l3lab/ntp-mathlib huggingface dataset:
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    

Proof state:
c : ℤ
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
h : Minimal (↑p * a1) (↑p * b1) c
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
⊢ ↑p ^ 2 ∣ c

Next tactic:
rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    

Proof state:
c : ℤ
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
h : Minimal (↑p * a1) (↑p * b1) c
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
⊢ (↑p ^ 2) ^ 2 ∣ (↑p * a1) ^ 4 + (↑p * b1) ^ 4

Next tactic:
apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    

Proof state:
c : ℤ
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
h : Minimal (↑p * a1) (↑p * b1) c
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
⊢ (↑p ^ 2) ^ 2 * (a1 ^ 4 + b1 ^ 4) = (↑p * a1) ^ 4 + (↑p * b1) ^ 4

Next tactic:
ring
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring
  

Proof state:
case intro.intro.intro.intro.intro
c : ℤ
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
h : Minimal (↑p * a1) (↑p * b1) c
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
hpc : ↑p ^ 2 ∣ c
⊢ False

Next tactic:
obtain ⟨c1, rfl⟩ := hpc
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring
  obtain ⟨c1, rfl⟩ := hpc
  

Proof state:
case intro.intro.intro.intro.intro.intro
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
c1 : ℤ
h : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)
⊢ False

Next tactic:
have hf : Fermat42 a1 b1 c1 :=
    (Fermat42.mul (Int.coe_nat_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring
  obtain ⟨c1, rfl⟩ := hpc
  have hf : Fermat42 a1 b1 c1 :=
    (Fermat42.mul (Int.coe_nat_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1
  

Proof state:
case intro.intro.intro.intro.intro.intro
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
c1 : ℤ
h : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)
hf : Fermat42 a1 b1 c1
⊢ False

Next tactic:
apply Nat.le_lt_asymm (h.2 _ _ _ hf)
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring
  obtain ⟨c1, rfl⟩ := hpc
  have hf : Fermat42 a1 b1 c1 :=
    (Fermat42.mul (Int.coe_nat_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1
  apply Nat.le_lt_asymm (h.2 _ _ _ hf)
  

Proof state:
case intro.intro.intro.intro.intro.intro
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
c1 : ℤ
h : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)
hf : Fermat42 a1 b1 c1
⊢ Int.natAbs c1 < Int.natAbs (↑p ^ 2 * c1)

Next tactic:
rw [Int.natAbs_mul, lt_mul_iff_one_lt_left, Int.natAbs_pow, Int.natAbs_ofNat]
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring
  obtain ⟨c1, rfl⟩ := hpc
  have hf : Fermat42 a1 b1 c1 :=
    (Fermat42.mul (Int.coe_nat_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1
  apply Nat.le_lt_asymm (h.2 _ _ _ hf)
  rw [Int.natAbs_mul, lt_mul_iff_one_lt_left, Int.natAbs_pow, Int.natAbs_ofNat]
  · 

Proof state:
case intro.intro.intro.intro.intro.intro
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
c1 : ℤ
h : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)
hf : Fermat42 a1 b1 c1
⊢ 1 < p ^ 2

Next tactic:
exact Nat.one_lt_pow _ _ two_ne_zero (Nat.Prime.one_lt hp)
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring
  obtain ⟨c1, rfl⟩ := hpc
  have hf : Fermat42 a1 b1 c1 :=
    (Fermat42.mul (Int.coe_nat_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1
  apply Nat.le_lt_asymm (h.2 _ _ _ hf)
  rw [Int.natAbs_mul, lt_mul_iff_one_lt_left, Int.natAbs_pow, Int.natAbs_ofNat]
  · exact Nat.one_lt_pow _ _ two_ne_zero (Nat.Prime.one_lt hp)
  · 

Proof state:
case intro.intro.intro.intro.intro.intro
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
c1 : ℤ
h : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)
hf : Fermat42 a1 b1 c1
⊢ 0 < Int.natAbs c1

Next tactic:
exact Nat.pos_of_ne_zero (Int.natAbs_ne_zero.2 (ne_zero hf))
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has positive `c`. -/
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
  

Proof state:
a b c : ℤ
⊢ Minimal a b c → Minimal a b (-c)

Next tactic:
rintro ⟨⟨ha, hb, heq⟩, h2⟩
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has positive `c`. -/
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
  rintro ⟨⟨ha, hb, heq⟩, h2⟩
  

Proof state:
case intro.intro.intro
a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs c ≤ Int.natAbs c1
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ Minimal a b (-c)

Next tactic:
constructor
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has positive `c`. -/
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
  rintro ⟨⟨ha, hb, heq⟩, h2⟩
  constructor
  · 

Proof state:
case intro.intro.intro.left
a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs c ≤ Int.natAbs c1
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ Fermat42 a b (-c)

