lean2/library/data/nat/order.lean

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/-
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 : } (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 : } (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 : ) : 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 : } (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 : ) : 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 + 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 ≤ m + n :=
!add.comm ▸ !le_add_right
theorem le.intro {n m k : } (h : n + k = m) : n ≤ m :=
h ▸ le_add_right n k
theorem le.elim {n m : } (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 : } : 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 : } (H : n ≤ m) (k : ) : k + n ≤ k + m :=
obtain (l : ) (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 : } (H : n ≤ m) (k : ) : n + k ≤ m + k :=
!add.comm ▸ !add.comm ▸ add_le_add_left H k
theorem le_of_add_le_add_left {k n m : } (H : k + n ≤ k + m) : n ≤ m :=
obtain (l : ) (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 : } (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 : } (H : n < m) (k : ) : 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 : } (H : n < m) (k : ) : n + k < m + k :=
!add.comm ▸ !add.comm ▸ add_lt_add_left H k
theorem lt_add_of_pos_right {n k : } (H : k > 0) : n < n + k :=
!add_zero ▸ add_lt_add_left H n
/- multiplication -/
theorem mul_le_mul_left {n m : } (k : ) (H : n ≤ m) : k * n ≤ k * m :=
obtain (l : ) (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 : } (k : ) (H : n ≤ m) : n * k ≤ m * k :=
!mul.comm ▸ !mul.comm ▸ !mul_le_mul_left H
theorem mul_le_mul {n m k l : } (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 : } (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 : } (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 : } (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 : ) : 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 : ) : 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 : } (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 : }, 0 < a → ∀b : , 0 < a + b :=
take a H b, @algebra.add_pos_of_pos_of_nonneg _ _ a b H !zero_le
theorem add_pos_right : ∀{a : }, 0 < a → ∀b : , 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 : },
a + b = 0 ↔ a = 0 ∧ b = 0 :=
take a b : ,
@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 : }, 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 : }, 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 : }, 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 : }, 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 : }, 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 : }, 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 : }, 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 : }, 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 : } (H : succ n + k = m) : n < m :=
lt_of_succ_le (le.intro H)
theorem lt_elim {n m : } (H : n < m) : ∃k, succ n + k = m :=
le.elim (succ_le_of_lt H)
theorem lt_add_succ (n m : ) : n < n + succ m :=
lt_intro !succ_add_eq_succ_add
theorem eq_zero_of_le_zero {n : } (H : n ≤ 0) : n = 0 :=
obtain (k : ) (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 ≤ succ n :=
le.intro !add_one
theorem succ_le_or_eq_of_le {n m : } (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 ≤ 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 : } : 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 ≤ 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 < 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 : } (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 : } (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 : ) : pred n ≤ n :=
nat.cases_on n
(pred_zero⁻¹ ▸ !le.refl)
(take k : , (!pred_succ)⁻¹ ▸ !self_le_succ)
theorem succ_pos (n : ) : 0 < succ n :=
!zero_lt_succ
theorem succ_pred_of_pos {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 : } (H : n < m) : exists k, m = succ k :=
discriminate
(take (Hm : m = 0), absurd (Hm ▸ H) !not_lt_zero)
(take (l : ) (Hm : m = succ l), exists.intro l Hm)
theorem lt_succ_self (n : ) : n < succ n :=
lt.base n
/- other forms of induction -/
protected theorem strong_induction_on {P : nat → Prop} (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 : → Prop} (y : ) (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 = 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 : } (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 : } (H : n > 0) : n ≠ 0 :=
ne.symm (ne_of_lt H)
theorem exists_eq_succ_of_pos {n : } (H : n > 0) : exists l, n = succ l :=
exists_eq_succ_of_lt H
theorem pos_of_dvd_of_pos {m 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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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)
theorem min_add_add_left (a b c : ) : min (a + b) (a + c) = a + min b c :=
decidable.by_cases
(assume H : b ≤ c,
assert H1 : a + b ≤ a + c, from add_le_add_left H _,
by rewrite [min_eq_left H, min_eq_left H1])
(assume H : ¬ b ≤ c,
assert H' : c ≤ b, from le_of_lt (lt_of_not_ge H),
assert H1 : a + c ≤ a + b, from add_le_add_left H' _,
by rewrite [min_eq_right H', min_eq_right H1])
theorem min_add_add_right (a b c : ) : min (a + c) (b + c) = min a b + c :=
by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply min_add_add_left
theorem max_add_add_left (a b c : ) : max (a + b) (a + c) = a + max b c :=
decidable.by_cases
(assume H : b ≤ c,
assert H1 : a + b ≤ a + c, from add_le_add_left H _,
by rewrite [max_eq_right H, max_eq_right H1])
(assume H : ¬ b ≤ c,
assert H' : c ≤ b, from le_of_lt (lt_of_not_ge H),
assert H1 : a + c ≤ a + b, from add_le_add_left H' _,
by rewrite [max_eq_left H', max_eq_left H1])
theorem max_add_add_right (a b c : ) : max (a + c) (b + c) = max a b + c :=
by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply max_add_add_left
end nat