Cauchy completion #

This file generalizes the Cauchy completion of (ℚ, abs) to the completion of a ring with absolute value.

def CauSeq.Completion.Cauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] (abv : β → α) [IsAbsoluteValue abv] :

Type u_2

The Cauchy completion of a ring with absolute value.

Equations

CauSeq.Completion.Cauchy abv = Quotient CauSeq.equiv

Instances For

source

def CauSeq.Completion.mk {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

CauSeq β abv → Cauchy abv

The map from Cauchy sequences into the Cauchy completion.

Equations

CauSeq.Completion.mk = Quotient.mk''

Instances For

source

@[simp]

theorem CauSeq.Completion.mk_eq_mk {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) :

⟦f⟧ = mk f

source

theorem CauSeq.Completion.mk_eq {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} :

mk f = mk g ↔ f ≈ g

source

def CauSeq.Completion.ofRat {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

Cauchy abv

The map from the original ring into the Cauchy completion.

Equations

CauSeq.Completion.ofRat x = CauSeq.Completion.mk (CauSeq.const abv x)

Instances For

source

instance CauSeq.Completion.instZeroCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Zero (Cauchy abv)

Equations

CauSeq.Completion.instZeroCauchy = { zero := CauSeq.Completion.ofRat 0 }

source

instance CauSeq.Completion.instOneCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

One (Cauchy abv)

Equations

CauSeq.Completion.instOneCauchy = { one := CauSeq.Completion.ofRat 1 }

source

instance CauSeq.Completion.instInhabitedCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Inhabited (Cauchy abv)

Equations

CauSeq.Completion.instInhabitedCauchy = { default := 0 }

source

theorem CauSeq.Completion.ofRat_zero {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

ofRat 0 = 0

source

theorem CauSeq.Completion.ofRat_one {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

ofRat 1 = 1

source

@[simp]

theorem CauSeq.Completion.mk_eq_zero {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} :

mk f = 0 ↔ f.LimZero

source

instance CauSeq.Completion.instAddCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Add (Cauchy abv)

Equations

CauSeq.Completion.instAddCauchy = { add := Quotient.map₂ (fun (x1 x2 : CauSeq β abv) => x1 + x2) ⋯ }

source

@[simp]

theorem CauSeq.Completion.mk_add {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv) :

mk f + mk g = mk (f + g)

source

instance CauSeq.Completion.instNegCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Neg (Cauchy abv)

Equations

CauSeq.Completion.instNegCauchy = { neg := Quotient.map Neg.neg ⋯ }

source

@[simp]

theorem CauSeq.Completion.mk_neg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) :

-mk f = mk (-f)

source

instance CauSeq.Completion.instMulCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Mul (Cauchy abv)

Equations

CauSeq.Completion.instMulCauchy = { mul := Quotient.map₂ (fun (x1 x2 : CauSeq β abv) => x1 * x2) ⋯ }

source

@[simp]

theorem CauSeq.Completion.mk_mul {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv) :

mk f * mk g = mk (f * g)

source

instance CauSeq.Completion.instSubCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Sub (Cauchy abv)

Equations

CauSeq.Completion.instSubCauchy = { sub := Quotient.map₂ Sub.sub ⋯ }

source

@[simp]

theorem CauSeq.Completion.mk_sub {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv) :

mk f - mk g = mk (f - g)

source

instance CauSeq.Completion.instSMulCauchyOfIsScalarTower {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {γ : Type u_3} [SMul γ β] [IsScalarTower γ β β] :

SMul γ (Cauchy abv)

Equations

CauSeq.Completion.instSMulCauchyOfIsScalarTower = { smul := fun (c : γ) => Quotient.map (fun (x : CauSeq β abv) => c • x) ⋯ }

source

@[simp]

theorem CauSeq.Completion.mk_smul {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {γ : Type u_3} [SMul γ β] [IsScalarTower γ β β] (c : γ) (f : CauSeq β abv) :

c • mk f = mk (c • f)

source

instance CauSeq.Completion.instPowCauchyNat {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Pow (Cauchy abv) ℕ

