lean2/library/data/nat/order.lean

/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad

The order relation on the natural numbers.
-/
import data.nat.basic algebra.ordered_ring
open eq.ops

namespace nat

/- lt and le -/

theorem le_of_lt_or_eq {m n : ℕ} (H : m < n ∨ m = n) : m ≤ n :=
or.elim H (take H1, le_of_lt H1) (take H1, H1 ▸ !le.refl)

theorem lt_or_eq_of_le {m n : ℕ} (H : m ≤ n) : m < n ∨ m = n :=
lt.by_cases
  (assume H1 : m < n, or.inl H1)
  (assume H1 : m = n, or.inr H1)
  (assume H1 : m > n, absurd (lt_of_le_of_lt H H1) !lt.irrefl)

theorem le_iff_lt_or_eq (m n : ℕ) : m ≤ n ↔ m < n ∨ m = n :=
iff.intro lt_or_eq_of_le le_of_lt_or_eq

theorem lt_of_le_and_ne {m n : ℕ} (H1 : m ≤ n) (H2 : m ≠ n) : m < n :=
or.elim (lt_or_eq_of_le H1)
  (take H3 : m < n, H3)
  (take H3 : m = n, by contradiction)

theorem lt_iff_le_and_ne (m n : ℕ) : m < n ↔ m ≤ n ∧ m ≠ n :=
iff.intro
  (take H, and.intro (le_of_lt H) (take H1, lt.irrefl _ (H1 ▸ H)))
  (take H, lt_of_le_and_ne (and.elim_left H) (and.elim_right H))

theorem le_add_right (n k : ℕ) : n ≤ n + k :=
nat.induction_on k
  (calc n ≤ n        : le.refl n
     ...  = n + zero : add_zero)
  (λ k (ih : n ≤ n + k), calc
     n   ≤ succ (n + k) : le_succ_of_le ih
     ... = n + succ k   : add_succ)

theorem le_add_left (n m : ℕ): n ≤ m + n :=
!add.comm ▸ !le_add_right

theorem le.intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
h ▸ le_add_right n k

theorem le.elim {n m : ℕ} (h : n ≤ m) : ∃k, n + k = m :=
by induction h with m h ih;existsi 0; reflexivity;
     cases ih with k H; existsi succ k; exact congr_arg succ H

theorem le.total {m n : ℕ} : m ≤ n ∨ n ≤ m :=
lt.by_cases
  (assume H : m < n, or.inl (le_of_lt H))
  (assume H : m = n, or.inl (by subst m))
  (assume H : m > n, or.inr (le_of_lt H))

/- addition -/

theorem add_le_add_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k + n ≤ k + m :=
obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
le.intro
  (calc
      k + n + l  = k + (n + l) : add.assoc
             ... = k + m       : by subst m)

theorem add_le_add_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n + k ≤ m + k :=
!add.comm ▸ !add.comm ▸ add_le_add_left H k

theorem le_of_add_le_add_left {k n m : ℕ} (H : k + n ≤ k + m) : n ≤ m :=
obtain (l : ℕ) (Hl : k + n + l = k + m), from (le.elim H),
le.intro (add.cancel_left
  (calc
      k + (n + l)  = k + n + l : add.assoc
               ... = k + m     : Hl))

theorem lt_of_add_lt_add_left {k n m : ℕ} (H : k + n < k + m) : n < m :=
let H' := le_of_lt H in
lt_of_le_and_ne (le_of_add_le_add_left H') (assume Heq, !lt.irrefl (Heq ▸ H))

theorem add_lt_add_left {n m : ℕ} (H : n < m) (k : ℕ) : k + n < k + m :=
lt_of_succ_le (!add_succ ▸ add_le_add_left (succ_le_of_lt H) k)

theorem add_lt_add_right {n m : ℕ} (H : n < m) (k : ℕ) : n + k < m + k :=
!add.comm ▸ !add.comm ▸ add_lt_add_left H k

theorem lt_add_of_pos_right {n k : ℕ} (H : k > 0) : n < n + k :=
!add_zero ▸ add_lt_add_left H n

