Turn binary ++⁺ and related operators infix.

src/Data/List/Base.agda

@@ -308,18 +308,17 @@ _─_ : (xs : List A) → Fin (length xs) → List A
 ------------------------------------------------------------------------
 -- Operations for reversing lists
 
-reverseAcc : List A → List A → List A
-reverseAcc = foldl (flip _∷_)
-
-reverse : List A → List A
-reverse = reverseAcc []
-
 -- "Reverse append" xs ʳ++ ys = reverse xs ++ ys
 
 infixr 5 _ʳ++_
 
 _ʳ++_ : List A → List A → List A
-_ʳ++_ = flip reverseAcc
+xs ʳ++ ys = foldl (flip _∷_) ys xs
+
+-- Reverse.
+
+reverse : List A → List A
+reverse = _ʳ++ []
 
 -- Snoc.
 
@@ -390,3 +389,13 @@ boolSpan p (x ∷ xs) with p x
 
 boolBreak : (A → Bool) → List A → (List A × List A)
 boolBreak p = boolSpan (not ∘ p)
+
+-- Version 1.2
+
+reverseAcc : List A → List A → List A
+reverseAcc = foldl (flip _∷_)
+
+{-# WARNING_ON_USAGE reverseAcc
+"Warning: reverseAcc was deprecated in v1.2.
+Please use _ʳ++_ instead."
+#-}

src/Data/List/Membership/Propositional/Properties.agda

@@ -103,8 +103,10 @@ module _ {v : A} where
   ∈-++⁺ˡ : ∀ {xs ys} → v ∈ xs → v ∈ xs ++ ys
   ∈-++⁺ˡ = Membershipₛ.∈-++⁺ˡ (P.setoid A)
 
-  ∈-++⁺ʳ : ∀ xs {ys} → v ∈ ys → v ∈ xs ++ ys
-  ∈-++⁺ʳ = Membershipₛ.∈-++⁺ʳ (P.setoid A)
+  infixr 5 _∈-++⁺ʳ_
+
+  _∈-++⁺ʳ_ : ∀ xs {ys} → v ∈ ys → v ∈ xs ++ ys
+  _∈-++⁺ʳ_ = Membershipₛ._∈-++⁺ʳ_ (P.setoid A)
 
   ∈-++⁻ : ∀ xs {ys} → v ∈ xs ++ ys → (v ∈ xs) ⊎ (v ∈ ys)
   ∈-++⁻ = Membershipₛ.∈-++⁻ (P.setoid A)

src/Data/List/Membership/Setoid/Properties.agda

@@ -149,8 +149,10 @@ module _ {c ℓ} (S : Setoid c ℓ) where
   ∈-++⁺ˡ : ∀ {v xs ys} → v ∈ xs → v ∈ xs ++ ys
   ∈-++⁺ˡ = Any.++⁺ˡ
 
-  ∈-++⁺ʳ : ∀ {v} xs {ys} → v ∈ ys → v ∈ xs ++ ys
-  ∈-++⁺ʳ = Any.++⁺ʳ
+  infixr 5 _∈-++⁺ʳ_
+
+  _∈-++⁺ʳ_ : ∀ {v} xs {ys} → v ∈ ys → v ∈ xs ++ ys
+  _∈-++⁺ʳ_ = Any._++⁺ʳ_
 
