l3lab/ntp-mathlib huggingface dataset:
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    

Proof state:
case h.inr.inr.intro
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; rwa [toNat_of_nonneg huv₀.le]
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by 

Proof state:
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
⊢ toNat (u - ⌊ξ⌋ * ↑v) < v

Next tactic:
zify
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; 

Proof state:
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
⊢ ↑(toNat (u - ⌊ξ⌋ * ↑v)) < ↑v

Next tactic:
rwa [toNat_of_nonneg huv₀.le]
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; rwa [toNat_of_nonneg huv₀.le]
    

Proof state:
case h.inr.inr.intro
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
huv₁' : toNat (u - ⌊ξ⌋ * ↑v) < v
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
have inv : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * ↑v).toNat :=
      (toNat_of_nonneg huv₀.le).symm ▸ invariant (Nat.cast_le.mpr ht) h
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; rwa [toNat_of_nonneg huv₀.le]
    have inv : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * ↑v).toNat :=
      (toNat_of_nonneg huv₀.le).symm ▸ invariant (Nat.cast_le.mpr ht) h
    

Proof state:
case h.inr.inr.intro
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
huv₁' : toNat (u - ⌊ξ⌋ * ↑v) < v
inv : ContfracLegendre.Ass (fract ξ)⁻¹ ↑v ↑(toNat (u - ⌊ξ⌋ * ↑v))
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
obtain ⟨n, hn⟩ := ih (u - ⌊ξ⌋ * v).toNat huv₁' inv
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; rwa [toNat_of_nonneg huv₀.le]
    have inv : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * ↑v).toNat :=
      (toNat_of_nonneg huv₀.le).symm ▸ invariant (Nat.cast_le.mpr ht) h
    obtain ⟨n, hn⟩ := ih (u - ⌊ξ⌋ * v).toNat huv₁' inv
    

Proof state:
case h.inr.inr.intro.intro
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
huv₁' : toNat (u - ⌊ξ⌋ * ↑v) < v
inv : ContfracLegendre.Ass (fract ξ)⁻¹ ↑v ↑(toNat (u - ⌊ξ⌋ * ↑v))
n : ℕ
hn : ↑↑v / ↑(toNat (u - ⌊ξ⌋ * ↑v)) = convergent (fract ξ)⁻¹ n
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
use n + 1
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h' : ContfracLegendre.Ass ξ u v) :
    ∃ n, (u / v : ℚ) = ξ.convergent n := by
  -- Porting note: `induction` got in trouble because of the irrelevant hypothesis `h`
  clear h; have h := h'; clear h'
  induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
  rcases lt_trichotomy v 1 with (ht | rfl | ht)
  · replace h := h.2.2
    simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at h
    exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
  · rw [Nat.cast_one, div_one]
    obtain ⟨_, h₁, h₂⟩ := h
    rcases le_or_lt (u : ℝ) ξ with ht | ht
    · use 0
      rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
      convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
    · replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
      have hξ₁ : ⌊ξ⌋ = u - 1 := by
        rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
        exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
      rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
      · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
        exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
      · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
          refine' ⟨(fract_lt_one ξ).le, _⟩
          rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
          convert h₁ using 1
          -- Porting note: added (`convert` handled this in lean 3)
          rw [sub_eq_add_neg]
          norm_num
        use 1
        simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
  · obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
    have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
    have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; rwa [toNat_of_nonneg huv₀.le]
    have inv : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * ↑v).toNat :=
      (toNat_of_nonneg huv₀.le).symm ▸ invariant (Nat.cast_le.mpr ht) h
    obtain ⟨n, hn⟩ := ih (u - ⌊ξ⌋ * v).toNat huv₁' inv
    use n + 1
    

Proof state:
case h
v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
ih : ∀ m < v, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
ξ : ℝ
u : ℤ
h : ContfracLegendre.Ass ξ u ↑v
ht : 1 < v
huv₀ : 0 < u - ⌊ξ⌋ * ↑v
huv₁ : u - ⌊ξ⌋ * ↑v < ↑v
Hv : ↑v ≠ 0
huv₁' : toNat (u - ⌊ξ⌋ * ↑v) < v
inv : ContfracLegendre.Ass (fract ξ)⁻¹ ↑v ↑(toNat (u - ⌊ξ⌋ * ↑v))
n : ℕ
hn : ↑↑v / ↑(toNat (u - ⌊ξ⌋ * ↑v)) = convergent (fract ξ)⁻¹ n
⊢ ↑u / ↑v = convergent ξ (n + 1)

