diff --git a/Batteries/Data/Nat/BinaryRec.lean b/Batteries/Data/Nat/BinaryRec.lean
index 91bef32a5a..f9f6556644 100644
--- a/Batteries/Data/Nat/BinaryRec.lean
+++ b/Batteries/Data/Nat/BinaryRec.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
+Copyright (c) 2017 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Praneeth Kolichala, Yuyang Zhao
+Authors: Mario Carneiro, Praneeth Kolichala, Yuyang Zhao
 -/
 
 /-!
@@ -16,74 +16,85 @@ This file defines binary recursion on `Nat`.
 * `Nat.binaryRecFromOne`: The same as `binaryRec`, but special casing both 0 and 1 as base cases.
 -/
 
+universe u
+
 namespace Nat
 
 /-- `bit b` appends the digit `b` to the little end of the binary representation of
   its natural number input. -/
 def bit (b : Bool) (n : Nat) : Nat :=
-  cond b (n + n + 1) (n + n)
+  cond b (2 * n + 1) (2 * n)
+
+private def bitImpl (b : Bool) (n : Nat) : Nat :=
+  cond b (n <<< 1 + 1) (n <<< 1)
+
+@[csimp] theorem bit_eq_bitImpl : bit = bitImpl := rfl
 
 theorem shiftRight_one (n) : n >>> 1 = n / 2 := rfl
 
+@[simp]
+theorem bit_decide_mod_two_eq_one_shiftRight_one (n : Nat) : bit (n % 2 = 1) (n >>> 1) = n := by
+  simp only [bit, shiftRight_one]
+  cases mod_two_eq_zero_or_one n with | _ h => simpa [h] using Nat.div_add_mod n 2
+
 theorem bit_testBit_zero_shiftRight_one (n : Nat) : bit (n.testBit 0) (n >>> 1) = n := by
-  simp only [bit, testBit_zero]
-  cases mod_two_eq_zero_or_one n with | _ h =>
-    simpa [h, shiftRight_one] using Eq.trans (by simp [h, Nat.two_mul]) (Nat.div_add_mod n 2)
+  simp
 
+@[simp]
 theorem bit_eq_zero_iff {n : Nat} {b : Bool} : bit b n = 0 ↔ n = 0 ∧ b = false := by
-  cases n <;> cases b <;> simp [bit, ← Nat.add_assoc]
+  cases n <;> cases b <;> simp [bit, Nat.shiftLeft_succ, Nat.two_mul, ← Nat.add_assoc]
 
-/-- For a predicate `C : Nat → Sort u`, if instances can be
+/-- For a predicate `motive : Nat → Sort u`, if instances can be
   constructed for natural numbers of the form `bit b n`,
   they can be constructed for any given natural number. -/
 @[inline]
-def bitCasesOn {C : Nat → Sort u} (n) (h : ∀ b n, C (bit b n)) : C n :=
+def bitCasesOn {motive : Nat → Sort u} (n) (h : ∀ b n, motive (bit b n)) : motive n :=
   -- `1 &&& n != 0` is faster than `n.testBit 0`. This may change when we have faster `testBit`.
   let x := h (1 &&& n != 0) (n >>> 1)
-  -- `congrArg C _` is `rfl` in non-dependent case
-  congrArg C n.bit_testBit_zero_shiftRight_one ▸ x
+  -- `congrArg motive _ ▸ x` is defeq to `x` in non-dependent case
+  congrArg motive n.bit_testBit_zero_shiftRight_one ▸ x
 
 /-- A recursion principle for `bit` representations of natural numbers.
-  For a predicate `C : Nat → Sort u`, if instances can be
+  For a predicate `motive : Nat → Sort u`, if instances can be
   constructed for natural numbers of the form `bit b n`,
   they can be constructed for all natural numbers. -/
 @[elab_as_elim, specialize]
-def binaryRec {C : Nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) (n : Nat) : C n :=
-  if n0 : n = 0 then congrArg C n0 ▸ z
+def binaryRec {motive : Nat → Sort u} (z : motive 0) (f : ∀ b n, motive n → motive (bit b n))
+    (n : Nat) : motive n :=
+  if n0 : n = 0 then congrArg motive n0 ▸ z
   else
     let x := f (1 &&& n != 0) (n >>> 1) (binaryRec z f (n >>> 1))
-    congrArg C n.bit_testBit_zero_shiftRight_one ▸ x
+    congrArg motive n.bit_testBit_zero_shiftRight_one ▸ x
 decreasing_by exact bitwise_rec_lemma n0
 
 /-- The same as `binaryRec`, but the induction step can assume that if `n=0`,
   the bit being appended is `true`-/
 @[elab_as_elim, specialize]
-def binaryRec' {C : Nat → Sort u} (z : C 0)
-    (f : ∀ b n, (n = 0 → b = true) → C n → C (bit b n)) : ∀ n, C n :=
+def binaryRec' {motive : Nat → Sort u} (z : motive 0)
+    (f : ∀ b n, (n = 0 → b = true) → motive n → motive (bit b n)) :
+    ∀ n, motive n :=
   binaryRec z fun b n ih =>
     if h : n = 0 → b = true then f b n h ih
     else
       have : bit b n = 0 := by
         rw [bit_eq_zero_iff]
-        cases n <;> cases b <;> simp at h <;> simp [h]
-      congrArg C this ▸ z
+        cases n <;> cases b <;> simp at h ⊢
+      congrArg motive this ▸ z
 
