lean2/library/data/real/complete.lean

/-
Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Robert Y. Lewis
The real numbers, constructed as equivalence classes of Cauchy sequences of rationals.
This construction follows Bishop and Bridges (1985).

At this point, we no longer proceed constructively: this file makes heavy use of decidability,
 excluded middle, and Hilbert choice.

Here, we show that ℝ is complete.
-/

import data.real.basic data.real.order data.real.division data.rat data.nat data.pnat
import logic.choice
open -[coercions] rat
local notation 0 := rat.of_num 0
local notation 1 := rat.of_num 1
open -[coercions] nat
open eq.ops
open pnat

local notation 2 := subtype.tag (nat.of_num 2) dec_trivial
local notation 3 := subtype.tag (nat.of_num 3) dec_trivial

namespace s

theorem rat_approx_l1 {s : seq} (H : regular s) :
        ∀ n : ℕ+, ∃ q : ℚ, ∃ N : ℕ+, ∀ m : ℕ+, m ≥ N → abs (s m - q) ≤ n⁻¹ :=
  begin
    intro n,
    existsi (s (2 * n)),
    existsi 2 * n,
    intro m Hm,
    apply rat.le.trans,
    apply H,
    rewrite -(add_halves n),
    apply rat.add_le_add_right,
    apply inv_ge_of_le Hm
  end

theorem rat_approx {s : seq} (H : regular s) :
        ∀ n : ℕ+, ∃ q : ℚ, s_le (s_abs (sadd s (sneg (const q)))) (const n⁻¹) :=
  begin
    intro m,
    rewrite ↑s_le,
    apply exists.elim (rat_approx_l1 H m),
    intro q Hq,
    apply exists.elim Hq,
    intro N HN,
    existsi q,
    apply nonneg_of_bdd_within,
    repeat (apply reg_add_reg | apply reg_neg_reg | apply abs_reg_of_reg | apply const_reg
             | assumption),
    intro n,
    existsi N,
    intro p Hp,
    rewrite ↑[sadd, sneg, s_abs, const],
    apply rat.le.trans,
    rotate 1,
    apply rat.sub_le_sub_left,
    apply HN,
    apply pnat.le.trans,
    apply Hp,
    rewrite -*pnat.mul.assoc,
    apply pnat.mul_le_mul_left,
    rewrite [sub_self, -neg_zero],
    apply neg_le_neg,
    apply rat.le_of_lt,
    apply inv_pos
  end

definition r_abs (s : reg_seq) : reg_seq :=
  reg_seq.mk (s_abs (reg_seq.sq s)) (abs_reg_of_reg (reg_seq.is_reg s))

theorem abs_well_defined {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s ≡ t) :
        s_abs s ≡ s_abs t :=
  begin
    rewrite [↑equiv at *],
    intro n,
    rewrite ↑s_abs,
    apply rat.le.trans,
    apply abs_abs_sub_abs_le_abs_sub,
    apply Heq
  end

theorem r_abs_well_defined {s t : reg_seq} (H : requiv s t) : requiv (r_abs s) (r_abs t) :=
  abs_well_defined (reg_seq.is_reg s) (reg_seq.is_reg t) H

theorem r_rat_approx (s : reg_seq) :
        ∀ n : ℕ+, ∃ q : ℚ, r_le (r_abs (radd s (rneg (r_const q)))) (r_const n⁻¹) :=
  rat_approx (reg_seq.is_reg s)

theorem const_bound {s : seq} (Hs : regular s) (n : ℕ+) :
        s_le (s_abs (sadd s (sneg (const (s n))))) (const n⁻¹) :=
  begin
    rewrite ↑[s_le, nonneg, s_abs, sadd, sneg, const],
    intro m,
    apply iff.mp !rat.le_add_iff_neg_le_sub_left,
    apply rat.le.trans,
    apply Hs,
    apply rat.add_le_add_right,
    rewrite -*pnat.mul.assoc,
    apply inv_ge_of_le,
    apply pnat.mul_le_mul_left
  end

theorem abs_const (a : ℚ) : const (abs a) ≡ s_abs (const a) :=
  begin
    rewrite [↑s_abs, ↑const],
    apply equiv.refl
  end

theorem r_abs_const (a : ℚ) : requiv (r_const (abs a) ) (r_abs (r_const a)) := abs_const a

theorem equiv_abs_of_ge_zero {s : seq} (Hs : regular s) (Hz : s_le zero s) : s_abs s ≡ s :=
  begin
    apply eq_of_bdd,
    apply abs_reg_of_reg Hs,
    apply Hs,
    intro j,
    rewrite ↑s_abs,
    let Hz' := s_nonneg_of_ge_zero Hs Hz,
    existsi 2 * j,
    intro n Hn,
    apply or.elim (em (s n ≥ 0)),
    intro Hpos,
    rewrite [rat.abs_of_nonneg Hpos, sub_self, abs_zero],
    apply rat.le_of_lt,
    apply inv_pos,
    intro Hneg,
    let Hneg' := lt_of_not_ge Hneg,
    have Hsn : -s n - s n > 0, from add_pos (neg_pos_of_neg Hneg') (neg_pos_of_neg Hneg'),
    rewrite [rat.abs_of_neg Hneg', rat.abs_of_pos Hsn],
    apply rat.le.trans,
    apply rat.add_le_add,
    repeat (apply rat.neg_le_neg; apply Hz'),
    rewrite *rat.neg_neg,
    apply rat.le.trans,
    apply rat.add_le_add,
    repeat (apply inv_ge_of_le; apply Hn),
    rewrite pnat.add_halves,
    apply rat.le.refl
  end

