mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Localization.NormTrace

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
Rₘ : Type u_3
inst✝¹¹ : CommRing Rₘ
inst✝¹⁰ : Algebra R Rₘ
M : Submonoid R
inst✝⁹ : IsLocalization M Rₘ
Sₘ : Type u_5
inst✝⁸ : CommRing Sₘ
inst✝⁷ : Algebra S Sₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : Algebra R Sₘ
inst✝⁴ : IsScalarTower R Rₘ Sₘ
inst✝³ : IsScalarTower R S Sₘ
inst✝² : IsLocalization (algebraMapSubmonoid S M) Sₘ
ι : Type u_6
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
this✝ : Module.Finite R S
this : Free R S
i j : ι
⊢ (trace Rₘ Sₘ) ((Basis.localizationLocalization Rₘ M Sₘ b) i * (Basis.localizationLocalization Rₘ M Sₘ b) j) =
    (algebraMap R Rₘ) ((trace R S) (b i * b j))

Tactic:
Basis.localizationLocalization_apply,

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
Rₘ : Type u_3
inst✝¹¹ : CommRing Rₘ
inst✝¹⁰ : Algebra R Rₘ
M : Submonoid R
inst✝⁹ : IsLocalization M Rₘ
Sₘ : Type u_5
inst✝⁸ : CommRing Sₘ
inst✝⁷ : Algebra S Sₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : Algebra R Sₘ
inst✝⁴ : IsScalarTower R Rₘ Sₘ
inst✝³ : IsScalarTower R S Sₘ
inst✝² : IsLocalization (algebraMapSubmonoid S M) Sₘ
ι : Type u_6
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
this✝ : Module.Finite R S
this : Free R S
i j : ι
⊢ (trace Rₘ Sₘ) ((algebraMap S Sₘ) (b i) * (algebraMap S Sₘ) (b j)) = (algebraMap R Rₘ) ((trace R S) (b i * b j))
Module: Mathlib.NumberTheory.NumberField.Discriminant.Defs

Goal state before:

Tactic:
NumberField.discr ℚ
    _ = Algebra.discr ℤ b := by convert! (discr_eq_discr ℚ b).symm
    _ = Algebra.trace ℤ (𝓞 ℚ) (b default * b default) := by
      rw [Algebra.discr_def, Matrix.det_unique, Algebra.traceMatrix_apply, Algebra.traceForm_apply]
    _ = Algebra.trace ℤ (𝓞 ℚ) 1 := by
      rw [Basis.map_apply, RingEquiv.toAddEquiv_eq_coe, ← AddEquiv.toIntLinearEquiv_symm,
        AddEquiv.coe_toIntLinearEquiv, Basis.singleton_apply,
        show (AddEquiv.symm ↑ringOfIntegersEquiv) (1 : ℤ) = ringOfIntegersEquiv.symm 1 by rfl,
        map_one, mul_one]
    _ = 1 := by rw [Algebra.trace_eq_matrix_trace b]; simp

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Defs

Goal state before:
pp: case e_a.inl
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
hunit : IsUnit r
hp : r = 1
⊢ (algebraMap ℤ ℚ) 1 = (algebraMap ℤ ℚ) r ^ 2

pp: case e_a.inr
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
hunit : IsUnit r
hm : r = -1
⊢ (algebraMap ℤ ℚ) 1 = (algebraMap ℤ ℚ) r ^ 2

Tactic:
·

Goal state after:
case e_a.inl
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
hunit : IsUnit r
hp : r = 1
⊢ (algebraMap ℤ ℚ) 1 = (algebraMap ℤ ℚ) r ^ 2
Module: Mathlib.RingTheory.Discriminant

Goal state before:
pp: case e_a.refine_2
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots :
  ∀ (σ : L →ₐ[K] E),
    Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
      eval (σ pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0
⊢ ∀ (a b : Fin pb.dim),
    ¬b = a →
      (e a) pb.gen ≠ (e b) pb.gen ∧
        Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
          eval ((e a) pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0

pp: case e_a.refine_3
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots✝ : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
hroots :
  ∀ (σ : L →ₐ[K] E),
    Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
      eval (σ pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0
⊢ ∀ (a : L →ₐ[K] E) (b : E)
    (ha :
      b ≠ a pb.gen ∧
        Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
          eval b (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0),
    ¬a = pb.lift b ⋯

pp: case e_a.refine_4
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
⊢ ∀ (a b : Fin pb.dim) (ha : ¬b = a), pb.lift ((e a) pb.gen) ⋯ = e a

Tactic:
·

Goal state after:
case e_a.refine_2
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots :
  ∀ (σ : L →ₐ[K] E),
    Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
      eval (σ pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0
⊢ ∀ (a b : Fin pb.dim),
    ¬b = a →
      (e a) pb.gen ≠ (e b) pb.gen ∧
        Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
          eval ((e a) pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0
Module: Mathlib.NumberTheory.NumberField.Discriminant.Defs

Goal state before:
pp: case e_a.inr
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
hunit : IsUnit r
hm : r = -1
⊢ (algebraMap ℤ ℚ) 1 = (algebraMap ℤ ℚ) r ^ 2

Tactic:
·

Goal state after:
case e_a.inr
ι : Type u_2
ι' : Type u_3
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : DecidableEq ι
inst✝³ : DecidableEq ι'
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
inst✝ : NumberField K
b : Basis ι ℚ K
b' : Basis ι' ℚ K
h : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)
h' : ∀ (i j : ι'), IsIntegral ℤ (b'.toMatrix (⇑(b.reindex (b.indexEquiv b'))) i j)
hint : IsIntegral ℤ ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
r : ℤ
hr : (algebraMap ℤ ℚ) r = ((b.reindex (b.indexEquiv b')).toMatrix ⇑b').det
hunit : IsUnit r
hm : r = -1
⊢ (algebraMap ℤ ℚ) 1 = (algebraMap ℤ ℚ) r ^ 2
Module: Mathlib.RingTheory.Discriminant

