lean2/library/data/pnat.lean

/-
Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Robert Y. Lewis

Basic facts about the positive natural numbers.

Developed primarily for use in the construction of ℝ. For the most part, the only theorems here
are those needed for that construction.

-/

import data.rat.order data.nat
open nat rat subtype eq.ops

namespace pnat

definition pnat := { n : ℕ | n > 0 }

notation `ℕ+` := pnat

definition pos (n : ℕ) (H : n > 0) : ℕ+ := tag n H

definition nat_of_pnat (p : ℕ+) : ℕ := elt_of p
reserve postfix `~`:std.prec.max_plus
local postfix ~ := nat_of_pnat

theorem pnat_pos (p : ℕ+) : p~ > 0 := has_property p

definition add (p q : ℕ+) : ℕ+ :=
  tag (p~ + q~) (nat.add_pos (pnat_pos p) (pnat_pos q))
infix `+` := add

definition mul (p q : ℕ+) : ℕ+ :=
  tag (p~ * q~) (nat.mul_pos (pnat_pos p) (pnat_pos q))
infix `*` := mul

definition le (p q : ℕ+) := p~ ≤ q~
infix `≤` := le
notation p `≥` q := q ≤ p

definition lt (p q : ℕ+) := p~ < q~
infix `<` := lt

protected theorem pnat.eq {p q : ℕ+} : p~ = q~ → p = q :=
  subtype.eq

definition pnat_le_decidable [instance] (p q : ℕ+) : decidable (p ≤ q) :=
  nat.decidable_le p~ q~

definition pnat_lt_decidable [instance] {p q : ℕ+} : decidable (p < q) :=
  nat.decidable_lt p~ q~

theorem le.trans {p q r : ℕ+} (H1 : p ≤ q) (H2 : q ≤ r) : p ≤ r := nat.le.trans H1 H2

definition max (p q : ℕ+) :=
  tag (nat.max p~ q~) (nat.lt_of_lt_of_le (!pnat_pos) (!le_max_right))

theorem max_right (a b : ℕ+) : max a b ≥ b := !le_max_right

theorem max_left (a b : ℕ+) : max a b ≥ a := !le_max_left

theorem max_eq_right {a b : ℕ+} (H : a < b) : max a b = b :=
  pnat.eq (nat.max_eq_right_of_lt H)

theorem max_eq_left {a b : ℕ+} (H : ¬ a < b) : max a b = a :=
  pnat.eq (nat.max_eq_left (le_of_not_gt H))

theorem le_of_lt {a b : ℕ+} : a < b → a ≤ b := nat.le_of_lt

theorem not_lt_of_ge {a b : ℕ+} : a ≤ b → ¬ (b < a) := nat.not_lt_of_ge

theorem le_of_not_gt {a b : ℕ+} : ¬ a < b → b ≤ a := nat.le_of_not_gt

theorem eq_of_le_of_ge {a b : ℕ+} (H1 : a ≤ b) (H2 : b ≤ a) : a = b :=
  pnat.eq (nat.eq_of_le_of_ge H1 H2)

theorem le.refl (a : ℕ+) : a ≤ a := !nat.le.refl

notation 2 := (tag 2 dec_trivial : ℕ+)
notation 3 := (tag 3 dec_trivial : ℕ+)
definition pone : ℕ+ := tag 1 dec_trivial

definition rat_of_pnat [reducible] (n : ℕ+) : ℚ := n~

theorem pnat.to_rat_of_nat (n : ℕ+) : rat_of_pnat n = of_nat n~ := rfl

-- these will come in rat

theorem rat_of_nat_nonneg (n : ℕ) : 0 ≤ of_nat n := trivial

theorem rat_of_pnat_ge_one (n : ℕ+) : rat_of_pnat n ≥ 1 :=
  (iff.mpr !of_nat_le_of_nat) (pnat_pos n)

theorem rat_of_pnat_is_pos (n : ℕ+) : rat_of_pnat n > 0 :=
  (iff.mpr !of_nat_pos) (pnat_pos n)