Next tactic:
apply And.intro ha (And.intro hb _)
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has positive `c`. -/
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
  rintro ⟨⟨ha, hb, heq⟩, h2⟩
  constructor
  · apply And.intro ha (And.intro hb _)
    

Proof state:
a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs c ≤ Int.natAbs c1
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ a ^ 4 + b ^ 4 = (-c) ^ 2

Next tactic:
rw [heq]
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has positive `c`. -/
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
  rintro ⟨⟨ha, hb, heq⟩, h2⟩
  constructor
  · apply And.intro ha (And.intro hb _)
    rw [heq]
    

Proof state:
a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs c ≤ Int.natAbs c1
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ c ^ 2 = (-c) ^ 2

Next tactic:
exact (neg_sq c).symm
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has positive `c`. -/
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
  rintro ⟨⟨ha, hb, heq⟩, h2⟩
  constructor
  · apply And.intro ha (And.intro hb _)
    rw [heq]
    exact (neg_sq c).symm
  

Proof state:
case intro.intro.intro.right
a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs c ≤ Int.natAbs c1
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs (-c) ≤ Int.natAbs c1

Next tactic:
rwa [Int.natAbs_neg c]
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1

Next tactic:
obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  

Proof state:
case intro.intro.intro
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1

Next tactic:
cases' Int.emod_two_eq_zero_or_one a0 with hap hap
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · 

Proof state:
case intro.intro.intro.inl
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1

Next tactic:
cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · 

Proof state:
case intro.intro.intro.inl.inl
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
hbp : b0 % 2 = 0
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1

Next tactic:
exfalso
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · exfalso
      

Proof state:
case intro.intro.intro.inl.inl.h
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
hbp : b0 % 2 = 0
⊢ False

Next tactic:
have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) :=
        Int.dvd_gcd (Int.dvd_of_emod_eq_zero hap) (Int.dvd_of_emod_eq_zero hbp)
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · exfalso
      have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) :=
        Int.dvd_gcd (Int.dvd_of_emod_eq_zero hap) (Int.dvd_of_emod_eq_zero hbp)
      

Proof state:
case intro.intro.intro.inl.inl.h
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
hbp : b0 % 2 = 0
h1 : 2 ∣ ↑(Int.gcd a0 b0)
⊢ False

Next tactic:
rw [Int.gcd_eq_one_iff_coprime.mpr (coprime_of_minimal hf)] at h1
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · exfalso
      have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) :=
        Int.dvd_gcd (Int.dvd_of_emod_eq_zero hap) (Int.dvd_of_emod_eq_zero hbp)
      rw [Int.gcd_eq_one_iff_coprime.mpr (coprime_of_minimal hf)] at h1
      

Proof state:
case intro.intro.intro.inl.inl.h
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
hbp : b0 % 2 = 0
h1 : 2 ∣ ↑1
⊢ False

Next tactic:
revert h1
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · exfalso
      have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) :=
        Int.dvd_gcd (Int.dvd_of_emod_eq_zero hap) (Int.dvd_of_emod_eq_zero hbp)
      rw [Int.gcd_eq_one_iff_coprime.mpr (coprime_of_minimal hf)] at h1
      revert h1
      

Proof state:
case intro.intro.intro.inl.inl.h
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
hbp : b0 % 2 = 0
⊢ 2 ∣ ↑1 → False

Next tactic:
decide
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · exfalso
      have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) :=
        Int.dvd_gcd (Int.dvd_of_emod_eq_zero hap) (Int.dvd_of_emod_eq_zero hbp)
      rw [Int.gcd_eq_one_iff_coprime.mpr (coprime_of_minimal hf)] at h1
      revert h1
      decide
    · 

Proof state:
case intro.intro.intro.inl.inr
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 0
hbp : b0 % 2 = 1
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1

Next tactic:
exact ⟨b0, ⟨a0, ⟨c0, minimal_comm hf, hbp⟩⟩⟩
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has `a` odd. -/
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 := by
  obtain ⟨a0, b0, c0, hf⟩ := exists_minimal h
  cases' Int.emod_two_eq_zero_or_one a0 with hap hap
  · cases' Int.emod_two_eq_zero_or_one b0 with hbp hbp
    · exfalso
      have h1 : 2 ∣ (Int.gcd a0 b0 : ℤ) :=
        Int.dvd_gcd (Int.dvd_of_emod_eq_zero hap) (Int.dvd_of_emod_eq_zero hbp)
      rw [Int.gcd_eq_one_iff_coprime.mpr (coprime_of_minimal hf)] at h1
      revert h1
      decide
    · exact ⟨b0, ⟨a0, ⟨c0, minimal_comm hf, hbp⟩⟩⟩
  

Proof state:
case intro.intro.intro.inr
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hap : a0 % 2 = 1
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1

