Archimedean groups and fields. #

This file defines the archimedean property for ordered groups and proves several results connected to this notion. Being archimedean means that for all elements x and y>0 there exists a natural number n such that x ≤ n • y.

Main definitions #

archimedean is a typeclass for an ordered additive commutative monoid to have the archimedean property.
archimedean.floor_ring defines a floor function on an archimedean linearly ordered ring making it into a floor_ring.
round defines a function rounding to the nearest integer for a linearly ordered field which is also a floor ring.

Main statements #

ℕ, ℤ, and ℚ are archimedean.

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@[class]

structure archimedean (α : Type u_2) [ordered_add_comm_monoid α] :

Prop

arch : ∀ (x : α) {y : α}, 0 < y → (∃ (n : ℕ), x ≤ n • y)

An ordered additive commutative monoid is called archimedean if for any two elements x, y such that 0 < y there exists a natural number n such that x ≤ n • y.

Instances

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@[protected, instance]

def order_dual.archimedean {α : Type u_1} [ordered_add_comm_group α] [archimedean α] :

archimedean (order_dual α)

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theorem exists_unique_zsmul_near_of_pos {α : Type u_1} [linear_ordered_add_comm_group α] [archimedean α] {a : α} (ha : 0 < a) (g : α) :

∃! (k : ℤ), k • a ≤ g ∧ g < (k + 1) • a

An archimedean decidable linearly ordered add_comm_group has a version of the floor: for a > 0, any g in the group lies between some two consecutive multiples of a.

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theorem exists_unique_zsmul_near_of_pos' {α : Type u_1} [linear_ordered_add_comm_group α] [archimedean α] {a : α} (ha : 0 < a) (g : α) :

∃! (k : ℤ), 0 ≤ g - k • a ∧ g - k • a < a

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theorem exists_unique_add_zsmul_mem_Ico {α : Type u_1} [linear_ordered_add_comm_group α] [archimedean α] {a : α} (ha : 0 < a) (b c : α) :

∃! (m : ℤ), b + m • a ∈ set.Ico c (c + a)

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theorem exists_unique_add_zsmul_mem_Ioc {α : Type u_1} [linear_ordered_add_comm_group α] [archimedean α] {a : α} (ha : 0 < a) (b c : α) :

∃! (m : ℤ), b + m • a ∈ set.Ioc c (c + a)

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theorem exists_nat_gt {α : Type u_1} [ordered_semiring α] [nontrivial α] [archimedean α] (x : α) :

∃ (n : ℕ), x < ↑n

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theorem exists_nat_ge {α : Type u_1} [ordered_semiring α] [archimedean α] (x : α) :

∃ (n : ℕ), x ≤ ↑n

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theorem add_one_pow_unbounded_of_pos {α : Type u_1} [ordered_semiring α] [nontrivial α] [archimedean α] (x : α) {y : α} (hy : 0 < y) :

∃ (n : ℕ), x < (y + 1) ^ n

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theorem pow_unbounded_of_one_lt {α : Type u_1} [linear_ordered_ring α] [archimedean α] (x : α) {y : α} (hy1 : 1 < y) :

∃ (n : ℕ), x < y ^ n

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theorem exists_nat_pow_near {α : Type u_1} [linear_ordered_ring α] [archimedean α] {x y : α} (hx : 1 ≤ x) (hy : 1 < y) :

∃ (n : ℕ), y ^ n ≤ x ∧ x < y ^ (n + 1)

Every x greater than or equal to 1 is between two successive natural-number powers of every y greater than one.

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theorem exists_int_gt {α : Type u_1} [linear_ordered_ring α] [archimedean α] (x : α) :

∃ (n : ℤ), x < ↑n

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theorem exists_int_lt {α : Type u_1} [linear_ordered_ring α] [archimedean α] (x : α) :

∃ (n : ℤ), ↑n < x

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theorem exists_floor {α : Type u_1} [linear_ordered_ring α] [archimedean α] (x : α) :

∃ (fl : ℤ), ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x

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theorem exists_mem_Ico_zpow {α : Type u_1} [linear_ordered_field α] [archimedean α] {x y : α} (hx : 0 < x) (hy : 1 < y) :

∃ (n : ℤ), x ∈ set.Ico (y ^ n) (y ^ (n + 1))

Every positive x is between two successive integer powers of another y greater than one. This is the same as exists_mem_Ioc_zpow, but with ≤ and < the other way around.

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theorem exists_mem_Ioc_zpow {α : Type u_1} [linear_ordered_field α] [archimedean α] {x y : α} (hx : 0 < x) (hy : 1 < y) :

∃ (n : ℤ), x ∈ set.Ioc (y ^ n) (y ^ (n + 1))

Every positive x is between two successive integer powers of another y greater than one. This is the same as exists_mem_Ico_zpow, but with ≤ and < the other way around.

