Analysis I, Section 4.4: gaps in the rational numbers

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

Irrationality of √2, and related facts about the rational numbers

Many of the results here can be established more quickly by relying more heavily on the Mathlib API; one can set oneself the exercise of doing so.

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Proposition 4.4.1 (Interspersing of integers by rationals) / Exercise 4.4.1

theorem declaration uses `sorry`Rat.between_int (x:ℚ) : ∃! n:ℤ, n ≤ x ∧ x < n+1 := byx:ℚ⊢ ∃! n, ↑n ≤ x ∧ x < ↑n + 1
  sorryAll goals completed! 🐙

theorem declaration uses `sorry`Nat.exists_gt (x:ℚ) : ∃ n:ℕ, n > x := byx:ℚ⊢ ∃ n, ↑n > x
  sorryAll goals completed! 🐙

Proposition 4.4.3 (Interspersing of rationals)

theorem Rat.exists_between_rat {x y:ℚ} (h: x < y) : ∃ z:ℚ, x < z ∧ z < y := byx:ℚy:ℚh:x < y⊢ ∃ z, x < z ∧ z < y
  -- This proof is written to follow the structure of the original text.
  -- The reader is encouraged to find shorter proofs, for instance
  -- using Mathlib's `linarith` tactic.
  use (x+y)/2hx:ℚy:ℚh:x < y⊢ x < (x + y) / 2 ∧ (x + y) / 2 < y
  have h' : x/2 < y/2 := byx:ℚy:ℚh:x < y⊢ ∃ z, x < z ∧ z < y
    rw [show x/2 = x*(1/2) by ring, show y/2 = y*(1/2) by ring]x:ℚy:ℚh:x < y⊢ x * (1 / 2) < y * (1 / 2)
    apply mul_lt_mul_of_pos_right hx:ℚy:ℚh:x < y⊢ 0 < 1 / 2; positivityAll goals completed! 🐙
  constructorh.leftx:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ x < (x + y) / 2h.rightx:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ (x + y) / 2 < y
  .h.leftx:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ x < (x + y) / 2 convert add_lt_add_right h' (x/2) using 1h.e'_3x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ x = x / 2 + x / 2h.e'_4x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ (x + y) / 2 = x / 2 + y / 2 <;>h.e'_3x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ x = x / 2 + x / 2h.e'_4x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ (x + y) / 2 = x / 2 + y / 2 ringAll goals completed! 🐙
  convert add_lt_add_right h' (y/2) using 1h.e'_3x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ (x + y) / 2 = y / 2 + x / 2h.e'_4x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ y = y / 2 + y / 2 <;>h.e'_3x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ (x + y) / 2 = y / 2 + x / 2h.e'_4x:ℚy:ℚh:x < yh':x / 2 < y / 2⊢ y = y / 2 + y / 2 ringAll goals completed! 🐙

Exercise 4.4.2 (a)

theorem declaration uses `sorry`Nat.no_infinite_descent : ¬ ∃ a:ℕ → ℕ, ∀ n, a (n+1) < a n := by⊢ ¬∃ a, ∀ (n : ℕ), a (n + 1) < a n
  sorryAll goals completed! 🐙

Exercise 4.4.2 (b)

def declaration uses `sorry`Int.infinite_descent : Decidable (∃ a:ℕ → ℤ, ∀ n, a (n+1) < a n) := by⊢ Decidable (∃ a, ∀ (n : ℕ), a (n + 1) < a n)
  -- the first line of this construction should be either `apply isTrue` or `apply isFalse`.
  sorryAll goals completed! 🐙