   ∈-++⁻ : ∀ {v} xs {ys} → v ∈ xs ++ ys → (v ∈ xs) ⊎ (v ∈ ys)
   ∈-++⁻ = Any.++⁻

src/Data/List/Relation/Binary/BagAndSetEquality.agda

@@ -145,9 +145,12 @@ module _ {ℓ k} {A B : Set ℓ} {f g : A → B} {xs ys} where
 
 module _ {a k} {A : Set a} {xs₁ xs₂ ys₁ ys₂ : List A} where
 
-  ++-cong : xs₁ ∼[ k ] xs₂ → ys₁ ∼[ k ] ys₂ →
-            xs₁ ++ ys₁ ∼[ k ] xs₂ ++ ys₂
-  ++-cong xs₁≈xs₂ ys₁≈ys₂ {x} =
+  infixr 5 _++-cong_
+
+  _++-cong_ : xs₁ ∼[ k ] xs₂ →
+              ys₁ ∼[ k ] ys₂ →
+              xs₁ ++ ys₁ ∼[ k ] xs₂ ++ ys₂
+  _++-cong_ xs₁≈xs₂ ys₁≈ys₂ {x} =
     x ∈ xs₁ ++ ys₁       ↔⟨ SK-sym $ ++↔ ⟩
     (x ∈ xs₁ ⊎ x ∈ ys₁)  ∼⟨ xs₁≈xs₂ ⊎-cong ys₁≈ys₂ ⟩
     (x ∈ xs₂ ⊎ x ∈ ys₂)  ↔⟨ ++↔ ⟩
@@ -220,7 +223,7 @@ commutativeMonoid {a} k A = record
     { isSemigroup = record
       { isMagma = record
         { isEquivalence = Eq.isEquivalence
-        ; ∙-cong        = ++-cong
+        ; ∙-cong        = _++-cong_
         }
       ; assoc         = λ xs ys zs →
                           Eq.reflexive (LP.++-assoc xs ys zs)

src/Data/List/Relation/Binary/Equality/Setoid.agda

@@ -85,8 +85,10 @@ module _ {b ℓ₂} (T : Setoid b ℓ₂) where
 ------------------------------------------------------------------------
 -- _++_
 
-++⁺ : ∀ {ws xs ys zs} → ws ≋ xs → ys ≋ zs → ws ++ ys ≋ xs ++ zs
-++⁺ = PW.++⁺
+infixr 5 _++⁺_
+
+_++⁺_ : ∀ {ws xs ys zs} → ws ≋ xs → ys ≋ zs → ws ++ ys ≋ xs ++ zs
+_++⁺_ = PW._++⁺_
 
 ++-cancelˡ : ∀ xs {ys zs} → xs ++ ys ≋ xs ++ zs → ys ≋ zs
 ++-cancelˡ = PW.++-cancelˡ
@@ -123,8 +125,10 @@ module _ {P : Pred A p} (P? : U.Decidable P) (resp : P Respects _≈_)
 ------------------------------------------------------------------------
 -- reverse
 
-ʳ++⁺ : ∀{xs xs′ ys ys′} → xs ≋ xs′ → ys ≋ ys′ → xs ʳ++ ys ≋ xs′ ʳ++ ys′
-ʳ++⁺ = PW.ʳ++⁺
+infixr 5 _ʳ++⁺_
+
+_ʳ++⁺_ : ∀{xs xs′ ys ys′} → xs ≋ xs′ → ys ≋ ys′ → xs ʳ++ ys ≋ xs′ ʳ++ ys′
+_ʳ++⁺_ = PW._ʳ++⁺_
 
 reverse⁺ : ∀ {xs ys} → xs ≋ ys → reverse xs ≋ reverse ys
 reverse⁺ = PW.reverse⁺

src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda

@@ -82,28 +82,36 @@ module _ {a b} {A : Set a} {B : Set b} (f : A → B) where
 
 module _ {a} {A : Set a} where
 
+  infixr 5 _++⁺ˡ_ _++⁺ʳ_
 
-  ++⁺ˡ : ∀ xs {ys zs : List A} → ys ↭ zs → xs ++ ys ↭ xs ++ zs
-  ++⁺ˡ []       ys↭zs = ys↭zs
-  ++⁺ˡ (x ∷ xs) ys↭zs = prep x (++⁺ˡ xs ys↭zs)
+  _++⁺ˡ_ : ∀ xs {ys zs : List A} →
+           ys ↭ zs →
+           xs ++ ys ↭ xs ++ zs
+  []       ++⁺ˡ ys↭zs = ys↭zs
+  (x ∷ xs) ++⁺ˡ ys↭zs = prep x (xs ++⁺ˡ ys↭zs)
 
