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feat: add (msb, getMsbD)_twoPow (#5851)

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Luisa Cicolini 2024-10-26 18:27:37 +01:00 • committed by GitHub
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src/Init/Data/BitVec/Lemmas.lean
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@ -2672,6 +2672,21 @@ theorem getElem_twoPow {i j : Nat} (h : j < w) : (twoPow w i)[j] = decide (j = i
simp [eq_comm]
omega
@[simp]
theorem getMsbD_twoPow {i j w: Nat} :
(twoPow w i).getMsbD j = (decide (i < w) && decide (j = w - i - 1)) := by
simp only [getMsbD_eq_getLsbD, getLsbD_twoPow]
by_cases h₀ : i < w <;> by_cases h₁ : j < w <;>
simp [h₀, h₁] <;> omega
@[simp]
theorem msb_twoPow {i w: Nat} :
(twoPow w i).msb = (decide (i < w) && decide (i = w - 1)) := by
simp only [BitVec.msb, getMsbD_eq_getLsbD, Nat.sub_zero, getLsbD_twoPow,
Bool.and_iff_right_iff_imp, Bool.and_eq_true, decide_eq_true_eq, and_imp]
intros
omega
theorem and_twoPow (x : BitVec w) (i : Nat) :
x &&& (twoPow w i) = if x.getLsbD i then twoPow w i else 0#w := by
ext j