Exercise 4.4.2 (b')

def declaration uses `sorry`Rat.pos_infinite_descent : Decidable (∃ a:ℕ → {x: ℚ // 0 < x}, ∀ n, a (n+1) < a n) := by⊢ Decidable (∃ a, ∀ (n : ℕ), a (n + 1) < a n)
  -- the first line of this construction should be either `apply isTrue` or `apply isFalse`.
  sorryAll goals completed! 🐙

even_iff_exists_two_mul.{u_2} {α : Type u_2} [Semiring α] {a : α} : Even a ↔ ∃ b, a = 2 * b#check even_iff_exists_two_mul

even_iff_exists_two_mul.{u_2} {α : Type u_2} [Semiring α] {a : α} : Even a ↔ ∃ b, a = 2 * b

odd_iff_exists_bit1.{u_2} {α : Type u_2} [Semiring α] {a : α} : Odd a ↔ ∃ b, a = 2 * b + 1#check odd_iff_exists_bit1

odd_iff_exists_bit1.{u_2} {α : Type u_2} [Semiring α] {a : α} : Odd a ↔ ∃ b, a = 2 * b + 1

theorem declaration uses `sorry`Nat.even_or_odd'' (n:ℕ) : Even n ∨ Odd n := byn:ℕ⊢ Even n ∨ Odd n
  sorryAll goals completed! 🐙

theorem declaration uses `sorry`Nat.not_even_and_odd (n:ℕ) : ¬ (Even n ∧ Odd n) := byn:ℕ⊢ ¬(Even n ∧ Odd n)
  sorryAll goals completed! 🐙

Nat.rec.{u} {motive : ℕ → Sort u} (zero : motive Nat.zero) (succ : (n : ℕ) → motive n → motive n.succ) (t : ℕ) :
  motive t#check Nat.rec

Nat.rec.{u} {motive : ℕ → Sort u} (zero : motive Nat.zero) (succ : (n : ℕ) → motive n → motive n.succ) (t : ℕ) :
  motive t

