diff --git a/Batteries/Data/Fin/Basic.lean b/Batteries/Data/Fin/Basic.lean
index 51fc68a28e..800d7c7c87 100644
--- a/Batteries/Data/Fin/Basic.lean
+++ b/Batteries/Data/Fin/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Robert Y. Lewis, Keeley Hoek, Mario Carneiro
+Authors: Robert Y. Lewis, Keeley Hoek, Mario Carneiro, François G. Dorais, Quang Dao
 -/
 import Batteries.Tactic.Alias
 import Batteries.Data.Array.Basic
@@ -18,20 +18,6 @@ alias enum := Array.finRange
 @[deprecated (since := "2024-11-15")]
 alias list := List.finRange
 
-/-- Heterogeneous fold over `Fin n` from the right: `foldr 3 f x = f 0 (f 1 (f 2 x))`, where
-`f 2 : α 3 → α 2`, `f 1 : α 2 → α 1`, etc.
-
-This is the dependent version of `Fin.foldr`. -/
-@[inline] def dfoldr (n : Nat) (α : Fin (n + 1) → Sort _)
-    (f : ∀ (i : Fin n), α i.succ → α i.castSucc) (init : α (last n)) : α 0 :=
-  loop n (Nat.lt_succ_self n) init where
-  /-- Inner loop for `Fin.dfoldr`.
-    `Fin.dfoldr.loop n α f i h x = f 0 (f 1 (... (f i x)))`  -/
-  @[specialize] loop (i : Nat) (h : i < n + 1) (x : α ⟨i, h⟩) : α 0 :=
-    match i with
-    | i + 1 => loop i (Nat.lt_of_succ_lt h) (f ⟨i, Nat.lt_of_succ_lt_succ h⟩ x)
-    | 0 => x
-
 /-- Heterogeneous monadic fold over `Fin n` from right to left:
 ```
 Fin.foldrM n f xₙ = do
@@ -41,25 +27,33 @@ Fin.foldrM n f xₙ = do
   let x₀ : α 0 ← f 0 x₁
   pure x₀
 ```
-This is the dependent version of `Fin.foldrM`, defined using `Fin.dfoldr`. -/
-@[inline] def dfoldrM [Monad m] (n : Nat) (α : Fin (n + 1) → Sort _)
+This is the dependent version of `Fin.foldrM`. -/
+@[inline] def dfoldrM [Monad m] (n : Nat) (α : Fin (n + 1) → Type _)
     (f : ∀ (i : Fin n), α i.succ → m (α i.castSucc)) (init : α (last n)) : m (α 0) :=
-  dfoldr n (fun i => m (α i)) (fun i x => x >>= f i) (pure init)
+  loop n (Nat.lt_succ_self n) init where
+  /--
+  Inner loop for `Fin.dfoldrM`.
+  ```
+  Fin.dfoldrM.loop n f i h xᵢ = do
+    let xᵢ₋₁ ← f (i-1) xᵢ
+    ...
+    let x₁ ← f 1 x₂
+    let x₀ ← f 0 x₁
+    pure x₀
+  ```
+  -/
+  @[specialize] loop (i : Nat) (h : i < n + 1) (x : α ⟨i, h⟩) : m (α 0) :=
+    match i with
+    | i + 1 => (f ⟨i, Nat.lt_of_succ_lt_succ h⟩ x) >>= loop i (Nat.lt_of_succ_lt h)
+    | 0 => pure x
 
-/-- Heterogeneous fold over `Fin n` from the left: `foldl 3 f x = f 0 (f 1 (f 2 x))`, where
-`f 0 : α 0 → α 1`, `f 1 : α 1 → α 2`, etc.
+/-- Heterogeneous fold over `Fin n` from the right: `foldr 3 f x = f 0 (f 1 (f 2 x))`, where
+`f 2 : α 3 → α 2`, `f 1 : α 2 → α 1`, etc.
 
