diff --git a/src/Init/Data/BitVec/Lemmas.lean b/src/Init/Data/BitVec/Lemmas.lean
index 2b668aacca..c383a2bee3 100644
--- a/src/Init/Data/BitVec/Lemmas.lean
+++ b/src/Init/Data/BitVec/Lemmas.lean
@@ -21,6 +21,9 @@ set_option linter.missingDocs true
 
 namespace BitVec
 
+@[simp] theorem mk_zero : BitVec.ofFin (w := w) ⟨0, h⟩ = 0#w := rfl
+@[simp] theorem ofNatLT_zero : BitVec.ofNatLT (w := w) 0 h = 0#w := rfl
+
 @[simp] theorem getLsbD_ofFin (x : Fin (2^n)) (i : Nat) :
     getLsbD (BitVec.ofFin x) i = x.val.testBit i := rfl
 
@@ -136,6 +139,8 @@ theorem toNat_ne_iff_ne {n} {x y : BitVec n} : x.toNat ≠ y.toNat ↔ x ≠ y :
 @[bitvec_to_nat] theorem toNat_eq {x y : BitVec n} : x = y ↔ x.toNat = y.toNat :=
   Iff.intro (congrArg BitVec.toNat) eq_of_toNat_eq
 
+theorem toNat_inj {x y : BitVec n} : x.toNat = y.toNat ↔ x = y := toNat_eq.symm
+
 @[bitvec_to_nat] theorem toNat_ne {x y : BitVec n} : x ≠ y ↔ x.toNat ≠ y.toNat := by
   rw [Ne, toNat_eq]
 
@@ -637,7 +642,7 @@ theorem toInt_nonneg_of_msb_false {x : BitVec w} (h : x.msb = false) : 0 ≤ x.t
   apply Nat.mod_eq_of_lt
   apply Nat.one_lt_two_pow (by omega)
 
-/-- Prove equality of bitvectors in terms of nat operations. -/
+/-- Prove equality of bitvectors in terms of integer operations. -/
 theorem eq_of_toInt_eq {x y : BitVec n} : x.toInt = y.toInt → x = y := by
   intro eq
   simp only [toInt_eq_toNat_cond] at eq
diff --git a/src/Init/Data/Int/Pow.lean b/src/Init/Data/Int/Pow.lean
index f3463f0e37..0b2f3aa699 100644
--- a/src/Init/Data/Int/Pow.lean
+++ b/src/Init/Data/Int/Pow.lean
@@ -67,4 +67,10 @@ theorem pow_lt_pow_of_lt {a : Int} {b c : Nat} (ha : 1 < a) (hbc : b < c):
   | 0 => rfl
   | k + 1 => by rw [Int.pow_succ, natAbs_mul, natAbs_pow, Nat.pow_succ]
 
+theorem toNat_pow_of_nonneg {x : Int} (h : 0 ≤ x) (k : Nat) : (x ^ k).toNat = x.toNat ^ k := by
+  induction k with
+  | zero => simp
+  | succ k ih =>
+    rw [Int.pow_succ, Int.toNat_mul (Int.pow_nonneg h) h, ih, Nat.pow_succ]
+
 end Int
diff --git a/src/Init/Grind/CommRing/Basic.lean b/src/Init/Grind/CommRing/Basic.lean
index 2d77d77407..5440d4c873 100644
--- a/src/Init/Grind/CommRing/Basic.lean
+++ b/src/Init/Grind/CommRing/Basic.lean
@@ -30,6 +30,7 @@ namespace Lean.Grind
 
 class CommRing (α : Type u) extends Add α, Mul α, Neg α, Sub α, HPow α Nat α where
   [ofNat : ∀ n, OfNat α n]
+  [intCast : IntCast α]
   add_assoc : ∀ a b c : α, a + b + c = a + (b + c)
   add_comm : ∀ a b : α, a + b = b + a
   add_zero : ∀ a : α, a + 0 = a
@@ -43,6 +44,8 @@ class CommRing (α : Type u) extends Add α, Mul α, Neg α, Sub α, HPow α Nat
   pow_zero : ∀ a : α, a ^ 0 = 1
   pow_succ : ∀ a : α, ∀ n : Nat, a ^ (n + 1) = (a ^ n) * a
   ofNat_succ : ∀ a : Nat, OfNat.ofNat (α := α) (a + 1) = OfNat.ofNat a + 1 := by intros; rfl
+  intCast_ofNat : ∀ n : Nat, Int.cast (OfNat.ofNat (α := Int) n) = OfNat.ofNat (α := α) n := by intros; rfl
+  intCast_neg : ∀ i : Int, Int.cast (R := α) (-i) = -Int.cast i := by intros; rfl
 
