bookshelf/Exercises/Apostol/Chapter_I_3.lean

/-
Chapter I 3

A Set of Axioms for the Real-Number System
-/
import Bookshelf.Real.Set

#check Archimedean
#check Real.exists_isLUB

namespace Real

-- ========================================
-- The least-upper-bound axiom (completeness axiom)
-- ========================================

/--
A property holds for the negation of elements in set `S` if and only if it also
holds for the elements of the negation of `S`.
-/
lemma set_neg_prop_iff_neg_set_prop (S : Set ℝ) (p : ℝ → Prop)
  : (∀ y, y ∈ S → p (-y)) ↔ (∀ y, y ∈ -S → p y) := by
  apply Iff.intro
  · intro h y hy
    rw [← neg_neg y, Set.neg_mem_neg] at hy
    have := h (-y) hy
    simp at this
    exact this
  · intro h y hy
    rw [← Set.neg_mem_neg] at hy
    exact h (-y) hy

/--
The upper bounds of the negation of a set is the negation of the lower bounds of
the set.
-/
lemma upper_bounds_neg_eq_neg_lower_bounds (S : Set ℝ)
  : upperBounds (-S) = -lowerBounds S := by
  suffices (∀ x, x ∈ upperBounds (-S) ↔ x ∈ -(lowerBounds S)) from
    Set.ext this
  intro x
  apply Iff.intro
  · intro hx
    unfold lowerBounds
    show -x ∈ { x | ∀ ⦃a : ℝ⦄, a ∈ S → x ≤ a }
    show ∀ ⦃a : ℝ⦄, a ∈ S → (-x) ≤ a
    intro a ha; rw [neg_le]; revert ha a
    rw [set_neg_prop_iff_neg_set_prop S (fun a => a ≤ x)]
    exact hx
  · intro hx
    unfold upperBounds
    show ∀ ⦃a : ℝ⦄, a ∈ -S → a ≤ x
    rw [← set_neg_prop_iff_neg_set_prop S (fun a => a ≤ x)]
    intro y hy; rw [neg_le]; revert hy y
    exact hx

/--
The negation of the upper bounds of a set is the lower bounds of the negation of
the set.
-/
lemma neg_upper_bounds_eq_lower_bounds_neg (S : Set ℝ)
  : -upperBounds S = lowerBounds (-S) := by
  suffices (∀ x, x ∈ -upperBounds S ↔ x ∈ lowerBounds (-S)) from
    Set.ext this
  intro x
  apply Iff.intro
  · intro hx
    unfold lowerBounds
    show ∀ ⦃a : ℝ⦄, a ∈ -S → x ≤ a
    rw [← set_neg_prop_iff_neg_set_prop S (fun a => x ≤ a)]
    intro y hy; rw [le_neg]; revert hy y
    exact hx
  · intro hx
    unfold upperBounds
    show -x ∈ { x | ∀ ⦃a : ℝ⦄, a ∈ S → a ≤ x }
    show ∀ ⦃a : ℝ⦄, a ∈ S → a ≤ (-x)
    intro a ha; rw [le_neg]; revert ha a
    rw [set_neg_prop_iff_neg_set_prop S (fun a => x ≤ a)]
    exact hx

/--
An element `x` is the least element of the negation of a set if and only if `-x`
if the greatest element of the set.
-/
lemma is_least_neg_set_eq_is_greatest_set_neq (S : Set ℝ)
  : IsLeast (-S) x = IsGreatest S (-x) := by
  unfold IsLeast IsGreatest
  rw [← neg_upper_bounds_eq_lower_bounds_neg S]
  rfl

/--
At least with respect to `ℝ`, `x` is the least upper bound of set `-S` if and
only if `-x` is the greatest lower bound of `S`.
-/
theorem is_lub_neg_set_iff_is_glb_set_neg (S : Set ℝ)
  : IsLUB (-S) x = IsGLB S (-x) :=
  calc IsLUB (-S) x
    _ = IsLeast (upperBounds (-S)) x := rfl
    _ = IsLeast (-lowerBounds S) x := by rw [upper_bounds_neg_eq_neg_lower_bounds S]
    _ = IsGreatest (lowerBounds S) (-x) := by rw [is_least_neg_set_eq_is_greatest_set_neq]
    _ = IsGLB S (-x) := rfl

