lean2/library/theories/analysis/real_limit.lean

/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad

Instantiates the reals as a metric space, and expresses completeness, sup, and inf in
a manner that is less constructive, but more convenient, than the way it is done in
data.real.complete.

The definitions here are noncomputable, for various reasons:

(1) We rely on the nonconstructive definition of abs.
(2) The theory of the reals uses the "some" operator e.g. to define the ceiling function.
    This can't be defined constructively as an operation on the quotient, because
    such a function is not continuous.
(3) We use "forall" and "exists" to say that a series converges, rather than carrying
    around rates of convergence explicitly. We then use "some" whenever we need to extract
    information, such as the limit.

These could be avoided in a constructive theory of analysis, but here we will not
follow that route.
-/
import .metric_space data.real.complete
open real eq.ops classical
noncomputable theory

namespace real

/- the reals form a metric space -/

protected definition to_metric_space [instance] : metric_space ℝ :=
⦃ metric_space,
  dist               := λ x y, abs (x - y),
  dist_self          := λ x, abstract by rewrite [sub_self, abs_zero] end,
  eq_of_dist_eq_zero := @eq_of_abs_sub_eq_zero,
  dist_comm          := abs_sub,
  dist_triangle      := abs_sub_le
⦄

open nat

definition converges_to_seq (X : ℕ → ℝ) (y : ℝ) : Prop :=
∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε

proposition converges_to_seq.intro {X : ℕ → ℝ} {y : ℝ}
    (H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) ≤ ε) :
  converges_to_seq X y :=
metric_space.converges_to_seq.intro H

notation X `⟶` y `in` `ℕ` := converges_to_seq X y

definition converges_seq [class] (X : ℕ → ℝ) : Prop := ∃ y, X ⟶ y in ℕ

definition limit_seq (X : ℕ → ℝ) [H : converges_seq X] : ℝ := some H

proposition converges_to_limit_seq (X : ℕ → ℝ) [H : converges_seq X] :
  (X ⟶ limit_seq X in ℕ) :=
some_spec H

proposition converges_to_seq_unique {X : ℕ → ℝ} {y₁ y₂ : ℝ}
  (H₁ : X ⟶ y₁ in ℕ) (H₂ : X ⟶ y₂ in ℕ) : y₁ = y₂ :=
metric_space.converges_to_seq_unique H₁ H₂

proposition eq_limit_of_converges_to_seq {X : ℕ → ℝ} (y : ℝ) (H : X ⟶ y in ℕ) :
  y = @limit_seq X (exists.intro y H) :=
converges_to_seq_unique H (@converges_to_limit_seq X (exists.intro y H))

proposition converges_to_seq_constant (y : ℝ) : (λn, y) ⟶ y in ℕ :=
metric_space.converges_to_seq_constant y

/- the completeness of the reals, "translated" from data.real.complete -/

definition cauchy (X : ℕ → ℝ) := metric_space.cauchy X

section
  open pnat subtype

  private definition pnat.succ (n : ℕ) : ℕ+ := tag (succ n) !succ_pos

  private definition r_seq_of (X : ℕ → ℝ) : r_seq := λ n, X (elt_of n)

  private lemma rate_of_cauchy_aux {X : ℕ → ℝ} (H : cauchy X) :
    ∀ k : ℕ+, ∃ N : ℕ+, ∀ m n : ℕ+,
      m ≥ N → n ≥ N → abs (X (elt_of m) - X (elt_of n)) ≤ of_rat k⁻¹ :=
  take k : ℕ+,
  have H1 : (rat.gt k⁻¹ (rat.of_num 0)), from !inv_pos,
  have H2 : (of_rat k⁻¹ > of_rat (rat.of_num 0)), from !of_rat_lt_of_rat_of_lt H1,
  obtain (N : ℕ) (H : ∀ m n, m ≥ N → n ≥ N → abs (X m - X n) < of_rat k⁻¹), from H _ H2,
  exists.intro (pnat.succ N)
    (take m n : ℕ+,
      assume Hm : m ≥ (pnat.succ N),
      assume Hn : n ≥ (pnat.succ N),
      have Hm' : elt_of m ≥ N, from nat.le.trans !le_succ Hm,
      have Hn' : elt_of n ≥ N, from nat.le.trans !le_succ Hn,
      show abs (X (elt_of m) - X (elt_of n)) ≤ of_rat k⁻¹, from le_of_lt (H _ _ Hm' Hn'))