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theorem exists_pow_lt_of_lt_one {α : Type u_1} [linear_ordered_field α] [archimedean α] {x y : α} (hx : 0 < x) (hy : y < 1) :

∃ (n : ℕ), y ^ n < x

For any y < 1 and any positive x, there exists n : ℕ with y ^ n < x.

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theorem exists_nat_pow_near_of_lt_one {α : Type u_1} [linear_ordered_field α] [archimedean α] {x y : α} (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1) :

∃ (n : ℕ), y ^ (n + 1) < x ∧ x ≤ y ^ n

Given x and y between 0 and 1, x is between two successive powers of y. This is the same as exists_nat_pow_near, but for elements between 0 and 1

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theorem sub_floor_div_mul_nonneg {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : α) {y : α} (hy : 0 < y) :

0 ≤ x - (↑⌊x / y⌋) * y

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theorem sub_floor_div_mul_lt {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : α) {y : α} (hy : 0 < y) :

x - (↑⌊x / y⌋) * y < y

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@[protected, instance]

def nat.archimedean :

archimedean ℕ

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@[protected, instance]

def int.archimedean :

archimedean ℤ

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noncomputable def archimedean.floor_ring (α : Type u_1) [linear_ordered_ring α] [archimedean α] :

floor_ring α

A linear ordered archimedean ring is a floor ring. This is not an instance because in some cases we have a computable floor function.

Equations

archimedean.floor_ring α = floor_ring.of_floor α (λ (a : α), classical.some _) _

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theorem archimedean_iff_nat_lt {α : Type u_1} [linear_ordered_field α] :

archimedean α ↔ ∀ (x : α), ∃ (n : ℕ), x < ↑n

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theorem archimedean_iff_nat_le {α : Type u_1} [linear_ordered_field α] :

archimedean α ↔ ∀ (x : α), ∃ (n : ℕ), x ≤ ↑n

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theorem exists_rat_gt {α : Type u_1} [linear_ordered_field α] [archimedean α] (x : α) :

∃ (q : ℚ), x < ↑q

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theorem archimedean_iff_rat_lt {α : Type u_1} [linear_ordered_field α] :

archimedean α ↔ ∀ (x : α), ∃ (q : ℚ), x < ↑q

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theorem archimedean_iff_rat_le {α : Type u_1} [linear_ordered_field α] :

archimedean α ↔ ∀ (x : α), ∃ (q : ℚ), x ≤ ↑q

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theorem exists_rat_lt {α : Type u_1} [linear_ordered_field α] [archimedean α] (x : α) :

∃ (q : ℚ), ↑q < x

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theorem exists_rat_btwn {α : Type u_1} [linear_ordered_field α] [archimedean α] {x y : α} (h : x < y) :

∃ (q : ℚ), x < ↑q ∧ ↑q < y

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theorem exists_nat_one_div_lt {α : Type u_1} [linear_ordered_field α] [archimedean α] {ε : α} (hε : 0 < ε) :

∃ (n : ℕ), 1 / (↑n + 1) < ε

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theorem exists_pos_rat_lt {α : Type u_1} [linear_ordered_field α] [archimedean α] {x : α} (x0 : 0 < x) :

∃ (q : ℚ), 0 < q ∧ ↑q < x

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def round {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : α) :

ℤ

round rounds a number to the nearest integer. round (1 / 2) = 1

Equations

round x = ⌊x + 1 / 2⌋

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@[simp]

theorem round_zero {α : Type u_1} [linear_ordered_field α] [floor_ring α] :

round 0 = 0

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@[simp]

theorem round_one {α : Type u_1} [linear_ordered_field α] [floor_ring α] :

round 1 = 1

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theorem abs_sub_round {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : α) :

|x - ↑(round x)| ≤ 1 / 2

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@[simp, norm_cast]

theorem rat.floor_cast {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : ℚ) :

⌊↑x⌋ = ⌊x⌋

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@[simp, norm_cast]

theorem rat.ceil_cast {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : ℚ) :

⌈↑x⌉ = ⌈x⌉

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@[simp, norm_cast]

theorem rat.round_cast {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : ℚ) :

round ↑x = round x

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@[simp, norm_cast]

theorem rat.cast_fract {α : Type u_1} [linear_ordered_field α] [floor_ring α] (x : ℚ) :

↑(int.fract x) = int.fract ↑x

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theorem exists_rat_near {α : Type u_1} [linear_ordered_field α] [archimedean α] (x : α) {ε : α} (ε0 : 0 < ε) :

∃ (q : ℚ), |x - ↑q| < ε

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@[protected, instance]

def rat.archimedean :

archimedean ℚ