l3lab/ntp-mathlib huggingface dataset:
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        

Proof state:
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m) ∧
    ↑m ∣ ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m)

Next tactic:
have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        

Proof state:
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m) ∧
    ↑m ∣ ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m)

Next tactic:
have := gcd_eq_gcd_ab n m
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        

Proof state:
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m) ∧
    ↑m ∣ ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m)

Next tactic:
constructor
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> 

Proof state:
case left
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m)

Next tactic:
rw [Int.emod_def, ← sub_add]
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> 

Proof state:
case right
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑m ∣ ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) % ↑(lcm n m)

Next tactic:
rw [Int.emod_def, ← sub_add]
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            

Proof state:
case left
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣
    ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) +
      ↑(lcm n m) * ((↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) / ↑(lcm n m))

Next tactic:
refine' dvd_add _ (dvd_mul_of_dvd_left _ _)
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            

Proof state:
case right
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑m ∣
    ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) +
      ↑(lcm n m) * ((↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m) / ↑(lcm n m))

Next tactic:
refine' dvd_add _ (dvd_mul_of_dvd_left _ _)
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          

Proof state:
case left.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m)

Next tactic:
try norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try 

Proof state:
case left.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m)

Next tactic:
norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          

Proof state:
case left.refine'_2
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑(lcm n m)

Next tactic:
try norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try 

Proof state:
case left.refine'_2
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑(lcm n m)

Next tactic:
norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          

Proof state:
case right.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑m ∣ ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m)

Next tactic:
try norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try 

Proof state:
case right.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑m ∣ ↑b - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m)

Next tactic:
norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          

Proof state:
case right.refine'_2
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑m ∣ ↑(lcm n m)

Next tactic:
try norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try 

Proof state:
case right.refine'_2
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑m ∣ ↑(lcm n m)

Next tactic:
norm_cast
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try norm_cast
        · 

Proof state:
case left.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) = ↑n * gcdA n m + ↑m * gcdB n m
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m)

Next tactic:
rw [← sub_eq_iff_eq_add'] at this
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try norm_cast
        · rw [← sub_eq_iff_eq_add'] at this
          

Proof state:
case left.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) - ↑n * gcdA n m = ↑m * gcdB n m
⊢ ↑n ∣ ↑a - (↑n * gcdA n m * ↑b + ↑m * gcdB n m * ↑a) / ↑(gcd n m)

Next tactic:
rw [← this, sub_mul, ← add_sub_assoc, add_comm, add_sub_assoc, ← mul_sub,
            Int.add_ediv_of_dvd_left, Int.mul_ediv_cancel_left _ hnonzero,
            Int.mul_ediv_assoc _ h.dvd, ← sub_sub, sub_self, zero_sub, dvd_neg, mul_assoc]
/-- The natural number less than `lcm n m` congruent to `a` mod `n` and `b` mod `m` -/
def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k ≡ b [MOD m] } :=
  if hn : n = 0 then ⟨a, by rw [hn, gcd_zero_left] at h; constructor; rfl; exact h⟩
  else
    if hm : m = 0 then ⟨b, by rw [hm, gcd_zero_right] at h; constructor; exact h.symm; rfl⟩
    else
      ⟨let (c, d) := xgcd n m; Int.toNat ((n * c * b + m * d * a) / gcd n m % lcm n m), by
        rw [xgcd_val]
        dsimp
        rw [modEq_iff_dvd, modEq_iff_dvd,
          Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]
        have hnonzero : (gcd n m : ℤ) ≠ 0 := by
          norm_cast
          rw [Nat.gcd_eq_zero_iff, not_and]
          exact fun _ => hm
        have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
        have := gcd_eq_gcd_ab n m
        constructor <;> rw [Int.emod_def, ← sub_add] <;>
            refine' dvd_add _ (dvd_mul_of_dvd_left _ _) <;>
          try norm_cast
        · rw [← sub_eq_iff_eq_add'] at this
          rw [← this, sub_mul, ← add_sub_assoc, add_comm, add_sub_assoc, ← mul_sub,
            Int.add_ediv_of_dvd_left, Int.mul_ediv_cancel_left _ hnonzero,
            Int.mul_ediv_assoc _ h.dvd, ← sub_sub, sub_self, zero_sub, dvd_neg, mul_assoc]
          

Proof state:
case left.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) - ↑n * gcdA n m = ↑m * gcdB n m
⊢ ↑n ∣ ↑n * (gcdA n m * ((↑b - ↑a) / ↑(gcd n m)))
case left.refine'_1
m n a b c d : ℕ
h : a ≡ b [MOD gcd n m]
hn : ¬n = 0
hm : ¬m = 0
hnonzero : ↑(gcd n m) ≠ 0
hcoedvd : ∀ (t : ℤ), ↑(gcd n m) ∣ t * (↑b - ↑a)
this : ↑(gcd n m) - ↑n * gcdA n m = ↑m * gcdB n m
⊢ ↑(gcd n m) ∣ ↑(gcd n m) * ↑a

Next tactic:
exact dvd_mul_right _ _