Documentation

Mathlib.Data.Real.Basic

Real numbers from Cauchy sequences #

This file defines ℝ as the type of equivalence classes of Cauchy sequences of rational numbers. This choice is motivated by how easy it is to prove that ℝ is a commutative ring, by simply lifting everything to ℚ.

The facts that the real numbers are an Archimedean floor ring, and a conditionally complete linear order, have been deferred to the file Mathlib/Data/Real/Archimedean.lean, in order to keep the imports here simple.

The fact that the real numbers are a (trivial) *-ring has similarly been deferred to Mathlib/Data/Real/Star.lean.

structure Real :

The type ℝ of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.

ofCauchy :: (

cauchy : CauSeq.Completion.Cauchy abs
The underlying Cauchy completion

)

Instances For

@[simp]

theorem CauSeq.Completion.ofRat_rat {abv : ℚ → ℚ} [IsAbsoluteValue abv] (q : ℚ) :

CauSeq.Completion.ofRat q = ↑q

theorem Real.ext_cauchy_iff {x : ℝ} {y : ℝ} :

x = y ↔ x.cauchy = y.cauchy

theorem Real.ext_cauchy {x : ℝ} {y : ℝ} :

x.cauchy = y.cauchy → x = y

def Real.equivCauchy :

ℝ ≃ CauSeq.Completion.Cauchy abs

The real numbers are isomorphic to the quotient of Cauchy sequences on the rationals.

Equations

Real.equivCauchy = { toFun := Real.cauchy, invFun := Real.ofCauchy, left_inv := Real.equivCauchy.proof_1, right_inv := Real.equivCauchy.proof_2 }

Instances For

instance Real.instZero :

Equations

Real.instZero = { zero := Real.zero }

instance Real.instOne :

Equations

Real.instOne = { one := Real.one }

instance Real.instAdd :

Equations

Real.instAdd = { add := Real.add }

instance Real.instNeg :

Equations

Real.instNeg = { neg := Real.neg }

instance Real.instMul :

Equations

Real.instMul = { mul := Real.mul }

instance Real.instSub :

Equations

Real.instSub = { sub := fun (a b : ℝ) => a + -b }

noncomputable instance Real.instInv :

Equations

Real.instInv = { inv := Real.inv' }

theorem Real.ofCauchy_zero :

{ cauchy := 0 } = 0

theorem Real.ofCauchy_one :

{ cauchy := 1 } = 1

theorem Real.ofCauchy_add (a : CauSeq.Completion.Cauchy abs) (b : CauSeq.Completion.Cauchy abs) :

{ cauchy := a + b } = { cauchy := a } + { cauchy := b }

theorem Real.ofCauchy_neg (a : CauSeq.Completion.Cauchy abs) :

{ cauchy := -a } = -{ cauchy := a }

theorem Real.ofCauchy_sub (a : CauSeq.Completion.Cauchy abs) (b : CauSeq.Completion.Cauchy abs) :

{ cauchy := a - b } = { cauchy := a } - { cauchy := b }

theorem Real.ofCauchy_mul (a : CauSeq.Completion.Cauchy abs) (b : CauSeq.Completion.Cauchy abs) :

{ cauchy := a * b } = { cauchy := a } * { cauchy := b }

theorem Real.ofCauchy_inv {f : CauSeq.Completion.Cauchy abs} :

{ cauchy := f⁻¹ } = { cauchy := f }⁻¹

theorem Real.cauchy_zero :

Real.cauchy 0 = 0

theorem Real.cauchy_one :

Real.cauchy 1 = 1

theorem Real.cauchy_add (a : ℝ) (b : ℝ) :

(a + b).cauchy = a.cauchy + b.cauchy

theorem Real.cauchy_neg (a : ℝ) :

(-a).cauchy = -a.cauchy

theorem Real.cauchy_mul (a : ℝ) (b : ℝ) :

(a * b).cauchy = a.cauchy * b.cauchy

theorem Real.cauchy_sub (a : ℝ) (b : ℝ) :

(a - b).cauchy = a.cauchy - b.cauchy

theorem Real.cauchy_inv (f : ℝ) :

f⁻¹.cauchy = f.cauchy⁻¹

instance Real.instNatCast :

Equations

Real.instNatCast = { natCast := fun (n : ℕ) => { cauchy := ↑n } }

instance Real.instIntCast :

Equations

Real.instIntCast = { intCast := fun (z : ℤ) => { cauchy := ↑z } }

instance Real.instNNRatCast :

Equations

Real.instNNRatCast = { nnratCast := fun (q : ℚ≥0) => { cauchy := ↑q } }

instance Real.instRatCast :

Equations

Real.instRatCast = { ratCast := fun (q : ℚ) => { cauchy := ↑q } }

theorem Real.ofCauchy_natCast (n : ℕ) :