 -- We reduce the priority of these parent instances,
 -- so that in downstream libraries with their own `CommRing` class,
@@ -53,6 +56,9 @@ attribute [instance 100] CommRing.toAdd CommRing.toMul CommRing.toNeg CommRing.t
 -- This is a low-priority instance, to avoid conflicts with existing `OfNat` instances.
 attribute [instance 100] CommRing.ofNat
 
+-- This is a low-priority instance, to avoid conflicts with existing `IntCast` instances.
+attribute [instance 100] CommRing.intCast
+
 namespace CommRing
 
 variable {α : Type u} [CommRing α]
@@ -69,6 +75,7 @@ theorem ofNat_add (a b : Nat) : OfNat.ofNat (α := α) (a + b) = OfNat.ofNat a +
   | succ b ih => rw [Nat.add_succ, ofNat_succ, ih, ofNat_succ b, add_assoc]
 
 theorem natCast_succ (n : Nat) : ((n + 1 : Nat) : α) = ((n : α) + 1) := ofNat_add _ _
+theorem natCast_add (a b : Nat) : ((a + b : Nat) : α) = ((a : α) + (b : α)) := ofNat_add _ _
 
 theorem zero_add (a : α) : 0 + a = a := by
   rw [add_comm, add_zero]
@@ -96,6 +103,9 @@ theorem ofNat_mul (a b : Nat) : OfNat.ofNat (α := α) (a * b) = OfNat.ofNat a *
   | zero => simp [Nat.mul_zero, mul_zero]
   | succ a ih => rw [Nat.mul_succ, ofNat_add, ih, ofNat_add, left_distrib, mul_one]
 
+theorem natCast_mul (a b : Nat) : ((a * b : Nat) : α) = ((a : α) * (b : α)) := by
+  rw [← ofNat_eq_natCast, ofNat_mul, ofNat_eq_natCast, ofNat_eq_natCast]
+
 theorem add_left_inj {a b : α} (c : α) : a + c = b + c ↔ a = b :=
   ⟨fun h => by simpa [add_assoc, add_neg_cancel, add_zero] using (congrArg (· + -c) h),
    fun g => congrArg (· + c) g⟩
@@ -134,25 +144,16 @@ theorem sub_eq_iff {a b c : α} : a - b = c ↔ a = c + b := by
 theorem sub_eq_zero_iff {a b : α} : a - b = 0 ↔ a = b := by
   simp [sub_eq_iff, zero_add]
 