  private definition rate_of_cauchy {X : ℕ → ℝ} (H : cauchy X) (k : ℕ+) : ℕ+ :=
  some (rate_of_cauchy_aux H k)

  private lemma cauchy_with_rate_of_cauchy {X : ℕ → ℝ} (H : cauchy X) :
    cauchy_with_rate (r_seq_of X) (rate_of_cauchy H) :=
  take k : ℕ+,
  some_spec (rate_of_cauchy_aux H k)

  private lemma converges_to_with_rate_of_cauchy {X : ℕ → ℝ} (H : cauchy X) :
    ∃ l Nb, converges_to_with_rate (r_seq_of X) l Nb :=
  begin
    apply exists.intro,
    apply exists.intro,
    apply converges_to_with_rate_of_cauchy_with_rate,
    exact cauchy_with_rate_of_cauchy H
  end

theorem converges_seq_of_cauchy {X : ℕ → ℝ} (H : cauchy X) : converges_seq X :=
obtain l Nb (conv : converges_to_with_rate (r_seq_of X) l Nb),
  from converges_to_with_rate_of_cauchy H,
exists.intro l
  (take ε : ℝ,
    suppose ε > 0,
    obtain (k' : ℕ) (Hn : 1 / succ k' < ε), from archimedean_small `ε > 0`,
    let k : ℕ+ := tag (succ k') !succ_pos,
        N : ℕ+ := Nb k in
    have Hk : real.of_rat k⁻¹ < ε,
      by rewrite [↑pnat.inv, of_rat_divide]; exact Hn,
    exists.intro (elt_of N)
      (take n : ℕ,
        assume Hn : n ≥ elt_of N,
        let n' : ℕ+ := tag n (nat.lt_of_lt_of_le (has_property N) Hn) in
        have abs (X n - l) ≤ real.of_rat k⁻¹, from conv k n' Hn,
        show abs (X n - l) < ε, from lt_of_le_of_lt this Hk))

open set

private definition exists_is_sup {X : set ℝ} (H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b)) :
  ∃ y, is_sup X y :=
let x := some (and.left H), b := some (and.right H) in
  exists_is_sup_of_inh_of_bdd X x (some_spec (and.left H)) b (some_spec (and.right H))

private definition sup_aux {X : set ℝ} (H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b)) :=
some (exists_is_sup H)

private definition sup_aux_spec {X : set ℝ} (H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b)) :
  is_sup X (sup_aux H) :=
some_spec (exists_is_sup H)

definition sup (X : set ℝ) : ℝ :=
if H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b) then sup_aux H else 0

proposition le_sup {x : ℝ} {X : set ℝ} (Hx : x ∈ X) {b : ℝ} (Hb : ∀ x, x ∈ X → x ≤ b) :
  x ≤ sup X :=
have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b),
  from and.intro (exists.intro x Hx) (exists.intro b Hb),
by+ rewrite [↑sup, dif_pos H]; exact and.left (sup_aux_spec H) x Hx

proposition sup_le {X : set ℝ} (HX : ∃ x, x ∈ X) {b : ℝ} (Hb : ∀ x, x ∈ X → x ≤ b) :
  sup X ≤ b :=
have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → x ≤ b),
  from and.intro HX (exists.intro b Hb),
by+ rewrite [↑sup, dif_pos H]; exact and.right (sup_aux_spec H) b Hb

private definition exists_is_inf {X : set ℝ} (H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x)) :
  ∃ y, is_inf X y :=
let x := some (and.left H), b := some (and.right H) in
  exists_is_inf_of_inh_of_bdd X x (some_spec (and.left H)) b (some_spec (and.right H))

private definition inf_aux {X : set ℝ} (H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x)) :=
some (exists_is_inf H)

private definition inf_aux_spec {X : set ℝ} (H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x)) :
  is_inf X (inf_aux H) :=
some_spec (exists_is_inf H)

definition inf (X : set ℝ) : ℝ :=
if H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x) then inf_aux H else 0

proposition inf_le {x : ℝ} {X : set ℝ} (Hx : x ∈ X) {b : ℝ} (Hb : ∀ x, x ∈ X → b ≤ x) :
  inf X ≤ x :=
have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x),
  from and.intro (exists.intro x Hx) (exists.intro b Hb),
by+ rewrite [↑inf, dif_pos H]; exact and.left (inf_aux_spec H) x Hx