-  ++⁺ʳ : ∀ {xs ys : List A} zs → xs ↭ ys → xs ++ zs ↭ ys ++ zs
-  ++⁺ʳ zs refl          = refl
-  ++⁺ʳ zs (prep x ↭)    = prep x (++⁺ʳ zs ↭)
-  ++⁺ʳ zs (swap x y ↭)  = swap x y (++⁺ʳ zs ↭)
-  ++⁺ʳ zs (trans ↭₁ ↭₂) = trans (++⁺ʳ zs ↭₁) (++⁺ʳ zs ↭₂)
+  _++⁺ʳ_ : ∀ {xs ys : List A} →
+           xs ↭ ys →
+           ∀ zs →
+           xs ++ zs ↭ ys ++ zs
+  refl        ++⁺ʳ zs = refl
+  prep x ↭    ++⁺ʳ zs = prep x (↭ ++⁺ʳ zs)
+  swap x y ↭  ++⁺ʳ zs = swap x y (↭ ++⁺ʳ zs)
+  trans ↭₁ ↭₂ ++⁺ʳ zs = trans (↭₁ ++⁺ʳ zs) (↭₂ ++⁺ʳ zs)
 
-  ++⁺ : _++_ Preserves₂ _↭_ ⟶ _↭_ ⟶ _↭_
-  ++⁺ ws↭xs ys↭zs = trans (++⁺ʳ _ ws↭xs) (++⁺ˡ _ ys↭zs)
+  infixr 5 _++⁺_
+
+  _++⁺_ : _++_ Preserves₂ _↭_ ⟶ _↭_ ⟶ _↭_
+  ws↭xs ++⁺ ys↭zs = trans (ws↭xs ++⁺ʳ _) (_ ++⁺ˡ ys↭zs)
 
   -- Some useful lemmas
 
   zoom : ∀ h {t xs ys : List A} → xs ↭ ys → h ++ xs ++ t ↭ h ++ ys ++ t
-  zoom h {t} = ++⁺ˡ h ∘ ++⁺ʳ t
+  zoom h {t} = (h ++⁺ˡ_) ∘ (_++⁺ʳ t)
 
   inject : ∀  (v : A) {ws xs ys zs} → ws ↭ ys → xs ↭ zs →
            ws ++ [ v ] ++ xs ↭ ys ++ [ v ] ++ zs
-  inject v ws↭ys xs↭zs = trans (++⁺ˡ _ (prep v xs↭zs)) (++⁺ʳ _ ws↭ys)
+  inject v ws↭ys xs↭zs = trans (_ ++⁺ˡ prep v xs↭zs) (ws↭ys ++⁺ʳ _)
 
   shift : ∀ v (xs ys : List A) → xs ++ [ v ] ++ ys ↭ v ∷ xs ++ ys
   shift v []       ys = refl
@@ -183,15 +191,15 @@ module _ {a} {A : Set a} where
   ++-comm (x ∷ xs) ys = begin
     x ∷ xs ++ ys         ↭⟨ prep x (++-comm xs ys) ⟩
     x ∷ ys ++ xs         ≡⟨ cong (λ v → x ∷ v ++ xs) (≡.sym (Lₚ.++-identityʳ _)) ⟩
-    (x ∷ ys ++ []) ++ xs ↭⟨ ++⁺ʳ xs (↭-sym (shift x ys [])) ⟩
+    (x ∷ ys ++ []) ++ xs ↭⟨ ↭-sym (shift x ys []) ++⁺ʳ xs ⟩
     (ys ++ [ x ]) ++ xs  ↭⟨ ++-assoc ys [ x ] xs ⟩
     ys ++ ([ x ] ++ xs)  ≡⟨⟩
     ys ++ (x ∷ xs)       ∎
 
