max/lean4-htt
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions 2

feat: add Bitvec.[(toInt, toFin)_twoPow, toNat_twoPow_of_le, toNat_twoPow_of_lt, toNat_twoPow_eq_ite] (#7225)

Browse source 
This PR contains `BitVec.(toInt, toFin)_twoPow` theorems, completing the
API for `BitVec.*_twoPow`. It also expands the `toNat_twoPow` API with
`toNat_twoPow_of_le`, `toNat_twoPow_of_lt`, as well as
`toNat_twoPow_eq_if` and moves `msb_twoPow` up, as it is used in the
`toInt_msb` proof.

---------

Co-authored-by: Henrik Böving <hargonix@gmail.com>
This commit is contained in:
Luisa Cicolini 2025-03-17 12:51:58 +00:00 • committed by GitHub
parent 6df6011641
commit 594587541c
No known key found for this signature in database
GPG key ID: B5690EEEBB952194
1 changed files with 43 additions and 8 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

51
src/Init/Data/BitVec/Lemmas.lean
View file

@ -4122,6 +4122,22 @@ theorem toNat_twoPow (w : Nat) (i : Nat) : (twoPow w i).toNat = 2^i % 2^w := by
have h1 : 1 < 2 ^ (w + 1) := Nat.one_lt_two_pow (by omega)
rw [Nat.mod_eq_of_lt h1, Nat.shiftLeft_eq, Nat.one_mul]
theorem toNat_twoPow_of_le {i w : Nat} (h : w ≤ i) : (twoPow w i).toNat = 0 := by
rw [toNat_twoPow]
apply Nat.mod_eq_zero_of_dvd
exact Nat.pow_dvd_pow_iff_le_right'.mpr h
theorem toNat_twoPow_of_lt {i w : Nat} (h : i < w) : (twoPow w i).toNat = 2^i := by
rw [toNat_twoPow]
apply Nat.mod_eq_of_lt
apply Nat.pow_lt_pow_of_lt (by omega) (by omega)
theorem toNat_twoPow_eq_ite {i w : Nat} : (twoPow w i).toNat = if i < w then 2^i else 0 := by
by_cases h : i < w
· simp only [h, toNat_twoPow_of_lt, if_true]
· simp only [h, if_false]
rw [toNat_twoPow_of_le (by omega)]
@[simp]
theorem getLsbD_twoPow (i j : Nat) : (twoPow w i).getLsbD j = ((i < w) && (i = j)) := by
rcases w with rfl | w
@ -4140,6 +4156,33 @@ theorem getLsbD_twoPow (i j : Nat) : (twoPow w i).getLsbD j = ((i < w) && (i = j
simp at hi
simp_all
@[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 toInt_twoPow {w i : Nat} :
(BitVec.twoPow w i).toInt = if w ≤ i then 0
else if i + 1 = w then (-(2^i : Nat) : Int) else 2^i := by
simp only [BitVec.toInt_eq_msb_cond, toNat_twoPow_eq_ite]
rcases w with _ | w
· simp
· by_cases h : i = w
· simp [h, show ¬ (w + 1 ≤ w) by omega]
omega
· by_cases h' : w + 1 ≤ i
· simp [h', show ¬ i < w + 1 by omega]
· simp [h, h', show i < w + 1 by omega, Int.natCast_pow]
theorem toFin_twoPow {w i : Nat} :
(BitVec.twoPow w i).toFin = Fin.ofNat' (2^w) (2^i) := by
rcases w with rfl | w
· simp [BitVec.twoPow, BitVec.toFin, toFin_shiftLeft, Fin.fin_one_eq_zero]
· simp [BitVec.twoPow, BitVec.toFin, toFin_shiftLeft, Nat.shiftLeft_eq]
@[simp]
theorem getElem_twoPow {i j : Nat} (h : j < w) : (twoPow w i)[j] = decide (j = i) := by
rw [←getLsbD_eq_getElem, getLsbD_twoPow]
@ -4153,14 +4196,6 @@ theorem getMsbD_twoPow {i j w: Nat} :
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 h