-instance intCastInst : IntCast α where
-  intCast n := match n with
-  | Int.ofNat n => OfNat.ofNat n
-  | Int.negSucc n => -OfNat.ofNat (n + 1)
-
-theorem intCast_zero : ((0 : Int) : α) = 0 := rfl
-theorem intCast_one : ((1 : Int) : α) = 1 := rfl
-theorem intCast_neg_one : ((-1 : Int) : α) = -1 := rfl
-theorem intCast_ofNat (n : Nat) : ((n : Int) : α) = (n : α) := rfl
-theorem intCast_ofNat_add_one (n : Nat) : ((n + 1 : Int) : α) = (n : α) + 1 := ofNat_add _ _
-theorem intCast_negSucc (n : Nat) : ((-(n + 1) : Int) : α) = -((n : α) + 1) := congrArg (- ·) (ofNat_add _ _)
-theorem intCast_neg (x : Int) : ((-x : Int) : α) = - (x : α) :=
-  match x with
-  | (0 : Nat) => neg_zero.symm
-  | (n + 1 : Nat) => by
-    rw [Int.natCast_add, Int.cast_ofNat_Int, intCast_negSucc, intCast_ofNat_add_one]
-  | -((n : Nat) + 1) => by
-    rw [Int.neg_neg, intCast_ofNat_add_one, intCast_negSucc, neg_neg]
-theorem intCast_nat_add {x y : Nat} : ((x + y : Int) : α) = ((x : α) + (y : α)) := ofNat_add _ _
+theorem intCast_zero : ((0 : Int) : α) = 0 := intCast_ofNat 0
+theorem intCast_one : ((1 : Int) : α) = 1 := intCast_ofNat 1
+theorem intCast_neg_one : ((-1 : Int) : α) = -1 := by rw [intCast_neg, intCast_ofNat]
+theorem intCast_natCast (n : Nat) : ((n : Int) : α) = (n : α) := intCast_ofNat n
+theorem intCast_natCast_add_one (n : Nat) : ((n + 1 : Int) : α) = (n : α) + 1 := by
+  rw [← Int.natCast_succ, intCast_natCast, natCast_add, ofNat_eq_natCast]
+theorem intCast_negSucc (n : Nat) : ((-(n + 1) : Int) : α) = -((n : α) + 1) := by
+  rw [intCast_neg, ← Int.natCast_succ, intCast_natCast, ofNat_eq_natCast, natCast_add]
+theorem intCast_nat_add {x y : Nat} : ((x + y : Int) : α) = ((x : α) + (y : α)) := by
+  rw [Int.ofNat_add_ofNat, intCast_natCast, natCast_add]
 theorem intCast_nat_sub {x y : Nat} (h : x ≥ y) : (((x - y : Nat) : Int) : α) = ((x : α) - (y : α)) := by
   induction x with
   | zero =>
@@ -162,29 +163,30 @@ theorem intCast_nat_sub {x y : Nat} (h : x ≥ y) : (((x - y : Nat) : Int) : α)
     by_cases h : x + 1 = y
     · simp [h, intCast_zero, sub_self]
     · have : ((x + 1 - y : Nat) : Int) = (x - y : Nat) + 1 := by omega
-      rw [this, intCast_ofNat_add_one]
+      rw [this, intCast_natCast_add_one]
       specialize ih (by omega)
-      rw [intCast_ofNat] at ih
+      rw [intCast_natCast] at ih
       rw [ih, natCast_succ, sub_eq_add_neg, sub_eq_add_neg, add_assoc, add_comm _ 1, ← add_assoc]
 theorem intCast_add (x y : Int) : ((x + y : Int) : α) = ((x : α) + (y : α)) :=
   match x, y with
-  | (x : Nat), (y : Nat) => ofNat_add _ _
+  | (x : Nat), (y : Nat) => by
+    rw [intCast_nat_add, intCast_natCast, intCast_natCast]
   | (x : Nat), (-(y + 1 : Nat)) => by
     by_cases h : x ≥ y + 1
     · have : (x + -(y+1 : Nat) : Int) = ((x - (y + 1) : Nat) : Int) := by omega
-      rw [this, intCast_neg, intCast_nat_sub h, intCast_ofNat, intCast_ofNat, sub_eq_add_neg]
+      rw [this, intCast_neg, intCast_nat_sub h, intCast_natCast, intCast_natCast, sub_eq_add_neg]
     · have : (x + -(y+1 : Nat) : Int) = (-(y + 1 - x : Nat) : Int) := by omega
-      rw [this, intCast_neg, intCast_nat_sub (by omega), intCast_ofNat, intCast_neg, intCast_ofNat,
+      rw [this, intCast_neg, intCast_nat_sub (by omega), intCast_natCast, intCast_neg, intCast_natCast,
         neg_sub, sub_eq_add_neg]
   | (-(x + 1 : Nat)), (y : Nat) => by
     by_cases h : y ≥ x+ 1
     · have : (-(x+1 : Nat) + y : Int) = ((y - (x + 1) : Nat) : Int) := by omega
-      rw [this, intCast_neg, intCast_nat_sub h, intCast_ofNat, intCast_ofNat, sub_eq_add_neg, add_comm]
+      rw [this, intCast_neg, intCast_nat_sub h, intCast_natCast, intCast_natCast, sub_eq_add_neg, add_comm]
     · have : (-(x+1 : Nat) + y : Int) = (-(x + 1 - y : Nat) : Int) := by omega
-      rw [this, intCast_neg, intCast_nat_sub (by omega), intCast_ofNat, intCast_neg, intCast_ofNat,
+      rw [this, intCast_neg, intCast_nat_sub (by omega), intCast_natCast, intCast_neg, intCast_natCast,
         neg_sub, sub_eq_add_neg, add_comm]
   | (-(x + 1 : Nat)), (-(y + 1 : Nat)) => by
-    rw [← Int.neg_add, intCast_neg, intCast_nat_add, neg_add, intCast_neg, intCast_neg, intCast_ofNat, intCast_ofNat]
+    rw [← Int.neg_add, intCast_neg, intCast_nat_add, neg_add, intCast_neg, intCast_neg, intCast_natCast, intCast_natCast]
 theorem intCast_sub (x y : Int) : ((x - y : Int) : α) = ((x : α) - (y : α)) := by
   rw [Int.sub_eq_add_neg, intCast_add, intCast_neg, sub_eq_add_neg]
 
@@ -200,17 +202,20 @@ theorem neg_mul (a b : α) : (-a) * b = -(a * b) := by
 theorem mul_neg (a b : α) : a * (-b) = -(a * b) := by
   rw [mul_comm, neg_mul, mul_comm]
 
