use it in proofs

doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda

@@ -20,6 +20,7 @@ open import Data.Nat.Properties
 open import Data.Vec.Base
 open import Data.Vec.Properties
 open import Data.Vec.Relation.Binary.Equality.Cast
+open import Function using (_∘_)
 open import Relation.Binary.PropositionalEquality
  using (_≡_; refl; sym; cong; module ≡-Reasoning)
 
@@ -187,7 +188,7 @@ example3a-fromList-++-++ {xs = xs} {ys} {zs} eq = begin
  fromList (xs List.++ ys List.++ zs)
  ≈⟨ fromList-++ xs ⟩
  fromList xs ++ fromList (ys List.++ zs)
- ≈⟨ ≈-cong (fromList xs ++_) (cast-++ʳ (List.length-++ ys) (fromList xs)) (fromList-++ ys) ⟩
+ ≈⟨ ≈-cong′ (fromList xs ++_) (fromList-++ ys) ⟩
  fromList xs ++ fromList ys ++ fromList zs
  ∎
  where open CastReasoning
@@ -218,9 +219,7 @@ example4-cong² : ∀ .(eq : (m + 1) + n ≡ n + suc m) a (xs : Vec A m) ys →
  cast eq (reverse ((xs ++ [ a ]) ++ ys)) ≡ ys ʳ++ reverse (xs ∷ʳ a)
 example4-cong² {m = m} {n} eq a xs ys = begin
  reverse ((xs ++ [ a ]) ++ ys)
- ≈⟨ ≈-cong reverse (cast-reverse (cong (_+ n) (+-comm 1 m)) ((xs ∷ʳ a) ++ ys))
- (≈-cong (_++ ys) (cast-++ˡ (+-comm 1 m) (xs ∷ʳ a))
- (unfold-∷ʳ _ a xs)) ⟨
+ ≈⟨ ≈-cong′ (reverse ∘ (_++ ys)) (unfold-∷ʳ (+-comm 1 m) a xs) ⟨
  reverse ((xs ∷ʳ a) ++ ys)
  ≈⟨ reverse-++ (+-comm (suc m) n) (xs ∷ʳ a) ys ⟩
  reverse ys ++ reverse (xs ∷ʳ a)
@@ -264,7 +263,7 @@ example6a-reverse-∷ʳ {n = n} x xs = begin-≡
  reverse (xs ∷ʳ x)
  ≡⟨ ≈-reflexive refl ⟨
  reverse (xs ∷ʳ x)
- ≈⟨ ≈-cong reverse (cast-reverse _ _) (unfold-∷ʳ (+-comm 1 n) x xs) ⟩
+ ≈⟨ ≈-cong′ reverse (unfold-∷ʳ (+-comm 1 n) x xs) ⟩
  reverse (xs ++ [ x ])
  ≈⟨ reverse-++ (+-comm n 1) xs [ x ] ⟩
  x ∷ reverse xs

src/Data/Vec/Properties.agda

@@ -1016,8 +1016,7 @@ reverse-++ : ∀ .(eq : m + n ≡ n + m) (xs : Vec A m) (ys : Vec A n) →
 reverse-++ {m = zero} {n = n} eq [] ys = ≈-sym (++-identityʳ (sym eq) (reverse ys))
 reverse-++ {m = suc m} {n = n} eq (x ∷ xs) ys = begin
  reverse (x ∷ xs ++ ys) ≂⟨ reverse-∷ x (xs ++ ys) ⟩
- reverse (xs ++ ys) ∷ʳ x ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (+-comm m n)) x (reverse (xs ++ ys)))
- (reverse-++ _ xs ys) ⟩
+ reverse (xs ++ ys) ∷ʳ x ≈⟨ ≈-cong′ (_∷ʳ x) (reverse-++ (+-comm m n) xs ys) ⟩
  (reverse ys ++ reverse xs) ∷ʳ x ≈⟨ ++-∷ʳ (sym (+-suc n m)) x (reverse ys) (reverse xs) ⟩
  reverse ys ++ (reverse xs ∷ʳ x) ≂⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟨
  reverse ys ++ (reverse (x ∷ xs)) ∎
@@ -1068,8 +1067,7 @@ map-ʳ++ {ys = ys} f xs = begin
  cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)
 ++-ʳ++ {m = m} {n} {o} eq xs {ys} {zs} = begin
  ((xs ++ ys) ʳ++ zs) ≂⟨ unfold-ʳ++ (xs ++ ys) zs ⟩
- reverse (xs ++ ys) ++ zs ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (xs ++ ys)))
- (reverse-++ (+-comm m n) xs ys) ⟩
+ reverse (xs ++ ys) ++ zs ≈⟨ ≈-cong′ (_++ zs) (reverse-++ (+-comm m n) xs ys) ⟩
  (reverse ys ++ reverse xs) ++ zs ≈⟨ ++-assoc (trans (cong (_+ o) (+-comm n m)) eq) (reverse ys) (reverse xs) zs ⟩
  reverse ys ++ (reverse xs ++ zs) ≂⟨ cong (reverse ys ++_) (unfold-ʳ++ xs zs) ⟨
  reverse ys ++ (xs ʳ++ zs) ≂⟨ unfold-ʳ++ ys (xs ʳ++ zs) ⟨
@@ -1081,8 +1079,7 @@ map-ʳ++ {ys = ys} f xs = begin
 ʳ++-ʳ++ {m = m} {n} {o} eq xs {ys} {zs} = begin
  (xs ʳ++ ys) ʳ++ zs ≂⟨ cong (_ʳ++ zs) (unfold-ʳ++ xs ys) ⟩
  (reverse xs ++ ys) ʳ++ zs ≂⟨ unfold-ʳ++ (reverse xs ++ ys) zs ⟩
- reverse (reverse xs ++ ys) ++ zs ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (reverse xs ++ ys)))
- (reverse-++ (+-comm m n) (reverse xs) ys) ⟩
+ reverse (reverse xs ++ ys) ++ zs ≈⟨ ≈-cong′ (_++ zs) (reverse-++ (+-comm m n) (reverse xs) ys) ⟩
  (reverse ys ++ reverse (reverse xs)) ++ zs ≂⟨ cong ((_++ zs) ∘ (reverse ys ++_)) (reverse-involutive xs) ⟩
  (reverse ys ++ xs) ++ zs ≈⟨ ++-assoc (+-assoc n m o) (reverse ys) xs zs ⟩
  reverse ys ++ (xs ++ zs) ≂⟨ unfold-ʳ++ ys (xs ++ zs) ⟨
@@ -1312,8 +1309,7 @@ fromList-reverse (x List.∷ xs) = begin
  fromList (List.reverse (x List.∷ xs)) ≈⟨ cast-fromList (List.ʳ++-defn xs) ⟩
  fromList (List.reverse xs List.++ List.[ x ]) ≈⟨ fromList-++ (List.reverse xs) ⟩
  fromList (List.reverse xs) ++ [ x ] ≈⟨ unfold-∷ʳ (+-comm 1 _) x (fromList (List.reverse xs)) ⟨
- fromList (List.reverse xs) ∷ʳ x ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (List.length-reverse xs)) _ _)
- (fromList-reverse xs) ⟩
+ fromList (List.reverse xs) ∷ʳ x ≈⟨ ≈-cong′ (_∷ʳ x) (fromList-reverse xs) ⟩
  reverse (fromList xs) ∷ʳ x ≂⟨ reverse-∷ x (fromList xs) ⟨
  reverse (x ∷ fromList xs) ≈⟨⟩
  reverse (fromList (x List.∷ xs)) ∎