proposition le_inf {X : set ℝ} (HX : ∃ x, x ∈ X) {b : ℝ} (Hb : ∀ x, x ∈ X → b ≤ x) :
  b ≤ inf X :=
have H : (∃ x, x ∈ X) ∧ (∃ b, ∀ x, x ∈ X → b ≤ x),
  from and.intro HX (exists.intro b Hb),
by+ rewrite [↑inf, dif_pos H]; exact and.right (inf_aux_spec H) b Hb

end

section inter_val
open set

-- this definition should be inherited from metric_space once a migrate is done.
definition continuous (f : ℝ → ℝ) :=
  ∀ x : ℝ, ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ, abs (x - x') < δ → abs (f x - f x') < ε

private definition inter_sup (a b : ℝ) (f : ℝ → ℝ) := sup {x | x < b ∧ f x < 0}

private theorem add_midpoint {a b : ℝ} (H : a < b) : a + (b - a) / 2 < b :=
  begin
    rewrite [-div_sub_div_same, sub_eq_add_neg, {b / 2 + _}add.comm, -add.assoc, -sub_eq_add_neg],
    apply add_lt_of_lt_sub_right,
    krewrite *sub_self_div_two,
    apply div_lt_div_of_lt_of_pos H two_pos
  end

section
parameters {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b) (Ha : f a < 0) (Hb : f b > 0)
include Hf Ha Hb Hab

private theorem Hinh : ∃ x, x ∈ {x | x < b ∧ f x < 0} := exists.intro a (and.intro Hab Ha)

private theorem Hmem : ∀ x, x ∈ {x | x < b ∧ f x < 0} → x ≤ b := λ x Hx, le_of_lt (and.left Hx)

private theorem Hsupleb : inter_sup a b f ≤ b := sup_le (Hinh) Hmem

private theorem sup_fn_interval_aux1 : f (inter_sup a b f) ≥ 0 :=
  have ¬ f (inter_sup a b f) < 0, from
  (suppose f (inter_sup a b f) < 0,
   have ∃ δ, δ > (0 : ℝ) ∧ inter_sup a b f + δ < b ∧ f (inter_sup a b f + δ) < 0, begin
     let Hcont := Hf (inter_sup a b f) (neg_pos_of_neg this),
     cases Hcont with δ Hδ,
     cases em (inter_sup a b f + δ / 2 < b),
     existsi δ / 2,
     split,
     apply div_pos_of_pos_of_pos (and.left Hδ) two_pos,
     split,
     assumption,
     have Habs : abs (inter_sup a b f - (inter_sup a b f + δ / 2)) < δ, begin
       krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
               abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
       apply div_two_lt_of_pos (and.left Hδ)
     end,
     let Hlt := and.right Hδ _ Habs,
     let Hlt' := sub_lt_of_abs_sub_lt_left Hlt,
     let Hlt'' := lt_add_of_sub_lt_right _ _ _ Hlt',
     rewrite [-sub_eq_add_neg at Hlt'', sub_self at Hlt''],
     assumption,
     let Hble := le_of_not_gt a_1,
     have Habs : abs (inter_sup a b f - b) < δ, begin
       apply abs_lt_of_lt_of_neg_lt,
       apply sub_lt_left_of_lt_add,
       apply lt_of_le_of_lt,
       apply Hsupleb,
       apply lt_add_of_pos_right (and.left Hδ),
       rewrite neg_sub,
       apply sub_lt_left_of_lt_add,
       apply lt_of_le_of_lt,
       apply Hble,
       apply add_lt_add_left,
       apply div_two_lt_of_pos (and.left Hδ)
     end,
     let Hlt := and.right Hδ _ Habs,
     let Hlt' := sub_lt_of_abs_sub_lt_left Hlt,
     let Hlt'' := lt_add_of_sub_lt_right _ _ _ Hlt',
     rewrite [-sub_eq_add_neg at Hlt'', sub_self at Hlt''],
     apply absurd Hb (not_lt_of_ge (le_of_lt Hlt''))
   end,
   obtain δ Hδ, from this,
   have inter_sup a b f + δ ∈ {x | x < b ∧ f x < 0},
     from and.intro (and.left (and.right Hδ)) (and.right (and.right Hδ)),
   have Hle : ¬ inter_sup a b f < inter_sup a b f + δ,
     from not_lt_of_ge (le_sup this Hmem),
   show false, from Hle (lt_add_of_pos_right (and.left Hδ))),
  le_of_not_gt this

