feat: add extern implementation for popcount

This commit introduces a native C++ implementation for `BitVec.popcount`
to significantly improve its performance, especially on large bitvectors.

- The `mpz` class is extended with a `popcount` method.
- A new extern function `lean_bitvec_popcount` is implemented in the
  runtime. It uses compiler intrinsics for hardware popcount instructions
  (e.g., `__builtin_popcountll`, `__popcnt64`) when available, and
  gracefully falls back to a generic implementation on other platforms.
- `BitVec.zerocount` is refactored to be a cheap calculation based on
  the now-fast `popcount`, rather than a separate fold.

Author: f64u

src/Init/Data/BitVec/Count.lean

@@ -52,10 +52,9 @@ Examples:
 * `(0b1111#4).popcount = 4`
 * `(0#8).popcount = 0`
 
-Note: This implementation could be optimized with a native `@[extern]` implementation
-using efficient CPU instructions (e.g., GMP's `gmp_popcount` or x86's `POPCNT`).
-See https://github.com/leanprover/lean4/issues/7887 for discussion of native implementations.
+This function uses a native implementation with CPU popcount instructions when available.
 -/
+@[extern "lean_bitvec_popcount"]
 def popcount (x : BitVec w) : Nat :=
   x.countP id
 
@@ -130,22 +129,27 @@ Count the number of `false` bits (zeros).
 This is the complement of `popcount`.
 -/
 def zerocount (x : BitVec w) : Nat :=
-  x.countP not
+  w - x.popcount
 
 @[simp]
 theorem zerocount_nil : zerocount nil = 0 := by
-  simp [zerocount, -ofNat_eq_ofNat]
+  simp [zerocount]
 
 @[simp]
 theorem zerocount_cons (b : Bool) (x : BitVec w) :
   zerocount (cons b x) = (!b).toNat + zerocount x := by
-    cases b <;> simp +arith [zerocount, countP]
+    cases b <;>
+      simp +arith [zerocount, Nat.sub_add_comm (popcount_le_width _)]
 
-theorem popcount_add_zerocount (x : BitVec w) :
-  x.popcount + x.zerocount = w := by
+theorem zerocount_eq_countP (x : BitVec w) :
+  x.zerocount = x.countP not := by
     induction x using BitVec.induction with
     | nil => simp [-ofNat_eq_ofNat]
-    | cons _ b => cases b <;> simp_all +arith
+    | cons _ b => cases b <;> simp_all
+
+theorem popcount_add_zerocount (x : BitVec w) :
+  x.popcount + x.zerocount = w := by
+    simp +arith [zerocount, popcount_le_width]
 
 @[simp]
 theorem zerocount_not {x : BitVec w} :
@@ -160,14 +164,14 @@ theorem popcount_not {x : BitVec w} :
 
 @[simp]
 theorem zerocount_zero : zerocount 0#w = w := by
-  simp [←popcount_add_zerocount 0#w, -ofNat_eq_ofNat]
+  simp [zerocount]
 
 @[simp]
 theorem zerocount_allOnes : zerocount (allOnes w) = 0 := by
-  simp [←not_zero]
+  simp [zerocount]
 
 theorem zerocount_le_width {x : BitVec w} : zerocount x ≤ w := by
-  simp [←popcount_add_zerocount x]
+  simp [zerocount]
 
 
 /--

src/runtime/mpz.cpp

@@ -222,6 +222,12 @@ size_t mpz::log2() const {
     return r - 1;
 }
 
+size_t mpz::popcount() const {
+    if (is_nonpos())
+        return 0;
+    return mpz_popcount(m_val);
+}
+
 mpz & mpz::operator&=(mpz const & o) {
     mpz_and(m_val, m_val, o.m_val);
     return *this;
@@ -856,6 +862,20 @@ size_t mpz::log2() const {
     return (m_size - 1)*sizeof(mpn_digit)*8 + log2_uint(m_digits[m_size - 1]);
 }
 
+size_t mpz::popcount() const {
+    if (is_nonpos())
+        return 0;
+    size_t count = 0;
+    for (size_t i = 0; i < m_size; i++) {
+        mpn_digit d = m_digits[i];
+        while (d) {
+            count += d & 1;
+            d >>= 1;
+        }
+    }
+    return count;
+}
+
 mpz & mpz::operator&=(mpz const & o) {
     digit_buffer r;
     size_t sz = std::max(m_size, o.m_size);

src/runtime/mpz.h

@@ -284,6 +284,12 @@ class LEAN_EXPORT mpz {
     */
     size_t log2() const;
 
+    /**
+       \brief Return the population count (number of 1 bits).
+       Return 0 if the number is negative
+    */
+    size_t popcount() const;
+
     friend void power(mpz & a, mpz const & b, unsigned k);
     friend void _power(mpz & a, mpz const & b, unsigned k) { power(a, b, k); }
     friend mpz pow(mpz a, unsigned k) { power(a, a, k); return a; }

src/runtime/object.cpp

@@ -1531,6 +1531,43 @@ extern "C" LEAN_EXPORT lean_obj_res lean_nat_log2(b_lean_obj_arg a) {
     }
 }
 
+extern "C" LEAN_EXPORT lean_obj_res lean_bitvec_popcount(b_lean_obj_arg /* w */, b_lean_obj_arg x) {
+    if (lean_is_scalar(x)) {
+        size_t n = lean_unbox(x);
+        unsigned count = 0;
+        bool builtin_used = false;
+
+        #if defined(__GNUC__) || defined(__clang__)
+            #if SIZE_MAX == UINT64_MAX
+                count = __builtin_popcountll(n);
+                builtin_used = true;
+            #elif SIZE_MAX == UINT32_MAX
+                count = __builtin_popcount(n);
+                builtin_used = true;
+            #endif
+        #elif defined(_MSC_VER)
+            #include <intrin.h>
+            #if SIZE_MAX == UINT64_MAX
+                count = __popcnt64(n);
+                builtin_used = true;
+            #elif SIZE_MAX == UINT32_MAX
+                count = __popcnt(n);
+                builtin_used = true;
+            #endif
+        #endif
+
+        if (!builtin_used) {
+            while (n) {
+                count += n & 1;
+                n >>= 1;
+            }
+        }
+        return lean_box(count);
+    } else {
+        return lean_box(mpz_value(x).popcount());
+    }
+}
+
 // =======================================
 // Integers