/--
Theorem I.27

Every nonempty set `S` that is bounded below has a greatest lower bound; that
is, there is a real number `L` such that `L = inf S`.
-/
theorem exists_isGLB (S : Set ℝ) (hne : S.Nonempty) (hbdd : BddBelow S)
  : ∃ x, IsGLB S x := by
  -- First we show the negation of a nonempty set bounded below is a nonempty
  -- set bounded above. In that case, we can then apply the completeness axiom
  -- to argue the existence of a supremum.
  have hne' : (-S).Nonempty := Set.nonempty_neg.mpr hne
  have hbdd' : ∃ x, ∀ (y : ℝ), y ∈ -S → y ≤ x := by
    rw [bddBelow_def] at hbdd
    let ⟨lb, lbp⟩ := hbdd
    refine ⟨-lb, ?_⟩
    rw [← set_neg_prop_iff_neg_set_prop S (fun y => y ≤ -lb)]
    intro y hy
    exact neg_le_neg (lbp y hy)
  rw [←bddAbove_def] at hbdd'
  -- Once we have found a supremum for `-S`, we argue the negation of this value
  -- is the same as the infimum of `S`.
  let ⟨ub, ubp⟩ := exists_isLUB (-S) hne' hbdd'
  exact ⟨-ub, (is_lub_neg_set_iff_is_glb_set_neg S).mp ubp⟩

/--
Every real should be less than or equal to the absolute value of its ceiling.
-/
lemma leq_nat_abs_ceil_self (x : ℝ) : x ≤ Int.natAbs ⌈x⌉ := by
  by_cases h : x ≥ 0
  · let k : ℤ := ⌈x⌉
    unfold Int.natAbs
    have k' : k = ⌈x⌉ := rfl
    rw [←k']
    have _ : k ≥ 0 := by  -- Hint for match below
      rw [k', ge_iff_le]
      exact Int.ceil_nonneg (ge_iff_le.mp h)
    match k with
    | Int.ofNat m => calc x
      _ ≤ ⌈x⌉ := Int.le_ceil x
      _ = Int.ofNat m := by rw [←k']
  · have h' : ((Int.natAbs ⌈x⌉) : ℝ) ≥ 0 := by simp
    calc x
      _ ≤ 0 := le_of_lt (lt_of_not_le h)
      _ ≤ ↑(Int.natAbs ⌈x⌉) := GE.ge.le h'    

-- ========================================
-- The Archimedean property of the real-number system
-- ========================================

/--
Theorem I.29

For every real `x` there exists a positive integer `n` such that `n > x`.
-/
theorem exists_pnat_geq_self (x : ℝ) : ∃ n : ℕ+, ↑n > x := by
  let x' : ℕ+ := ⟨Int.natAbs ⌈x⌉ + 1, by simp⟩
  have h : x < x' := calc x
    _ ≤ Int.natAbs ⌈x⌉ := leq_nat_abs_ceil_self x
    _ < ↑↑(Int.natAbs ⌈x⌉ + 1) := by simp
    _ = x' := rfl
  exact ⟨x', h⟩

/--
Theorem I.30

If `x > 0` and if `y` is an arbitrary real number, there exists a positive
integer `n` such that `nx > y`.

This is known as the *Archimedean Property of the Reals*.
-/
theorem exists_pnat_mul_self_geq_of_pos {x y : ℝ}
  : x > 0 → ∃ n : ℕ+, n * x > y := by
  intro hx
  let ⟨n, p⟩ := exists_pnat_geq_self (y / x)
  have p' := mul_lt_mul_of_pos_right p hx
  rw [div_mul, div_self (show x ≠ 0 from LT.lt.ne' hx), div_one] at p'
  exact ⟨n, p'⟩

