feat: add BitVec.neg_neg (#4977)

.. as well as neg_neq_iff_neq_neg.

---------

Co-authored-by: Henrik Böving <hargonix@gmail.com>

Author: tobiasgrosser

src/Init/Data/BitVec/Lemmas.lean

@@ -162,6 +162,16 @@ theorem toNat_zero (n : Nat) : (0#n).toNat = 0 := by trivial
 @[simp] theorem toNat_mod_cancel (x : BitVec n) : x.toNat % (2^n) = x.toNat :=
   Nat.mod_eq_of_lt x.isLt
 
+@[simp] theorem sub_toNat_mod_cancel {x : BitVec w} (h : ¬ x = 0#w) :
+    (2 ^ w - x.toNat) % 2 ^ w = 2 ^ w - x.toNat := by
+  simp only [toNat_eq, toNat_ofNat, Nat.zero_mod] at h
+  rw [Nat.mod_eq_of_lt (by omega)]
+
+@[simp] theorem sub_sub_toNat_cancel {x : BitVec w} :
+    2 ^ w - (2 ^ w - x.toNat) = x.toNat := by
+  simp [Nat.sub_sub_eq_min, Nat.min_eq_right]
+  omega
+
 private theorem lt_two_pow_of_le {x m n : Nat} (lt : x < 2 ^ m) (le : m ≤ n) : x < 2 ^ n :=
   Nat.lt_of_lt_of_le lt (Nat.pow_le_pow_of_le_right (by trivial : 0 < 2) le)
 
@@ -300,8 +310,7 @@ theorem truncate_eq_zeroExtend {v : Nat} {x : BitVec w} :
 
 @[simp, bv_toNat] theorem toNat_zeroExtend' {m n : Nat} (p : m ≤ n) (x : BitVec m) :
     (zeroExtend' p x).toNat = x.toNat := by
-  unfold zeroExtend'
-  simp [p, x.isLt, Nat.mod_eq_of_lt]
+  simp [zeroExtend']
 
 @[bv_toNat] theorem toNat_zeroExtend (i : Nat) (x : BitVec n) :
     BitVec.toNat (zeroExtend i x) = x.toNat % 2^i := by
@@ -1264,6 +1273,19 @@ theorem neg_eq_not_add (x : BitVec w) : -x = ~~~x + 1 := by
   have hx : x.toNat < 2^w := x.isLt
   rw [Nat.sub_sub, Nat.add_comm 1 x.toNat, ← Nat.sub_sub, Nat.sub_add_cancel (by omega)]
 
+@[simp]
+theorem neg_neg {x : BitVec w} : - - x = x := by
+  by_cases h : x = 0#w
+  · simp [h]
+  · simp [bv_toNat, h]
+
+theorem neg_ne_iff_ne_neg {x y : BitVec w} : -x ≠ y ↔ x ≠ -y := by
+  constructor
+  all_goals
+    intro h h'
+    subst h'
+    simp at h
+
 /-! ### mul -/
 
 theorem mul_def {n} {x y : BitVec n} : x * y = (ofFin <| x.toFin * y.toFin) := by rfl