bookshelf/one-variable-calculus/Apostol/Chapter_I_3_10.lean

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import Mathlib.Data.PNat.Basic
import Mathlib.Data.Real.Basic
#check Archimedean
namespace Real
/--
Every real should be less than or equal to the absolute value of its ceiling.
-/
lemma leq_nat_abs_ceil_self (x : ) : x ≤ Int.natAbs ⌈x⌉ := by
by_cases h : x ≥ 0
· let k : := ⌈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⌉) : ) ≥ 0 := by simp
calc x
_ ≤ 0 := le_of_lt (lt_of_not_le h)
_ ≤ ↑(Int.natAbs ⌈x⌉) := GE.ge.le h'
/--
Theorem I.29
For every real `x` there exists a positive integer `n` such that `n > x`.
-/
theorem exists_pnat_geq_self (x : ) : ∃ n : +, ↑n > x := by
let x' : + := ⟨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 : }
: x > 0 → ∃ 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_iff_eq {x y a : }
: (∀ n : +, a ≤ x ∧ x ≤ a + (y / n)) → x = a := by
intro h
match @trichotomous LT.lt _ x a with
| -- x = a
Or.inr (Or.inl r) => exact r
| -- x < a
Or.inl r =>
have z : a < a := lt_of_le_of_lt (h 1).left r
simp at z
| -- x > a
Or.inr (Or.inr r) =>
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 : ) := by simp
show 0 < ((↑↑n)⁻¹ : ) from inv_pos.mpr hp
rw [inv_mul_eq_div, ←mul_assoc, mul_comm (n⁻¹ : ), ←one_div, mul_one_div] at hn
simp at hn
have hn := add_lt_add_left hn a
have := calc a + y / ↑↑n
_ < a + c := hn
_ = x := hc.right
_ ≤ a + y / ↑↑n := (h n).right
simp at this
end Real