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)
      · 

Proof state:
case h.inr.inl.intro.intro.inr.inr
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
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
/-- 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
          

Proof state:
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ ⌊(fract ξ)⁻¹⌋ = 1

Next tactic:
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]
/-- 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]
          

Proof state:
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ fract ξ ≤ 1 ∧ 2⁻¹ < fract ξ

Next tactic:
refine' ⟨(fract_lt_one ξ).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, _⟩
          

Proof state:
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ 2⁻¹ < fract ξ

Next tactic:
rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
/-- 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]
          

Proof state:
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ ↑u - (1 - 2⁻¹) < ξ

Next tactic:
convert h₁ using 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)
          

Proof state:
case h.e'_3
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ ↑u - (1 - 2⁻¹) = ↑u + -(1 / 2)

Next tactic:
rw [sub_eq_add_neg]
/-- 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]
          

Proof state:
case h.e'_3
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
⊢ ↑u + -(1 - 2⁻¹) = ↑u + -(1 / 2)

Next tactic:
norm_num
/-- 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
        

Proof state:
case h.inr.inl.intro.intro.inr.inr
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
use 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
        

Proof state:
case h
v : ℤ
hv : 2 ≤ v
ξ : ℝ
u : ℤ
ih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = convergent ξ n
left✝ : IsCoprime u ↑1
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
h₁ : ↑u + -(1 / 2) < ξ
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ ≠ ↑⌊ξ⌋
hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1
⊢ ↑u = convergent ξ 1

Next tactic:
simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
/-- 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]
  · 

Proof state:
case h.inr.inr
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
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
obtain ⟨huv₀, huv₁⟩ := aux₂ (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
    

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
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'