Archimedean groups and fields #

This file proves several results connected to the notion of Archimedean groups. 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.floorRing defines a floor function on an archimedean linearly ordered ring making it into a floorRing.

Main statements #

ℕ, ℤ, and ℚ are archimedean.

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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

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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

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

MulArchimedean Gᵒᵈ

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

Archimedean Gᵒᵈ

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

Archimedean (Additive G)

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

MulArchimedean (Multiplicative G)

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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.

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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.

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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

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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

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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@[instance 100]

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

IsDirectedOrder R

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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

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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

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@[instance 100]

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

IsCodirectedOrder R

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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.

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

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theorem archimedean_iff_nat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] :

Archimedean K ↔ ∀ (x : K), ∃ (n : ℕ), x < ↑n

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theorem archimedean_iff_nat_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] :

Archimedean K ↔ ∀ (x : K), ∃ (n : ℕ), x ≤ ↑n

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theorem archimedean_iff_int_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] :

Archimedean K ↔ ∀ (x : K), ∃ (n : ℤ), x < ↑n

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theorem archimedean_iff_int_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] :

Archimedean K ↔ ∀ (x : K), ∃ (n : ℤ), x ≤ ↑n

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theorem archimedean_iff_rat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] :

Archimedean K ↔ ∀ (x : K), ∃ (q : ℚ), x < ↑q

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theorem archimedean_iff_rat_le {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] :

Archimedean K ↔ ∀ (x : K), ∃ (q : ℚ), x ≤ ↑q

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instance instArchimedeanRat :

Archimedean ℚ

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theorem exists_rat_gt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] (x : K) :

∃ (q : ℚ), x < ↑q

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theorem exists_rat_lt {K : Type u_4} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] (x : K) :

∃ (q : ℚ), ↑q < x

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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instance instArchimedeanNat :

Archimedean ℕ

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instance instArchimedeanInt :

Archimedean ℤ

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

Archimedean { x : M // 0 ≤ x }

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instance Nonneg.instMulArchimedean {R : Type u_3} [CommSemiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] [ExistsAddOfLE R] :

MulArchimedean { x : R // 0 ≤ x }

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instance instArchimedeanNNRat :

Archimedean ℚ≥0

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instance instMulArchimedeanNNRat :

MulArchimedean ℚ≥0

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@[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 ⋯

Instances For

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@[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.

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

MulArchimedean Mˣ

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

Archimedean (AddUnits M)

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

Archimedean (WithBot M)

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

MulArchimedean (WithZero M)

Documentation

Mathlib.Algebra.Order.Archimedean.Basic

Archimedean groups and fields #

Main definitions #

Main statements #