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.
• MulArchimedean is a typeclass for an ordered commutative monoid to have the "mul-archimedean property" where for x and y > 1, there exists a natural number n such that x ≤ y ^ n.
• Archimedean.floorRing defines a floor function on an archimedean linearly ordered ring making it into a floorRing.

Main statements

• ℕ, ℤ, and ℚ are archimedean.

class Archimedean (M : Type u_5) [AddCommMonoid M] [PartialOrder M] : Prop

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.

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

  For any two elements x, y such that 0 < y, there exists a natural number n such that x ≤ n • y.

class MulArchimedean (M : Type u_5) [CommMonoid M] [PartialOrder M] : Prop

An ordered commutative monoid is called MulArchimedean if for any two elements x, y such that 1 < y, there exists a natural number n such that x ≤ y ^ n.

• arch (x : M) {y : M} : 1 < y → ∃ (n : ℕ), x ≤ y ^ n

  For any two elements x, y such that 1 < y, there exists a natural number n such that x ≤ y ^ n.

theorem MulArchimedean.comap {G : Type u_1} {M : Type u_2} [CommMonoid G] [LinearOrder G] [CommMonoid M] [PartialOrder M] [MulArchimedean M] (f : G →* M) (hf : StrictMono ⇑f) : MulArchimedean G

theorem Archimedean.comap {G : Type u_1} {M : Type u_2} [AddCommMonoid G] [LinearOrder G] [AddCommMonoid M] [PartialOrder M] [Archimedean M] (f : G →+ M) (hf : StrictMono ⇑f) : Archimedean G

instance OrderDual.instMulArchimedean {G : Type u_1} [CommGroup G] [PartialOrder G] [IsOrderedMonoid G] [MulArchimedean G] : MulArchimedean Gᵒᵈ

instance OrderDual.instAddArchimedean {G : Type u_1} [AddCommGroup G] [PartialOrder G] [IsOrderedAddMonoid G] [Archimedean G] : Archimedean Gᵒᵈ

instance Additive.instArchimedean {G : Type u_1} [CommGroup G] [PartialOrder G] [MulArchimedean G] : Archimedean (Additive G)

instance Multiplicative.instMulArchimedean {G : Type u_1} [AddCommGroup G] [PartialOrder G] [Archimedean G] : MulArchimedean (Multiplicative G)

theorem exists_lt_pow {M : Type u_2} [CommMonoid M] [PartialOrder M] [MulArchimedean M] [MulLeftStrictMono M] {a : M} (ha : 1 < a) (b : M) : ∃ (n : ℕ), b < a ^ n

theorem exists_lt_nsmul {M : Type u_2} [AddCommMonoid M] [PartialOrder M] [Archimedean M] [AddLeftStrictMono M] {a : M} (ha : 0 < a) (b : M) : ∃ (n : ℕ), b < n • a

theorem existsUnique_zpow_near_of_one_lt {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : 1 < a) (g : G) : ∃! k : ℤ, a ^ k ≤ g ∧ g < a ^ (k + 1)

An archimedean decidable linearly ordered CommGroup has a version of the floor: for a > 1, any g in the group lies between some two consecutive powers of a.

theorem existsUnique_zsmul_near_of_pos {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : 0 < a) (g : G) : ∃! k : ℤ, k • a ≤ g ∧ g < (k + 1) • a

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

theorem existsUnique_zpow_near_of_one_lt' {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : 1 < a) (g : G) : ∃! k : ℤ, 1 ≤ g / a ^ k ∧ g / a ^ k < a

theorem existsUnique_zsmul_near_of_pos' {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : 0 < a) (g : G) : ∃! k : ℤ, 0 ≤ g - k • a ∧ g - k • a < a

theorem existsUnique_div_zpow_mem_Ico {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : 1 < a) (b c : G) : ∃! m : ℤ, b / a ^ m ∈ Set.Ico c (c * a)

theorem existsUnique_sub_zsmul_mem_Ico {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : 0 < a) (b c : G) : ∃! m : ℤ, b - m • a ∈ Set.Ico c (c + a)

theorem existsUnique_mul_zpow_mem_Ico {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : 1 < a) (b c : G) : ∃! m : ℤ, b * a ^ m ∈ Set.Ico c (c * a)

theorem existsUnique_add_zsmul_mem_Ico {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : 0 < a) (b c : G) : ∃! m : ℤ, b + m • a ∈ Set.Ico c (c + a)

theorem existsUnique_add_zpow_mem_Ioc {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : 1 < a) (b c : G) : ∃! m : ℤ, b * a ^ m ∈ Set.Ioc c (c * a)

theorem existsUnique_add_zsmul_mem_Ioc {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : 0 < a) (b c : G) : ∃! m : ℤ, b + m • a ∈ Set.Ioc c (c + a)

