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feat: BitVec.toNat theorems for rotateLeft and rotateRight #6347

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feat: BitVec.toNat theorems for rotateLeft and rotateRight #6347

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4f21275
toNat thms
mhk119 Dec 8, 2024
e854510
tonat thms
mhk119 Dec 8, 2024
f6f9895
add def thms
mhk119 Dec 16, 2024
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22 changes: 21 additions & 1 deletion src/Init/Data/BitVec/Lemmas.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -2745,7 +2745,12 @@ theorem smod_zero {x : BitVec n} : x.smod 0#n = x := by

/-! # Rotate Left -/

/-- rotateLeft is invariant under `mod` by the bitwidth. -/
/--`rotateLeft` is defined in terms of left and right shifts. -/
theorem rotateLeft_def {x : BitVec w} {r : Nat} :
x.rotateLeft r = (x <<< (r % w)) ||| (x >>> (w - r % w)) := by
simp only [rotateLeft, rotateLeftAux]

/-- `rotateLeft` is invariant under `mod` by the bitwidth. -/
@[simp]
theorem rotateLeft_mod_eq_rotateLeft {x : BitVec w} {r : Nat} :
x.rotateLeft (r % w) = x.rotateLeft r := by
Expand Down Expand Up @@ -2889,8 +2894,18 @@ theorem msb_rotateLeft {m w : Nat} {x : BitVec w} :
· simp
omega

@[simp]
theorem toNat_rotateLeft {x : BitVec w} {r : Nat} :
(x.rotateLeft r).toNat = (x.toNat <<< (r % w)) % (2^w) ||| x.toNat >>> (w - r % w) := by
simp only [rotateLeft_def, toNat_shiftLeft, toNat_ushiftRight, toNat_or]

/-! ## Rotate Right -/

/--`rotateRight` is defined in terms of left and right shifts. -/
theorem rotateRight_def {x : BitVec w} {r : Nat} :
x.rotateRight r = (x >>> (r % w)) ||| (x <<< (w - r % w)) := by
simp only [rotateRight, rotateRightAux]

/--
Accessing bits in `x.rotateRight r` the range `[0, w-r)` is equal to
accessing bits `x` in the range `[r, w)`.
Expand Down Expand Up @@ -3026,6 +3041,11 @@ theorem msb_rotateRight {r w : Nat} {x : BitVec w} :
simp [h₁]
· simp [show w = 0 by omega]

@[simp]
theorem toNat_rotateRight {x : BitVec w} {r : Nat} :
(x.rotateRight r).toNat = (x.toNat >>> (r % w)) ||| x.toNat <<< (w - r % w) % (2^w) := by
simp only [rotateRight_def, toNat_shiftLeft, toNat_ushiftRight, toNat_or]

/- ## twoPow -/

theorem twoPow_eq (w : Nat) (i : Nat) : twoPow w i = 1#w <<< i := by
Expand Down
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