git a/src/Data/Nat/Instances.agda b/src/Data/Nat/Instances.agda
index 9c886a19b0..50835edbe4 100644
--- a/src/Data/Nat/Instances.agda
+++ b/src/Data/Nat/Instances.agda
@@ -8,10 +8,10 @@
 
 module Data.Nat.Instances where
 
-open import Data.Nat.Properties using (≤-isDecTotalOrder; _≟_)
+open import Data.Nat.Properties using (≤-isDecTotalOrder; _≡?_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  ℕ-≡-isDecEquivalence = isDecEquivalence _≟_
+  ℕ-≡-isDecEquivalence = isDecEquivalence _≡?_
   ℕ-≤-isDecTotalOrder = ≤-isDecTotalOrder
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 88c36f0d5c..73080b48d3 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -286,7 +286,7 @@ euclidsLemma m n {p} pp@(prime pr) p∣m*n = result
 
   -- if the GCD of m and p is greater than one, then it must be p and
   -- hence p ∣ m.
-  ... | Bézout.result d@(2+ _) g _ with d ≟ p
+  ... | Bézout.result d@(2+ _) g _ with d ≡? p
   ...   | yes d≡p@refl = inj₁ (GCD.gcd∣m g)
   ...   | no  d≢p = contradiction (composite-≢ d d≢p (GCD.gcd∣n g)) pr
 
@@ -374,7 +374,7 @@ irreducible? n@(suc _) =
   -- Decidability
   irreducibleUpTo? : Decidable IrreducibleUpTo
   irreducibleUpTo? n = allUpTo?
-    (λ m → (m ∣? n) →? (m ≟ 1 ⊎? m ≟ n)) n
+    (λ m → (m ∣? n) →? (m ≡? 1 ⊎? m ≡? n)) n
 
 -- Relationship between primality and irreducibility.
 prime⇒irreducible : Prime p → Irreducible p
diff --git a/src/Data/Nat/Solver.agda b/src/Data/Nat/Solver.agda
index efd14902b4..31f89dfb40 100644
--- a/src/Data/Nat/Solver.agda
+++ b/src/Data/Nat/Solver.agda
@@ -19,4 +19,4 @@ open import Data.Nat.Properties
 -- containing _+_ and _*_
 
 module +-*-Solver =
-  Solver (ACR.fromCommutativeSemiring +-*-commutativeSemiring) _≟_
+  Solver (ACR.fromCommutativeSemiring +-*-commutativeSemiring) _≡?_
diff --git a/src/Data/Product/Instances.agda b/src/Data/Product/Instances.agda
index 63e931d182..b5226d8368 100644
--- a/src/Data/Product/Instances.agda
+++ b/src/Data/Product/Instances.agda
@@ -24,4 +24,4 @@ private
 
 instance
   Σ-≡-isDecEquivalence : ∀ {B : A → Set b} {{_ : IsDecEquivalence {A = A} _≡_}} {{_ : ∀ {a} → IsDecEquivalence {A = B a} _≡_}} → IsDecEquivalence {A = Σ A B} _≡_
-  Σ-≡-isDecEquivalence = isDecEquivalence (≡-dec _≟_ _≟_)
+  Σ-≡-isDecEquivalence = isDecEquivalence (≡-dec _≈?_ _≈?_)
diff --git a/src/Data/Product/Relation/Binary/Lex/NonStrict.agda b/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
index 599a7d011c..5dba1ee4fb 100644
--- a/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
@@ -157,8 +157,8 @@ module _ {_≈₁_ : Rel A ℓ₁} {_≤₁_ : Rel A ℓ₂}
     { isTotalOrder = ×-isTotalOrder (_≟_ to₁)
                                     (isTotalOrder to₁)
                                     (isTotalOrder to₂)
-    ; _≟_          = Pointwise.×-decidable (_≟_ to₁) (_≟_ to₂)
-    ; _≤?_         = ×-decidable (_≟_ to₁) (_≤?_ to₁) (_≤?_ to₂)
+    ; _≈?_         = Pointwise.×-decidable (_≈?_ to₁) (_≈?_ to₂)
+    ; _≤?_         = ×-decidable (_≈?_ to₁) (_≤?_ to₁) (_≤?_ to₂)
     }
     where open IsDecTotalOrder
 
@@ -174,7 +174,7 @@ module _ {_≈₁_ : Rel A ℓ₁} {_≤₁_ : Rel A ℓ₂}
                TotalOrder b ℓ₃ ℓ₄ →
                TotalOrder _ _ _
 ×-totalOrder t₁ t₂ = record
-  { isTotalOrder = ×-isTotalOrder T₁._≟_ T₁.isTotalOrder T₂.isTotalOrder
+  { isTotalOrder = ×-isTotalOrder T₁._≈?_ T₁.isTotalOrder T₂.isTotalOrder
   } where module T₁ = DecTotalOrder t₁; module T₂ = TotalOrder t₂
 
 ×-decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ →
diff --git a/src/Data/Rational/Instances.agda b/src/Data/Rational/Instances.agda
index 90bfe7f498..267f89eb38 100644
--- a/src/Data/Rational/Instances.agda
+++ b/src/Data/Rational/Instances.agda
@@ -8,10 +8,10 @@
 
 module Data.Rational.Instances where
 
-open import Data.Rational.Properties using (_≟_; ≤-isDecTotalOrder)
+open import Data.Rational.Properties using (_≡?_; ≤-isDecTotalOrder)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  ℚ-≡-isDecEquivalence = isDecEquivalence _≟_
+  ℚ-≡-isDecEquivalence = isDecEquivalence _≡?_
   ℚ-≤-isDecTotalOrder = ≤-isDecTotalOrder
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 2e12751416..687aa714cd 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -91,19 +91,19 @@ mkℚ-injective : ∀ {n₁ n₂ d₁ d₂}
                 mkℚ n₁ d₁ c₁ ≡ mkℚ n₂ d₂ c₂ → n₁ ≡ n₂ × d₁ ≡ d₂
 mkℚ-injective refl = refl , refl
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality ℚ
-mkℚ n₁ d₁ _ ≟ mkℚ n₂ d₂ _ = map′
+_≡?_ : DecidableEquality ℚ
+mkℚ n₁ d₁ _ ≡? mkℚ n₂ d₂ _ = map′
   (uncurry mkℚ-cong)
   mkℚ-injective
-  (n₁ ℤ.≟ n₂ ×-dec d₁ ℕ.≟ d₂)
+  (n₁ ℤ.≡? n₂ ×-dec d₁ ℕ.≡? d₂)
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid ℚ
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 1≢0 : 1ℚ ≢ 0ℚ
 1≢0 = λ ()
@@ -560,7 +560,7 @@ _≥?_ = flip _≤?_
 ≤-isDecTotalOrder : IsDecTotalOrder _≡_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≟_
+  ; _≈?_         = _≡?_
   ; _≤?_         = _≤?_
   }
 
@@ -858,7 +858,7 @@ private
 open Definitions ℚ ℚᵘ ℚᵘ._≃_
 
 toℚᵘ-homo-+ : Homomorphic₂ toℚᵘ _+_ ℚᵘ._+_
-toℚᵘ-homo-+ p@record{} q@record{} with +-nf p q ℤ.≟ 0ℤ
+toℚᵘ-homo-+ p@record{} q@record{} with +-nf p q ℤ.≡? 0ℤ
 ... | yes nf[p,q]≡0 = *≡* $ begin
   ↥ᵘ (toℚᵘ (p + q)) ℤ.* ↧+ᵘ p q   ≡⟨ cong (ℤ._* ↧+ᵘ p q) (↥ᵘ-toℚᵘ (p + q)) ⟩
   ↥ (p + q) ℤ.* ↧+ᵘ p q           ≡⟨ cong (ℤ._* ↧+ᵘ p q) eq ⟩
@@ -1094,7 +1094,7 @@ private
 -- Monomorphic to unnormalised _*_
 
 toℚᵘ-homo-* : Homomorphic₂ toℚᵘ _*_ ℚᵘ._*_
-toℚᵘ-homo-* p@record{} q@record{} with *-nf p q ℤ.≟ 0ℤ
+toℚᵘ-homo-* p@record{} q@record{} with *-nf p q ℤ.≡? 0ℤ
 ... | yes nf[p,q]≡0 = *≡* $ begin
   ↥ᵘ (toℚᵘ (p * q)) ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) (↥ᵘ-toℚᵘ (p * q)) ⟩
   ↥ (p * q)         ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) eq ⟩
@@ -1913,3 +1913,10 @@ nonPos*nonPos⇒nonPos = nonPos*nonPos⇒nonNeg
 "Warning: nonPos*nonPos⇒nonPos was deprecated in v2.4.
 Please use nonPos*nonPos⇒nonNeg instead."
 #-}
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Rational/Solver.agda b/src/Data/Rational/Solver.agda
index 3298efd560..d1391915f6 100644
--- a/src/Data/Rational/Solver.agda
+++ b/src/Data/Rational/Solver.agda
@@ -10,11 +10,11 @@ module Data.Rational.Solver where
 
 import Algebra.Solver.Ring.Simple as Solver
 import Algebra.Solver.Ring.AlmostCommutativeRing as ACR
-open import Data.Rational.Properties using (_≟_; +-*-commutativeRing)
+open import Data.Rational.Properties using (_≡?_; +-*-commutativeRing)
 
 ------------------------------------------------------------------------
 -- A module for automatically solving propositional equivalences
 -- containing _+_ and _*_
 
 module +-*-Solver =
-  Solver (ACR.fromCommutativeRing +-*-commutativeRing) _≟_
+  Solver (ACR.fromCommutativeRing +-*-commutativeRing) _≡?_
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 24fe04681a..53278f57b5 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -119,7 +119,7 @@ drop-*≡* (*≡* eq) = eq
 infix 4 _≃?_
 
 _≃?_ : Decidable _≃_
-p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* ↧ p)
+p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≡? ↥ q ℤ.* ↧ p)
 
 0≄1 : 0ℚᵘ ≄ 1ℚᵘ
 0≄1 = Dec.from-no (0ℚᵘ ≃? 1ℚᵘ)
@@ -146,7 +146,7 @@ p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* 
 ≃-isDecEquivalence : IsDecEquivalence _≃_
 ≃-isDecEquivalence = record
   { isEquivalence = ≃-isEquivalence
-  ; _≟_           = _≃?_
+  ; _≈?_          = _≃?_
   }
 
 ≄-isApartnessRelation : IsApartnessRelation _≃_ _≄_
@@ -321,7 +321,7 @@ _≥?_ = flip _≤?_
 ≤-isDecTotalOrder : IsDecTotalOrder _≃_ _≤_
 ≤-isDecTotalOrder = record
   { isTotalOrder = ≤-isTotalOrder
-  ; _≟_          = _≃?_
+  ; _≈?_         = _≃?_
   ; _≤?_         = _≤?_
   }
 
diff --git a/src/Data/String/Instances.agda b/src/Data/String/Instances.agda
index 5a4b8b70ba..2467c16aba 100644
--- a/src/Data/String/Instances.agda
+++ b/src/Data/String/Instances.agda
@@ -8,9 +8,9 @@
 
 module Data.String.Instances where
 
-open import Data.String.Properties using (_≟_)
+open import Data.String.Properties using (_≡?_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  String-≡-isDecEquivalence = isDecEquivalence _≟_
+  String-≡-isDecEquivalence = isDecEquivalence _≡?_
diff --git a/src/Data/Sum/Instances.agda b/src/Data/Sum/Instances.agda
index 8b9a8d142e..9b1732b385 100644
--- a/src/Data/Sum/Instances.agda
+++ b/src/Data/Sum/Instances.agda
@@ -14,7 +14,7 @@ open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
-open import Relation.Binary.TypeClasses using (IsDecEquivalence; _≟_)
+open import Relation.Binary.TypeClasses using (IsDecEquivalence; _≈?_)
 
 private
   variable
@@ -24,4 +24,4 @@ private
 
 instance
   ⊎-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → {{IsDecEquivalence {A = B} _≡_}} → IsDecEquivalence {A = A ⊎ B} _≡_
-  ⊎-≡-isDecEquivalence = isDecEquivalence (≡-dec _≟_ _≟_)
+  ⊎-≡-isDecEquivalence = isDecEquivalence (≡-dec _≈?_ _≈?_)
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index f162dd56dc..3194b05fc4 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -203,7 +203,7 @@ module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
                         IsDecTotalOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)
   ⊎-<-isDecTotalOrder to₁ to₂ = record
     { isTotalOrder = ⊎-<-isTotalOrder (isTotalOrder to₁) (isTotalOrder to₂)
-    ; _≟_          = PW.⊎-decidable (_≟_  to₁) (_≟_  to₂)
+    ; _≈?_         = PW.⊎-decidable (_≈?_  to₁) (_≈?_  to₂)
     ; _≤?_         = ⊎-<-decidable (_≤?_ to₁) (_≤?_ to₂)
     }
     where open IsDecTotalOrder
diff --git a/src/Data/These/Instances.agda b/src/Data/These/Instances.agda
index 49208291e8..c3b816b465 100644
--- a/src/Data/These/Instances.agda
+++ b/src/Data/These/Instances.agda
@@ -14,7 +14,7 @@ open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
-open import Relation.Binary.TypeClasses using (IsDecEquivalence; _≟_)
+open import Relation.Binary.TypeClasses using (IsDecEquivalence; _≈?_)
 
 private
   variable
@@ -24,4 +24,4 @@ private
 
 instance
   These-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → {{IsDecEquivalence {A = B} _≡_}} → IsDecEquivalence {A = These A B} _≡_
-  These-≡-isDecEquivalence = isDecEquivalence (≡-dec _≟_ _≟_)
+  These-≡-isDecEquivalence = isDecEquivalence (≡-dec _≈?_ _≈?_)
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index 5e472cfe2b..d405fc20f8 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -92,7 +92,7 @@ private
   k′≉key-p k′≈key-p = k≉k′ (Eq.trans (≈-lookup (tree p)) (Eq.sym k′≈key-p))
 
 ∈ₖᵥ-insert⁺⁺ : (k , x) ∈ₖᵥ insert k x m
-∈ₖᵥ-insert⁺⁺ {k = k} {m = tree t} with IAny.any? ((k ≟_) ∘ key) t
+∈ₖᵥ-insert⁺⁺ {k = k} {m = tree t} with IAny.any? ((k ≈?_) ∘ key) t
 ... | yes k∈ = tree (IAnyₚ.Any-insert-just _ _ _ _ (λ k′ → _, ≡-refl) k∈)
 ... | no ¬k∈ = tree (IAnyₚ.Any-insert-nothing _ _ _ _ (refl , ≡-refl) ¬k∈)
 
diff --git a/src/Data/Tree/AVL/Sets/Membership/Properties.agda b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
index 8b46933af5..93e4643ad0 100644
--- a/src/Data/Tree/AVL/Sets/Membership/Properties.agda
+++ b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
@@ -59,7 +59,7 @@ private
 -- insert
 
 ∈-insert⁺ : x ∈ s → x ∈ insert y s
-∈-insert⁺ {x = x} {s = s} {y = y} x∈s with x ≟ y
+∈-insert⁺ {x = x} {s = s} {y = y} x∈s with x ≈? y
 ... | yes x≈y = ∈fromMap (Map.∈ₖᵥ-Respectsˡ (Eq.sym x≈y , refl) Map.∈ₖᵥ-insert⁺⁺)
 ... | no x≉y = ∈fromMap (Map.∈ₖᵥ-insert⁺ x≉y (∈toMap x∈s))
 
diff --git a/src/Data/Unit/Instances.agda b/src/Data/Unit/Instances.agda
index 00bdf10f88..3bcb3635a4 100644
--- a/src/Data/Unit/Instances.agda
+++ b/src/Data/Unit/Instances.agda
@@ -8,10 +8,10 @@
 
 module Data.Unit.Instances where
 
-open import Data.Unit.Properties using (_≟_; ≡-isDecTotalOrder)
+open import Data.Unit.Properties using (_≡?_; ≡-isDecTotalOrder)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  ⊤-≡-isDecEquivalence = isDecEquivalence _≟_
+  ⊤-≡-isDecEquivalence = isDecEquivalence _≡?_
   ⊤-≤-isDecTotalOrder = ≡-isDecTotalOrder
diff --git a/src/Data/Unit/Polymorphic/Instances.agda b/src/Data/Unit/Polymorphic/Instances.agda
index a9c5908fa8..01a210776a 100644
--- a/src/Data/Unit/Polymorphic/Instances.agda
+++ b/src/Data/Unit/Polymorphic/Instances.agda
@@ -9,7 +9,7 @@
 module Data.Unit.Polymorphic.Instances where
 
 open import Data.Unit.Polymorphic.Base using (⊤)
-open import Data.Unit.Polymorphic.Properties using (_≟_; ≡-isDecTotalOrder)
+open import Data.Unit.Polymorphic.Properties using (_≡?_; ≡-isDecTotalOrder)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Binary.PropositionalEquality.Properties
@@ -23,7 +23,7 @@ private
 
 instance
   ⊤-≡-isDecEquivalence : IsDecEquivalence {A = ⊤ {a}} _≡_
-  ⊤-≡-isDecEquivalence = isDecEquivalence _≟_
+  ⊤-≡-isDecEquivalence = isDecEquivalence _≡?_
 
   ⊤-≤-isDecTotalOrder : IsDecTotalOrder {A = ⊤ {a}} _≡_ _≡_
   ⊤-≤-isDecTotalOrder = ≡-isDecTotalOrder _
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
index 1ba6aed44f..afac44fb06 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
@@ -68,7 +68,7 @@ module _ {R : Rel A ℓ} where
                      ∀ n → IsDecEquivalence (Pointwise R {n})
   isDecEquivalence isDecEq n = record
     { isEquivalence = isEquivalence Eq.isEquivalence n
-    ; _≟_           = decidable Eq._≟_
+    ; _≈?_          = decidable Eq._≈?_
     } where module Eq = IsDecEquivalence isDecEq
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Instances.agda b/src/Data/Vec/Instances.agda
index 9dd3936187..836f1984e9 100644
--- a/src/Data/Vec/Instances.agda
+++ b/src/Data/Vec/Instances.agda
@@ -15,7 +15,7 @@ open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
-open import Relation.Binary.TypeClasses using (IsDecEquivalence; _≟_)
+open import Relation.Binary.TypeClasses using (IsDecEquivalence; _≈?_)
 
 private
   variable
@@ -27,4 +27,4 @@ instance
   vecApplicative = applicative
 
   Vec-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → ∀ {n} → IsDecEquivalence {A = Vec A n} _≡_
-  Vec-≡-isDecEquivalence = isDecEquivalence (≡-dec _≟_)
+  Vec-≡-isDecEquivalence = isDecEquivalence (≡-dec _≈?_)
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index f2223d62f7..38594a4f69 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Relation.Binary.Lex.Core {a} {A : Set a} where
 
 open import Data.Nat.Base using (ℕ; suc)
-import Data.Nat.Properties as ℕ using (_≟_; ≡-irrelevant)
+import Data.Nat.Properties as ℕ using (_≡?_; ≡-irrelevant)
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
@@ -136,7 +136,7 @@ module _ {P : Set} {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   module _ (P? : Dec P) (_≈?_ : Decidable _≈_) (_≺?_ : Decidable _≺_) where
 
     decidable : ∀ {m n} → Decidable (_<ₗₑₓ_ {m} {n})
-    decidable {m} {n} xs ys with m ℕ.≟ n
+    decidable {m} {n} xs ys with m ℕ.≡? n
     decidable {_} {_} []       []       | yes refl = Dec.map P⇔[]<[] P?
     decidable {_} {_} (x ∷ xs) (y ∷ ys) | yes refl = Dec.map ∷<∷-⇔ ((x ≺? y) ⊎? (x ≈? y) ×? (decidable xs ys))
     decidable {_} {_} _        _        | no  m≢n    = no (λ xs<ys → contradiction (length-equal xs<ys) m≢n)
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index b48244820c..f0bbb5b0c8 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -167,8 +167,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
                               ∀ {n} → IsDecStrictPartialOrder (_≋_ {n} {n}) _<_
   <-isDecStrictPartialOrder ≺-isDecStrictPartialOrder = record
     { isStrictPartialOrder = <-isStrictPartialOrder O.isStrictPartialOrder
-    ; _≟_                  = Pointwise.decidable O._≟_
-    ; _<?_                 = <-decidable O._≟_ O._<?_
+    ; _≈?_                 = Pointwise.decidable O._≈?_
+    ; _<?_                 = <-decidable O._≈?_ O._<?_
     } where module O = IsDecStrictPartialOrder ≺-isDecStrictPartialOrder
 
   <-isStrictTotalOrder : IsStrictTotalOrder _≈_ _≺_ →
@@ -279,8 +279,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
                         ∀ {n} → IsDecPartialOrder (_≋_ {n} {n}) _≤_
   ≤-isDecPartialOrder ≺-isDecStrictPartialOrder = record
     { isPartialOrder = ≤-isPartialOrder isStrictPartialOrder
-    ; _≟_            = Pointwise.decidable _≟_
-    ; _≤?_           = ≤-dec _≟_ _<?_
+    ; _≈?_           = Pointwise.decidable _≈?_
+    ; _≤?_           = ≤-dec _≈?_ _<?_
     } where open IsDecStrictPartialOrder ≺-isDecStrictPartialOrder
 
   ≤-isTotalOrder : IsStrictTotalOrder _≈_ _≺_ →
@@ -294,8 +294,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
                       ∀ {n} → IsDecTotalOrder (_≋_ {n} {n}) _≤_
   ≤-isDecTotalOrder ≺-isStrictTotalOrder = record
     { isTotalOrder = ≤-isTotalOrder ≺-isStrictTotalOrder
-    ; _≟_          = Pointwise.decidable _≟_
-    ; _≤?_         = ≤-dec _≟_ _<?_
+    ; _≈?_         = Pointwise.decidable _≈?_
+    ; _≤?_         = ≤-dec _≈?_ _<?_
     } where open IsStrictTotalOrder ≺-isStrictTotalOrder
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
index 832bba496c..a6e081eb91 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
@@ -132,7 +132,7 @@ isDecEquivalence : ∀ {_∼_ : Rel A ℓ} {n} →
                    IsDecEquivalence (Pointwise _∼_ {n = n})
 isDecEquivalence decEquiv = record
   { isEquivalence = isEquivalence DecEq.isEquivalence
-  ; _≟_           = decidable DecEq._≟_
+  ; _≈?_          = decidable DecEq._≈?_
   } where module DecEq = IsDecEquivalence decEquiv
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
index 50fe5fc824..93ffc15d2b 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
@@ -139,7 +139,7 @@ module _ {_∼_ : Rel A ℓ} where
                      IsDecEquivalence (Pointwise _∼_ {n})
   isDecEquivalence decEquiv n = record
     { isEquivalence = isEquivalence Eq.isEquivalence n
-    ; _≟_           = decidable Eq._≟_
+    ; _≈?_          = decidable Eq._≈?_
     } where module Eq = IsDecEquivalence decEquiv
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Word64/Instances.agda b/src/Data/Word64/Instances.agda
index db8737dffc..2dfae48966 100644
--- a/src/Data/Word64/Instances.agda
+++ b/src/Data/Word64/Instances.agda
@@ -8,9 +8,9 @@
 
 module Data.Word64.Instances where
 
-open import Data.Word64.Properties using (_≟_)
+open import Data.Word64.Properties using (_≡?_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (isDecEquivalence)
 
 instance
-  Word64-≡-isDecEquivalence = isDecEquivalence _≟_
+  Word64-≡-isDecEquivalence = isDecEquivalence _≡?_
diff --git a/src/Data/Word64/Properties.agda b/src/Data/Word64/Properties.agda
index 7ca0f4918c..4bcbe3b000 100644
--- a/src/Data/Word64/Properties.agda
+++ b/src/Data/Word64/Properties.agda
@@ -53,7 +53,7 @@ open import Agda.Builtin.Word.Properties
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-x ≈? y = toℕ x ℕ.≟ toℕ y
+x ≈? y = toℕ x ℕ.≡? toℕ y
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -70,7 +70,7 @@ x ≈? y = toℕ x ℕ.≟ toℕ y
 ≈-isDecEquivalence : IsDecEquivalence _≈_
 ≈-isDecEquivalence = record
   { isEquivalence = ≈-isEquivalence
-  ; _≟_           = _≈?_
+  ; _≈?_          = _≈?_
   }
 
 ≈-decSetoid : DecSetoid _ _
@@ -80,22 +80,22 @@ x ≈? y = toℕ x ℕ.≟ toℕ y
 ------------------------------------------------------------------------
 -- Properties of _≡_
 
-infix 4 _≟_
-_≟_ : DecidableEquality Word64
-x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
+infix 4 _≡?_
+_≡?_ : DecidableEquality Word64
+x ≡? y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
 
 ≡-setoid : Setoid _ _
 ≡-setoid = setoid Word64
 
 ≡-decSetoid : DecSetoid _ _
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ------------------------------------------------------------------------
 -- Boolean equality test.
 
 infix 4 _==_
 _==_ : Word64 → Word64 → Bool
-w₁ == w₂ = ⌊ w₁ ≟ w₂ ⌋
+w₁ == w₂ = ⌊ w₁ ≡? w₂ ⌋
 
 ------------------------------------------------------------------------
 -- Properties of _<_
@@ -106,3 +106,18 @@ _<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
 
 <-strictTotalOrder-≈ : StrictTotalOrder _ _ _
 <-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection.agda b/src/Reflection.agda
index fd3476cebb..64aa468c25 100644
--- a/src/Reflection.agda
+++ b/src/Reflection.agda
@@ -48,46 +48,46 @@ Please use Reflection.AST.Argument.Information's ArgInfo instead."
 infix 4 _≟-Lit_ _≟-Name_ _≟-Meta_ _≟-Visibility_ _≟-Relevance_ _≟-Arg-info_
         _≟-Pattern_ _≟-ArgPatterns_
 
-_≟-Lit_ = Literal._≟_
+_≟-Lit_ = Literal._≡?_
 {-# WARNING_ON_USAGE _≟-Lit_
 "Warning: _≟-Lit_ was deprecated in v1.3.
-Please use Reflection.AST.Literal's _≟_ instead."
+Please use Reflection.AST.Literal's _≡?_ instead."
 #-}
 
-_≟-Name_ = Name._≟_
+_≟-Name_ = Name._≡?_
 {-# WARNING_ON_USAGE _≟-Name_
 "Warning: _≟-Name_ was deprecated in v1.3.
-Please use Reflection.AST.Name's _≟_ instead."
+Please use Reflection.AST.Name's _≡?_ instead."
 #-}
 
-_≟-Meta_ = Meta._≟_
+_≟-Meta_ = Meta._≡?_
 {-# WARNING_ON_USAGE _≟-Meta_
 "Warning: _≟-Meta_ was deprecated in v1.3.
-Please use Reflection.AST.Meta's _≟_ instead."
+Please use Reflection.AST.Meta's _≡?_ instead."
 #-}
 
-_≟-Visibility_ = Visibility._≟_
+_≟-Visibility_ = Visibility._≡?_
 {-# WARNING_ON_USAGE _≟-Visibility_
 "Warning: _≟-Visibility_ was deprecated in v1.3.
-Please use Reflection.AST.Argument.Visibility's _≟_ instead."
+Please use Reflection.AST.Argument.Visibility's _≡?_ instead."
 #-}
 
-_≟-Relevance_ = Relevance._≟_
+_≟-Relevance_ = Relevance._≡?_
 {-# WARNING_ON_USAGE _≟-Relevance_
 "Warning: _≟-Relevance_ was deprecated in v1.3.
-Please use Reflection.AST.Argument.Relevance's _≟_ instead."
+Please use Reflection.AST.Argument.Relevance's _≡?_ instead."
 #-}
 
-_≟-Arg-info_ = Information._≟_
+_≟-Arg-info_ = Information._≡?_
 {-# WARNING_ON_USAGE _≟-Arg-info_
 "Warning: _≟-Arg-info_ was deprecated in v1.3.
-Please use Reflection.AST.Argument.Information's _≟_ instead."
+Please use Reflection.AST.Argument.Information's _≡?_ instead."
 #-}
 
