Documentation

Mathlib.Data.Real.Basic

Real numbers from Cauchy sequences #

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

structure Real :

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

Instances For

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

    Equations
    @[simp]
    theorem CauSeq.Completion.ofRat_rat {abv : } [inst : IsAbsoluteValue abv] (q : ) :
    theorem Real.ext_cauchy_iff {x : } {y : } :
    x = y x.cauchy = y.cauchy
    theorem Real.ext_cauchy {x : } {y : } :
    x.cauchy = y.cauchyx = y

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

    Equations
    theorem Real.zero_def :
    Real.zero = { cauchy := 0 }
    theorem Real.one_def :
    Real.one = { cauchy := 1 }
    theorem Real.add_def :
    ∀ (x x_1 : ), Real.add x x_1 = match x, x_1 with | { cauchy := a }, { cauchy := b } => { cauchy := a + b }
    theorem Real.neg_def :
    ∀ (x : ), Real.neg x = match x with | { cauchy := a } => { cauchy := -a }
    theorem Real.mul_def :
    ∀ (x x_1 : ), Real.mul x x_1 = match x, x_1 with | { cauchy := a }, { cauchy := b } => { cauchy := a * b }
    Equations
    noncomputable instance Real.instInvReal :
    Equations
    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 :
    0.cauchy = 0
    theorem Real.cauchy_one :
    1.cauchy = 1
    theorem Real.cauchy_add (a : ) (b : ) :
    (a + b).cauchy = a.cauchy + b.cauchy
    theorem Real.cauchy_neg (a : ) :
    (-a).cauchy = -a.cauchy
    theorem Real.cauchy_mul (a : ) (b : ) :
    (a * b).cauchy = a.cauchy * b.cauchy
    theorem Real.cauchy_sub (a : ) (b : ) :
    (a - b).cauchy = a.cauchy - b.cauchy
    theorem Real.cauchy_inv (f : ) :
    f⁻¹.cauchy = f.cauchy⁻¹
    Equations
    Equations
    Equations
    theorem Real.ofCauchy_natCast (n : ) :
    { cauchy := n } = n
    theorem Real.ofCauchy_intCast (z : ) :
    { cauchy := z } = z
    theorem Real.ofCauchy_ratCast (q : ) :
    { cauchy := q } = q
    theorem Real.cauchy_natCast (n : ) :
    (n).cauchy = n
    theorem Real.cauchy_intCast (z : ) :
    (z).cauchy = z
    theorem Real.cauchy_ratCast (q : ) :
    (q).cauchy = q
    @[simp]
    theorem Real.ringEquivCauchy_apply (self : ) :
    Real.ringEquivCauchy self = self.cauchy

    real.equiv_Cauchy as a ring equivalence.

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

    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 is found first, then decaying it to these typeclasses would result in a noncomputable version of them.

    Equations
    Equations
    def Real.mk (x : CauSeq abs) :

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

    Equations
    theorem Real.mk_eq {f : CauSeq abs} {g : CauSeq abs} :
    theorem Real.lt_def :
    ∀ (x x_1 : ), Real.lt x x_1 = match x, x_1 with | { cauchy := x }, { cauchy := y } => Quotient.liftOn₂ x y (fun x x_2 => x < x_2) Real.definition.proof_1✝
    theorem Real.lt_cauchy {f : CauSeq abs} {g : CauSeq abs} :
    { cauchy := Quotient.mk CauSeq.equiv f } < { cauchy := Quotient.mk CauSeq.equiv g } f < g
    @[simp]
    theorem Real.mk_lt {f : CauSeq abs} {g : CauSeq abs} :
    theorem Real.mk_add {f : CauSeq abs} {g : CauSeq abs} :
    theorem Real.mk_mul {f : CauSeq abs} {g : CauSeq abs} :
    @[simp]
    theorem Real.mk_pos {f : CauSeq abs} :
    theorem Real.le_def (x : ) (y : ) :
    Real.le x y = (x < y x = y)
    @[simp]
    theorem Real.mk_le {f : CauSeq abs} {g : CauSeq abs} :
    theorem Real.ind_mk {C : Prop} (x : ) (h : (y : CauSeq abs) → C (Real.mk y)) :
    C x
    theorem Real.add_lt_add_iff_left {a : } {b : } (c : ) :
    c + a < c + b a < b
    theorem Real.ratCast_lt {x : } {y : } :
    x < y x < y
    theorem Real.mul_pos {a : } {b : } :
    0 < a0 < b0 < a * b
    theorem Real.sup_def :
    ∀ (x x_1 : ), Real.sup x x_1 = match x, x_1 with | { cauchy := x }, { cauchy := y } => { cauchy := Quotient.map₂ (fun x x_2 => x x_2) Real.definition.proof_2✝ x y }
    theorem Real.ofCauchy_sup (a : CauSeq abs) (b : CauSeq abs) :
    { cauchy := Quotient.mk CauSeq.equiv (a b) } = { cauchy := Quotient.mk CauSeq.equiv a } { cauchy := Quotient.mk CauSeq.equiv b }
    @[simp]
    theorem Real.mk_sup (a : CauSeq abs) (b : CauSeq abs) :
    theorem Real.inf_def :
    ∀ (x x_1 : ), Real.inf x x_1 = match x, x_1 with | { cauchy := x }, { cauchy := y } => { cauchy := Quotient.map₂ (fun x x_2 => x x_2) Real.definition.proof_2✝ x y }
    theorem Real.ofCauchy_inf (a : CauSeq abs) (b : CauSeq abs) :
    { cauchy := Quotient.mk CauSeq.equiv (a b) } = { cauchy := Quotient.mk CauSeq.equiv a } { cauchy := Quotient.mk CauSeq.equiv b }
    @[simp]
    theorem Real.mk_inf (a : CauSeq abs) (b : CauSeq abs) :
    Equations
    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.
    noncomputable instance Real.field :
    Equations
    Equations
    noncomputable instance Real.decidableLT (a : ) (b : ) :
    Decidable (a < b)
    Equations
    noncomputable instance Real.decidableLE (a : ) (b : ) :
    Equations
    noncomputable instance Real.decidableEq (a : ) (b : ) :
    Decidable (a = b)
    Equations
    unsafe instance Real.instReprReal :

    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.