-This is the dependent version of `Fin.foldl`. -/
-@[inline] def dfoldl (n : Nat) (α : Fin (n + 1) → Sort _)
-    (f : ∀ (i : Fin n), α i.castSucc → α i.succ) (init : α 0) : α (last n) :=
-  loop 0 (Nat.zero_lt_succ n) init where
-  /-- Inner loop for `Fin.dfoldl`. `Fin.dfoldl.loop n α f i h x = f n (f (n-1) (... (f i x)))` -/
-  @[semireducible, specialize] loop (i : Nat) (h : i < n + 1) (x : α ⟨i, h⟩) : α (last n) :=
-    if h' : i < n then
-      loop (i + 1) (Nat.succ_lt_succ h') (f ⟨i, h'⟩ x)
-    else
-      haveI : ⟨i, h⟩ = last n := by ext; simp; omega
-      _root_.cast (congrArg α this) x
+This is the dependent version of `Fin.foldr`. -/
+@[inline] def dfoldr (n : Nat) (α : Fin (n + 1) → Type _)
+    (f : ∀ (i : Fin n), α i.succ → α i.castSucc) (init : α (last n)) : α 0 :=
+  dfoldrM (m := Id) n α f init
 
 /-- Heterogeneous monadic fold over `Fin n` from left to right:
 ```
@@ -71,7 +65,7 @@ Fin.foldlM n f x₀ = do
   pure xₙ
 ```
 This is the dependent version of `Fin.foldlM`. -/
-@[inline] def dfoldlM [Monad m] (n : Nat) (α : Fin (n + 1) → Sort _)
+@[inline] def dfoldlM [Monad m] (n : Nat) (α : Fin (n + 1) → Type _)
     (f : ∀ (i : Fin n), α i.castSucc → m (α i.succ)) (init : α 0) : m (α (last n)) :=
   loop 0 (Nat.zero_lt_succ n) init where
   /-- Inner loop for `Fin.dfoldlM`.
@@ -89,3 +83,11 @@ This is the dependent version of `Fin.foldlM`. -/
     else
       haveI : ⟨i, h⟩ = last n := by ext; simp; omega
       _root_.cast (congrArg (fun i => m (α i)) this) (pure x)
+
+/-- Heterogeneous fold over `Fin n` from the left: `foldl 3 f x = f 0 (f 1 (f 2 x))`, where
+`f 0 : α 0 → α 1`, `f 1 : α 1 → α 2`, etc.
+
+This is the dependent version of `Fin.foldl`. -/
+@[inline] def dfoldl (n : Nat) (α : Fin (n + 1) → Type _)
+    (f : ∀ (i : Fin n), α i.castSucc → α i.succ) (init : α 0) : α (last n) :=
+  dfoldlM (m := Id) n α f init
diff --git a/Batteries/Data/Fin/Fold.lean b/Batteries/Data/Fin/Fold.lean
index 51d2f0c53c..9d2d821c2f 100644
--- a/Batteries/Data/Fin/Fold.lean
+++ b/Batteries/Data/Fin/Fold.lean
@@ -8,98 +8,61 @@ import Batteries.Data.Fin.Basic
 
 namespace Fin
 
-/-! ### dfoldr -/
+/-! ### dfoldrM -/
 
-theorem dfoldr_loop_zero (f : (i : Fin n) → α i.succ → α i.castSucc) (x) :
-    dfoldr.loop n α f 0 (Nat.zero_lt_succ n) x = x := rfl
+theorem dfoldrM_loop_zero [Monad m] (f : (i : Fin n) → α i.succ → m (α i.castSucc)) (x) :
+    dfoldrM.loop n α f 0 h x = pure x := rfl
 
-theorem dfoldr_loop_succ (f : (i : Fin n) → α i.succ → α i.castSucc) (h : i < n) (x) :
-      dfoldr.loop n α f (i+1) (Nat.add_lt_add_right h 1) x =
-        dfoldr.loop n α f i (Nat.lt_add_right 1 h) (f ⟨i, h⟩ x) := rfl
+theorem dfoldrM_loop_succ [Monad m] (f : (i : Fin n) → α i.succ → m (α i.castSucc)) (x) :
+    dfoldrM.loop n α f (i+1) h x = f ⟨i, by omega⟩ x >>= dfoldrM.loop n α f i (by omega) := rfl
 
