Documentation

Mathlib.Algebra.Order.CauSeq.Basic

Cauchy sequences #

A basic theory of Cauchy sequences, used in the construction of the reals and p-adic numbers. Where applicable, lemmas that will be reused in other contexts have been stated in extra generality. There are other "versions" of Cauchyness in the library, in particular Cauchy filters in topology. This is a concrete implementation that is useful for simplicity and computability reasons.

Important definitions #

IsCauSeq: a predicate that says f : ℕ → β is Cauchy.
CauSeq: the type of Cauchy sequences valued in type β with respect to an absolute value function abv.

Tags #

sequence, cauchy, abs val, absolute value

theorem rat_add_continuous_lemma {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] (abv : β → α) [IsAbsoluteValue abv] {ε : α} (ε0 : 0 < ε) :

∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ + a₂ - (b₁ + b₂)) < ε

theorem rat_mul_continuous_lemma {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] (abv : β → α) [IsAbsoluteValue abv] {ε : α} {K₁ : α} {K₂ : α} (ε0 : 0 < ε) :

∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε

theorem rat_inv_continuous_lemma {α : Type u_1} [LinearOrderedField α] {β : Type u_3} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv] {ε : α} {K : α} (ε0 : 0 < ε) (K0 : 0 < K) :

∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε

def IsCauSeq {α : Type u_3} [LinearOrderedField α] {β : Type u_4} [Ring β] (abv : β → α) (f : ℕ → β) :

A sequence is Cauchy if the distance between its entries tends to zero.

Equations

IsCauSeq abv f = ∀ ε > 0, ∃ (i : ℕ), ∀ j ≥ i, abv (f j - f i) < ε

theorem IsCauSeq.cauchy₂ {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :

∃ (i : ℕ), ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε

theorem IsCauSeq.cauchy₃ {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :

∃ (i : ℕ), ∀ j ≥ i, ∀ k ≥ j, abv (f k - f j) < ε

theorem IsCauSeq.bounded {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) :

∃ (r : α), ∀ (i : ℕ), abv (f i) < r

theorem IsCauSeq.bounded' {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) (x : α) :

∃ r > x, ∀ (i : ℕ), abv (f i) < r

theorem IsCauSeq.const {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

IsCauSeq abv fun (x_1 : ℕ) => x

theorem IsCauSeq.add {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} {g : ℕ → β} (hf : IsCauSeq abv f) (hg : IsCauSeq abv g) :

IsCauSeq abv (f + g)

theorem IsCauSeq.mul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} {g : ℕ → β} (hf : IsCauSeq abv f) (hg : IsCauSeq abv g) :

IsCauSeq abv (f * g)

@[simp]

theorem isCauSeq_neg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} :

IsCauSeq abv (-f) ↔ IsCauSeq abv f

theorem IsCauSeq.of_neg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} :

IsCauSeq abv (-f) → IsCauSeq abv f

Alias of the forward direction of isCauSeq_neg.

theorem IsCauSeq.neg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} :

IsCauSeq abv f → IsCauSeq abv (-f)

Alias of the reverse direction of isCauSeq_neg.

def CauSeq {α : Type u_3} [LinearOrderedField α] (β : Type u_4) [Ring β] (abv : β → α) :

Type u_4

CauSeq β abv is the type of β-valued Cauchy sequences, with respect to the absolute value function abv.

Equations

CauSeq β abv = { f : ℕ → β // IsCauSeq abv f }

instance CauSeq.instCoeFunForallNat {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} :

CoeFun (CauSeq β abv) fun (x : CauSeq β abv) => ℕ → β

Equations

CauSeq.instCoeFunForallNat = { coe := Subtype.val }

theorem CauSeq.ext_iff {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} {f : CauSeq β abv} {g : CauSeq β abv} :

f = g ↔ ∀ (i : ℕ), ↑f i = ↑g i

theorem CauSeq.ext {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} {f : CauSeq β abv} {g : CauSeq β abv} (h : ∀ (i : ℕ), ↑f i = ↑g i) :

f = g

theorem CauSeq.isCauSeq {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} (f : CauSeq β abv) :

IsCauSeq abv ↑f

theorem CauSeq.cauchy {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} (f : CauSeq β abv) {ε : α} :

0 < ε → ∃ (i : ℕ), ∀ j ≥ i, abv (↑f j - ↑f i) < ε

def CauSeq.ofEq {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} (f : CauSeq β abv) (g : ℕ → β) (e : ∀ (i : ℕ), ↑f i = g i) :

CauSeq β abv

Given a Cauchy sequence f, create a Cauchy sequence from a sequence g with the same values as f.