/- multiplication -/

theorem mul_le_mul_left {n m : ℕ} (k : ℕ) (H : n ≤ m) : k * n ≤ k * m :=
obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
have H2 : k * n + k * l = k * m, by rewrite [-mul.left_distrib, Hl],
le.intro H2

theorem mul_le_mul_right {n m : ℕ} (k : ℕ) (H : n ≤ m) : n * k ≤ m * k :=
!mul.comm ▸ !mul.comm ▸ !mul_le_mul_left H

theorem mul_le_mul {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n * m ≤ k * l :=
le.trans (!mul_le_mul_right H1) (!mul_le_mul_left H2)

theorem mul_lt_mul_of_pos_left {n m k : ℕ} (H : n < m) (Hk : k > 0) : k * n < k * m :=
have H2 : k * n < k * n + k, from lt_add_of_pos_right Hk,
have H3 : k * n + k ≤ k * m, from !mul_succ ▸ mul_le_mul_left k (succ_le_of_lt H),
lt_of_lt_of_le H2 H3

theorem mul_lt_mul_of_pos_right {n m k : ℕ} (H : n < m) (Hk : k > 0) : n * k < m * k :=
!mul.comm ▸ !mul.comm ▸ mul_lt_mul_of_pos_left H Hk

/- min and max -/

-- Because these are defined in init/nat.lean, we cannot use the definitions in algebra.

theorem max_le {n m k : ℕ} (H₁ : n ≤ k) (H₂ : m ≤ k) : max n m ≤ k :=
decidable.by_cases
  (assume H : n < m, by rewrite [↑max, if_pos H]; apply H₂)
  (assume H : ¬ n < m, by rewrite [↑max, if_neg H]; apply H₁)

theorem min_le_left (n m : ℕ) : min n m ≤ n :=
decidable.by_cases
  (assume H : n < m, by rewrite [↑min, if_pos H])
  (assume H : ¬ n < m,
    assert H' : m ≤ n, from or_resolve_right !lt_or_ge H,
    by rewrite [↑min, if_neg H]; apply H')

theorem min_le_right (n m : ℕ) : min n m ≤ m :=
decidable.by_cases
  (assume H : n < m, by rewrite [↑min, if_pos H]; apply le_of_lt H)
  (assume H : ¬ n < m,
    assert H' : m ≤ n, from or_resolve_right !lt_or_ge H,
    by rewrite [↑min, if_neg H])

theorem le_min {n m k : ℕ} (H₁ : k ≤ n) (H₂ : k ≤ m) : k ≤ min n m :=
decidable.by_cases
  (assume H : n < m, by rewrite [↑min, if_pos H]; apply H₁)
  (assume H : ¬ n < m, by rewrite [↑min, if_neg H]; apply H₂)

/- nat is an instance of a linearly ordered semiring and a lattice-/

section migrate_algebra
  open [classes] algebra
  local attribute nat.comm_semiring [instance]

  protected definition linear_ordered_semiring [reducible] :
    algebra.linear_ordered_semiring nat :=
  ⦃ algebra.linear_ordered_semiring, nat.comm_semiring,
    add_left_cancel            := @add.cancel_left,
    add_right_cancel           := @add.cancel_right,
    lt                         := lt,
    le                         := le,
    le_refl                    := le.refl,
    le_trans                   := @le.trans,
    le_antisymm                := @le.antisymm,
    le_total                   := @le.total,
    le_iff_lt_or_eq            := @le_iff_lt_or_eq,
    le_of_lt                   := @le_of_lt,
    lt_irrefl                  := @lt.irrefl,
    lt_of_lt_of_le             := @lt_of_lt_of_le,
    lt_of_le_of_lt             := @lt_of_le_of_lt,
    lt_of_add_lt_add_left      := @lt_of_add_lt_add_left,
    add_lt_add_left            := @add_lt_add_left,
    add_le_add_left            := @add_le_add_left,
    le_of_add_le_add_left      := @le_of_add_le_add_left,
    zero_ne_one                := ne.symm (succ_ne_zero zero),
    mul_le_mul_of_nonneg_left  := (take a b c H1 H2, mul_le_mul_left c H1),
    mul_le_mul_of_nonneg_right := (take a b c H1 H2, mul_le_mul_right c H1),
    mul_lt_mul_of_pos_left     := @mul_lt_mul_of_pos_left,
    mul_lt_mul_of_pos_right    := @mul_lt_mul_of_pos_right ⦄