Next tactic:
exact ⟨a0, ⟨b0, ⟨c0, hf, hap⟩⟩⟩
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0

Next tactic:
obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
  

Proof state:
case intro.intro.intro.intro
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hc : a0 % 2 = 1
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0

Next tactic:
rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
  rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
  · 

Proof state:
case intro.intro.intro.intro.inl
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hc : a0 % 2 = 1
h1 : 0 < c0
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0

Next tactic:
use a0, b0, c0
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
  rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
  · use a0, b0, c0
  · 

Proof state:
case intro.intro.intro.intro.inr.inl
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hc : a0 % 2 = 1
h1 : 0 = c0
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0

Next tactic:
exfalso
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
  rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
  · use a0, b0, c0
  · exfalso
    

Proof state:
case intro.intro.intro.intro.inr.inl.h
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hc : a0 % 2 = 1
h1 : 0 = c0
⊢ False

Next tactic:
exact ne_zero hf.1 h1.symm
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
  rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
  · use a0, b0, c0
  · exfalso
    exact ne_zero hf.1 h1.symm
  · 

Proof state:
case intro.intro.intro.intro.inr.inr
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hc : a0 % 2 = 1
h1 : c0 < 0
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0

Next tactic:
use a0, b0, -c0, neg_of_minimal hf, hc
/-- We can assume that a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` has
`a` odd and `c` positive. -/
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by
  obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
  rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
  · use a0, b0, c0
  · exfalso
    exact ne_zero hf.1 h1.symm
  · use a0, b0, -c0, neg_of_minimal hf, hc
    

Proof state:
case right
a b c : ℤ
h : Fermat42 a b c
a0 b0 c0 : ℤ
hf : Minimal a0 b0 c0
hc : a0 % 2 = 1
h1 : c0 < 0
⊢ 0 < -c0

Next tactic:
exact neg_pos.mpr h1
theorem Int.coprime_of_sq_sum {r s : ℤ} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r := by
  

Proof state:
r s : ℤ
h2 : IsCoprime s r
⊢ IsCoprime (r ^ 2 + s ^ 2) r

Next tactic:
rw [sq, sq]
theorem Int.coprime_of_sq_sum {r s : ℤ} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r := by
  rw [sq, sq]
  

Proof state:
r s : ℤ
h2 : IsCoprime s r
⊢ IsCoprime (r * r + s * s) r

Next tactic:
exact (IsCoprime.mul_left h2 h2).mul_add_left_left r
theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
    IsCoprime (r ^ 2 + s ^ 2) (r * s) := by
  

Proof state:
r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (r ^ 2 + s ^ 2) (r * s)

Next tactic:
apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
    IsCoprime (r ^ 2 + s ^ 2) (r * s) := by
  apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
  

Proof state:
r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (r ^ 2 + s ^ 2) s

Next tactic:
rw [add_comm]
theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
    IsCoprime (r ^ 2 + s ^ 2) (r * s) := by
  apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
  rw [add_comm]; 

Proof state:
r s : ℤ
h : IsCoprime r s
⊢ IsCoprime (s ^ 2 + r ^ 2) s

Next tactic:
apply Int.coprime_of_sq_sum h
theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False := by
  -- Use the fact that a ^ 2, b ^ 2, c form a pythagorean triple to obtain m and n such that
  -- a ^ 2 = m ^ 2 - n ^ 2, b ^ 2 = 2 * m * n and c = m ^ 2 + n ^ 2
  -- first the formula:
  

Proof state:
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
⊢ False

Next tactic:
have ht : PythagoreanTriple (a ^ 2) (b ^ 2) c := by
    delta PythagoreanTriple
    linear_combination h.1.2.2
theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False := by
  -- Use the fact that a ^ 2, b ^ 2, c form a pythagorean triple to obtain m and n such that
  -- a ^ 2 = m ^ 2 - n ^ 2, b ^ 2 = 2 * m * n and c = m ^ 2 + n ^ 2
  -- first the formula:
  have ht : PythagoreanTriple (a ^ 2) (b ^ 2) c := by
    

Proof state:
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
⊢ PythagoreanTriple (a ^ 2) (b ^ 2) c

Next tactic:
delta PythagoreanTriple
theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False := by
  -- Use the fact that a ^ 2, b ^ 2, c form a pythagorean triple to obtain m and n such that
  -- a ^ 2 = m ^ 2 - n ^ 2, b ^ 2 = 2 * m * n and c = m ^ 2 + n ^ 2
  -- first the formula:
  have ht : PythagoreanTriple (a ^ 2) (b ^ 2) c := by
    delta PythagoreanTriple
    

Proof state:
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
⊢ a ^ 2 * a ^ 2 + b ^ 2 * b ^ 2 = c * c

Next tactic:
linear_combination h.1.2.2