   ++-isMagma : IsMagma _↭_ _++_
   ++-isMagma = record
     { isEquivalence = ↭-isEquivalence
-    ; ∙-cong        = ++⁺
+    ; ∙-cong        = _++⁺_
     }
 
   ++-magma : Magma _ _
@@ -238,7 +246,7 @@ module _ {a} {A : Set a} where
   shifts : ∀ xs ys {zs : List A} → xs ++ ys ++ zs ↭ ys ++ xs ++ zs
   shifts xs ys {zs} = begin
      xs ++ ys  ++ zs ↭˘⟨ ++-assoc xs ys zs ⟩
-    (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩
+    (xs ++ ys) ++ zs ↭⟨ ++-comm xs ys ++⁺ʳ zs ⟩
     (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩
      ys ++ xs  ++ zs ∎
 

src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda

@@ -138,18 +138,20 @@ shift {v} {w} v≈w (x ∷ xs) ys = begin
   x ∷ w ∷ xs ++ ys        <<⟨ refl ⟩
   w ∷ x ∷ xs ++ ys        ∎
 
-++⁺ˡ : ∀ xs {ys zs : List A} → ys ↭ zs → xs ++ ys ↭ xs ++ zs
-++⁺ˡ []       ys↭zs = ys↭zs
-++⁺ˡ (x ∷ xs) ys↭zs = prep ≈-refl (++⁺ˡ xs ys↭zs)
+infixr 5 _++⁺ˡ_ _++⁺ʳ_ _++⁺_
 
-++⁺ʳ : ∀ {xs ys : List A} zs → xs ↭ ys → xs ++ zs ↭ ys ++ zs
-++⁺ʳ zs refl          = refl
-++⁺ʳ zs (prep x ↭)    = prep x (++⁺ʳ zs ↭)
-++⁺ʳ zs (swap x y ↭)  = swap x y (++⁺ʳ zs ↭)
-++⁺ʳ zs (trans ↭₁ ↭₂) = trans (++⁺ʳ zs ↭₁) (++⁺ʳ zs ↭₂)
+_++⁺ˡ_ : ∀ xs {ys zs : List A} → ys ↭ zs → xs ++ ys ↭ xs ++ zs
+[]       ++⁺ˡ ys↭zs = ys↭zs
+(x ∷ xs) ++⁺ˡ ys↭zs = prep ≈-refl (xs ++⁺ˡ ys↭zs)
 
-++⁺ : _++_ Preserves₂ _↭_ ⟶ _↭_ ⟶ _↭_
-++⁺ ws↭xs ys↭zs = trans (++⁺ʳ _ ws↭xs) (++⁺ˡ _ ys↭zs)
+_++⁺ʳ_ : ∀ {xs ys : List A} → xs ↭ ys → ∀ zs → xs ++ zs ↭ ys ++ zs
+refl        ++⁺ʳ zs = refl
+prep x ↭    ++⁺ʳ zs = prep x (↭ ++⁺ʳ zs)
+swap x y ↭  ++⁺ʳ zs = swap x y (↭ ++⁺ʳ zs)
+trans ↭₁ ↭₂ ++⁺ʳ zs = trans (↭₁ ++⁺ʳ zs) (↭₂ ++⁺ʳ zs)
+
+_++⁺_ : _++_ Preserves₂ _↭_ ⟶ _↭_ ⟶ _↭_
+ws↭xs ++⁺ ys↭zs = trans (ws↭xs ++⁺ʳ _ ) (_ ++⁺ˡ ys↭zs)
 