theorem equiv_neg_abs_of_le_zero {s : seq} (Hs : regular s) (Hz : s_le s zero) : s_abs s ≡ sneg s :=
  begin
    apply eq_of_bdd,
    apply abs_reg_of_reg Hs,
    apply reg_neg_reg Hs,
    intro j,
    rewrite [↑s_abs, ↑s_le at Hz],
    have Hz' : nonneg (sneg s), begin
      apply nonneg_of_nonneg_equiv,
      rotate 3,
      apply Hz,
      rotate 2,
      apply s_zero_add,
      repeat (apply Hs | apply zero_is_reg | apply reg_neg_reg | apply reg_add_reg)
    end,
    existsi 2 * j,
    intro n Hn,
    apply or.elim (em (s n ≥ 0)),
    intro Hpos,
    have Hsn : s n + s n ≥ 0, from add_nonneg Hpos Hpos,
    rewrite [rat.abs_of_nonneg Hpos, ↑sneg, rat.sub_neg_eq_add, rat.abs_of_nonneg Hsn],
    rewrite [↑nonneg at Hz', ↑sneg at Hz'],
    apply rat.le.trans,
    apply rat.add_le_add,
    repeat apply (rat.le_of_neg_le_neg !Hz'),
    apply rat.le.trans,
    apply rat.add_le_add,
    repeat (apply inv_ge_of_le; apply Hn),
    rewrite pnat.add_halves,
    apply rat.le.refl,
    intro Hneg,
    let Hneg' := lt_of_not_ge Hneg,
    rewrite [rat.abs_of_neg Hneg', ↑sneg, rat.sub_neg_eq_add, rat.neg_add_eq_sub, rat.sub_self,
                abs_zero],
    apply rat.le_of_lt,
    apply inv_pos
  end

theorem r_equiv_abs_of_ge_zero {s : reg_seq} (Hz : r_le r_zero s) : requiv (r_abs s) s :=
  equiv_abs_of_ge_zero (reg_seq.is_reg s) Hz

theorem r_equiv_neg_abs_of_le_zero {s : reg_seq} (Hz : r_le s r_zero) : requiv (r_abs s) (-s) :=
  equiv_neg_abs_of_le_zero (reg_seq.is_reg s) Hz

end s

namespace real
open [classes] s

theorem p_add_fractions (n : ℕ+) : (2 * n)⁻¹ + (2 * 3 * n)⁻¹ + (3 * n)⁻¹ = n⁻¹ :=
  assert T : 2⁻¹ + 2⁻¹ * 3⁻¹ + 3⁻¹ = 1, from dec_trivial,
  by rewrite[*inv_mul_eq_mul_inv,-*rat.right_distrib,T,rat.one_mul]

theorem rewrite_helper9 (a b c : ℝ) : b - c = (b - a) - (c - a) :=
  by rewrite[-sub_add_eq_sub_sub_swap,sub_add_cancel]

theorem rewrite_helper10 (a b c d : ℝ) : c - d = (c - a) + (a - b) + (b - d) :=
  by rewrite[*add_sub,*sub_add_cancel]

noncomputable definition rep (x : ℝ) : s.reg_seq := some (quot.exists_rep x)

definition re_abs (x : ℝ) : ℝ :=
  quot.lift_on x (λ a, quot.mk (s.r_abs a)) (take a b Hab, quot.sound (s.r_abs_well_defined Hab))

theorem r_abs_nonneg {x : ℝ} : zero ≤ x → re_abs x = x :=
  quot.induction_on x (λ a Ha, quot.sound  (s.r_equiv_abs_of_ge_zero Ha))

theorem r_abs_nonpos {x : ℝ} : x ≤ zero → re_abs x = -x :=
  quot.induction_on x (λ a Ha, quot.sound (s.r_equiv_neg_abs_of_le_zero Ha))

theorem abs_const' (a : ℚ) : of_rat (rat.abs a) = re_abs (of_rat a) := quot.sound (s.r_abs_const a)

theorem re_abs_is_abs : re_abs = real.abs := funext
  (begin
    intro x,
    apply eq.symm,
    let Hor := em (zero ≤ x),
    apply or.elim Hor,
    intro Hor1,
    rewrite [abs_of_nonneg Hor1, r_abs_nonneg Hor1],
    intro Hor2,
    have Hor2' : x ≤ zero, from le_of_lt (lt_of_not_ge Hor2),
    rewrite [abs_of_neg (lt_of_not_ge Hor2), r_abs_nonpos Hor2']
  end)

theorem abs_const (a : ℚ) : of_rat (rat.abs a) = abs (of_rat a) :=
  by rewrite -re_abs_is_abs -- ????

theorem rat_approx' (x : ℝ) : ∀ n : ℕ+, ∃ q : ℚ, re_abs (x - of_rat q) ≤ of_rat n⁻¹ :=
  quot.induction_on x (λ s n, s.r_rat_approx s n)

theorem rat_approx (x : ℝ) : ∀ n : ℕ+, ∃ q : ℚ, abs (x - of_rat q) ≤ of_rat n⁻¹ :=
  by rewrite -re_abs_is_abs; apply rat_approx'

noncomputable definition approx (x : ℝ) (n : ℕ+) := some (rat_approx x n)

theorem approx_spec (x : ℝ) (n : ℕ+) : abs (x - (of_rat (approx x n))) ≤ of_rat n⁻¹ :=
  some_spec (rat_approx x n)

theorem approx_spec' (x : ℝ) (n : ℕ+) : abs ((of_rat (approx x n)) - x) ≤ of_rat n⁻¹ :=
  by rewrite abs_sub; apply approx_spec

notation `r_seq` := ℕ+ → ℝ

noncomputable definition converges_to (X : r_seq) (a : ℝ) (N : ℕ+ → ℕ+) :=
  ∀ k : ℕ+, ∀ n : ℕ+, n ≥ N k → abs (X n - a) ≤ of_rat k⁻¹