Equations

f.ofEq g e = ⟨g, ⋯⟩

theorem CauSeq.cauchy₂ {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) {ε : α} :

0 < ε → ∃ (i : ℕ), ∀ j ≥ i, ∀ k ≥ i, abv (↑f j - ↑f k) < ε

theorem CauSeq.cauchy₃ {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) {ε : α} :

0 < ε → ∃ (i : ℕ), ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑f j) < ε

theorem CauSeq.bounded {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) :

∃ (r : α), ∀ (i : ℕ), abv (↑f i) < r

theorem CauSeq.bounded' {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (x : α) :

∃ r > x, ∀ (i : ℕ), abv (↑f i) < r

instance CauSeq.instAdd {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Add (CauSeq β abv)

Equations

CauSeq.instAdd = { add := fun (f g : CauSeq β abv) => ⟨↑f + ↑g, ⋯⟩ }

@[simp]

theorem CauSeq.coe_add {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (g : CauSeq β abv) :

↑(f + g) = ↑f + ↑g

@[simp]

theorem CauSeq.add_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (g : CauSeq β abv) (i : ℕ) :

↑(f + g) i = ↑f i + ↑g i

def CauSeq.const {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] (abv : β → α) [IsAbsoluteValue abv] (x : β) :

CauSeq β abv

The constant Cauchy sequence.

Equations

CauSeq.const abv x = ⟨fun (x_1 : ℕ) => x, ⋯⟩

@[simp]

theorem CauSeq.coe_const {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

↑(CauSeq.const abv x) = Function.const ℕ x

@[simp]

theorem CauSeq.const_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (i : ℕ) :

↑(CauSeq.const abv x) i = x

theorem CauSeq.const_inj {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {x : β} {y : β} :

CauSeq.const abv x = CauSeq.const abv y ↔ x = y

instance CauSeq.instZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Zero (CauSeq β abv)

Equations

CauSeq.instZero = { zero := CauSeq.const abv 0 }

instance CauSeq.instOne {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

One (CauSeq β abv)

Equations

CauSeq.instOne = { one := CauSeq.const abv 1 }

instance CauSeq.instInhabited {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Inhabited (CauSeq β abv)

Equations

CauSeq.instInhabited = { default := 0 }

@[simp]

theorem CauSeq.coe_zero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

↑0 = 0

@[simp]

theorem CauSeq.coe_one {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

↑1 = 1

@[simp]

theorem CauSeq.zero_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (i : ℕ) :

↑0 i = 0

@[simp]

theorem CauSeq.one_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (i : ℕ) :

↑1 i = 1

@[simp]

theorem CauSeq.const_zero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

CauSeq.const abv 0 = 0

@[simp]

theorem CauSeq.const_one {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

CauSeq.const abv 1 = 1

theorem CauSeq.const_add {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (y : β) :

CauSeq.const abv (x + y) = CauSeq.const abv x + CauSeq.const abv y

instance CauSeq.instMul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Mul (CauSeq β abv)

Equations

CauSeq.instMul = { mul := fun (f g : CauSeq β abv) => ⟨↑f * ↑g, ⋯⟩ }

@[simp]

theorem CauSeq.coe_mul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (g : CauSeq β abv) :

↑(f * g) = ↑f * ↑g

@[simp]

theorem CauSeq.mul_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (g : CauSeq β abv) (i : ℕ) :

↑(f * g) i = ↑f i * ↑g i

theorem CauSeq.const_mul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (y : β) :

CauSeq.const abv (x * y) = CauSeq.const abv x * CauSeq.const abv y

instance CauSeq.instNeg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Neg (CauSeq β abv)

Equations

CauSeq.instNeg = { neg := fun (f : CauSeq β abv) => ⟨-↑f, ⋯⟩ }

@[simp]

theorem CauSeq.coe_neg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) :

↑(-f) = -↑f

@[simp]

theorem CauSeq.neg_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (i : ℕ) :

