diff --git a/src/Init/Data/BitVec/Lemmas.lean b/src/Init/Data/BitVec/Lemmas.lean
index 8c7471551e0f..2b025732571d 100644
--- a/src/Init/Data/BitVec/Lemmas.lean
+++ b/src/Init/Data/BitVec/Lemmas.lean
@@ -206,6 +206,7 @@ theorem eq_of_getMsbD_eq {x y : BitVec w}
 theorem of_length_zero {x : BitVec 0} : x = 0#0 := by ext; simp
 
 theorem toNat_zero_length (x : BitVec 0) : x.toNat = 0 := by simp [of_length_zero]
+theorem toInt_zero_length (x : BitVec 0) : x.toInt = 0 := by simp [of_length_zero]
 theorem getLsbD_zero_length (x : BitVec 0) : x.getLsbD i = false := by simp
 theorem getMsbD_zero_length (x : BitVec 0) : x.getMsbD i = false := by simp
 theorem msb_zero_length (x : BitVec 0) : x.msb = false := by simp [BitVec.msb, of_length_zero]
@@ -386,7 +387,19 @@ theorem msb_eq_getLsbD_last (x : BitVec w) :
 @[bv_toNat] theorem getLsbD_succ_last (x : BitVec (w + 1)) :
     x.getLsbD w = decide (2 ^ w ≤ x.toNat) := getLsbD_last x
 
-@[bv_toNat] theorem msb_eq_decide (x : BitVec w) : BitVec.msb x = decide (2 ^ (w-1) ≤ x.toNat) := by
+
+/-- 
+An alternative to `msb_eq_decide` in terms of `2 * x.toNat`,
+in order to avoid `2 ^ (w - 1)`.
+-/
+@[bv_toNat] theorem msb_eq_decide_le_mul_two (x : BitVec w) : 
+    BitVec.msb x = decide (2 ^ w ≤ 2 * x.toNat) := by
+  rw [x.msb_eq_getLsbD_last, x.getLsbD_last]
+  simp
+  rcases w with rfl | w  <;> simp <;> omega
+
+@[bv_toNat, deprecated msb_eq_decide_le_mul_two (since := "21-10-2024") ]
+theorem msb_eq_decide (x : BitVec w) : BitVec.msb x = decide (2 ^ (w-1) ≤ x.toNat) := by
   simp [msb_eq_getLsbD_last, getLsbD_last]
 
 theorem toNat_ge_of_msb_true {x : BitVec n} (p : BitVec.msb x = true) : x.toNat ≥ 2^(n-1) := by
@@ -496,6 +509,38 @@ theorem toInt_pos_iff {w : Nat} {x : BitVec w} :
     0 ≤ BitVec.toInt x ↔ 2 * x.toNat < 2 ^ w := by
   simp [toInt_eq_toNat_cond]; omega
 
+/-
+If `x.msb` is false, then the value of `x` when interpreted as a 2s complement
+integer is between `[0..2^n/2)`. 
+To avoid the corner case at `n = 0`, we phrase the bounds as `2 * x < 2^n` instead of `x < 2^(n-1)`.
+-/
+theorem toInt_bounds_of_msb_eq_false {x : BitVec n} (hmsb : x.msb = false) :
+    0 ≤ x.toInt ∧ 2 * x.toInt < 2^n := by
+  have := x.msb_eq_decide_le_mul_two
+  rw [hmsb] at this
+  simp only [false_eq_decide_iff, Nat.not_le] at this
+  rw [BitVec.toInt_eq_toNat_cond]
+  simp [this]
+  apply And.intro
+  · omega
+  · norm_cast
+
+/-
+If `x.msb` is true, then the value of `x` when interpreted as a 2s complement
+integer is between `[-2^n..0).
+-/
+theorem toInt_bounds_of_msb_eq_true {x : BitVec n} (hmsb : x.msb = true) :
+    -2^n ≤ x.toInt ∧ x.toInt < 0 := by
+  have := x.msb_eq_decide_le_mul_two
+  rw [hmsb] at this
+  simp only [true_eq_decide_iff] at this
+  rw [BitVec.toInt_eq_toNat_cond]
+  simp [show ¬ 2 * x.toNat < 2 ^ n by omega]
+  apply And.intro
+  · norm_cast
+    omega
+  · omega
+
 theorem eq_zero_or_eq_one (a : BitVec 1) : a = 0#1 ∨ a = 1#1 := by
   obtain ⟨a, ha⟩ := a
   simp only [Nat.reducePow]
@@ -2111,16 +2156,6 @@ theorem not_neg (x : BitVec w) : ~~~(-x) = x + -1#w := by
         show (_ - x.toNat) % _ = _ by rw [Nat.mod_eq_of_lt (by omega)]]
       omega
 
