lean2/library/data/nat/parity.lean

/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura

Parity
-/
import data.nat.div logic.identities

namespace nat
open decidable

definition even (n : nat) := n mod 2 = 0

definition decidable_even [instance] : ∀ n, decidable (even n) :=
λ n, !nat.has_decidable_eq

definition odd (n : nat) := ¬even n

definition decidable_odd [instance] : ∀ n, decidable (odd n) :=
λ n, decidable_not

lemma even_of_dvd {n} : 2 ∣ n → even n :=
mod_eq_zero_of_dvd

lemma dvd_of_even {n} : even n → 2 ∣ n :=
dvd_of_mod_eq_zero

lemma not_odd_zero : ¬ odd 0 :=
dec_trivial

lemma even_zero : even 0 :=
dec_trivial

lemma odd_one : odd 1 :=
dec_trivial

lemma not_even_one : ¬ even 1 :=
dec_trivial

lemma odd_eq_not_even : ∀ n, odd n = ¬ even n :=
λ n, rfl

lemma odd_iff_not_even : ∀ n, odd n ↔ ¬ even n :=
λ n, !iff.refl

lemma odd_of_not_even {n} : ¬ even n → odd n :=
λ h, iff.mp' !odd_iff_not_even h

lemma even_of_not_odd {n} : ¬ odd n → even n :=
λ h, not_not_elim (iff.mp (not_iff_not_of_iff !odd_iff_not_even) h)

lemma not_odd_of_even {n} : even n → ¬ odd n :=
λ h, iff.mp' (not_iff_not_of_iff !odd_iff_not_even) (not_not_intro h)

lemma not_even_of_odd {n} : odd n → ¬ even n :=
λ h, iff.mp !odd_iff_not_even h

lemma odd_succ_of_even {n} : even n → odd (succ n) :=
λ h, by_contradiction (λ hn : ¬ odd (succ n),
  assert 0 = 1, from calc
    0  = (n+1) mod 2 : even_of_not_odd hn
   ... = 1 mod 2     : add_mod_eq_add_mod_right 1 h,
  by contradiction)

lemma eq_1_of_ne_0_lt_2 : ∀ {n : nat}, n ≠ 0 → n < 2 → n = 1
| 0     h₁ h₂ := absurd rfl h₁
| 1     h₁ h₂ := rfl
| (n+2) h₁ h₂ := absurd (lt_of_succ_lt_succ (lt_of_succ_lt_succ h₂)) !not_lt_zero

lemma mod_eq_of_odd {n} : odd n → n mod 2 = 1 :=
λ h,
  have h₁ : ¬ n mod 2 = 0, from h,
  have h₂ : n mod 2 < 2, from mod_lt n dec_trivial,
  eq_1_of_ne_0_lt_2 h₁ h₂

lemma odd_of_mod_eq {n} : n mod 2 = 1 → odd n :=
λ h, by_contradiction (λ hn,
  assert h₁ : n mod 2 = 0, from even_of_not_odd hn,
  by rewrite h at h₁; contradiction)

lemma even_succ_of_odd {n} : odd n → even (succ n) :=
λ h,
  have h₁ : n mod 2 = 1,       from mod_eq_of_odd h,
  have h₂ : n mod 2 = 1 mod 2, from mod_eq_of_odd h,
  have h₃ : (n+1) mod 2 = 0,   from add_mod_eq_add_mod_right 1 h₂,
  h₃

lemma odd_succ_succ_of_odd {n} : odd n → odd (succ (succ n)) :=
λ h, odd_succ_of_even (even_succ_of_odd h)

lemma even_succ_succ_of_even {n} : even n → even (succ (succ n)) :=
λ h, even_succ_of_odd (odd_succ_of_even h)

lemma even_of_odd_succ {n} : odd (succ n) → even n :=
λ h, by_contradiction (λ he,
  have h₁ : odd n, from odd_of_not_even he,
  have h₂ : even (succ n), from even_succ_of_odd h₁,
  absurd h₂ (not_even_of_odd h))

lemma odd_of_even_succ {n} : even (succ n) → odd n :=
λ h, by_contradiction (λ he,
  have h₁ : even n, from even_of_not_odd he,
  have h₂ : odd (succ n), from odd_succ_of_even h₁,
  absurd h (not_even_of_odd h₂))

lemma even_of_even_succ_succ {n} : even (succ (succ n)) → even n :=
λ h, even_of_odd_succ (odd_of_even_succ h)

lemma odd_of_odd_succ_succ {n} : odd (succ (succ n)) → odd n :=
λ h, odd_of_even_succ (even_of_odd_succ h)

lemma dvd_of_odd {n} : odd n → 2 ∣ n+1 :=
λ h, dvd_of_even (even_succ_of_odd h)

lemma odd_of_dvd {n} : 2 ∣ n+1 → odd n :=
λ h, odd_of_even_succ (even_of_dvd h)