↑(-f) i = -↑f i

theorem CauSeq.const_neg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

CauSeq.const abv (-x) = -CauSeq.const abv x

instance CauSeq.instSub {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Sub (CauSeq β abv)

Equations

CauSeq.instSub = { sub := fun (f g : CauSeq β abv) => (f + -g).ofEq (fun (x : ℕ) => ↑f x - ↑g x) ⋯ }

@[simp]

theorem CauSeq.coe_sub {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (g : CauSeq β abv) :

↑(f - g) = ↑f - ↑g

@[simp]

theorem CauSeq.sub_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (g : CauSeq β abv) (i : ℕ) :

↑(f - g) i = ↑f i - ↑g i

theorem CauSeq.const_sub {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (y : β) :

CauSeq.const abv (x - y) = CauSeq.const abv x - CauSeq.const abv y

instance CauSeq.instSMul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] :

SMul G (CauSeq β abv)

Equations

CauSeq.instSMul = { smul := fun (a : G) (f : CauSeq β abv) => (CauSeq.const abv (a • 1) * f).ofEq (a • ↑f) ⋯ }

@[simp]

theorem CauSeq.coe_smul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] (a : G) (f : CauSeq β abv) :

↑(a • f) = a • ↑f

@[simp]

theorem CauSeq.smul_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] (a : G) (f : CauSeq β abv) (i : ℕ) :

↑(a • f) i = a • ↑f i

theorem CauSeq.const_smul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] (a : G) (x : β) :

CauSeq.const abv (a • x) = a • CauSeq.const abv x

instance CauSeq.instIsScalarTower {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] :

IsScalarTower G (CauSeq β abv) (CauSeq β abv)

Equations

⋯ = ⋯

instance CauSeq.addGroup {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

AddGroup (CauSeq β abv)

Equations

CauSeq.addGroup = Function.Injective.addGroup Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance CauSeq.instNatCast {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

NatCast (CauSeq β abv)

Equations

CauSeq.instNatCast = { natCast := fun (n : ℕ) => CauSeq.const abv ↑n }

instance CauSeq.instIntCast {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

IntCast (CauSeq β abv)

Equations

CauSeq.instIntCast = { intCast := fun (n : ℤ) => CauSeq.const abv ↑n }

instance CauSeq.addGroupWithOne {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

AddGroupWithOne (CauSeq β abv)

Equations

CauSeq.addGroupWithOne = Function.Injective.addGroupWithOne Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance CauSeq.instPowNat {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Pow (CauSeq β abv) ℕ

Equations

CauSeq.instPowNat = { pow := fun (f : CauSeq β abv) (n : ℕ) => (npowRec n f).ofEq (fun (i : ℕ) => ↑f i ^ n) ⋯ }

@[simp]

theorem CauSeq.coe_pow {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (n : ℕ) :

↑(f ^ n) = ↑f ^ n

@[simp]

theorem CauSeq.pow_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (n : ℕ) (i : ℕ) :

↑(f ^ n) i = ↑f i ^ n

theorem CauSeq.const_pow {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (n : ℕ) :

CauSeq.const abv (x ^ n) = CauSeq.const abv x ^ n

instance CauSeq.ring {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Ring (CauSeq β abv)

Equations

CauSeq.ring = Function.Injective.ring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance CauSeq.instCommRingOfIsAbsoluteValue {α : Type u_1} [LinearOrderedField α] {β : Type u_3} [CommRing β] {abv : β → α} [IsAbsoluteValue abv] :

CommRing (CauSeq β abv)

Equations

CauSeq.instCommRingOfIsAbsoluteValue = CommRing.mk ⋯

def CauSeq.LimZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} (f : CauSeq β abv) :

LimZero f holds when f approaches 0.