    Equations
    theorem Real.le_mk_of_forall_le {x : } {f : CauSeq abs} :
    (i, ∀ (j : ), j ix ↑(f j)) → x Real.mk f
    theorem Real.mk_le_of_forall_le {f : CauSeq abs} {x : } (h : i, ∀ (j : ), j i↑(f j) x) :
    theorem Real.mk_near_of_forall_near {f : CauSeq abs} {x : } {ε : } (H : i, ∀ (j : ), j iabs (↑(f j) - x) ε) :
    abs (Real.mk f - x) ε
    theorem Real.isCauSeq_iff_lift {f : } :
    IsCauSeq abs f IsCauSeq abs fun i => ↑(f i)
    theorem Real.of_near (f : ) (x : ) (h : ∀ (ε : ), ε > 0i, ∀ (j : ), j iabs (↑(f j) - x) < ε) :
    h', Real.mk { val := f, property := h' } = x
    theorem Real.exists_floor (x : ) :
    ub, ub x ∀ (z : ), z xz ub
    theorem Real.exists_isLUB (S : Set ) (hne : Set.Nonempty S) (hbdd : BddAbove S) :
    x, IsLUB S x
    noncomputable instance Real.instSupSetReal :
    Equations
    theorem Real.supₛ_def (S : Set ) :
    supₛ S = if h : Set.Nonempty S BddAbove S then Classical.choose (_ : x, IsLUB S x) else 0
    theorem Real.isLUB_supₛ (S : Set ) (h₁ : Set.Nonempty S) (h₂ : BddAbove S) :
    noncomputable instance Real.instInfSetReal :
    Equations
    theorem Real.is_glb_infₛ (S : Set ) (h₁ : Set.Nonempty S) (h₂ : BddBelow S) :
    Equations
    • One or more equations did not get rendered due to their size.
    theorem Real.lt_infₛ_add_pos {s : Set } (h : Set.Nonempty s) {ε : } (hε : 0 < ε) :
    a, a s a < infₛ s + ε
    theorem Real.add_neg_lt_supₛ {s : Set } (h : Set.Nonempty s) {ε : } (hε : ε < 0) :
    a, a s supₛ s + ε < a
    theorem Real.infₛ_le_iff {s : Set } (h : BddBelow s) (h' : Set.Nonempty s) {a : } :
    infₛ s a ∀ (ε : ), 0 < εx, x s x < a + ε
    theorem Real.le_supₛ_iff {s : Set } (h : BddAbove s) (h' : Set.Nonempty s) {a : } :
    a supₛ s ∀ (ε : ), ε < 0x, x s a + ε < x
    theorem Real.csupᵢ_empty {α : Sort u_1} [inst : IsEmpty α] (f : α) :
    (i, f i) = 0
    @[simp]
    theorem Real.csupᵢ_const_zero {α : Sort u_1} :
    (_i, 0) = 0
    theorem Real.supᵢ_of_not_bddAbove {α : Sort u_1} {f : α} (hf : ¬BddAbove (Set.range f)) :
    (i, f i) = 0
    theorem Real.supₛ_univ :
    supₛ Set.univ = 0
    theorem Real.cinfᵢ_empty {α : Sort u_1} [inst : IsEmpty α] (f : α) :
    (i, f i) = 0
    @[simp]
    theorem Real.cinfᵢ_const_zero {α : Sort u_1} :
    (_i, 0) = 0
    theorem Real.infᵢ_of_not_bddBelow {α : Sort u_1} {f : α} (hf : ¬BddBelow (Set.range f)) :
    (i, f i) = 0
    theorem Real.supₛ_nonneg (S : Set ) (hS : ∀ (x : ), x S0 x) :

    As 0 is the default value for Real.supₛ of the empty set or sets which are not bounded above, it suffices to show that S is bounded below by 0 to show that 0 ≤ infₛ S.

    theorem Real.supₛ_nonpos (S : Set ) (hS : ∀ (x : ), x Sx 0) :

    As 0 is the default value for Real.supₛ of the empty set, it suffices to show that S is bounded above by 0 to show that supₛ S ≤ 0.

    theorem Real.infₛ_nonneg (S : Set ) (hS : ∀ (x : ), x S0 x) :

    As 0 is the default value for Real.infₛ of the empty set, it suffices to show that S is bounded below by 0 to show that 0 ≤ infₛ S.

    theorem Real.infᵢ_nonneg {ι : Sort u_1} {f : ι} (hf : ∀ (i : ι), 0 f i) :

    As 0 is the default value for Real.infₛ of the empty set, it suffices to show that f i is bounded below by 0 to show that 0 ≤ infᵢ f.

    theorem Real.infₛ_nonpos (S : Set ) (hS : ∀ (x : ), x Sx 0) :

    As 0 is the default value for Real.infₛ of the empty set or sets which are not bounded below, it suffices to show that S is bounded above by 0 to show that infₛ S ≤ 0.

    theorem Real.infₛ_le_supₛ (s : Set ) (h₁ : BddBelow s) (h₂ : BddAbove s) :
    theorem Real.cauSeq_converges (f : CauSeq abs) :
    x, f CauSeq.const abs x