Next tactic:
rw [convergent_succ, ← hn,
      (mod_cast toNat_of_nonneg huv₀.le : ((u - ⌊ξ⌋ * v).toNat : ℚ) = u - ⌊ξ⌋ * v),
      cast_ofNat, inv_div, sub_div, mul_div_cancel _ Hv, add_sub_cancel'_right]
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  

Proof state:
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
⊢ ∃ n, q = convergent ξ n

Next tactic:
refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
  · 

Proof state:
case refine'_1
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
⊢ IsCoprime q.num ↑q.den

Next tactic:
exact coprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
  · exact coprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
  · 

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
hd : ↑q.den = 1
⊢ -(1 / 2) < ξ - ↑q.num

Next tactic:
rw [← q.den_eq_one_iff.mp (Nat.cast_eq_one.mp hd)] at h
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
  · exact coprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
  · rw [← q.den_eq_one_iff.mp (Nat.cast_eq_one.mp hd)] at h
    

Proof state:
case refine'_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑↑q.num| < 1 / (2 * ↑(↑q.num).den ^ 2)
hd : ↑q.den = 1
⊢ -(1 / 2) < ξ - ↑q.num

Next tactic:
simpa only [Rat.coe_int_den, Nat.cast_one, one_pow, mul_one] using (abs_lt.mp h).1
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
  · exact coprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
  · rw [← q.den_eq_one_iff.mp (Nat.cast_eq_one.mp hd)] at h
    simpa only [Rat.coe_int_den, Nat.cast_one, one_pow, mul_one] using (abs_lt.mp h).1
  · 

Proof state:
case refine'_3
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
⊢ |ξ - ↑q.num / ↑↑q.den| < (↑↑q.den * (2 * ↑↑q.den - 1))⁻¹

Next tactic:
obtain ⟨hq₀, hq₁⟩ := aux₀ (Nat.cast_pos.mpr q.pos)
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
  · exact coprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
  · rw [← q.den_eq_one_iff.mp (Nat.cast_eq_one.mp hd)] at h
    simpa only [Rat.coe_int_den, Nat.cast_one, one_pow, mul_one] using (abs_lt.mp h).1
  · obtain ⟨hq₀, hq₁⟩ := aux₀ (Nat.cast_pos.mpr q.pos)
    

Proof state:
case refine'_3.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
hq₀ : 0 < ↑↑q.den
hq₁ : 0 < 2 * ↑↑q.den - 1
⊢ |ξ - ↑q.num / ↑↑q.den| < (↑↑q.den * (2 * ↑↑q.den - 1))⁻¹

Next tactic:
replace hq₁ := mul_pos hq₀ hq₁
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
    ∃ n, q = ξ.convergent n := by
  refine' q.num_div_den ▸ exists_rat_eq_convergent' ⟨_, fun hd => _, _⟩
  · exact coprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
  · rw [← q.den_eq_one_iff.mp (Nat.cast_eq_one.mp hd)] at h
    simpa only [Rat.coe_int_den, Nat.cast_one, one_pow, mul_one] using (abs_lt.mp h).1
  · obtain ⟨hq₀, hq₁⟩ := aux₀ (Nat.cast_pos.mpr q.pos)
    replace hq₁ := mul_pos hq₀ hq₁
    

Proof state:
case refine'_3.intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h✝ : ContfracLegendre.Ass ξ u v
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
hq₀ : 0 < ↑↑q.den
hq₁ : 0 < ↑↑q.den * (2 * ↑↑q.den - 1)
⊢ |ξ - ↑q.num / ↑↑q.den| < (↑↑q.den * (2 * ↑↑q.den - 1))⁻¹

Next tactic:
have hq₂ : (0 : ℝ) < 2 * (q.den * q.den) := mul_pos zero_lt_two (mul_pos hq₀ hq₀)