-theorem intCast_nat_mul (x y : Nat) : ((x * y : Int) : α) = ((x : α) * (y : α)) := ofNat_mul _ _
+theorem intCast_nat_mul (x y : Nat) : ((x * y : Int) : α) = ((x : α) * (y : α)) := by
+  rw [Int.ofNat_mul_ofNat, intCast_natCast, natCast_mul]
+
 theorem intCast_mul (x y : Int) : ((x * y : Int) : α) = ((x : α) * (y : α)) :=
   match x, y with
-  | (x : Nat), (y : Nat) => ofNat_mul _ _
+  | (x : Nat), (y : Nat) => by
+    rw [intCast_nat_mul, intCast_natCast, intCast_natCast]
   | (x : Nat), (-(y + 1 : Nat)) => by
-    rw [Int.mul_neg, intCast_neg, intCast_nat_mul, intCast_neg, mul_neg, intCast_ofNat, intCast_ofNat]
+    rw [Int.mul_neg, intCast_neg, intCast_nat_mul, intCast_neg, mul_neg, intCast_natCast, intCast_natCast]
   | (-(x + 1 : Nat)), (y : Nat) => by
-    rw [Int.neg_mul, intCast_neg, intCast_nat_mul, intCast_neg, neg_mul, intCast_ofNat, intCast_ofNat]
+    rw [Int.neg_mul, intCast_neg, intCast_nat_mul, intCast_neg, neg_mul, intCast_natCast, intCast_natCast]
   | (-(x + 1 : Nat)), (-(y + 1 : Nat)) => by
     rw [Int.neg_mul_neg, intCast_neg, intCast_neg, neg_mul, mul_neg, neg_neg, intCast_nat_mul,
-      intCast_ofNat, intCast_ofNat]
+      intCast_natCast, intCast_natCast]
 
 theorem intCast_pow (x : Int) (k : Nat) : ((x ^ k : Int) : α) = (x : α) ^ k := by
   induction k
@@ -240,10 +245,10 @@ theorem intCast_eq_zero_iff (x : Int) : (x : α) = 0 ↔ x % p = 0 :=
   match x with
   | (x : Nat) => by
     have := ofNat_eq_zero_iff (α := α) p (x := x)
-    rw [Int.ofNat_mod_ofNat]
+    rw [Int.ofNat_mod_ofNat, intCast_natCast]
     norm_cast
   | -(x + 1 : Nat) => by
-    rw [Int.neg_emod, Int.ofNat_mod_ofNat, intCast_neg, intCast_ofNat, neg_eq_zero]
+    rw [Int.neg_emod, Int.ofNat_mod_ofNat, intCast_neg, intCast_natCast, neg_eq_zero]
     have := ofNat_eq_zero_iff (α := α) p (x := x + 1)
     rw [ofNat_eq_natCast] at this
     rw [this]
@@ -273,7 +278,7 @@ theorem intCast_ext_iff {x y : Int} : (x : α) = (y : α) ↔ x % p = y % p := b
 
 theorem ofNat_ext_iff {x y : Nat} : OfNat.ofNat (α := α) x = OfNat.ofNat (α := α) y ↔ x % p = y % p := by
   have := intCast_ext_iff (α := α) p (x := x) (y := y)
-  simp only [intCast_ofNat, ← Int.ofNat_emod] at this
+  simp only [intCast_natCast, ← Int.ofNat_emod] at this
   simp only [ofNat_eq_natCast]
   norm_cast at this
 
@@ -288,7 +293,7 @@ theorem intCast_emod (x : Int) : ((x % p : Int) : α) = (x : α) := by
   rw [intCast_ext_iff p, Int.emod_emod]
 
 theorem natCast_emod (x : Nat) : ((x % p : Nat) : α) = (x : α) := by
-  simp only [← intCast_ofNat]
+  simp only [← intCast_natCast]
   rw [Int.ofNat_emod, intCast_emod]
 
 theorem ofNat_emod (x : Nat) : OfNat.ofNat (α := α) (x % p) = OfNat.ofNat x :=
diff --git a/src/Init/Grind/CommRing/BitVec.lean b/src/Init/Grind/CommRing/BitVec.lean
index 85cb0590b2..0af5910691 100644
--- a/src/Init/Grind/CommRing/BitVec.lean
+++ b/src/Init/Grind/CommRing/BitVec.lean
@@ -23,6 +23,7 @@ instance : CommRing (BitVec w) where
   pow_zero _ := BitVec.pow_zero
   pow_succ _ _ := BitVec.pow_succ
   ofNat_succ x := BitVec.ofNat_add x 1
+  intCast_neg _ := BitVec.ofInt_neg
 
 instance : IsCharP (BitVec w) (2 ^ w) where
   ofNat_eq_zero_iff {x} := by simp [BitVec.ofInt, BitVec.toNat_eq]
diff --git a/src/Init/Grind/CommRing/Poly.lean b/src/Init/Grind/CommRing/Poly.lean
index 2f4673add3..b9e74b1cd2 100644
--- a/src/Init/Grind/CommRing/Poly.lean
+++ b/src/Init/Grind/CommRing/Poly.lean
@@ -30,12 +30,18 @@ abbrev Context (α : Type u) := RArray α
 def Var.denote {α} (ctx : Context α) (v : Var) : α :=
   ctx.get v
 