Equations

CauSeq.Completion.instPowCauchyNat = { pow := fun (x : CauSeq.Completion.Cauchy abv) (n : ℕ) => Quotient.map (fun (x : CauSeq β abv) => x ^ n) ⋯ x }

source

@[simp]

theorem CauSeq.Completion.mk_pow {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (n : ℕ) (f : CauSeq β abv) :

mk f ^ n = mk (f ^ n)

source

instance CauSeq.Completion.instNatCastCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

NatCast (Cauchy abv)

Equations

CauSeq.Completion.instNatCastCauchy = { natCast := fun (n : ℕ) => CauSeq.Completion.mk ↑n }

source

instance CauSeq.Completion.instIntCastCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

IntCast (Cauchy abv)

Equations

CauSeq.Completion.instIntCastCauchy = { intCast := fun (n : ℤ) => CauSeq.Completion.mk ↑n }

source

@[simp]

theorem CauSeq.Completion.ofRat_natCast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (n : ℕ) :

ofRat ↑n = ↑n

source

@[simp]

theorem CauSeq.Completion.ofRat_intCast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (z : ℤ) :

ofRat ↑z = ↑z

source

theorem CauSeq.Completion.ofRat_add {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β) :

ofRat (x + y) = ofRat x + ofRat y

source

theorem CauSeq.Completion.ofRat_neg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

ofRat (-x) = -ofRat x

source

theorem CauSeq.Completion.ofRat_mul {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β) :

ofRat (x * y) = ofRat x * ofRat y

source

instance CauSeq.Completion.Cauchy.ring {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

Ring (Cauchy abv)

Equations

One or more equations did not get rendered due to their size.

source

def CauSeq.Completion.ofRatRingHom {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] :

β →+* Cauchy abv

CauSeq.Completion.ofRat as a RingHom

Equations

CauSeq.Completion.ofRatRingHom = { toFun := CauSeq.Completion.ofRat, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

Instances For

source

@[simp]

theorem CauSeq.Completion.ofRatRingHom_apply {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

ofRatRingHom x = ofRat x

source

theorem CauSeq.Completion.ofRat_sub {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β) :

ofRat (x - y) = ofRat x - ofRat y

source

instance CauSeq.Completion.Cauchy.commRing {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [CommRing β] {abv : β → α} [IsAbsoluteValue abv] :

CommRing (Cauchy abv)

Equations

CauSeq.Completion.Cauchy.commRing = { toRing := CauSeq.Completion.Cauchy.ring, mul_comm := ⋯ }

source

instance CauSeq.Completion.instNNRatCast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

NNRatCast (Cauchy abv)

Equations

CauSeq.Completion.instNNRatCast = { nnratCast := fun (q : ℚ≥0) => CauSeq.Completion.ofRat ↑q }

source

instance CauSeq.Completion.instRatCast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

RatCast (Cauchy abv)

Equations

CauSeq.Completion.instRatCast = { ratCast := fun (q : ℚ) => CauSeq.Completion.ofRat ↑q }

source

@[simp]

theorem CauSeq.Completion.ofRat_nnratCast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (q : ℚ≥0) :

ofRat ↑q = ↑q

source

@[simp]

theorem CauSeq.Completion.ofRat_ratCast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (q : ℚ) :

ofRat ↑q = ↑q

source

noncomputable instance CauSeq.Completion.instInvCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

Inv (Cauchy abv)

Equations

One or more equations did not get rendered due to their size.

source

theorem CauSeq.Completion.inv_zero {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

0⁻¹ = 0

source

@[simp]

theorem CauSeq.Completion.inv_mk {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) :

(mk f)⁻¹ = mk (f.inv hf)

source

theorem CauSeq.Completion.cau_seq_zero_ne_one {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

¬0 ≈ 1

source

theorem CauSeq.Completion.zero_ne_one {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

0 ≠ 1

source

theorem CauSeq.Completion.inv_mul_cancel {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {x : Cauchy abv} :

x ≠ 0 → x⁻¹ * x = 1

source

theorem CauSeq.Completion.mul_inv_cancel {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {x : Cauchy abv} :

x ≠ 0 → x * x⁻¹ = 1

source

theorem CauSeq.Completion.ofRat_inv {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (x : β) :

ofRat x⁻¹ = (ofRat x)⁻¹

source

noncomputable instance CauSeq.Completion.instDivInvMonoid {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

DivInvMonoid (Cauchy abv)

Equations

One or more equations did not get rendered due to their size.

source

theorem CauSeq.Completion.ofRat_div {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (x y : β) :

ofRat (x / y) = ofRat x / ofRat y

source

noncomputable instance CauSeq.Completion.Cauchy.divisionRing {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] :

DivisionRing (Cauchy abv)

The Cauchy completion forms a division ring.