private theorem sup_fn_interval_aux2 : f (inter_sup a b f) ≤ 0 :=
  have ¬ f (inter_sup a b f) > 0, from
  (suppose f (inter_sup a b f) > 0,
    have ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ, abs (inter_sup a b f - x') < δ → f x' > 0, begin
      let Hcont := Hf (inter_sup a b f) this,
      cases Hcont with δ Hδ,
      existsi δ,
      split,
      exact and.left Hδ,
      intro x Hx,
      let Hlt := and.right Hδ _ Hx,
      let Hlt' := sub_lt_of_abs_sub_lt_right Hlt,
      rewrite sub_self at Hlt',
      exact Hlt'
    end,
    obtain δ Hδ, from this,
    have ∀ x, x ∈ {x | x < b ∧ f x < 0} → x ≤ inter_sup a b f - δ / 2, from
     (take x, suppose x ∈ {x | x < b ∧ f x < 0},
      have x ≤ inter_sup a b f, from le_sup this Hmem,
      have ¬ x > inter_sup a b f - δ / 2, from
        (assume Hngt,
         have abs (inter_sup a b f - x) < δ, begin
           apply abs_lt_of_lt_of_neg_lt,
           apply sub_lt_of_sub_lt,
           apply gt.trans,
           exact Hngt,
           apply sub_lt_sub_left,
           exact div_two_lt_of_pos (and.left Hδ),
           rewrite neg_sub,
           apply lt_of_le_of_lt,
           rotate 1,
           apply and.left Hδ,
           apply sub_nonpos_of_le,
           apply le_sup,
           exact this,
           exact Hmem
         end,
         have f x > 0, from and.right Hδ _ this,
         show false, from (not_lt_of_gt this) (and.right `x ∈ {x | x < b ∧ f x < 0}`)),
      le_of_not_gt this),
    have Hle : inter_sup a b f ≤ inter_sup a b f - δ / 2, from sup_le Hinh this,
    show false, from not_le_of_gt
                     (sub_lt_of_pos _ (div_pos_of_pos_of_pos (and.left Hδ) (two_pos))) Hle),
  le_of_not_gt this

private theorem sup_fn_interval : f (inter_sup a b f) = 0 :=
  eq_of_le_of_ge sup_fn_interval_aux2 sup_fn_interval_aux1

private theorem intermediate_value_incr_aux1 : ∃ δ : ℝ, δ > 0 ∧ ∀ y, abs (b - y) < δ → f y > 0 :=
  begin
    let Hcont := Hf b Hb,
    cases Hcont with δ Hδ,
    existsi δ,
    split,
    exact and.left Hδ,
    intro y Hy,
    let Hy' := and.right Hδ _ Hy,
    let Hlt := sub_lt_of_abs_sub_lt_right Hy',
    rewrite sub_self at Hlt,
    assumption
  end

private theorem intermediate_value_incr_aux2 : ∃ δ : ℝ, δ > 0 ∧ a + δ < b ∧ f (a + δ) < 0 :=
  begin
    have Hnfa : - (f a) > 0, from neg_pos_of_neg Ha,
    let Hcont := Hf a Hnfa,
    cases Hcont with δ Hδ,
    cases em (a + δ < b) with Haδ Haδ,
    existsi δ / 2,
    split,
    exact div_pos_of_pos_of_pos (and.left Hδ) two_pos,
    split,
    apply lt.trans,
    apply add_lt_add_left,
    exact div_two_lt_of_pos (and.left Hδ),
    assumption,
    have Habs : abs (a - (a + δ / 2)) < δ, begin
      krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
               abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
      exact div_two_lt_of_pos (and.left Hδ)
    end,
    let Hlt := and.right Hδ _ Habs,
    let Hlt' := sub_lt_of_abs_sub_lt_left Hlt,
    let Hlt'' := lt_add_of_sub_lt_right _ _ _ Hlt',
    rewrite [-sub_eq_add_neg at Hlt'', sub_self at Hlt''],
    assumption,
    existsi (b - a) / 2,
    split,
    apply div_pos_of_pos_of_pos,
    exact sub_pos_of_lt _ _ Hab,
    exact two_pos,
    split,
    apply add_midpoint Hab,
    have Habs : abs (a - (a + (b - a) / 2)) < δ, begin
      krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
                abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt _ _ Hab) two_pos)],
      apply lt_of_lt_of_le,
      apply div_two_lt_of_pos (sub_pos_of_lt _ _ Hab),
      apply sub_left_le_of_le_add,
      apply le_of_not_gt Haδ
    end,
    let Hlt := and.right Hδ _ Habs,
    let Hlt' := sub_lt_of_abs_sub_lt_left Hlt,
    let Hlt'' := lt_add_of_sub_lt_right _ _ _ Hlt',
    rewrite [-sub_eq_add_neg at Hlt'', sub_self at Hlt''],
    assumption
  end