-_≟-Pattern_ = Pattern._≟_
+_≟-Pattern_ = Pattern._≡?_
 {-# WARNING_ON_USAGE _≟-Pattern_
 "Warning: _≟-Pattern_ was deprecated in v1.3.
-Please use Reflection.AST.Pattern's _≟_ instead."
+Please use Reflection.AST.Pattern's _≡?_ instead."
 #-}
 
 _≟-ArgPatterns_ = Pattern._≟s_
@@ -130,56 +130,56 @@ infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_
         _≟-Clause_ _≟-Clauses_ _≟_
         _≟-Sort_
 
-_≟-AbsTerm_ = Term._≟-AbsTerm_
+_≟-AbsTerm_ = Term._≡?-AbsTerm_
 {-# WARNING_ON_USAGE _≟-AbsTerm_
 "Warning: _≟-AbsTerm_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-AbsTerm_ instead."
+Please use Reflection.AST.Term's _≡?-AbsTerm_ instead."
 #-}
 
-_≟-AbsType_ = Term._≟-AbsType_
+_≟-AbsType_ = Term._≡?-AbsType_
 {-# WARNING_ON_USAGE _≟-AbsType_
 "Warning: _≟-AbsType_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-AbsType_ instead."
+Please use Reflection.AST.Term's _≡?-AbsType_ instead."
 #-}
 
-_≟-ArgTerm_ = Term._≟-ArgTerm_
+_≟-ArgTerm_ = Term._≡?-ArgTerm_
 {-# WARNING_ON_USAGE _≟-ArgTerm_
 "Warning: _≟-ArgTerm_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-ArgTerm_ instead."
+Please use Reflection.AST.Term's _≡?-ArgTerm_ instead."
 #-}
 
-_≟-ArgType_ = Term._≟-ArgType_
+_≟-ArgType_ = Term._≡?-ArgType_
 {-# WARNING_ON_USAGE _≟-ArgType_
 "Warning: _≟-ArgType_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-ArgType_ instead."
+Please use Reflection.AST.Term's _≡?-ArgType_ instead."
 #-}
 
-_≟-Args_    = Term._≟-Args_
+_≟-Args_    = Term._≡?-Args_
 {-# WARNING_ON_USAGE _≟-Args_
 "Warning: _≟-Args_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-Args_ instead."
+Please use Reflection.AST.Term's _≡?-Args_ instead."
 #-}
 
-_≟-Clause_  = Term._≟-Clause_
+_≟-Clause_  = Term._≡?-Clause_
 {-# WARNING_ON_USAGE _≟-Clause_
 "Warning: _≟-Clause_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-Clause_ instead."
+Please use Reflection.AST.Term's _≡?-Clause_ instead."
 #-}
 
-_≟-Clauses_ = Term._≟-Clauses_
+_≟-Clauses_ = Term._≡?-Clauses_
 {-# WARNING_ON_USAGE _≟-Clauses_
 "Warning: _≟-Clauses_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-Clauses_ instead."
+Please use Reflection.AST.Term's _≡?-Clauses_ instead."
 #-}
 
-_≟_         = Term._≟_
+_≟_         = Term._≡?_
 {-# WARNING_ON_USAGE _≟_
 "Warning: _≟_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟_ instead."
+Please use Reflection.AST.Term's _≡?_ instead."
 #-}
 
-_≟-Sort_    = Term._≟-Sort_
+_≟-Sort_    = Term._≡?-Sort_
 {-# WARNING_ON_USAGE _≟-Sort_
 "Warning: _≟-Sort_ was deprecated in v1.3.
-Please use Reflection.AST.Term's _≟-Sort_ instead."
+Please use Reflection.AST.Term's _≡?-Sort_ instead."
 #-}
diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index e91ae8d00c..3084ca87a2 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Abstraction where
 
 open import Data.String.Base as String using (String)
-open import Data.String.Properties as String using (_≟_)
+open import Data.String.Properties as String using (_≡?_)
 open import Data.Product.Base using (_×_; <_,_>; uncurry)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (Dec; map′; _×?_)
@@ -58,8 +58,8 @@ unAbs : Abs A → A
 unAbs (abs s a) = a
 
 unAbs-dec : {abs1 abs2 : Abs A} → Dec (unAbs abs1 ≡ unAbs abs2) → Dec (abs1 ≡ abs2)
-unAbs-dec {abs1 = abs s a} {abs t a′} a≟a′ =
-  map′ (uncurry (cong₂ abs)) abs-injective (s String.≟ t ×? a≟a′)
+unAbs-dec {abs1 = abs s a} {abs t a′} a≡?a′ =
+  map′ (uncurry (cong₂ abs)) abs-injective (s String.≡? t ×? a≡?a′)
 
 ≡-dec : DecidableEquality A → DecidableEquality (Abs A)
-≡-dec _≟_ x y = unAbs-dec (unAbs x ≟ unAbs y)
+≡-dec _≡?_ x y = unAbs-dec (unAbs x ≡? unAbs y)
diff --git a/src/Reflection/AST/AlphaEquality.agda b/src/Reflection/AST/AlphaEquality.agda
index fc2e3bfcd5..f6d1a7b1b5 100644
--- a/src/Reflection/AST/AlphaEquality.agda
+++ b/src/Reflection/AST/AlphaEquality.agda
@@ -55,22 +55,22 @@ open AlphaEquality {{...}} public
 
 instance
   α-Visibility : AlphaEquality Visibility
-  α-Visibility = ≟⇒α Visibility._≟_
+  α-Visibility = ≟⇒α Visibility._≡?_
 
   α-Modality : AlphaEquality Modality
-  α-Modality = ≟⇒α Modality._≟_
+  α-Modality = ≟⇒α Modality._≡?_
 
   α-ArgInfo : AlphaEquality ArgInfo
-  α-ArgInfo = ≟⇒α ArgInfo._≟_
+  α-ArgInfo = ≟⇒α ArgInfo._≡?_
 
   α-Literal : AlphaEquality Literal
-  α-Literal = ≟⇒α Literal._≟_
+  α-Literal = ≟⇒α Literal._≡?_
 
   α-Meta : AlphaEquality Meta
-  α-Meta = ≟⇒α Meta._≟_
+  α-Meta = ≟⇒α Meta._≡?_
 
   α-Name : AlphaEquality Name
-  α-Name = ≟⇒α Name._≟_
+  α-Name = ≟⇒α Name._≡?_
 
 ------------------------------------------------------------------------
 -- Interesting cases
diff --git a/src/Reflection/AST/Argument.agda b/src/Reflection/AST/Argument.agda
index 9fd29e7330..5582e5b4a9 100644
--- a/src/Reflection/AST/Argument.agda
+++ b/src/Reflection/AST/Argument.agda
@@ -86,7 +86,7 @@ unArg (arg i a) = a
 
 unArg-dec : {arg1 arg2 : Arg A} → Dec (unArg arg1 ≡ unArg arg2) → Dec (arg1 ≡ arg2)
 unArg-dec {arg1 = arg i x} {arg j y} arg1≟arg2 =
-  map′ (uncurry (cong₂ arg)) arg-injective (i Information.≟ j ×? arg1≟arg2)
+  map′ (uncurry (cong₂ arg)) arg-injective (i Information.≡? j ×? arg1≟arg2)
 
 ≡-dec : DecidableEquality A → DecidableEquality (Arg A)
-≡-dec _≟_ x y = unArg-dec (unArg x ≟ unArg y)
+≡-dec _≡?_ x y = unArg-dec (unArg x ≡? unArg y)
diff --git a/src/Reflection/AST/Argument/Information.agda b/src/Reflection/AST/Argument/Information.agda
index 307a1b1976..6940c3c492 100644
--- a/src/Reflection/AST/Argument/Information.agda
+++ b/src/Reflection/AST/Argument/Information.agda
@@ -47,11 +47,27 @@ arg-info-injective₂ refl = refl
 arg-info-injective : arg-info v m ≡ arg-info v′ m′ → v ≡ v′ × m ≡ m′
 arg-info-injective = < arg-info-injective₁ , arg-info-injective₂ >
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality ArgInfo
-arg-info v m ≟ arg-info v′ m′ =
+_≡?_ : DecidableEquality ArgInfo
+arg-info v m ≡? arg-info v′ m′ =
   map′
     (uncurry (cong₂ arg-info))
     arg-info-injective
-    (v Visibility.≟ v′ ×? m Modality.≟ m′)
+    (v Visibility.≡? v′ ×? m Modality.≡? m′)
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Argument/Modality.agda b/src/Reflection/AST/Argument/Modality.agda
index e71b0f2710..61886cad95 100644
--- a/src/Reflection/AST/Argument/Modality.agda
+++ b/src/Reflection/AST/Argument/Modality.agda
@@ -47,11 +47,28 @@ modality-injective₂ refl = refl
 modality-injective : modality r q ≡ modality r′ q′ → r ≡ r′ × q ≡ q′
 modality-injective = < modality-injective₁ , modality-injective₂ >
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality Modality
-modality r q ≟ modality r′ q′ =
+_≡?_ : DecidableEquality Modality
+modality r q ≡? modality r′ q′ =
   map′
     (uncurry (cong₂ modality))
     modality-injective
-    (r Relevance.≟ r′ ×? q Quantity.≟ q′)
+    (r Relevance.≡? r′ ×? q Quantity.≡? q′)
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Argument/Quantity.agda b/src/Reflection/AST/Argument/Quantity.agda
index a042218660..6d1a0a7fc2 100644
--- a/src/Reflection/AST/Argument/Quantity.agda
+++ b/src/Reflection/AST/Argument/Quantity.agda
@@ -21,10 +21,27 @@ open Quantity public
 ------------------------------------------------------------------------
 -- Decidable equality
 
+infix 4 _≡?_
+
+_≡?_ : DecidableEquality Quantity
+quantity-ω ≡? quantity-ω = yes refl
+quantity-0 ≡? quantity-0 = yes refl
+quantity-ω ≡? quantity-0 = no λ()
+quantity-0 ≡? quantity-ω = no λ()
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
 infix 4 _≟_
 
-_≟_ : DecidableEquality Quantity
-quantity-ω ≟ quantity-ω = yes refl
-quantity-0 ≟ quantity-0 = yes refl
-quantity-ω ≟ quantity-0 = no λ()
-quantity-0 ≟ quantity-ω = no λ()
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Argument/Relevance.agda b/src/Reflection/AST/Argument/Relevance.agda
index a06095dc8b..c4020c350e 100644
--- a/src/Reflection/AST/Argument/Relevance.agda
+++ b/src/Reflection/AST/Argument/Relevance.agda
@@ -21,10 +21,27 @@ open Relevance public
 ------------------------------------------------------------------------
 -- Decidable equality
 
+infix 4 _≡?_
+
+_≡?_ : DecidableEquality Relevance
+relevant   ≡? relevant   = yes refl
+irrelevant ≡? irrelevant = yes refl
+relevant   ≡? irrelevant = no λ()
+irrelevant ≡? relevant   = no λ()
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
 infix 4 _≟_
 
-_≟_ : DecidableEquality Relevance
-relevant   ≟ relevant   = yes refl
-irrelevant ≟ irrelevant = yes refl
-relevant   ≟ irrelevant = no λ()
-irrelevant ≟ relevant   = no λ()
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Argument/Visibility.agda b/src/Reflection/AST/Argument/Visibility.agda
index db68cb4b27..35f7133346 100644
--- a/src/Reflection/AST/Argument/Visibility.agda
+++ b/src/Reflection/AST/Argument/Visibility.agda
@@ -21,15 +21,31 @@ open Visibility public
 ------------------------------------------------------------------------
 -- Decidable equality
 
-infix 4 _≟_
+infix 4 _≡?_
+
+_≡?_ : DecidableEquality Visibility
+visible   ≡? visible   = yes refl
+hidden    ≡? hidden    = yes refl
+instance′ ≡? instance′ = yes refl
+visible   ≡? hidden    = no λ()
+visible   ≡? instance′ = no λ()
+hidden    ≡? visible   = no λ()
+hidden    ≡? instance′ = no λ()
+instance′ ≡? visible   = no λ()
+instance′ ≡? hidden    = no λ()
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
 
-_≟_ : DecidableEquality Visibility
-visible   ≟ visible   = yes refl
-hidden    ≟ hidden    = yes refl
-instance′ ≟ instance′ = yes refl
-visible   ≟ hidden    = no λ()
-visible   ≟ instance′ = no λ()
-hidden    ≟ visible   = no λ()
-hidden    ≟ instance′ = no λ()
-instance′ ≟ visible   = no λ()
-instance′ ≟ hidden    = no λ()
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index 400499afd3..75c9447bec 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -9,15 +9,15 @@
 module Reflection.AST.Definition where
 
 import Data.List.Properties as List using (≡-dec)
-import Data.Nat.Properties as ℕ using (_≟_)
+import Data.Nat.Properties as ℕ using (_≡?_)
 open import Data.Product.Base using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable.Core using (map′; _×?_; yes; no)
 open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 import Reflection.AST.Argument as Arg using (Arg; ≡-dec)
-import Reflection.AST.Argument.Quantity as Quantity using (Quantity; _≟_)
-import Reflection.AST.Name as Name using (Name; _≟_)
-import Reflection.AST.Term as Term using (Term; _≟-Clauses_)
+import Reflection.AST.Argument.Quantity as Quantity using (Quantity; _≡?_)
+import Reflection.AST.Name as Name using (Name; _≡?_)
+import Reflection.AST.Term as Term using (Term; _≡?-Clauses_)
 
 ------------------------------------------------------------------------
 -- Re-exporting type publicly
@@ -65,51 +65,51 @@ constructor′-injective₂ refl = refl
 constructor′-injective : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → c ≡ c′ × q ≡ q′
 constructor′-injective = < constructor′-injective₁ , constructor′-injective₂ >
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality Definition
-function cs       ≟ function cs′        =
-  map′ (cong function) function-injective (cs Term.≟-Clauses cs′)
-data-type pars cs ≟ data-type pars′ cs′ =
+_≡?_ : DecidableEquality Definition
+function cs       ≡? function cs′        =
+  map′ (cong function) function-injective (cs Term.≡?-Clauses cs′)
+data-type pars cs ≡? data-type pars′ cs′ =
   map′ (uncurry (cong₂ data-type)) data-type-injective
-           (pars ℕ.≟ pars′ ×? List.≡-dec Name._≟_ cs cs′)
-record′ c fs      ≟ record′ c′ fs′      =
+           (pars ℕ.≡? pars′ ×? List.≡-dec Name._≡?_ cs cs′)
+record′ c fs      ≡? record′ c′ fs′      =
   map′ (uncurry (cong₂ record′)) record′-injective
-           (c Name.≟ c′ ×? List.≡-dec (Arg.≡-dec Name._≟_) fs fs′)
-constructor′ d q  ≟ constructor′ d′ q′  =
+           (c Name.≡? c′ ×? List.≡-dec (Arg.≡-dec Name._≡?_) fs fs′)
+constructor′ d q  ≡? constructor′ d′ q′  =
   map′ (uncurry (cong₂ constructor′)) constructor′-injective
-           (d Name.≟ d′ ×? q Quantity.≟ q′)
-axiom             ≟ axiom               = yes refl
-primitive′        ≟ primitive′          = yes refl
+           (d Name.≡? d′ ×? q Quantity.≡? q′)
+axiom             ≡? axiom               = yes refl
+primitive′        ≡? primitive′          = yes refl
 
 -- antidiagonal
-function cs ≟ data-type pars cs₁ = no (λ ())
-function cs ≟ record′ c fs = no (λ ())
-function cs ≟ constructor′ d q = no (λ ())
-function cs ≟ axiom = no (λ ())
-function cs ≟ primitive′ = no (λ ())
-data-type pars cs ≟ function cs₁ = no (λ ())
-data-type pars cs ≟ record′ c fs = no (λ ())
-data-type pars cs ≟ constructor′ d q = no (λ ())
-data-type pars cs ≟ axiom = no (λ ())
-data-type pars cs ≟ primitive′ = no (λ ())
-record′ c fs ≟ function cs = no (λ ())
-record′ c fs ≟ data-type pars cs = no (λ ())
-record′ c fs ≟ constructor′ d q = no (λ ())
-record′ c fs ≟ axiom = no (λ ())
-record′ c fs ≟ primitive′ = no (λ ())
-constructor′ d q ≟ function cs = no (λ ())
-constructor′ d q ≟ data-type pars cs = no (λ ())
-constructor′ d q ≟ record′ c fs = no (λ ())
-constructor′ d q ≟ axiom = no (λ ())
-constructor′ d q ≟ primitive′ = no (λ ())
-axiom ≟ function cs = no (λ ())
-axiom ≟ data-type pars cs = no (λ ())
-axiom ≟ record′ c fs = no (λ ())
-axiom ≟ constructor′ d q = no (λ ())
-axiom ≟ primitive′ = no (λ ())
-primitive′ ≟ function cs = no (λ ())
-primitive′ ≟ data-type pars cs = no (λ ())
-primitive′ ≟ record′ c fs = no (λ ())
-primitive′ ≟ constructor′ d q = no (λ ())
-primitive′ ≟ axiom = no (λ ())
+function cs ≡? data-type pars cs₁ = no (λ ())
+function cs ≡? record′ c fs = no (λ ())
+function cs ≡? constructor′ d q = no (λ ())
+function cs ≡? axiom = no (λ ())
+function cs ≡? primitive′ = no (λ ())
+data-type pars cs ≡? function cs₁ = no (λ ())
+data-type pars cs ≡? record′ c fs = no (λ ())
+data-type pars cs ≡? constructor′ d q = no (λ ())
+data-type pars cs ≡? axiom = no (λ ())
+data-type pars cs ≡? primitive′ = no (λ ())
+record′ c fs ≡? function cs = no (λ ())
+record′ c fs ≡? data-type pars cs = no (λ ())
+record′ c fs ≡? constructor′ d q = no (λ ())
+record′ c fs ≡? axiom = no (λ ())
+record′ c fs ≡? primitive′ = no (λ ())
+constructor′ d q ≡? function cs = no (λ ())
+constructor′ d q ≡? data-type pars cs = no (λ ())
+constructor′ d q ≡? record′ c fs = no (λ ())
+constructor′ d q ≡? axiom = no (λ ())
+constructor′ d q ≡? primitive′ = no (λ ())
+axiom ≡? function cs = no (λ ())
+axiom ≡? data-type pars cs = no (λ ())
+axiom ≡? record′ c fs = no (λ ())
+axiom ≡? constructor′ d q = no (λ ())
+axiom ≡? primitive′ = no (λ ())
+primitive′ ≡? function cs = no (λ ())
+primitive′ ≡? data-type pars cs = no (λ ())
+primitive′ ≡? record′ c fs = no (λ ())
+primitive′ ≡? constructor′ d q = no (λ ())
+primitive′ ≡? axiom = no (λ ())
diff --git a/src/Reflection/AST/Instances.agda b/src/Reflection/AST/Instances.agda
index 27a766076d..8d0971d71d 100644
--- a/src/Reflection/AST/Instances.agda
+++ b/src/Reflection/AST/Instances.agda
@@ -33,20 +33,20 @@ private
     A : Set a
 
 instance
-  Lit-≡-isDecEquivalence = isDecEquivalence Literal._≟_
-  Name-≡-isDecEquivalence = isDecEquivalence Name._≟_
-  Meta-≡-isDecEquivalence = isDecEquivalence Meta._≟_
-  Visibility-≡-isDecEquivalence = isDecEquivalence Visibility._≟_
-  Relevance-≡-isDecEquivalence = isDecEquivalence Relevance._≟_
-  Quantity-≡-isDecEquivalence = isDecEquivalence Quantity._≟_
-  ArgInfo-≡-isDecEquivalence = isDecEquivalence Information._≟_
-  Pattern-≡-isDecEquivalence = isDecEquivalence Pattern._≟_
-  Clause-≡-isDecEquivalence = isDecEquivalence Term._≟-Clause_
-  Term-≡-isDecEquivalence = isDecEquivalence Term._≟_
-  Sort-≡-isDecEquivalence = isDecEquivalence Term._≟-Sort_
+  Lit-≡-isDecEquivalence = isDecEquivalence Literal._≡?_
+  Name-≡-isDecEquivalence = isDecEquivalence Name._≡?_
+  Meta-≡-isDecEquivalence = isDecEquivalence Meta._≡?_
+  Visibility-≡-isDecEquivalence = isDecEquivalence Visibility._≡?_
+  Relevance-≡-isDecEquivalence = isDecEquivalence Relevance._≡?_
+  Quantity-≡-isDecEquivalence = isDecEquivalence Quantity._≡?_
+  ArgInfo-≡-isDecEquivalence = isDecEquivalence Information._≡?_
+  Pattern-≡-isDecEquivalence = isDecEquivalence Pattern._≡?_
+  Clause-≡-isDecEquivalence = isDecEquivalence Term._≡?-Clause_
+  Term-≡-isDecEquivalence = isDecEquivalence Term._≡?_
+  Sort-≡-isDecEquivalence = isDecEquivalence Term._≡?-Sort_
 
   Abs-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → IsDecEquivalence {A = Abstraction.Abs A} _≡_
-  Abs-≡-isDecEquivalence = isDecEquivalence (Abstraction.≡-dec _≟_)
+  Abs-≡-isDecEquivalence = isDecEquivalence (Abstraction.≡-dec _≈?_)
 
   Arg-≡-isDecEquivalence : {{IsDecEquivalence {A = A} _≡_}} → IsDecEquivalence {A = Argument.Arg A} _≡_
-  Arg-≡-isDecEquivalence = isDecEquivalence (Argument.≡-dec _≟_)
+  Arg-≡-isDecEquivalence = isDecEquivalence (Argument.≡-dec _≈?_)
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index 9048e75ede..27e7c87816 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -51,59 +51,74 @@ string-injective refl = refl
 name-injective : ∀ {x y} → name x ≡ name y → x ≡ y
 name-injective refl = refl
 
-infix 4 _≟_
-_≟_ : DecidableEquality Literal
-nat x ≟ nat x₁ = map′ (cong nat) nat-injective (x ℕ.≟ x₁)
-nat x ≟ word64 x₁ = no (λ ())
-nat x ≟ float x₁ = no (λ ())
-nat x ≟ char x₁ = no (λ ())
-nat x ≟ string x₁ = no (λ ())
-nat x ≟ name x₁ = no (λ ())
-nat x ≟ meta x₁ = no (λ ())
-word64 x ≟ word64 x₁ = map′ (cong word64) word64-injective (x Word64.≟ x₁)
-word64 x ≟ nat x₁ = no (λ ())
-word64 x ≟ float x₁ = no (λ ())
-word64 x ≟ char x₁ = no (λ ())
-word64 x ≟ string x₁ = no (λ ())
-word64 x ≟ name x₁ = no (λ ())
-word64 x ≟ meta x₁ = no (λ ())
-float x ≟ nat x₁ = no (λ ())
-float x ≟ word64 x₁ = no (λ ())
-float x ≟ float x₁ = map′ (cong float) float-injective (x Float.≟ x₁)
-float x ≟ char x₁ = no (λ ())
-float x ≟ string x₁ = no (λ ())
-float x ≟ name x₁ = no (λ ())
-float x ≟ meta x₁ = no (λ ())
-char x ≟ nat x₁ = no (λ ())
-char x ≟ word64 x₁ = no (λ ())
-char x ≟ float x₁ = no (λ ())
-char x ≟ char x₁ = map′ (cong char) char-injective (x Char.≟ x₁)
-char x ≟ string x₁ = no (λ ())
-char x ≟ name x₁ = no (λ ())
-char x ≟ meta x₁ = no (λ ())
-string x ≟ nat x₁ = no (λ ())
-string x ≟ word64 x₁ = no (λ ())
-string x ≟ float x₁ = no (λ ())
-string x ≟ char x₁ = no (λ ())
-string x ≟ string x₁ = map′ (cong string) string-injective (x String.≟ x₁)
-string x ≟ name x₁ = no (λ ())
-string x ≟ meta x₁ = no (λ ())
-name x ≟ nat x₁ = no (λ ())
-name x ≟ word64 x₁ = no (λ ())
-name x ≟ float x₁ = no (λ ())
-name x ≟ char x₁ = no (λ ())
-name x ≟ string x₁ = no (λ ())
-name x ≟ name x₁ = map′ (cong name) name-injective (x Name.≟ x₁)
-name x ≟ meta x₁ = no (λ ())
-meta x ≟ nat x₁ = no (λ ())
-meta x ≟ word64 x₁ = no (λ ())
-meta x ≟ float x₁ = no (λ ())
-meta x ≟ char x₁ = no (λ ())
-meta x ≟ string x₁ = no (λ ())
-meta x ≟ name x₁ = no (λ ())
-meta x ≟ meta x₁ = map′ (cong meta) meta-injective (x Meta.≟ x₁)
+infix 4 _≡?_
+_≡?_ : DecidableEquality Literal
+nat x ≡? nat x₁ = map′ (cong nat) nat-injective (x ℕ.≡? x₁)
+nat x ≡? word64 x₁ = no (λ ())
+nat x ≡? float x₁ = no (λ ())
+nat x ≡? char x₁ = no (λ ())
+nat x ≡? string x₁ = no (λ ())
+nat x ≡? name x₁ = no (λ ())
+nat x ≡? meta x₁ = no (λ ())
+word64 x ≡? word64 x₁ = map′ (cong word64) word64-injective (x Word64.≡? x₁)
+word64 x ≡? nat x₁ = no (λ ())
+word64 x ≡? float x₁ = no (λ ())
+word64 x ≡? char x₁ = no (λ ())
+word64 x ≡? string x₁ = no (λ ())
+word64 x ≡? name x₁ = no (λ ())
+word64 x ≡? meta x₁ = no (λ ())
+float x ≡? nat x₁ = no (λ ())
+float x ≡? word64 x₁ = no (λ ())
+float x ≡? float x₁ = map′ (cong float) float-injective (x Float.≡? x₁)
+float x ≡? char x₁ = no (λ ())
+float x ≡? string x₁ = no (λ ())
+float x ≡? name x₁ = no (λ ())
+float x ≡? meta x₁ = no (λ ())
+char x ≡? nat x₁ = no (λ ())
+char x ≡? word64 x₁ = no (λ ())
+char x ≡? float x₁ = no (λ ())
+char x ≡? char x₁ = map′ (cong char) char-injective (x Char.≡? x₁)
+char x ≡? string x₁ = no (λ ())
+char x ≡? name x₁ = no (λ ())
+char x ≡? meta x₁ = no (λ ())
+string x ≡? nat x₁ = no (λ ())
+string x ≡? word64 x₁ = no (λ ())
+string x ≡? float x₁ = no (λ ())
+string x ≡? char x₁ = no (λ ())
+string x ≡? string x₁ = map′ (cong string) string-injective (x String.≡? x₁)
+string x ≡? name x₁ = no (λ ())
+string x ≡? meta x₁ = no (λ ())
+name x ≡? nat x₁ = no (λ ())
+name x ≡? word64 x₁ = no (λ ())
+name x ≡? float x₁ = no (λ ())
+name x ≡? char x₁ = no (λ ())
+name x ≡? string x₁ = no (λ ())
+name x ≡? name x₁ = map′ (cong name) name-injective (x Name.≡? x₁)
+name x ≡? meta x₁ = no (λ ())
+meta x ≡? nat x₁ = no (λ ())
+meta x ≡? word64 x₁ = no (λ ())
+meta x ≡? float x₁ = no (λ ())
+meta x ≡? char x₁ = no (λ ())
+meta x ≡? string x₁ = no (λ ())
+meta x ≡? name x₁ = no (λ ())
+meta x ≡? meta x₁ = map′ (cong meta) meta-injective (x Meta.≡? x₁)
 
 infix 4 _≡ᵇ_
 
 _≡ᵇ_ : Literal → Literal → Bool
-l ≡ᵇ l′ = isYes (l ≟ l′)
+l ≡ᵇ l′ = isYes (l ≡? l′)
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index bef152149f..4aaa009fa1 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -8,7 +8,7 @@
 
 module Reflection.AST.Meta where
 
-import Data.Nat.Properties as ℕ using (_≟_)
+import Data.Nat.Properties as ℕ using (_≡?_)
 open import Function.Base using (_on_)
 open import Relation.Nullary.Decidable.Core using (map′)
 open import Relation.Binary.Core using (Rel)
@@ -24,13 +24,29 @@ open import Agda.Builtin.Reflection.Properties public
 
 -- Equality of metas is decidable.
 