  protected definition lattice [reducible] : algebra.lattice nat :=
  ⦃ algebra.lattice, nat.linear_ordered_semiring,
    min          := min,
    max          := max,
    min_le_left  := min_le_left,
    min_le_right := min_le_right,
    le_min       := @le_min,
    le_max_left  := le_max_left,
    le_max_right := le_max_right,
    max_le       := @max_le ⦄

  local attribute nat.linear_ordered_semiring [instance]
  local attribute nat.lattice [instance]

  migrate from algebra with nat
    replacing dvd → dvd, has_le.ge → ge, has_lt.gt → gt, min → min, max → max
    hiding add_pos_of_pos_of_nonneg,  add_pos_of_nonneg_of_pos,
      add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg, le_add_of_nonneg_of_le,
      le_add_of_le_of_nonneg, lt_add_of_nonneg_of_lt, lt_add_of_lt_of_nonneg,
      lt_of_mul_lt_mul_left, lt_of_mul_lt_mul_right, pos_of_mul_pos_left, pos_of_mul_pos_right,
      mul_lt_mul

  attribute le.trans ge.trans lt.trans gt.trans [trans]
  attribute lt_of_lt_of_le lt_of_le_of_lt gt_of_gt_of_ge gt_of_ge_of_gt [trans]

  theorem add_pos_left : ∀{a : ℕ}, 0 < a → ∀b : ℕ, 0 < a + b :=
    take a H b, @algebra.add_pos_of_pos_of_nonneg _ _ a b H !zero_le
  theorem add_pos_right : ∀{a : ℕ}, 0 < a → ∀b : ℕ, 0 < b + a :=
    take a H b, !add.comm ▸ add_pos_left H b
  theorem add_eq_zero_iff_eq_zero_and_eq_zero : ∀{a b : ℕ},
    a + b = 0 ↔ a = 0 ∧ b = 0 :=
    take a b : ℕ,
      @algebra.add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg _ _ a b !zero_le !zero_le
  theorem le_add_of_le_left : ∀{a b c : ℕ}, b ≤ c → b ≤ a + c :=
    take a b c H, @algebra.le_add_of_nonneg_of_le _ _ a b c !zero_le H
  theorem le_add_of_le_right : ∀{a b c : ℕ}, b ≤ c → b ≤ c + a :=
    take a b c H, @algebra.le_add_of_le_of_nonneg _ _ a b c H !zero_le
  theorem lt_add_of_lt_left : ∀{b c : ℕ}, b < c → ∀a, b < a + c :=
    take b c H a, @algebra.lt_add_of_nonneg_of_lt _ _ a b c !zero_le H
  theorem lt_add_of_lt_right : ∀{b c : ℕ}, b < c → ∀a, b < c + a :=
    take b c H a, @algebra.lt_add_of_lt_of_nonneg _ _ a b c H !zero_le
  theorem lt_of_mul_lt_mul_left : ∀{a b c : ℕ}, c * a < c * b → a < b :=
    take a b c H, @algebra.lt_of_mul_lt_mul_left _ _ a b c H !zero_le
  theorem lt_of_mul_lt_mul_right : ∀{a b c : ℕ}, a * c < b * c → a < b :=
    take a b c H, @algebra.lt_of_mul_lt_mul_right _ _ a b c H !zero_le
  theorem pos_of_mul_pos_left : ∀{a b : ℕ}, 0 < a * b → 0 < b :=
    take a b H, @algebra.pos_of_mul_pos_left _ _ a b H !zero_le
  theorem pos_of_mul_pos_right : ∀{a b : ℕ}, 0 < a * b → 0 < a :=
    take a b H, @algebra.pos_of_mul_pos_right _ _ a b H !zero_le
end migrate_algebra

theorem zero_le_one : 0 ≤ 1 := dec_trivial
theorem zero_lt_one : 0 < 1 := dec_trivial