noncomputable definition cauchy (X : r_seq) (M : ℕ+ → ℕ+) :=
  ∀ k : ℕ+, ∀ m n : ℕ+, m ≥ M k → n ≥ M k → abs (X m - X n) ≤ of_rat k⁻¹

theorem cauchy_of_converges_to {X : r_seq} {a : ℝ} {N : ℕ+ → ℕ+} (Hc : converges_to X a N) :
        cauchy X (λ k, N (2 * k)) :=
  begin
    intro k m n Hm Hn,
    rewrite (rewrite_helper9 a),
    apply le.trans,
    apply abs_add_le_abs_add_abs,
    apply le.trans,
    apply add_le_add,
    apply Hc,
    apply Hm,
    krewrite abs_neg,
    apply Hc,
    apply Hn,
    xrewrite of_rat_add,
    apply of_rat_le_of_rat_of_le,
    rewrite pnat.add_halves,
    apply rat.le.refl
  end

definition Nb (M : ℕ+ → ℕ+) := λ k, pnat.max (3 * k) (M (2 * k))

theorem Nb_spec_right (M : ℕ+ → ℕ+) (k : ℕ+) : M (2 * k) ≤ Nb M k := !max_right

theorem Nb_spec_left (M : ℕ+ → ℕ+) (k : ℕ+) : 3 * k ≤ Nb M k := !max_left

noncomputable definition lim_seq {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) : ℕ+ → ℚ :=
  λ k, approx (X (Nb M k)) (2 * k)

theorem lim_seq_reg_helper {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) {m n : ℕ+}
        (Hmn : M (2 * n) ≤M (2 * m)) :
           abs (of_rat (lim_seq Hc m) - X (Nb M m)) + abs (X (Nb M m) - X (Nb M n)) + abs
            (X (Nb M n) - of_rat (lim_seq Hc n)) ≤ of_rat (m⁻¹ + n⁻¹) :=
  begin
    apply le.trans,
    apply add_le_add_three,
    apply approx_spec',
    rotate 1,
    apply approx_spec,
    rotate 1,
    apply Hc,
    rotate 1,
    apply Nb_spec_right,
    rotate 1,
    apply pnat.le.trans,
    apply Hmn,
    apply Nb_spec_right,
    rewrite [*of_rat_add, rat.add.assoc, pnat.add_halves],
    apply of_rat_le_of_rat_of_le,
    apply rat.add_le_add_right,
    apply inv_ge_of_le,
    apply pnat.mul_le_mul_left
  end

theorem lim_seq_reg {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) : s.regular (lim_seq Hc) :=
  begin
    rewrite ↑s.regular,
    intro m n,
    apply le_of_rat_le_of_rat,
    rewrite [abs_const, -of_rat_sub, (rewrite_helper10 (X (Nb M m)) (X (Nb M n)))],
    apply real.le.trans,
    apply abs_add_three,
    let Hor := em (M (2 * m) ≥ M (2 * n)),
    apply or.elim Hor,
    intro Hor1,
    apply lim_seq_reg_helper Hc Hor1,
    intro Hor2,
    let Hor2' := pnat.le_of_lt (pnat.lt_of_not_le Hor2),
    rewrite [real.abs_sub (X (Nb M n)), abs_sub (X (Nb M m)), abs_sub, -- ???
             rat.add.comm, add_comm_three],
    apply lim_seq_reg_helper Hc Hor2'
  end

theorem lim_seq_spec {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) (k : ℕ+) :
        s.s_le (s.s_abs (s.sadd (lim_seq Hc) (s.sneg (s.const (lim_seq Hc k))) )) (s.const k⁻¹) :=
  begin
    apply s.const_bound,
    apply lim_seq_reg
  end

noncomputable definition r_lim_seq {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) : s.reg_seq :=
  s.reg_seq.mk (lim_seq Hc) (lim_seq_reg Hc)

theorem r_lim_seq_spec {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) (k : ℕ+) :
        s.r_le (s.r_abs (( s.radd (r_lim_seq Hc) (s.rneg (s.r_const ((s.reg_seq.sq (r_lim_seq Hc)) k)))))) (s.r_const (k)⁻¹) :=
  lim_seq_spec Hc k

noncomputable definition lim {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) : ℝ :=
  quot.mk (r_lim_seq Hc)

theorem re_lim_spec {x : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy x M) (k : ℕ+) :
        re_abs ((lim Hc) - (of_rat ((lim_seq Hc) k))) ≤ of_rat k⁻¹ :=
  r_lim_seq_spec Hc k

theorem lim_spec' {x : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy x M) (k : ℕ+) :
        abs ((lim Hc) - (of_rat ((lim_seq Hc) k))) ≤ of_rat k⁻¹ :=
  by rewrite -re_abs_is_abs; apply re_lim_spec

theorem lim_spec {x : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy x M) (k : ℕ+) :
        abs ((of_rat ((lim_seq Hc) k)) - (lim Hc)) ≤ of_rat (k)⁻¹ :=
  by rewrite abs_sub; apply lim_spec'