-/-! ### abs -/
-
-@[simp, bv_toNat]
-theorem toNat_abs {x : BitVec w} : x.abs.toNat = if x.msb then 2^w - x.toNat else x.toNat := by
-  simp only [BitVec.abs, neg_eq]
-  by_cases h : x.msb = true
-  · simp only [h, ↓reduceIte, toNat_neg]
-    have : 2 * x.toNat ≥ 2 ^ w := BitVec.msb_eq_true_iff_two_mul_ge.mp h
-    rw [Nat.mod_eq_of_lt (by omega)]
-  · simp [h]
 
 /-! ### mul -/
 
@@ -2860,6 +2895,52 @@ theorem toInt_neg_of_ne_intMin {x : BitVec w} (rs : x ≠ intMin w) :
   have := @Nat.two_pow_pred_mul_two w (by omega)
   split <;> split <;> omega
 
+/-- The msb of `intMin w` is `true` for all  `w > 0` -/
+@[simp] theorem msb_intMin : (intMin w).msb = decide (w > 0) := by
+  rw [intMin]
+  rw [msb_eq_decide]
+  simp
+  rcases w with rfl | w 
+  · rfl
+  · simp
+    have : 0 < 2^w := Nat.pow_pos (by decide)
+    have : 2^w < 2^(w + 1) := by
+      rw [Nat.pow_succ]
+      omega
+    rw [Nat.mod_eq_of_lt (by omega)]
+    simp
+
+/--
+If the width is zero, then `intMin` is `0`,
+and otherwise it is `-2^(n - 1)`.
+-/
+theorem toInt_intMin_eq_if (n : Nat) : (BitVec.intMin n).toInt =
+  if n = 0 then 0 else - 2^(n - 1) := by
+  simp [BitVec.toInt_intMin]
+  rcases n with rfl | n
+  · simp
+  · simp
+    norm_cast
+    have : 2^n > 0 := by exact Nat.two_pow_pos n
+    have : 2^n < 2^(n + 1) := by
+      simp [Nat.pow_succ]
+      omega
+    rw [Nat.mod_eq_of_lt (by omega)]
+
+/--
+Negating `intMin` returns `intMin`.
+Thus, converting `(-x)` to an `Int` return `- x.toInt` for all bitvectors other than `intMin`.
+-/
+theorem toInt_neg_eq_if {x : BitVec n} : 
+  (-x).toInt = 
+  if x = intMin n
+  then x.toInt
+  else - x.toInt := by
+  by_cases hx : x = intMin n
+  · simp [hx]
+  · simp [hx]
+    rw [toInt_neg_of_ne_intMin hx]
+
 /-! ### intMax -/
 
 /-- The bitvector of width `w` that has the largest value when interpreted as an integer. -/
@@ -2891,6 +2972,42 @@ theorem getLsbD_intMax (w : Nat) : (intMax w).getLsbD i = decide (i + 1 < w) :=
   · rw [Nat.sub_add_cancel (Nat.two_pow_pos (w - 1)), Nat.two_pow_pred_mod_two_pow (by omega)]
 