-theorem dfoldr_loop (f : (i : Fin (n+1)) → α i.succ → α i.castSucc) (h : i+1 ≤ n+1) (x) :
-      dfoldr.loop (n+1) α f (i+1) (Nat.add_lt_add_right h 1) x =
-        f 0 (dfoldr.loop n (α ∘ succ) (f ·.succ) i h x) := by
+theorem dfoldrM_loop [Monad m] [LawfulMonad m] (f : (i : Fin (n+1)) → α i.succ → m (α i.castSucc))
+    (x) : dfoldrM.loop (n+1) α f (i+1) h x =
+      dfoldrM.loop n (α ∘ succ) (f ·.succ) i (by omega) x >>= f 0 := by
   induction i with
-  | zero => rfl
-  | succ i ih => exact ih ..
-
-@[simp] theorem dfoldr_zero (f : (i : Fin 0) → α i.succ → α i.castSucc) (x) :
-    dfoldr 0 α f x = x := rfl
-
-theorem dfoldr_succ (f : (i : Fin (n+1)) → α i.succ → α i.castSucc) (x) :
-    dfoldr (n+1) α f x = f 0 (dfoldr n (α ∘ succ) (f ·.succ) x) := dfoldr_loop ..
-
-theorem dfoldr_succ_last (f : (i : Fin (n+1)) → α i.succ → α i.castSucc) (x) :
-    dfoldr (n+1) α f x = dfoldr n (α ∘ castSucc) (f ·.castSucc) (f (last n) x) := by
-  induction n with
-  | zero => simp only [dfoldr_succ, dfoldr_zero, last, zero_eta]
-  | succ n ih => rw [dfoldr_succ, ih (α := α ∘ succ) (f ·.succ), dfoldr_succ]; congr
-
-theorem dfoldr_eq_dfoldrM (f : (i : Fin n) → α i.succ → α i.castSucc) (x) :
-    dfoldr n α f x = dfoldrM (m:=Id) n α f x := rfl
-
-theorem dfoldr_eq_foldr (f : Fin n → α → α) (x : α) : dfoldr n (fun _ => α) f x = foldr n f x := by
-  induction n with
-  | zero => simp only [dfoldr_zero, foldr_zero]
-  | succ n ih => simp only [dfoldr_succ, foldr_succ, Function.comp_apply, Function.comp_def, ih]
-
-/-! ### dfoldrM -/
+  | zero =>
+    rw [dfoldrM_loop_zero, dfoldrM_loop_succ, pure_bind]
+    conv => rhs; rw [← bind_pure (f 0 x)]
+    rfl
+  | succ i ih =>
+    rw [dfoldrM_loop_succ, dfoldrM_loop_succ, bind_assoc]
+    congr; funext; exact ih ..
 
 @[simp] theorem dfoldrM_zero [Monad m] (f : (i : Fin 0) → α i.succ → m (α i.castSucc)) (x) :
     dfoldrM 0 α f x = pure x := rfl
 
-theorem dfoldrM_succ [Monad m] (f : (i : Fin (n+1)) → α i.succ → m (α i.castSucc))
-    (x) : dfoldrM (n+1) α f x = dfoldrM n (α ∘ succ) (f ·.succ) x >>= f 0 := dfoldr_succ ..
+theorem dfoldrM_succ [Monad m] [LawfulMonad m] (f : (i : Fin (n+1)) → α i.succ → m (α i.castSucc))
+    (x) : dfoldrM (n+1) α f x = dfoldrM n (α ∘ succ) (f ·.succ) x >>= f 0 := dfoldrM_loop ..
 