theorem converges_of_cauchy {X : r_seq} {M : ℕ+ → ℕ+} (Hc : cauchy X M) :
        converges_to X (lim Hc) (Nb M) :=
  begin
    intro k n Hn,
    rewrite (rewrite_helper10 (X (Nb M n)) (of_rat (lim_seq Hc n))),
    apply le.trans,
    apply abs_add_three,
    apply le.trans,
    apply add_le_add_three,
    apply Hc,
    apply pnat.le.trans,
    rotate 1,
    apply Hn,
    rotate_right 1,
    apply Nb_spec_right,
    have HMk : M (2 * k) ≤ Nb M n, begin
      apply pnat.le.trans,
      apply Nb_spec_right,
      apply pnat.le.trans,
      apply Hn,
      apply pnat.le.trans,
      apply mul_le_mul_left 3,
      apply Nb_spec_left
    end,
    apply HMk,
    rewrite ↑lim_seq,
    apply approx_spec,
    apply lim_spec,
    rewrite 2 of_rat_add,
    apply of_rat_le_of_rat_of_le,
    apply rat.le.trans,
    apply rat.add_le_add_three,
    apply rat.le.refl,
    apply inv_ge_of_le,
    apply pnat_mul_le_mul_left',
    apply pnat.le.trans,
    rotate 1,
    apply Hn,
    rotate_right 1,
    apply Nb_spec_left,
    apply inv_ge_of_le,
    apply pnat.le.trans,
    rotate 1,
    apply Hn,
    rotate_right 1,
    apply Nb_spec_left,
    rewrite [-*pnat.mul.assoc, p_add_fractions],
    apply rat.le.refl
  end

-------------------------------------------
-- int embedding theorems
-- archimedean properties, integer floor and ceiling

section ints

open int

theorem archimedean (x : ℝ) : ∃ z : ℤ, x ≤ of_rat (of_int z) :=
  begin
    apply quot.induction_on x,
    intro s,
    cases (s.bdd_of_regular (s.reg_seq.is_reg s)) with [b, Hb],
    existsi (ubound b),
    have H : s.s_le (s.reg_seq.sq s) (s.const (rat.of_nat (ubound b))), begin
      apply s.s_le_of_le_pointwise (s.reg_seq.is_reg s),
      apply s.const_reg,
      intro n,
      apply rat.le.trans,
      apply Hb,
      apply ubound_ge
    end,
    apply H
  end

theorem archimedean_strict (x : ℝ) : ∃ z : ℤ, x < of_rat (of_int z) :=
  begin
    cases archimedean x with [z, Hz],
    existsi z + 1,
    apply lt_of_le_of_lt,
    apply Hz,
    apply of_rat_lt_of_rat_of_lt,
    apply iff.mpr !of_int_lt_of_int,
    apply int.lt_add_of_pos_right,
    apply dec_trivial
  end

theorem archimedean' (x : ℝ) : ∃ z : ℤ, x ≥ of_rat (of_int z) :=
  begin
    cases archimedean (-x) with [z, Hz],
    existsi -z,
    rewrite [of_int_neg, -of_rat_neg], -- change the direction of of_rat_neg
    apply iff.mp !neg_le_iff_neg_le Hz
  end

theorem archimedean_strict' (x : ℝ) : ∃ z : ℤ, x > of_rat (of_int z) :=
  begin
    cases archimedean_strict (-x) with [z, Hz],
    existsi -z,
    rewrite [of_int_neg, -of_rat_neg],
    apply iff.mp !neg_lt_iff_neg_lt Hz
  end

theorem ex_smallest_of_bdd {P : ℤ → Prop} (Hbdd : ∃ b : ℤ, ∀ z : ℤ, z ≤ b → ¬ P z)
        (Hinh : ∃ z : ℤ, P z) : ∃ lb : ℤ, P lb ∧ (∀ z : ℤ, z < lb → ¬ P z) :=
  begin
    cases Hbdd with [b, Hb],
    cases Hinh with [elt, Helt],
    existsi b + of_nat (least (λ n, P (b + of_nat n)) (succ (nat_abs (elt - b)))),
    have Heltb : elt > b, begin
      apply int.lt_of_not_ge,
      intro Hge,
      apply false.elim ((Hb _ Hge) Helt)
    end,
    have H' : P (b + of_nat (nat_abs (elt - b))), begin
      rewrite [of_nat_nat_abs_of_nonneg (int.le_of_lt (iff.mpr !int.sub_pos_iff_lt Heltb)),
              int.add.comm, int.sub_add_cancel],
      apply Helt
    end,
    apply and.intro,
    apply least_of_lt _ !lt_succ_self H',
    intros z Hz,
    cases (em (z ≤ b)) with [Hzb, Hzb],
    apply Hb _ Hzb,
    let Hzb' := int.lt_of_not_ge Hzb,
    let Hpos := iff.mpr !int.sub_pos_iff_lt Hzb',
    have Hzbk : z = b + of_nat (nat_abs (z - b)),
      by rewrite [of_nat_nat_abs_of_nonneg (int.le_of_lt Hpos), int.add.comm, int.sub_add_cancel],
    have Hk : nat_abs (z - b) < least (λ n, P (b + of_nat n)) (succ (nat_abs (elt - b))), begin
     let Hz' := iff.mp !int.lt_add_iff_sub_lt_left Hz,
     rewrite [-of_nat_nat_abs_of_nonneg (int.le_of_lt Hpos) at Hz'],
     apply iff.mp !int.of_nat_lt_of_nat Hz'
    end,
    let Hk' := nat.not_le_of_gt Hk,
    rewrite Hzbk,
    apply λ p, mt (ge_least_of_lt _ p) Hk',
    apply nat.lt.trans Hk,
    apply least_lt _ !lt_succ_self H'
  end