-infix 4 _≈?_ _≟_ _≈_
+infix 4 _≈?_ _≡?_ _≈_
 
 _≈_ : Rel Meta _
 _≈_ = _≡_ on toℕ
 
 _≈?_ : Decidable _≈_
-_≈?_ = On.decidable toℕ _≡_ ℕ._≟_
+_≈?_ = On.decidable toℕ _≡_ ℕ._≡?_
 
-_≟_ : DecidableEquality Meta
-m ≟ n = map′ (toℕ-injective _ _) (cong toℕ) (m ≈? n)
+_≡?_ : DecidableEquality Meta
+m ≡? n = map′ (toℕ-injective _ _) (cong toℕ) (m ≈? n)
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index 867d14bd6e..055d5df0d1 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Name where
 
 open import Data.List.Base using (List)
 import Data.Product.Properties as Prodₚ using (≡-dec)
-import Data.Word64.Properties as Wₚ using (_≟_)
+import Data.Word64.Properties as Wₚ using (_≡?_)
 open import Function.Base using (_on_)
 open import Relation.Nullary.Decidable.Core using (map′)
 open import Relation.Binary.Core using (Rel)
@@ -38,13 +38,29 @@ Names = List Name
 -- Decidable equality for names
 ------------------------------------------------------------------------
 
-infix 4 _≈?_ _≟_ _≈_
+infix 4 _≈?_ _≡?_ _≈_
 
 _≈_ : Rel Name _
 _≈_ = _≡_ on toWords
 
 _≈?_ : Decidable _≈_
-_≈?_ = decidable toWords _≡_ (Prodₚ.≡-dec Wₚ._≟_ Wₚ._≟_)
+_≈?_ = decidable toWords _≡_ (Prodₚ.≡-dec Wₚ._≡?_ Wₚ._≡?_)
 
-_≟_ : DecidableEquality Name
-m ≟ n = map′ (toWords-injective _ _) (cong toWords) (m ≈? n)
+_≡?_ : DecidableEquality Name
+m ≡? n = map′ (toWords-injective _ _) (cong toWords) (m ≈? n)
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Reflection/AST/Pattern.agda b/src/Reflection/AST/Pattern.agda
index 814baf1038..ea8e41899f 100644
--- a/src/Reflection/AST/Pattern.agda
+++ b/src/Reflection/AST/Pattern.agda
@@ -24,6 +24,6 @@ open import Reflection.AST.Term public
            ; pat-con-injective  to con-injective
            ; pat-var-injective  to var-injective
            ; pat-lit-injective  to lit-injective
-           ; _≟-Patterns_       to _≟s_
-           ; _≟-Pattern_        to _≟_
+           ; _≡?-Patterns_      to _≟s_
+           ; _≡?-Pattern_       to _≡?_
            )
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index 0a1b9e3b53..9a3dc369b5 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -11,21 +11,21 @@ module Reflection.AST.Term where
 open import Data.List.Base as List hiding (_++_)
 open import Data.List.Properties using (∷-dec)
 open import Data.Nat.Base using (ℕ; zero; suc)
-import Data.Nat.Properties as ℕ using (_≟_)
+import Data.Nat.Properties as ℕ using (_≡?_)
 open import Data.Product.Base using (_×_; _,_; <_,_>; uncurry; map₁)
 open import Data.Product.Properties using (,-injective)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.String.Base using (String)
-open import Data.String.Properties as String hiding (_≟_)
+open import Data.String.Properties as String hiding (_≟_; _≡?_)
 open import Relation.Nullary.Decidable using (map′; _×?_; yes; no)
 open import Relation.Binary.Definitions using (Decidable; DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 open import Reflection.AST.Abstraction using (Abs; abs; unAbs-dec)
 open import Reflection.AST.Argument as Arg
 open import Reflection.AST.Argument.Information using (visibility)
-open import Reflection.AST.Argument.Visibility as Visibility hiding (_≟_)
-import Reflection.AST.Literal as Literal using (Literal; _≟_)
-import Reflection.AST.Meta as Meta using (Meta; _≟_)
+open import Reflection.AST.Argument.Visibility as Visibility hiding (_≟_; _≡?_)
+import Reflection.AST.Literal as Literal using (Literal; _≡?_)
+import Reflection.AST.Meta as Meta using (Meta; _≡?_)
 open import Reflection.AST.Name as Name using (Name)
 
 ------------------------------------------------------------------------
@@ -127,60 +127,60 @@ absurd-clause-injective₂ refl = refl
 absurd-clause-injective : ∀ {tel tel′ ps ps′} → absurd-clause tel ps ≡ absurd-clause tel′ ps′ → tel ≡ tel′ × ps ≡ ps′
 absurd-clause-injective = < absurd-clause-injective₁ , absurd-clause-injective₂ >
 
-infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_
-        _≟-Clause_ _≟-Clauses_ _≟_
-        _≟-Sort_ _≟-Pattern_ _≟-Patterns_ _≟-Telescope_
-
-_≟-AbsTerm_  : DecidableEquality (Abs Term)
-_≟-AbsType_  : DecidableEquality (Abs Type)
-_≟-ArgTerm_  : DecidableEquality (Arg Term)
-_≟-ArgType_  : DecidableEquality (Arg Type)
-_≟-Args_     : DecidableEquality (Args Term)
-_≟-Clause_   : DecidableEquality Clause
-_≟-Clauses_  : DecidableEquality Clauses
-_≟_          : DecidableEquality Term
-_≟-Sort_     : DecidableEquality Sort
-_≟-Patterns_ : DecidableEquality (Args Pattern)
-_≟-Pattern_  : DecidableEquality Pattern
+infix 4 _≡?-AbsTerm_ _≡?-AbsType_ _≡?-ArgTerm_ _≡?-ArgType_ _≡?-Args_
+        _≡?-Clause_ _≡?-Clauses_ _≡?_
+        _≡?-Sort_ _≡?-Pattern_ _≡?-Patterns_ _≡?-Telescope_
+
+_≡?-AbsTerm_  : DecidableEquality (Abs Term)
+_≡?-AbsType_  : DecidableEquality (Abs Type)
+_≡?-ArgTerm_  : DecidableEquality (Arg Term)
+_≡?-ArgType_  : DecidableEquality (Arg Type)
+_≡?-Args_     : DecidableEquality (Args Term)
+_≡?-Clause_   : DecidableEquality Clause
+_≡?-Clauses_  : DecidableEquality Clauses
+_≡?_          : DecidableEquality Term
+_≡?-Sort_     : DecidableEquality Sort
+_≡?-Patterns_ : DecidableEquality (Args Pattern)
+_≡?-Pattern_  : DecidableEquality Pattern
 
 -- Decidable equality 'transformers'
 -- We need to inline these because the terms are not sized so
 -- termination would not obvious if we were to use higher-order
 -- functions such as Data.List.Properties' ≡-dec
 
-abs s a ≟-AbsTerm abs s′ a′ = unAbs-dec (a ≟ a′)
-abs s a ≟-AbsType abs s′ a′ = unAbs-dec (a ≟ a′)
-arg i a ≟-ArgTerm arg i′ a′ = unArg-dec (a ≟ a′)
-arg i a ≟-ArgType arg i′ a′ = unArg-dec (a ≟ a′)
-
-[]       ≟-Args []       = yes refl
-(x ∷ xs) ≟-Args (y ∷ ys) = ∷-dec (x ≟-ArgTerm y) (xs ≟-Args ys)
-[]       ≟-Args (_ ∷ _)  = no λ()
-(_ ∷ _)  ≟-Args []       = no λ()
-
-[]       ≟-Clauses []       = yes refl
-(x ∷ xs) ≟-Clauses (y ∷ ys) = ∷-dec (x ≟-Clause y) (xs ≟-Clauses ys)
-[]       ≟-Clauses (_ ∷ _)  = no λ()
-(_ ∷ _)  ≟-Clauses []       = no λ()
-
-_≟-Telescope_ : DecidableEquality Telescope
-[] ≟-Telescope [] = yes refl
-((x , t) ∷ tel) ≟-Telescope ((x′ , t′) ∷ tel′) = ∷-dec
-  (map′ (uncurry (cong₂ _,_)) ,-injective ((x String.≟ x′) ×? (t ≟-ArgTerm t′)))
-  (tel ≟-Telescope tel′)
-[] ≟-Telescope (_ ∷ _) = no λ ()
-(_ ∷ _) ≟-Telescope [] = no λ ()
-
-clause tel ps b ≟-Clause clause tel′ ps′ b′ =
+abs s a ≡?-AbsTerm abs s′ a′ = unAbs-dec (a ≡? a′)
+abs s a ≡?-AbsType abs s′ a′ = unAbs-dec (a ≡? a′)
+arg i a ≡?-ArgTerm arg i′ a′ = unArg-dec (a ≡? a′)
+arg i a ≡?-ArgType arg i′ a′ = unArg-dec (a ≡? a′)
+
+[]       ≡?-Args []       = yes refl
+(x ∷ xs) ≡?-Args (y ∷ ys) = ∷-dec (x ≡?-ArgTerm y) (xs ≡?-Args ys)
+[]       ≡?-Args (_ ∷ _)  = no λ()
+(_ ∷ _)  ≡?-Args []       = no λ()
+
+[]       ≡?-Clauses []       = yes refl
+(x ∷ xs) ≡?-Clauses (y ∷ ys) = ∷-dec (x ≡?-Clause y) (xs ≡?-Clauses ys)
+[]       ≡?-Clauses (_ ∷ _)  = no λ()
+(_ ∷ _)  ≡?-Clauses []       = no λ()
+
+_≡?-Telescope_ : DecidableEquality Telescope
+[] ≡?-Telescope [] = yes refl
+((x , t) ∷ tel) ≡?-Telescope ((x′ , t′) ∷ tel′) = ∷-dec
+  (map′ (uncurry (cong₂ _,_)) ,-injective ((x String.≡? x′) ×? (t ≡?-ArgTerm t′)))
+  (tel ≡?-Telescope tel′)
+[] ≡?-Telescope (_ ∷ _) = no λ ()
+(_ ∷ _) ≡?-Telescope [] = no λ ()
+
+clause tel ps b ≡?-Clause clause tel′ ps′ b′ =
   map′ (λ (tel≡tel′ , ps≡ps′ , b≡b′) → cong₂ (uncurry clause) (cong₂ _,_ tel≡tel′ ps≡ps′) b≡b′)
            clause-injective
-           (tel ≟-Telescope tel′ ×? ps ≟-Patterns ps′ ×? b ≟ b′)
-absurd-clause tel ps ≟-Clause absurd-clause tel′ ps′ =
+           (tel ≡?-Telescope tel′ ×? ps ≡?-Patterns ps′ ×? b ≡? b′)
+absurd-clause tel ps ≡?-Clause absurd-clause tel′ ps′ =
   map′ (uncurry (cong₂ absurd-clause))
            absurd-clause-injective
-           (tel ≟-Telescope tel′ ×? ps ≟-Patterns ps′)
-clause _ _ _      ≟-Clause absurd-clause _ _ = no λ()
-absurd-clause _ _ ≟-Clause clause _ _ _      = no λ()
+           (tel ≡?-Telescope tel′ ×? ps ≡?-Patterns ps′)
+clause _ _ _      ≡?-Clause absurd-clause _ _ = no λ()
+absurd-clause _ _ ≡?-Clause clause _ _ _      = no λ()
 
 var-injective₁ : ∀ {x x′ args args′} → Term.var x args ≡ var x′ args′ → x ≡ x′
 var-injective₁ refl = refl
@@ -266,151 +266,151 @@ propLit-injective refl = refl
 inf-injective : ∀ {x y} → inf x ≡ inf y → x ≡ y
 inf-injective refl = refl
 
-var x args ≟ var x′ args′ =
-  map′ (uncurry (cong₂ var)) var-injective (x ℕ.≟ x′ ×? args ≟-Args args′)
-con c args ≟ con c′ args′ =
-  map′ (uncurry (cong₂ con)) con-injective (c Name.≟ c′ ×? args ≟-Args args′)
-def f args ≟ def f′ args′ =
-  map′ (uncurry (cong₂ def)) def-injective (f Name.≟ f′ ×? args ≟-Args args′)
-meta x args ≟ meta x′ args′ =
-  map′ (uncurry (cong₂ meta)) meta-injective (x Meta.≟ x′   ×? args ≟-Args args′)
-lam v t    ≟ lam v′ t′    =
-  map′ (uncurry (cong₂ lam)) lam-injective (v Visibility.≟ v′ ×? t ≟-AbsTerm t′)
-pat-lam cs args ≟ pat-lam cs′ args′ =
-  map′ (uncurry (cong₂ pat-lam)) pat-lam-injective (cs ≟-Clauses cs′ ×? args ≟-Args args′)
-pi t₁ t₂   ≟ pi t₁′ t₂′   =
-  map′ (uncurry (cong₂ pi))  pi-injective (t₁ ≟-ArgType t₁′  ×? t₂ ≟-AbsType t₂′)
-sort s     ≟ sort s′      = map′ (cong sort)  sort-injective (s ≟-Sort s′)
-lit l      ≟ lit l′       = map′ (cong lit)   lit-injective (l Literal.≟ l′)
-unknown    ≟ unknown      = yes refl
-
-var x args ≟ con c args′ = no λ()
-var x args ≟ def f args′ = no λ()
-var x args ≟ lam v t     = no λ()
-var x args ≟ pi t₁ t₂    = no λ()
-var x args ≟ sort _      = no λ()
-var x args ≟ lit _      = no λ()
-var x args ≟ meta _ _    = no λ()
-var x args ≟ unknown     = no λ()
-con c args ≟ var x args′ = no λ()
-con c args ≟ def f args′ = no λ()
-con c args ≟ lam v t     = no λ()
-con c args ≟ pi t₁ t₂    = no λ()
-con c args ≟ sort _      = no λ()
-con c args ≟ lit _      = no λ()
-con c args ≟ meta _ _    = no λ()
-con c args ≟ unknown     = no λ()
-def f args ≟ var x args′ = no λ()
-def f args ≟ con c args′ = no λ()
-def f args ≟ lam v t     = no λ()
-def f args ≟ pi t₁ t₂    = no λ()
-def f args ≟ sort _      = no λ()
-def f args ≟ lit _      = no λ()
-def f args ≟ meta _ _    = no λ()
-def f args ≟ unknown     = no λ()
-lam v t    ≟ var x args  = no λ()
-lam v t    ≟ con c args  = no λ()
-lam v t    ≟ def f args  = no λ()
-lam v t    ≟ pi t₁ t₂    = no λ()
-lam v t    ≟ sort _      = no λ()
-lam v t    ≟ lit _      = no λ()
-lam v t    ≟ meta _ _    = no λ()
-lam v t    ≟ unknown     = no λ()
-pi t₁ t₂   ≟ var x args  = no λ()
-pi t₁ t₂   ≟ con c args  = no λ()
-pi t₁ t₂   ≟ def f args  = no λ()
-pi t₁ t₂   ≟ lam v t     = no λ()
-pi t₁ t₂   ≟ sort _      = no λ()
-pi t₁ t₂   ≟ lit _      = no λ()
-pi t₁ t₂   ≟ meta _ _    = no λ()
-pi t₁ t₂   ≟ unknown     = no λ()
-sort _     ≟ var x args  = no λ()
-sort _     ≟ con c args  = no λ()
-sort _     ≟ def f args  = no λ()
-sort _     ≟ lam v t     = no λ()
-sort _     ≟ pi t₁ t₂    = no λ()
-sort _     ≟ lit _       = no λ()
-sort _     ≟ meta _ _    = no λ()
-sort _     ≟ unknown     = no λ()
-lit _     ≟ var x args  = no λ()
-lit _     ≟ con c args  = no λ()
-lit _     ≟ def f args  = no λ()
-lit _     ≟ lam v t     = no λ()
-lit _     ≟ pi t₁ t₂    = no λ()
-lit _     ≟ sort _      = no λ()
-lit _     ≟ meta _ _    = no λ()
-lit _     ≟ unknown     = no λ()
-meta _ _   ≟ var x args  = no λ()
-meta _ _   ≟ con c args  = no λ()
-meta _ _   ≟ def f args  = no λ()
-meta _ _   ≟ lam v t     = no λ()
-meta _ _   ≟ pi t₁ t₂    = no λ()
-meta _ _   ≟ sort _      = no λ()
-meta _ _   ≟ lit _       = no λ()
-meta _ _   ≟ unknown     = no λ()
-unknown    ≟ var x args  = no λ()
-unknown    ≟ con c args  = no λ()
-unknown    ≟ def f args  = no λ()
-unknown    ≟ lam v t     = no λ()
-unknown    ≟ pi t₁ t₂    = no λ()
-unknown    ≟ sort _      = no λ()
-unknown    ≟ lit _       = no λ()
-unknown    ≟ meta _ _    = no λ()
-pat-lam _ _ ≟ var x args  = no λ()
-pat-lam _ _ ≟ con c args  = no λ()
-pat-lam _ _ ≟ def f args  = no λ()
-pat-lam _ _ ≟ lam v t     = no λ()
-pat-lam _ _ ≟ pi t₁ t₂    = no λ()
-pat-lam _ _ ≟ sort _      = no λ()
-pat-lam _ _ ≟ lit _       = no λ()
-pat-lam _ _ ≟ meta _ _    = no λ()
-pat-lam _ _ ≟ unknown     = no λ()
-var x args  ≟ pat-lam _ _ = no λ()
-con c args  ≟ pat-lam _ _ = no λ()
-def f args  ≟ pat-lam _ _ = no λ()
-lam v t     ≟ pat-lam _ _ = no λ()
-pi t₁ t₂    ≟ pat-lam _ _ = no λ()
-sort _      ≟ pat-lam _ _ = no λ()
-lit _       ≟ pat-lam _ _ = no λ()
-meta _ _    ≟ pat-lam _ _ = no λ()
-unknown     ≟ pat-lam _ _ = no λ()
-
-set t   ≟-Sort set t′  = map′ (cong set) set-injective (t ≟ t′)
-lit n   ≟-Sort lit n′  = map′ (cong lit) slit-injective (n ℕ.≟ n′)
-prop t   ≟-Sort prop t′  = map′ (cong prop) prop-injective (t ≟ t′)
-propLit n   ≟-Sort propLit n′  = map′ (cong propLit) propLit-injective (n ℕ.≟ n′)
-inf n   ≟-Sort inf n′  = map′ (cong inf) inf-injective (n ℕ.≟ n′)
-unknown ≟-Sort unknown = yes refl
-set _   ≟-Sort lit _   = no λ()
-set _   ≟-Sort prop _   = no λ()
-set _   ≟-Sort propLit _   = no λ()
-set _   ≟-Sort inf _   = no λ()
-set _   ≟-Sort unknown = no λ()
-lit _   ≟-Sort set _   = no λ()
-lit _   ≟-Sort prop _   = no λ()
-lit _   ≟-Sort propLit _   = no λ()
-lit _   ≟-Sort inf _   = no λ()
-lit _   ≟-Sort unknown = no λ()
-prop _  ≟-Sort set _   = no λ()
-prop _  ≟-Sort lit _   = no λ()
-prop _  ≟-Sort propLit _   = no λ()
-prop _  ≟-Sort inf _   = no λ()
-prop _  ≟-Sort unknown   = no λ()
-propLit _  ≟-Sort set _   = no λ()
-propLit _  ≟-Sort lit _   = no λ()
-propLit _  ≟-Sort prop _   = no λ()
-propLit _  ≟-Sort inf _   = no λ()
-propLit _  ≟-Sort unknown   = no λ()
-inf _  ≟-Sort set _   = no λ()
-inf _  ≟-Sort lit _   = no λ()
-inf _  ≟-Sort prop _   = no λ()
-inf _  ≟-Sort propLit _   = no λ()
-inf _  ≟-Sort unknown   = no λ()
-unknown ≟-Sort set _   = no λ()
-unknown ≟-Sort lit _   = no λ()
-unknown ≟-Sort prop _   = no λ()
-unknown ≟-Sort propLit _   = no λ()
-unknown ≟-Sort inf _   = no λ()
+var x args ≡? var x′ args′ =
+  map′ (uncurry (cong₂ var)) var-injective (x ℕ.≡? x′ ×? args ≡?-Args args′)
+con c args ≡? con c′ args′ =
+  map′ (uncurry (cong₂ con)) con-injective (c Name.≡? c′ ×? args ≡?-Args args′)
+def f args ≡? def f′ args′ =
+  map′ (uncurry (cong₂ def)) def-injective (f Name.≡? f′ ×? args ≡?-Args args′)
+meta x args ≡? meta x′ args′ =
+  map′ (uncurry (cong₂ meta)) meta-injective (x Meta.≡? x′   ×? args ≡?-Args args′)
+lam v t    ≡? lam v′ t′    =
+  map′ (uncurry (cong₂ lam)) lam-injective (v Visibility.≡? v′ ×? t ≡?-AbsTerm t′)
+pat-lam cs args ≡? pat-lam cs′ args′ =
+  map′ (uncurry (cong₂ pat-lam)) pat-lam-injective (cs ≡?-Clauses cs′ ×? args ≡?-Args args′)
+pi t₁ t₂   ≡? pi t₁′ t₂′   =
+  map′ (uncurry (cong₂ pi))  pi-injective (t₁ ≡?-ArgType t₁′  ×? t₂ ≡?-AbsType t₂′)
+sort s     ≡? sort s′      = map′ (cong sort)  sort-injective (s ≡?-Sort s′)
+lit l      ≡? lit l′       = map′ (cong lit)   lit-injective (l Literal.≡? l′)
+unknown    ≡? unknown      = yes refl
+
+var x args ≡? con c args′ = no λ()
+var x args ≡? def f args′ = no λ()
+var x args ≡? lam v t     = no λ()
+var x args ≡? pi t₁ t₂    = no λ()
+var x args ≡? sort _      = no λ()
+var x args ≡? lit _      = no λ()
+var x args ≡? meta _ _    = no λ()
+var x args ≡? unknown     = no λ()
+con c args ≡? var x args′ = no λ()
+con c args ≡? def f args′ = no λ()
+con c args ≡? lam v t     = no λ()
+con c args ≡? pi t₁ t₂    = no λ()
+con c args ≡? sort _      = no λ()
+con c args ≡? lit _      = no λ()
+con c args ≡? meta _ _    = no λ()
+con c args ≡? unknown     = no λ()
+def f args ≡? var x args′ = no λ()
+def f args ≡? con c args′ = no λ()
+def f args ≡? lam v t     = no λ()
+def f args ≡? pi t₁ t₂    = no λ()
+def f args ≡? sort _      = no λ()
+def f args ≡? lit _      = no λ()
+def f args ≡? meta _ _    = no λ()
+def f args ≡? unknown     = no λ()
+lam v t    ≡? var x args  = no λ()
+lam v t    ≡? con c args  = no λ()
+lam v t    ≡? def f args  = no λ()
+lam v t    ≡? pi t₁ t₂    = no λ()
+lam v t    ≡? sort _      = no λ()
+lam v t    ≡? lit _      = no λ()
+lam v t    ≡? meta _ _    = no λ()
+lam v t    ≡? unknown     = no λ()
+pi t₁ t₂   ≡? var x args  = no λ()
+pi t₁ t₂   ≡? con c args  = no λ()
+pi t₁ t₂   ≡? def f args  = no λ()
+pi t₁ t₂   ≡? lam v t     = no λ()
+pi t₁ t₂   ≡? sort _      = no λ()
+pi t₁ t₂   ≡? lit _      = no λ()
+pi t₁ t₂   ≡? meta _ _    = no λ()
+pi t₁ t₂   ≡? unknown     = no λ()
+sort _     ≡? var x args  = no λ()
+sort _     ≡? con c args  = no λ()
+sort _     ≡? def f args  = no λ()
+sort _     ≡? lam v t     = no λ()
+sort _     ≡? pi t₁ t₂    = no λ()
+sort _     ≡? lit _       = no λ()
+sort _     ≡? meta _ _    = no λ()
+sort _     ≡? unknown     = no λ()
+lit _     ≡? var x args  = no λ()
+lit _     ≡? con c args  = no λ()
+lit _     ≡? def f args  = no λ()
+lit _     ≡? lam v t     = no λ()
+lit _     ≡? pi t₁ t₂    = no λ()
+lit _     ≡? sort _      = no λ()
+lit _     ≡? meta _ _    = no λ()
+lit _     ≡? unknown     = no λ()
+meta _ _   ≡? var x args  = no λ()
+meta _ _   ≡? con c args  = no λ()
+meta _ _   ≡? def f args  = no λ()
+meta _ _   ≡? lam v t     = no λ()
+meta _ _   ≡? pi t₁ t₂    = no λ()
+meta _ _   ≡? sort _      = no λ()
+meta _ _   ≡? lit _       = no λ()
+meta _ _   ≡? unknown     = no λ()
+unknown    ≡? var x args  = no λ()
+unknown    ≡? con c args  = no λ()
+unknown    ≡? def f args  = no λ()
+unknown    ≡? lam v t     = no λ()
+unknown    ≡? pi t₁ t₂    = no λ()
+unknown    ≡? sort _      = no λ()
+unknown    ≡? lit _       = no λ()
+unknown    ≡? meta _ _    = no λ()
+pat-lam _ _ ≡? var x args  = no λ()
+pat-lam _ _ ≡? con c args  = no λ()
+pat-lam _ _ ≡? def f args  = no λ()
+pat-lam _ _ ≡? lam v t     = no λ()
+pat-lam _ _ ≡? pi t₁ t₂    = no λ()
+pat-lam _ _ ≡? sort _      = no λ()
+pat-lam _ _ ≡? lit _       = no λ()
+pat-lam _ _ ≡? meta _ _    = no λ()
+pat-lam _ _ ≡? unknown     = no λ()
+var x args  ≡? pat-lam _ _ = no λ()
+con c args  ≡? pat-lam _ _ = no λ()
+def f args  ≡? pat-lam _ _ = no λ()
+lam v t     ≡? pat-lam _ _ = no λ()
+pi t₁ t₂    ≡? pat-lam _ _ = no λ()
+sort _      ≡? pat-lam _ _ = no λ()
+lit _       ≡? pat-lam _ _ = no λ()
+meta _ _    ≡? pat-lam _ _ = no λ()
+unknown     ≡? pat-lam _ _ = no λ()
+
+set t   ≡?-Sort set t′  = map′ (cong set) set-injective (t ≡? t′)
+lit n   ≡?-Sort lit n′  = map′ (cong lit) slit-injective (n ℕ.≡? n′)
+prop t   ≡?-Sort prop t′  = map′ (cong prop) prop-injective (t ≡? t′)
+propLit n   ≡?-Sort propLit n′  = map′ (cong propLit) propLit-injective (n ℕ.≡? n′)
+inf n   ≡?-Sort inf n′  = map′ (cong inf) inf-injective (n ℕ.≡? n′)
+unknown ≡?-Sort unknown = yes refl
+set _   ≡?-Sort lit _   = no λ()
+set _   ≡?-Sort prop _   = no λ()
+set _   ≡?-Sort propLit _   = no λ()
+set _   ≡?-Sort inf _   = no λ()
+set _   ≡?-Sort unknown = no λ()
+lit _   ≡?-Sort set _   = no λ()
+lit _   ≡?-Sort prop _   = no λ()
+lit _   ≡?-Sort propLit _   = no λ()
+lit _   ≡?-Sort inf _   = no λ()
+lit _   ≡?-Sort unknown = no λ()
+prop _  ≡?-Sort set _   = no λ()
+prop _  ≡?-Sort lit _   = no λ()
+prop _  ≡?-Sort propLit _   = no λ()
+prop _  ≡?-Sort inf _   = no λ()
+prop _  ≡?-Sort unknown   = no λ()
+propLit _  ≡?-Sort set _   = no λ()
+propLit _  ≡?-Sort lit _   = no λ()
+propLit _  ≡?-Sort prop _   = no λ()
+propLit _  ≡?-Sort inf _   = no λ()
+propLit _  ≡?-Sort unknown   = no λ()
+inf _  ≡?-Sort set _   = no λ()
+inf _  ≡?-Sort lit _   = no λ()
+inf _  ≡?-Sort prop _   = no λ()
+inf _  ≡?-Sort propLit _   = no λ()
+inf _  ≡?-Sort unknown   = no λ()
+unknown ≡?-Sort set _   = no λ()
+unknown ≡?-Sort lit _   = no λ()
+unknown ≡?-Sort prop _   = no λ()
+unknown ≡?-Sort propLit _   = no λ()
+unknown ≡?-Sort inf _   = no λ()
 
 
 pat-con-injective₁ : ∀ {c c′ args args′} → Pattern.con c args ≡ con c′ args′ → c ≡ c′
@@ -437,46 +437,120 @@ dot-injective refl = refl
 absurd-injective : ∀ {x y} → absurd x ≡ absurd y → x ≡ y
 absurd-injective refl = refl
 