{ cauchy := ↑n } = ↑n

theorem Real.ofCauchy_intCast (z : ℤ) :

{ cauchy := ↑z } = ↑z

theorem Real.ofCauchy_nnratCast (q : ℚ≥0) :

{ cauchy := ↑q } = ↑q

theorem Real.ofCauchy_ratCast (q : ℚ) :

{ cauchy := ↑q } = ↑q

theorem Real.cauchy_natCast (n : ℕ) :

(↑n).cauchy = ↑n

theorem Real.cauchy_intCast (z : ℤ) :

(↑z).cauchy = ↑z

theorem Real.cauchy_nnratCast (q : ℚ≥0) :

(↑q).cauchy = ↑q

theorem Real.cauchy_ratCast (q : ℚ) :

(↑q).cauchy = ↑q

instance Real.commRing :

Equations

Real.commRing = CommRing.mk Real.commRing.proof_25

def Real.ringEquivCauchy :

ℝ ≃+* CauSeq.Completion.Cauchy abs

Real.equivCauchy as a ring equivalence.

Equations

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

Instances For

@[simp]

theorem Real.ringEquivCauchy_apply (self : ℝ) :

Real.ringEquivCauchy self = self.cauchy

@[simp]

theorem Real.ringEquivCauchy_symm_apply_cauchy (cauchy : CauSeq.Completion.Cauchy abs) :

(Real.ringEquivCauchy.symm cauchy).cauchy = cauchy

Extra instances to short-circuit type class resolution.

These short-circuits have an additional property of ensuring that a computable path is found; if Field ℝ is found first, then decaying it to these typeclasses would result in a noncomputable version of them.

instance Real.instRing :

Equations

Real.instRing = inferInstance

instance Real.instCommSemiring :

CommSemiring ℝ

Equations

Real.instCommSemiring = inferInstance

instance Real.semiring :

Equations

Real.semiring = inferInstance

instance Real.instCommMonoidWithZero :

CommMonoidWithZero ℝ

Equations

Real.instCommMonoidWithZero = inferInstance

instance Real.instMonoidWithZero :

MonoidWithZero ℝ

Equations

Real.instMonoidWithZero = inferInstance

instance Real.instAddCommGroup :

AddCommGroup ℝ

Equations

Real.instAddCommGroup = inferInstance

instance Real.instAddGroup :

Equations

Real.instAddGroup = inferInstance

instance Real.instAddCommMonoid :

AddCommMonoid ℝ

Equations

Real.instAddCommMonoid = inferInstance

instance Real.instAddMonoid :

Equations

Real.instAddMonoid = inferInstance

instance Real.instAddLeftCancelSemigroup :

AddLeftCancelSemigroup ℝ

Equations

Real.instAddLeftCancelSemigroup = inferInstance

instance Real.instAddRightCancelSemigroup :

AddRightCancelSemigroup ℝ

Equations

Real.instAddRightCancelSemigroup = inferInstance

instance Real.instAddCommSemigroup :

AddCommSemigroup ℝ

Equations

Real.instAddCommSemigroup = inferInstance

instance Real.instAddSemigroup :

AddSemigroup ℝ

Equations

Real.instAddSemigroup = inferInstance

instance Real.instCommMonoid :

Equations

Real.instCommMonoid = inferInstance

instance Real.instMonoid :

Equations

Real.instMonoid = inferInstance

instance Real.instCommSemigroup :

CommSemigroup ℝ

Equations

Real.instCommSemigroup = inferInstance

instance Real.instSemigroup :

Equations

Real.instSemigroup = inferInstance

instance Real.instInhabited :

Equations

Real.instInhabited = { default := 0 }

def Real.mk (x : CauSeq ℚ abs) :

Make a real number from a Cauchy sequence of rationals (by taking the equivalence class).

Equations

Real.mk x = { cauchy := CauSeq.Completion.mk x }

Instances For

theorem Real.mk_eq {f : CauSeq ℚ abs} {g : CauSeq ℚ abs} :

Real.mk f = Real.mk g ↔ f ≈ g

instance Real.instLT :

Equations

Real.instLT = { lt := Real.lt }

theorem Real.lt_cauchy {f : CauSeq ℚ abs} {g : CauSeq ℚ abs} :

{ cauchy := ⟦f⟧ } < { cauchy := ⟦g⟧ } ↔ f < g

@[simp]

theorem Real.mk_lt {f : CauSeq ℚ abs} {g : CauSeq ℚ abs} :

Real.mk f < Real.mk g ↔ f < g

theorem Real.mk_zero :

theorem Real.mk_one :

theorem Real.mk_add {f : CauSeq ℚ abs} {g : CauSeq ℚ abs} :