+def denoteInt {α} [CommRing α] (k : Int) : α :=
+  bif k < 0 then
+    - OfNat.ofNat (α := α) k.natAbs
+  else
+    OfNat.ofNat (α := α) k.natAbs
+
 def Expr.denote {α} [CommRing α] (ctx : Context α) : Expr → α
   | .add a b  => denote ctx a + denote ctx b
   | .sub a b  => denote ctx a - denote ctx b
   | .mul a b  => denote ctx a * denote ctx b
   | .neg a    => -denote ctx a
-  | .num k    => k
+  | .num k    => denoteInt k
   | .var v    => v.denote ctx
   | .pow a k  => denote ctx a ^ k
 
@@ -498,6 +504,11 @@ def NullCert.toPolyC (nc : NullCert) (c : Nat) : Poly :=
 Theorems for justifying the procedure for commutative rings in `grind`.
 -/
 
+theorem denoteInt_eq {α} [CommRing α] (k : Int) : denoteInt (α := α) k = k := by
+  simp [denoteInt, cond_eq_if] <;> split
+  next h => rw [ofNat_eq_natCast, ← intCast_natCast, ← intCast_neg, ← Int.eq_neg_natAbs_of_nonpos (Int.le_of_lt h)]
+  next h => rw [ofNat_eq_natCast, ← intCast_natCast, ← Int.eq_natAbs_of_nonneg (Int.le_of_not_gt h)]
+
 theorem Power.denote_eq {α} [CommRing α] (ctx : Context α) (p : Power)
     : p.denote ctx = p.x.denote ctx ^ p.k := by
   cases p <;> simp [Power.denote] <;> split <;> simp [pow_zero, pow_succ, one_mul]
@@ -677,7 +688,7 @@ theorem Expr.denote_toPoly {α} [CommRing α] (ctx : Context α) (e : Expr)
   fun_induction toPoly
     <;> simp [toPoly, denote, Poly.denote, Poly.denote_ofVar, Poly.denote_combine,
           Poly.denote_mul, Poly.denote_mulConst, Poly.denote_pow, intCast_pow, intCast_neg, intCast_one,
-          neg_mul, one_mul, sub_eq_add_neg, *]
+          neg_mul, one_mul, sub_eq_add_neg, denoteInt_eq, *]
   next => simp [Poly.denote_ofMon, Mon.denote, Power.denote_eq, mul_one]
 
 theorem Expr.eq_of_toPoly_eq {α} [CommRing α] (ctx : Context α) (a b : Expr) (h : a.toPoly == b.toPoly) : a.denote ctx = b.denote ctx := by
@@ -845,7 +856,7 @@ theorem Expr.denote_toPolyC {α c} [CommRing α] [IsCharP α c] (ctx : Context 
   unfold toPolyC
   fun_induction toPolyC.go
     <;> simp [toPolyC.go, denote, Poly.denote, Poly.denote_ofVar, Poly.denote_combineC,
-          Poly.denote_mulC, Poly.denote_mulConstC, Poly.denote_powC, *]
+          Poly.denote_mulC, Poly.denote_mulConstC, Poly.denote_powC, denoteInt_eq, *]
   next => rw [IsCharP.intCast_emod]
   next => rw [intCast_neg, neg_mul, intCast_one, one_mul]
   next => rw [intCast_neg, neg_mul, intCast_one, one_mul, sub_eq_add_neg]
diff --git a/src/Init/Grind/CommRing/SInt.lean b/src/Init/Grind/CommRing/SInt.lean
index b56da453cd..c8d8881450 100644
--- a/src/Init/Grind/CommRing/SInt.lean
+++ b/src/Init/Grind/CommRing/SInt.lean
@@ -26,6 +26,7 @@ instance : CommRing Int8 where
   pow_zero := Int8.pow_zero
   pow_succ := Int8.pow_succ
   ofNat_succ x := Int8.ofNat_add x 1
+  intCast_neg := Int8.ofInt_neg
 