-con c ps ≟-Pattern con c′ ps′ = map′ (uncurry (cong₂ con)) pat-con-injective (c Name.≟ c′ ×? ps ≟-Patterns ps′)
-var x    ≟-Pattern var x′     = map′ (cong var) pat-var-injective (x ℕ.≟ x′)
-lit l    ≟-Pattern lit l′     = map′ (cong lit) pat-lit-injective (l Literal.≟ l′)
-proj a   ≟-Pattern proj a′    = map′ (cong proj) proj-injective (a Name.≟ a′)
-dot t    ≟-Pattern dot t′     = map′ (cong dot) dot-injective (t ≟ t′)
-absurd x ≟-Pattern absurd x′  = map′ (cong absurd) absurd-injective (x ℕ.≟ x′)
-
-con x x₁ ≟-Pattern dot x₂ = no (λ ())
-con x x₁ ≟-Pattern var x₂ = no (λ ())
-con x x₁ ≟-Pattern lit x₂ = no (λ ())
-con x x₁ ≟-Pattern proj x₂ = no (λ ())
-con x x₁ ≟-Pattern absurd x₂ = no (λ ())
-dot x ≟-Pattern con x₁ x₂ = no (λ ())
-dot x ≟-Pattern var x₁ = no (λ ())
-dot x ≟-Pattern lit x₁ = no (λ ())
-dot x ≟-Pattern proj x₁ = no (λ ())
-dot x ≟-Pattern absurd x₁ = no (λ ())
-var s ≟-Pattern con x x₁ = no (λ ())
-var s ≟-Pattern dot x = no (λ ())
-var s ≟-Pattern lit x = no (λ ())
-var s ≟-Pattern proj x = no (λ ())
-var s ≟-Pattern absurd x = no (λ ())
-lit x ≟-Pattern con x₁ x₂ = no (λ ())
-lit x ≟-Pattern dot x₁ = no (λ ())
-lit x ≟-Pattern var _ = no (λ ())
-lit x ≟-Pattern proj x₁ = no (λ ())
-lit x ≟-Pattern absurd x₁ = no (λ ())
-proj x ≟-Pattern con x₁ x₂ = no (λ ())
-proj x ≟-Pattern dot x₁ = no (λ ())
-proj x ≟-Pattern var _ = no (λ ())
-proj x ≟-Pattern lit x₁ = no (λ ())
-proj x ≟-Pattern absurd x₁ = no (λ ())
-absurd x ≟-Pattern con x₁ x₂ = no (λ ())
-absurd x ≟-Pattern dot x₁ = no (λ ())
-absurd x ≟-Pattern var _ = no (λ ())
-absurd x ≟-Pattern lit x₁ = no (λ ())
-absurd x ≟-Pattern proj x₁ = no (λ ())
-
-[]             ≟-Patterns []             = yes refl
-(arg i p ∷ xs) ≟-Patterns (arg j q ∷ ys) = ∷-dec (unArg-dec (p ≟-Pattern q)) (xs ≟-Patterns ys)
-
-[]      ≟-Patterns (_ ∷ _) = no λ()
-(_ ∷ _) ≟-Patterns []      = no λ()
+con c ps ≡?-Pattern con c′ ps′ = map′ (uncurry (cong₂ con)) pat-con-injective (c Name.≡? c′ ×? ps ≡?-Patterns ps′)
+var x    ≡?-Pattern var x′     = map′ (cong var) pat-var-injective (x ℕ.≡? x′)
+lit l    ≡?-Pattern lit l′     = map′ (cong lit) pat-lit-injective (l Literal.≡? l′)
+proj a   ≡?-Pattern proj a′    = map′ (cong proj) proj-injective (a Name.≡? a′)
+dot t    ≡?-Pattern dot t′     = map′ (cong dot) dot-injective (t ≡? t′)
+absurd x ≡?-Pattern absurd x′  = map′ (cong absurd) absurd-injective (x ℕ.≡? x′)
+
+con x x₁ ≡?-Pattern dot x₂ = no (λ ())
+con x x₁ ≡?-Pattern var x₂ = no (λ ())
+con x x₁ ≡?-Pattern lit x₂ = no (λ ())
+con x x₁ ≡?-Pattern proj x₂ = no (λ ())
+con x x₁ ≡?-Pattern absurd x₂ = no (λ ())
+dot x ≡?-Pattern con x₁ x₂ = no (λ ())
+dot x ≡?-Pattern var x₁ = no (λ ())
+dot x ≡?-Pattern lit x₁ = no (λ ())
+dot x ≡?-Pattern proj x₁ = no (λ ())
+dot x ≡?-Pattern absurd x₁ = no (λ ())
+var s ≡?-Pattern con x x₁ = no (λ ())
+var s ≡?-Pattern dot x = no (λ ())
+var s ≡?-Pattern lit x = no (λ ())
+var s ≡?-Pattern proj x = no (λ ())
+var s ≡?-Pattern absurd x = no (λ ())
+lit x ≡?-Pattern con x₁ x₂ = no (λ ())
+lit x ≡?-Pattern dot x₁ = no (λ ())
+lit x ≡?-Pattern var _ = no (λ ())
+lit x ≡?-Pattern proj x₁ = no (λ ())
+lit x ≡?-Pattern absurd x₁ = no (λ ())
+proj x ≡?-Pattern con x₁ x₂ = no (λ ())
+proj x ≡?-Pattern dot x₁ = no (λ ())
+proj x ≡?-Pattern var _ = no (λ ())
+proj x ≡?-Pattern lit x₁ = no (λ ())
+proj x ≡?-Pattern absurd x₁ = no (λ ())
+absurd x ≡?-Pattern con x₁ x₂ = no (λ ())
+absurd x ≡?-Pattern dot x₁ = no (λ ())
+absurd x ≡?-Pattern var _ = no (λ ())
+absurd x ≡?-Pattern lit x₁ = no (λ ())
+absurd x ≡?-Pattern proj x₁ = no (λ ())
+
+[]             ≡?-Patterns []             = yes refl
+(arg i p ∷ xs) ≡?-Patterns (arg j q ∷ ys) = ∷-dec (unArg-dec (p ≡?-Pattern q)) (xs ≡?-Patterns ys)
+
+[]      ≡?-Patterns (_ ∷ _) = no λ()
+(_ ∷ _) ≡?-Patterns []      = no λ()
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_
+        _≟-Clause_ _≟-Clauses_ _≟_
+        _≟-Sort_ _≟-Pattern_ _≟-Patterns_ _≟-Telescope_
+
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
+_≟-AbsTerm_ = _≡?-AbsTerm_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟-AbsTerm_ was deprecated in v2.4.
+Please use _≡?-AbsTerm_ instead."
+#-}
+_≟-AbsType_ = _≡?-AbsType_
+{-# WARNING_ON_USAGE _≟-AbsType_
+"Warning: _≟-AbsType_ was deprecated in v2.4.
+Please use _≡?-AbsType_ instead."
+#-}
+_≟-ArgTerm_ = _≡?-ArgTerm_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟-ArgTerm_ was deprecated in v2.4.
+Please use _≡?-ArgTerm_ instead."
+#-}
+_≟-ArgType_ = _≡?-ArgType_
+{-# WARNING_ON_USAGE _≟-ArgType_
+"Warning: _≟-ArgType_ was deprecated in v2.4.
+Please use _≡?-ArgType_ instead."
+#-}
+_≟-Args_ = _≡?-Args_
+{-# WARNING_ON_USAGE _≟-Args_
+"Warning: _≟-Args_ was deprecated in v2.4.
+Please use _≡?-Args_ instead."
+#-}
+_≟-Clause_ = _≡?-Clause_
+{-# WARNING_ON_USAGE _≟-Clause_
+"Warning: _≟-Clause_ was deprecated in v2.4.
+Please use _≡?-Clause_ instead."
+#-}
+_≟-Clauses_ = _≡?-Clauses_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟-Clauses_ was deprecated in v2.4.
+Please use _≡?-Clauses_ instead."
+#-}
+_≟-Sort_ = _≡?-Sort_
+{-# WARNING_ON_USAGE _≟-Sort_
+"Warning: _≟-Sort_ was deprecated in v2.4.
+Please use _≡?-Sort_ instead."
+#-}
+_≟-Pattern_ = _≡?-Pattern_
+{-# WARNING_ON_USAGE _≟-Pattern_
+"Warning: _≟-Pattern_ was deprecated in v2.4.
+Please use _≡?-Pattern_ instead."
+#-}
+_≟-Patterns_ = _≡?-Patterns_
+{-# WARNING_ON_USAGE _≟-Patterns_
+"Warning: _≟-Patterns_ was deprecated in v2.4.
+Please use _≡?-Patterns_ instead."
+#-}
+_≟-Telescope_ = _≡?-Telescope_
+{-# WARNING_ON_USAGE _≟-Telescope_
+"Warning: _≟-Telescope_ was deprecated in v2.4.
+Please use _≡?-Telescope_ instead."
+#-}
diff --git a/src/Relation/Binary/Construct/Flip/Ord.agda b/src/Relation/Binary/Construct/Flip/Ord.agda
index 3a2a5f3524..8b799779a5 100644
--- a/src/Relation/Binary/Construct/Flip/Ord.agda
+++ b/src/Relation/Binary/Construct/Flip/Ord.agda
@@ -101,7 +101,7 @@ isEquivalence {≈ = ≈} eq = record
 isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)
 isDecEquivalence {≈ = ≈} dec = record
   { isEquivalence = isEquivalence Dec.isEquivalence
-  ; _≟_           = decidable ≈ Dec._≟_
+  ; _≈?_          = decidable ≈ Dec._≈?_
   } where module Dec = IsDecEquivalence dec
 
 isPreorder : IsPreorder ≈ ∼ → IsPreorder (flip ≈) (flip ∼)
@@ -127,7 +127,7 @@ isTotalOrder {≈ = ≈} {≤ = ≤} O = record
 isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder (flip ≈) (flip ≤)
 isDecTotalOrder {≈ = ≈} {≤ = ≤} O = record
   { isTotalOrder = isTotalOrder O.isTotalOrder
-  ; _≟_          = decidable ≈ O._≟_
+  ; _≈?_         = decidable ≈ O._≈?_
   ; _≤?_         = decidable ≤ O._≤?_
   } where module O = IsDecTotalOrder O
 
diff --git a/src/Relation/Binary/Construct/Interior/Symmetric.agda b/src/Relation/Binary/Construct/Interior/Symmetric.agda
index 4f4eb561a7..e061a893d4 100644
--- a/src/Relation/Binary/Construct/Interior/Symmetric.agda
+++ b/src/Relation/Binary/Construct/Interior/Symmetric.agda
@@ -111,7 +111,7 @@ module _ {R : Rel A ℓ} (refl : Reflexive R) (trans : Transitive R) where
     isDecPartialOrder : IsDecPartialOrder (SymInterior R) R
     isDecPartialOrder = record
       { isPartialOrder = isPartialOrder
-      ; _≟_ = decidable R?
+      ; _≈?_ = decidable R?
       ; _≤?_ = R?
       }
 
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index d974768350..3623097128 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -113,10 +113,10 @@ isPartialOrder semilattice = record
 
 isDecPartialOrder : IsSemilattice _∙_ → Decidable _≈_ →
                     IsDecPartialOrder _≈_ _≤_
-isDecPartialOrder semilattice _≟_ = record
+isDecPartialOrder semilattice _≈?_ = record
   { isPartialOrder = isPartialOrder semilattice
-  ; _≟_            = _≟_
-  ; _≤?_           = dec _≟_
+  ; _≈?_           = _≈?_
+  ; _≤?_           = dec _≈?_
   }
 
 isTotalOrder : IsSemilattice _∙_ → Selective _∙_ → IsTotalOrder _≈_ _≤_
@@ -128,10 +128,10 @@ isTotalOrder latt sel  = record
 
 isDecTotalOrder : IsSemilattice _∙_ → Selective _∙_ →
                   Decidable _≈_ → IsDecTotalOrder _≈_ _≤_
-isDecTotalOrder latt sel _≟_ = record
+isDecTotalOrder latt sel _≈?_ = record
   { isTotalOrder = isTotalOrder latt sel
-  ; _≟_          = _≟_
-  ; _≤?_         = dec _≟_
+  ; _≈?_         = _≈?_
+  ; _≤?_         = dec _≈?_
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Construct/StrictToNonStrict.agda b/src/Relation/Binary/Construct/StrictToNonStrict.agda
index c103cc7bab..8e94efe242 100644
--- a/src/Relation/Binary/Construct/StrictToNonStrict.agda
+++ b/src/Relation/Binary/Construct/StrictToNonStrict.agda
@@ -145,7 +145,7 @@ isTotalOrder STO = record
 isDecTotalOrder : IsStrictTotalOrder _≈_ _<_ → IsDecTotalOrder _≈_ _≤_
 isDecTotalOrder STO = record
   { isTotalOrder = isTotalOrder STO
-  ; _≟_          = S._≟_
+  ; _≈?_         = S._≈?_
   ; _≤?_         = decidable′ S.compare
   }
   where module S = IsStrictTotalOrder STO
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 6761bea53a..1be883007c 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -198,7 +198,7 @@ decSetoid dec = record
   { _≈_              = λ x y → x ≅ y
   ; isDecEquivalence = record
       { isEquivalence = isEquivalence
-      ; _≟_           = dec
+      ; _≈?_           = dec
       }
   }
 
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda b/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
index b3ad700105..39d2bf3d0d 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
@@ -37,7 +37,7 @@ isDecEquivalence : ∀ {_≈_ : IRel Aᵢ ℓ} → IsIndexedDecEquivalence Aᵢ
                    (index : I) → IsDecEquivalence (_≈_ {index})
 isDecEquivalence isEq index = record
   { isEquivalence = isEquivalence E.isEquivalenceᵢ index
-  ; _≟_           = E._≟ᵢ_
+  ; _≈?_           = E._≈ᵢ?_
   } where module E = IsIndexedDecEquivalence isEq
 
 isPreorder : ∀ {_≈_ : IRel Aᵢ ℓ₁} {_∼_ : IRel Aᵢ ℓ₂} →
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
index 48265dbef5..575f681d4b 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
@@ -69,13 +69,19 @@ record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where
 
 
 record IsIndexedDecEquivalence : Set (i ⊔ a ⊔ ℓ) where
-  infix 4 _≟ᵢ_
+  infix 4 _≟ᵢ_ _≈ᵢ?_
   field
-    _≟ᵢ_           : Decidable A _≈ᵢ_
+    _≈ᵢ?_          : Decidable A _≈ᵢ_
     isEquivalenceᵢ : IsIndexedEquivalence
 
   open IsIndexedEquivalence isEquivalenceᵢ public
 
+  _≟ᵢ_ = _≈ᵢ?_
+  {-# WARNING_ON_USAGE _≟ᵢ_
+  "Warning: _≟ᵢ_ was deprecated in v2.4.
+  Please use _≈ᵢ?_ instead."
+  #-}
+
 
 ------------------------------------------------------------------------
 -- Preorders
diff --git a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
index 90ea114cb5..493b1d9896 100644
--- a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
@@ -119,8 +119,8 @@ dualMeetSemilattice = record
 ... | no  x∨y≉y = no (contraposition x≤y⇒x∨y≈y x∨y≉y)
 
 ≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_
-≈-dec⇒isDecPartialOrder _≟_ = record
+≈-dec⇒isDecPartialOrder _≈?_ = record
   { isPartialOrder = isPartialOrder
-  ; _≟_            = _≟_
-  ; _≤?_           = ≈-dec⇒≤-dec _≟_
+  ; _≈?_           = _≈?_
+  ; _≤?_           = ≈-dec⇒≤-dec _≈?_
   }
diff --git a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
index e5b77d4092..875412ebf9 100644
--- a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
@@ -49,11 +49,11 @@ open J dualJoinSemilattice public
 -- If ≈ is decidable then so is ≤
 
 ≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_
-≈-dec⇒≤-dec _≟_ = flip (≈-dec⇒≥-dec _≟_)
+≈-dec⇒≤-dec _≈?_ = flip (≈-dec⇒≥-dec _≈?_)
 
 ≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_
-≈-dec⇒isDecPartialOrder _≟_ = record
+≈-dec⇒isDecPartialOrder _≈?_ = record
   { isPartialOrder = isPartialOrder
-  ; _≟_            = _≟_
-  ; _≤?_           = ≈-dec⇒≤-dec _≟_
+  ; _≈?_           = _≈?_
+  ; _≤?_           = ≈-dec⇒≤-dec _≈?_
   }
diff --git a/src/Relation/Binary/Morphism/OrderMonomorphism.agda b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
index faa81dd372..82116420a8 100644
--- a/src/Relation/Binary/Morphism/OrderMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
@@ -92,7 +92,7 @@ isTotalOrder O = record
 isDecTotalOrder : IsDecTotalOrder _≈₂_ _≲₂_ → IsDecTotalOrder _≈₁_ _≲₁_
 isDecTotalOrder O = record
   { isTotalOrder = isTotalOrder O.isTotalOrder
-  ; _≟_          = EqM.dec O._≟_
+  ; _≈?_         = EqM.dec O._≈?_
   ; _≤?_         = dec O._≤?_
   } where module O = IsDecTotalOrder O
 
diff --git a/src/Relation/Binary/Morphism/RelMonomorphism.agda b/src/Relation/Binary/Morphism/RelMonomorphism.agda
index 693e6ebf80..1b4bbe3e36 100644
--- a/src/Relation/Binary/Morphism/RelMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/RelMonomorphism.agda
@@ -61,5 +61,5 @@ isEquivalence isEq = record
 isDecEquivalence : IsDecEquivalence _∼₂_ → IsDecEquivalence _∼₁_
 isDecEquivalence isDecEq = record
   { isEquivalence  = isEquivalence E.isEquivalence
-  ; _≟_            = dec E._≟_
+  ; _≈?_           = dec E._≈?_
   } where module E = IsDecEquivalence isDecEq
diff --git a/src/Relation/Binary/TypeClasses.agda b/src/Relation/Binary/TypeClasses.agda
index 48f0271f6a..a958d6f915 100644
--- a/src/Relation/Binary/TypeClasses.agda
+++ b/src/Relation/Binary/TypeClasses.agda
@@ -10,5 +10,5 @@ module Relation.Binary.TypeClasses where
 
 open import Relation.Binary.Structures using (IsDecEquivalence; IsDecTotalOrder) public
 
-open IsDecEquivalence {{...}} using (_≟_) public
+open IsDecEquivalence {{...}} using (_≈?_) public
 open IsDecTotalOrder {{...}} using (_≤?_) public
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index ceffe6587d..d7e6a14571 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -108,7 +108,7 @@ private
     term' ← term
     symTerm' ← symTerm
     -- Don't show the same term twice.
-    let symErr = case term' Term.≟ symTerm' of λ where
+    let symErr = case term' Term.≡? symTerm' of λ where
       (yes _) → []
       (no _) → strErr "\n" ∷ termErr symTerm' ∷ []
     typeError (strErr "cong! failed, tried:\n" ∷ termErr term' ∷ symErr)
diff --git a/src/Test/Golden.agda b/src/Test/Golden.agda
index cc3198269c..1533f09752 100644
--- a/src/Test/Golden.agda
+++ b/src/Test/Golden.agda
@@ -92,7 +92,7 @@ open import Data.Nat.ListAction using (sum)
 import Data.Nat.Show as ℕ using (show)
 open import Data.Product.Base using (_×_; _,_)
 open import Data.String.Base as String using (String; lines; unlines; unwords; concat)
-open import Data.String.Properties as String using (_≟_)
+open import Data.String.Properties as String using (_≡?_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Unit.Base using (⊤)
 
@@ -183,7 +183,7 @@ options(exe ∷ rest) = mkOptions exe rest where
     inj₂ (mfp , opts) ← pure $ go rest nothing (initOptions exe)
       where inj₁ arg → pure (inj₁ (InvalidArgument arg))
     term ← fromMaybe "" <$> lookupEnv "TERM"
-    let opts = if does (term ≟ "DUMB")
+    let opts = if does (term ≡? "DUMB")
                then record opts { colour = false }
                else opts
     just fp ← pure mfp
@@ -221,7 +221,7 @@ runTest opts testPath = do
                          else putStrLn (fileNotFound "expected")
                        pure (inj₁ testPath)
 
-  let result = does (out String.≟ exp)
+  let result = does (out String.≡? exp)
 
   if result
     then printTiming (opts .timing) time
@@ -354,7 +354,7 @@ filterTests : Options → List String → List String
 filterTests opts = case onlyNames opts of λ where
   [] → id
   xs → let names = List.map String.toList xs in
-       filter (λ n → any? (λ m → infix? Char._≟_ m (String.toList n)) names)
+       filter (λ n → any? (λ m → infix? Char._≡?_ m (String.toList n)) names)
 
 poolRunner : Options → TestPool → IO Summary
 poolRunner opts pool = do
diff --git a/src/Text/Regex/Properties.agda b/src/Text/Regex/Properties.agda
index b384207d7f..ce76e26375 100644
--- a/src/Text/Regex/Properties.agda
+++ b/src/Text/Regex/Properties.agda
@@ -52,7 +52,7 @@ open import Text.Regex.Properties.Core preorder public
 infix 4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]
 
 _∈ᴿ?_ : Decidable _∈ᴿ_
-c ∈ᴿ? [ a ]     = map′ [_] (λ where [ eq ] → eq) (c ≟ a)
+c ∈ᴿ? [ a ]     = map′ [_] (λ where [ eq ] → eq) (c ≈? a)
 c ∈ᴿ? (lb ─ ub) = map′ (uncurry _─_) (λ where (ge ─ le) → ge , le)
                 $ (lb ≤? c) ×? (c ≤? ub)
 

From f7dee08b1b142bb1acf0841d00630f840b72403b Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 12:18:54 +0000
Subject: [PATCH 05/20] fix: whitespace

---
 src/Algebra/Solver/Monoid/Normal.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Algebra/Solver/Monoid/Normal.agda b/src/Algebra/Solver/Monoid/Normal.agda
index 2195eeafac..4554873b6e 100644
--- a/src/Algebra/Solver/Monoid/Normal.agda
+++ b/src/Algebra/Solver/Monoid/Normal.agda
@@ -99,7 +99,7 @@ correct (e₁ ⊕ e₂) ρ = begin
 
 infix 4 _≟_
 
-_≟_ = _≡?_ 
+_≟_ = _≡?_
 {-# WARNING_ON_USAGE _≟_
 "Warning: _≟_ was deprecated in v2.4.
 Please use _≡?_ instead."

From 845628712b7d38f6ff604614640b0e1ba41e0cd3 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 12:49:01 +0000
Subject: [PATCH 06/20] add: `style-guide` entry

---
 doc/style-guide.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/doc/style-guide.md b/doc/style-guide.md
index cfecfbd699..9ded0f896f 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -586,6 +586,12 @@ word within a compound word is capitalized except for the first word.
   converse relations are systematically introduced, eg `≥` as `\ge`
   and `≱` as `\gen`.
 
+* Decidable predicates and relations should typically be written as `R?`,
+  where `R` is the underlying property being asserted to be `Decidable`,
+  moreover typically sharing the same fixity and precedence as `R`: thus
+  - `_≡?_` (at `infix 4`) for `DecidableEquality`
+  - `_≈?_` for the fieldname of the general `IsDecEquivalence`
+
 * Any exceptions to these conventions should be flagged on the GitHub
   `agda-stdlib` issue tracker in the usual way.
 