/--
Theorem I.31

If three real numbers `a`, `x`, and `y` satisfy the inequalities
`a ≤ x ≤ a + y / n` for every integer `n ≥ 1`, then `x = a`.
-/
theorem forall_pnat_leq_self_leq_frac_imp_eq {x y a : ℝ}
  : (∀ n : ℕ+, a ≤ x ∧ x ≤ a + (y / n)) → x = a := by
  intro h
  match @trichotomous ℝ LT.lt _ x a with
  | Or.inr (Or.inl r) =>  -- x = a
    exact r
  | Or.inl r =>  -- x < a
    have z : a < a := lt_of_le_of_lt (h 1).left r
    simp at z
  | Or.inr (Or.inr r) =>  -- x > a
    let ⟨c, hc⟩ := exists_pos_add_of_lt' r
    let ⟨n, hn⟩ := @exists_pnat_mul_self_geq_of_pos c y hc.left
    have hn := mul_lt_mul_of_pos_left hn $
      have hp : 0 < (↑↑n : ℝ) := by simp
      show 0 < ((↑↑n)⁻¹ : ℝ) from inv_pos.mpr hp
    rw [
      inv_mul_eq_div,
      ← mul_assoc, mul_comm (n⁻¹ : ℝ),
      ← one_div,
      mul_one_div
    ] at hn
    simp at hn
    have := calc a + y / ↑↑n
      _ < a + c := add_lt_add_left hn a
      _ = x := hc.right
      _ ≤ a + y / ↑↑n := (h n).right
    simp at this

/--
If three real numbers `a`, `x`, and `y` satisfy the inequalities
`a - y / n ≤ x ≤ a` for every integer `n ≥ 1`, then `x = a`.
-/
theorem forall_pnat_frac_leq_self_leq_imp_eq {x y a : ℝ}
  : (∀ n : ℕ+, a - (y / n) ≤ x ∧ x ≤ a) → x = a := by
  intro h
  match @trichotomous ℝ LT.lt _ x a with
  | Or.inr (Or.inl r) =>  -- x = a
    exact r
  | Or.inl r =>  -- x < a
    let ⟨c, hc⟩ := exists_pos_add_of_lt' r
    let ⟨n, hn⟩ := @exists_pnat_mul_self_geq_of_pos c y hc.left
    have hn := mul_lt_mul_of_pos_left hn $
      have hp : 0 < (↑↑n : ℝ) := by simp
      show 0 < ((↑↑n)⁻¹ : ℝ) from inv_pos.mpr hp
    rw [
      inv_mul_eq_div,
      ← mul_assoc, mul_comm (n⁻¹ : ℝ),
      ← one_div,
      mul_one_div
    ] at hn
    simp at hn
    have := calc a - y / ↑↑n
      _ > a - c := sub_lt_sub_left hn a
      _ = x := sub_eq_of_eq_add (Eq.symm hc.right)
      _ ≥ a - y / ↑↑n := (h n).left
    simp at this
  | Or.inr (Or.inr r) =>  -- x > a
    have z : x < x := lt_of_le_of_lt (h 1).right r
    simp at z

-- ========================================
-- Fundamental properties of the supremum and infimum
-- ========================================

/--
Every member of a set `S` is less than or equal to some value `ub` if and only
if `ub` is an upper bound of `S`. 
-/
lemma mem_upper_bounds_iff_forall_le {S : Set ℝ}
  : ub ∈ upperBounds S ↔ ∀ x ∈ S, x ≤ ub := by
  apply Iff.intro
  · intro h _ hx
    exact (h hx)
  · exact id

/--
If a set `S` has a least upper bound, it follows every member of `S` is less
than or equal to that value.
-/
lemma forall_lub_imp_forall_le {S : Set ℝ}
  : IsLUB S ub → ∀ x ∈ S, x ≤ ub := by
  intro h
  rw [← mem_upper_bounds_iff_forall_le]
  exact h.left

/--
Any member of the upper bounds of a set must be greater than or equal to the
least member of that set.
-/
lemma mem_imp_ge_lub {x : ℝ} (h : IsLUB S s) : x ∈ upperBounds S → x ≥ s := by
  intro hx
  exact h.right hx