 -- Algebraic properties
 
@@ -170,7 +172,7 @@ shift {v} {w} v≈w (x ∷ xs) ys = begin
 ++-comm (x ∷ xs) ys = begin
   x ∷ xs ++ ys         ↭⟨ prep ≈-refl (++-comm xs ys) ⟩
   x ∷ ys ++ xs         ≡⟨ cong (λ v → x ∷ v ++ xs) (≡.sym (Lₚ.++-identityʳ _)) ⟩
-  (x ∷ ys ++ []) ++ xs ↭⟨ ++⁺ʳ xs (↭-sym (shift ≈-refl ys [])) ⟩
+  (x ∷ ys ++ []) ++ xs ↭⟨ ↭-sym (shift ≈-refl ys []) ++⁺ʳ xs ⟩
   (ys ++ [ x ]) ++ xs  ↭⟨ ++-assoc ys [ x ] xs ⟩
   ys ++ ([ x ] ++ xs)  ≡⟨⟩
   ys ++ (x ∷ xs)       ∎
@@ -180,7 +182,7 @@ shift {v} {w} v≈w (x ∷ xs) ys = begin
 ++-isMagma : IsMagma _↭_ _++_
 ++-isMagma = record
   { isEquivalence = ↭-isEquivalence
-  ; ∙-cong        = ++⁺
+  ; ∙-cong        = _++⁺_
   }
 
 ++-isSemigroup : IsSemigroup _↭_ _++_
@@ -227,16 +229,16 @@ shift {v} {w} v≈w (x ∷ xs) ys = begin
 -- Some other useful lemmas
 
 zoom : ∀ h {t xs ys : List A} → xs ↭ ys → h ++ xs ++ t ↭ h ++ ys ++ t
-zoom h {t} = ++⁺ˡ h ∘ ++⁺ʳ t
+zoom h {t} = (h ++⁺ˡ_) ∘ (_++⁺ʳ t)
 
 inject : ∀  (v : A) {ws xs ys zs} → ws ↭ ys → xs ↭ zs →
          ws ++ [ v ] ++ xs ↭ ys ++ [ v ] ++ zs
-inject v ws↭ys xs↭zs = trans (++⁺ˡ _ (prep ≈-refl xs↭zs)) (++⁺ʳ _ ws↭ys)
+inject v ws↭ys xs↭zs = trans (_ ++⁺ˡ prep ≈-refl xs↭zs) (ws↭ys ++⁺ʳ _)
 
 shifts : ∀ xs ys {zs : List A} → xs ++ ys ++ zs ↭ ys ++ xs ++ zs
 shifts xs ys {zs} = begin
    xs ++ ys  ++ zs ↭˘⟨ ++-assoc xs ys zs ⟩
-  (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩
+  (xs ++ ys) ++ zs ↭⟨ ++-comm xs ys ++⁺ʳ zs ⟩
   (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩
    ys ++ xs  ++ zs ∎
 

src/Data/List/Relation/Binary/Pointwise.agda

@@ -203,10 +203,14 @@ module _ {_∼_ : REL A B ℓ} where
 
 module _ {_∼_ : REL A B ℓ} where
 
-  ++⁺ : ∀ {ws xs ys zs} → Pointwise _∼_ ws xs → Pointwise _∼_ ys zs →
-        Pointwise _∼_ (ws ++ ys) (xs ++ zs)
-  ++⁺ []            ys∼zs = ys∼zs
-  ++⁺ (w∼x ∷ ws∼xs) ys∼zs = w∼x ∷ ++⁺ ws∼xs ys∼zs
+  infixr 5 _++⁺_
+
+  _++⁺_ : ∀ {ws xs ys zs} →
+          Pointwise _∼_ ws xs →
+          Pointwise _∼_ ys zs →
+          Pointwise _∼_ (ws ++ ys) (xs ++ zs)
+  []            ++⁺ ys∼zs = ys∼zs
+  (w∼x ∷ ws∼xs) ++⁺ ys∼zs = w∼x ∷ ws∼xs ++⁺ ys∼zs
 