From 412eb939bb7d17637426961a6f8e0f76c064d36e Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 13:27:13 +0000
Subject: [PATCH 07/20] fix: `test/reflection/assumption`

---
 tests/reflection/assumption/Main.agda | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/tests/reflection/assumption/Main.agda b/tests/reflection/assumption/Main.agda
index cf0d21b514..bdc397613a 100644
--- a/tests/reflection/assumption/Main.agda
+++ b/tests/reflection/assumption/Main.agda
@@ -13,7 +13,7 @@ open import Data.Unit.Base using (⊤)
 open import Function.Base using (case_of_; _$_)
 open import Reflection.TCM hiding (pure)
 open import Reflection.AST.Term
-open import Reflection.AST.Literal hiding (_≟_)
+open import Reflection.AST.Literal hiding (_≟_; _≡?_)
 open import Reflection.AST.Argument using (Arg; unArg; arg)
 open import Reflection.AST.DeBruijn
 open import Reflection.AST.Show
@@ -44,7 +44,7 @@ searchEntry : ℕ → Type → List (String × Arg Type) → Maybe ℕ
 searchEntry n ty [] = nothing
 searchEntry n ty ((_ , e) ∷ es) = do
   ty ← strengthen ty
-  if does (ty ≟ unArg e)
+  if does (ty ≡? unArg e)
     then just n
     else searchEntry (suc n) ty es
 

From 377d56ebb6b1c37a594d18967398839949fc20f5 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 13:31:16 +0000
Subject: [PATCH 08/20] fix: `data/trie` plus exports from `Data.String`

---
 src/Data/String.agda      | 2 +-
 tests/data/trie/Main.agda | 4 ++--
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/Data/String.agda b/src/Data/String.agda
index 9186c514d1..e5717fff09 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -27,7 +27,7 @@ open import Data.List.Membership.DecPropositional Char._≡?_
 -- Re-export contents of base, and decidability of equality
 
 open import Data.String.Base public
-open import Data.String.Properties using (_≈?_; _≡?_; _<?_; _==_) public
+open import Data.String.Properties using (_≟_; _≈?_; _≡?_; _<?_; _==_) public
 
 ------------------------------------------------------------------------
 -- Conversion functions
diff --git a/tests/data/trie/Main.agda b/tests/data/trie/Main.agda
index 3ce2ad3ecb..91246b5a93 100644
--- a/tests/data/trie/Main.agda
+++ b/tests/data/trie/Main.agda
@@ -35,7 +35,7 @@ record Lexer t : Set (suc t) where
   Keyword = String × Tok
 
   Distinct : Rel Keyword 0ℓ
-  Distinct a b = ⌊ ¬? ((proj₁ a) String.≟ (proj₁ b)) ⌋
+  Distinct a b = ⌊ ¬? ((proj₁ a) String.≡? (proj₁ b)) ⌋
 
   field
     keywords : List# Keyword Distinct
@@ -90,7 +90,7 @@ module LetIn where
   show RPAR   = "RPAR"
   show (ID x) = "ID \"" ++ x ++ "\""
 
-  keywords : List# (String × TOK) (λ a b → ⌊ ¬? ((proj₁ a) String.≟ (proj₁ b)) ⌋)
+  keywords : List# (String × TOK) (λ a b → ⌊ ¬? ((proj₁ a) String.≡? (proj₁ b)) ⌋)
   keywords =  ("let" , LET)
            ∷# ("="   , EQ)
            ∷# ("in"  , IN)

From 355a427426d20a7151aea8833a86e42c8bd05766 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 19:35:52 +0000
Subject: [PATCH 09/20] fix: renaming and deprecation for `Fin` and `Nat`

---
 CHANGELOG.md                 | 19 ++++++++++++++-----
 src/Data/Nat/Properties.agda | 19 +++++++++++--------
 2 files changed, 25 insertions(+), 13 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index f83411bc22..ec0466e228 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -87,6 +87,15 @@ Deprecated names
   ```agda
   ¬∀⟶∃¬-smallest  ↦   ¬∀⇒∃¬-smallest
   ¬∀⟶∃¬-          ↦   ¬∀⇒∃¬
+  _≟_       ↦   _≡?_
+  inj⇒≟     ↦   inj⇒≡?
+  ```
+
+* In `Data.Nat.Properties`:
+  ```agda
+  _≟_     ↦   _≡?_
+  ≟-diag  ↦   ≡?-≡
+  ≟-≡     ↦   ≡?-≢
   ```
 
 * In `Data.Rational.Properties`:
@@ -239,10 +248,10 @@ Additions to existing modules
 * In `Data.Fin.Properties`:
   ```agda
   ≡-irrelevant : Irrelevant {A = Fin n} _≡_
-  ≟-≡          : (eq : i ≡ j) → (i ≟ j) ≡ yes eq
-  ≟-≡-refl     : (i : Fin n) → (i ≟ i) ≡ yes refl
-  ≟-≢          : (i≢j : i ≢ j) → (i ≟ j) ≡ no i≢j
-  inject-<     : inject j < i
+  ≡?-≡          : (eq : i ≡ j) → (i ≡? j) ≡ yes eq
+  ≡?-≡-refl     : (i : Fin n) → (i ≡? i) ≡ yes refl
+  ≡?-≢          : (i≢j : i ≢ j) → (i ≡? j) ≡ no i≢j
+  inject-<      : inject j < i
 
   record Least⟨_⟩ (P : Pred (Fin n) p) : Set p where
     constructor least
@@ -305,7 +314,7 @@ Additions to existing modules
 
 * In `Data.Nat.Properties`:
   ```agda
-  ≟-≢   : (m≢n : m ≢ n) → (m ≟ n) ≡ no m≢n
+  ≡?-≡-refl : ∀ n  → (n ≡? n) ≡ yes refl
   ∸-suc : .(m ≤ n) → suc n ∸ m ≡ suc (n ∸ m)
   ^-distribʳ-* : ∀ m n o → (n * o) ^ m ≡ n ^ m * o ^ m
   2*suc[n]≡2+n+n : ∀ n → 2 * (suc n) ≡ 2 + (n + n)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 1ddc89e2d6..b0bc4d77c6 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -111,11 +111,14 @@ m ≡? n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 ≡-irrelevant : Irrelevant {A = ℕ} _≡_
 ≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≡?_
 
-≡?-diag : (eq : m ≡ n) → (m ≡? n) ≡ yes eq
-≡?-diag = ≡-≡?-identity _≡?_
+≡?-≡ : (eq : m ≡ n) → (m ≡? n) ≡ yes eq
+≡?-≡ = ≡-≡?-identity _≡?_
 
-≡?-≡ : (m≢n : m ≢ n) → (m ≡? n) ≡ no m≢n
-≡?-≡ = ≢-≡?-identity _≡?_
+≡?-≡-refl : ∀ n  → (n ≡? n) ≡ yes refl
+≡?-≡-refl _ = ≡?-≡ refl
+
+≡?-≢ : (m≢n : m ≢ n) → (m ≡? n) ≡ no m≢n
+≡?-≢ = ≢-≡?-identity _≡?_
 
 ≡-isDecEquivalence : IsDecEquivalence (_≡_ {A = ℕ})
 ≡-isDecEquivalence = record
@@ -2470,14 +2473,14 @@ _≟_ = _≡?_
 Please use _≡?_ instead."
 #-}
 
-≟-diag = ≡?-diag
+≟-diag = ≡?-≡
 {-# WARNING_ON_USAGE ≟-diag
 "Warning: ≟-diag was deprecated in v2.4.
-Please use ≡?-diag instead."
+Please use ≡?-≡ instead."
 #-}
 
-≟-≡ = ≡?-≡
+≟-≡ = ≡?-≢
 {-# WARNING_ON_USAGE ≟-≡
 "Warning: ≟-≡ was deprecated in v2.4.
-Please use ≡?-≡ instead."
+Please use ≡?-≢ instead."
 #-}

From ed8643672764340c27d46d5477cf3ece3777daa1 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 19:45:43 +0000
Subject: [PATCH 10/20] fix: more deprecations

---
 CHANGELOG.md | 7 +++++++
 1 file changed, 7 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ec0466e228..eef65a4f16 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -108,6 +108,12 @@ Deprecated names
   truncate-irrelevant  ↦  Relation.Binary.PropositionalEquality.Core.refl
   ```
 
+* In `Relation.Binary.PropositionalEquality`:
+  ```agda
+  ≡-≟-identity     ↦   ≡-≡?-identity
+  ≢-≟-identity     ↦   ≢-≡?-identity
+  ```
+
 * In `Relation.Nullary.Decidable.Core`:
   ```agda
   ⊤-dec     ↦   ⊤?
@@ -115,6 +121,7 @@ Deprecated names
   _×-dec_  ↦   _×?_
   _⊎-dec_  ↦   _⊎?_
   _→-dec_  ↦   _→?_
+  ```
 
 * In `Relation.Nullary.Negation`:
   ```agda

From a6143ab2d3247e3a56719372bd6ecbd91c7390c6 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Wed, 25 Feb 2026 20:54:52 +0000
Subject: [PATCH 11/20] fix: cosmetic renaming

---
 src/Axiom/UniquenessOfIdentityProofs.agda | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/Axiom/UniquenessOfIdentityProofs.agda b/src/Axiom/UniquenessOfIdentityProofs.agda
index 752b9c444d..cb1ec76231 100644
--- a/src/Axiom/UniquenessOfIdentityProofs.agda
+++ b/src/Axiom/UniquenessOfIdentityProofs.agda
@@ -66,14 +66,14 @@ module Constant⇒UIP
 -- function over proofs of equality which is constant: it returns the
 -- proof produced by the decision procedure.
 
-module Decidable⇒UIP (_≟_ : DecidableEquality A)
+module Decidable⇒UIP (_≡?_ : DecidableEquality A)
   where
 
   ≡-normalise : _≡_ {A = A} ⇒ _≡_
-  ≡-normalise {x} {y} x≡y = recompute (x ≟ y) x≡y
+  ≡-normalise {x} {y} x≡y = recompute (x ≡? y) x≡y
 
   ≡-normalise-constant : (p q : x ≡ y) → ≡-normalise p ≡ ≡-normalise q
-  ≡-normalise-constant {x = x} {y = y} = recompute-constant (x ≟ y)
+  ≡-normalise-constant {x = x} {y = y} = recompute-constant (x ≡? y)
 
   ≡-irrelevant : UIP A
   ≡-irrelevant = Constant⇒UIP.≡-irrelevant ≡-normalise ≡-normalise-constant

From 36754caca27b8f7b4593c4a3064082cad26290cd Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Thu, 26 Feb 2026 10:41:44 +0000
Subject: [PATCH 12/20] add: better entry in `CHANGELOG`, plus tweak to
 `style-guide`

---
 CHANGELOG.md       | 19 +++++++++++++++----
 doc/style-guide.md |  2 +-
 2 files changed, 16 insertions(+), 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index eef65a4f16..77883ee67b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -21,6 +21,17 @@ Bug-fixes
 Non-backwards compatible changes
 --------------------------------
 
+* The notation for `Decidable` relations has been (partially) standardised: thus
+  - `_≡?_` (at `infix 4`) for `DecidableEquality`
+  - `_≈?_` (ditto.) for the fieldname of the general `IsDecEquivalence`
+
+  Despite being non-backwards compatible, because a fieldname has changed, the
+  old notation `_≟_` (which was used for both of the above) has been retained,
+  but deprecated. This leads to a large amount of (trivial) deprecations, in
+  addition to the substantive one under `Relation.Binary.Structures`, and in
+  `Data.{Nat|Fin}.Properties` for the concrete datatypes. These deprecations
+  are summarised below, but are not each documented for each affected module.
+
 Minor improvements
 ------------------
 
@@ -255,10 +266,10 @@ Additions to existing modules
 * In `Data.Fin.Properties`:
   ```agda
   ≡-irrelevant : Irrelevant {A = Fin n} _≡_
-  ≡?-≡          : (eq : i ≡ j) → (i ≡? j) ≡ yes eq
-  ≡?-≡-refl     : (i : Fin n) → (i ≡? i) ≡ yes refl
-  ≡?-≢          : (i≢j : i ≢ j) → (i ≡? j) ≡ no i≢j
-  inject-<      : inject j < i
+  ≡?-≡         : (eq : i ≡ j) → (i ≡? j) ≡ yes eq
+  ≡?-≡-refl    : (i : Fin n) → (i ≡? i) ≡ yes refl
+  ≡?-≢         : (i≢j : i ≢ j) → (i ≡? j) ≡ no i≢j
+  inject-<     : inject j < i
 
   record Least⟨_⟩ (P : Pred (Fin n) p) : Set p where
     constructor least
diff --git a/doc/style-guide.md b/doc/style-guide.md
index 9ded0f896f..2714a94d4c 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -590,7 +590,7 @@ word within a compound word is capitalized except for the first word.
   where `R` is the underlying property being asserted to be `Decidable`,
   moreover typically sharing the same fixity and precedence as `R`: thus
   - `_≡?_` (at `infix 4`) for `DecidableEquality`
-  - `_≈?_` for the fieldname of the general `IsDecEquivalence`
+  - `_≈?_` (ditto.) for the fieldname of the general `IsDecEquivalence`
 
 * Any exceptions to these conventions should be flagged on the GitHub
   `agda-stdlib` issue tracker in the usual way.

From 6a895bd86ee4c6b56b24ca15eea1f63beff6c971 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 13:17:49 +0000
Subject: [PATCH 13/20] tidy up `Relation.*`

---
 CHANGELOG.md                                  |  5 +++
 src/Data/Fin/Properties.agda                  |  1 -
 src/Relation/Binary/Consequences.agda         |  4 +--
 .../Binary/Construct/Add/Point/Equality.agda  |  8 ++---
 .../Closure/Reflexive/Properties.agda         |  4 +--
 .../Binary/Construct/NaturalOrder/Right.agda  |  2 +-
 .../Lattice/Properties/JoinSemilattice.agda   |  2 +-
 .../PropositionalEquality/Properties.agda     |  4 +--
 src/Relation/Nary.agda                        | 36 +++++++++++++------
 src/Relation/Nullary/Decidable.agda           |  2 +-
 10 files changed, 43 insertions(+), 25 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 77883ee67b..5dd4295fe5 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -125,6 +125,11 @@ Deprecated names
   ≢-≟-identity     ↦   ≢-≡?-identity
   ```
 
+* In `Relation.Nary`:
+  ```agda
+  ≟-mapₙ     ↦   ≡?-mapₙ
+  ```
+
 * In `Relation.Nullary.Decidable.Core`:
   ```agda
   ⊤-dec     ↦   ⊤?
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index f3ea2a31b9..6f0d9dab42 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -1337,7 +1337,6 @@ _≟_ = _≡?_
 Please use _≡?_ instead."
 #-}
 
-
 ≟-≡-refl = ≡?-≡-refl
 {-# WARNING_ON_USAGE ≟-≡-refl
 "Warning: ≟-≡-refl was deprecated in v2.4.
diff --git a/src/Relation/Binary/Consequences.agda b/src/Relation/Binary/Consequences.agda
index 43d91631ec..ee183c2ea5 100644
--- a/src/Relation/Binary/Consequences.agda
+++ b/src/Relation/Binary/Consequences.agda
@@ -79,9 +79,9 @@ module _ {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} where
 
   total∧dec⇒dec : _≈_ ⇒ _≤_ → Antisymmetric _≈_ _≤_ →
                   Total _≤_ → Decidable _≈_ → Decidable _≤_
-  total∧dec⇒dec refl antisym total _≟_ x y with total x y
+  total∧dec⇒dec refl antisym total _≈?_ x y with total x y
   ... | inj₁ x≤y = yes x≤y
-  ... | inj₂ y≤x = map′ refl (flip antisym y≤x) (x ≟ y)
+  ... | inj₂ y≤x = map′ refl (flip antisym y≤x) (x ≈? y)
 
 module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {≤₁ : Rel A ℓ₃} {≤₂ : Rel B ℓ₄} where
 
diff --git a/src/Relation/Binary/Construct/Add/Point/Equality.agda b/src/Relation/Binary/Construct/Add/Point/Equality.agda
index fdb3d9fee0..25609fffe6 100644
--- a/src/Relation/Binary/Construct/Add/Point/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Point/Equality.agda
@@ -53,10 +53,10 @@ data _≈∙_ : Rel (Pointed A) (a ⊔ ℓ) where
 ≈∙-trans ≈-trans [ x≈y ] [ y≈z ] = [ ≈-trans x≈y y≈z ]
 
 ≈∙-dec : Decidable _≈_ → Decidable _≈∙_
-≈∙-dec _≟_ ∙     ∙     = yes ∙≈∙
-≈∙-dec _≟_ ∙     [ l ] = no (λ ())
-≈∙-dec _≟_ [ k ] ∙     = no (λ ())
-≈∙-dec _≟_ [ k ] [ l ] = Dec.map′ [_] [≈]-injective (k ≟ l)
+≈∙-dec _≈?_ ∙     ∙     = yes ∙≈∙
+≈∙-dec _≈?_ ∙     [ l ] = no λ()
+≈∙-dec _≈?_ [ k ] ∙     = no λ()
+≈∙-dec _≈?_ [ k ] [ l ] = Dec.map′ [_] [≈]-injective (k ≈? l)
 
 ≈∙-irrelevant : Irrelevant _≈_ → Irrelevant _≈∙_
 ≈∙-irrelevant ≈-irr ∙≈∙   ∙≈∙   = ≡.refl
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index dce7099889..713ef58f3e 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -135,7 +135,7 @@ module _ {_~_ : Rel A ℓ} where
   isDecPartialOrder O = record
     { isPartialOrder = isPartialOrder O.isStrictPartialOrder
     ; _≈?_           = O._≈?_
-    ; _≤?_           = dec O._≟_ O._<?_
+    ; _≤?_           = dec O._≈?_ O._<?_
     } where module O = IsDecStrictPartialOrder O
 
   isTotalOrder : IsStrictTotalOrder _≡_ _~_ → IsTotalOrder _≡_ _~ᵒ_
@@ -148,5 +148,5 @@ module _ {_~_ : Rel A ℓ} where
   isDecTotalOrder O = record
     { isTotalOrder = isTotalOrder O
     ; _≈?_         = _≈?_
-    ; _≤?_         = dec _≟_ _<?_
+    ; _≤?_         = dec _≈?_ _<?_
     } where open IsStrictTotalOrder O
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index 3623097128..48067d73a8 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -91,7 +91,7 @@ resp₂ : IsMagma _∙_ →  _≤_ Respects₂ _≈_
 resp₂ magma = respʳ magma , respˡ magma
 
 dec : Decidable _≈_ → Decidable _≤_
-dec _≟_ x y = x ≟ (y ∙ x)
+dec _≈?_ x y = x ≈? (y ∙ x)
 
 ------------------------------------------------------------------------
 -- Structures
diff --git a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
index 493b1d9896..b86f871298 100644
--- a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
@@ -114,7 +114,7 @@ dualMeetSemilattice = record
 -- If ≈ is decidable then so is ≤
 
 ≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_
-≈-dec⇒≤-dec _≟_ x y with (x ∨ y) ≟ y
+≈-dec⇒≤-dec _≈?_ x y with (x ∨ y) ≈? y
 ... | yes x∨y≈y = yes (trans (x≤x∨y x y) (reflexive x∨y≈y))
 ... | no  x∨y≉y = no (contraposition x≤y⇒x∨y≈y x∨y≉y)
 
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index e9af17a238..2cfb46564f 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -172,9 +172,9 @@ setoid A = record
   }
 
 decSetoid : DecidableEquality A → DecSetoid _ _
-decSetoid _≟_ = record
+decSetoid _≡?_ = record
   { _≈_              = _≡_
-  ; isDecEquivalence = isDecEquivalence _≟_
+  ; isDecEquivalence = isDecEquivalence _≡?_
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Nary.agda b/src/Relation/Nary.agda
index 54adb5369c..e9c821f390 100644
--- a/src/Relation/Nary.agda
+++ b/src/Relation/Nary.agda
@@ -16,7 +16,6 @@ module Relation.Nary where
 
 open import Data.Unit.Base using (⊤)
 open import Data.Bool.Base using (true; false)
-open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Nat.Base using (zero; suc)
 open import Data.Product.Base as Product using (_×_; _,_)
 open import Data.Product.Nary.NonDependent
@@ -25,7 +24,7 @@ open import Function.Base using (_$_; _∘′_)
 open import Function.Nary.NonDependent
 open import Level using (Level; _⊔_; Lift)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
-open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction′)
 open import Relation.Nullary.Decidable.Core as Dec
   using (Dec; yes; no; _because_; _×-dec_)
 import Relation.Unary as Unary
@@ -77,15 +76,15 @@ infix 5 ∃⟨_⟩ Π[_] ∀[_]
 ∀[_] : ∀ {n ls r} {as : Sets n ls} → as ⇉ Set r → Set (r ⊔ (⨆ n ls))
 ∀[_] = quantₙ Unary.IUniversal _
 
--- ≟-mapₙ : ∀ n. (con : A₁ → ⋯ → Aₙ → R) →
---               Injectiveₙ n con →
---               ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →
---               Dec (a₁₁ ≡ a₁₂) → ⋯ → Dec (aₙ₁ ≡ aₙ₂) →
---               Dec (con a₁₁ ⋯ aₙ₁ ≡ con a₁₂ ⋯ aₙ₂)
+-- ≡?-mapₙ : ∀ n. (con : A₁ → ⋯ → Aₙ → R) →
+--                Injectiveₙ n con →
+--                ∀ a₁₁ a₁₂ ⋯ aₙ₁ aₙ₂ →
+--                Dec (a₁₁ ≡ a₁₂) → ⋯ → Dec (aₙ₁ ≡ aₙ₂) →
+--                Dec (con a₁₁ ⋯ aₙ₁ ≡ con a₁₂ ⋯ aₙ₂)
 
-≟-mapₙ : ∀ n {ls} {as : Sets n ls} (con : Arrows n as R) → Injectiveₙ n con →
-         ∀ {l r} → Arrows n (Dec <$> Equalₙ n l r) (Dec (uncurryₙ n con l ≡ uncurryₙ n con r))
-≟-mapₙ n con con-inj =
+≡?-mapₙ : ∀ n {ls} {as : Sets n ls} (con : Arrows n as R) → Injectiveₙ n con →
+          ∀ {l r} → Arrows n (Dec <$> Equalₙ n l r) (Dec (uncurryₙ n con l ≡ uncurryₙ n con r))
+≡?-mapₙ n con con-inj =
   curryₙ n λ a?s → let as? = Product-dec n a?s in
   Dec.map′ (cong (uncurryₙ n con) ∘′ fromEqualₙ n) con-inj as?
 
@@ -205,5 +204,20 @@ toWitness : ∀ {n ls r} {as : Sets n ls} (R : as ⇉ Set r) (R? : Decidable R)
               ∀[ R ⇒ ⌊ R? ⌋ ]
 toWitness {zero} R R? with R?
 ... | true because _ = _
-... | no          ¬r = ⊥-elim ∘′ ¬r
+... | no          ¬r = contradiction′ ¬r
 toWitness {suc n} R R? = toWitness (R _) (R? _)
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+≟-mapₙ = ≡?-mapₙ
+{-# WARNING_ON_USAGE ≟-mapₙ
+"Warning: ≟-mapₙ was deprecated in v2.4.
+Please use ≡?-mapₙ instead."
+#-}
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index e9e16a8c88..6caf11d5a1 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -47,7 +47,7 @@ module _ {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} (injection : Injection S T)
   open Injection injection
 
   via-injection : Decidable Eq₂._≈_ → Decidable Eq₁._≈_
-  via-injection _≟_ x y = map′ injective cong (to x ≟ to y)
+  via-injection _≈?_ x y = map′ injective cong (to x ≈? to y)
 
 ------------------------------------------------------------------------
 -- A lemma relating True and Dec

From 93a74c6a4ff2318bdbd4b007356b60aa38f5732f Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 13:22:40 +0000
Subject: [PATCH 14/20] tidy up `Reflection.*`

---
 src/Reflection/AST/AlphaEquality.agda | 31 ++++++++++++++++++++-------
 1 file changed, 23 insertions(+), 8 deletions(-)

diff --git a/src/Reflection/AST/AlphaEquality.agda b/src/Reflection/AST/AlphaEquality.agda
index f6d1a7b1b5..dc9a7d3e5f 100644
--- a/src/Reflection/AST/AlphaEquality.agda
+++ b/src/Reflection/AST/AlphaEquality.agda
@@ -44,8 +44,8 @@ open AlphaEquality {{...}} public
 ------------------------------------------------------------------------
 -- Utilities
 
-≟⇒α : DecidableEquality A → AlphaEquality A
-≟⇒α _≟_ = mkAlphaEquality (λ x y → ⌊ x ≟ y ⌋)
+≡?⇒α : DecidableEquality A → AlphaEquality A
+≡?⇒α _≡?_ = mkAlphaEquality (λ x y → ⌊ x ≡? y ⌋)
 
 ------------------------------------------------------------------------
 -- Propositional cases
@@ -55,22 +55,22 @@ open AlphaEquality {{...}} public
 
 instance
   α-Visibility : AlphaEquality Visibility
-  α-Visibility = ≟⇒α Visibility._≡?_
+  α-Visibility = ≡?⇒α Visibility._≡?_
 
   α-Modality : AlphaEquality Modality
-  α-Modality = ≟⇒α Modality._≡?_
+  α-Modality = ≡?⇒α Modality._≡?_
 
   α-ArgInfo : AlphaEquality ArgInfo
-  α-ArgInfo = ≟⇒α ArgInfo._≡?_
+  α-ArgInfo = ≡?⇒α ArgInfo._≡?_
 
   α-Literal : AlphaEquality Literal
-  α-Literal = ≟⇒α Literal._≡?_
+  α-Literal = ≡?⇒α Literal._≡?_
 
   α-Meta : AlphaEquality Meta
-  α-Meta = ≟⇒α Meta._≡?_
+  α-Meta = ≡?⇒α Meta._≡?_
 
   α-Name : AlphaEquality Name
-  α-Name = ≟⇒α Name._≡?_
+  α-Name = ≡?⇒α Name._≡?_
 
 ------------------------------------------------------------------------
 -- Interesting cases
@@ -344,3 +344,18 @@ instance
 
   α-ArgsPattern : AlphaEquality (Args Pattern)
   α-ArgsPattern = mkAlphaEquality _=α=-ArgsPattern_
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+≟⇒α = ≡?⇒α
+{-# WARNING_ON_USAGE ≟⇒α
+"Warning: ≟⇒α was deprecated in v2.4.
+Please use ≡?⇒α instead."
+#-}

From 1b273e95baaa0eddf07c8efdff58d4fa46d55fc6 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 13:24:46 +0000
Subject: [PATCH 15/20] tidy up `Reflection.*`

---
 CHANGELOG.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5dd4295fe5..5a7e1bb4c1 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -119,6 +119,11 @@ Deprecated names
   truncate-irrelevant  ↦  Relation.Binary.PropositionalEquality.Core.refl
   ```
 
+* In `Reflection.AST.AlphaEquality`:
+  ```agda
+  ≟⇒α     ↦   ≡?⇒α
+  ```
+
 * In `Relation.Binary.PropositionalEquality`:
   ```agda
   ≡-≟-identity     ↦   ≡-≡?-identity

From d0946ba86115c90b90c74dd5758bbad277ef711b Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 13:32:09 +0000
Subject: [PATCH 16/20] tidy up `Effect.Monad.Partiality`

---
 CHANGELOG.md                     |  5 +++++
 src/Effect/Monad/Partiality.agda | 35 +++++++++++++++++++++++---------
 2 files changed, 30 insertions(+), 10 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5a7e1bb4c1..e1c3c16a15 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -119,6 +119,11 @@ Deprecated names
   truncate-irrelevant  ↦  Relation.Binary.PropositionalEquality.Core.refl
   ```
 