theorem ex_largest_of_bdd {P : ℤ → Prop} (Hbdd : ∃ b : ℤ, ∀ z : ℤ, z ≥ b → ¬ P z)
        (Hinh : ∃ z : ℤ, P z) : ∃ ub : ℤ, P ub ∧ (∀ z : ℤ, z > ub → ¬ P z) :=
  begin
    cases Hbdd with [b, Hb],
    cases Hinh with [elt, Helt],
    existsi b - of_nat (least (λ n, P (b - of_nat n)) (succ (nat_abs (b - elt)))),
    have Heltb : elt < b, begin
      apply int.lt_of_not_ge,
      intro Hge,
      apply false.elim ((Hb _ Hge) Helt)
    end,
    have H' : P (b - of_nat (nat_abs (b - elt))), begin
      rewrite [of_nat_nat_abs_of_nonneg (int.le_of_lt (iff.mpr !int.sub_pos_iff_lt Heltb)),
              int.sub_sub_self],
      apply Helt
    end,
    apply and.intro,
    apply least_of_lt _ !lt_succ_self H',
    intros z Hz,
    cases (em (z ≥ b)) with [Hzb, Hzb],
    apply Hb _ Hzb,
    let Hzb' := int.lt_of_not_ge Hzb,
    let Hpos := iff.mpr !int.sub_pos_iff_lt Hzb',
    have Hzbk : z = b - of_nat (nat_abs (b - z)),
      by rewrite [of_nat_nat_abs_of_nonneg (int.le_of_lt Hpos), int.sub_sub_self],
    have Hk : nat_abs (b - z) < least (λ n, P (b - of_nat n)) (succ (nat_abs (b - elt))), begin
      let Hz' := iff.mp !int.lt_add_iff_sub_lt_left (iff.mpr !int.lt_add_iff_sub_lt_right Hz),
      rewrite [-of_nat_nat_abs_of_nonneg (int.le_of_lt Hpos) at Hz'],
      apply iff.mp !int.of_nat_lt_of_nat Hz'
    end,
    let Hk' := nat.not_le_of_gt Hk,
    rewrite Hzbk,
    apply λ p, mt (ge_least_of_lt _ p) Hk',
    apply nat.lt.trans Hk,
    apply least_lt _ !lt_succ_self H'
  end

definition ex_floor (x : ℝ) :=
  (@ex_largest_of_bdd (λ z, x ≥ of_rat (of_int z))
    (begin
      existsi (some (archimedean_strict x)),
      let Har := some_spec (archimedean_strict x),
      intros z Hz,
      apply not_le_of_gt,
      apply lt_of_lt_of_le,
      apply Har,
      have H : of_rat (of_int (some (archimedean_strict x))) ≤ of_rat (of_int z), begin
        apply of_rat_le_of_rat_of_le,
        apply iff.mpr !of_int_le_of_int,
        apply Hz
      end,
      exact H
    end)
    (begin
      existsi some (archimedean' x),
      apply some_spec (archimedean' x)
    end))

noncomputable definition floor (x : ℝ) : ℤ :=
  some (ex_floor x)

noncomputable definition ceil (x : ℝ) : ℤ := - floor (-x)

theorem floor_spec (x : ℝ) : of_rat (of_int (floor x)) ≤ x :=
  and.left (some_spec (ex_floor x))

theorem floor_largest {x : ℝ} {z : ℤ} (Hz : z > floor x) : x < of_rat (of_int z) :=
  begin
    apply lt_of_not_ge,
    cases some_spec (ex_floor x),
    apply a_1 _ Hz
  end

theorem ceil_spec (x : ℝ) : of_rat (of_int (ceil x)) ≥ x :=
  begin
    rewrite [↑ceil, of_int_neg, -of_rat_neg],
    apply iff.mp !le_neg_iff_le_neg,
    apply floor_spec
  end

theorem ceil_smallest {x : ℝ} {z : ℤ} (Hz : z < ceil x) : x > of_rat (of_int z) :=
  begin
    rewrite ↑ceil at Hz,
    let Hz' := floor_largest (iff.mp !int.lt_neg_iff_lt_neg Hz),
    rewrite [of_int_neg at Hz', -of_rat_neg at Hz'],
    apply lt_of_neg_lt_neg Hz'
  end

theorem floor_succ (x : ℝ) : (floor x) + 1 = floor (x + 1) :=
  begin
    apply by_contradiction,
    intro H,
    cases int.lt_or_gt_of_ne H with [Hlt, Hgt],
    let Hl := floor_largest (iff.mp !int.add_lt_add_iff_lt_sub_right Hlt),
    rewrite [of_int_sub at Hl, -of_rat_sub at Hl],
    let Hl' := iff.mpr !add_lt_add_iff_lt_sub_right Hl,
    apply (not_le_of_gt Hl') !floor_spec,
    let Hl := floor_largest Hgt,
    rewrite [of_int_add at Hl, -of_rat_add at Hl],
    let Hl' := lt_of_add_lt_add_right Hl,
    apply (not_le_of_gt Hl') !floor_spec
  end

theorem floor_succ_lt (x : ℝ) : floor (x - 1) < floor x :=
  begin
    apply @int.lt_of_add_lt_add_right _ 1,
    rewrite [floor_succ (x - 1), sub_add_cancel],
    apply int.lt_add_of_pos_right dec_trivial
  end

theorem ceil_succ (x : ℝ) : ceil x < ceil (x + 1) :=
  begin
    rewrite [↑ceil, neg_add],
    apply int.neg_lt_neg,
    apply floor_succ_lt
  end

end ints
--------------------------------------------------
-- supremum property
-- this development roughly follows the proof of completeness done in Isabelle.
-- It does not depend on the previous proof of Cauchy completeness. Much of the same
-- machinery can be used to show that Cauchy completeness implies the supremum property.

section supremum
open prod nat
local postfix `~` := nat_of_pnat

-- The top part of this section could be refactored. What is the appropriate place to define
-- bounds, supremum, etc? In algebra/ordered_field? They potentially apply to more than just ℝ.

