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feat: Nat.and_le_(left|right) #4597

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Jul 2, 2024
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feat: Nat.and_le_(left|right) #4597

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24 changes: 24 additions & 0 deletions src/Init/Data/Nat/Bitwise/Lemmas.lean
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Expand Up @@ -504,3 +504,27 @@ theorem mul_add_lt_is_or {b : Nat} (b_lt : b < 2^i) (a : Nat) : 2^i * a + b = 2^

@[simp] theorem testBit_shiftRight (x : Nat) : testBit (x >>> i) j = testBit x (i+j) := by
simp [testBit, ←shiftRight_add]

/-! ### le -/

theorem le_of_testBit {n m : Nat} (h : ∀ i, n.testBit i = true → m.testBit i = true) : n ≤ m := by
induction n using div2Induction generalizing m
next n ih =>
have : n / 2 ≤ m / 2 := by
rcases n with (_|n)
· simp
· exact ih (Nat.succ_pos _) fun i => by simpa using h (i + 1)
rw [← div_add_mod n 2, ← div_add_mod m 2]
cases hn : n.testBit 0
· have hn2 : n % 2 = 0 := by simp at hn; omega
rw [hn2]
omega
· have hn2 : n % 2 = 1 := by simpa using hn
have hm2 : m % 2 = 1 := by simpa using h _ hn
omega

theorem and_le_left {n m : Nat} : n &&& m ≤ n :=
le_of_testBit (by simpa using fun i x _ => x)

theorem and_le_right {n m : Nat} : n &&& m ≤ m :=
le_of_testBit (by simp)
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