Proposition 4.4.4 / Exercise 4.4.3

theorem declaration uses `sorry`Rat.not_exist_sqrt_two : ¬ ∃ x:ℚ, x^2 = 2 := by⊢ ¬∃ x, x ^ 2 = 2
  -- This proof is written to follow the structure of the original text.
  by_contra hh:∃ x, x ^ 2 = 2⊢ False; choose x hx using hx:ℚhx:x ^ 2 = 2⊢ False
  have hnon : x ≠ 0 := by⊢ ¬∃ x, x ^ 2 = 2 aesopAll goals completed! 🐙
  wlog hpos : x > 0inrx:ℚhx:x ^ 2 = 2hnon:x ≠ 0this:∀ (x : ℚ), x ^ 2 = 2 → x ≠ 0 → x > 0 → Falsehpos:¬x > 0⊢ Falsex:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0⊢ False
  .inrx:ℚhx:x ^ 2 = 2hnon:x ≠ 0this:∀ (x : ℚ), x ^ 2 = 2 → x ≠ 0 → x > 0 → Falsehpos:¬x > 0⊢ False apply this _ _ _ (show -x>0 by⊢ ¬∃ x, x ^ 2 = 2 simpx:ℚhx:x ^ 2 = 2hnon:x ≠ 0this:∀ (x : ℚ), x ^ 2 = 2 → x ≠ 0 → x > 0 → Falsehpos:¬x > 0⊢ x < 0; orderAll goals completed! 🐙) <;>x:ℚhx:x ^ 2 = 2hnon:x ≠ 0this:∀ (x : ℚ), x ^ 2 = 2 → x ≠ 0 → x > 0 → Falsehpos:¬x > 0⊢ (-x) ^ 2 = 2x:ℚhx:x ^ 2 = 2hnon:x ≠ 0this:∀ (x : ℚ), x ^ 2 = 2 → x ≠ 0 → x > 0 → Falsehpos:¬x > 0⊢ -x ≠ 0 grindAll goals completed! 🐙
  have hrep : ∃ p q:ℕ, p > 0 ∧ q > 0 ∧ p^2 = 2*q^2 := by⊢ ¬∃ x, x ^ 2 = 2
    use x.num.toNat, x.denhx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0⊢ x.num.toNat > 0 ∧ x.den > 0 ∧ x.num.toNat ^ 2 = 2 * x.den ^ 2
    observe hnum_pos : x.num > 0hx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.num⊢ x.num.toNat > 0 ∧ x.den > 0 ∧ x.num.toNat ^ 2 = 2 * x.den ^ 2
    observe hden_pos : x.den > 0hx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.den⊢ x.num.toNat > 0 ∧ x.den > 0 ∧ x.num.toNat ^ 2 = 2 * x.den ^ 2
    refine ⟨ byx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.den⊢ x.num.toNat > 0 simp [hpos]All goals completed! 🐙, hden_pos, ?_ ⟩
    rw [←num_div_den x] at hxhx:ℚhx:(↑x.num / ↑x.den) ^ 2 = 2hnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.den⊢ x.num.toNat ^ 2 = 2 * x.den ^ 2; field_simp at hxhx:ℚhnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.denhx:↑x.num ^ 2 = ↑x.den ^ 2 * 2⊢ x.num.toNat ^ 2 = 2 * x.den ^ 2
    have hnum_cast : x.num = x.num.toNat := Int.eq_natCast_toNat.mpr (byx:ℚhnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.denhx:↑x.num ^ 2 = ↑x.den ^ 2 * 2⊢ 0 ≤ x.num positivityAll goals completed! 🐙)
    rw [hnum_cast] at hxhx:ℚhnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.denhx:↑↑x.num.toNat ^ 2 = ↑x.den ^ 2 * 2hnum_cast:x.num = ↑x.num.toNat⊢ x.num.toNat ^ 2 = 2 * x.den ^ 2; norm_cast at hxhx:ℚhnon:x ≠ 0hpos:x > 0hnum_pos:0 < x.numhden_pos:0 < x.denhnum_cast:x.num = ↑x.num.toNathx:x.num.toNat ^ 2 = x.den ^ 2 * 2⊢ x.num.toNat ^ 2 = 2 * x.den ^ 2; grindAll goals completed! 🐙
  set P : ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p^2 = 2*q^2x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2⊢ False
  have hP : ∃ p, P p := by⊢ ¬∃ x, x ^ 2 = 2 aesopAll goals completed! 🐙
  have hiter (p:ℕ) (hPp: P p) : ∃ q, q < p ∧ P q := by⊢ ¬∃ x, x ^ 2 = 2
    obtain hp | hp := p.even_or_odd''inlx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:Even p⊢ ∃ q < p, P qinrx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:Odd p⊢ ∃ q < p, P q
    .inlx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:Even p⊢ ∃ q < p, P q rw [even_iff_exists_two_mul] at hpinlx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:∃ b, p = 2 * b⊢ ∃ q < p, P q
      obtain ⟨ k, rfl ⟩ := hpinlx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)⊢ ∃ q < 2 * k, P q
      choose q hpos hq using hPp.2inlx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)q:ℕhpos:q > 0hq:(2 * k) ^ 2 = 2 * q ^ 2⊢ ∃ q < 2 * k, P q