 instance : IsCharP Int8 (2 ^ 8) where
   ofNat_eq_zero_iff {x} := by
@@ -51,7 +52,7 @@ instance : CommRing Int16 where
   pow_zero := Int16.pow_zero
   pow_succ := Int16.pow_succ
   ofNat_succ x := Int16.ofNat_add x 1
-
+  intCast_neg := Int16.ofInt_neg
 instance : IsCharP Int16 (2 ^ 16) where
   ofNat_eq_zero_iff {x} := by
     have : OfNat.ofNat x = Int16.ofInt x := rfl
@@ -76,7 +77,7 @@ instance : CommRing Int32 where
   pow_zero := Int32.pow_zero
   pow_succ := Int32.pow_succ
   ofNat_succ x := Int32.ofNat_add x 1
-
+  intCast_neg := Int32.ofInt_neg
 instance : IsCharP Int32 (2 ^ 32) where
   ofNat_eq_zero_iff {x} := by
     have : OfNat.ofNat x = Int32.ofInt x := rfl
@@ -101,7 +102,7 @@ instance : CommRing Int64 where
   pow_zero := Int64.pow_zero
   pow_succ := Int64.pow_succ
   ofNat_succ x := Int64.ofNat_add x 1
-
+  intCast_neg := Int64.ofInt_neg
 instance : IsCharP Int64 (2 ^ 64) where
   ofNat_eq_zero_iff {x} := by
     have : OfNat.ofNat x = Int64.ofInt x := rfl
@@ -126,7 +127,7 @@ instance : CommRing ISize where
   pow_zero := ISize.pow_zero
   pow_succ := ISize.pow_succ
   ofNat_succ x := ISize.ofNat_add x 1
-
+  intCast_neg := ISize.ofInt_neg
 open System.Platform (numBits)
 
 instance : IsCharP ISize (2 ^ numBits) where
diff --git a/src/Init/Grind/CommRing/UInt.lean b/src/Init/Grind/CommRing/UInt.lean
index 5385c65710..0777e83351 100644
--- a/src/Init/Grind/CommRing/UInt.lean
+++ b/src/Init/Grind/CommRing/UInt.lean
@@ -7,7 +7,6 @@ prelude
 import Init.Grind.CommRing.Basic
 import Init.Data.UInt.Lemmas
 
-
 namespace UInt8
 
 /-- Variant of `UInt8.ofNat_mod_size` replacing `2 ^ 8` with `256`.-/
@@ -16,6 +15,16 @@ theorem ofNat_mod_size' : ofNat (x % 256) = ofNat x := ofNat_mod_size
 instance : IntCast UInt8 where
   intCast x := UInt8.ofInt x
 
+theorem intCast_ofNat (x : Nat) : (OfNat.ofNat (α := Int) x : UInt8) = OfNat.ofNat x := by
+    -- A better proof would be welcome!
+    simp only [Int.cast, IntCast.intCast]
+    rw [UInt8.ofInt]
+    rw [Int.toNat_emod (Int.zero_le_ofNat x) (by decide)]
+    erw [Int.toNat_natCast]
+    rw [Int.toNat_pow_of_nonneg (by decide)]
+    simp only [ofNat, BitVec.ofNat, Fin.ofNat', Int.reduceToNat, Nat.dvd_refl,
+      Nat.mod_mod_of_dvd, instOfNat]
+
 end UInt8
 
 namespace UInt16
@@ -26,6 +35,16 @@ theorem ofNat_mod_size' : ofNat (x % 65536) = ofNat x := ofNat_mod_size
 instance : IntCast UInt16 where
   intCast x := UInt16.ofInt x
 
+theorem intCast_ofNat (x : Nat) : (OfNat.ofNat (α := Int) x : UInt16) = OfNat.ofNat x := by
+    -- A better proof would be welcome!
+    simp only [Int.cast, IntCast.intCast]
+    rw [UInt16.ofInt]
+    rw [Int.toNat_emod (Int.zero_le_ofNat x) (by decide)]
+    erw [Int.toNat_natCast]
+    rw [Int.toNat_pow_of_nonneg (by decide)]
+    simp only [ofNat, BitVec.ofNat, Fin.ofNat', Int.reduceToNat, Nat.dvd_refl,
+      Nat.mod_mod_of_dvd, instOfNat]
+
 end UInt16
 
 namespace UInt32
@@ -36,6 +55,16 @@ theorem ofNat_mod_size' : ofNat (x % 4294967296) = ofNat x := ofNat_mod_size
 instance : IntCast UInt32 where
   intCast x := UInt32.ofInt x
 