/- properties specific to nat -/

theorem lt_intro {n m k : ℕ} (H : succ n + k = m) : n < m :=
lt_of_succ_le (le.intro H)

theorem lt_elim {n m : ℕ} (H : n < m) : ∃k, succ n + k = m :=
le.elim (succ_le_of_lt H)

theorem lt_add_succ (n m : ℕ) : n < n + succ m :=
lt_intro !succ_add_eq_succ_add

theorem eq_zero_of_le_zero {n : ℕ} (H : n ≤ 0) : n = 0 :=
obtain (k : ℕ) (Hk : n + k = 0), from le.elim H,
eq_zero_of_add_eq_zero_right Hk

/- succ and pred -/

theorem le_of_lt_succ {m n : nat} (H : m < succ n) : m ≤ n :=
le_of_succ_le_succ H

theorem lt_iff_succ_le (m n : nat) : m < n ↔ succ m ≤ n :=
iff.rfl

theorem lt_succ_iff_le (m n : nat) : m < succ n ↔ m ≤ n :=
iff.intro le_of_lt_succ lt_succ_of_le

theorem self_le_succ (n : ℕ) : n ≤ succ n :=
le.intro !add_one

theorem succ_le_or_eq_of_le {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m :=
or.elim (lt_or_eq_of_le H)
  (assume H1 : n < m, or.inl (succ_le_of_lt H1))
  (assume H1 : n = m, or.inr H1)

theorem pred_le_of_le_succ {n m : ℕ} : n ≤ succ m → pred n ≤ m :=
nat.cases_on n
  (assume H, !pred_zero⁻¹ ▸ zero_le m)
  (take n',
    assume H : succ n' ≤ succ m,
    have H1 : n' ≤ m, from le_of_succ_le_succ H,
    !pred_succ⁻¹ ▸ H1)

theorem succ_le_of_le_pred {n m : ℕ} : succ n ≤ m → n ≤ pred m :=
nat.cases_on m
  (assume H, absurd H !not_succ_le_zero)
  (take m',
    assume H : succ n ≤ succ m',
    have H1 : n ≤ m', from le_of_succ_le_succ H,
    !pred_succ⁻¹ ▸ H1)

theorem pred_le_pred_of_le {n m : ℕ} : n ≤ m → pred n ≤ pred m :=
nat.cases_on n
  (assume H, pred_zero⁻¹ ▸ zero_le (pred m))
  (take n',
    assume H : succ n' ≤ m,
    !pred_succ⁻¹ ▸ succ_le_of_le_pred H)

theorem pre_lt_of_lt : ∀ {n m : ℕ}, n < m → pred n < m
| 0        m h := h
| (succ n) m h := lt_of_succ_lt h

theorem lt_of_pred_lt_pred {n m : ℕ} (H : pred n < pred m) : n < m :=
lt_of_not_ge
  (take H1 : m ≤ n,
    not_lt_of_ge (pred_le_pred_of_le H1) H)

theorem le_or_eq_succ_of_le_succ {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m :=
or_of_or_of_imp_left (succ_le_or_eq_of_le H)
   (take H2 : succ n ≤ succ m, show n ≤ m, from le_of_succ_le_succ H2)

theorem le_pred_self (n : ℕ) : pred n ≤ n :=
nat.cases_on n
  (pred_zero⁻¹ ▸ !le.refl)
  (take k : ℕ, (!pred_succ)⁻¹ ▸ !self_le_succ)