Equations

f.LimZero = ∀ ε > 0, ∃ (i : ℕ), ∀ j ≥ i, abv (↑f j) < ε

theorem CauSeq.add_limZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (hf : f.LimZero) (hg : g.LimZero) :

(f + g).LimZero

theorem CauSeq.mul_limZero_right {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) {g : CauSeq β abv} (hg : g.LimZero) :

(f * g).LimZero

theorem CauSeq.mul_limZero_left {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (g : CauSeq β abv) (hg : f.LimZero) :

(f * g).LimZero

theorem CauSeq.neg_limZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : f.LimZero) :

(-f).LimZero

theorem CauSeq.sub_limZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (hf : f.LimZero) (hg : g.LimZero) :

(f - g).LimZero

theorem CauSeq.limZero_sub_rev {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (hfg : (f - g).LimZero) :

(g - f).LimZero

theorem CauSeq.zero_limZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

CauSeq.LimZero 0

theorem CauSeq.const_limZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {x : β} :

(CauSeq.const abv x).LimZero ↔ x = 0

instance CauSeq.equiv {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Setoid (CauSeq β abv)

Equations

CauSeq.equiv = { r := fun (f g : CauSeq β abv) => (f - g).LimZero, iseqv := ⋯ }

theorem CauSeq.add_equiv_add {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 : CauSeq β abv} {f2 : CauSeq β abv} {g1 : CauSeq β abv} {g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :

f1 + g1 ≈ f2 + g2

theorem CauSeq.neg_equiv_neg {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (hf : f ≈ g) :

theorem CauSeq.sub_equiv_sub {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 : CauSeq β abv} {f2 : CauSeq β abv} {g1 : CauSeq β abv} {g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :

f1 - g1 ≈ f2 - g2

theorem CauSeq.equiv_def₃ {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :

∃ (i : ℕ), ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑g j) < ε

theorem CauSeq.limZero_congr {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (h : f ≈ g) :

f.LimZero ↔ g.LimZero

theorem CauSeq.abv_pos_of_not_limZero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) :

∃ K > 0, ∃ (i : ℕ), ∀ j ≥ i, K ≤ abv (↑f j)

theorem CauSeq.of_near {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ (i : ℕ), ∀ j ≥ i, abv (f j - ↑g j) < ε) :

theorem CauSeq.not_limZero_of_not_congr_zero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f ≈ 0) :

¬f.LimZero

theorem CauSeq.mul_equiv_zero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (g : CauSeq β abv) {f : CauSeq β abv} (hf : f ≈ 0) :

g * f ≈ 0

theorem CauSeq.mul_equiv_zero' {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (g : CauSeq β abv) {f : CauSeq β abv} (hf : f ≈ 0) :

f * g ≈ 0

theorem CauSeq.mul_not_equiv_zero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} {g : CauSeq β abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) :

¬f * g ≈ 0

theorem CauSeq.const_equiv {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {x : β} {y : β} :

CauSeq.const abv x ≈ CauSeq.const abv y ↔ x = y

theorem CauSeq.mul_equiv_mul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 : CauSeq β abv} {f2 : CauSeq β abv} {g1 : CauSeq β abv} {g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :

f1 * g1 ≈ f2 * g2

theorem CauSeq.smul_equiv_smul {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] {f1 : CauSeq β abv} {f2 : CauSeq β abv} (c : G) (hf : f1 ≈ f2) :

c • f1 ≈ c • f2

theorem CauSeq.pow_equiv_pow {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 : CauSeq β abv} {f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) :

f1 ^ n ≈ f2 ^ n

theorem CauSeq.one_not_equiv_zero {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] [IsDomain β] (abv : β → α) [IsAbsoluteValue abv] :

¬CauSeq.const abv 1 ≈ CauSeq.const abv 0

theorem CauSeq.inv_aux {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) (ε : α) :

ε > 0 → ∃ (i : ℕ), ∀ j ≥ i, abv ((↑f j)⁻¹ - (↑f i)⁻¹) < ε

def CauSeq.inv {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (hf : ¬f.LimZero) :

CauSeq β abv

Given a Cauchy sequence f with nonzero limit, create a Cauchy sequence with values equal to the inverses of the values of f.

Equations

f.inv hf = ⟨fun (j : ℕ) => (↑f j)⁻¹, ⋯⟩

@[simp]

theorem CauSeq.coe_inv {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) :

↑(f.inv hf) = (↑f)⁻¹

@[simp]

theorem CauSeq.inv_apply {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) (i : ℕ) :

↑(f.inv hf) i = (↑f i)⁻¹

theorem CauSeq.inv_mul_cancel {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) :

f.inv hf * f ≈ 1

theorem CauSeq.mul_inv_cancel {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) :

f * f.inv hf ≈ 1

theorem CauSeq.const_inv {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {x : β} (hx : x ≠ 0) :

CauSeq.const abv x⁻¹ = (CauSeq.const abv x).inv ⋯

def CauSeq.Pos {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) :

The entries of a positive Cauchy sequence eventually have a positive lower bound.