 
+/-! ### abs -/
+
+theorem abs_def {x : BitVec w} : x.abs = if x.msb then .neg x else x := rfl
+
+theorem abs_eq_if (x : BitVec w) : x.abs = 
+    if x.msb = true then
+      if x = BitVec.intMin w then (BitVec.intMin w) else -x
+    else x := by
+  · rw [BitVec.abs_def]
+    by_cases hx : x.msb = true <;> by_cases hx' : x = BitVec.intMin w <;> simp [hx, hx']
+
+theorem toInt_abs (x : BitVec w) :
+    x.abs.toInt = if x = (intMin w) then if w = 0 then 0 else - 2^(w - 1) else x.toInt.abs := by
+  rcases w with rfl | w 
+  · simp [toInt_zero_length]
+  · simp only [gt_iff_lt, Nat.zero_lt_succ, Nat.add_one_ne_zero, ↓reduceIte]
+    rw [BitVec.abs_eq_if]
+    by_cases hx : x = intMin (w + 1)
+    · simp only [hx, reduceIte]
+      have := BitVec.msb_intMin (w := w + 1)
+      rw [this]
+      simp only [gt_iff_lt, Nat.zero_lt_succ, decide_True, ↓reduceIte]
+      rw [BitVec.toInt_intMin_eq_if]
+      simp
+    · simp only [hx, reduceIte]
+      rcases hmsb : x.msb
+      · simp only [Bool.false_eq_true, ↓reduceIte]
+        have := BitVec.toInt_bounds_of_msb_eq_false hmsb
+        rw [Int.abs_eq_self]
+        omega
+      · simp only [reduceIte]
+        have hxbounds := BitVec.toInt_bounds_of_msb_eq_true hmsb
+        rw [BitVec.toInt_neg_eq_if]
+        simp only [hx, reduceIte]
+        rw [Int.abs_eq_neg (by omega)]
+
 /-! ### Non-overflow theorems -/
 
 /-- If `x.toNat * y.toNat < 2^w`, then the multiplication `(x * y)` does not overflow. -/
diff --git a/src/Init/Data/Int/Basic.lean b/src/Init/Data/Int/Basic.lean
index dbf661c4be1b..fed9719af55b 100644
--- a/src/Init/Data/Int/Basic.lean
+++ b/src/Init/Data/Int/Basic.lean
@@ -333,6 +333,13 @@ instance : Min Int := minOfLe
 
 instance : Max Int := maxOfLe
 
+/--
+Return the absolute value of an integer.
+-/
+def abs : Int → Int
+  | ofNat n   => .ofNat n
+  | negSucc n => .ofNat n.succ
+
 end Int
 
 /--
diff --git a/src/Init/Data/Int/Lemmas.lean b/src/Init/Data/Int/Lemmas.lean
index 4b0e560fb00d..4b555d825373 100644
--- a/src/Init/Data/Int/Lemmas.lean
+++ b/src/Init/Data/Int/Lemmas.lean
@@ -531,4 +531,28 @@ theorem natCast_one : ((1 : Nat) : Int) = (1 : Int) := rfl
 @[simp] theorem natCast_mul (a b : Nat) : ((a * b : Nat) : Int) = (a : Int) * (b : Int) := by
   simp
 
+/-! abs lemmas -/
+
+@[simp]
+theorem abs_eq_self {x : Int} (h : x ≥ 0) : x.abs = x := by
+  cases x 
+  case ofNat h => 
+    rfl
+  case negSucc h =>
+    contradiction
+
+@[simp]
+theorem Int.abs_zero : Int.abs 0 = 0 := rfl
+
+@[simp]
+theorem abs_eq_neg {x : Int} (h : x < 0) : x.abs = -x := by
+  cases x
+  case ofNat h => 
+    contradiction
+  case negSucc n =>
+    rfl
+
+@[simp]
+theorem ofNat_abs (x : Nat) : (x : Int).abs = (x : Int) := rfl
+
 end Int