/--
Theorem I.32a

Let `h` be a given positive number and let `S` be a set of real numbers. If `S`
has a supremum, then for some `x` in `S` we have `x > sup S - h`.
-/
theorem sup_imp_exists_gt_sup_sub_delta {S : Set ℝ} {s h : ℝ} (hp : h > 0)
  : IsLUB S s → ∃ x ∈ S, x > s - h := by
  intro hb
  -- Suppose all members of our set was less than `s - h`. Then `s` couldn't be
  -- the *least* upper bound.
  by_contra nb
  suffices s - h ∈ upperBounds S by
    have h' : h < h := calc h
      _ ≤ 0 := (le_sub_self_iff s).mp (hb.right this)
      _ < h := hp
    simp at h'
  rw [not_exists] at nb
  have nb' : ∀ x ∈ S, x ≤ s - h := by
    intro x hx
    exact le_of_not_gt (not_and.mp (nb x) hx)
  rwa [← mem_upper_bounds_iff_forall_le] at nb'

/--
Every member of a set `S` is greater than or equal to some value `lb` if and
only if `lb` is a lower bound of `S`. 
-/
lemma mem_lower_bounds_iff_forall_ge {S : Set ℝ}
  : lb ∈ lowerBounds S ↔ ∀ x ∈ S, x ≥ lb := by
  apply Iff.intro
  · intro h _ hx
    exact (h hx)
  · exact id

/--
If a set `S` has a greatest lower bound, it follows every member of `S` is
greater than or equal to that value.
-/
lemma forall_glb_imp_forall_ge {S : Set ℝ}
  : IsGLB S lb → ∀ x ∈ S, x ≥ lb := by
  intro h
  rw [← mem_lower_bounds_iff_forall_ge]
  exact h.left

/--
Any member of the lower bounds of a set must be less than or equal to the
greatest member of that set.
-/
lemma mem_imp_le_glb {x : ℝ} (h : IsGLB S s) : x ∈ lowerBounds S → x ≤ s := by
  intro hx
  exact h.right hx

/--
Theorem I.32b

Let `h` be a given positive number and let `S` be a set of real numbers. If `S`
has an infimum, then for some `x` in `S` we have `x < inf S + h`.
-/
theorem inf_imp_exists_lt_inf_add_delta {S : Set ℝ} {s h : ℝ} (hp : h > 0)
  : IsGLB S s → ∃ x ∈ S, x < s + h := by
  intro hb
  -- Suppose all members of our set was greater than `s + h`. Then `s` couldn't
  -- be the *greatest* lower bound.
  by_contra nb
  suffices s + h ∈ lowerBounds S by
    have h' : h < h := calc h
      _ ≤ 0 := (add_le_iff_nonpos_right s).mp (hb.right this)
      _ < h := hp
    simp at h'
  rw [not_exists] at nb
  have nb' : ∀ x ∈ S, x ≥ s + h := by
    intro x hx
    exact le_of_not_gt (not_and.mp (nb x) hx)
  rwa [← mem_lower_bounds_iff_forall_ge] at nb'

/--
Theorem I.33a (Additive Property)

