-- Mathlib/NumberTheory/FLT/Three.lean
theorem fermatLastTheoremThree : FermatLastTheoremFor 3

Proof state:
K : Type := CyclotomicField 3 ℚ
⊢ FermatLastTheoremFor 3

Tactic:
let hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K

Result:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
⊢ FermatLastTheoremFor 3

Proof state:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
⊢ False

Tactic:
let S' : FermatLastTheoremForThreeGen.Solution' hζ :=
{ a := a
  b := b
  c := c
  u := u
  ha := ha
  hb := hb
  hc := hc
  coprime := coprime
  hcdvd := hcdvd
  H := H }

Result:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
⊢ False

Proof state:
⊢ FermatLastTheoremFor 3

Tactic:
classical
let K := CyclotomicField 3 ℚ
let hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K
have : NumberField K := IsCyclotomicExtension.numberField {3} ℚ _
apply FermatLastTheoremForThree_of_FermatLastTheoremThreeGen hζ
intro a b c u hc ha hb hcdvd coprime H
let S' : FermatLastTheoremForThreeGen.Solution' hζ :=
{ a := a
  b := b
  c := c
  u := u
  ha := ha
  hb := hb
  hc := hc
  coprime := coprime
  hcdvd := hcdvd
  H := H }
obtain ⟨S, -⟩ := FermatLastTheoremForThreeGen.exists_Solution_of_Solution' S'
obtain ⟨Smin, hSmin⟩ := S.exists_minimal
obtain ⟨Sfin, hSfin⟩ := Smin.exists_Solution_multiplicity_lt
linarith [hSmin Sfin]

Result:
no goals

Proof state:
case intro
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S : FermatLastTheoremForThreeGen.Solution hζ
⊢ False

Tactic:
obtain ⟨Smin, hSmin⟩ := S.exists_minimal

Result:
case intro.intro
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S Smin : FermatLastTheoremForThreeGen.Solution hζ
hSmin : Smin.isMinimal
⊢ False

Proof state:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
⊢ FermatLastTheoremFor 3

Tactic:
apply FermatLastTheoremForThree_of_FermatLastTheoremThreeGen hζ

Result:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
⊢ FermatLastTheoremForThreeGen hζ

Proof state:
case intro.intro.intro
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S Smin : FermatLastTheoremForThreeGen.Solution hζ
hSmin : Smin.isMinimal
Sfin : FermatLastTheoremForThreeGen.Solution hζ
hSfin : Sfin.multiplicity < Smin.multiplicity
⊢ False

Tactic:
linarith [hSmin Sfin]

Result:
no goals

Proof state:
⊢ FermatLastTheoremFor 3

Tactic:
let K := CyclotomicField 3 ℚ

Result:
K : Type := CyclotomicField 3 ℚ
⊢ FermatLastTheoremFor 3

Proof state:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
⊢ FermatLastTheoremFor 3

Tactic:
have : NumberField K := IsCyclotomicExtension.numberField {3} ℚ _

Result:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
⊢ FermatLastTheoremFor 3

Proof state:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
⊢ FermatLastTheoremForThreeGen hζ

Tactic:
intro a b c u hc ha hb hcdvd coprime H

Result:
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
⊢ False

Proof state:
case intro.intro
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S Smin : FermatLastTheoremForThreeGen.Solution hζ
hSmin : Smin.isMinimal
⊢ False

Tactic:
obtain ⟨Sfin, hSfin⟩ := Smin.exists_Solution_multiplicity_lt

Result:
case intro.intro.intro
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
  { a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S Smin : FermatLastTheoremForThreeGen.Solution hζ
hSmin : Smin.isMinimal
Sfin : FermatLastTheoremForThreeGen.Solution hζ
hSfin : Sfin.multiplicity < Smin.multiplicity
⊢ False