      have : q^2 = 2 * k^2 := by⊢ ¬∃ x, x ^ 2 = 2 linarithAll goals completed! 🐙
      use qhx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)q:ℕhpos:q > 0hq:(2 * k) ^ 2 = 2 * q ^ 2this:q ^ 2 = 2 * k ^ 2⊢ q < 2 * k ∧ P q; constructorh.leftx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)q:ℕhpos:q > 0hq:(2 * k) ^ 2 = 2 * q ^ 2this:q ^ 2 = 2 * k ^ 2⊢ q < 2 * kh.rightx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)q:ℕhpos:q > 0hq:(2 * k) ^ 2 = 2 * q ^ 2this:q ^ 2 = 2 * k ^ 2⊢ P q
      .h.leftx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)q:ℕhpos:q > 0hq:(2 * k) ^ 2 = 2 * q ^ 2this:q ^ 2 = 2 * k ^ 2⊢ q < 2 * k sorryAll goals completed! 🐙
      exact ⟨ hpos, k, byx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pk:ℕhPp:P (2 * k)q:ℕhpos:q > 0hq:(2 * k) ^ 2 = 2 * q ^ 2this:q ^ 2 = 2 * k ^ 2⊢ k > 0 linarith [hPp.1]All goals completed! 🐙, this ⟩
    have h1 : Odd (p^2) := by⊢ ¬∃ x, x ^ 2 = 2
      sorryAll goals completed! 🐙
    have h2 : Even (p^2) := by⊢ ¬∃ x, x ^ 2 = 2
      choose q hpos hq using hPp.2x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:Odd ph1:Odd (p ^ 2)q:ℕhpos:q > 0hq:p ^ 2 = 2 * q ^ 2⊢ Even (p ^ 2)
      rw [even_iff_exists_two_mul]x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos✝:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:Odd ph1:Odd (p ^ 2)q:ℕhpos:q > 0hq:p ^ 2 = 2 * q ^ 2⊢ ∃ b, p ^ 2 = 2 * b
      use q^2All goals completed! 🐙
    observe : ¬(Even (p ^ 2) ∧ Odd (p ^ 2))inrx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P pp:ℕhPp:P php:Odd ph1:Odd (p ^ 2)h2:Even (p ^ 2)this:¬(Even (p ^ 2) ∧ Odd (p ^ 2))⊢ ∃ q < p, P q
    tautoAll goals completed! 🐙
  classical
  set f : ℕ → ℕ := fun p ↦ if hPp: P p then (hiter p hPp).choose else 0x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P phiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0⊢ False
  have hf (p:ℕ) (hPp: P p) : (f p < p) ∧ P (f p) := by⊢ ¬∃ x, x ^ 2 = 2
    simp [f, hPp]x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hP:∃ p, P phiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0p:ℕhPp:P p⊢ ⋯.choose < p ∧ P ⋯.choose; exact (hiter p hPp).choose_specAll goals completed! 🐙
  choose p hP using hPx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0hf:∀ (p : ℕ), P p → f p < p ∧ P (f p)p:ℕhP:P p⊢ False
  set a : ℕ → ℕ := Nat.rec p (fun n p ↦ f p)x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0hf:∀ (p : ℕ), P p → f p < p ∧ P (f p)p:ℕhP:P pa:ℕ → ℕ := fun t ↦ Nat.rec p (fun n p ↦ f p) t⊢ False
  have ha (n:ℕ) : P (a n) := by⊢ ¬∃ x, x ^ 2 = 2
    induction n with
    | zero =>zerox:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0hf:∀ (p : ℕ), P p → f p < p ∧ P (f p)p:ℕhP:P pa:ℕ → ℕ := fun t ↦ Nat.rec p (fun n p ↦ f p) t⊢ P (a 0) exact hPAll goals completed! 🐙
    | succ n ih =>succx:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0hf:∀ (p : ℕ), P p → f p < p ∧ P (f p)p:ℕhP:P pa:ℕ → ℕ := fun t ↦ Nat.rec p (fun n p ↦ f p) tn:ℕih:P (a n)⊢ P (a (n + 1)) exact (hf _ ih).2All goals completed! 🐙
  have hlt (n:ℕ) : a (n+1) < a n := by⊢ ¬∃ x, x ^ 2 = 2
    have : a (n+1) = f (a n) := n.rec_add_one p (fun n p ↦ f p)x:ℚhx:x ^ 2 = 2hnon:x ≠ 0hpos:x > 0hrep:∃ p q, p > 0 ∧ q > 0 ∧ p ^ 2 = 2 * q ^ 2P:ℕ → Prop := fun p ↦ p > 0 ∧ ∃ q > 0, p ^ 2 = 2 * q ^ 2hiter:∀ (p : ℕ), P p → ∃ q < p, P qf:ℕ → ℕ := fun p ↦ if hPp : P p then ⋯.choose else 0hf:∀ (p : ℕ), P p → f p < p ∧ P (f p)p:ℕhP:P pa:ℕ → ℕ := fun t ↦ Nat.rec p (fun n p ↦ f p) tha:∀ (n : ℕ), P (a n)n:ℕthis:a (n + 1) = f (a n)⊢ a (n + 1) < a n
    grindAll goals completed! 🐙
  exact Nat.no_infinite_descent ⟨ a, hlt ⟩All goals completed! 🐙