local notation 2 := (1 : ℚ) + 1
parameter X : ℝ → Prop

definition rpt {A : Type} (op : A → A) : ℕ → A → A
 | rpt 0 := λ a, a
 | rpt (succ k) := λ a, op (rpt k a)


definition ub (x : ℝ) := ∀ y : ℝ, X y → y ≤ x
definition bounded := ∃ x : ℝ, ub x
definition sup (x : ℝ) := ub x ∧ ∀ y : ℝ, ub y → x ≤ y

parameter elt : ℝ
hypothesis inh :  X elt
parameter bound : ℝ
hypothesis bdd : ub bound

include inh bdd

-- this should exist somewhere, no? I can't find it
theorem not_forall_of_exists_not {A : Type} {P : A → Prop} (H : ∃ a : A, ¬ P a) :
        ¬ ∀ a : A, P a :=
  begin
    intro Hall,
    cases H with [a, Ha],
    apply Ha (Hall a)
  end

definition avg (a b : ℚ) := a / 2 + b / 2

noncomputable definition bisect (ab : ℚ × ℚ) :=
  if ub (avg (pr1 ab) (pr2 ab)) then
    (pr1 ab, (avg (pr1 ab) (pr2 ab)))
  else
    (avg (pr1 ab) (pr2 ab), pr2 ab)

noncomputable definition under : ℚ := of_int (floor (elt - 1))

theorem under_spec1 : of_rat under < elt :=
  have H : of_rat under < of_rat (of_int (floor elt)), begin
    apply of_rat_lt_of_rat_of_lt,
    apply iff.mpr !of_int_lt_of_int,
    apply floor_succ_lt
  end,
  lt_of_lt_of_le H !floor_spec

theorem under_spec : ¬ ub under :=
  begin
    rewrite ↑ub,
    apply not_forall_of_exists_not,
    existsi elt,
    apply iff.mpr not_implies_iff_and_not,
    apply and.intro,
    apply inh,
    apply not_le_of_gt under_spec1
  end

noncomputable definition over : ℚ := of_int (ceil (bound + 1)) -- b

theorem over_spec1 : bound < of_rat over :=
  have H : of_rat (of_int (ceil bound)) < of_rat over, begin
    apply of_rat_lt_of_rat_of_lt,
    apply iff.mpr !of_int_lt_of_int,
    apply ceil_succ
  end,
  lt_of_le_of_lt !ceil_spec H

theorem over_spec : ub over :=
  begin
    rewrite ↑ub,
    intro y Hy,
    apply le_of_lt,
    apply lt_of_le_of_lt,
    apply bdd,
    apply Hy,
    apply over_spec1
  end

noncomputable definition under_seq := λ n : ℕ, pr1 (rpt bisect n (under, over)) -- A

noncomputable definition over_seq := λ n : ℕ, pr2 (rpt bisect n (under, over)) -- B

noncomputable definition avg_seq := λ n : ℕ, avg (over_seq n) (under_seq n) -- C

theorem avg_symm (n : ℕ) : avg_seq n = avg (under_seq n) (over_seq n) :=
  by rewrite [↑avg_seq, ↑avg, rat.add.comm]

theorem over_0 : over_seq 0 = over := rfl
theorem under_0 : under_seq 0 = under := rfl

theorem succ_helper (n : ℕ) : avg (pr1 (rpt bisect n (under, over))) (pr2 (rpt bisect n (under, over))) = avg_seq n :=
   by rewrite avg_symm

theorem under_succ (n : ℕ) : under_seq (succ n) =
        (if ub (avg_seq n) then under_seq n else avg_seq n) :=
  begin
    cases (em (ub (avg_seq n))) with [Hub, Hub],
    rewrite [if_pos Hub],
    have H :  pr1 (bisect (rpt bisect n (under, over))) = under_seq n, by
      rewrite [↑under_seq, ↑bisect at {2}, -succ_helper at Hub, if_pos Hub],
    apply H,
    rewrite [if_neg Hub],
    have H : pr1 (bisect (rpt bisect n (under, over))) = avg_seq n, by
      rewrite [↑bisect at {2}, -succ_helper at Hub, if_neg Hub, avg_symm],
    apply H
  end

theorem over_succ (n : ℕ) : over_seq (succ n) =
        (if ub (avg_seq n) then avg_seq n else over_seq n) :=
  begin
    cases (em (ub (avg_seq n))) with [Hub, Hub],
    rewrite [if_pos Hub],
    have H : pr2 (bisect (rpt bisect n (under, over))) = avg_seq n, by
      rewrite [↑bisect at {2}, -succ_helper at Hub, if_pos Hub, avg_symm],
    apply H,
    rewrite [if_neg Hub],
    have H : pr2 (bisect (rpt bisect n (under, over))) = over_seq n, by
      rewrite [↑over_seq, ↑bisect at {2}, -succ_helper at Hub, if_neg Hub],
    apply H
  end

theorem width (n : ℕ) : over_seq n - under_seq n = (over - under) / (rat.pow 2 n) :=
  nat.induction_on n
    (by xrewrite [over_0, under_0, rat.pow_zero, rat.div_one])
    (begin
      intro a Ha,
      rewrite [over_succ, under_succ],
      let Hou := calc
        (over_seq a) / 2 - (under_seq a) / 2 = ((over - under) / rat.pow 2 a) / 2 : by rewrite [rat.div_sub_div_same, Ha]
        ... = (over - under) / (rat.pow 2 a * 2) : rat.div_div_eq_div_mul (rat.ne_of_gt (rat.pow_pos dec_trivial _)) dec_trivial
        ... = (over - under) / rat.pow 2 (a + 1) : by rewrite rat.pow_add,
      cases (em (ub (avg_seq a))),
      rewrite [*if_pos a_1, -add_one, -Hou, ↑avg_seq, ↑avg, rat.add.assoc, rat.div_two_sub_self],
      rewrite [*if_neg a_1, -add_one, -Hou, ↑avg_seq, ↑avg, rat.sub_add_eq_sub_sub, rat.sub_self_div_two]
    end)

