-- Mathlib/NumberTheory/DiophantineApproximation/Basic.lean
theorem exists_rat_abs_sub_lt_and_lt_of_irrational {ξ : ℝ} (hξ : Irrational ξ) (q : ℚ) :
    ∃ q' : ℚ, |ξ - q'| < 1 / (q'.den : ℝ) ^ 2 ∧ |ξ - q'| < |ξ - q|

Proof state:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ ↑q'.den < ↑m + 1

Tactic:
exact lt_add_of_le_of_pos (Nat.cast_le.mpr hden) zero_lt_one

Result:
no goals

Proof state:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
refine
  ⟨q', lt_of_le_of_lt hbd ?_,
    lt_of_le_of_lt hbd <|
      (one_div_lt md_pos h).mpr <|
        hm.trans <|
          lt_of_lt_of_le (lt_add_one _) <|
            (le_mul_iff_one_le_right <| add_pos m_pos zero_lt_one).mpr <|
              mod_cast (q'.pos : 1 ≤ q'.den)⟩

Result:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ 1 / ((↑m + 1) * ↑q'.den) < 1 / ↑q'.den ^ 2

Proof state:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
have den_pos : (0 : ℝ) < q'.den := Nat.cast_pos.mpr q'.pos

Result:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Proof state:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
have md_pos := mul_pos (add_pos m_pos zero_lt_one) den_pos

Result:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Proof state:
case intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
obtain ⟨q', hbd, hden⟩ := exists_rat_abs_sub_le_and_den_le ξ (Nat.cast_pos.mp m_pos)

Result:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Proof state:
ξ : ℝ
hξ : Irrational ξ
q : ℚ
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
have h := abs_pos.mpr (sub_ne_zero.mpr <| Irrational.ne_rat hξ q)

Result:
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Proof state:
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
obtain ⟨m, hm⟩ := exists_nat_gt (1 / |ξ - q|)

Result:
case intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Proof state:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ 1 / ((↑m + 1) * ↑q'.den) < 1 / ↑q'.den ^ 2

Tactic:
rw [sq, one_div_lt_one_div md_pos (mul_pos den_pos den_pos), mul_lt_mul_right den_pos]

Result:
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ ↑q'.den < ↑m + 1

Proof state:
case intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|

Tactic:
have m_pos : (0 : ℝ) < m := (one_div_pos.mpr h).trans hm

Result:
case intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
⊢ ∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|