theorem intermediate_value_incr : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
  begin
    existsi inter_sup a b f,
    split,
    cases intermediate_value_incr_aux2 with δ Hδ,
    apply lt_of_lt_of_le,
    apply lt_add_of_pos_right,
    exact and.left Hδ,
    apply le_sup,
    exact and.right Hδ,
    intro x Hx,
    apply le_of_lt,
    exact and.left Hx,
    split,
    cases intermediate_value_incr_aux1 with δ Hδ,
    apply lt_of_le_of_lt,
    rotate 1,
    apply sub_lt_of_pos,
    exact and.left Hδ,
    exact sup_fn_interval,
    apply sup_le,
    exact exists.intro a (and.intro Hab Ha),
    intro x Hx,
    apply le_of_not_gt,
    intro Hxgt,
    have Hxgt' : b - x < δ, from sub_lt_of_sub_lt _ _ _ Hxgt,
    rewrite -(abs_of_pos (sub_pos_of_lt _ _ (and.left Hx))) at Hxgt',
    let Hxgt'' := and.right Hδ _ Hxgt',
    exact not_lt_of_ge (le_of_lt Hxgt'') (and.right Hx)
  end

end

private definition neg_f (f : ℝ → ℝ) := λ x, - f x

private theorem neg_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) : continuous (neg_f f) :=
  begin
    intros x ε Hε,
    cases Hcon x Hε with δ Hδ,
    cases Hδ with Hδ₁ Hδ₂,
    existsi δ,
    split,
    assumption,
    intros x' Hx',
    let HD := Hδ₂ x' Hx',
    rewrite [-abs_neg, ↑neg_f, neg_neg_sub_neg],
    assumption
  end

theorem intermediate_value_decr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b)
        (Ha : f a > 0) (Hb : f b < 0) : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
  begin
    have Ha' : (neg_f f) a < 0, from neg_neg_of_pos Ha,
    have Hb' : (neg_f f) b > 0, from neg_pos_of_neg Hb,
    have Hcon : continuous (neg_f f), from neg_continuous_of_continuous Hf,
    let Hiv := intermediate_value_incr Hcon Hab Ha' Hb',
    cases Hiv with c Hc,
    existsi c,
    split,
    exact and.left Hc,
    split,
    exact and.left (and.right Hc),
    apply eq_zero_of_neg_eq_zero,
    apply and.right (and.right Hc)
  end

end inter_val

/-
proposition converges_to_at_unique {f : M → N} {y₁ y₂ : N} {x : M}
  (H₁ : f ⟶ y₁ '[at x]) (H₂ : f ⟶ y₂ '[at x]) : y₁ = y₂ :=
eq_of_forall_dist_le
  (take ε, suppose ε > 0,
    have e2pos : ε / 2 > 0, from  div_pos_of_pos_of_pos `ε > 0` two_pos,
    obtain δ₁ [(δ₁pos : δ₁ > 0) (Hδ₁ : ∀ x', x ≠ x' ∧ dist x x' < δ₁ → dist (f x') y₁ < ε / 2)],
      from H₁ e2pos,
    obtain δ₂ [(δ₂pos : δ₂ > 0) (Hδ₂ : ∀ x', x ≠ x' ∧ dist x x' < δ₂ → dist (f x') y₂ < ε / 2)],
      from H₂ e2pos,
    let δ := min δ₁ δ₂ in
    have δ > 0, from lt_min δ₁pos δ₂pos,

-/

end real