Real.mk (f + g) = Real.mk f + Real.mk g

theorem Real.mk_mul {f : CauSeq ℚ abs} {g : CauSeq ℚ abs} :

Real.mk (f * g) = Real.mk f * Real.mk g

theorem Real.mk_neg {f : CauSeq ℚ abs} :

Real.mk (-f) = -Real.mk f

@[simp]

theorem Real.mk_pos {f : CauSeq ℚ abs} :

0 < Real.mk f ↔ f.Pos

instance Real.instLE :

Equations

Real.instLE = { le := Real.le }

@[simp]

theorem Real.mk_le {f : CauSeq ℚ abs} {g : CauSeq ℚ abs} :

Real.mk f ≤ Real.mk g ↔ f ≤ g

theorem Real.ind_mk {C : ℝ → Prop} (x : ℝ) (h : ∀ (y : CauSeq ℚ abs), C (Real.mk y)) :

C x

theorem Real.add_lt_add_iff_left {a : ℝ} {b : ℝ} (c : ℝ) :

c + a < c + b ↔ a < b

instance Real.partialOrder :

PartialOrder ℝ

Equations

Real.partialOrder = PartialOrder.mk Real.partialOrder.proof_4

instance Real.instPreorder :

Equations

Real.instPreorder = inferInstance

theorem Real.ratCast_lt {x : ℚ} {y : ℚ} :

↑x < ↑y ↔ x < y

theorem Real.zero_lt_one :

0 < 1

theorem Real.fact_zero_lt_one :

Fact (0 < 1)

@[deprecated mul_pos]

theorem Real.mul_pos {a : ℝ} {b : ℝ} :

0 < a → 0 < b → 0 < a * b

instance Real.instStrictOrderedCommRing :

StrictOrderedCommRing ℝ

Equations

Real.instStrictOrderedCommRing = StrictOrderedCommRing.mk ⋯

instance Real.strictOrderedRing :

StrictOrderedRing ℝ

Equations

Real.strictOrderedRing = inferInstance

instance Real.strictOrderedCommSemiring :

StrictOrderedCommSemiring ℝ

Equations

Real.strictOrderedCommSemiring = inferInstance

instance Real.strictOrderedSemiring :

StrictOrderedSemiring ℝ

Equations

Real.strictOrderedSemiring = inferInstance

instance Real.orderedRing :

OrderedRing ℝ

Equations

Real.orderedRing = inferInstance

instance Real.orderedSemiring :

OrderedSemiring ℝ

Equations

Real.orderedSemiring = inferInstance

instance Real.orderedAddCommGroup :

OrderedAddCommGroup ℝ

Equations

Real.orderedAddCommGroup = inferInstance

instance Real.orderedCancelAddCommMonoid :

OrderedCancelAddCommMonoid ℝ

Equations

Real.orderedCancelAddCommMonoid = inferInstance

instance Real.orderedAddCommMonoid :

OrderedAddCommMonoid ℝ

Equations

Real.orderedAddCommMonoid = inferInstance

instance Real.nontrivial :

Equations

Real.nontrivial = ⋯

instance Real.instSup :

Equations

Real.instSup = { sup := Real.sup }

theorem Real.ofCauchy_sup (a : CauSeq ℚ abs) (b : CauSeq ℚ abs) :

{ cauchy := ⟦a ⊔ b⟧ } = { cauchy := ⟦a⟧ } ⊔ { cauchy := ⟦b⟧ }

@[simp]

theorem Real.mk_sup (a : CauSeq ℚ abs) (b : CauSeq ℚ abs) :

Real.mk (a ⊔ b) = Real.mk a ⊔ Real.mk b

instance Real.instInf :

Equations

Real.instInf = { inf := Real.inf }

theorem Real.ofCauchy_inf (a : CauSeq ℚ abs) (b : CauSeq ℚ abs) :

{ cauchy := ⟦a ⊓ b⟧ } = { cauchy := ⟦a⟧ } ⊓ { cauchy := ⟦b⟧ }

@[simp]

theorem Real.mk_inf (a : CauSeq ℚ abs) (b : CauSeq ℚ abs) :

Real.mk (a ⊓ b) = Real.mk a ⊓ Real.mk b

instance Real.instDistribLattice :

DistribLattice ℝ

Equations

Real.instDistribLattice = DistribLattice.mk Real.instDistribLattice.proof_10

instance Real.lattice :

Equations

Real.lattice = inferInstance

instance Real.instSemilatticeInf :

SemilatticeInf ℝ

Equations

Real.instSemilatticeInf = inferInstance

instance Real.instSemilatticeSup :

SemilatticeSup ℝ

Equations

Real.instSemilatticeSup = inferInstance

instance Real.instIsTotalLe :

