mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
x : ↥(torsion K)
hc : 2 < orderOf ↑↑↑x
⊢ ↑x = 1 ∨ ↑x = -1

Tactic:
]

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
x : ↥(torsion K)
hc : 2 < orderOf ↑↑↑x
⊢ ↑x = 1 ∨ ↑x = -1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
this : (starRingEnd ℂ) (w.embedding ↑x) = ψ ↑x
⊢ ψ ↑x = w.embedding ↑x

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
this : (starRingEnd ℂ) (w.embedding ↑x) = ψ ↑x
⊢ ψ ↑x = w.embedding ↑x
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: case neg.«2»
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
x : ↥(torsion K)
hi : ↑↑x = -1
hc : 2 ≤ 2
⊢ ↑x = 1 ∨ ↑x = -1

Tactic:
]

Goal state after:
case neg.«2»
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
x : ↥(torsion K)
hi : ↑↑x = -1
hc : 2 ≤ 2
⊢ ↑x = 1 ∨ ↑x = -1
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
this : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)
⊢ Fintype.card ↥(torsion K) = 2

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
this : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)
⊢ Fintype.card ↥(torsion K) = 2
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ ‖(w.embedding ↑x).re‖
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
this : (w.embedding ↑x).im = 0
⊢ 1 ≤ |(w.embedding ↑x).re|

Tactic:
]

Goal state after:
case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ ‖(w.embedding ↑x).re‖
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
this : (w.embedding ↑x).im = 0
⊢ 1 ≤ |(w.embedding ↑x).re|
Module: Mathlib.NumberTheory.NumberField.Units.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
this : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)
x : ↥(torsion K)
x✝ : x ∈ Finset.univ
⊢ ↑x = ↑1 ∨ ↑x = ↑⟨-1, ⋯⟩

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (Module.finrank ℚ K)
this : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)
x : ↥(torsion K)
x✝ : x ∈ Finset.univ
⊢ ↑x = ↑1 ∨ ↑x = ↑⟨-1, ⋯⟩
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ℚ⟮↑x⟯.toSubalgebra = ⊤

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ℚ⟮↑x⟯.toSubalgebra = ⊤
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2
⊢ ∑ j, ∑ i with mkComplex i = j, 1 = 2 * nrComplexPlaces K

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2
⊢ ∑ j, ∑ i with mkComplex i = j, 1 = 2 * nrComplexPlaces K
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case e'_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
⊢ card { x // mkComplex x = ⟨w, hw⟩ } = card { x // mk x = w }

Tactic:
]

Goal state after:
case e'_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
⊢ card { x // mkComplex x = ⟨w, hw⟩ } = card { x // mk x = w }
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝ : { x // mkComplex x = ⟨w, hw⟩ }
φ : { φ // ¬ComplexEmbedding.IsReal φ }
hφ : ↑(mkComplex φ) = ↑⟨w, hw⟩
⊢ mk ↑φ = w

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝ : { x // mkComplex x = ⟨w, hw⟩ }
φ : { φ // ¬ComplexEmbedding.IsReal φ }
hφ : ↑(mkComplex φ) = ↑⟨w, hw⟩
⊢ mk ↑φ = w
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case e'_2.refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝¹ x✝ : { x // mkComplex x = ⟨w, hw⟩ }
h : x✝¹ = x✝
⊢ x✝¹ = x✝

Tactic:
]

Goal state after:
case e'_2.refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝¹ x✝ : { x // mkComplex x = ⟨w, hw⟩ }
h : x✝¹ = x✝
⊢ x✝¹ = x✝
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case e'_2.refine_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝ : { x // mk x = w }
φ : K →+* ℂ
hφ : mk φ = w
⊢ mk ↑⟨φ, ⋯⟩ = ↑⟨w, hw⟩

Tactic:
]

Goal state after:
case e'_2.refine_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝ : { x // mk x = w }
φ : K →+* ℂ
hφ : mk φ = w
⊢ mk ↑⟨φ, ⋯⟩ = ↑⟨w, hw⟩
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
heq :
  (expand k ch) (C u) * (expand k ch) ca ^ p + (expand k ch) (C v) * (expand k ch) cb ^ q +
      (expand k ch) (C w) * (expand k ch) cc ^ r =
    0
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
⊢ C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0

Tactic:
]

Goal state after:
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
heq :
  (expand k ch) (C u) * (expand k ch) ca ^ p + (expand k ch) (C v) * (expand k ch) cb ^ q +
      (expand k ch) (C w) * (expand k ch) cc ^ r =
    0
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
⊢ C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ card (K →+* ℂ) = card (K →+* ℂ)

pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ card { φ // ComplexEmbedding.IsReal φ } ≤ card (K →+* ℂ)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ card { φ // ComplexEmbedding.IsReal φ } ≤ card (K →+* ℂ)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ (-1) ^ nrComplexPlaces K = (-1) ^ nrComplexPlaces K

pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Function.Involutive.toPerm ComplexEmbedding.conjugate ⋯ ^ 2 = 1

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Function.Involutive.toPerm ComplexEmbedding.conjugate ⋯ ^ 2 = 1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : finrank ℚ K = 1
this : nrRealPlaces K = 1
⊢ nrRealPlaces K = 1

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : finrank ℚ K = 1
this : nrRealPlaces K = 1
⊢ nrRealPlaces K = 1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (finrank ℚ K)
hc : nrRealPlaces K = 0
⊢ Even (2 * nrComplexPlaces K)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Odd (finrank ℚ K)
hc : nrRealPlaces K = 0
⊢ Even (2 * nrComplexPlaces K)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
⊢ False

