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feat: add Nat.[shiftLeft_or_distrib, shiftLeft_xor_distrib, shiftLeft_and_distrib, testBit_mul_two_pow, bitwise_mul_two_pow, shiftLeft_bitwise_distrib] (#6630)

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This PR adds theorems `Nat.[shiftLeft_or_distrib`,
shiftLeft_xor_distrib`, shiftLeft_and_distrib`, `testBit_mul_two_pow`,
`bitwise_mul_two_pow`, `shiftLeft_bitwise_distrib]`, to prove
`Nat.shiftLeft_or_distrib` by emulating the proof strategy of
`shiftRight_and_distrib`.

In particular, `Nat.shiftLeft_or_distrib` is necessary to simplify the
proofs in #6476.

---------

Co-authored-by: Alex Keizer <alex@keizer.dev>
This commit is contained in:
Luisa Cicolini 2025-01-16 10:59:00 +00:00 • committed by GitHub
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src/Init/Data/Nat/Bitwise/Lemmas.lean
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@ -711,6 +711,32 @@ theorem mul_add_lt_is_or {b : Nat} (b_lt : b < 2^i) (a : Nat) : 2^i * a + b = 2^
rw [mod_two_eq_one_iff_testBit_zero, testBit_shiftLeft]
simp
theorem testBit_mul_two_pow (x i n : Nat) :
(x * 2 ^ n).testBit i = (decide (n ≤ i) && x.testBit (i - n)) := by
rw [← testBit_shiftLeft, shiftLeft_eq]
theorem bitwise_mul_two_pow (of_false_false : f false false = false := by rfl) :
(bitwise f x y) * 2 ^ n = bitwise f (x * 2 ^ n) (y * 2 ^ n) := by
apply Nat.eq_of_testBit_eq
simp only [testBit_mul_two_pow, testBit_bitwise of_false_false, Bool.if_false_right]
intro i
by_cases hn : n ≤ i
· simp [hn]
· simp [hn, of_false_false]
theorem shiftLeft_bitwise_distrib {a b : Nat} (of_false_false : f false false = false := by rfl) :
(bitwise f a b) <<< i = bitwise f (a <<< i) (b <<< i) := by
simp [shiftLeft_eq, bitwise_mul_two_pow of_false_false]
theorem shiftLeft_and_distrib {a b : Nat} : (a &&& b) <<< i = a <<< i &&& b <<< i :=
shiftLeft_bitwise_distrib
theorem shiftLeft_or_distrib {a b : Nat} : (a ||| b) <<< i = a <<< i ||| b <<< i :=
shiftLeft_bitwise_distrib
theorem shiftLeft_xor_distrib {a b : Nat} : (a ^^^ b) <<< i = a <<< i ^^^ b <<< i :=
shiftLeft_bitwise_distrib
@[simp] theorem decide_shiftRight_mod_two_eq_one :
decide (x >>> i % 2 = 1) = x.testBit i := by
simp only [testBit, one_and_eq_mod_two, mod_two_bne_zero]