theorem binary_nat_bound (a : ℕ) : of_nat a ≤ (rat.pow 2 a) :=
  nat.induction_on a (rat.zero_le_one)
   (take n, assume Hn,
    calc
     of_nat (succ n) = (of_nat n) + 1 : of_nat_add
       ... ≤ rat.pow 2 n + 1 : rat.add_le_add_right Hn
       ... ≤ rat.pow 2 n + rat.pow 2 n : rat.add_le_add_left (rat.pow_ge_one_of_ge_one rat.two_ge_one _)
       ... = rat.pow 2 (succ n) : rat.pow_two_add)

theorem binary_bound (a : ℚ) : ∃ n : ℕ, a ≤ rat.pow 2 n :=
  exists.intro (ubound a) (calc
      a ≤ of_nat (ubound a) : ubound_ge
    ... ≤ rat.pow 2 (ubound a) : binary_nat_bound)

theorem rat_power_two_le (k : ℕ+) : rat_of_pnat k ≤ rat.pow 2 k~ :=
  !binary_nat_bound

theorem width_narrows : ∃ n : ℕ, over_seq n - under_seq n ≤ 1 :=
  begin
    cases binary_bound (over - under) with [a, Ha],
    existsi a,
    rewrite (width a),
    apply rat.div_le_of_le_mul,
    apply rat.pow_pos dec_trivial,
    rewrite rat.mul_one,
    apply Ha
  end

noncomputable definition over' := over_seq (some width_narrows)

noncomputable definition under' := under_seq (some width_narrows)

noncomputable definition over_seq' := λ n, over_seq (n + some width_narrows)

noncomputable definition under_seq' := λ n, under_seq (n + some width_narrows)

theorem over_seq'0 : over_seq' 0 = over' :=
  by rewrite [↑over_seq', nat.zero_add]

theorem under_seq'0 : under_seq' 0 = under' :=
  by rewrite [↑under_seq', nat.zero_add]

theorem under_over' : over' - under' ≤ 1 := some_spec width_narrows

theorem width' (n : ℕ) : over_seq' n - under_seq' n ≤ 1 / rat.pow 2 n :=
  nat.induction_on n
    (begin
      xrewrite [over_seq'0, under_seq'0, rat.pow_zero, rat.div_one],
      apply under_over'
    end)
    (begin
      intros a Ha,
      rewrite [↑over_seq' at *, ↑under_seq' at *, *succ_add at *, width at *,
              -add_one, -(add_one a), rat.pow_add, rat.pow_add _ a 1, *rat.pow_one],
      apply rat.div_mul_le_div_mul_of_div_le_div_pos' Ha dec_trivial
    end)

theorem PA (n : ℕ) : ¬ ub (under_seq n) :=
  nat.induction_on n
    (by rewrite under_0; apply under_spec)
    (begin
      intro a Ha,
      rewrite under_succ,
      cases (em (ub (avg_seq a))),
      rewrite (if_pos a_1),
      assumption,
      rewrite (if_neg a_1),
      assumption
    end)

theorem PB (n : ℕ) : ub (over_seq n) :=
  nat.induction_on n
    (by rewrite over_0; apply over_spec)
    (begin
      intro a Ha,
      rewrite over_succ,
      cases (em (ub (avg_seq a))),
      rewrite (if_pos a_1),
      assumption,
      rewrite (if_neg a_1),
      assumption
    end)

theorem under_lt_over : under < over :=
  begin
    cases (exists_not_of_not_forall under_spec) with [x, Hx],
    cases ((iff.mp not_implies_iff_and_not) Hx) with [HXx, Hxu],
    apply lt_of_rat_lt_of_rat,
    apply lt_of_lt_of_le,
    apply lt_of_not_ge Hxu,
    apply over_spec _ HXx
  end

theorem under_seq_lt_over_seq : ∀ m n : ℕ, under_seq m < over_seq n :=
  begin
    intros,
    cases (exists_not_of_not_forall (PA m)) with [x, Hx],
    cases ((iff.mp not_implies_iff_and_not) Hx) with [HXx, Hxu],
    apply lt_of_rat_lt_of_rat,
    apply lt_of_lt_of_le,
    apply lt_of_not_ge Hxu,
    apply PB,
    apply HXx
  end

theorem under_seq_lt_over_seq_single : ∀ n : ℕ, under_seq n < over_seq n :=
  by intros; apply under_seq_lt_over_seq

theorem under_seq'_lt_over_seq' : ∀ m n : ℕ, under_seq' m < over_seq' n :=
  by intros; apply under_seq_lt_over_seq

theorem under_seq'_lt_over_seq'_single : ∀ n : ℕ, under_seq' n < over_seq' n :=
  by intros; apply under_seq_lt_over_seq

theorem under_seq_mono_helper (i k : ℕ) : under_seq i ≤ under_seq (i + k) :=
  (nat.induction_on k
    (by rewrite nat.add_zero; apply rat.le.refl)
    (begin
      intros a Ha,
      rewrite [add_succ, under_succ],
      cases (em (ub (avg_seq (i + a)))) with [Havg, Havg],
      rewrite (if_pos Havg),
      apply Ha,
      rewrite [if_neg Havg, ↑avg_seq, ↑avg],
      apply rat.le.trans,
      apply Ha,
      rewrite -rat.add_halves at {1},
      apply rat.add_le_add_right,
      apply rat.div_le_div_of_le_of_pos,
      apply rat.le_of_lt,
      apply under_seq_lt_over_seq,
      apply dec_trivial
    end))