theorem of_nat_le_of_nat_of_le {m n : ℕ} (H : m ≤ n) : of_nat m ≤ of_nat n :=
  (iff.mpr !of_nat_le_of_nat) H

theorem of_nat_lt_of_nat_of_lt {m n : ℕ} (H : m < n) : of_nat m < of_nat n :=
  (iff.mpr !of_nat_lt_of_nat) H

theorem rat_of_pnat_le_of_pnat_le {m n : ℕ+} (H : m ≤ n) : rat_of_pnat m ≤ rat_of_pnat n :=
  of_nat_le_of_nat_of_le H

theorem rat_of_pnat_lt_of_pnat_lt {m n : ℕ+} (H : m < n) : rat_of_pnat m < rat_of_pnat n :=
  of_nat_lt_of_nat_of_lt H

theorem pnat_le_of_rat_of_pnat_le {m n : ℕ+} (H : rat_of_pnat m ≤ rat_of_pnat n) : m ≤ n :=
  (iff.mp !of_nat_le_of_nat) H

definition inv (n : ℕ+) : ℚ := (1 : ℚ) / rat_of_pnat n
postfix `⁻¹` := inv

theorem inv_pos (n : ℕ+) : n⁻¹ > 0 := div_pos_of_pos !rat_of_pnat_is_pos

theorem inv_le_one (n : ℕ+) : n⁻¹ ≤ (1 : ℚ) :=
  begin
    rewrite [↑inv, -one_div_one],
    apply div_le_div_of_le,
    apply rat.zero_lt_one,
    apply rat_of_pnat_ge_one
  end

theorem inv_lt_one_of_gt {n : ℕ+} (H : n~ > 1) : n⁻¹ < (1 : ℚ) :=
  begin
    rewrite [↑inv, -one_div_one],
    apply div_lt_div_of_lt,
    apply rat.zero_lt_one,
    rewrite pnat.to_rat_of_nat,
    apply (of_nat_lt_of_nat_of_lt H)
  end

theorem pone_inv : pone⁻¹ = 1 := rfl

theorem add_invs_nonneg (m n : ℕ+) : 0 ≤ m⁻¹ + n⁻¹ :=
  begin
    apply rat.le_of_lt,
    apply rat.add_pos,
    repeat apply inv_pos
  end

theorem one_mul (n : ℕ+) : pone * n = n :=
  begin
    apply pnat.eq,
    rewrite [↑pone, ↑mul, ↑nat_of_pnat, one_mul]
  end

theorem pone_le (n : ℕ+) : pone ≤ n :=
  succ_le_of_lt (pnat_pos n)

theorem pnat_to_rat_mul (a b : ℕ+) : rat_of_pnat (a * b) = rat_of_pnat a * rat_of_pnat b := rfl

theorem mul_lt_mul_left (a b c : ℕ+) (H : a < b) : a * c < b * c :=
   nat.mul_lt_mul_of_pos_right H !pnat_pos

theorem one_lt_two : pone < 2 := !nat.le.refl

theorem inv_two_mul_lt_inv (n : ℕ+) : (2 * n)⁻¹ < n⁻¹ :=
  begin
    rewrite ↑inv,
    apply div_lt_div_of_lt,
    apply rat_of_pnat_is_pos,
    have H : n~ < (2 * n)~, begin
      rewrite -one_mul at {1},
      apply mul_lt_mul_left,
      apply one_lt_two
    end,
    apply of_nat_lt_of_nat_of_lt,
    apply H
  end

theorem inv_two_mul_le_inv (n : ℕ+) : (2 * n)⁻¹ ≤ n⁻¹ := rat.le_of_lt !inv_two_mul_lt_inv

theorem inv_ge_of_le {p q : ℕ+} (H : p ≤ q) : q⁻¹ ≤ p⁻¹ :=
  div_le_div_of_le !rat_of_pnat_is_pos (rat_of_pnat_le_of_pnat_le H)