Equations

f.Pos = ∃ K > 0, ∃ (i : ℕ), ∀ j ≥ i, K ≤ ↑f j

theorem CauSeq.not_limZero_of_pos {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} :

f.Pos → ¬f.LimZero

theorem CauSeq.const_pos {α : Type u_1} [LinearOrderedField α] {x : α} :

(CauSeq.const abs x).Pos ↔ 0 < x

theorem CauSeq.add_pos {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} :

f.Pos → g.Pos → (f + g).Pos

theorem CauSeq.pos_add_limZero {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} :

f.Pos → g.LimZero → (f + g).Pos

theorem CauSeq.mul_pos {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} :

f.Pos → g.Pos → (f * g).Pos

theorem CauSeq.trichotomy {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) :

f.Pos ∨ f.LimZero ∨ (-f).Pos

instance CauSeq.instLTAbs {α : Type u_1} [LinearOrderedField α] :

LT (CauSeq α abs)

Equations

CauSeq.instLTAbs = { lt := fun (f g : CauSeq α abs) => (g - f).Pos }

instance CauSeq.instLEAbs {α : Type u_1} [LinearOrderedField α] :

LE (CauSeq α abs)

Equations

CauSeq.instLEAbs = { le := fun (f g : CauSeq α abs) => f < g ∨ f ≈ g }

theorem CauSeq.lt_of_lt_of_eq {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} {h : CauSeq α abs} (fg : f < g) (gh : g ≈ h) :

f < h

theorem CauSeq.lt_of_eq_of_lt {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} {h : CauSeq α abs} (fg : f ≈ g) (gh : g < h) :

f < h

theorem CauSeq.lt_trans {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} {h : CauSeq α abs} (fg : f < g) (gh : g < h) :

f < h

theorem CauSeq.lt_irrefl {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} :

¬f < f

theorem CauSeq.le_of_eq_of_le {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} {h : CauSeq α abs} (hfg : f ≈ g) (hgh : g ≤ h) :

f ≤ h

theorem CauSeq.le_of_le_of_eq {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} {h : CauSeq α abs} (hfg : f ≤ g) (hgh : g ≈ h) :

f ≤ h

instance CauSeq.instPreorderAbs {α : Type u_1} [LinearOrderedField α] :

Preorder (CauSeq α abs)

Equations

CauSeq.instPreorderAbs = Preorder.mk ⋯ ⋯ ⋯

theorem CauSeq.le_antisymm {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} (fg : f ≤ g) (gf : g ≤ f) :

f ≈ g

theorem CauSeq.lt_total {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) (g : CauSeq α abs) :

f < g ∨ f ≈ g ∨ g < f

theorem CauSeq.le_total {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) (g : CauSeq α abs) :

f ≤ g ∨ g ≤ f

theorem CauSeq.const_lt {α : Type u_1} [LinearOrderedField α] {x : α} {y : α} :

CauSeq.const abs x < CauSeq.const abs y ↔ x < y

theorem CauSeq.const_le {α : Type u_1} [LinearOrderedField α] {x : α} {y : α} :

CauSeq.const abs x ≤ CauSeq.const abs y ↔ x ≤ y

theorem CauSeq.le_of_exists {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} (h : ∃ (i : ℕ), ∀ j ≥ i, ↑f j ≤ ↑g j) :

f ≤ g

theorem CauSeq.exists_gt {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) :

∃ (a : α), f < CauSeq.const abs a

theorem CauSeq.exists_lt {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) :

∃ (a : α), CauSeq.const abs a < f

theorem CauSeq.rat_sup_continuous_lemma {α : Type u_1} [LinearOrderedField α] {ε : α} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :

|a₁ - b₁| < ε → |a₂ - b₂| < ε → |a₁ ⊔ a₂ - b₁ ⊔ b₂| < ε

theorem CauSeq.rat_inf_continuous_lemma {α : Type u_1} [LinearOrderedField α] {ε : α} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :

|a₁ - b₁| < ε → |a₂ - b₂| < ε → |a₁ ⊓ a₂ - b₁ ⊓ b₂| < ε

instance CauSeq.instSupAbs {α : Type u_1} [LinearOrderedField α] :