+theorem intCast_ofNat (x : Nat) : (OfNat.ofNat (α := Int) x : UInt32) = OfNat.ofNat x := by
+    -- A better proof would be welcome!
+    simp only [Int.cast, IntCast.intCast]
+    rw [UInt32.ofInt]
+    rw [Int.toNat_emod (Int.zero_le_ofNat x) (by decide)]
+    erw [Int.toNat_natCast]
+    rw [Int.toNat_pow_of_nonneg (by decide)]
+    simp only [ofNat, BitVec.ofNat, Fin.ofNat', Int.reduceToNat, Nat.dvd_refl,
+      Nat.mod_mod_of_dvd, instOfNat]
+
 end UInt32
 
 namespace UInt64
@@ -46,6 +75,16 @@ theorem ofNat_mod_size' : ofNat (x % 18446744073709551616) = ofNat x := ofNat_mo
 instance : IntCast UInt64 where
   intCast x := UInt64.ofInt x
 
+theorem intCast_ofNat (x : Nat) : (OfNat.ofNat (α := Int) x : UInt64) = OfNat.ofNat x := by
+    -- A better proof would be welcome!
+    simp only [Int.cast, IntCast.intCast]
+    rw [UInt64.ofInt]
+    rw [Int.toNat_emod (Int.zero_le_ofNat x) (by decide)]
+    erw [Int.toNat_natCast]
+    rw [Int.toNat_pow_of_nonneg (by decide)]
+    simp only [ofNat, BitVec.ofNat, Fin.ofNat', Int.reduceToNat, Nat.dvd_refl,
+      Nat.mod_mod_of_dvd, instOfNat]
+
 end UInt64
 
 namespace USize
@@ -53,9 +92,21 @@ namespace USize
 instance : IntCast USize where
   intCast x := USize.ofInt x
 
+theorem intCast_ofNat (x : Nat) : (OfNat.ofNat (α := Int) x : USize) = OfNat.ofNat x := by
+    -- A better proof would be welcome!
+    simp only [Int.cast, IntCast.intCast]
+    rw [USize.ofInt]
+    rw [Int.toNat_emod (Int.zero_le_ofNat x)]
+    · erw [Int.toNat_natCast]
+      rw [Int.toNat_pow_of_nonneg (by decide)]
+      simp only [ofNat, BitVec.ofNat, Fin.ofNat', Int.reduceToNat, Nat.dvd_refl,
+        Nat.mod_mod_of_dvd, instOfNat]
+    · obtain _ | _ := System.Platform.numBits_eq <;> simp_all
+
 end USize
 namespace Lean.Grind
 
+
 instance : CommRing UInt8 where
   add_assoc := UInt8.add_assoc
   add_comm := UInt8.add_comm
@@ -70,6 +121,8 @@ instance : CommRing UInt8 where
   pow_zero := UInt8.pow_zero
   pow_succ := UInt8.pow_succ
   ofNat_succ x := UInt8.ofNat_add x 1
+  intCast_neg := UInt8.ofInt_neg
+  intCast_ofNat := UInt8.intCast_ofNat
 
 instance : IsCharP UInt8 256 where
   ofNat_eq_zero_iff {x} := by
@@ -90,6 +143,8 @@ instance : CommRing UInt16 where
   pow_zero := UInt16.pow_zero
   pow_succ := UInt16.pow_succ
   ofNat_succ x := UInt16.ofNat_add x 1
+  intCast_neg := UInt16.ofInt_neg
+  intCast_ofNat := UInt16.intCast_ofNat
 
 instance : IsCharP UInt16 65536 where
   ofNat_eq_zero_iff {x} := by
@@ -110,6 +165,8 @@ instance : CommRing UInt32 where
   pow_zero := UInt32.pow_zero
   pow_succ := UInt32.pow_succ
   ofNat_succ x := UInt32.ofNat_add x 1
+  intCast_neg := UInt32.ofInt_neg
+  intCast_ofNat := UInt32.intCast_ofNat
 
 instance : IsCharP UInt32 4294967296 where
   ofNat_eq_zero_iff {x} := by
@@ -130,6 +187,8 @@ instance : CommRing UInt64 where
   pow_zero := UInt64.pow_zero
   pow_succ := UInt64.pow_succ
   ofNat_succ x := UInt64.ofNat_add x 1
+  intCast_neg := UInt64.ofInt_neg
+  intCast_ofNat := UInt64.intCast_ofNat
 
 instance : IsCharP UInt64 18446744073709551616 where
   ofNat_eq_zero_iff {x} := by
@@ -150,6 +209,8 @@ instance : CommRing USize where
   pow_zero := USize.pow_zero
   pow_succ := USize.pow_succ
   ofNat_succ x := USize.ofNat_add x 1
+  intCast_neg := USize.ofInt_neg
+  intCast_ofNat := USize.intCast_ofNat
 
 open System.Platform
 
diff --git a/src/Init/Prelude.lean b/src/Init/Prelude.lean
index 05b7cea128..6784e6d7df 100644
--- a/src/Init/Prelude.lean
+++ b/src/Init/Prelude.lean
@@ -2037,23 +2037,23 @@ structure BitVec (w : Nat) where
 /--
 Bitvectors have decidable equality.
 