+* In `Effect.Monad.Partiality`:
+  ```agda
+  _≟-Kind_     ↦   _≡?-Kind_
+  ```
+
 * In `Reflection.AST.AlphaEquality`:
   ```agda
   ≟⇒α     ↦   ≡?⇒α
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index d2bf58cac5..f1700c7b63 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -113,22 +113,22 @@ data Kind : Set where
 
 -- Kind equality is decidable.
 
-infix 4 _≟-Kind_
+infix 4 _≡?-Kind_
 
-_≟-Kind_ : DecidableEquality Kind
-_≟-Kind_ strong       strong       = yes ≡.refl
-_≟-Kind_ strong       (other k)    = no λ()
-_≟-Kind_ (other k)    strong       = no λ()
-_≟-Kind_ (other geq)  (other geq)  = yes ≡.refl
-_≟-Kind_ (other geq)  (other weak) = no λ()
-_≟-Kind_ (other weak) (other geq)  = no λ()
-_≟-Kind_ (other weak) (other weak) = yes ≡.refl
+_≡?-Kind_ : DecidableEquality Kind
+_≡?-Kind_ strong       strong       = yes ≡.refl
+_≡?-Kind_ strong       (other k)    = no λ()
+_≡?-Kind_ (other k)    strong       = no λ()
+_≡?-Kind_ (other geq)  (other geq)  = yes ≡.refl
+_≡?-Kind_ (other geq)  (other weak) = no λ()
+_≡?-Kind_ (other weak) (other geq)  = no λ()
+_≡?-Kind_ (other weak) (other weak) = yes ≡.refl
 
 -- A predicate which is satisfied only for equalities. Note that, for
 -- concrete inputs, this predicate evaluates to ⊤ or ⊥.
 
 Equality : Kind → Set
-Equality k = False (k ≟-Kind other geq)
+Equality k = False (k ≡?-Kind other geq)
 
 ------------------------------------------------------------------------
 -- Equality/ordering
@@ -933,3 +933,18 @@ idempotent {A = A} B x f = sound (idem x)
                                                      laterˡ (refl (Setoid.refl B))) ⟩
     (♭ x >>= λ y′ →     ♭ x >>= λ y″ → f y′ y″)  ≳⟨ idem (♭ x) ⟩≅
     (♭ x >>= λ y′ → f y′ y′)                     ∎))
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+_≟-Kind_ = _≡?-Kind_
+{-# WARNING_ON_USAGE _≟-Kind_
+"Warning: _≟-Kind_ was deprecated in v2.4.
+Please use _≡?-Kind_ instead."
+#-}

From 2f6403920785e62addbeec0e1c12caaab1a4feb3 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 13:58:35 +0000
Subject: [PATCH 17/20] tidy up `Algebra`

---
 src/Algebra/Construct/LexProduct.agda         |  8 +-
 src/Algebra/Construct/LexProduct/Base.agda    |  4 +-
 src/Algebra/Construct/LexProduct/Inner.agda   | 22 ++---
 .../CancellativeCommutativeSemiring.agda      |  4 +-
 src/Algebra/Solver/Monoid/Expression.agda     |  4 +-
 src/Algebra/Solver/Monoid/ExpressionBAK.agda  | 97 +++++++++++++++++++
 src/Algebra/Solver/Monoid/Solver.agda         |  2 +-
 7 files changed, 119 insertions(+), 22 deletions(-)
 create mode 100644 src/Algebra/Solver/Monoid/ExpressionBAK.agda

diff --git a/src/Algebra/Construct/LexProduct.agda b/src/Algebra/Construct/LexProduct.agda
index 3fbcf86c80..8a4cd45e47 100644
--- a/src/Algebra/Construct/LexProduct.agda
+++ b/src/Algebra/Construct/LexProduct.agda
@@ -21,7 +21,7 @@ import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 module Algebra.Construct.LexProduct
   {ℓ₁ ℓ₂ ℓ₃ ℓ₄} (M : Magma ℓ₁ ℓ₂) (N : Magma ℓ₃ ℓ₄)
-  (_≟₁_ : Decidable (Magma._≈_ M))
+  (_≈₁?_ : Decidable (Magma._≈_ M))
   where
 
 open Magma M using (_∙_ ; ∙-cong)
@@ -39,7 +39,7 @@ open Magma N using ()
   ; refl    to ≈₂-refl
   )
 
-import Algebra.Construct.LexProduct.Inner M N _≟₁_ as InnerLex
+import Algebra.Construct.LexProduct.Inner M N _≈₁?_ as InnerLex
 
 private
   infix 4 _≋_
@@ -53,7 +53,7 @@ private
 -- Definition
 ------------------------------------------------------------------------
 
-open import Algebra.Construct.LexProduct.Base _∙_ _◦_ _≟₁_ public
+open import Algebra.Construct.LexProduct.Base _∙_ _◦_ _≈₁?_ public
   renaming (lex to _⊕_)
 
 ------------------------------------------------------------------------
@@ -103,7 +103,7 @@ identityʳ : ∀ {e f} → RightIdentity _≈₁_ e _∙_ → RightIdentity _≈
 identityʳ id₁ id₂ (x , a) = id₁ x , InnerLex.identityʳ id₁ id₂
 
 sel : Selective _≈₁_ _∙_ → Selective _≈₂_ _◦_ → Selective _≋_ _⊕_
-sel ∙-sel ◦-sel (a , x) (b , y) with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b
+sel ∙-sel ◦-sel (a , x) (b , y) with (a ∙ b) ≈₁? a | (a ∙ b) ≈₁? b
 ... | no  ab≉a | no  ab≉b  = contradiction₂ (∙-sel a b) ab≉a ab≉b
 ... | yes ab≈a | no  _     = inj₁ (ab≈a , ≈₂-refl)
 ... | no  _    | yes ab≈b  = inj₂ (ab≈b , ≈₂-refl)
diff --git a/src/Algebra/Construct/LexProduct/Base.agda b/src/Algebra/Construct/LexProduct/Base.agda
index 85686fd20c..585c8d730d 100644
--- a/src/Algebra/Construct/LexProduct/Base.agda
+++ b/src/Algebra/Construct/LexProduct/Base.agda
@@ -16,7 +16,7 @@ open import Relation.Nullary.Decidable.Core using (does; yes; no)
 module Algebra.Construct.LexProduct.Base
   {a b ℓ} {A : Set a} {B : Set b}
   (_∙_ : Op₂ A) (_◦_ : Op₂ B)
-  {_≈₁_ : Rel A ℓ} (_≟₁_ : Decidable _≈₁_)
+  {_≈₁_ : Rel A ℓ} (_≈₁?_ : Decidable _≈₁_)
   where
 
 ------------------------------------------------------------------------
@@ -27,7 +27,7 @@ module Algebra.Construct.LexProduct.Base
 -- operator that only calculates the second component of product.
 
 innerLex : A → A → B → B → B
-innerLex a b x y with does ((a ∙ b) ≟₁ a) | does ((a ∙ b) ≟₁ b)
+innerLex a b x y with does ((a ∙ b) ≈₁? a) | does ((a ∙ b) ≈₁? b)
 ... | true  | false = x
 ... | false | true  = y
 ... |     _ |     _ = x ◦ y
diff --git a/src/Algebra/Construct/LexProduct/Inner.agda b/src/Algebra/Construct/LexProduct/Inner.agda
index beff9e9e3a..b6755e2528 100644
--- a/src/Algebra/Construct/LexProduct/Inner.agda
+++ b/src/Algebra/Construct/LexProduct/Inner.agda
@@ -21,7 +21,7 @@ import Algebra.Construct.LexProduct.Base as Base
 
 module Algebra.Construct.LexProduct.Inner
   {ℓ₁ ℓ₂ ℓ₃ ℓ₄} (M : Magma ℓ₁ ℓ₂) (N : Magma ℓ₃ ℓ₄)
-  (_≟₁_ : Decidable (Magma._≈_ M))
+  (_≈₁?_ : Decidable (Magma._≈_ M))
   where
 
 open module M = Magma M
@@ -48,7 +48,7 @@ private
 ------------------------------------------------------------------------
 -- Base definition
 
-open Base _∙_ _◦_ _≟₁_ public
+open Base _∙_ _◦_ _≈₁?_ public
   using (innerLex)
 
 -- Save ourselves some typing in this file
@@ -122,19 +122,19 @@ open SetoidReasoning N.setoid
 open NaturalOrder
 
 case₁ : a ∙ b ≈₁ a → a ∙ b ≉₁ b → lex a b x y ≈₂ x
-case₁ {a} {b} ab≈a ab≉b with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b
+case₁ {a} {b} ab≈a ab≉b with (a ∙ b) ≈₁? a | (a ∙ b) ≈₁? b
 ... | no  ab≉a | _        = contradiction ab≈a ab≉a
 ... | yes _    | yes ab≈b = contradiction ab≈b ab≉b
 ... | yes _    | no  _    = N.refl
 
 case₂ : a ∙ b ≉₁ a → a ∙ b ≈₁ b → lex a b x y ≈₂ y
-case₂ {a} {b} ab≉a ab≈b with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b
+case₂ {a} {b} ab≉a ab≈b with (a ∙ b) ≈₁? a | (a ∙ b) ≈₁? b
 ... | yes ab≈a | _        = contradiction ab≈a ab≉a
 ... | no _     | no  ab≉b = contradiction ab≈b ab≉b
 ... | no _     | yes _    = N.refl
 
 case₃ : a ∙ b ≈₁ a → a ∙ b ≈₁ b → lex a b x y ≈₂ (x ◦ y)
-case₃ {a} {b} ab≈a ab≈b with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b
+case₃ {a} {b} ab≈a ab≈b with (a ∙ b) ≈₁? a | (a ∙ b) ≈₁? b
 ... | no  ab≉a | _        = contradiction ab≈a ab≉a
 ... | yes _    | no  ab≉b = contradiction ab≈b ab≉b
 ... | yes _    | yes _    = N.refl
@@ -144,7 +144,7 @@ case₃ {a} {b} ab≈a ab≈b with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b
 
 cong : a ≈₁ b → c ≈₁ d → w ≈₂ x → y ≈₂ z → lex a c w y ≈₂ lex b d x z
 cong {a} {b} {c} {d} a≈b c≈d w≈x y≈z
-  with (a ∙ c) ≟₁ a | (a ∙ c) ≟₁ c | (b ∙ d) ≟₁ b | (b ∙ d) ≟₁ d
+  with (a ∙ c) ≈₁? a | (a ∙ c) ≈₁? c | (b ∙ d) ≈₁? b | (b ∙ d) ≈₁? d
 ... | yes _    | yes _    | yes _    | yes _    = ◦-cong w≈x y≈z
 ... | yes _    | yes _    | no  _    | no  _    = ◦-cong w≈x y≈z
 ... | no  _    | no  _    | yes _    | yes _    = ◦-cong w≈x y≈z
@@ -172,7 +172,7 @@ assoc : Associative _≈₁_ _∙_ → Commutative _≈₁_ _∙_ →
         Selective _≈₁_ _∙_ → Associative _≈₂_ _◦_ →
         ∀ a b c x y z  → lex (a ∙ b) c (lex a b x y) z  ≈₂ lex a (b ∙ c) x (lex b c y z)
 assoc ∙-assoc ∙-comm ∙-sel ◦-assoc a b c x y z
-  with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b | (b ∙ c) ≟₁ b | (b ∙ c) ≟₁ c
+  with (a ∙ b) ≈₁? a | (a ∙ b) ≈₁? b | (b ∙ c) ≈₁? b | (b ∙ c) ≈₁? c
 ... | _        | _        | no  bc≉b | no  bc≉c = contradiction₂ (∙-sel b c) bc≉b bc≉c
 ... | no  ab≉a | no  ab≉b | _        | _        = contradiction₂ (∙-sel a b) ab≉a ab≉b
 ... | yes ab≈a | no  ab≉b | no  bc≉b | yes bc≈c = cong₁₂ ab≈a (M.sym bc≈c)
@@ -229,7 +229,7 @@ assoc ∙-assoc ∙-comm ∙-sel ◦-assoc a b c x y z
 comm : Commutative _≈₁_ _∙_ → Commutative _≈₂_ _◦_ →
        ∀ a b x y → lex a b x y ≈₂ lex b a y x
 comm ∙-comm ◦-comm a b x y
-  with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b | (b ∙ a) ≟₁ b | (b ∙ a) ≟₁ a
+  with (a ∙ b) ≈₁? a | (a ∙ b) ≈₁? b | (b ∙ a) ≈₁? b | (b ∙ a) ≈₁? a
 ... | yes ab≈a | _        | _        | no  ba≉a = contradiction (M.trans (∙-comm b a) ab≈a) ba≉a
 ... | no  ab≉a | _        | _        | yes ba≈a = contradiction (M.trans (∙-comm a b) ba≈a) ab≉a
 ... | _        | yes ab≈b | no  ba≉b | _        = contradiction (M.trans (∙-comm b a) ab≈b) ba≉b
@@ -240,7 +240,7 @@ comm ∙-comm ◦-comm a b x y
 ... | no  _    | no  _    | no  _    | no  _    = ◦-comm x y
 
 idem : Idempotent _≈₂_ _◦_ → ∀ a b x → lex a b x x ≈₂ x
-idem ◦-idem a b x with does ((a ∙ b) ≟₁ a) | does ((a ∙ b) ≟₁ b)
+idem ◦-idem a b x with does ((a ∙ b) ≈₁? a) | does ((a ∙ b) ≈₁? b)
 ... | false | false = ◦-idem x
 ... | false | true  = N.refl
 ... | true  | false = N.refl
@@ -248,14 +248,14 @@ idem ◦-idem a b x with does ((a ∙ b) ≟₁ a) | does ((a ∙ b) ≟₁ b)
 
 zeroʳ : ∀ {e f} → RightZero _≈₁_ e _∙_ → RightZero _≈₂_ f _◦_ →
         lex a e x f ≈₂ f
-zeroʳ {a} {x} {e} {f} ze₁ ze₂ with (a ∙ e) ≟₁ a | (a ∙ e) ≟₁ e
+zeroʳ {a} {x} {e} {f} ze₁ ze₂ with (a ∙ e) ≈₁? a | (a ∙ e) ≈₁? e
 ... | _     | no  a∙e≉e = contradiction (ze₁ a) a∙e≉e
 ... | no  _ | yes _     = N.refl
 ... | yes _ | yes _     = ze₂ x
 
 identityʳ : ∀ {e f} → RightIdentity _≈₁_ e _∙_ → RightIdentity _≈₂_ f _◦_ →
             lex a e x f ≈₂ x
-identityʳ {a} {x} {e} {f} id₁ id₂ with (a ∙ e) ≟₁ a | (a ∙ e) ≟₁ e
+identityʳ {a} {x} {e} {f} id₁ id₂ with (a ∙ e) ≈₁? a | (a ∙ e) ≈₁? e
 ... | no  a∙e≉a | _     = contradiction (id₁ a) a∙e≉a
 ... | yes _     | no  _ = N.refl
 ... | yes _     | yes _ = id₂ x
diff --git a/src/Algebra/Properties/CancellativeCommutativeSemiring.agda b/src/Algebra/Properties/CancellativeCommutativeSemiring.agda
index 2062c33237..5d696058fd 100644
--- a/src/Algebra/Properties/CancellativeCommutativeSemiring.agda
+++ b/src/Algebra/Properties/CancellativeCommutativeSemiring.agda
@@ -26,7 +26,7 @@ open import Relation.Binary.Reasoning.Setoid setoid
 *-almostCancelʳ = comm∧almostCancelˡ⇒almostCancelʳ *-comm *-cancelˡ-nonZero
 
 xy≈0⇒x≈0∨y≈0 : Decidable _≈_ → ∀ {x y} → x * y ≈ 0# → x ≈ 0# ⊎ y ≈ 0#
-xy≈0⇒x≈0∨y≈0 _≟_ {x} {y} xy≈0 with x ≟ 0# | y ≟ 0#
+xy≈0⇒x≈0∨y≈0 _≈?_ {x} {y} xy≈0 with x ≈? 0# | y ≈? 0#
 ... | yes x≈0 | _       = inj₁ x≈0
 ... | no  _   | yes y≈0 = inj₂ y≈0
 ... | no  x≉0 | no  y≉0 = contradiction y≈0 y≉0
@@ -35,6 +35,6 @@ xy≈0⇒x≈0∨y≈0 _≟_ {x} {y} xy≈0 with x ≟ 0# | y ≟ 0#
   y≈0    = *-cancelˡ-nonZero _ y 0# x≉0 xy≈x*0
 
 x≉0∧y≉0⇒xy≉0 : Decidable _≈_ → ∀ {x y} → x ≉ 0# → y ≉ 0# → x * y ≉ 0#
-x≉0∧y≉0⇒xy≉0 _≟_ x≉0 y≉0 xy≈0 with xy≈0⇒x≈0∨y≈0 _≟_ xy≈0
+x≉0∧y≉0⇒xy≉0 _≈?_ x≉0 y≉0 xy≈0 with xy≈0⇒x≈0∨y≈0 _≈?_ xy≈0
 ... | inj₁ x≈0 = x≉0 x≈0
 ... | inj₂ y≈0 = y≉0 y≈0
diff --git a/src/Algebra/Solver/Monoid/Expression.agda b/src/Algebra/Solver/Monoid/Expression.agda
index 7b71c9e9c2..3c28c07ffc 100644
--- a/src/Algebra/Solver/Monoid/Expression.agda
+++ b/src/Algebra/Solver/Monoid/Expression.agda
@@ -52,14 +52,14 @@ Env n = Vec Carrier n
 -- API for normal expressions
 
 record NormalAPI a : Set (suc a ⊔ c ⊔ ℓ) where
-  infix 5 _≟_
+  infix 4 _≡?_
 
   field
 
 -- The type of normal forms.
     Normal    : ℕ → Set a
 -- We can decide if two normal forms are /syntactically/ equal.
-    _≟_       : DecidableEquality (Normal n)
+    _≡?_      : DecidableEquality (Normal n)
 -- A normaliser.
     normalise : Expr n → Normal n
 -- The semantics of a normal form.
diff --git a/src/Algebra/Solver/Monoid/ExpressionBAK.agda b/src/Algebra/Solver/Monoid/ExpressionBAK.agda
new file mode 100644
index 0000000000..74bbe4405e
--- /dev/null
+++ b/src/Algebra/Solver/Monoid/ExpressionBAK.agda
@@ -0,0 +1,97 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- A solver for equations over monoids
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Algebra.Bundles.Raw using (RawMonoid)
+
+module Algebra.Solver.Monoid.ExpressionBAK {c ℓ} (M : RawMonoid c ℓ) where
+
+open import Algebra.Morphism.Structures using (IsMonoidHomomorphism)
+open import Data.Fin.Base using (Fin)
+open import Data.Maybe.Base as Maybe using (Maybe)
+open import Data.Nat.Base using (ℕ)
+open import Data.Vec.Base using (Vec; lookup)
+open import Level using (suc; _⊔_)
+open import Relation.Binary.Consequences using (dec⇒weaklyDec)
+open import Relation.Binary.Definitions using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+
+private
+  variable
+    n : ℕ
+
+  module M = RawMonoid M
+
+
+------------------------------------------------------------------------
+-- Monoid expressions
+
+-- There is one constructor for every operation, plus one for
+-- variables; there may be at most n variables.
+
+infixr 5 _⊕_
+
+data Expr (n : ℕ) : Set where
+  var : Fin n → Expr n
+  id  : Expr n
+  _⊕_ : Expr n → Expr n → Expr n
+
+-- An environment contains one value for every variable.
+
+Env : ℕ → Set _
+Env n = Vec M.Carrier n
+
+-- The semantics of an expression is a function from an environment to
+-- a value.
+
+⟦_⟧ : Expr n → Env n → M.Carrier
+⟦ var x   ⟧ ρ = lookup ρ x
+⟦ id      ⟧ ρ = M.ε
+⟦ e₁ ⊕ e₂ ⟧ ρ = ⟦ e₁ ⟧ ρ M.∙ ⟦ e₂ ⟧ ρ
+
+------------------------------------------------------------------------
+-- API for normal expressions
+
+record Normal {a r} : Set (suc (a ⊔ r) ⊔ c ⊔ ℓ) where
+  infix 5 _≟_
+
+  field
+
+    NF        : ℕ → RawMonoid a r
+
+  private
+    N : ℕ → Set a
+    N n = RawMonoid.Carrier (NF n)
+
+  field
+
+    ⟦var_⟧    : Fin n → N n
+
+-- We can decide if two normal forms are /syntactically/ equal.
+
+    _≟_       : DecidableEquality (N n)
+
+-- The semantics of a normal form.
+    ⟦_⟧⇓      : N n → Env n → M.Carrier
+
+    ⟦var_⟧⇓   : ∀ x ρ → ⟦ ⟦var x ⟧ ⟧⇓ ρ M.≈ lookup {n = n} ρ x
+
+    ⟦_⟧⇓-homo : ∀ ρ → IsMonoidHomomorphism (NF n) M λ e → ⟦ e ⟧⇓ ρ
+
+-- A normaliser.
+
+  module _ {n} where
+
+    open RawMonoid (NF n)
+
+    normalise : Expr n → N n
+    normalise (var x)   = ⟦var x ⟧
+    normalise id        = ε
+    normalise (e₁ ⊕ e₂) = (normalise e₁) ∙ (normalise e₂)
+
+    _semi≟_ : ∀ e₁ e₂ → Maybe (normalise e₁ ≡ normalise e₂)
+    e₁ semi≟ e₂ = dec⇒weaklyDec _≟_ (normalise e₁) (normalise e₂)
diff --git a/src/Algebra/Solver/Monoid/Solver.agda b/src/Algebra/Solver/Monoid/Solver.agda
index a1281e9639..ab77687ed2 100644
--- a/src/Algebra/Solver/Monoid/Solver.agda
+++ b/src/Algebra/Solver/Monoid/Solver.agda
@@ -43,7 +43,7 @@ open R public
 -- for determining if two expressions have the same semantics.
 
 prove′ : ∀ e₁ e₂ → Maybe ((ρ : Env n) → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
-prove′ e₁ e₂ = Maybe.map lemma $ dec⇒weaklyDec _≟_ (normalise e₁) (normalise e₂)
+prove′ e₁ e₂ = Maybe.map lemma $ dec⇒weaklyDec _≡?_ (normalise e₁) (normalise e₂)
   where
   open import Relation.Binary.Reasoning.Setoid setoid
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ

From b1762896f2003bec3c249f044402a3556ed55eac Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 15:31:37 +0000
Subject: [PATCH 18/20] refactor: tidy up the `Data.*` hierarchy

---
 src/Data/Fin/Induction.agda                   |  2 +-
 src/Data/List/Countdown.agda                  | 27 ++++++++--------
 src/Data/List/Fresh/Membership/DecSetoid.agda |  4 +--
 .../List/Membership/DecPropositional.agda     |  4 +--
 src/Data/List/Membership/DecSetoid.agda       |  6 ++--
 src/Data/List/Properties.agda                 | 10 +++---
 .../Binary/Disjoint/DecPropositional.agda     |  4 +--
 .../Disjoint/Propositional/Properties.agda    |  6 ++--
 .../Binary/Disjoint/Setoid/Properties.agda    |  8 ++---
 .../Binary/Equality/DecPropositional.agda     |  6 ++--
 .../Relation/Binary/Equality/DecSetoid.agda   |  2 +-
 .../List/Relation/Binary/Lex/NonStrict.agda   |  8 ++---
 src/Data/List/Relation/Binary/Lex/Strict.agda |  6 ++--
 .../Propositional/Properties/WithK.agda       | 10 +++---
 .../Sublist/DecPropositional/Solver.agda      |  4 +--
 .../Binary/Sublist/DecSetoid/Solver.agda      |  2 +-
 .../Binary/Subset/DecPropositional.agda       |  4 +--
 .../Unary/Enumerates/Setoid/Properties.agda   |  4 +--
 src/Data/List/Relation/Unary/Grouped.agda     |  8 ++---
 .../Unary/Unique/DecPropositional.agda        |  4 +--
 .../Unique/DecPropositional/Properties.agda   |  8 ++---
 .../Unary/Unique/DecSetoid/Properties.agda    |  4 +--
 src/Data/Maybe/Properties.agda                |  8 ++---
 src/Data/Parity/Properties.agda               | 32 ++++++++++++++-----
 src/Data/Record.agda                          | 28 ++++++++--------
 src/Data/Refinement.agda                      |  2 +-
 src/Data/Refinement/Properties.agda           | 24 +++++++++++---
 src/Data/Sign/Properties.agda                 | 32 ++++++++++++++-----
 src/Data/Sum/Relation/Binary/Pointwise.agda   |  8 ++---
 src/Data/Unit/Polymorphic.agda                |  2 +-
 src/Data/Vec/Functional/Properties.agda       |  6 ++--
 src/Data/Vec/Membership/DecPropositional.agda |  4 +--
 src/Data/Vec/Membership/DecSetoid.agda        |  2 +-
 src/Data/Vec/Properties.agda                  |  6 ++--
 .../Binary/Equality/DecPropositional.agda     |  6 ++--
 .../Relation/Binary/Equality/DecSetoid.agda   |  2 +-
 .../Vec/Relation/Binary/Lex/NonStrict.agda    | 22 ++++++-------
 37 files changed, 187 insertions(+), 138 deletions(-)

diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index 6d0a9f9d72..81d9a47435 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -97,7 +97,7 @@ private
 >-weakInduction {n = n} P Pₙ Pᵢ₊₁⇒Pᵢ i = induct (>-wellFounded i)
   where
   induct : ∀ {i} → Acc _>_ i → P i
-  induct {i} (acc rec) with n ℕ.≟ toℕ i
+  induct {i} (acc rec) with n ℕ.≡? toℕ i
   ... | yes n≡i = subst P (toℕ-injective (trans (toℕ-fromℕ n) n≡i)) Pₙ
   ... | no  n≢i = subst P (inject₁-lower₁ i n≢i) (Pᵢ₊₁⇒Pᵢ _ Pᵢ₊₁)
     where Pᵢ₊₁ = induct (rec (ℕ.≤-reflexive (cong suc (sym (toℕ-lower₁ i n≢i)))))
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index 746d91902c..4858be7337 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -36,8 +36,9 @@ open ≡.≡-Reasoning
 
 private
   open module D = DecSetoid D
-    hiding (refl) renaming (Carrier to Elem)
-  open import Data.List.Membership.Setoid D.setoid
+    using (_≈_; sym; trans; _≈?_; setoid) renaming (Carrier to Elem)
+  open import Data.List.Membership.DecSetoid D
+    using (_∈_; _∉_; _∈?_)
 
 ------------------------------------------------------------------------
 -- Helper functions
@@ -48,7 +49,7 @@ private
 
   first-occurrence : ∀ {xs} x → x ∈ xs → x ∈ xs
   first-occurrence x (here x≈y)           = here x≈y
-  first-occurrence x (there {x = y} x∈xs) with x ≟ y
+  first-occurrence x (there {x = y} x∈xs) with x ≈? y
   ... | true  because [x≈y] = here (invert [x≈y])
   ... | false because   _   = there $ first-occurrence x x∈xs
 
@@ -67,12 +68,12 @@ private
              first-index x₁ x₁∈xs ≡ first-index x₂ x₂∈xs
     helper (here x₁≈x) (here x₂≈x)           = refl
     helper (here x₁≈x) (there {x = x} x₂∈xs)
-      with x₂ ≟ x | dec-true (x₂ ≟ x) (trans (sym x₁≈x₂) x₁≈x)
+      with x₂ ≈? x | dec-true (x₂ ≈? x) (trans (sym x₁≈x₂) x₁≈x)
     ... | _ | refl = refl
     helper (there {x = x} x₁∈xs) (here x₂≈x)
-      with x₁ ≟ x | dec-true (x₁ ≟ x) (trans x₁≈x₂ x₂≈x)
+      with x₁ ≈? x | dec-true (x₁ ≈? x) (trans x₁≈x₂ x₂≈x)
     ... | _ | refl = refl
-    helper (there {x = x} x₁∈xs) (there x₂∈xs) with x₁ ≟ x | x₂ ≟ x
+    helper (there {x = x} x₁∈xs) (there x₂∈xs) with x₁ ≈? x | x₂ ≈? x
     ... | true  because _ | true  because _ = refl
     ... | false because _ | false because _ = cong suc $ helper x₁∈xs x₂∈xs
     ... | yes x₁≈x | no  x₂≉x = contradiction (trans (sym x₁≈x₂) x₁≈x) x₂≉x
@@ -88,13 +89,13 @@ private
     helper : ∀ {xs} (x₁∈xs : x₁ ∈ xs) (x₂∈xs : x₂ ∈ xs) →
              first-index x₁ x₁∈xs ≡ first-index x₂ x₂∈xs → x₁ ≈ x₂
     helper (here x₁≈x) (here x₂≈x)             _  = trans x₁≈x (sym x₂≈x)
-    helper (here x₁≈x) (there {x = x} x₂∈xs)   _  with x₂ ≟ x
+    helper (here x₁≈x) (there {x = x} x₂∈xs)   _  with x₂ ≈? x
     helper (here x₁≈x) (there {x = x} x₂∈xs)   _  | yes x₂≈x = trans x₁≈x (sym x₂≈x)
     helper (here x₁≈x) (there {x = x} x₂∈xs)   () | no  x₂≉x
-    helper (there {x = x} x₁∈xs) (here x₂≈x)   _  with x₁ ≟ x
+    helper (there {x = x} x₁∈xs) (here x₂≈x)   _  with x₁ ≈? x
     helper (there {x = x} x₁∈xs) (here x₂≈x)   _  | yes x₁≈x = trans x₁≈x (sym x₂≈x)
     helper (there {x = x} x₁∈xs) (here x₂≈x)   () | no  x₁≉x
-    helper (there {x = x} x₁∈xs) (there x₂∈xs) _  with x₁ ≟ x | x₂ ≟ x
+    helper (there {x = x} x₁∈xs) (there x₂∈xs) _  with x₁ ≈? x | x₂ ≈? x
     helper (there {x = x} x₁∈xs) (there x₂∈xs) _  | yes x₁≈x | yes x₂≈x = trans x₁≈x (sym x₂≈x)
     helper (there {x = x} x₁∈xs) (there x₂∈xs) () | yes x₁≈x | no  x₂≉x
     helper (there {x = x} x₁∈xs) (there x₂∈xs) () | no  x₁≉x | yes x₂≈x
@@ -124,7 +125,7 @@ record _⊕_ (counted : List Elem) (n : ℕ) : Set where
 
 -- A countdown can be initialised by proving that Elem is finite.
 
-empty : ∀ {n} → Injection D.setoid (≡.setoid (Fin n)) → [] ⊕ n
+empty : ∀ {n} → Injection setoid (≡.setoid (Fin n)) → [] ⊕ n
 empty inj =
   record { kind      = inj₂ ∘ to
          ; injective = λ {x} {y} {i} eq₁ eq₂ → injective (begin
@@ -147,7 +148,7 @@ emptyFromList counted complete = empty record
 -- Finds out if an element has been counted yet.
 
 lookup : ∀ {counted n} → counted ⊕ n → ∀ x → Dec (x ∈ counted)
-lookup {counted} _ x = Any.any? (_≟_ x) counted
+lookup {counted} _ x = x ∈? counted
 
 -- When no element remains to be counted all elements have been
 -- counted.