-theorem dfoldrM_eq_foldrM [Monad m] [LawfulMonad m] (f : (i : Fin n) → α → m α) (x : α) :
+theorem dfoldrM_eq_foldrM [Monad m] [LawfulMonad m] (f : (i : Fin n) → α → m α) (x) :
     dfoldrM n (fun _ => α) f x = foldrM n f x := by
-  induction n generalizing x with
+  induction n with
   | zero => simp only [dfoldrM_zero, foldrM_zero]
   | succ n ih => simp only [dfoldrM_succ, foldrM_succ, Function.comp_def, ih]
 
-/-! ### dfoldl -/
-
-theorem dfoldl_loop_lt (f : ∀ (i : Fin n), α i.castSucc → α i.succ) (h : i < n) (x) :
-    dfoldl.loop n α f i (Nat.lt_add_right 1 h) x =
-      dfoldl.loop n α f (i+1) (Nat.add_lt_add_right h 1) (f ⟨i, h⟩ x) := by
-  rw [dfoldl.loop, dif_pos h]
-
-theorem dfoldl_loop_eq (f : ∀ (i : Fin n), α i.castSucc → α i.succ) (x) :
-    dfoldl.loop n α f n (Nat.le_refl _) x = x := by
-  rw [dfoldl.loop, dif_neg (Nat.lt_irrefl _), cast_eq]
+theorem dfoldr_eq_dfoldrM (f : (i : Fin n) → α i.succ → α i.castSucc) (x) :
+    dfoldr n α f x = dfoldrM (m:=Id) n α f x := rfl
 
-@[simp] theorem dfoldl_zero (f : (i : Fin 0) → α i.castSucc → α i.succ) (x) :
-    dfoldl 0 α f x = x := dfoldl_loop_eq ..
+/-! ### dfoldr -/
 
-theorem dfoldl_loop (f : (i : Fin (n+1)) → α i.castSucc → α i.succ) (h : i < n+1) (x) :
-    dfoldl.loop (n+1) α f i (Nat.lt_add_right 1 h) x =
-      dfoldl.loop n (α ∘ succ) (f ·.succ ·) i h (f ⟨i, h⟩ x) := by
-  if h' : i < n then
-    rw [dfoldl_loop_lt _ h _]
-    rw [dfoldl_loop_lt _ h' _, dfoldl_loop]; rfl
-  else
-    cases Nat.le_antisymm (Nat.le_of_lt_succ h) (Nat.not_lt.1 h')
-    rw [dfoldl_loop_lt]
-    rw [dfoldl_loop_eq, dfoldl_loop_eq]
+@[simp] theorem dfoldr_zero (f : (i : Fin 0) → α i.succ → α i.castSucc) (x) :
+    dfoldr 0 α f x = x := rfl
 
-theorem dfoldl_succ (f : (i : Fin (n+1)) → α i.castSucc → α i.succ) (x) :
-    dfoldl (n+1) α f x = dfoldl n (α ∘ succ) (f ·.succ ·) (f 0 x) := dfoldl_loop ..
+theorem dfoldr_succ (f : (i : Fin (n+1)) → α i.succ → α i.castSucc) (x) :
+    dfoldr (n+1) α f x = f 0 (dfoldr n (α ∘ succ) (f ·.succ) x) := dfoldrM_succ ..
 
-theorem dfoldl_succ_last (f : (i : Fin (n+1)) → α i.castSucc → α i.succ) (x) :
-    dfoldl (n+1) α f x = f (last n) (dfoldl n (α ∘ castSucc) (f ·.castSucc ·) x) := by
-  rw [dfoldl_succ]
+theorem dfoldr_succ_last {n : Nat} {α : Fin (n+2) → Sort _}
+    (f : (i : Fin (n+1)) → α i.succ → α i.castSucc) (x : α (last (n+1))) :
+      dfoldr (n+1) α f x = dfoldr n (α ∘ castSucc) (f ·.castSucc) (f (last n) x) := by
   induction n with
-  | zero => simp [dfoldl_succ, last]
-  | succ n ih => rw [dfoldl_succ, @ih (α ∘ succ) (f ·.succ ·), dfoldl_succ]; congr
+  | zero => simp only [dfoldr_succ, dfoldr_zero, last, zero_eta]
+  | succ n ih => rw [dfoldr_succ, ih (α := α ∘ succ) (f ·.succ), dfoldr_succ]; congr
 