-This should be used via the instance `DecidableEq (BitVec n)`.
+This should be used via the instance `DecidableEq (BitVec w)`.
 -/
 -- We manually derive the `DecidableEq` instances for `BitVec` because
 -- we want to have builtin support for bit-vector literals, and we
 -- need a name for this function to implement `canUnfoldAtMatcher` at `WHNF.lean`.
-def BitVec.decEq (x y : BitVec n) : Decidable (Eq x y) :=
+def BitVec.decEq (x y : BitVec w) : Decidable (Eq x y) :=
   match x, y with
   | ⟨n⟩, ⟨m⟩ =>
     dite (Eq n m)
       (fun h => isTrue (h ▸ rfl))
       (fun h => isFalse (fun h' => BitVec.noConfusion h' (fun h' => absurd h' h)))
 
-instance : DecidableEq (BitVec n) := BitVec.decEq
+instance : DecidableEq (BitVec w) := BitVec.decEq
 
-/-- The `BitVec` with value `i`, given a proof that `i < 2^n`. -/
+/-- The `BitVec` with value `i`, given a proof that `i < 2^w`. -/
 @[match_pattern]
-protected def BitVec.ofNatLT {n : Nat} (i : Nat) (p : LT.lt i (hPow 2 n)) : BitVec n where
+protected def BitVec.ofNatLT {w : Nat} (i : Nat) (p : LT.lt i (hPow 2 w)) : BitVec w where
   toFin := ⟨i, p⟩
 
 /--
@@ -2061,14 +2061,14 @@ Return the underlying `Nat` that represents a bitvector.
 
 This is O(1) because `BitVec` is a (zero-cost) wrapper around a `Nat`.
 -/
-protected def BitVec.toNat (x : BitVec n) : Nat := x.toFin.val
+protected def BitVec.toNat (x : BitVec w) : Nat := x.toFin.val
 
-instance : LT (BitVec n) where lt := (LT.lt ·.toNat ·.toNat)
-instance (x y : BitVec n) : Decidable (LT.lt x y) :=
+instance : LT (BitVec w) where lt := (LT.lt ·.toNat ·.toNat)
+instance (x y : BitVec w) : Decidable (LT.lt x y) :=
   inferInstanceAs (Decidable (LT.lt x.toNat y.toNat))
 
-instance : LE (BitVec n) where le := (LE.le ·.toNat ·.toNat)
-instance (x y : BitVec n) : Decidable (LE.le x y) :=
+instance : LE (BitVec w) where le := (LE.le ·.toNat ·.toNat)
+instance (x y : BitVec w) : Decidable (LE.le x y) :=
   inferInstanceAs (Decidable (LE.le x.toNat y.toNat))
 
 /-- The number of distinct values representable by `UInt8`, that is, `2^8 = 256`. -/
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/RingId.lean b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/RingId.lean
index 96eaa23b4e..d5b45be7d1 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/RingId.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/RingId.lean
@@ -57,16 +57,13 @@ private def getPowFn (type : Expr) (u : Level) (commRingInst : Expr) : GoalM Exp
     throwError "instance for power operator{indentExpr inst}\nis not definitionally equal to the `Grind.CommRing` one{indentExpr inst'}"
   internalizeFn <| mkApp4 (mkConst ``HPow.hPow [u, 0, u]) type Nat.mkType type inst
 
-private def getIntCastFn (type : Expr) (u : Level) (_commRingInst : Expr) : GoalM Expr := do
+private def getIntCastFn (type : Expr) (u : Level) (commRingInst : Expr) : GoalM Expr := do
   let instType := mkApp (mkConst ``IntCast [u]) type
   let .some inst ← trySynthInstance instType |
     throwError "failed to find instance for ring intCast{indentExpr instType}"
-  -- TODO uncomment after we fix `CommRing` definition
-  /-
-  let inst' := mkApp2 (mkConst ``Grind.CommRing.intCastInst [u]) type commRingInst
+  let inst' := mkApp2 (mkConst ``Grind.CommRing.intCast [u]) type commRingInst
   unless (← withDefault <| isDefEq inst inst') do
     throwError "instance for intCast{indentExpr inst}\nis not definitionally equal to the `Grind.CommRing` one{indentExpr inst'}"
-  -/
   internalizeFn <| mkApp2 (mkConst ``IntCast.intCast [u]) type inst
 
 private def getNatCastFn (type : Expr) (u : Level) (commRingInst : Expr) : GoalM Expr := do