Documentation

Mathlib.Data.Real.CauSeq

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→ 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 exists_forall_ge_and {α : Type u_1} [inst : ] {P : αProp} {Q : αProp} :
(i, (j : α) → j iP j) → (i, (j : α) → j iQ j) → i, ∀ (j : α), j iP j Q j
theorem rat_add_continuous_lemma {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] (abv : βα) [inst : ] {ε : α} (ε0 : 0 < ε) :
δ, δ > 0 ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δabv (a₂ - b₂) < δabv (a₁ + a₂ - (b₁ + b₂)) < ε
theorem rat_mul_continuous_lemma {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] (abv : βα) [inst : ] {ε : α} {K₁ : α} {K₂ : α} (ε0 : 0 < ε) :
δ, δ > 0 ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁abv b₂ < K₂abv (a₁ - b₁) < δabv (a₂ - b₂) < δabv (a₁ * a₂ - b₁ * b₂) < ε
theorem rat_inv_continuous_lemma {α : Type u_2} [inst : ] {β : Type u_1} [inst : ] (abv : βα) [inst : ] {ε : α} {K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
δ, δ > 0 ∀ {a b : β}, K abv aK abv babv (a - b) < δabv (a⁻¹ - b⁻¹) < ε
def IsCauSeq {α : Type u_1} [inst : ] {β : Type u_2} [inst : Ring β] (abv : βα) (f : β) :

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

Equations
• IsCauSeq abv f = ∀ (ε : α), ε > 0i, ∀ (j : ), j iabv (f j - f i) < ε
theorem IsCauSeq.cauchy₂ {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : β} (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
i, ∀ (j : ), j i∀ (k : ), k iabv (f j - f k) < ε
theorem IsCauSeq.cauchy₃ {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : β} (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
i, ∀ (j : ), j i∀ (k : ), k jabv (f k - f j) < ε
theorem IsCauSeq.add {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : β} {g : β} (hf : IsCauSeq abv f) (hg : IsCauSeq abv g) :
IsCauSeq abv (f + g)
def CauSeq {α : Type u_1} [inst : ] (β : Type u_2) [inst : Ring β] (abv : βα) :
Type u_2

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

Equations
instance CauSeq.instCoeFunCauSeqForAllNat {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} :
CoeFun (CauSeq β abv) fun x => β
Equations
• CauSeq.instCoeFunCauSeqForAllNat = { coe := Subtype.val }
theorem CauSeq.ext {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} {f : CauSeq β abv} {g : CauSeq β abv} (h : ∀ (i : ), f i = g i) :
f = g
theorem CauSeq.isCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} (f : CauSeq β abv) :
IsCauSeq abv f
theorem CauSeq.cauchy {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} (f : CauSeq β abv) {ε : α} :
0 < εi, ∀ (j : ), j iabv (f j - f i) < ε
def CauSeq.ofEq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} (f : CauSeq β abv) (g : β) (e : ∀ (i : ), 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
• CauSeq.ofEq f g e = { val := g, property := (_ : ∀ (ε : α), ε > 0i, ∀ (j : ), j iabv (g j - g i) < ε) }
theorem CauSeq.cauchy₂ {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) {ε : α} :
0 < εi, ∀ (j : ), j i∀ (k : ), k iabv (f j - f k) < ε
theorem CauSeq.cauchy₃ {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) {ε : α} :
0 < εi, ∀ (j : ), j i∀ (k : ), k jabv (f k - f j) < ε
theorem CauSeq.bounded {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) :
r, ∀ (i : ), abv (f i) < r
theorem CauSeq.bounded' {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (x : α) :
r, r > x ∀ (i : ), abv (f i) < r
instance CauSeq.instAddCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Equations
• CauSeq.instAddCauSeq = { add := fun f g => { val := f + g, property := (_ : IsCauSeq abv (f + g)) } }
@[simp]
theorem CauSeq.coe_add {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (g : CauSeq β abv) (i : ) :
↑(f + g) i = f i + g i
def CauSeq.const {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] (abv : βα) [inst : ] (x : β) :
CauSeq β abv

The constant Cauchy sequence.