Given nonempty subsets `A` and `B` of `ℝ`, let `C` denote the set
`C = {a + b : a ∈ A, b ∈ B}`. If each of `A` and `B` has a supremum, then `C`
has a supremum, and `sup C = sup A + sup B`.
-/
theorem sup_minkowski_sum_eq_sup_add_sup (A B : Set ℝ) (a b : ℝ)
  (hA : A.Nonempty) (hB : B.Nonempty)
  (ha : IsLUB A a) (hb : IsLUB B b)
  : IsLUB (Real.minkowski_sum A B) (a + b) := by
  let C := Real.minkowski_sum A B
  -- First we show `a + b` is an upper bound of `C`.
  have hub : a + b ∈ upperBounds C := by
    rw [mem_upper_bounds_iff_forall_le]
    intro x hx
    have ⟨a', ⟨ha', ⟨b', ⟨hb', hxab⟩⟩⟩⟩: ∃ a ∈ A, ∃ b ∈ B, x = a + b := hx
    have hs₁ : a' ≤ a := (forall_lub_imp_forall_le ha) a' ha'
    have hs₂ : b' ≤ b := (forall_lub_imp_forall_le hb) b' hb'
    exact calc x
      _ = a' + b' := hxab
      _ ≤ a + b := add_le_add hs₁ hs₂
  -- Now we show `a + b` is the *least* upper bound of `C`. We know a least
  -- upper bound `c` exists; show that `c = a + b`.
  have ⟨c, hc⟩ := exists_isLUB C
    (Real.nonempty_minkowski_sum_iff_nonempty_add_nonempty.mpr ⟨hA, hB⟩)
    ⟨a + b, hub⟩
  suffices (∀ n : ℕ+, c ≤ a + b ∧ a + b ≤ c + (1 / n)) by
    rwa [← forall_pnat_leq_self_leq_frac_imp_eq this] at hc
  intro n
  apply And.intro
  · exact hc.right hub
  · have hd : 1 / (2 * n) > (0 : ℝ) := by norm_num
    have ⟨a', ha'⟩ := sup_imp_exists_gt_sup_sub_delta hd ha
    have ⟨b', hb'⟩ := sup_imp_exists_gt_sup_sub_delta hd hb
    have hab' : a + b < a' + b' + 1 / n := by
      have ha'' := add_lt_add_right ha'.right (1 / (2 * ↑↑n))
      have hb'' := add_lt_add_right hb'.right (1 / (2 * ↑↑n))
      rw [sub_add_cancel] at ha'' hb''
      have hr := add_lt_add ha'' hb''
      ring_nf at hr
      ring_nf
      rwa [add_assoc, add_comm b' (↑↑n)⁻¹, ← add_assoc]
    have hc' : a' + b' ≤ c := by
      refine forall_lub_imp_forall_le hc (a' + b') ?_
      show ∃ a ∈ A, ∃ b ∈ B, a' + b' = a + b
      exact ⟨a', ⟨ha'.left, ⟨b', ⟨hb'.left, rfl⟩⟩⟩⟩
    calc a + b
      _ ≤ a' + b' + 1 / n := le_of_lt hab'
      _ ≤ c + 1 / n := add_le_add_right hc' (1 / n)

/--
Theorem I.33b (Additive Property)

Given nonempty subsets `A` and `B` of `ℝ`, let `C` denote the set
`C = {a + b : a ∈ A, b ∈ B}`. If each of `A` and `B` has an infimum, then `C`
has an infimum, and `inf C = inf A + inf B`.
-/
theorem inf_minkowski_sum_eq_inf_add_inf (A B : Set ℝ)
  (hA : A.Nonempty) (hB : B.Nonempty)
  (ha : IsGLB A a) (hb : IsGLB B b)
  : IsGLB (Real.minkowski_sum A B) (a + b) := by
  let C := Real.minkowski_sum A B
  -- First we show `a + b` is a lower bound of `C`.
  have hlb : a + b ∈ lowerBounds C := by
    rw [mem_lower_bounds_iff_forall_ge]
    intro x hx
    have ⟨a', ⟨ha', ⟨b', ⟨hb', hxab⟩⟩⟩⟩: ∃ a ∈ A, ∃ b ∈ B, x = a + b := hx
    have hs₁ : a' ≥ a := (forall_glb_imp_forall_ge ha) a' ha'
    have hs₂ : b' ≥ b := (forall_glb_imp_forall_ge hb) b' hb'
    exact calc x
      _ = a' + b' := hxab
      _ ≥ a + b := add_le_add hs₁ hs₂
  -- Now we show `a + b` is the *greatest* lower bound of `C`. We know a
  -- greatest lower bound `c` exists; show that `c = a + b`.
  have ⟨c, hc⟩ := exists_isGLB C
    (Real.nonempty_minkowski_sum_iff_nonempty_add_nonempty.mpr ⟨hA, hB⟩)
    ⟨a + b, hlb⟩
  suffices (∀ n : ℕ+, c - (1 / n) ≤ a + b ∧ a + b ≤ c) by
    rwa [← forall_pnat_frac_leq_self_leq_imp_eq this] at hc
  intro n
  apply And.intro
  · have hd : 1 / (2 * n) > (0 : ℝ) := by norm_num
    have ⟨a', ha'⟩ := inf_imp_exists_lt_inf_add_delta hd ha
    have ⟨b', hb'⟩ := inf_imp_exists_lt_inf_add_delta hd hb
    have hab' : a' + b' - 1 / n < a + b := by
      have ha'' := sub_lt_sub_right ha'.right (1 / (2 * ↑↑n))
      have hb'' := sub_lt_sub_right hb'.right (1 / (2 * ↑↑n))
      rw [add_sub_cancel] at ha'' hb''
      have hr := add_lt_add ha'' hb''
      ring_nf at hr
      ring_nf
      rwa [← add_sub_assoc, add_sub_right_comm]
    have hc' : c ≤ a' + b' := by
      refine forall_glb_imp_forall_ge hc (a' + b') ?_
      show ∃ a ∈ A, ∃ b ∈ B, a' + b' = a + b
      exact ⟨a', ⟨ha'.left, ⟨b', ⟨hb'.left, rfl⟩⟩⟩⟩
    calc c - 1 / n
      _ ≤ a' + b' - 1 / n := sub_le_sub_right hc' (1 / n)
      _ ≤ a + b := le_of_lt hab'
  · exact hc.right hlb