Proposition 4.4.5

theorem Rat.exist_approx_sqrt_two {ε:ℚ} (hε:ε>0) : ∃ x ≥ (0:ℚ), x^2 < 2 ∧ 2 < (x+ε)^2 := byε:ℚhε:ε > 0⊢ ∃ x ≥ 0, x ^ 2 < 2 ∧ 2 < (x + ε) ^ 2
  -- This proof is written to follow the structure of the original text.
  by_contra! hε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2⊢ False
  have (n:ℕ): (n*ε)^2 < 2 := byε:ℚhε:ε > 0⊢ ∃ x ≥ 0, x ^ 2 < 2 ∧ 2 < (x + ε) ^ 2
    induction' n with n hnzeroε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2⊢ (↑0 * ε) ^ 2 < 2succε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2n:ℕhn:(↑n * ε) ^ 2 < 2⊢ (↑(n + 1) * ε) ^ 2 < 2; simpsuccε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2n:ℕhn:(↑n * ε) ^ 2 < 2⊢ (↑(n + 1) * ε) ^ 2 < 2
    simp [add_mul]succε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2n:ℕhn:(↑n * ε) ^ 2 < 2⊢ (↑n * ε + ε) ^ 2 < 2
    apply lt_of_le_of_ne (h (n*ε) (byε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2n:ℕhn:(↑n * ε) ^ 2 < 2⊢ ↑n * ε ≥ 0 positivityAll goals completed! 🐙) hn)
    have := not_exist_sqrt_twosuccε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2n:ℕhn:(↑n * ε) ^ 2 < 2this:¬∃ x, x ^ 2 = 2⊢ (↑n * ε + ε) ^ 2 ≠ 2
    aesopAll goals completed! 🐙
  choose n hn using Nat.exists_gt (2/ε)ε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:↑n > 2 / ε⊢ False
  rw [gt_iff_lt, div_lt_iff₀', mul_comm, ←sq_lt_sq₀] at hnε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 ^ 2 < (↑n * ε) ^ 2⊢ Falsehaε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 < ↑n * ε⊢ 0 ≤ 2hbε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 < ↑n * ε⊢ 0 ≤ ↑n * εε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 / ε < ↑n⊢ 0 < ε <;>ε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 ^ 2 < (↑n * ε) ^ 2⊢ Falsehaε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 < ↑n * ε⊢ 0 ≤ 2hbε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 < ↑n * ε⊢ 0 ≤ ↑n * εε:ℚhε:ε > 0h:∀ x ≥ 0, x ^ 2 < 2 → (x + ε) ^ 2 ≤ 2this:∀ (n : ℕ), (↑n * ε) ^ 2 < 2n:ℕhn:2 / ε < ↑n⊢ 0 < ε try positivityAll goals completed! 🐙
  grindAll goals completed! 🐙

/-- Example 4.4.6 -/
example :
  let ε:ℚ := 1/1000
  let x:ℚ := 1414/1000
  x^2 < 2 ∧ 2 < (x+ε)^2 := by⊢ let ε := 1 / 1000;
let x := 1414 / 1000;
x ^ 2 < 2 ∧ 2 < (x + ε) ^ 2 norm_numAll goals completed! 🐙