Sup (CauSeq α abs)

Equations

CauSeq.instSupAbs = { sup := fun (f g : CauSeq α abs) => ⟨↑f ⊔ ↑g, ⋯⟩ }

instance CauSeq.instInfAbs {α : Type u_1} [LinearOrderedField α] :

Inf (CauSeq α abs)

Equations

CauSeq.instInfAbs = { inf := fun (f g : CauSeq α abs) => ⟨↑f ⊓ ↑g, ⋯⟩ }

@[simp]

theorem CauSeq.coe_sup {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) (g : CauSeq α abs) :

↑(f ⊔ g) = ↑f ⊔ ↑g

@[simp]

theorem CauSeq.coe_inf {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs) (g : CauSeq α abs) :

↑(f ⊓ g) = ↑f ⊓ ↑g

theorem CauSeq.sup_limZero {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} (hf : f.LimZero) (hg : g.LimZero) :

(f ⊔ g).LimZero

theorem CauSeq.inf_limZero {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs} {g : CauSeq α abs} (hf : f.LimZero) (hg : g.LimZero) :

(f ⊓ g).LimZero

theorem CauSeq.sup_equiv_sup {α : Type u_1} [LinearOrderedField α] {a₁ : CauSeq α abs} {b₁ : CauSeq α abs} {a₂ : CauSeq α abs} {b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :

a₁ ⊔ b₁ ≈ a₂ ⊔ b₂

theorem CauSeq.inf_equiv_inf {α : Type u_1} [LinearOrderedField α] {a₁ : CauSeq α abs} {b₁ : CauSeq α abs} {a₂ : CauSeq α abs} {b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :

a₁ ⊓ b₁ ≈ a₂ ⊓ b₂

theorem CauSeq.sup_lt {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} {c : CauSeq α abs} (ha : a < c) (hb : b < c) :

a ⊔ b < c

theorem CauSeq.lt_inf {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} {c : CauSeq α abs} (hb : a < b) (hc : a < c) :

a < b ⊓ c

@[simp]

theorem CauSeq.sup_idem {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs) :

a ⊔ a = a

@[simp]

theorem CauSeq.inf_idem {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs) :

a ⊓ a = a

theorem CauSeq.sup_comm {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs) (b : CauSeq α abs) :

a ⊔ b = b ⊔ a

theorem CauSeq.inf_comm {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs) (b : CauSeq α abs) :

a ⊓ b = b ⊓ a

theorem CauSeq.sup_eq_right {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} (h : a ≤ b) :

a ⊔ b ≈ b

theorem CauSeq.inf_eq_right {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} (h : b ≤ a) :

a ⊓ b ≈ b

theorem CauSeq.sup_eq_left {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} (h : b ≤ a) :

a ⊔ b ≈ a

theorem CauSeq.inf_eq_left {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} (h : a ≤ b) :

a ⊓ b ≈ a

theorem CauSeq.le_sup_left {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} :

a ≤ a ⊔ b

theorem CauSeq.inf_le_left {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} :

a ⊓ b ≤ a

theorem CauSeq.le_sup_right {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} :

b ≤ a ⊔ b

theorem CauSeq.inf_le_right {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} :

a ⊓ b ≤ b

theorem CauSeq.sup_le {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} {c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) :

a ⊔ b ≤ c

theorem CauSeq.le_inf {α : Type u_1} [LinearOrderedField α] {a : CauSeq α abs} {b : CauSeq α abs} {c : CauSeq α abs} (hb : a ≤ b) (hc : a ≤ c) :

a ≤ b ⊓ c

Note that DistribLattice (CauSeq α abs) is not true because there is no PartialOrder.

theorem CauSeq.sup_inf_distrib_left {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs) (b : CauSeq α abs) (c : CauSeq α abs) :

a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)

theorem CauSeq.sup_inf_distrib_right {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs) (b : CauSeq α abs) (c : CauSeq α abs) :

a ⊓ b ⊔ c = (a ⊔ c) ⊓ (b ⊔ c)