 module _ {_∼_ : Rel₂ A ℓ} where
 
@@ -228,29 +232,28 @@ module _ {_∼_ : Rel₂ A ℓ} where
 
 module _ {_∼_ : REL A B ℓ} where
 
-  concat⁺ : ∀ {xss yss} → Pointwise (Pointwise _∼_) xss yss →
+  concat⁺ : ∀ {xss yss} →
+            Pointwise (Pointwise _∼_) xss yss →
             Pointwise _∼_ (concat xss) (concat yss)
   concat⁺ []                = []
-  concat⁺ (xs∼ys ∷ xss∼yss) = ++⁺ xs∼ys (concat⁺ xss∼yss)
+  concat⁺ (xs∼ys ∷ xss∼yss) = xs∼ys ++⁺ concat⁺ xss∼yss
 
 ------------------------------------------------------------------------
 -- reverse
 
 module _ {R : REL A B ℓ} where
 
-  reverseAcc⁺ : ∀ {as bs as′ bs′} → Pointwise R as′ bs′ → Pointwise R as bs →
-                Pointwise R (reverseAcc as′ as) (reverseAcc bs′ bs)
-  reverseAcc⁺ rs′ []       = rs′
-  reverseAcc⁺ rs′ (r ∷ rs) = reverseAcc⁺ (r ∷ rs′) rs
+  infixr 5 _ʳ++⁺_
 
-  ʳ++⁺ : ∀ {as bs as′ bs′} →
+  _ʳ++⁺_ : ∀ {as bs as′ bs′} →
            Pointwise R as bs →
            Pointwise R as′ bs′ →
            Pointwise R (as ʳ++ as′) (bs ʳ++ bs′)
-  ʳ++⁺ rs rs′ = reverseAcc⁺ rs′ rs
+  []       ʳ++⁺ rs′ = rs′
+  (r ∷ rs) ʳ++⁺ rs′ = rs ʳ++⁺ (r ∷ rs′)
 
   reverse⁺ : ∀ {as bs} → Pointwise R as bs → Pointwise R (reverse as) (reverse bs)
-  reverse⁺ = reverseAcc⁺ []
+  reverse⁺ = _ʳ++⁺ []
 
 ------------------------------------------------------------------------
 -- map
@@ -364,3 +367,16 @@ decidable-≡ = ≡-dec
 "Warning: decidable-≡ was deprecated in v1.0.
 Please use ≡-dec from `Data.List.Properties` instead."
 #-}
+
+-- Version 1.2
+
+reverseAcc⁺ : ∀ {R : REL A B ℓ} {as bs as′ bs′} →
+              Pointwise R as′ bs′ →
+              Pointwise R as bs →
+              Pointwise R (as ʳ++ as′) (bs ʳ++ bs′)  -- Pointwise R (reverseAcc as′ as) (reverseAcc bs′ bs)
+reverseAcc⁺ rs′ = _ʳ++⁺ rs′
+
+{-# WARNING_ON_USAGE reverseAcc⁺
+"Warning: reverseAcc⁺ was deprecated in v1.2.
+Please use _ʳ++⁺_ instead."
+#-}

src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda

@@ -69,10 +69,14 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
-  ++⁺ : ∀ {as bs cs ds} → Pointwise R as bs →
-        Prefix R cs ds → Prefix R (as ++ cs) (bs ++ ds)
-  ++⁺ []       cs⊆ds = cs⊆ds
-  ++⁺ (r ∷ rs) cs⊆ds = r ∷ (++⁺ rs cs⊆ds)
+  infixr 5 _++⁺_
+
+  _++⁺_ : ∀ {as bs cs ds} →
+          Pointwise R as bs →
+          Prefix R cs ds →
+          Prefix R (as ++ cs) (bs ++ ds)
+  []       ++⁺ cs⊆ds = cs⊆ds
+  (r ∷ rs) ++⁺ cs⊆ds = r ∷ (rs ++⁺ cs⊆ds)
 
   ++⁻ : ∀ {as bs cs ds} → length as ≡ length bs →
         Prefix R (as ++ cs) (bs ++ ds) → Prefix R cs ds