l3lab/ntp-mathlib huggingface dataset:
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      

Proof state:
case tfae_1_to_2.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ ↑(Int.natAbs a ^ n + Int.natAbs b ^ n) = ↑(Int.natAbs c ^ n)

Next tactic:
push_cast
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      

Proof state:
case tfae_1_to_2.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ |a| ^ n + |b| ^ n = |c| ^ n

Next tactic:
simp only [hn.pow_abs, habc]
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    

Proof state:
case tfae_1_to_2.inr
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
⊢ False

Next tactic:
obtain ha | ha := ha.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> 

Proof state:
case tfae_1_to_2.inr.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
⊢ False

Next tactic:
obtain hb | hb := hb.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> 

Proof state:
case tfae_1_to_2.inr.inr
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
⊢ False

Next tactic:
obtain hb | hb := hb.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      

Proof state:
case tfae_1_to_2.inr.inl.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
⊢ False

Next tactic:
obtain hc | hc := hc.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      

Proof state:
case tfae_1_to_2.inr.inl.inr
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
⊢ False

Next tactic:
obtain hc | hc := hc.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      

Proof state:
case tfae_1_to_2.inr.inr.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
hb : b < 0
⊢ False

Next tactic:
obtain hc | hc := hc.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      

Proof state:
case tfae_1_to_2.inr.inr.inr
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : 0 < a
hb : 0 < b
⊢ False

Next tactic:
obtain hc | hc := hc.lt_or_lt
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · 

Proof state:
case tfae_1_to_2.inr.inl.inl.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ False

Next tactic:
refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ Int.natAbs a ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ Int.natAbs b ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ Int.natAbs c ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      

Proof state:
case tfae_1_to_2.inr.inl.inl.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ ↑(Int.natAbs a ^ n + Int.natAbs b ^ n) = ↑(Int.natAbs c ^ n)

Next tactic:
push_cast
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      

Proof state:
case tfae_1_to_2.inr.inl.inl.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : c < 0
⊢ |a| ^ n + |b| ^ n = |c| ^ n

Next tactic:
simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · 

Proof state:
case tfae_1_to_2.inr.inl.inl.inr
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : 0 < c
⊢ False

Next tactic:
exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : b < 0
hc : 0 < c
⊢ 0 < c ^ n

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · 

Proof state:
case tfae_1_to_2.inr.inl.inr.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ False

Next tactic:
refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · refine' h b.natAbs c.natAbs a.natAbs (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ Int.natAbs b ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ Int.natAbs c ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ Int.natAbs a ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      

Proof state:
case tfae_1_to_2.inr.inl.inr.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ ↑(Int.natAbs b ^ n + Int.natAbs c ^ n) = ↑(Int.natAbs a ^ n)

Next tactic:
push_cast
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      

Proof state:
case tfae_1_to_2.inr.inl.inr.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : c < 0
⊢ |b| ^ n + |c| ^ n = |a| ^ n

Next tactic:
simp only [abs_of_pos, abs_of_neg, hn.neg_pow, habc, add_neg_eq_iff_eq_add,
        eq_neg_add_iff_add_eq, *]
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [hn.pow_abs, habc]
    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>
      obtain hc | hc := hc.lt_or_lt
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, ← mul_add, habc, *]
    · exact (by positivity : 0 < c ^ n).not_lt $ habc.symm.trans_lt $ add_neg (hn.pow_neg ha) $
        hn.pow_neg hb
    · refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
      push_cast
      simp only [abs_of_pos, abs_of_neg, hn.neg_pow, habc, add_neg_eq_iff_eq_add,
        eq_neg_add_iff_add_eq, *]
    · 

Proof state:
case tfae_1_to_2.inr.inl.inr.inr
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha✝ : a ≠ 0
hb✝ : b ≠ 0
hc✝ : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Odd n
ha : a < 0
hb : 0 < b
hc : 0 < c
⊢ False

Next tactic:
refine' h a.natAbs c.natAbs b.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)