theorem existsUnique_sub_zpow_mem_Ioc {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : 1 < a) (b c : G) : ∃! m : ℤ, b / a ^ m ∈ Set.Ioc c (c * a)

theorem existsUnique_sub_zsmul_mem_Ioc {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : 0 < a) (b c : G) : ∃! m : ℤ, b - m • a ∈ Set.Ioc c (c + a)

theorem exists_pow_lt {G : Type u_1} [CommGroup G] [LinearOrder G] [IsOrderedMonoid G] [MulArchimedean G] {a : G} (ha : a < 1) (b : G) : ∃ (n : ℕ), a ^ n < b

theorem exists_nsmul_lt {G : Type u_1} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G] [Archimedean G] {a : G} (ha : a < 0) (b : G) : ∃ (n : ℕ), n • a < b

theorem exists_nat_ge {R : Type u_3} [Semiring R] [PartialOrder R] [IsOrderedRing R] [Archimedean R] (x : R) : ∃ (n : ℕ), x ≤ ↑n

@[instance 100]

instance instIsDirectedOrder {R : Type u_3} [Semiring R] [PartialOrder R] [IsOrderedRing R] [Archimedean R] : IsDirectedOrder R

theorem exists_nat_gt {R : Type u_3} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] (x : R) : ∃ (n : ℕ), x < ↑n

theorem add_one_pow_unbounded_of_pos {R : Type u_3} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] {y : R} (x : R) (hy : 0 < y) : ∃ (n : ℕ), x < (y + 1) ^ n

theorem pow_unbounded_of_one_lt {R : Type u_3} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] {y : R} [ExistsAddOfLE R] (x : R) (hy1 : 1 < y) : ∃ (n : ℕ), x < y ^ n

theorem exists_int_ge {R : Type u_3} [Ring R] [PartialOrder R] [IsOrderedRing R] [Archimedean R] (x : R) : ∃ (n : ℤ), x ≤ ↑n

theorem exists_int_le {R : Type u_3} [Ring R] [PartialOrder R] [IsOrderedRing R] [Archimedean R] (x : R) : ∃ (n : ℤ), ↑n ≤ x

@[instance 100]

instance instIsCodirectedOrder {R : Type u_3} [Ring R] [PartialOrder R] [IsOrderedRing R] [Archimedean R] : IsCodirectedOrder R

theorem exists_int_gt {R : Type u_3} [Ring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] (x : R) : ∃ (n : ℤ), x < ↑n

theorem exists_int_lt {R : Type u_3} [Ring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] (x : R) : ∃ (n : ℤ), ↑n < x

theorem exists_floor {R : Type u_3} [Ring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] (x : R) : ∃ (fl : ℤ), ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x

See exists_floor' for a more general version which only assumes the element is bounded by two integers.

theorem exists_nat_pow_near {R : Type u_3} [Semiring R] [LinearOrder R] [IsStrictOrderedRing R] [Archimedean R] [ExistsAddOfLE R] {x y : R} (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.

theorem exists_nat_one_div_lt {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {ε : K} (hε : 0 < ε) : ∃ (n : ℕ), 1 / (↑n + 1) < ε

theorem exists_mem_Ico_zpow {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (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.

theorem exists_mem_Ioc_zpow {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (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.

theorem exists_pow_lt_of_lt_one {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (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.

theorem exists_nat_pow_near_of_lt_one {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} [ExistsAddOfLE K] (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

theorem exists_pow_btwn_of_lt_mul {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] [ExistsAddOfLE K] {a b c : K} (h : a < b * c) (hb₀ : 0 < b) (hb₁ : b ≤ 1) (hc₀ : 0 < c) (hc₁ : c < 1) : ∃ (n : ℕ), a < c ^ n ∧ c ^ n < b

If a < b * c, 0 < b ≤ 1 and 0 < c < 1, then there is a power c ^ n with n : ℕ strictly between a and b.

theorem exists_zpow_btwn_of_lt_mul {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] [ExistsAddOfLE K] {a b c : K} (h : a < b * c) (hb₀ : 0 < b) (hc₀ : 0 < c) (hc₁ : c < 1) : ∃ (n : ℤ), a < c ^ n ∧ c ^ n < b

If a < b * c, b is positive and 0 < c < 1, then there is a power c ^ n with n : ℤ strictly between a and b.

