Theorems for ushiftRight

src/Init/Data/BitVec/Lemmas.lean

@@ -1240,6 +1240,37 @@ theorem toNat_ushiftRight_lt (x : BitVec w) (n : Nat) (hn : n ≤ w) :
     · apply hn
   · apply Nat.pow_pos (by decide)
 
+theorem toNat_ushiftRight_eq_zero (x : BitVec w) (n : Nat) (hn : w ≤ n) :
+    (x >>> n).toNat = 0 := by
+  rw [toNat_ushiftRight, Nat.shiftRight_eq_div_pow, Nat.div_eq_of_lt]
+  exact Nat.lt_of_lt_of_le x.isLt (Nat.pow_le_pow_of_le Nat.one_lt_two hn)
+
+@[simp]
+theorem toInt_ushiftRight_zero (x : BitVec w) :
+    (x >>> 0).toInt = x.toInt := by simp
+
+
+@[simp]
+theorem toInt_ushiftRight_pos (x : BitVec w) (n : Nat) (hn : 0 < n) :
+    (x >>> n).toInt = x.toNat >>> n := by
+  rw [toInt_eq_toNat_cond]
+  by_cases hn: n ≤ w
+  · have h1 := Nat.mul_lt_mul_of_pos_left (toNat_ushiftRight_lt x n hn) Nat.two_pos
+    rw [← Nat.pow_succ'] at h1
+    simp only [Nat.lt_of_lt_of_le h1 (Nat.pow_le_pow_of_le (Nat.one_lt_two) (show _ ≤ w by omega))]
+    simp
+  · rw [← toNat_ushiftRight]
+    simp [toNat_ushiftRight_eq_zero x n (by omega), Nat.two_pow_pos w]
+
+@[simp]
+theorem toFin_uShiftRight (x : BitVec w) (n : Nat) :
+    (x >>> n).toFin = x.toFin / (Fin.ofNat' (2^w) (2^n)) := by
+    apply Fin.eq_of_val_eq
+    by_cases hn : n < w
+    · simp [Nat.shiftRight_eq_div_pow, Nat.mod_eq_of_lt (Nat.pow_lt_pow_of_lt Nat.one_lt_two hn)]
+    · simp only [Nat.not_lt] at hn
+      simp [toNat_ushiftRight_eq_zero x n hn, Nat.dvd_iff_mod_eq_zero.mp (Nat.pow_dvd_pow 2 hn)]
+
 @[simp]
 theorem getMsbD_ushiftRight {x : BitVec w} {i n : Nat} :
     (x >>> n).getMsbD i = (decide (i < w) && (!decide (i < n) && x.getMsbD (i - n))) := by