Equations
• CauSeq.const abv x = { val := fun x => x, property := (_ : ∀ (ε : α), ε > 0i, ∀ (j : ), j iabv ((fun x => x) j - (fun x => x) i) < ε) }
@[simp]
theorem CauSeq.coe_const {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) :
↑(CauSeq.const abv x) =
@[simp]
theorem CauSeq.const_apply {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) (i : ) :
↑(CauSeq.const abv x) i = x
theorem CauSeq.const_inj {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {x : β} {y : β} :
CauSeq.const abv x = CauSeq.const abv y x = y
instance CauSeq.instZeroCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Zero (CauSeq β abv)
Equations
instance CauSeq.instOneCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
One (CauSeq β abv)
Equations
instance CauSeq.instInhabitedCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Inhabited (CauSeq β abv)
Equations
• CauSeq.instInhabitedCauSeq = { default := 0 }
@[simp]
theorem CauSeq.coe_zero {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
0 = 0
@[simp]
theorem CauSeq.coe_one {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
1 = 1
@[simp]
theorem CauSeq.zero_apply {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (i : ) :
0 i = 0
@[simp]
theorem CauSeq.one_apply {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (i : ) :
1 i = 1
@[simp]
theorem CauSeq.const_zero {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
CauSeq.const abv 0 = 0
@[simp]
theorem CauSeq.const_one {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
CauSeq.const abv 1 = 1
theorem CauSeq.const_add {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) (y : β) :
CauSeq.const abv (x + y) = CauSeq.const abv x + CauSeq.const abv y
instance CauSeq.instMulCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Mul (CauSeq β abv)
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem CauSeq.coe_mul {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (g : CauSeq β abv) (i : ) :
↑(f * g) i = f i * g i
theorem CauSeq.const_mul {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) (y : β) :
CauSeq.const abv (x * y) = CauSeq.const abv x * CauSeq.const abv y
instance CauSeq.instNegCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Neg (CauSeq β abv)
Equations
@[simp]
theorem CauSeq.coe_neg {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) :
↑(-f) = -f
@[simp]
theorem CauSeq.neg_apply {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (i : ) :
↑(-f) i = -f i
theorem CauSeq.const_neg {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) :
instance CauSeq.instSubCauSeq {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Sub (CauSeq β abv)
Equations
• CauSeq.instSubCauSeq = { sub := fun f g => CauSeq.ofEq (f + -g) (fun x => f x - g x) (_ : ∀ (i : ), f i + -g i = f i - g i) }
@[simp]
theorem CauSeq.coe_sub {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (g : CauSeq β abv) (i : ) :
↑(f - g) i = f i - g i
theorem CauSeq.const_sub {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) (y : β) :
CauSeq.const abv (x - y) = CauSeq.const abv x - CauSeq.const abv y
instance CauSeq.instSMulCauSeq {α : Type u_1} {β : Type u_2} {G : Type u_3} [inst : ] [inst : Ring β] {abv : βα} [inst : ] [inst : SMul G β] [inst : ] :
SMul G (CauSeq β abv)
Equations
@[simp]
theorem CauSeq.coe_smul {α : Type u_1} {β : Type u_2} {G : Type u_3} [inst : ] [inst : Ring β] {abv : βα} [inst : ] [inst : SMul G β] [inst : ] (a : G) (f : CauSeq β abv) :
↑(a f) = a f
@[simp]
theorem CauSeq.smul_apply {α : Type u_1} {β : Type u_2} {G : Type u_3} [inst : ] [inst : Ring β] {abv : βα} [inst : ] [inst : SMul G β] [inst : ] (a : G) (f : CauSeq β abv) (i : ) :
↑(a f) i = a f i
theorem CauSeq.const_smul {α : Type u_2} {β : Type u_1} {G : Type u_3} [inst : ] [inst : Ring β] {abv : βα} [inst : ] [inst : SMul G β] [inst : ] (a : G) (x : β) :
CauSeq.const abv (a x) = a CauSeq.const abv x
instance CauSeq.instIsScalarTowerCauSeqInstSMulCauSeqToSMulInstMulCauSeq {α : Type u_1} {β : Type u_2} {G : Type u_3} [inst : ] [inst : Ring β] {abv : βα} [inst : ] [inst : SMul G β] [inst : ] :
IsScalarTower G (CauSeq β abv) (CauSeq β abv)
Equations
instance CauSeq.addGroup {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Equations
• One or more equations did not get rendered due to their size.
instance CauSeq.instNatCast {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
NatCast (CauSeq β abv)
Equations
• CauSeq.instNatCast = { natCast := fun n => CauSeq.const abv n }
instance CauSeq.instIntCast {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
IntCast (CauSeq β abv)
Equations
• CauSeq.instIntCast = { intCast := fun n => CauSeq.const abv n }
instance CauSeq.addGroupWithOne {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Equations
• One or more equations did not get rendered due to their size.
instance CauSeq.instPowCauSeqNat {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Pow (CauSeq β abv)
Equations
@[simp]
theorem CauSeq.coe_pow {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (n : ) :
↑(f ^ n) = f ^ n
@[simp]
theorem CauSeq.pow_apply {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) (n : ) (i : ) :
↑(f ^ n) i = f i ^ n
theorem CauSeq.const_pow {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (x : β) (n : ) :
CauSeq.const abv (x ^ n) = CauSeq.const abv x ^ n
instance CauSeq.ring {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Ring (CauSeq β abv)
Equations
• One or more equations did not get rendered due to their size.
instance CauSeq.instCommRingCauSeqToRing {α : Type u_1} [inst : ] {β : Type u_2} [inst : ] {abv : βα} [inst : ] :
CommRing (CauSeq β abv)
Equations
• CauSeq.instCommRingCauSeqToRing = let src := CauSeq.ring; CommRing.mk (_ : ∀ (a b : CauSeq β abv), a * b = b * a)
def CauSeq.LimZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} (f : CauSeq β abv) :