lemma even_two_mul : ∀ n, even (2 * n) :=
λ n, even_of_dvd (dvd_mul_right 2 n)

lemma odd_two_mul_plus_one : ∀ n, odd (2 * n + 1) :=
λ n, odd_succ_of_even (even_two_mul n)

lemma not_even_two_mul_plus_one : ∀ n, ¬ even (2 * n + 1) :=
λ n, not_even_of_odd (odd_two_mul_plus_one n)

lemma not_odd_two_mul : ∀ n, ¬ odd (2 * n) :=
λ n, not_odd_of_even (even_two_mul n)

lemma even_pred_of_odd : ∀ {n}, odd n → even (pred n)
| 0     h := absurd h not_odd_zero
| (n+1) h := even_of_odd_succ h

lemma even_or_odd : ∀ n, even n ∨ odd n :=
λ n, by_cases
  (λ h : even n,   or.inl h)
  (λ h : ¬ even n, or.inr (odd_of_not_even h))

lemma exists_of_even {n} : even n → ∃ k, n = 2*k :=
λ h, exists_eq_mul_right_of_dvd (dvd_of_even h)

lemma exists_of_odd : ∀ {n}, odd n → ∃ k, n = 2*k + 1
| 0 h     := absurd h not_odd_zero
| (n+1) h :=
  obtain k (hk : n = 2*k), from exists_of_even (even_of_odd_succ h),
  exists.intro k (by subst n)

lemma even_of_exists {n} : (∃ k, n = 2 * k) → even n :=
λ h, obtain k (hk : n = 2 * k), from h,
 have h₁ : 2 ∣ n, by subst n; apply dvd_mul_right,
 even_of_dvd h₁

lemma odd_of_exists {n} : (∃ k, n = 2 * k + 1) → odd n :=
λ h, by_contradiction (λ hn,
  have h₁ : even n, from even_of_not_odd hn,
  have h₂ : ∃ k, n = 2 * k, from exists_of_even h₁,
  obtain k₁ (hk₁ : n = 2 * k₁ + 1), from h,
  obtain k₂ (hk₂ : n = 2 * k₂), from h₂,
  assert h₃ : (2 * k₁ + 1) mod 2 = (2 * k₂) mod 2, by rewrite [-hk₁, -hk₂],
  begin
    rewrite [mul_mod_right at h₃, add.comm at h₃, add_mul_mod_self_left at h₃],
    contradiction
  end)

lemma even_add_of_even_of_even {n m} : even n → even m → even (n+m) :=
λ h₁ h₂,
obtain k₁ (hk₁ : n = 2 * k₁), from exists_of_even h₁,
obtain k₂ (hk₂ : m = 2 * k₂), from exists_of_even h₂,
even_of_exists (exists.intro (k₁+k₂) (by rewrite [hk₁, hk₂, mul.left_distrib]))

lemma even_add_of_odd_of_odd {n m} : odd n → odd m → even (n+m) :=
λ h₁ h₂,
  assert h₃ : even (succ n + succ m), from even_add_of_even_of_even (even_succ_of_odd h₁) (even_succ_of_odd h₂),
  have h₄ : even(succ (succ (n + m))), by rewrite [add_succ at h₃, succ_add at h₃]; exact h₃,
  even_of_even_succ_succ h₄

lemma odd_add_of_even_of_odd {n m} : even n → odd m → odd (n+m) :=
λ h₁ h₂,
  assert h₃ : even (n + succ m), from even_add_of_even_of_even h₁ (even_succ_of_odd h₂),
  odd_of_even_succ h₃

lemma odd_add_of_odd_of_even {n m} : odd n → even m → odd (n+m) :=
λ h₁ h₂,
  assert h₃ : odd (m+n), from odd_add_of_even_of_odd h₂ h₁,
  by rewrite add.comm at h₃; exact h₃

lemma even_mul_of_even_left {n} (m) : even n → even (n*m) :=
λ h,
  obtain k (hk : n = 2*k), from exists_of_even h,
  even_of_exists (exists.intro (k*m) (by rewrite [hk, mul.assoc]))

lemma even_mul_of_even_right {n} (m) : even n → even (m*n) :=
λ h₁,
  assert h₂ : even (n*m), from even_mul_of_even_left _ h₁,
  by rewrite mul.comm at h₂; exact h₂

lemma odd_mul_of_odd_of_odd {n m} : odd n → odd m → odd (n*m) :=
λ h₁ h₂,
  assert h₃ : even (n * succ m), from even_mul_of_even_right _ (even_succ_of_odd h₂),
  assert h₄ : even (n * m + n), by rewrite mul_succ at h₃; exact h₃,
  by_contradiction (λ hn,
    assert h₅ : even (n*m), from even_of_not_odd hn,
    absurd h₄ (not_even_of_odd (odd_add_of_even_of_odd h₅ h₁)))
end nat