mathlib documentation

data.nat.parity

Parity of natural numbers #

This file contains theorems about the even and odd predicates on the natural numbers.

Tags #

even, odd

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@[simp]
theorem nat.mod_two_ne_one {n : ℕ} :
¬n % 2 = 1 ↔ n % 2 = 0
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@[simp]
theorem nat.mod_two_ne_zero {n : ℕ} :
¬n % 2 = 0 ↔ n % 2 = 1
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theorem nat.even_iff {n : ℕ} :
even n ↔ n % 2 = 0
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theorem nat.odd_iff {n : ℕ} :
odd n ↔ n % 2 = 1
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theorem nat.not_even_iff {n : ℕ} :
¬even n ↔ n % 2 = 1
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theorem nat.not_odd_iff {n : ℕ} :
¬odd n ↔ n % 2 = 0
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theorem nat.even_iff_not_odd {n : ℕ} :
even n ↔ ¬odd n
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@[simp]
theorem nat.odd_iff_not_even {n : ℕ} :
odd n ↔ ¬even n
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theorem nat.is_compl_even_odd  :
is_compl {n : ℕ | even n} {n : ℕ | odd n}
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theorem nat.even_or_odd (n : ℕ) :
even n ∨ odd n
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theorem nat.even_or_odd' (n : ℕ) :
∃ (k : ℕ), n = 2 * k ∨ n = 2 * k + 1
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theorem nat.even_xor_odd (n : ℕ) :
xor (even n) (odd n)
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theorem nat.even_xor_odd' (n : ℕ) :
∃ (k : ℕ), xor (n = 2 * k) (n = 2 * k + 1)
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theorem nat.odd_gt_zero {n : ℕ} (h : odd n) :
0 < n
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@[simp]
theorem nat.two_dvd_ne_zero {n : ℕ} :
¬2 ∣ n ↔ n % 2 = 1
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@[protected, instance]
def nat.even.decidable_pred  :
decidable_pred even
Equations
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@[protected, instance]
def nat.odd.decidable_pred  :
decidable_pred odd
Equations
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meta def simp_attr.parity_simps  :
(user_attribute simp_lemmas)

Simp attribute for lemmas about even

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@[simp]
theorem nat.not_even_one  :
¬even 1
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theorem nat.even_add {m n : ℕ} :
even (m + n) ↔ (even m ↔ even n)
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theorem nat.even_add' {m n : ℕ} :
even (m + n) ↔ (odd m ↔ odd n)
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@[simp]
theorem nat.not_even_bit1 (n : ℕ) :
¬even (bit1 n)
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theorem nat.two_not_dvd_two_mul_add_one (n : ℕ) :
¬2 ∣ 2 * n + 1
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theorem nat.two_not_dvd_two_mul_sub_one {n : ℕ} (w : 0 < n) :
¬2 ∣ 2 * n - 1
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theorem nat.even_sub {m n : ℕ} (h : n ≤ m) :
even (m - n) ↔ (even m ↔ even n)
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theorem nat.even_sub' {m n : ℕ} (h : n ≤ m) :
even (m - n) ↔ (odd m ↔ odd n)
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theorem nat.odd.sub_odd {m n : ℕ} (hm : odd m) (hn : odd n) :
even (m - n)
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theorem nat.even_succ {n : ℕ} :
even n.succ ↔ ¬even n
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theorem nat.even_mul {m n : ℕ} :
even (m * n) ↔ even m ∨ even n
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theorem nat.odd_mul {m n : ℕ} :
odd (m * n) ↔ odd m ∧ odd n
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theorem nat.odd.of_mul_left {m n : ℕ} (h : odd (m * n)) :
odd m
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theorem nat.odd.of_mul_right {m n : ℕ} (h : odd (m * n)) :
odd n
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theorem nat.even_pow {m n : ℕ} :
even (m ^ n) ↔ even m ∧ n ≠ 0

If m and n are natural numbers, then the natural number m^n is even if and only if m is even and n is positive.

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theorem nat.even_pow' {m n : ℕ} (h : n ≠ 0) :
even (m ^ n) ↔ even m
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theorem nat.even_div {m n : ℕ} :
even (m / n) ↔ m % 2 * n / n = 0
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theorem nat.odd_add {m n : ℕ} :
odd (m + n) ↔ (odd m ↔ even n)
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theorem nat.odd_add' {m n : ℕ} :
odd (m + n) ↔ (odd n ↔ even m)
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theorem nat.ne_of_odd_add {m n : ℕ} (h : odd (m + n)) :
m ≠ n
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theorem nat.odd_sub {m n : ℕ} (h : n ≤ m) :
odd (m - n) ↔ (odd m ↔ even n)
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theorem nat.odd.sub_even {m n : ℕ} (h : n ≤ m) (hm : odd m) (hn : even n) :
odd (m - n)
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theorem nat.odd_sub' {m n : ℕ} (h : n ≤ m) :
odd (m - n) ↔ (odd n ↔ even m)
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theorem nat.even.sub_odd {m n : ℕ} (h : n ≤ m) (hm : even m) (hn : odd n) :
odd (m - n)
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theorem nat.even_mul_succ_self (n : ℕ) :
even (n * (n + 1))
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theorem nat.even_mul_self_pred (n : ℕ) :
even (n * (n - 1))
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theorem nat.even_sub_one_of_prime_ne_two {p : ℕ} (hp : nat.prime p) (hodd : p ≠ 2) :
even (p - 1)
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theorem nat.two_mul_div_two_of_even {n : ℕ} :
even n → 2 * (n / 2) = n
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theorem nat.div_two_mul_two_of_even {n : ℕ} :
even n → (n / 2) * 2 = n
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theorem nat.two_mul_div_two_add_one_of_odd {n : ℕ} (h : odd n) :
2 * (n / 2) + 1 = n
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theorem nat.div_two_mul_two_add_one_of_odd {n : ℕ} (h : odd n) :
(n / 2) * 2 + 1 = n
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theorem nat.one_add_div_two_mul_two_of_odd {n : ℕ} (h : odd n) :
1 + (n / 2) * 2 = n
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theorem neg_one_pow_eq_one_iff_even {R : Type u_1} [monoid R] [has_distrib_neg R] {n : ℕ} (h : -1 ≠ 1) :
(-1) ^ n = 1 ↔ even n