theorem under_seq_mono (i j : ℕ) (H : i ≤ j) : under_seq i ≤ under_seq j :=
  begin
    cases le.elim H with [k, Hk'],
    rewrite -Hk',
    apply under_seq_mono_helper
  end

theorem over_seq_mono_helper (i k : ℕ) : over_seq (i + k) ≤ over_seq i :=
  nat.induction_on k
    (by rewrite nat.add_zero; apply rat.le.refl)
    (begin
      intros a Ha,
      rewrite [add_succ, over_succ],
      cases (em (ub (avg_seq (i + a)))) with [Havg, Havg],
      rewrite [if_pos Havg, ↑avg_seq, ↑avg],
      apply rat.le.trans,
      rotate 1,
      apply Ha,
      rotate 1,
      rewrite -{over_seq (i + a)}rat.add_halves at {2},
      apply rat.add_le_add_left,
      apply rat.div_le_div_of_le_of_pos,
      apply rat.le_of_lt,
      apply under_seq_lt_over_seq,
      apply dec_trivial,
      rewrite [if_neg Havg],
      apply Ha
    end)

theorem over_seq_mono (i j : ℕ) (H : i ≤ j) : over_seq j ≤ over_seq i :=
  begin
    cases le.elim H with [k, Hk'],
    rewrite -Hk',
    apply over_seq_mono_helper
  end

theorem rat_power_two_inv_ge (k : ℕ+) : 1 / rat.pow 2 k~ ≤ k⁻¹ :=
  rat.div_le_div_of_le !rat_of_pnat_is_pos !rat_power_two_le

open s
theorem regular_lemma_helper {s : seq} {m n : ℕ+} (Hm : m ≤ n)
        (H : ∀ n i : ℕ+, i ≥ n → under_seq' n~ ≤ s i ∧ s i ≤ over_seq' n~) :
        rat.abs (s m - s n) ≤ m⁻¹ + n⁻¹ :=
  begin
    cases (H m n Hm) with [T1under, T1over],
    cases (H m m (!pnat.le.refl)) with [T2under, T2over],
    apply rat.le.trans,
    apply rat.dist_bdd_within_interval,
    apply under_seq'_lt_over_seq'_single,
    rotate 1,
    repeat assumption,
    apply rat.le.trans,
    apply width',
    apply rat.le.trans,
    apply rat_power_two_inv_ge,
    apply rat.le_add_of_nonneg_right,
    apply rat.le_of_lt (!inv_pos)
  end

theorem regular_lemma (s : seq) (H : ∀ n i : ℕ+, i ≥ n → under_seq' n~ ≤ s i ∧ s i ≤ over_seq' n~) :
        regular s :=
  begin
    rewrite ↑regular,
    intros,
    cases (em (m ≤ n)) with [Hm, Hn],
    apply regular_lemma_helper Hm H,
    let T := regular_lemma_helper (pnat.le_of_lt (pnat.lt_of_not_le Hn)) H,
    rewrite [rat.abs_sub at T, {n⁻¹ + _}rat.add.comm at T],
    exact T
  end

noncomputable definition p_under_seq : seq := λ n : ℕ+, under_seq' n~

noncomputable definition p_over_seq : seq := λ n : ℕ+, over_seq' n~

theorem under_seq_regular : regular p_under_seq :=
  begin
    apply regular_lemma,
    intros n i Hni,
    apply and.intro,
    apply under_seq_mono,
    apply nat.add_le_add_right Hni,
    apply rat.le_of_lt,
    apply under_seq_lt_over_seq
  end

theorem over_seq_regular : regular p_over_seq :=
  begin
    apply regular_lemma,
    intros n i Hni,
    apply and.intro,
    apply rat.le_of_lt,
    apply under_seq_lt_over_seq,
    apply over_seq_mono,
    apply nat.add_le_add_right Hni
  end

noncomputable definition sup_over : ℝ := quot.mk (reg_seq.mk p_over_seq over_seq_regular)

noncomputable definition sup_under : ℝ := quot.mk (reg_seq.mk p_under_seq under_seq_regular)

theorem over_bound : ub sup_over :=
  begin
    rewrite ↑ub,
    intros y Hy,
    apply le_of_le_reprs,
    intro n,
    apply PB,
    apply Hy
  end

theorem under_lowest_bound : ∀ y : ℝ, ub y → sup_under ≤ y :=
  begin
    intros y Hy,
    apply le_of_reprs_le,
    intro n,
    cases (exists_not_of_not_forall (PA _)) with [x, Hx],
    cases (iff.mp not_implies_iff_and_not Hx) with [HXx, Hxn],
    apply le.trans,
    apply le_of_lt,
    apply lt_of_not_ge Hxn,
    apply Hy,
    apply HXx
  end

theorem under_over_equiv : p_under_seq ≡ p_over_seq :=
  begin
    rewrite ↑equiv,
    intros,
    apply rat.le.trans,
    have H : p_under_seq n < p_over_seq n, from !under_seq_lt_over_seq,
    rewrite [rat.abs_of_neg (iff.mpr !rat.sub_neg_iff_lt H), rat.neg_sub],
    apply width',
    apply rat.le.trans,
    apply rat_power_two_inv_ge,
    apply rat.le_add_of_nonneg_left,
    apply rat.le_of_lt !inv_pos
  end

theorem under_over_eq : sup_under = sup_over := quot.sound under_over_equiv

theorem ex_sup_of_inh_of_bdd : ∃ x : ℝ, sup x :=
  exists.intro sup_over (and.intro over_bound (under_over_eq ▸ under_lowest_bound))

end supremum

end real