theorem inv_gt_of_lt {p q : ℕ+} (H : p < q) : q⁻¹ < p⁻¹ :=
  div_lt_div_of_lt !rat_of_pnat_is_pos (rat_of_pnat_lt_of_pnat_lt H)

theorem ge_of_inv_le {p q : ℕ+} (H : p⁻¹ ≤ q⁻¹) : q ≤ p :=
  pnat_le_of_rat_of_pnat_le (le_of_div_le !rat_of_pnat_is_pos H)

theorem two_mul (p : ℕ+) : rat_of_pnat (2 * p) = (1 + 1) * rat_of_pnat p :=
  by rewrite pnat_to_rat_mul

theorem add_halves (p : ℕ+) : (2 * p)⁻¹ + (2 * p)⁻¹ = p⁻¹ :=
  begin
    rewrite [↑inv, -(@add_halves (1 / (rat_of_pnat p))), *div_div_eq_div_mul'],
    have H : rat_of_pnat (2 * p) = rat_of_pnat p * (1 + 1), by rewrite [rat.mul.comm, two_mul],
    rewrite *H
  end

theorem add_halves_double (m n : ℕ+) :
        m⁻¹ + n⁻¹ = ((2 * m)⁻¹ + (2 * n)⁻¹) + ((2 * m)⁻¹ + (2 * n)⁻¹) :=
  have hsimp [visible] : ∀ a b : ℚ, (a + a) + (b + b) = (a + b) + (a + b),
    by intros; rewrite [rat.add.assoc, -(rat.add.assoc a b b), {_+b}rat.add.comm, -*rat.add.assoc],
  by rewrite [-add_halves m, -add_halves n, hsimp]

theorem inv_mul_eq_mul_inv {p q : ℕ+} : (p * q)⁻¹ = p⁻¹ * q⁻¹ :=
  by rewrite [↑inv, pnat_to_rat_mul, one_div_mul_one_div''']

theorem inv_mul_le_inv (p q : ℕ+) : (p * q)⁻¹ ≤ q⁻¹ :=
  begin
    rewrite [inv_mul_eq_mul_inv, -{q⁻¹}rat.one_mul at {2}],
    apply rat.mul_le_mul,
    apply inv_le_one,
    apply rat.le.refl,
    apply rat.le_of_lt,
    apply inv_pos,
    apply rat.le_of_lt rat.zero_lt_one
  end

theorem pnat_mul_le_mul_left' (a b c : ℕ+) (H : a ≤ b) : c * a ≤ c * b :=
  nat.mul_le_mul_of_nonneg_left H (nat.le_of_lt !pnat_pos)

theorem mul.assoc (a b c : ℕ+) : a * b * c = a * (b * c) :=
  pnat.eq !nat.mul.assoc

theorem mul.comm (a b : ℕ+) : a * b = b * a :=
  pnat.eq !nat.mul.comm

theorem add.assoc (a b c : ℕ+) : a + b + c = a + (b + c) :=
  pnat.eq !nat.add.assoc

theorem mul_le_mul_left (p q : ℕ+) : q ≤ p * q :=
  begin
    rewrite [-one_mul at {1}, mul.comm, mul.comm p],
    apply pnat_mul_le_mul_left',
    apply pone_le
  end

theorem mul_le_mul_right (p q : ℕ+) : p ≤ p * q :=
  by rewrite mul.comm; apply mul_le_mul_left

theorem pnat.lt_of_not_le {p q : ℕ+} (H : ¬ p ≤ q) : q < p :=
  nat.lt_of_not_ge H

theorem inv_cancel_left (p : ℕ+) : rat_of_pnat p * p⁻¹ = (1 : ℚ) :=
  mul_one_div_cancel (ne.symm (rat.ne_of_lt !rat_of_pnat_is_pos))