@@ -181,7 +182,7 @@ insert {counted} {n} counted⊕1+n x x∉counted =
   helper _ _ _ eq₁ eq₂ refl = injective eq₁ eq₂
 
   kind′ : ∀ y → y ∈ x ∷ counted ⊎ Fin n
-  kind′  y with y ≟ x | kind x | kind y | helper x y
+  kind′  y with y ≈? x | kind x | kind y | helper x y
   kind′  y | yes y≈x | _              | _              | _   = inj₁ (here y≈x)
   kind′  y | _       | inj₁ x∈counted | _              | _   = contradiction x∈counted x∉counted
   kind′  y | _       | _              | inj₁ y∈counted | _   = inj₁ (there y∈counted)
@@ -189,7 +190,7 @@ insert {counted} {n} counted⊕1+n x x∉counted =
     inj₂ (punchOut (y≉x ∘ sym ∘ hlp _ refl refl))
 
   inj : ∀ {y z i} → kind′ y ≡ inj₂ i → kind′ z ≡ inj₂ i → y ≈ z
-  inj {y} {z} eq₁ eq₂ with y ≟ x | z ≟ x | kind x | kind y | kind z
+  inj {y} {z} eq₁ eq₂ with y ≈? x | z ≈? x | kind x | kind y | kind z
                          | helper x y | helper x z | helper y z
   inj ()  _   | yes _ | _     | _              | _      | _      | _ | _ | _
   inj _   ()  | _     | yes _ | _              | _      | _      | _ | _ | _
diff --git a/src/Data/List/Fresh/Membership/DecSetoid.agda b/src/Data/List/Fresh/Membership/DecSetoid.agda
index 6a119d5d43..ddbf7ed495 100644
--- a/src/Data/List/Fresh/Membership/DecSetoid.agda
+++ b/src/Data/List/Fresh/Membership/DecSetoid.agda
@@ -17,7 +17,7 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary.Decidable using (¬?)
 
-open DecSetoid DS using (setoid; _≟_) renaming (Carrier to A)
+open DecSetoid DS using (setoid; _≈?_) renaming (Carrier to A)
 
 private
   variable
@@ -34,7 +34,7 @@ open MembershipSetoid setoid public
 infix 4 _∈?_ _∉?_
 
 _∈?_ : Decidable (_∈_ {R = R})
-x ∈? xs = any? (x ≟_) xs
+x ∈? xs = any? (x ≈?_) xs
 
 _∉?_ : Decidable (_∉_ {R = R})
 x ∉? xs = ¬? (x ∈? xs)
diff --git a/src/Data/List/Membership/DecPropositional.agda b/src/Data/List/Membership/DecPropositional.agda
index 3c81c0ff1e..559aa7aeda 100644
--- a/src/Data/List/Membership/DecPropositional.agda
+++ b/src/Data/List/Membership/DecPropositional.agda
@@ -9,7 +9,7 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Membership.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+  {a} {A : Set a} (_≡?_ : DecidableEquality A) where
 
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
@@ -17,5 +17,5 @@ open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 -- Re-export contents of propositional membership
 
 open import Data.List.Membership.Propositional {A = A} public
-open import Data.List.Membership.DecSetoid (decSetoid _≟_) public
+open import Data.List.Membership.DecSetoid (decSetoid _≡?_) public
   using (_∈?_; _∉?_)
diff --git a/src/Data/List/Membership/DecSetoid.agda b/src/Data/List/Membership/DecSetoid.agda
index 29fe88adb6..4b3d52cfde 100644
--- a/src/Data/List/Membership/DecSetoid.agda
+++ b/src/Data/List/Membership/DecSetoid.agda
@@ -14,12 +14,12 @@ open import Data.List.Relation.Unary.Any using (any?)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary.Decidable using (¬?)
 
-open DecSetoid DS
+open DecSetoid DS using (_≈?_; setoid)
 
 ------------------------------------------------------------------------
 -- Re-export contents of propositional membership
 
-open import Data.List.Membership.Setoid (DecSetoid.setoid DS) public
+open import Data.List.Membership.Setoid setoid public
 
 ------------------------------------------------------------------------
 -- Other operations
@@ -27,7 +27,7 @@ open import Data.List.Membership.Setoid (DecSetoid.setoid DS) public
 infix 4 _∈?_ _∉?_
 
 _∈?_ : Decidable _∈_
-x ∈? xs = any? (x ≟_) xs
+x ∈? xs = any? (x ≈?_) xs
 
 _∉?_ : Decidable _∉_
 x ∉? xs = ¬? (x ∈? xs)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 370676e601..0674b2fff1 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -76,13 +76,13 @@ private
 ∷-injectiveʳ refl = refl
 
 ∷-dec : Dec (x ≡ y) → Dec (xs ≡ ys) → Dec (x ∷ xs ≡ y List.∷ ys)
-∷-dec x≟y xs≟ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≟y ×-dec xs≟ys)
+∷-dec x≡?y xs≡?ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≡?y ×-dec xs≡?ys)
 
 ≡-dec : DecidableEquality A → DecidableEquality (List A)
-≡-dec _≟_ []       []       = yes refl
-≡-dec _≟_ (x ∷ xs) []       = no λ()
-≡-dec _≟_ []       (y ∷ ys) = no λ()
-≡-dec _≟_ (x ∷ xs) (y ∷ ys) = ∷-dec (x ≟ y) (≡-dec _≟_ xs ys)
+≡-dec _≈?_ []       []       = yes refl
+≡-dec _≈?_ (x ∷ xs) []       = no λ()
+≡-dec _≈?_ []       (y ∷ ys) = no λ()
+≡-dec _≈?_ (x ∷ xs) (y ∷ ys) = ∷-dec (x ≈? y) (≡-dec _≈?_ xs ys)
 
 ------------------------------------------------------------------------
 -- map
diff --git a/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda b/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda
index 82d39c261f..e4667a4905 100644
--- a/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda
@@ -9,7 +9,7 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Disjoint.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A)
+  {a} {A : Set a} (_≡?_ : DecidableEquality A)
   where
 
 ------------------------------------------------------------------------
@@ -17,5 +17,5 @@ module Data.List.Relation.Binary.Disjoint.DecPropositional
 
 open import Data.List.Relation.Binary.Disjoint.Propositional {A = A} public
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
-open import Data.List.Relation.Binary.Disjoint.DecSetoid (decSetoid _≟_) public
+open import Data.List.Relation.Binary.Disjoint.DecSetoid (decSetoid _≡?_) public
   using (disjoint?)
diff --git a/src/Data/List/Relation/Binary/Disjoint/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Disjoint/Propositional/Properties.agda
index 23021265d2..33af44c765 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Propositional/Properties.agda
@@ -61,6 +61,6 @@ module _ {a} {A : Set a} where
   concat⁺ʳ = S.concat⁺ʳ (setoid _)
 
   -- deduplicate
-  module _ (_≟_ : DecidableEquality A) where
-    deduplicate⁺ : Disjoint xs ys → Disjoint (deduplicate _≟_ xs) (deduplicate _≟_ ys)
-    deduplicate⁺ = S.deduplicate⁺ Sᴬ _≟_
+  module _ (_≡?_ : DecidableEquality A) where
+    deduplicate⁺ : Disjoint xs ys → Disjoint (deduplicate _≡?_ xs) (deduplicate _≡?_ ys)
+    deduplicate⁺ = S.deduplicate⁺ Sᴬ _≡?_
diff --git a/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
index 3416fc31e8..9d439af381 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
@@ -22,7 +22,7 @@ open import Data.Sum.Base using (inj₁; inj₂)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (Symmetric; DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 module _ {c ℓ} (S : Setoid c ℓ) where
 
@@ -56,9 +56,9 @@ module _ {c ℓ} (S : Setoid c ℓ) where
   ... | inj₂ v∈xss = concat⁺ʳ vs#xss (v∈vs , v∈xss)
 
   -- deduplicate
-  module _ (_≟_ : DecidableEquality A) where
+  module _ (_≡?_ : DecidableEquality A) where
 
     deduplicate⁺ : ∀ {xs ys} → Disjoint S xs ys →
-                   Disjoint S (deduplicate _≟_ xs) (deduplicate _≟_ ys)
-    deduplicate⁺ = let ∈-dedup⁻ = Mem.∈-deduplicate⁻ S _≟_ in
+                   Disjoint S (deduplicate _≡?_ xs) (deduplicate _≡?_ ys)
+    deduplicate⁺ = let ∈-dedup⁻ = Mem.∈-deduplicate⁻ S _≡?_ in
       _∘ Product.map (∈-dedup⁻ _) (∈-dedup⁻ _)
diff --git a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
index 8f6b20ec99..ad15a890f1 100644
--- a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
@@ -13,7 +13,7 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Equality.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+  {a} {A : Set a} (_≈?_ : DecidableEquality A) where
 
 open import Data.List.Base using (List)
 open import Data.List.Properties using (≡-dec)
@@ -26,7 +26,7 @@ open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 -- equality
 
 open PropositionalEq public
-open DecSetoidEq (decSetoid _≟_) public
+open DecSetoidEq (decSetoid _≈?_) public
   using (_≋?_; ≋-isDecEquivalence; ≋-decSetoid)
 
 ------------------------------------------------------------------------
@@ -35,4 +35,4 @@ open DecSetoidEq (decSetoid _≟_) public
 infix 4 _≡?_
 
 _≡?_ : DecidableEquality (List A)
-_≡?_ = ≡-dec _≟_
+_≡?_ = ≡-dec _≈?_
diff --git a/src/Data/List/Relation/Binary/Equality/DecSetoid.agda b/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
index d0c4ab6923..2efa6dc71e 100644
--- a/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
@@ -23,7 +23,7 @@ open DecSetoid DS using (setoid)
 
 open DecSetoid ≋-decSetoid public
   using ()
-  renaming (isDecEquivalence to ≋-isDecEquivalence; _≟_ to _≋?_)
+  renaming (isDecEquivalence to ≋-isDecEquivalence; _≈?_ to _≋?_)
 
 ------------------------------------------------------------------------
 -- Make all definitions from setoid equality available
diff --git a/src/Data/List/Relation/Binary/Lex/NonStrict.agda b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
index 893620d3ff..c500292644 100644
--- a/src/Data/List/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
@@ -83,12 +83,12 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
 
     <-compare : Symmetric _≈_ → Decidable _≈_ → Antisymmetric _≈_ _≼_ →
                 Total _≼_ → Trichotomous _≋_ _<_
-    <-compare sym _≟_ antisym tot =
-      Strict.<-compare sym (Conv.<-trichotomous _ _ sym _≟_ antisym tot)
+    <-compare sym _≈?_ antisym tot =
+      Strict.<-compare sym (Conv.<-trichotomous _ _ sym _≈?_ antisym tot)
 
     <-decidable : Decidable _≈_ → Decidable _≼_ → Decidable _<_
-    <-decidable _≟_ _≼?_ =
-      Core.decidable (no id) _≟_ (Conv.<-decidable _ _ _≟_ _≼?_)
+    <-decidable _≈?_ _≼?_ =
+      Core.decidable (no id) _≈?_ (Conv.<-decidable _ _ _≈?_ _≼?_)
 
     <-isStrictPartialOrder : IsPartialOrder _≈_ _≼_ →
                              IsStrictPartialOrder _≋_ _<_
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index f274241522..301144d23c 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -204,7 +204,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
     ≤-isDecPartialOrder sto = record
       { isPartialOrder = ≤-isPartialOrder isStrictPartialOrder
       ; _≈?_            = Pointwise.decidable _≈?_
-      ; _≤?_           = ≤-decidable _≟_ _<?_
+      ; _≤?_           = ≤-decidable _≈?_ _<?_
       } where open IsStrictTotalOrder sto
 
     ≤-isTotalOrder : IsStrictTotalOrder _≈_ _≺_ → IsTotalOrder _≋_ _≤_
@@ -218,8 +218,8 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
                         IsDecTotalOrder _≋_ _≤_
     ≤-isDecTotalOrder sto = record
       { isTotalOrder = ≤-isTotalOrder sto
-      ; _≈?_          = Pointwise.decidable _≈?_
-      ; _≤?_         = ≤-decidable _≟_ _<?_
+      ; _≈?_         = Pointwise.decidable _≈?_
+      ; _≤?_         = ≤-decidable _≈?_ _<?_
       }
       where open IsStrictTotalOrder sto
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties/WithK.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties/WithK.agda
index 3bed7656fc..3202241446 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties/WithK.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties/WithK.agda
@@ -31,13 +31,13 @@ open import Relation.Binary.Definitions using (DecidableEquality)
 ------------------------------------------------------------------------
 -- deduplicate
 
-module _ {a} {A : Set a} (_≟_ : DecidableEquality A) where
+module _ {a} {A : Set a} (_≡?_ : DecidableEquality A) where
 
   private
-    dedup≡   = deduplicate    _≟_
-    ∈-dedup≡ = deduplicate-∈⇔ _≟_
+    dedup≡   = deduplicate    _≡?_
+    ∈-dedup≡ = deduplicate-∈⇔ _≡?_
 
-  open import Data.List.Relation.Unary.Unique.DecPropositional.Properties _≟_
+  open import Data.List.Relation.Unary.Unique.DecPropositional.Properties _≡?_
     using (deduplicate-!)
 
   dedup-++-↭ : ∀ {xs ys} → Disjoint xs ys → dedup≡ (xs ++ ys) ↭ dedup≡ xs ++ dedup≡ ys
@@ -45,7 +45,7 @@ module _ {a} {A : Set a} (_≟_ : DecidableEquality A) where
     = ∼bag⇒↭
     $ unique∧set⇒bag
         (deduplicate-! _)
-        (++⁺ (deduplicate-! _) (deduplicate-! _) (deduplicate⁺ _≟_ disj))
+        (++⁺ (deduplicate-! _) (deduplicate-! _) (deduplicate⁺ _≡?_ disj))
     λ {x} → begin
     x ∈ dedup≡ (xs ++ ys)           ∼⟨ SK-sym ∈-dedup≡ ⟩
     x ∈ xs ++ ys                    ∼⟨ ++-∈⇔ ⟩
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
index a68c5a2b2e..0d4a0192ae 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
@@ -10,9 +10,9 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional.Solver
-       {a} {A : Set a} (_≟_ : DecidableEquality A)
+       {a} {A : Set a} (_≡?_ : DecidableEquality A)
        where
 
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
-open import Data.List.Relation.Binary.Sublist.DecSetoid.Solver (decSetoid _≟_) public
+open import Data.List.Relation.Binary.Sublist.DecSetoid.Solver (decSetoid _≡?_) public
diff --git a/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda
index 7564052444..f74650285a 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda
@@ -12,5 +12,5 @@ module Data.List.Relation.Binary.Sublist.DecSetoid.Solver {c ℓ} (S : DecSetoid
 
 open DecSetoid S
 
-open import Data.List.Relation.Binary.Sublist.Heterogeneous.Solver _≈_ refl _≟_
+open import Data.List.Relation.Binary.Sublist.Heterogeneous.Solver _≈_ refl _≈?_
   using (Item; module Item; TList; module TList; prove) public
diff --git a/src/Data/List/Relation/Binary/Subset/DecPropositional.agda b/src/Data/List/Relation/Binary/Subset/DecPropositional.agda
index a183d298f3..921aa07643 100644
--- a/src/Data/List/Relation/Binary/Subset/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Subset/DecPropositional.agda
@@ -9,7 +9,7 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Subset.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A)
+  {a} {A : Set a} (_≡?_ : DecidableEquality A)
   where
 
 ------------------------------------------------------------------------
@@ -17,5 +17,5 @@ module Data.List.Relation.Binary.Subset.DecPropositional
 
 open import Data.List.Relation.Binary.Subset.Propositional {A = A} public
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
-open import Data.List.Relation.Binary.Subset.DecSetoid (decSetoid _≟_) public
+open import Data.List.Relation.Binary.Subset.DecSetoid (decSetoid _≡?_) public
   using (_⊆?_)
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index b91e409d90..7afd5b280d 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -78,8 +78,8 @@ module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) where
 module _ (S? : DecSetoid a ℓ₁) where
   open DecSetoid S? renaming (setoid to S)
 
-  deduplicate⁺ : IsEnumeration S xs → IsEnumeration S (deduplicate _≟_ xs)
-  deduplicate⁺ = ∈-deduplicate⁺ S _≟_ (respʳ-flip S) ∘_
+  deduplicate⁺ : IsEnumeration S xs → IsEnumeration S (deduplicate _≈?_ xs)
+  deduplicate⁺ = ∈-deduplicate⁺ S _≈?_ (respʳ-flip S) ∘_
 
 ------------------------------------------------------------------------
 -- lookup
diff --git a/src/Data/List/Relation/Unary/Grouped.agda b/src/Data/List/Relation/Unary/Grouped.agda
index 8ee0d50f0c..5a14f76d24 100644
--- a/src/Data/List/Relation/Unary/Grouped.agda
+++ b/src/Data/List/Relation/Unary/Grouped.agda
@@ -42,10 +42,10 @@ data Grouped {A : Set a} (_≈_ : Rel A ℓ) : Pred (List A) (a ⊔ ℓ) where
 module _ {_≈_ : Rel A ℓ} where
 
   grouped? : B.Decidable _≈_ → U.Decidable (Grouped _≈_)
-  grouped? _≟_ [] = yes []
-  grouped? _≟_ (x ∷ []) = yes ([] ∷≉ [])
-  grouped? _≟_ (x ∷ y ∷ xs) =
-    Dec.map′ from to ((x ≟ y ⊎? all? (λ z → ¬? (x ≟ z)) (y ∷ xs)) ×? (grouped? _≟_ (y ∷ xs)))
+  grouped? _≈?_ [] = yes []
+  grouped? _≈?_ (x ∷ []) = yes ([] ∷≉ [])
+  grouped? _≈?_ (x ∷ y ∷ xs) =
+    Dec.map′ from to ((x ≈? y ⊎? all? (λ z → ¬? (x ≈? z)) (y ∷ xs)) ×? (grouped? _≈?_ (y ∷ xs)))
     where
     from : ((x ≈ y) ⊎ All (λ z → ¬ (x ≈ z)) (y ∷ xs)) × Grouped _≈_ (y ∷ xs) → Grouped _≈_ (x ∷ y ∷ xs)
     from (inj₁ x≈y          , grouped[y∷xs]) = x≈y          ∷≈ grouped[y∷xs]
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
index f2e3692103..c0e11d3a50 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
@@ -11,9 +11,9 @@ open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Relation.Unary.Unique.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+  {a} {A : Set a} (_≡?_ : DecidableEquality A) where
 
 ------------------------------------------------------------------------
 -- Re-export setoid definition
 
-open import Data.List.Relation.Unary.Unique.DecSetoid (decSetoid _≟_) public
+open import Data.List.Relation.Unary.Unique.DecSetoid (decSetoid _≡?_) public
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
index 44039ab152..4a5aeb717f 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
@@ -9,11 +9,11 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Unary.Unique.DecPropositional.Properties
-  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+  {a} {A : Set a} (_≡?_ : DecidableEquality A) where
 
 open import Data.List.Base using (deduplicate)
 open import Data.List.Relation.Unary.All.Properties using (all-filter)
-open import Data.List.Relation.Unary.Unique.DecPropositional _≟_
+open import Data.List.Relation.Unary.Unique.DecPropositional _≡?_
 import Data.List.Relation.Unary.Unique.DecSetoid.Properties as Setoid
 open import Level
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
@@ -26,5 +26,5 @@ open import Data.List.Relation.Unary.Unique.Propositional.Properties public
 ------------------------------------------------------------------------
 -- deduplicate
 
-deduplicate-! : ∀ xs → Unique (deduplicate _≟_ xs)
-deduplicate-! = Setoid.deduplicate-! (decSetoid _≟_)
+deduplicate-! : ∀ xs → Unique (deduplicate _≡?_ xs)
+deduplicate-! = Setoid.deduplicate-! (decSetoid _≡?_)
diff --git a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
index 73d42b7b8a..cf91d5b539 100644
--- a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
@@ -27,6 +27,6 @@ module _ (DS : DecSetoid a ℓ) where
   open DecSetoid DS renaming (setoid to S)
   open Unique DS
 
-  deduplicate-! : ∀ xs → Unique (deduplicate _≟_ xs)
+  deduplicate-! : ∀ xs → Unique (deduplicate _≈?_ xs)
   deduplicate-! []       = []
-  deduplicate-! (x ∷ xs) = all-filter _ (deduplicate _≟_ xs) ∷ filter⁺ S _ (deduplicate-! xs)
+  deduplicate-! (x ∷ xs) = all-filter _ (deduplicate _≈?_ xs) ∷ filter⁺ S _ (deduplicate-! xs)
diff --git a/src/Data/Maybe/Properties.agda b/src/Data/Maybe/Properties.agda
index 479e08856a..079f945f88 100644
--- a/src/Data/Maybe/Properties.agda
+++ b/src/Data/Maybe/Properties.agda
@@ -40,10 +40,10 @@ just-injective : ∀ {x y} → (Maybe A ∋ just x) ≡ just y → x ≡ y
 just-injective refl = refl
 
 ≡-dec : DecidableEquality A → DecidableEquality (Maybe A)
-≡-dec _≟_ nothing  nothing  = yes refl
-≡-dec _≟_ (just x) nothing  = no λ()
-≡-dec _≟_ nothing  (just y) = no λ()
-≡-dec _≟_ (just x) (just y) = map′ (cong just) just-injective (x ≟ y)
+≡-dec _≡?_ nothing  nothing  = yes refl
+≡-dec _≡?_ (just x) nothing  = no λ()
+≡-dec _≡?_ nothing  (just y) = no λ()
+≡-dec _≡?_ (just x) (just y) = map′ (cong just) just-injective (x ≡? y)
 
 ------------------------------------------------------------------------
 -- map
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index 180b706cda..50558c3b94 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -38,22 +38,22 @@ open import Algebra.Morphism.Structures
 ------------------------------------------------------------------------
 -- Equality
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality Parity
-1ℙ ≟ 1ℙ = yes refl
-1ℙ ≟ 0ℙ = no λ()
-0ℙ ≟ 1ℙ = no λ()
-0ℙ ≟ 0ℙ = yes refl
+_≡?_ : DecidableEquality Parity
+1ℙ ≡? 1ℙ = yes refl
+1ℙ ≡? 0ℙ = no λ()
+0ℙ ≡? 1ℙ = no λ()
+0ℙ ≡? 0ℙ = yes refl
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid Parity
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ≡-isDecEquivalence : IsDecEquivalence _≡_
-≡-isDecEquivalence = isDecEquivalence _≟_
+≡-isDecEquivalence = isDecEquivalence _≡?_
 
 ------------------------------------------------------------------------
 -- _⁻¹
@@ -554,3 +554,19 @@ parity-isSemiringHomomorphism = record
   { isNearSemiringHomomorphism = parity-isNearSemiringHomomorphism
   ; 1#-homo = refl
   }
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Record.agda b/src/Data/Record.agda
index 709ce69a67..66b74bca87 100644
--- a/src/Data/Record.agda
+++ b/src/Data/Record.agda
@@ -22,7 +22,7 @@ open import Relation.Nullary.Decidable using (does)
 -- The module is parametrised by the type of labels, which should come
 -- with decidable equality.
 
-module Data.Record {ℓ} (Label : Set ℓ) (_≟_ : DecidableEquality Label) where
+module Data.Record {ℓ} (Label : Set ℓ) (_≡?_ : DecidableEquality Label) where
 
 ------------------------------------------------------------------------
 -- A Σ-type with a manifest field
@@ -89,7 +89,7 @@ infix 4 _∈_
 
 _∈_ : ∀ {s} → Label → Signature s → Set
 ℓ ∈ Sig =
-  foldr (λ ℓ′ → if does (ℓ ≟ ℓ′) then (λ _ → ⊤) else id) ⊥ (labels Sig)
+  foldr (λ ℓ′ → if does (ℓ ≡? ℓ′) then (λ _ → ⊤) else id) ⊥ (labels Sig)
 
 ------------------------------------------------------------------------
 -- Projections
@@ -100,10 +100,10 @@ _∈_ : ∀ {s} → Label → Signature s → Set
 Restrict : ∀ {s} (Sig : Signature s) (ℓ : Label) → ℓ ∈ Sig →
            Signature s
 Restrict ∅              ℓ ()
-Restrict (Sig , ℓ′ ∶ A) ℓ ℓ∈ with does (ℓ ≟ ℓ′)
+Restrict (Sig , ℓ′ ∶ A) ℓ ℓ∈ with does (ℓ ≡? ℓ′)
 ... | true  = Sig
 ... | false = Restrict Sig ℓ ℓ∈
-Restrict (Sig , ℓ′ ≔ a) ℓ ℓ∈ with does (ℓ ≟ ℓ′)
+Restrict (Sig , ℓ′ ≔ a) ℓ ℓ∈ with does (ℓ ≡? ℓ′)
 ... | true  = Sig
 ... | false = Restrict Sig ℓ ℓ∈
 
@@ -113,10 +113,10 @@ Restricted Sig ℓ ℓ∈ = Record (Restrict Sig ℓ ℓ∈)
 Proj : ∀ {s} (Sig : Signature s) (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} →
        Restricted Sig ℓ ℓ∈ → Set s
 Proj ∅              ℓ {}
-Proj (Sig , ℓ′ ∶ A) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)
+Proj (Sig , ℓ′ ∶ A) ℓ {ℓ∈} with does (ℓ ≡? ℓ′)
 ... | true  = A
 ... | false = Proj Sig ℓ {ℓ∈}
-Proj (_,_≔_ Sig ℓ′ {A = A} a) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)
+Proj (_,_≔_ Sig ℓ′ {A = A} a) ℓ {ℓ∈} with does (ℓ ≡? ℓ′)
 ... | true  = A
 ... | false = Proj Sig ℓ {ℓ∈}
 
@@ -127,10 +127,10 @@ infixl 5 _∣_