Equations

One or more equations did not get rendered due to their size.

source

unsafe instance CauSeq.Completion.instReprCauchy {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] [Repr β] :

Repr (Cauchy abv)

Show the first 10 items of a representative of this equivalence class of cauchy sequences.

The representative chosen is the one passed in the VM to Quot.mk, so two cauchy sequences converging to the same number may be printed differently.

Equations

One or more equations did not get rendered due to their size.

source

noncomputable instance CauSeq.Completion.Cauchy.field {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Field β] {abv : β → α} [IsAbsoluteValue abv] :

Field (Cauchy abv)

The Cauchy completion forms a field.

Equations

One or more equations did not get rendered due to their size.

source

class CauSeq.IsComplete {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (β : Type u_2) [Ring β] (abv : β → α) [IsAbsoluteValue abv] :

Prop

A class stating that a ring with an absolute value is complete, i.e. every Cauchy sequence has a limit.

isComplete (s : CauSeq β abv) : ∃ (b : β), s ≈ const abv b
Every Cauchy sequence has a limit.

Instances

source

theorem CauSeq.complete {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (s : CauSeq β abv) :

∃ (b : β), s ≈ const abv b

source

noncomputable def CauSeq.lim {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (s : CauSeq β abv) :

The limit of a Cauchy sequence in a complete ring. Chosen non-computably.

Equations

s.lim = Classical.choose ⋯

Instances For

source

theorem CauSeq.equiv_lim {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (s : CauSeq β abv) :

s ≈ const abv s.lim

source

theorem CauSeq.eq_lim_of_const_equiv {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] {f : CauSeq β abv} {x : β} (h : const abv x ≈ f) :

x = f.lim

source

theorem CauSeq.lim_eq_of_equiv_const {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] {f : CauSeq β abv} {x : β} (h : f ≈ const abv x) :

f.lim = x

source

theorem CauSeq.lim_eq_lim_of_equiv {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] {f g : CauSeq β abv} (h : f ≈ g) :

f.lim = g.lim

source

@[simp]

theorem CauSeq.lim_const {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (x : β) :

(const abv x).lim = x

source

theorem CauSeq.lim_add {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (f g : CauSeq β abv) :

f.lim + g.lim = (f + g).lim

source

theorem CauSeq.lim_mul_lim {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (f g : CauSeq β abv) :

f.lim * g.lim = (f * g).lim

source

theorem CauSeq.lim_mul {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (f : CauSeq β abv) (x : β) :

f.lim * x = (f * const abv x).lim

source

theorem CauSeq.lim_neg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (f : CauSeq β abv) :

(-f).lim = -f.lim

source

theorem CauSeq.lim_eq_zero_iff {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] (f : CauSeq β abv) :

f.lim = 0 ↔ f.LimZero

source

theorem CauSeq.lim_inv {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {β : Type u_2} [Field β] {abv : β → α} [IsAbsoluteValue abv] [IsComplete β abv] {f : CauSeq β abv} (hf : ¬f.LimZero) :

(f.inv hf).lim = f.lim⁻¹

source

theorem CauSeq.lim_le {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [IsComplete α abs] {f : CauSeq α abs} {x : α} (h : f ≤ const abs x) :

f.lim ≤ x

source

theorem CauSeq.le_lim {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [IsComplete α abs] {f : CauSeq α abs} {x : α} (h : const abs x ≤ f) :

x ≤ f.lim

source

theorem CauSeq.lt_lim {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [IsComplete α abs] {f : CauSeq α abs} {x : α} (h : const abs x < f) :

x < f.lim

source

theorem CauSeq.lim_lt {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [IsComplete α abs] {f : CauSeq α abs} {x : α} (h : f < const abs x) :

f.lim < x

Documentation

Mathlib.Algebra.Order.CauSeq.Completion

Cauchy completion #