theorem succ_pos (n : ℕ) : 0 < succ n :=
!zero_lt_succ

theorem succ_pred_of_pos {n : ℕ} (H : n > 0) : succ (pred n) = n :=
(or_resolve_right (eq_zero_or_eq_succ_pred n) (ne.symm (ne_of_lt H)))⁻¹

theorem exists_eq_succ_of_lt {n m : ℕ} (H : n < m) : exists k, m = succ k :=
discriminate
  (take (Hm : m = 0), absurd (Hm ▸ H) !not_lt_zero)
  (take (l : ℕ) (Hm : m = succ l), exists.intro l Hm)

theorem lt_succ_self (n : ℕ) : n < succ n :=
lt.base n

/- other forms of induction -/

protected theorem strong_induction_on {P : nat → Prop} (n : ℕ) (H : ∀n, (∀m, m < n → P m) → P n) :
    P n :=
have H1 : ∀ {n m : nat}, m < n → P m, from
  take n,
  nat.induction_on n
    (show ∀m, m < 0 → P m, from take m H, absurd H !not_lt_zero)
    (take n',
      assume IH : ∀ {m : nat}, m < n' → P m,
      assert H2: P n', from H n' @IH,
      show ∀m, m < succ n' → P m, from
        take m,
        assume H3 : m < succ n',
        or.elim (lt_or_eq_of_le (le_of_lt_succ H3))
          (assume H4: m < n', IH H4)
          (assume H4: m = n', by subst m; assumption)),
H1 !lt_succ_self

protected theorem case_strong_induction_on {P : nat → Prop} (a : nat) (H0 : P 0)
  (Hind : ∀(n : nat), (∀m, m ≤ n → P m) → P (succ n)) : P a :=
nat.strong_induction_on a
  (take n,
   show (∀ m, m < n → P m) → P n, from
     nat.cases_on n
       (assume H : (∀m, m < 0 → P m), show P 0, from H0)
       (take n,
         assume H : (∀m, m < succ n → P m),
         show P (succ n), from
           Hind n (take m, assume H1 : m ≤ n, H _ (lt_succ_of_le H1))))

/- pos -/

theorem by_cases_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀ {y : nat}, y > 0 → P y) :
  P y :=
nat.cases_on y H0 (take y, H1 !succ_pos)

theorem eq_zero_or_pos (n : ℕ) : n = 0 ∨ n > 0 :=
or_of_or_of_imp_left
  (or.swap (lt_or_eq_of_le !zero_le))
  (take H : 0 = n, by subst n)

theorem pos_of_ne_zero {n : ℕ} (H : n ≠ 0) : n > 0 :=
or.elim !eq_zero_or_pos (take H2 : n = 0, by contradiction) (take H2 : n > 0, H2)

theorem ne_zero_of_pos {n : ℕ} (H : n > 0) : n ≠ 0 :=
ne.symm (ne_of_lt H)

theorem exists_eq_succ_of_pos {n : ℕ} (H : n > 0) : exists l, n = succ l :=
exists_eq_succ_of_lt H

theorem pos_of_dvd_of_pos {m n : ℕ} (H1 : m ∣ n) (H2 : n > 0) : m > 0 :=
pos_of_ne_zero
  (assume H3 : m = 0,
    assert H4 : n = 0, from eq_zero_of_zero_dvd (H3 ▸ H1),
    ne_of_lt H2 (by subst n))

/- multiplication -/

theorem mul_lt_mul_of_le_of_lt {n m k l : ℕ} (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) :
  n * m < k * l :=
lt_of_le_of_lt (mul_le_mul_right m H1) (mul_lt_mul_of_pos_left H2 Hk)

theorem mul_lt_mul_of_lt_of_le {n m k l : ℕ} (Hl : l > 0) (H1 : n < k) (H2 : m ≤ l) :
  n * m < k * l :=
lt_of_le_of_lt (mul_le_mul_left n H2) (mul_lt_mul_of_pos_right H1 Hl)