-theorem dfoldl_eq_foldl (f : Fin n → α → α) (x : α) :
-    dfoldl n (fun _ => α) f x = foldl n (fun x i => f i x) x := by
-  induction n generalizing x with
-  | zero => simp only [dfoldl_zero, foldl_zero]
-  | succ n ih =>
-    simp only [dfoldl_succ, foldl_succ, Function.comp_apply, Function.comp_def]
-    congr; simp only [ih]
+theorem dfoldr_eq_foldr (f : (i : Fin n) → α → α) (x) :
+    dfoldr n (fun _ => α) f x = foldr n f x := by
+  induction n with
+  | zero => simp only [dfoldr_zero, foldr_zero]
+  | succ n ih => simp only [dfoldr_succ, foldr_succ, Function.comp_def, ih]
 
 /-! ### dfoldlM -/
 
@@ -113,7 +76,7 @@ theorem dfoldlM_loop_eq [Monad m] (f : ∀ (i : Fin n), α i.castSucc → m (α
   rw [dfoldlM.loop, dif_neg (Nat.lt_irrefl _), cast_eq]
 
 @[simp] theorem dfoldlM_zero [Monad m] (f : (i : Fin 0) → α i.castSucc → m (α i.succ)) (x) :
-    dfoldlM 0 α f x = pure x := dfoldlM_loop_eq ..
+    dfoldlM 0 α f x = pure x := rfl
 
 theorem dfoldlM_loop [Monad m] (f : (i : Fin (n+1)) → α i.castSucc → m (α i.succ)) (h : i < n+1)
     (x) : dfoldlM.loop (n+1) α f i (Nat.lt_add_right 1 h) x =
@@ -140,6 +103,32 @@ theorem dfoldlM_eq_foldlM [Monad m] (f : (i : Fin n) → α → m α) (x : α) :
     simp only [dfoldlM_succ, foldlM_succ, Function.comp_apply, Function.comp_def]
     congr; ext; simp only [ih]
 
+/-! ### dfoldl -/
+
+@[simp] theorem dfoldl_zero (f : (i : Fin 0) → α i.castSucc → α i.succ) (x) :
+    dfoldl 0 α f x = x := rfl
+
+theorem dfoldl_succ (f : (i : Fin (n+1)) → α i.castSucc → α i.succ) (x) :
+    dfoldl (n+1) α f x = dfoldl n (α ∘ succ) (f ·.succ ·) (f 0 x) := dfoldlM_succ ..
+
+theorem dfoldl_succ_last (f : (i : Fin (n+1)) → α i.castSucc → α i.succ) (x) :
+    dfoldl (n+1) α f x = f (last n) (dfoldl n (α ∘ castSucc) (f ·.castSucc ·) x) := by
+  rw [dfoldl_succ]
+  induction n with
+  | zero => simp [dfoldl_succ, last]
+  | succ n ih => rw [dfoldl_succ, @ih (α ∘ succ) (f ·.succ ·), dfoldl_succ]; congr
+
+theorem dfoldl_eq_dfoldlM (f : (i : Fin n) → α i.castSucc → α i.succ) (x) :
+    dfoldl n α f x = dfoldlM (m := Id) n α f x := rfl
+
+theorem dfoldl_eq_foldl (f : Fin n → α → α) (x : α) :
+    dfoldl n (fun _ => α) f x = foldl n (fun x i => f i x) x := by
+  induction n generalizing x with
+  | zero => simp only [dfoldl_zero, foldl_zero]
+  | succ n ih =>
+    simp only [dfoldl_succ, foldl_succ, Function.comp_apply, Function.comp_def]
+    congr; simp only [ih]
+
 /-! ### `Fin.fold{l/r}{M}` equals `List.fold{l/r}{M}` -/
 
 theorem foldlM_eq_foldlM_finRange [Monad m] (f : α → Fin n → m α) (x) :