 _∣_ : ∀ {s} {Sig : Signature s} → Record Sig →
       (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} → Restricted Sig ℓ ℓ∈
 _∣_ {Sig = ∅}            r       ℓ {}
-_∣_ {Sig = Sig , ℓ′ ∶ A} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)
+_∣_ {Sig = Sig , ℓ′ ∶ A} (rec r) ℓ {ℓ∈} with does (ℓ ≡? ℓ′)
 ... | true  = Σ.proj₁ r
 ... | false = _∣_ (Σ.proj₁ r) ℓ {ℓ∈}
-_∣_ {Sig = Sig , ℓ′ ≔ a} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)
+_∣_ {Sig = Sig , ℓ′ ≔ a} (rec r) ℓ {ℓ∈} with does (ℓ ≡? ℓ′)
 ... | true  = Manifest-Σ.proj₁ r
 ... | false = _∣_ (Manifest-Σ.proj₁ r) ℓ {ℓ∈}
 
@@ -140,10 +140,10 @@ _·_ : ∀ {s} {Sig : Signature s} (r : Record Sig)
       (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} →
       Proj Sig ℓ {ℓ∈} (r ∣ ℓ)
 _·_ {Sig = ∅}            r       ℓ {}
-_·_ {Sig = Sig , ℓ′ ∶ A} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)
+_·_ {Sig = Sig , ℓ′ ∶ A} (rec r) ℓ {ℓ∈} with does (ℓ ≡? ℓ′)
 ... | true  = Σ.proj₂ r
 ... | false = _·_ (Σ.proj₁ r) ℓ {ℓ∈}
-_·_ {Sig = Sig , ℓ′ ≔ a} (rec r) ℓ {ℓ∈} with does (ℓ ≟ ℓ′)
+_·_ {Sig = Sig , ℓ′ ≔ a} (rec r) ℓ {ℓ∈} with does (ℓ ≡? ℓ′)
 ... | true  = Manifest-Σ.proj₂ r
 ... | false = _·_ (Manifest-Σ.proj₁ r) ℓ {ℓ∈}
 
@@ -159,10 +159,10 @@ mutual
   _With_≔_ : ∀ {s} (Sig : Signature s) (ℓ : Label) {ℓ∈ : ℓ ∈ Sig} →
              ((r : Restricted Sig ℓ ℓ∈) → Proj Sig ℓ r) → Signature s
   _With_≔_ ∅ ℓ {} a
-  _With_≔_ (Sig , ℓ′ ∶ A)   ℓ {ℓ∈} a with does (ℓ ≟ ℓ′)
+  _With_≔_ (Sig , ℓ′ ∶ A)   ℓ {ℓ∈} a with does (ℓ ≡? ℓ′)
   ... | true  = Sig                   , ℓ′ ≔ a
   ... | false = _With_≔_ Sig ℓ {ℓ∈} a , ℓ′ ∶ A ∘ drop-With
-  _With_≔_  (Sig , ℓ′ ≔ a′) ℓ {ℓ∈} a with does (ℓ ≟ ℓ′)
+  _With_≔_  (Sig , ℓ′ ≔ a′) ℓ {ℓ∈} a with does (ℓ ≡? ℓ′)
   ... | true  = Sig                   , ℓ′ ≔ a
   ... | false = _With_≔_ Sig ℓ {ℓ∈} a , ℓ′ ≔ a′ ∘ drop-With
 
@@ -170,9 +170,9 @@ mutual
               {a : (r : Restricted Sig ℓ ℓ∈) → Proj Sig ℓ r} →
               Record (_With_≔_ Sig ℓ {ℓ∈} a) → Record Sig
   drop-With {Sig = ∅} {ℓ∈ = ()}      r
-  drop-With {Sig = Sig , ℓ′ ∶ A} {ℓ} (rec r) with does (ℓ ≟ ℓ′)
+  drop-With {Sig = Sig , ℓ′ ∶ A} {ℓ} (rec r) with does (ℓ ≡? ℓ′)
   ... | true  = rec (Manifest-Σ.proj₁ r , Manifest-Σ.proj₂ r)
   ... | false = rec (drop-With (Σ.proj₁ r) , Σ.proj₂ r)
-  drop-With {Sig = Sig , ℓ′ ≔ a} {ℓ} (rec r) with does (ℓ ≟ ℓ′)
+  drop-With {Sig = Sig , ℓ′ ≔ a} {ℓ} (rec r) with does (ℓ ≡? ℓ′)
   ... | true  = rec (Manifest-Σ.proj₁ r ,)
   ... | false = rec (drop-With (Manifest-Σ.proj₁ r) ,)
diff --git a/src/Data/Refinement.agda b/src/Data/Refinement.agda
index 4fc812677e..07825f5690 100644
--- a/src/Data/Refinement.agda
+++ b/src/Data/Refinement.agda
@@ -17,4 +17,4 @@ open import Data.Refinement.Base public
 -- Publicly re-export queries
 
 open import Data.Refinement.Properties public
-  using (value-injective; _≟_)
+  using (value-injective; _≟_; _≡?_)
diff --git a/src/Data/Refinement/Properties.agda b/src/Data/Refinement/Properties.agda
index 4dff634964..34fc50dc46 100644
--- a/src/Data/Refinement/Properties.agda
+++ b/src/Data/Refinement/Properties.agda
@@ -28,8 +28,24 @@ private
 value-injective : value v ≡ value w → v ≡ w
 value-injective refl = refl
 
-module _ (eq? : DecidableEquality A) where
+module _ (_≈?_ : DecidableEquality A) where
 
-  infix 4 _≟_
-  _≟_ : DecidableEquality [ x ∈ A ∣ P x ]
-  v ≟ w = Dec.map′ value-injective (cong value) (eq? (value v) (value w))
+  infix 4 _≡?_
+  _≡?_ : DecidableEquality [ x ∈ A ∣ P x ]
+  v ≡? w = Dec.map′ value-injective (cong value) (value v ≈? value w)
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Sign/Properties.agda b/src/Data/Sign/Properties.agda
index 4d66eb399b..ba2757ea8f 100644
--- a/src/Data/Sign/Properties.agda
+++ b/src/Data/Sign/Properties.agda
@@ -34,22 +34,22 @@ open import Algebra.Consequences.Propositional
 ------------------------------------------------------------------------
 -- Equality
 
-infix 4 _≟_
+infix 4 _≡?_
 
-_≟_ : DecidableEquality Sign
-- ≟ - = yes refl
-- ≟ + = no λ()
-+ ≟ - = no λ()
-+ ≟ + = yes refl
+_≡?_ : DecidableEquality Sign
+- ≡? - = yes refl
+- ≡? + = no λ()
++ ≡? - = no λ()
++ ≡? + = yes refl
 
 ≡-setoid : Setoid 0ℓ 0ℓ
 ≡-setoid = setoid Sign
 
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
-≡-decSetoid = decSetoid _≟_
+≡-decSetoid = decSetoid _≡?_
 
 ≡-isDecEquivalence : IsDecEquivalence _≡_
-≡-isDecEquivalence = isDecEquivalence _≟_
+≡-isDecEquivalence = isDecEquivalence _≡?_
 
 ------------------------------------------------------------------------
 -- opposite
@@ -209,3 +209,19 @@ s*opposite[s]≡- - = refl
 opposite[s]*s≡- : ∀ s → opposite s * s ≡ -
 opposite[s]*s≡- + = refl
 opposite[s]*s≡- - = refl
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.4
+
+infix 4 _≟_
+_≟_ = _≡?_
+{-# WARNING_ON_USAGE _≟_
+"Warning: _≟_ was deprecated in v2.4.
+Please use _≡?_ instead."
+#-}
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index 9acf4961f5..337857b238 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -79,10 +79,10 @@ drop-inj₂ (inj₂ x) = x
 ⊎-substitutive subst₁ subst₂ P (inj₂ x) = subst₂ (P ∘ inj₂) x
 
 ⊎-decidable : Decidable R → Decidable S → Decidable (Pointwise R S)
-⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₁ y) = Dec.map′ inj₁ drop-inj₁ (x ≟₁ y)
-⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₂ y) = no λ()
-⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₁ y) = no λ()
-⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₂ y) = Dec.map′ inj₂ drop-inj₂ (x ≟₂ y)
+⊎-decidable _R?_ _S?_ (inj₁ x) (inj₁ y) = Dec.map′ inj₁ drop-inj₁ (x R? y)
+⊎-decidable _R?_ _S?_ (inj₁ x) (inj₂ y) = no λ()
+⊎-decidable _R?_ _S?_ (inj₂ x) (inj₁ y) = no λ()
+⊎-decidable _R?_ _S?_ (inj₂ x) (inj₂ y) = Dec.map′ inj₂ drop-inj₂ (x S? y)
 
 ⊎-reflexive : ≈₁ ⇒ R → ≈₂ ⇒ S →
               (Pointwise ≈₁ ≈₂) ⇒ (Pointwise R S)
diff --git a/src/Data/Unit/Polymorphic.agda b/src/Data/Unit/Polymorphic.agda
index 3d828e5a50..e68273656e 100644
--- a/src/Data/Unit/Polymorphic.agda
+++ b/src/Data/Unit/Polymorphic.agda
@@ -16,4 +16,4 @@ open import Data.Unit.Polymorphic.Base public
 ------------------------------------------------------------------------
 -- Re-export query operations
 
-open import Data.Unit.Polymorphic.Properties public using (_≟_)
+open import Data.Unit.Polymorphic.Properties public using (_≟_; _≡?_)
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index 7987473e7b..bcdd8245fb 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -50,10 +50,10 @@ module _ {xs ys : Vector A (suc n)} where
   ∷-injective eq = eq zero , eq ∘ suc
 
 ≗-dec : DecidableEquality A → Decidable {A = Vector A n} _≗_
-≗-dec {n = zero}  _≟_ xs ys = yes λ ()
-≗-dec {n = suc n} _≟_ xs ys =
+≗-dec {n = zero}  _≡?_ xs ys = yes λ()
+≗-dec {n = suc n} _≡?_ xs ys =
   map′ (Product.uncurry ∷-cong) ∷-injective
-       (head xs ≟ head ys ×? ≗-dec _≟_ (tail xs) (tail ys))
+       (head xs ≡? head ys ×? ≗-dec _≡?_ (tail xs) (tail ys))
 
 ------------------------------------------------------------------------
 -- updateAt
diff --git a/src/Data/Vec/Membership/DecPropositional.agda b/src/Data/Vec/Membership/DecPropositional.agda
index f7b028e11f..2ec4d5d43c 100644
--- a/src/Data/Vec/Membership/DecPropositional.agda
+++ b/src/Data/Vec/Membership/DecPropositional.agda
@@ -9,7 +9,7 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.Vec.Membership.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+  {a} {A : Set a} (_≡?_ : DecidableEquality A) where
 
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
@@ -17,5 +17,5 @@ open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 -- Re-export contents of propositional membership
 
 open import Data.Vec.Membership.Propositional {A = A} public
-open import Data.Vec.Membership.DecSetoid (decSetoid _≟_) public
+open import Data.Vec.Membership.DecSetoid (decSetoid _≡?_) public
   using (_∈?_)
diff --git a/src/Data/Vec/Membership/DecSetoid.agda b/src/Data/Vec/Membership/DecSetoid.agda
index fac9138660..0c0915f17a 100644
--- a/src/Data/Vec/Membership/DecSetoid.agda
+++ b/src/Data/Vec/Membership/DecSetoid.agda
@@ -26,4 +26,4 @@ open import Data.Vec.Membership.Setoid setoid public
 infix 4 _∈?_
 
 _∈?_ : ∀ x {n} (xs : Vec A n) → Dec (x ∈ xs)
-x ∈? xs = any? (x ≟_) xs
+x ∈? xs = any? (x ≈?_) xs
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index bdf8a08f7e..471e36c413 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -75,9 +75,9 @@ toList-injective m≡n (x ∷ xs) (y ∷ ys) xs=ys =
 ∷-injective refl = refl , refl
 
 ≡-dec : DecidableEquality A → DecidableEquality (Vec A n)
-≡-dec _≟_ []       []       = yes refl
-≡-dec _≟_ (x ∷ xs) (y ∷ ys) = map′ (uncurry (cong₂ _∷_))
-  ∷-injective (x ≟ y ×? ≡-dec _≟_ xs ys)
+≡-dec _≈?_ []       []       = yes refl
+≡-dec _≈?_ (x ∷ xs) (y ∷ ys) = map′ (uncurry (cong₂ _∷_))
+  ∷-injective (x ≈? y ×? ≡-dec _≈?_ xs ys)
 
 ------------------------------------------------------------------------
 -- _[_]=_
diff --git a/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda b/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
index e6dbf84f81..5d43db4e33 100644
--- a/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
@@ -9,7 +9,7 @@
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.Vec.Relation.Binary.Equality.DecPropositional
-  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+  {a} {A : Set a} (_≈?_ : DecidableEquality A) where
 
 open import Data.Vec.Base using (Vec)
 open import Data.Vec.Properties using (≡-dec)
@@ -22,7 +22,7 @@ open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 -- equality
 
 open PEq public
-open DSEq (decSetoid _≟_) public
+open DSEq (decSetoid _≈?_) public
   using (_≋?_; ≋-isDecEquivalence; ≋-decSetoid)
 
 ------------------------------------------------------------------------
@@ -31,4 +31,4 @@ open DSEq (decSetoid _≟_) public
 infix 4 _≡?_
 
 _≡?_ : ∀ {n} → DecidableEquality (Vec A n)
-_≡?_ = ≡-dec _≟_
+_≡?_ = ≡-dec _≈?_
diff --git a/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda b/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
index 01fd39b051..0abe2e8d05 100644
--- a/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
@@ -31,7 +31,7 @@ open Equality setoid public
 infix 4 _≋?_
 
 _≋?_ : ∀ {m n} → Decidable (_≋_ {m} {n})
-_≋?_ = PW.decidable _≟_
+_≋?_ = PW.decidable _≈?_
 
 ≋-isDecEquivalence : ∀ n → IsDecEquivalence (_≋_ {n})
 ≋-isDecEquivalence = PW.isDecEquivalence isDecEquivalence
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index 24bee076ef..8475b41a75 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -100,12 +100,12 @@ module _ {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂} where
 
   <-cmp : Symmetric _≈_ → Decidable _≈_ → Antisymmetric _≈_ _≼_ → Total _≼_ →
           ∀ {n} → Trichotomous (_≋_ {n} {n}) _<_
-  <-cmp ≈-sym _≟_ ≼-antisym ≼-total = Strict.<-cmp ≈-sym
-    (Conv.<-trichotomous _ _ ≈-sym _≟_ ≼-antisym ≼-total)
+  <-cmp ≈-sym _≈?_ ≼-antisym ≼-total = Strict.<-cmp ≈-sym
+    (Conv.<-trichotomous _ _ ≈-sym _≈?_ ≼-antisym ≼-total)
 
   <-dec : Decidable _≈_ → Decidable _≼_ → ∀ {m n} → Decidable (_<_ {m} {n})
-  <-dec _≟_ _≼?_ = Core.decidable (no id) _≟_
-    (Conv.<-decidable _ _ _≟_ _≼?_)
+  <-dec _≈?_ _≼?_ = Core.decidable (no id) _≈?_
+    (Conv.<-decidable _ _ _≈?_ _≼?_)
 
 ------------------------------------------------------------------------
 -- Structures
@@ -186,13 +186,13 @@ module _ {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂} where
 
   ≤-total : Symmetric _≈_ → Decidable _≈_ → Antisymmetric _≈_ _≼_ → Total _≼_ →
             ∀ {n} → Total (_≤_ {n})
-  ≤-total ≈-sym _≟_ ≼-antisym ≼-total = Strict.≤-total ≈-sym
-    (Conv.<-trichotomous _ _ ≈-sym _≟_ ≼-antisym ≼-total)
+  ≤-total ≈-sym _≈?_ ≼-antisym ≼-total = Strict.≤-total ≈-sym
+    (Conv.<-trichotomous _ _ ≈-sym _≈?_ ≼-antisym ≼-total)
 
   ≤-dec : Decidable _≈_ → Decidable _≼_ →
           ∀ {m n} → Decidable (_≤_ {m} {n})
-  ≤-dec _≟_ _≼?_ = Core.decidable (yes tt) _≟_
-    (Conv.<-decidable _ _ _≟_ _≼?_)
+  ≤-dec _≈?_ _≼?_ = Core.decidable (yes tt) _≈?_
+    (Conv.<-decidable _ _ _≈?_ _≼?_)
 
   ≤-resp₂ : IsEquivalence _≈_ → _≼_ Respects₂ _≈_ →
             ∀ {n} → _Respects₂_ (_≤_ {n} {n}) _≋_
@@ -218,7 +218,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂} where
 
   ≤-isTotalOrder : Decidable _≈_ → IsTotalOrder _≈_ _≼_ →
                    ∀ {n} → IsTotalOrder (_≋_ {n} {n}) _≤_
-  ≤-isTotalOrder _≟_ ≼-isTotalOrder = Strict.≤-isTotalOrder (Conv.<-isStrictTotalOrder₁ _ _ _≟_ ≼-isTotalOrder)
+  ≤-isTotalOrder _≈?_ ≼-isTotalOrder = Strict.≤-isTotalOrder (Conv.<-isStrictTotalOrder₁ _ _ _≈?_ ≼-isTotalOrder)
 
   ≤-isDecTotalOrder : IsDecTotalOrder _≈_ _≼_ →
                       ∀ {n} → IsDecTotalOrder (_≋_ {n} {n}) _≤_
@@ -243,8 +243,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂} where
   } where open DecPoset ≼-dpo
 
 ≤-totalOrder : (≼-dto : TotalOrder a ℓ₁ ℓ₂) → Decidable (TotalOrder._≈_ ≼-dto) → ℕ → TotalOrder _ _ _
-≤-totalOrder ≼-dto _≟_ n = record
-  { isTotalOrder = ≤-isTotalOrder _≟_ isTotalOrder {n = n}
+≤-totalOrder ≼-dto _≈?_ n = record
+  { isTotalOrder = ≤-isTotalOrder _≈?_ isTotalOrder {n = n}
   } where open TotalOrder ≼-dto
 
 ≤-decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ → ℕ → DecTotalOrder _ _ _

From 651afe1d77c37baabb21036b1b2d85b9cd75ca0b Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Fri, 27 Feb 2026 16:00:37 +0000
Subject: [PATCH 19/20] fix: `README.Design.Decidability`

---
 doc/README/Design/Decidability.agda | 84 ++++++++++++++---------------
 1 file changed, 42 insertions(+), 42 deletions(-)

diff --git a/doc/README/Design/Decidability.agda b/doc/README/Design/Decidability.agda
index 63ab9f2021..a32fb8782f 100644
--- a/doc/README/Design/Decidability.agda
+++ b/doc/README/Design/Decidability.agda
@@ -24,21 +24,21 @@ open import Relation.Nary
 
 open import Relation.Nullary.Reflects
 
-infix 4 _≟₀_ _≟₁_ _≟₂_
+infix 4 _≡?₀_ _≡?₁_ _≡?₂_
 
 -- A proof of `Reflects P b` shows that a proposition `P` has the truth value of
 -- the boolean `b`. A proof of `Reflects P true` amounts to a proof of `P`, and
 -- a proof of `Reflects P false` amounts to a refutation of `P`.
 
 ex₀ : (n : ℕ) → Reflects (n ≡ n) true
-ex₀ n = ofʸ refl
+ex₀ n = of refl
 
 ex₁ : (n : ℕ) → Reflects (zero ≡ suc n) false
-ex₁ n = ofⁿ λ ()
+ex₁ n = of λ()
 
 ex₂ : (b : Bool) → Reflects (T b) b
-ex₂ false = ofⁿ id
-ex₂ true  = ofʸ tt
+ex₂ false = of id
+ex₂ true  = of tt
 
 ------------------------------------------------------------------------
 -- Dec
@@ -64,71 +64,71 @@ ex₄ n = no λ ()
 -- only the `yes` and `no` patterns. The following procedure decides whether two
 -- given natural numbers are equal.
 
-_≟₀_ : (m n : ℕ) → Dec (m ≡ n)
-zero  ≟₀ zero  = yes refl
-zero  ≟₀ suc n = no λ ()
-suc m ≟₀ zero  = no λ ()
-suc m ≟₀ suc n with m ≟₀ n
+_≡?₀_ : (m n : ℕ) → Dec (m ≡ n)
+zero  ≡?₀ zero  = yes refl
+zero  ≡?₀ suc n = no λ()
+suc m ≡?₀ zero  = no λ()
+suc m ≡?₀ suc n with m ≡?₀ n
 ... | yes p = yes (cong suc p)
 ... | no ¬p = no (¬p ∘ suc-injective)
 
--- In this case, we can see that `does (suc m ≟ suc n)` should be equal to
--- `does (m ≟ n)`, because a `yes` from `m ≟ n` gives rise to a `yes` from the
+-- In this case, we can see that `does (suc m ≡? suc n)` should be equal to
+-- `does (m ≡? n)`, because a `yes` from `m ≡? n` gives rise to a `yes` from the
 -- result, and similarly for `no`. However, in the above definition, this
 -- equality does not hold definitionally, because we always do a case split
 -- before returning a result. To avoid this, we can return the `does` part
 -- separately, before any pattern matching.
 
-_≟₁_ : (m n : ℕ) → Dec (m ≡ n)
-zero  ≟₁ zero  = yes refl
-zero  ≟₁ suc n = no λ ()
-suc m ≟₁ zero  = no λ ()
-does  (suc m ≟₁ suc n) = does (m ≟₁ n)
-proof (suc m ≟₁ suc n) with m ≟₁ n
-... | yes p = ofʸ (cong suc p)
-... | no ¬p = ofⁿ (¬p ∘ suc-injective)
+_≡?₁_ : (m n : ℕ) → Dec (m ≡ n)
+zero  ≡?₁ zero  = yes refl
+zero  ≡?₁ suc n = no λ()
+suc m ≡?₁ zero  = no λ()
+does  (suc m ≡?₁ suc n) = does (m ≡?₁ n)
+proof (suc m ≡?₁ suc n) with m ≡?₁ n
+... | yes p = of (cong suc p)
+... | no ¬p = of (¬p ∘ suc-injective)
 
 -- We now get definitional equalities such as the following.
 
-_ : (m n : ℕ) → does (5 + m ≟₁ 3 + n) ≡ does (2 + m ≟₁ n)
+_ : (m n : ℕ) → does (5 + m ≡?₁ 3 + n) ≡ does (2 + m ≡?₁ n)
 _ = λ m n → refl
 
 -- Even better, from a maintainability point of view, is to use `map` or `map′`,
 -- both of which capture the pattern of the `does` field remaining the same, but
 -- the `proof` field being updated.
 
-_≟₂_ : (m n : ℕ) → Dec (m ≡ n)
-zero  ≟₂ zero  = yes refl
-zero  ≟₂ suc n = no λ ()
-suc m ≟₂ zero  = no λ ()
-suc m ≟₂ suc n = map′ (cong suc) suc-injective (m ≟₂ n)
+_≡?₂_ : (m n : ℕ) → Dec (m ≡ n)
+zero  ≡?₂ zero  = yes refl
+zero  ≡?₂ suc n = no λ()
+suc m ≡?₂ zero  = no λ()
+suc m ≡?₂ suc n = map′ (cong suc) suc-injective (m ≡?₂ n)
 
-_ : (m n : ℕ) → does (5 + m ≟₂ 3 + n) ≡ does (2 + m ≟₂ n)
+_ : (m n : ℕ) → does (5 + m ≡?₂ 3 + n) ≡ does (2 + m ≡?₂ n)
 _ = λ m n → refl
 
 -- `map′` can be used in conjunction with combinators such as `_⊎?_` and
 -- `_×?_` to build complex (simply typed) decision procedures.
 
-module ListDecEq₀ {a} {A : Set a} (_≟ᴬ_ : (x y : A) → Dec (x ≡ y)) where
+module ListDecEq₀ {a} {A : Set a} (_≡?ᴬ_ : (x y : A) → Dec (x ≡ y)) where
 
-  _≟ᴸᴬ_ : (xs ys : List A) → Dec (xs ≡ ys)
-  []       ≟ᴸᴬ []       = yes refl
-  []       ≟ᴸᴬ (y ∷ ys) = no λ ()
-  (x ∷ xs) ≟ᴸᴬ []       = no λ ()
-  (x ∷ xs) ≟ᴸᴬ (y ∷ ys) =
-    map′ (uncurry (cong₂ _∷_)) ∷-injective (x ≟ᴬ y ×? xs ≟ᴸᴬ ys)
+  _≡?ᴸᴬ_ : (xs ys : List A) → Dec (xs ≡ ys)
+  []       ≡?ᴸᴬ []       = yes refl
+  []       ≡?ᴸᴬ (y ∷ ys) = no λ()
+  (x ∷ xs) ≡?ᴸᴬ []       = no λ()
+  (x ∷ xs) ≡?ᴸᴬ (y ∷ ys) =
+    map′ (uncurry (cong₂ _∷_)) ∷-injective (x ≡?ᴬ y ×? xs ≡?ᴸᴬ ys)
 
 -- The final case says that `x ∷ xs ≡ y ∷ ys` exactly when `x ≡ y` *and*
 -- `xs ≡ ys`. The proofs are updated by the first two arguments to `map′`.
 
 -- In the case of ≡-equality tests, the pattern
--- `map′ (congₙ c) c-injective (x₀ ≟ y₀ ×? ... ×? xₙ₋₁ ≟ yₙ₋₁)`
--- is captured by `≟-mapₙ n c c-injective (x₀ ≟ y₀) ... (xₙ₋₁ ≟ yₙ₋₁)`.
+-- `map′ (congₙ c) c-injective (x₀ ≡? y₀ ×? ... ×? xₙ₋₁ ≡? yₙ₋₁)`
+-- is captured by `≡?-mapₙ n c c-injective (x₀ ≡? y₀) ... (xₙ₋₁ ≡? yₙ₋₁)`.
 
-module ListDecEq₁ {a} {A : Set a} (_≟ᴬ_ : (x y : A) → Dec (x ≡ y)) where
+module ListDecEq₁ {a} {A : Set a} (_≡?ᴬ_ : (x y : A) → Dec (x ≡ y)) where
 
-  _≟ᴸᴬ_ : (xs ys : List A) → Dec (xs ≡ ys)
-  []       ≟ᴸᴬ []       = yes refl
-  []       ≟ᴸᴬ (y ∷ ys) = no λ ()
-  (x ∷ xs) ≟ᴸᴬ []       = no λ ()
-  (x ∷ xs) ≟ᴸᴬ (y ∷ ys) = ≟-mapₙ 2 _∷_ ∷-injective (x ≟ᴬ y) (xs ≟ᴸᴬ ys)
+  _≡?ᴸᴬ_ : (xs ys : List A) → Dec (xs ≡ ys)
+  []       ≡?ᴸᴬ []       = yes refl
+  []       ≡?ᴸᴬ (y ∷ ys) = no λ()
+  (x ∷ xs) ≡?ᴸᴬ []       = no λ()
+  (x ∷ xs) ≡?ᴸᴬ (y ∷ ys) = ≡?-mapₙ 2 _∷_ ∷-injective (x ≡?ᴬ y) (xs ≡?ᴸᴬ ys)

From 00c55e19374f1bcbd9ed3beed71936564d714931 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sat, 28 Feb 2026 03:14:05 +0000
Subject: [PATCH 20/20] fix: remove module added in error

---
 src/Algebra/Solver/Monoid/ExpressionBAK.agda | 97 --------------------
 1 file changed, 97 deletions(-)
 delete mode 100644 src/Algebra/Solver/Monoid/ExpressionBAK.agda

diff --git a/src/Algebra/Solver/Monoid/ExpressionBAK.agda b/src/Algebra/Solver/Monoid/ExpressionBAK.agda
deleted file mode 100644
index 74bbe4405e..0000000000
--- a/src/Algebra/Solver/Monoid/ExpressionBAK.agda
+++ /dev/null
@@ -1,97 +0,0 @@
-------------------------------------------------------------------------
--- The Agda standard library
---
--- A solver for equations over monoids
-------------------------------------------------------------------------
-
-{-# OPTIONS --cubical-compatible --safe #-}
-
-open import Algebra.Bundles.Raw using (RawMonoid)
-
-module Algebra.Solver.Monoid.ExpressionBAK {c ℓ} (M : RawMonoid c ℓ) where
-
-open import Algebra.Morphism.Structures using (IsMonoidHomomorphism)
-open import Data.Fin.Base using (Fin)
-open import Data.Maybe.Base as Maybe using (Maybe)
-open import Data.Nat.Base using (ℕ)
-open import Data.Vec.Base using (Vec; lookup)
-open import Level using (suc; _⊔_)
-open import Relation.Binary.Consequences using (dec⇒weaklyDec)
-open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_)
-
-private
-  variable
-    n : ℕ
-
-  module M = RawMonoid M
-
-
-------------------------------------------------------------------------
--- Monoid expressions
-
--- There is one constructor for every operation, plus one for
--- variables; there may be at most n variables.
-
-infixr 5 _⊕_
-
-data Expr (n : ℕ) : Set where
-  var : Fin n → Expr n
-  id  : Expr n
-  _⊕_ : Expr n → Expr n → Expr n
-
--- An environment contains one value for every variable.
-
-Env : ℕ → Set _
-Env n = Vec M.Carrier n
-
--- The semantics of an expression is a function from an environment to
--- a value.
-
-⟦_⟧ : Expr n → Env n → M.Carrier
-⟦ var x   ⟧ ρ = lookup ρ x
-⟦ id      ⟧ ρ = M.ε
-⟦ e₁ ⊕ e₂ ⟧ ρ = ⟦ e₁ ⟧ ρ M.∙ ⟦ e₂ ⟧ ρ
-
-------------------------------------------------------------------------
--- API for normal expressions
-
-record Normal {a r} : Set (suc (a ⊔ r) ⊔ c ⊔ ℓ) where
-  infix 5 _≟_
-
-  field
-
-    NF        : ℕ → RawMonoid a r
-
-  private
-    N : ℕ → Set a
-    N n = RawMonoid.Carrier (NF n)
-
-  field
-
-    ⟦var_⟧    : Fin n → N n
-
--- We can decide if two normal forms are /syntactically/ equal.
-
-    _≟_       : DecidableEquality (N n)
-
--- The semantics of a normal form.
-    ⟦_⟧⇓      : N n → Env n → M.Carrier
-
-    ⟦var_⟧⇓   : ∀ x ρ → ⟦ ⟦var x ⟧ ⟧⇓ ρ M.≈ lookup {n = n} ρ x
-
-    ⟦_⟧⇓-homo : ∀ ρ → IsMonoidHomomorphism (NF n) M λ e → ⟦ e ⟧⇓ ρ
-
--- A normaliser.
-
-  module _ {n} where
-
-    open RawMonoid (NF n)
-
-    normalise : Expr n → N n
-    normalise (var x)   = ⟦var x ⟧
-    normalise id        = ε
-    normalise (e₁ ⊕ e₂) = (normalise e₁) ∙ (normalise e₂)
-
-    _semi≟_ : ∀ e₁ e₂ → Maybe (normalise e₁ ≡ normalise e₂)
-    e₁ semi≟ e₂ = dec⇒weaklyDec _≟_ (normalise e₁) (normalise e₂)