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feat: add (msb, getMsbD)_twoPow #5851

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Oct 26, 2024
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feat: add (msb, getMsbD)_twoPow #5851

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msb-getmsbd-twopow
luisacicolini Oct 26, 2024
9df17ff
moved theorems closer to getelem
luisacicolini Oct 26, 2024
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15 changes: 15 additions & 0 deletions src/Init/Data/BitVec/Lemmas.lean
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Expand Up @@ -2704,6 +2704,21 @@ theorem shiftLeft_eq_mul_twoPow (x : BitVec w) (n : Nat) :
ext i
simp [getLsbD_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, mul_twoPow_eq_shiftLeft]

@[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

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/- ### cons -/

@[simp] theorem true_cons_zero : cons true 0#w = twoPow (w + 1) w := by
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