theorem archimedean_iff_nat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] : Archimedean K ↔ ∀ (x : K), ∃ (n : ℕ), x < ↑n

theorem archimedean_iff_nat_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] : Archimedean K ↔ ∀ (x : K), ∃ (n : ℕ), x ≤ ↑n

theorem archimedean_iff_int_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] : Archimedean K ↔ ∀ (x : K), ∃ (n : ℤ), x < ↑n

theorem archimedean_iff_int_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] : Archimedean K ↔ ∀ (x : K), ∃ (n : ℤ), x ≤ ↑n

theorem archimedean_iff_rat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] : Archimedean K ↔ ∀ (x : K), ∃ (q : ℚ), x < ↑q

theorem archimedean_iff_rat_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] : Archimedean K ↔ ∀ (x : K), ∃ (q : ℚ), x ≤ ↑q

instance instArchimedeanRat : Archimedean ℚ

theorem exists_rat_gt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] (x : K) : ∃ (q : ℚ), x < ↑q

theorem exists_rat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] (x : K) : ∃ (q : ℚ), ↑q < x

theorem exists_div_btwn {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} {n : ℕ} (h : x < y) (nh : (y - x)⁻¹ < ↑n) : ∃ (z : ℤ), x < ↑z / ↑n ∧ ↑z / ↑n < y

theorem exists_rat_btwn {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} (h : x < y) : ∃ (q : ℚ), x < ↑q ∧ ↑q < y

theorem exists_rat_mem_uIoo {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} (h : x ≠ y) : ∃ (q : ℚ), ↑q ∈ Set.uIoo x y

theorem exists_pow_btwn {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {n : ℕ} (hn : n ≠ 0) {x y : K} (h : x < y) (hy : 0 < y) : ∃ (q : K), 0 < q ∧ x < q ^ n ∧ q ^ n < y

theorem exists_rat_pow_btwn {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {n : ℕ} (hn : n ≠ 0) {x y : K} (h : x < y) (hy : 0 < y) : ∃ (q : ℚ), 0 < q ∧ x < ↑q ^ n ∧ ↑q ^ n < y

There is a rational power between any two positive elements of an archimedean ordered field.

theorem le_of_forall_rat_lt_imp_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} (h : ∀ (q : ℚ), ↑q < x → ↑q ≤ y) : x ≤ y

theorem le_of_forall_lt_rat_imp_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} (h : ∀ (q : ℚ), y < ↑q → x ≤ ↑q) : x ≤ y

theorem le_iff_forall_rat_lt_imp_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} : x ≤ y ↔ ∀ (q : ℚ), ↑q < x → ↑q ≤ y

theorem le_iff_forall_lt_rat_imp_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} : x ≤ y ↔ ∀ (q : ℚ), y < ↑q → x ≤ ↑q

theorem eq_of_forall_rat_lt_iff_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} (h : ∀ (q : ℚ), ↑q < x ↔ ↑q < y) : x = y

theorem eq_of_forall_lt_rat_iff_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x y : K} (h : ∀ (q : ℚ), x < ↑q ↔ y < ↑q) : x = y

theorem exists_pos_rat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {x : K} (x0 : 0 < x) : ∃ (q : ℚ), 0 < q ∧ ↑q < x

theorem exists_rat_near {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {ε : K} (x : K) (ε0 : 0 < ε) : ∃ (q : ℚ), |x - ↑q| < ε

instance instArchimedeanNat : Archimedean ℕ

instance instArchimedeanInt : Archimedean ℤ

instance Nonneg.instArchimedean {M : Type u_2} [AddCommMonoid M] [PartialOrder M] [IsOrderedAddMonoid M] [Archimedean M] : Archimedean { x : M // 0 ≤ x }

instance Nonneg.instMulArchimedean {R : Type u_3} [CommSemiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] [ExistsAddOfLE R] : MulArchimedean { x : R // 0 ≤ x }

instance instArchimedeanNNRat : Archimedean ℚ≥0

instance instMulArchimedeanNNRat : MulArchimedean ℚ≥0

@[implicit_reducible]

noncomputable def Archimedean.floorRing (R : Type u_5) [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [Archimedean R] : FloorRing R

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.floorRing R = FloorRing.ofBounded R ⋯

@[instance 100]

instance FloorRing.archimedean (K : Type u_5) [Field K] [LinearOrder K] [IsStrictOrderedRing K] [FloorRing K] : Archimedean K

A linear ordered field that is a floor ring is archimedean.

instance Units.instMulArchimedean (M : Type u_5) [CommMonoid M] [PartialOrder M] [MulArchimedean M] : MulArchimedean Mˣ

instance AddUnits.instAddArchimedean (M : Type u_5) [AddCommMonoid M] [PartialOrder M] [Archimedean M] : Archimedean (AddUnits M)

instance WithBot.instArchimedean (M : Type u_5) [AddCommMonoid M] [PartialOrder M] [Archimedean M] : Archimedean (WithBot M)

instance WithZero.instMulArchimedean (M : Type u_5) [CommMonoid M] [PartialOrder M] [MulArchimedean M] : MulArchimedean (WithZero M)