IsTotal ℝ fun (x1 x2 : ℝ) => x1 ≤ x2

Equations

Real.instIsTotalLe = ⋯

noncomputable instance Real.linearOrder :

LinearOrder ℝ

Equations

Real.linearOrder = Lattice.toLinearOrder ℝ

noncomputable instance Real.linearOrderedCommRing :

LinearOrderedCommRing ℝ

Equations

Real.linearOrderedCommRing = LinearOrderedCommRing.mk ⋯

noncomputable instance Real.instLinearOrderedRing :

LinearOrderedRing ℝ

Equations

Real.instLinearOrderedRing = inferInstance

noncomputable instance Real.instLinearOrderedSemiring :

LinearOrderedSemiring ℝ

Equations

Real.instLinearOrderedSemiring = inferInstance

instance Real.instIsDomain :

Equations

Real.instIsDomain = ⋯

noncomputable instance Real.instDivInvMonoid :

DivInvMonoid ℝ

Equations

Real.instDivInvMonoid = DivInvMonoid.mk Real.instDivInvMonoid.proof_1 zpowRec Real.instDivInvMonoid.proof_2 Real.instDivInvMonoid.proof_3 Real.instDivInvMonoid.proof_4

theorem Real.ofCauchy_div (f : CauSeq.Completion.Cauchy abs) (g : CauSeq.Completion.Cauchy abs) :

{ cauchy := f / g } = { cauchy := f } / { cauchy := g }

noncomputable instance Real.instLinearOrderedField :

LinearOrderedField ℝ

Equations

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

noncomputable instance Real.instLinearOrderedAddCommGroup :

LinearOrderedAddCommGroup ℝ

Equations

Real.instLinearOrderedAddCommGroup = inferInstance

noncomputable instance Real.field :

Equations

Real.field = inferInstance

noncomputable instance Real.instDivisionRing :

DivisionRing ℝ

Equations

Real.instDivisionRing = inferInstance

noncomputable instance Real.decidableLT (a : ℝ) (b : ℝ) :

Decidable (a < b)

Equations

a.decidableLT b = inferInstance

noncomputable instance Real.decidableLE (a : ℝ) (b : ℝ) :

Decidable (a ≤ b)

Equations

a.decidableLE b = inferInstance

noncomputable instance Real.decidableEq (a : ℝ) (b : ℝ) :

Decidable (a = b)

Equations

a.decidableEq b = inferInstance

unsafe instance Real.instRepr :

Show an underlying cauchy sequence for real numbers.

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.

theorem Real.le_mk_of_forall_le {x : ℝ} {f : CauSeq ℚ abs} :

(∃ (i : ℕ), ∀ j ≥ i, x ≤ ↑(↑f j)) → x ≤ Real.mk f

theorem Real.mk_le_of_forall_le {f : CauSeq ℚ abs} {x : ℝ} (h : ∃ (i : ℕ), ∀ j ≥ i, ↑(↑f j) ≤ x) :

Real.mk f ≤ x

theorem Real.mk_near_of_forall_near {f : CauSeq ℚ abs} {x : ℝ} {ε : ℝ} (H : ∃ (i : ℕ), ∀ j ≥ i, |↑(↑f j) - x| ≤ ε) :

|Real.mk f - x| ≤ ε

theorem Real.mul_add_one_le_add_one_pow {a : ℝ} (ha : 0 ≤ a) (b : ℕ) :

a * ↑b + 1 ≤ (a + 1) ^ b

def IsNonarchimedean {A : Type u_1} [Add A] (f : A → ℝ) :

A function f : R → ℝ≥0 is nonarchimedean if it satisfies the strong triangle inequality f (r + s) ≤ max (f r) (f s) for all r s : R.

Equations

IsNonarchimedean f = ∀ (r s : A), f (r + s) ≤ max (f r) (f s)

Instances For

def IsPowMul {R : Type u_1} [Pow R ℕ] (f : R → ℝ) :

A function f : R → ℝ is power-multiplicative if for all r ∈ R and all positive n ∈ ℕ, f (r ^ n) = (f r) ^ n.

Equations

IsPowMul f = ∀ (a : R) {n : ℕ}, 1 ≤ n → f (a ^ n) = f a ^ n

Instances For

def RingHom.IsBoundedWrt {α : Type u_1} [Ring α] {β : Type u_2} [Ring β] (nα : α → ℝ) (nβ : β → ℝ) (f : α →+* β) :

A ring homomorphism f : α →+* β is bounded with respect to the functions nα : α → ℝ and nβ : β → ℝ if there exists a positive constant C such that for all x in α, nβ (f x) ≤ C * nα x.

Equations

RingHom.IsBoundedWrt nα nβ f = ∃ (C : ℝ), 0 < C ∧ ∀ (x : α), nβ (f x) ≤ C * nα x

Instances For