 /-- The same as `binaryRec`, but special casing both 0 and 1 as base cases -/
 @[elab_as_elim, specialize]
-def binaryRecFromOne {C : Nat → Sort u} (z₀ : C 0) (z₁ : C 1)
-    (f : ∀ b n, n ≠ 0 → C n → C (bit b n)) : ∀ n, C n :=
+def binaryRecFromOne {motive : Nat → Sort u} (z₀ : motive 0) (z₁ : motive 1)
+    (f : ∀ b n, n ≠ 0 → motive n → motive (bit b n)) :
+    ∀ n, motive n :=
   binaryRec' z₀ fun b n h ih =>
     if h' : n = 0 then
       have : bit b n = bit true 0 := by
         rw [h', h h']
-      congrArg C this ▸ z₁
+      congrArg motive this ▸ z₁
     else f b n h' ih
 
 theorem bit_val (b n) : bit b n = 2 * n + b.toNat := by
-  rw [Nat.mul_comm]
-  induction b with
-  | false => exact congrArg (· + n) n.zero_add.symm
-  | true => exact congrArg (· + n + 1) n.zero_add.symm
+  cases b <;> rfl
 
 @[simp]
 theorem bit_div_two (b n) : bit b n / 2 = n := by
@@ -95,40 +106,50 @@ theorem bit_div_two (b n) : bit b n / 2 = n := by
 theorem bit_mod_two (b n) : bit b n % 2 = b.toNat := by
   cases b <;> simp [bit_val, mul_add_mod]
 
-@[simp] theorem bit_shiftRight_one (b n) : bit b n >>> 1 = n :=
+@[simp]
+theorem bit_shiftRight_one (b n) : bit b n >>> 1 = n :=
   bit_div_two b n
 
-theorem bit_testBit_zero (b n) : (bit b n).testBit 0 = b := by
+theorem testBit_bit_zero (b n) : (bit b n).testBit 0 = b := by
   simp
 
+variable {motive : Nat → Sort u}
+
 @[simp]
-theorem bitCasesOn_bit {C : Nat → Sort u} (h : ∀ b n, C (bit b n)) (b : Bool) (n : Nat) :
+theorem bitCasesOn_bit (h : ∀ b n, motive (bit b n)) (b : Bool) (n : Nat) :
     bitCasesOn (bit b n) h = h b n := by
-  change congrArg C (bit b n).bit_testBit_zero_shiftRight_one ▸ h _ _ = h b n
-  generalize congrArg C (bit b n).bit_testBit_zero_shiftRight_one = e; revert e
-  rw [bit_testBit_zero, bit_shiftRight_one]
+  change congrArg motive (bit b n).bit_testBit_zero_shiftRight_one ▸ h _ _ = h b n
+  generalize congrArg motive (bit b n).bit_testBit_zero_shiftRight_one = e; revert e
+  rw [testBit_bit_zero, bit_shiftRight_one]
   intros; rfl
 
 unseal binaryRec in
 @[simp]
-theorem binaryRec_zero {C : Nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) :
+theorem binaryRec_zero (z : motive 0) (f : ∀ b n, motive n → motive (bit b n)) :
     binaryRec z f 0 = z :=
   rfl
 
-theorem binaryRec_eq {C : Nat → Sort u} {z : C 0} {f : ∀ b n, C n → C (bit b n)} (b n)
-    (h : f false 0 z = z ∨ (n = 0 → b = true)) :
+@[simp]
+theorem binaryRec_one (z : motive 0) (f : ∀ b n, motive n → motive (bit b n)) :
+    binaryRec (motive := motive) z f 1 = f true 0 z := by
+  rw [binaryRec]
+  simp only [add_one_ne_zero, ↓reduceDIte, Nat.reduceShiftRight, binaryRec_zero]
+  rfl
+
+theorem binaryRec_eq {z : motive 0} {f : ∀ b n, motive n → motive (bit b n)}
+    (b n) (h : f false 0 z = z ∨ (n = 0 → b = true)) :
     binaryRec z f (bit b n) = f b n (binaryRec z f n) := by
   by_cases h' : bit b n = 0
   case pos =>
     obtain ⟨rfl, rfl⟩ := bit_eq_zero_iff.mp h'
-    simp only [forall_const, or_false] at h
+    simp only [Bool.false_eq_true, imp_false, not_true_eq_false, or_false] at h
     unfold binaryRec
     exact h.symm
   case neg =>
     rw [binaryRec, dif_neg h']
-    change congrArg C (bit b n).bit_testBit_zero_shiftRight_one ▸ f _ _ _ = _
-    generalize congrArg C (bit b n).bit_testBit_zero_shiftRight_one = e; revert e
-    rw [bit_testBit_zero, bit_shiftRight_one]
+    change congrArg motive (bit b n).bit_testBit_zero_shiftRight_one ▸ f _ _ _ = _
+    generalize congrArg motive (bit b n).bit_testBit_zero_shiftRight_one = e; revert e
+    rw [testBit_bit_zero, bit_shiftRight_one]
     intros; rfl
 
 end Nat