theorem inv_cancel_right (p : ℕ+) : p⁻¹ * rat_of_pnat p = (1 : ℚ) :=
  by rewrite rat.mul.comm; apply inv_cancel_left

theorem lt_add_left (p q : ℕ+) : p < p + q :=
  begin
    have H : p~ < p~ + q~, begin
      rewrite -nat.add_zero at {1},
      apply nat.add_lt_add_left,
      apply pnat_pos
    end,
    apply H
  end

theorem inv_add_lt_left (p q : ℕ+) : (p + q)⁻¹ < p⁻¹ :=
  by apply inv_gt_of_lt; apply lt_add_left

theorem div_le_pnat (q : ℚ) (n : ℕ+) (H : q ≥ n⁻¹) : 1 / q ≤ rat_of_pnat n :=
  begin
    apply rat.div_le_of_le_mul,
    apply rat.lt_of_lt_of_le,
    apply inv_pos,
    rotate 1,
    apply H,
    apply rat.le_mul_of_div_le,
    apply rat_of_pnat_is_pos,
    apply H
  end

theorem pnat_cancel' (n m : ℕ+) : (n * n * m)⁻¹ * (rat_of_pnat n * rat_of_pnat n) = m⁻¹ :=
  assert hsimp : ∀ a b c : ℚ, (a * a * (b * b * c)) = (a * b) * (a * b) * c, from
    λa b c, by rewrite[-*rat.mul.assoc]; exact (!rat.mul.right_comm ▸ rfl),
  by rewrite [rat.mul.comm, *inv_mul_eq_mul_inv, hsimp, *inv_cancel_left, *rat.one_mul]

definition pceil (a : ℚ) : ℕ+ := tag (ubound a) !ubound_pos

theorem pceil_helper {a : ℚ} {n : ℕ+} (H : pceil a ≤ n) (Ha : a > 0) : n⁻¹ ≤ 1 / a :=
  rat.le.trans (inv_ge_of_le H) (div_le_div_of_le Ha (ubound_ge a))

theorem inv_pceil_div (a b : ℚ) (Ha : a > 0) (Hb : b > 0) : (pceil (a / b))⁻¹ ≤ b / a :=
  div_div' ▸ div_le_div_of_le
    (div_pos_of_pos (pos_div_of_pos_of_pos Hb Ha))
    ((div_div_eq_mul_div (ne_of_gt Hb) (ne_of_gt Ha))⁻¹ ▸
      !rat.one_mul⁻¹ ▸ !ubound_ge)

theorem sep_by_inv {a b : ℚ} (H : a > b) : ∃ N : ℕ+, a > (b + N⁻¹ + N⁻¹) :=
  begin
    apply exists.elim (find_midpoint H),
    intro c Hc,
    existsi (pceil ((1 + 1 + 1) / c)),
    apply rat.lt.trans,
    rotate 1,
    apply and.left Hc,
    rewrite rat.add.assoc,
    apply rat.add_lt_add_left,
    rewrite -(@rat.add_halves c) at {3},
    apply rat.add_lt_add,
    repeat (apply rat.lt_of_le_of_lt;
    apply inv_pceil_div;
    apply dec_trivial;
    apply and.right Hc;
    apply div_lt_div_of_pos_of_lt_of_pos;
    repeat (apply two_pos);
    apply and.right Hc)
  end

theorem nonneg_of_ge_neg_invs (a : ℚ) (H : ∀ n : ℕ+, -n⁻¹ ≤ a) : 0 ≤ a :=
  rat.le_of_not_gt (suppose a < 0,
    have H2 : 0 < -a, from neg_pos_of_neg this,
   (rat.not_lt_of_ge !H) (iff.mp !lt_neg_iff_lt_neg (calc
       (pceil (of_num 2 / -a))⁻¹ ≤ -a / of_num 2
          : !inv_pceil_div dec_trivial H2
                             ... < -a / 1
          : div_lt_div_of_pos_of_lt_of_pos dec_trivial dec_trivial H2
                             ... = -a : div_one)))

end pnat