LimZero f holds when f approaches 0.

Equations
• = ∀ (ε : α), ε > 0i, ∀ (j : ), j iabv (f j) < ε
theorem CauSeq.add_limZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (hf : ) (hg : ) :
theorem CauSeq.mul_limZero_right {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : CauSeq β abv) {g : CauSeq β abv} (hg : ) :
theorem CauSeq.mul_limZero_left {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} (g : CauSeq β abv) (hg : ) :
theorem CauSeq.neg_limZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) :
theorem CauSeq.sub_limZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (hf : ) (hg : ) :
theorem CauSeq.limZero_sub_rev {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (hfg : CauSeq.LimZero (f - g)) :
theorem CauSeq.zero_limZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
theorem CauSeq.const_limZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {x : β} :
instance CauSeq.equiv {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] :
Setoid (CauSeq β abv)
Equations
theorem CauSeq.add_equiv_add {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (hf : f g) :
-f -g
theorem CauSeq.sub_equiv_sub {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (h : f g) {ε : α} (ε0 : 0 < ε) :
i, ∀ (j : ), j i∀ (k : ), k jabv (f k - g j) < ε
theorem CauSeq.limZero_congr {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (h : f g) :
theorem CauSeq.abv_pos_of_not_limZero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) :
K, K > 0 i, ∀ (j : ), j iK abv (f j)
theorem CauSeq.of_near {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (f : β) (g : CauSeq β abv) (h : ∀ (ε : α), ε > 0i, ∀ (j : ), j iabv (f j - g j) < ε) :
IsCauSeq abv f
theorem CauSeq.not_limZero_of_not_congr_zero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ¬f 0) :
theorem CauSeq.mul_equiv_zero {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] (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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f : CauSeq β abv} {g : CauSeq β abv} (hf : ¬f 0) (hg : ¬g 0) :
¬f * g 0
theorem CauSeq.const_equiv {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {x : β} {y : β} :
theorem CauSeq.mul_equiv_mul {α : Type u_1} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {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_3} {β : Type u_2} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {G : Type u_1} [inst : SMul G β] [inst : ] {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} [inst : ] [inst : Ring β] {abv : βα} [inst : ] {f1 : CauSeq β abv} {f2 : CauSeq β abv} (hf : f1 f2) (n : ) :
f1 ^ n f2 ^ n
theorem CauSeq.one_not_equiv_zero {α : Type u_2} {β : Type u_1} [inst : ] [inst : Ring β] [inst : ] (abv : βα) [inst : ] :
theorem CauSeq.inv_aux {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) (ε : α) :
ε > 0i, ∀ (j : ), j iabv ((f j)⁻¹ - (f i)⁻¹) < ε
def CauSeq.inv {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {abv : βα} [inst : ] (f : CauSeq β abv) (hf : ) :
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
@[simp]
theorem CauSeq.coe_inv {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) :
↑(CauSeq.inv f hf) = (f)⁻¹
@[simp]
theorem CauSeq.inv_apply {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) (i : ) :
↑(CauSeq.inv f hf) i = (f i)⁻¹
theorem CauSeq.inv_mul_cancel {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) :
CauSeq.inv f hf * f 1
theorem CauSeq.mul_inv_cancel {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] {abv : βα} [inst : ] {f : CauSeq β abv} (hf : ) :
f * CauSeq.inv f hf 1
theorem CauSeq.const_inv {α : Type u_2} {β : Type u_1} [inst : ] [inst : ] {abv : βα} [inst : ] {x : β} (hx : x 0) :
def CauSeq.Pos {α : Type u_1} [inst : ] (f : CauSeq α abs) :

