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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 .

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.

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.

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    Instances For
      @[simp]
      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.

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      Instances For
        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
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        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.equivCauchy as a ring equivalence.

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

          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.

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          def Real.mk (x : CauSeq abs) :

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

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          Instances For
            theorem Real.mk_eq {f : CauSeq abs} {g : CauSeq abs} :
            Equations
            theorem Real.lt_cauchy {f : CauSeq abs} {g : CauSeq abs} :
            { cauchy := f } < { cauchy := 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} :
            theorem Real.mk_neg {f : CauSeq abs} :
            @[simp]
            theorem Real.mk_pos {f : CauSeq abs} :
            Equations
            @[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
            Equations
            theorem Real.ofCauchy_sup (a : CauSeq abs) (b : CauSeq abs) :
            { cauchy := a b } = { cauchy := a } { cauchy := b }
            @[simp]
            theorem Real.mk_sup (a : CauSeq abs) (b : CauSeq abs) :
            theorem Real.ofCauchy_inf (a : CauSeq abs) (b : CauSeq abs) :
            { cauchy := a b } = { cauchy := a } { cauchy := 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.
            noncomputable instance Real.field :
            Equations
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            noncomputable instance Real.decidableLT (a : ) (b : ) :
            Decidable (a < b)
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            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 : ), ji, x (f j))x Real.mk f
            theorem Real.mk_le_of_forall_le {f : CauSeq abs} {x : } (h : ∃ (i : ), ji, (f j) x) :
            theorem Real.mk_near_of_forall_near {f : CauSeq abs} {x : } {ε : } (H : ∃ (i : ), ji, |(f j) - x| ε) :
            |Real.mk f - x| ε