feat: add BitVec.msb_(smod, srem) (#8974)

This PR adds `BitVec.msb_(smod, srem)`. 

co-authored with @tobiasgrosser and @bollu

---------

Co-authored-by: Tobias Grosser <github@grosser.es>
Co-authored-by: Siddharth <siddu.druid@gmail.com>

src/Init/Data/BitVec/Bitblast.lean

@@ -1930,6 +1930,44 @@ theorem toInt_sub_neg_umod {x y : BitVec w} (hxmsb : x.msb = true) (hymsb : y.ms
         Int.dvd_neg] at hdvd
       simp only [hdvd, ↓reduceIte, Int.natAbs_cast]
 
+theorem srem_zero_of_dvd {x y : BitVec w} (h : y.toInt ∣ x.toInt) :
+    x.srem y = 0#w := by
+  have := toInt_dvd_toInt_iff (x := x) (y := y)
+  by_cases hx : x.msb <;> by_cases hy : y.msb
+  <;> simp only [h, hx, reduceIte, hy, false_eq_true, true_iff] at this
+  <;> simp [srem, hx, hy, this]
+
+/--
+The remainder for `srem`, i.e. division with rounding to zero is negative
+iff `x` is negative and `y` does not divide `x`.
+
+We can eventually build fast circuits for the divisibility test `x.srem y = 0`.
+-/ 
+theorem msb_srem {x y : BitVec w} : (x.srem y).msb =
+    (x.msb && decide (x.srem y ≠ 0)) := by
+  rw [msb_eq_toInt]
+  by_cases hx : x.msb
+  · by_cases hsrem : x.srem y = 0#w
+    · simp [hsrem]
+    · have := toInt_neg_of_msb_true hx
+      by_cases hdvd : y.toInt ∣ x.toInt
+      · simp [BitVec.srem_zero_of_dvd hdvd] at hsrem
+      · simp only [toInt_srem, Int.tmod_eq_emod, show ¬0 ≤ x.toInt by omega, hdvd, _root_.or_self,
+          reduceIte, hx, ofNat_eq_ofNat, ne_eq, hsrem, not_false_eq_true, decide_true, Bool.and_self,
+          decide_eq_true_eq, gt_iff_lt]
+        have hlt := Int.emod_lt (a := x.toInt) (b := y.toInt)
+        by_cases hy0 : y = 0#w
+        · simp only [hy0, toInt_zero, Int.emod_zero, Int.natAbs_zero, Int.cast_ofNat_Int,
+            Int.sub_zero, gt_iff_lt]
+          exact toInt_neg_of_msb_true hx
+        · simp only [← toInt_inj, toInt_zero] at hy0
+          simp only [ne_eq, hy0, not_false_eq_true, forall_const] at hlt
+          have := Int.le_natAbs (a := y.toInt)
+          omega
+  · simp only [toInt_srem, hx, ofNat_eq_ofNat, ne_eq, decide_not, Bool.false_and,
+      decide_eq_false_iff_not, Int.not_lt]
+    apply Int.tmod_nonneg y.toInt (by exact toInt_nonneg_of_msb_false (by simp at hx; exact hx))
+
 theorem toInt_smod {x y : BitVec w} :
     (x.smod y).toInt = x.toInt.fmod y.toInt := by
   rcases w with _|w
@@ -1998,6 +2036,30 @@ theorem getMsbD_smod {x y : BitVec w} :
   by_cases hx : x.msb <;> by_cases hy : y.msb
   <;> simp [hx, hy]
 
+theorem msb_smod {x y : BitVec w} :
+    (x.smod y).msb = (x.msb && y = 0) || (y.msb && (x.smod y) ≠ 0) := by
+  rw [msb_eq_toInt]
+  by_cases hx : x.msb <;> by_cases hy : y.msb
+  · by_cases hsmod : x.smod y = 0#w <;> simp [hx, hy, hsmod]
+  · simp only [hx, ofNat_eq_ofNat, Bool.true_and, decide_eq_decide, decide_iff_dist, hy, ne_eq,
+      decide_not, Bool.false_and, Bool.or_false, beq_iff_eq]
+    constructor
+    · intro h
+      apply Classical.byContradiction
+      intro hcontra
+      rw [toInt_smod] at h
+      have := toInt_nonneg_of_msb_false (by simp at hy; exact hy)
+      have := Int.fmod_nonneg_of_pos (a := x.toInt) (b := y.toInt) (by simp [← toInt_inj] at hcontra; omega)
+      omega
+    · intro h
+      simp only [h, smod_zero]
+      exact toInt_neg_of_msb_true hx
+  · by_cases hsmod : x.smod y = 0#w <;> simp [hx, hy, hsmod]
+  · simp only [toInt_smod, hx, ofNat_eq_ofNat, Bool.false_and, decide_eq_false_iff_not, Int.not_lt,
+      hy, ne_eq, decide_not, Bool.or_false, decide_eq_true_eq]
+    simp only [not_eq_true] at hx hy
+    apply Int.fmod_nonneg (by exact toInt_nonneg_of_msb_false hx) (by exact toInt_nonneg_of_msb_false hy)
+
 /-! ### Lemmas that use bit blasting circuits -/
 
 theorem add_sub_comm {x y : BitVec w} : x + y - z = x - z + y := by