theorem mul_lt_mul_of_le_of_le {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n * m < k * l :=
have H3 : n * m ≤ k * m, from mul_le_mul_right m (le_of_lt H1),
have H4 : k * m < k * l, from mul_lt_mul_of_pos_left H2 (lt_of_le_of_lt !zero_le H1),
lt_of_le_of_lt H3 H4

theorem eq_of_mul_eq_mul_left {m k n : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k :=
have H2 : n * m ≤ n * k, by rewrite H,
have H3 : n * k ≤ n * m, by rewrite H,
have H4 : m ≤ k, from le_of_mul_le_mul_left H2 Hn,
have H5 : k ≤ m, from le_of_mul_le_mul_left H3 Hn,
le.antisymm H4 H5

theorem eq_of_mul_eq_mul_right {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k :=
eq_of_mul_eq_mul_left Hm (!mul.comm ▸ !mul.comm ▸ H)

theorem eq_zero_or_eq_of_mul_eq_mul_left {n m k : ℕ} (H : n * m = n * k) : n = 0 ∨ m = k :=
or_of_or_of_imp_right !eq_zero_or_pos
  (assume Hn : n > 0, eq_of_mul_eq_mul_left Hn H)

theorem eq_zero_or_eq_of_mul_eq_mul_right  {n m k : ℕ} (H : n * m = k * m) : m = 0 ∨ n = k :=
eq_zero_or_eq_of_mul_eq_mul_left (!mul.comm ▸ !mul.comm ▸ H)

theorem eq_one_of_mul_eq_one_right {n m : ℕ} (H : n * m = 1) : n = 1 :=
have H2 : n * m > 0, by rewrite H; apply succ_pos,
have H3 : n > 0, from pos_of_mul_pos_right H2,
have H4 : m > 0, from pos_of_mul_pos_left H2,
or.elim (le_or_gt n 1)
  (assume H5 : n ≤ 1,
    show n = 1, from le.antisymm H5 (succ_le_of_lt H3))
  (assume H5 : n > 1,
    have H6 : n * m ≥ 2 * 1, from mul_le_mul (succ_le_of_lt H5) (succ_le_of_lt H4),
    have H7 : 1 ≥ 2, from !mul_one ▸ H ▸ H6,
    absurd !lt_succ_self (not_lt_of_ge H7))

theorem eq_one_of_mul_eq_one_left {n m : ℕ} (H : n * m = 1) : m = 1 :=
eq_one_of_mul_eq_one_right (!mul.comm ▸ H)

theorem eq_one_of_mul_eq_self_left {n m : ℕ} (Hpos : n > 0) (H : m * n = n) : m = 1 :=
eq_of_mul_eq_mul_right Hpos (H ⬝ !one_mul⁻¹)

theorem eq_one_of_mul_eq_self_right {n m : ℕ} (Hpos : m > 0) (H : m * n = m) : n = 1 :=
eq_one_of_mul_eq_self_left Hpos (!mul.comm ▸ H)

theorem eq_one_of_dvd_one {n : ℕ} (H : n ∣ 1) : n = 1 :=
dvd.elim H
  (take m,
    assume H1 : 1 = n * m,
    eq_one_of_mul_eq_one_right H1⁻¹)

/- min and max -/

theorem lt_min {a b c : ℕ} (H₁ : a < b) (H₂ : a < c) : a < min b c :=
decidable.by_cases
  (assume H : b ≤ c, by rewrite (min_eq_left H); apply H₁)
  (assume H : ¬ b ≤ c,
    assert H' : c ≤ b, from le_of_lt (lt_of_not_ge H),
    by rewrite (min_eq_right H'); apply H₂)

theorem max_lt {a b c : ℕ} (H₁ : a < c) (H₂ : b < c) : max a b < c :=
decidable.by_cases
  (assume H : a ≤ b, by rewrite (max_eq_right H); apply H₂)
  (assume H : ¬ a ≤ b,
    assert H' : b ≤ a, from le_of_lt (lt_of_not_ge H),
    by rewrite (max_eq_left H'); apply H₁)

end nat