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`
  

Proof state:
ξ : ℝ
u v✝ : ℤ
hv : 2 ≤ v✝
h : ContfracLegendre.Ass ξ u v✝
v : ℕ
h' : ContfracLegendre.Ass ξ u ↑v
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
clear 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; 

Proof state:
ξ : ℝ
u v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
h' : ContfracLegendre.Ass ξ u ↑v
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
have h := 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'; 

Proof state:
ξ : ℝ
u v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
h' h : ContfracLegendre.Ass ξ u ↑v
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
clear 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'
  

Proof state:
ξ : ℝ
u v✝ : ℤ
hv : 2 ≤ v✝
v : ℕ
h : ContfracLegendre.Ass ξ u ↑v
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
/-- 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 => ?_
  

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
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
rcases lt_trichotomy v 1 with (ht | rfl | ht)
/-- 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)
  · 

Proof state:
case h.inl
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 : v < 1
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
replace h := h.2.2
/-- 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
    

Proof state:
case h.inl
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
ht : v < 1
h : |ξ - ↑u / ↑↑v| < (↑↑v * (2 * ↑↑v - 1))⁻¹
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
      cast_zero, inv_zero] at 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
    

Proof state:
case h.inl
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
ht : v < 1
h : |ξ| < 0
⊢ ∃ n, ↑u / ↑v = convergent ξ n

Next tactic:
exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt 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)
  · 

Proof state:
case h.inr.inl
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
h : ContfracLegendre.Ass ξ u ↑1
⊢ ∃ n, ↑u / ↑1 = convergent ξ n

Next tactic:
rw [Nat.cast_one, div_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]
    

Proof state:
case h.inr.inl
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
h : ContfracLegendre.Ass ξ u ↑1
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
obtain ⟨_, h₁, h₂⟩ := 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
    

Proof state:
case h.inr.inl.intro.intro
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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
rcases le_or_lt (u : ℝ) ξ with ht | ht
/-- 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
    · 

Proof state:
case h.inr.inl.intro.intro.inl
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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ↑u ≤ ξ
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
use 0
/-- 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
      

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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ↑u ≤ ξ
⊢ ↑u = convergent ξ 0

Next tactic:
rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
/-- 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]
      

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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ↑u ≤ ξ
⊢ ↑u ≤ ξ ∧ ξ < ↑u + 1

Next tactic:
convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2)
/-- 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) <;> 

Proof state:
case h.e'_2.h.e'_4.h.e'_5
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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ↑u ≤ ξ
⊢ ↑u = ↑u / ↑↑1

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) <;> 

Proof state:
case h.e'_2.h.e'_4.h.e'_6
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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ↑u ≤ ξ
⊢ 1 = (↑↑1 * (2 * ↑↑1 - 1))⁻¹

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
    · 

Proof state:
case h.inr.inl.intro.intro.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
h₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u
h₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹
ht : ξ < ↑u
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
/-- 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)
      

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

Next tactic:
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⟩
/-- 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
        

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
⊢ ⌊ξ⌋ = u - 1

Next tactic:
rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
/-- 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]
        

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
⊢ ↑u - 1 ≤ ξ ∧ ξ < ↑u

Next tactic:
exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
/-- 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⟩
      

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

Next tactic:
rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
/-- 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
      · 

Proof state:
case h.inr.inl.intro.intro.inr.inl
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:
rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
/-- 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
        

Proof state:
case h.inr.inl.intro.intro.inr.inl
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 : 1 < 1 / 2
hξ₁ : ⌊ξ⌋ = u - 1
Hξ : ξ = ↑⌊ξ⌋
⊢ ∃ n, ↑u = convergent ξ n

Next tactic:
exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)