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

Equations
theorem CauSeq.not_limZero_of_pos {α : Type u_1} [inst : ] {f : CauSeq α abs} :
theorem CauSeq.const_pos {α : Type u_1} [inst : ] {x : α} :
theorem CauSeq.add_pos {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} :
CauSeq.Pos (f + g)
theorem CauSeq.pos_add_limZero {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} :
CauSeq.Pos (f + g)
theorem CauSeq.mul_pos {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} :
CauSeq.Pos (f * g)
theorem CauSeq.trichotomy {α : Type u_1} [inst : ] (f : CauSeq α abs) :
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• One or more equations did not get rendered due to their size.
theorem CauSeq.lt_of_lt_of_eq {α : Type u_1} [inst : ] {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} [inst : ] {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} [inst : ] {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} [inst : ] {f : CauSeq α abs} :
¬f < f
theorem CauSeq.le_of_eq_of_le {α : Type u_1} [inst : ] {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} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} {h : CauSeq α abs} (hfg : f g) (hgh : g h) :
f h
Equations
• One or more equations did not get rendered due to their size.
theorem CauSeq.le_antisymm {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} (fg : f g) (gf : g f) :
f g
theorem CauSeq.lt_total {α : Type u_1} [inst : ] (f : CauSeq α abs) (g : CauSeq α abs) :
f < g f g g < f
theorem CauSeq.le_total {α : Type u_1} [inst : ] (f : CauSeq α abs) (g : CauSeq α abs) :
f g g f
theorem CauSeq.const_lt {α : Type u_1} [inst : ] {x : α} {y : α} :
CauSeq.const abs x < CauSeq.const abs y x < y
theorem CauSeq.const_le {α : Type u_1} [inst : ] {x : α} {y : α} :
theorem CauSeq.le_of_exists {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} (h : i, ∀ (j : ), j if j g j) :
f g
theorem CauSeq.exists_gt {α : Type u_1} [inst : ] (f : CauSeq α abs) :
a, f < CauSeq.const abs a
theorem CauSeq.exists_lt {α : Type u_1} [inst : ] (f : CauSeq α abs) :
a, CauSeq.const abs a < f
theorem CauSeq.rat_sup_continuous_lemma {α : Type u_1} [inst : ] {ε : α} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
abs (a₁ - b₁) < εabs (a₂ - b₂) < εabs (a₁ a₂ - b₁ b₂) < ε
theorem CauSeq.rat_inf_continuous_lemma {α : Type u_1} [inst : ] {ε : α} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} :
abs (a₁ - b₁) < εabs (a₂ - b₂) < εabs (a₁ a₂ - b₁ b₂) < ε
Equations
• One or more equations did not get rendered due to their size.
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem CauSeq.coe_sup {α : Type u_1} [inst : ] (f : CauSeq α abs) (g : CauSeq α abs) :
↑(f g) = f g
@[simp]
theorem CauSeq.coe_inf {α : Type u_1} [inst : ] (f : CauSeq α abs) (g : CauSeq α abs) :
↑(f g) = f g
theorem CauSeq.sup_limZero {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} (hf : ) (hg : ) :
theorem CauSeq.inf_limZero {α : Type u_1} [inst : ] {f : CauSeq α abs} {g : CauSeq α abs} (hf : ) (hg : ) :
theorem CauSeq.sup_equiv_sup {α : Type u_1} [inst : ] {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} [inst : ] {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} [inst : ] {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} [inst : ] {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} [inst : ] (a : CauSeq α abs) :
a a = a
@[simp]
theorem CauSeq.inf_idem {α : Type u_1} [inst : ] (a : CauSeq α abs) :
a a = a
theorem CauSeq.sup_comm {α : Type u_1} [inst : ] (a : CauSeq α abs) (b : CauSeq α abs) :
a b = b a
theorem CauSeq.inf_comm {α : Type u_1} [inst : ] (a : CauSeq α abs) (b : CauSeq α abs) :
a b = b a
theorem CauSeq.sup_eq_right {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} (h : a b) :
a b b
theorem CauSeq.inf_eq_right {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} (h : b a) :
a b b
theorem CauSeq.sup_eq_left {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} (h : b a) :
a b a
theorem CauSeq.inf_eq_left {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} (h : a b) :
a b a
theorem CauSeq.le_sup_left {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} :
a a b
theorem CauSeq.inf_le_left {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} :
a b a
theorem CauSeq.le_sup_right {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} :
b a b
theorem CauSeq.inf_le_right {α : Type u_1} [inst : ] {a : CauSeq α abs} {b : CauSeq α abs} :
a b b
theorem CauSeq.sup_le {α : Type u_1} [inst : ] {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} [inst : ] {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} [inst : ] (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} [inst : ] (a : CauSeq α abs) (b : CauSeq α abs) (c : CauSeq α abs) :
a b c = (a c) (b c)