/--
Theorem I.34

Given two nonempty subsets `S` and `T` of `ℝ` such that `s ≤ t` for every `s` in
`S` and every `t` in `T`. Then `S` has a supremum, and `T` has an infimum, and
they satisfy the inequality `sup S ≤ inf T`.
-/
theorem forall_mem_le_forall_mem_imp_sup_le_inf (S T : Set ℝ)
  (hS : S.Nonempty) (hT : T.Nonempty)
  (p : ∀ s ∈ S, ∀ t ∈ T, s ≤ t)
  : ∃ (s : ℝ), IsLUB S s ∧ ∃ (t : ℝ), IsGLB T t ∧ s ≤ t := by
  -- Sshow a supremum of `S` and an infimum of `T` exists (since each set bounds
  -- above and below the other, respectively).
  let ⟨s, hs⟩ := hS
  let ⟨t, ht⟩ := hT
  have ps : t ∈ upperBounds S := by
    intro x hx
    exact p x hx t ht
  have pt : s ∈ lowerBounds T  := by
    intro x hx
    exact p s hs x hx
  have ⟨S_lub, hS_lub⟩ := Real.exists_isLUB S hS ⟨t, ps⟩
  have ⟨T_glb, hT_glb⟩ := Real.exists_isGLB T hT ⟨s, pt⟩
  refine ⟨S_lub, ⟨hS_lub, ⟨T_glb, ⟨hT_glb, ?_⟩⟩⟩⟩
  -- Assume `T_glb < S_lub`. Then `∃ c, T_glb + c < S_lub` which in turn implies
  -- existence of some `x ∈ S` such that `T_glb < S_lub - c / 2 < x < S_lub`.
  by_contra nr
  rw [not_le] at nr
  have ⟨c, hc⟩ := exists_pos_add_of_lt' nr
  have c_div_two_gt_zero : c / 2 > 0 := by
    have hr := div_lt_div_of_lt (show (0 : ℝ) < 2 by simp) hc.left
    rwa [zero_div] at hr
  have T_glb_lt_S_lub_sub_c_div_two : T_glb < S_lub - c / 2 := by
    have hr := congrFun (congrArg HSub.hSub hc.right) (c / 2)
    rw [add_sub_assoc, sub_half c] at hr
    calc T_glb
      _ < T_glb + c / 2 := (lt_add_iff_pos_right T_glb).mpr c_div_two_gt_zero
      _ = S_lub - c / 2 := hr
  -- Since `x ∈ S`, `p` implies `x ≤ t` for all `t ∈ T`. So `x ≤ T_glb`. But the
  -- above implies `T_glb < x`.
  have ⟨x, hx⟩ := sup_imp_exists_gt_sup_sub_delta c_div_two_gt_zero hS_lub
  have : x < x := calc x
    _ ≤ T_glb := mem_imp_le_glb hT_glb (p x hx.left)
    _ < S_lub - c / 2 := T_glb_lt_S_lub_sub_c_div_two
    _ < x := hx.right
  simp at this

end Real