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
⊢ False
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (f ζ) k
⊢ (ComplexEmbedding.conjugate f) ζ = f ζ

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (f ζ) k
⊢ (ComplexEmbedding.conjugate f) ζ = f ζ
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
heq : C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0
⊢ C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0

Tactic:
]

Goal state after:
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
heq : C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0
⊢ C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (f ζ) k
him : |(f ζ).re| = ‖f ζ‖
⊢ (f ζ).re = 1 ∨ (f ζ).re = -1

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (f ζ) k
him : |(f ζ).re| = ‖f ζ‖
⊢ (f ζ).re = 1 ∨ (f ζ).re = -1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (f ζ) k
him : |(f ζ).re| = ‖f ζ‖
this : (f ζ).re = 1 ∨ (f ζ).re = -1
⊢ (f ζ).re = 1 ∨ (f ζ).re = -1

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x✝ : { w // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (f ζ) k
him : |(f ζ).re| = ‖f ζ‖
this : (f ζ).re = 1 ∨ (f ζ).re = -1
⊢ (f ζ).re = 1 ∨ (f ζ).re = -1
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
⊢ C v * b ^ q + C w * c ^ r + C u * a ^ p = 0

Tactic:
]

Goal state after:
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
⊢ C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: v : InfinitePlace ℚ
x : ℚ
⊢ ↑v x = ↑|x|

Tactic:
]

Goal state after:
v : InfinitePlace ℚ
x : ℚ
⊢ ↑v x = ↑|x|
Module: Mathlib.NumberTheory.NumberField.EquivReindex

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℚ K = finrank ℚ K

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.EquivReindex

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : DecidableEq (K →+* ℂ)
⊢ (Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑(equivReindex K)) (RingHom.equivRatAlgHom K ℂ)).det ^ 2 ≠
    0

Tactic:
]

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : DecidableEq (K →+* ℂ)
⊢ (Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑(equivReindex K)) (RingHom.equivRatAlgHom K ℂ)).det ^ 2 ≠
    0
Module: Mathlib.NumberTheory.NumberField.EquivReindex

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : DecidableEq (K →+* ℂ)
⊢ (Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑(equivReindex K)) (RingHom.equivRatAlgHom K ℂ)).det ^ 2 =
    (algebraMap ℚ ℂ) (Algebra.discr ℚ (⇑(integralBasis K) ∘ ⇑(equivReindex K).symm.symm))

Tactic:
]

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : DecidableEq (K →+* ℂ)
⊢ (Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑(equivReindex K)) (RingHom.equivRatAlgHom K ℂ)).det ^ 2 =
    (algebraMap ℚ ℂ) (Algebra.discr ℚ (⇑(integralBasis K) ∘ ⇑(equivReindex K).symm.symm))
Module: Mathlib.NumberTheory.NumberField.EquivReindex

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : K
i : K →+* ℂ
⊢ (∑ i, ((latticeBasis K).repr ((canonicalEmbedding K) α)) ((equivReindex K) i) • (latticeBasis K) ((equivReindex K) i))
      i =
    ((basisMatrix K)ᵀ *ᵥ fun i ↦ ↑((((integralBasis K).reindex (equivReindex K).symm).repr α) i)) i

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : K
i : K →+* ℂ
⊢ (∑ i, ((latticeBasis K).repr ((canonicalEmbedding K) α)) ((equivReindex K) i) • (latticeBasis K) ((equivReindex K) i))
      i =
    ((basisMatrix K)ᵀ *ᵥ fun i ↦ ↑((((integralBasis K).reindex (equivReindex K).symm).repr α) i)) i
Module: Mathlib.NumberTheory.NumberField.EquivReindex

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : DecidableEq (K →+* ℂ)
α : 𝓞 K
i : K →+* ℂ
⊢ (fun i ↦ ↑((((integralBasis K).reindex (equivReindex K).symm).repr ((algebraMap (𝓞 K) K) α)) i)) i =
    ↑((((integralBasis K).reindex (equivReindex K).symm).repr ↑α) i)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case refine_2
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : r * p + p * q + q * r ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ b.natDegree = 0

Tactic:
]

Goal state after:
case refine_2
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : r * p + p * q + q * r ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ b.natDegree = 0
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case refine_2
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : r * p + p * q + q * r ≤ q * r * p
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ b.natDegree = 0

Tactic:
]

Goal state after:
case refine_2
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : r * p + p * q + q * r ≤ q * r * p
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ b.natDegree = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
w : InfinitePlace K
hw : w ∈ Finset.univ.erase w₁
⊢ f w ^ w.mult = ?m.102 w

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case refine_3
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : p * q + q * r + r * p ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C w * c ^ r + C u * a ^ p + C v * b ^ q = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ c.natDegree = 0

Tactic:
]

Goal state after:
case refine_3
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : p * q + q * r + r * p ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C w * c ^ r + C u * a ^ p + C v * b ^ q = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ c.natDegree = 0
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case refine_3
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : p * q + q * r + r * p ≤ r * p * q
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C w * c ^ r + C u * a ^ p + C v * b ^ q = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ c.natDegree = 0

Tactic:
]

Goal state after:
case refine_3
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : p * q + q * r + r * p ≤ r * p * q
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C w * c ^ r + C u * a ^ p + C v * b ^ q = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ c.natDegree = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
⊢ B = B

pp: case refine_2
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
⊢ S ≠ 0

pp: case refine_2
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
⊢ ↑w₁.mult ≠ 0

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
⊢ S ≠ 0

case refine_2
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℝ≥0
hf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
S : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult
⊢ ↑w₁.mult ≠ 0