Goal state before:
pp: case e_a.refine_3
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots✝ : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
hroots :
  ∀ (σ : L →ₐ[K] E),
    Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
      eval (σ pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0
⊢ ∀ (a : L →ₐ[K] E) (b : E)
    (ha :
      b ≠ a pb.gen ∧
        Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
          eval b (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0),
    ¬a = pb.lift b ⋯

pp: case e_a.refine_4
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
⊢ ∀ (a b : Fin pb.dim) (ha : ¬b = a), pb.lift ((e a) pb.gen) ⋯ = e a

Tactic:
·

Goal state after:
case e_a.refine_3
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots✝ : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
hroots :
  ∀ (σ : L →ₐ[K] E),
    Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
      eval (σ pb.gen) (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0
⊢ ∀ (a : L →ₐ[K] E) (b : E)
    (ha :
      b ≠ a pb.gen ∧
        Polynomial.map (algebraMap K E) (minpoly K pb.gen) ≠ 0 ∧
          eval b (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) = 0),
    ¬a = pb.lift b ⋯
Module: Mathlib.RingTheory.Discriminant

Goal state before:
pp: case e_a.refine_4
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
⊢ ∀ (a b : Fin pb.dim) (ha : ¬b = a), pb.lift ((e a) pb.gen) ⋯ = e a

Tactic:
·

Goal state after:
case e_a.refine_4
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Module.Finite K L
pb : PowerBasis K L
inst✝ : Algebra.IsSeparable K L
E : Type v := AlgebraicClosure L
this : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)
e : Fin pb.dim ≃ (L →ₐ[K] E)
hnodup : ((minpoly K pb.gen).aroots E).Nodup
hroots : ∀ (σ : L →ₐ[K] E), σ pb.gen ∈ (minpoly K pb.gen).aroots E
⊢ ∀ (a b : Fin pb.dim) (ha : ¬b = a), pb.lift ((e a) pb.gen) ⋯ = e a
Module: Mathlib.NumberTheory.Padics.RingHoms

Goal state before:
pp: case e'_3
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n k : ℕ
hk : ∀ n_1 ≥ k, ‖limNthHom f_compat r - ↑(nthHom f r n_1)‖ < ↑p ^ (-↑n)
this : limNthHom f_compat r - ↑(nthHom f r (max n k)) ∈ Ideal.span {↑p ^ n}
⊢ ↑p ^ n = (Int.castRingHom ℤ_[p]) (↑p ^ n)

pp: case e'_4
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n k : ℕ
hk : ∀ n_1 ≥ k, ‖limNthHom f_compat r - ↑(nthHom f r n_1)‖ < ↑p ^ (-↑n)
this : limNthHom f_compat r - ↑(nthHom f r (max n k)) ∈ Ideal.span {↑p ^ n}
⊢ ↑(nthHom f r (max n k)) - ↑((f n) r).val = (Int.castRingHom ℤ_[p]) (nthHom f r (max n k) - nthHom f r n)

Tactic:
·

Goal state after:
case e'_3
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n k : ℕ
hk : ∀ n_1 ≥ k, ‖limNthHom f_compat r - ↑(nthHom f r n_1)‖ < ↑p ^ (-↑n)
this : limNthHom f_compat r - ↑(nthHom f r (max n k)) ∈ Ideal.span {↑p ^ n}
⊢ ↑p ^ n = (Int.castRingHom ℤ_[p]) (↑p ^ n)
Module: Mathlib.NumberTheory.Padics.RingHoms

Goal state before:
pp: case e'_4
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n k : ℕ
hk : ∀ n_1 ≥ k, ‖limNthHom f_compat r - ↑(nthHom f r n_1)‖ < ↑p ^ (-↑n)
this : limNthHom f_compat r - ↑(nthHom f r (max n k)) ∈ Ideal.span {↑p ^ n}
⊢ ↑(nthHom f r (max n k)) - ↑((f n) r).val = (Int.castRingHom ℤ_[p]) (nthHom f r (max n k) - nthHom f r n)

Tactic:
·

Goal state after:
case e'_4
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n k : ℕ
hk : ∀ n_1 ≥ k, ‖limNthHom f_compat r - ↑(nthHom f r n_1)‖ < ↑p ^ (-↑n)
this : limNthHom f_compat r - ↑(nthHom f r (max n k)) ∈ Ideal.span {↑p ^ n}
⊢ ↑(nthHom f r (max n k)) - ↑((f n) r).val = (Int.castRingHom ℤ_[p]) (nthHom f r (max n k) - nthHom f r n)
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: case refine_1
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
j : Fin (subOnePowerBasis ℚ hζ).dim
⊢ IsIntegral ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen

pp: case refine_2
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
j : Fin (subOnePowerBasis ℚ hζ).dim
⊢ minpoly ℚ (IsPrimitiveRoot.powerBasis ℚ hζ).gen =
    Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen)

pp: case refine_3
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ IsIntegral ℤ (subOnePowerBasis ℚ hζ).gen

pp: case refine_4
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ minpoly ℚ (subOnePowerBasis ℚ hζ).gen = Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (subOnePowerBasis ℚ hζ).gen)

Tactic:
·

Goal state after:
case refine_1
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
j : Fin (subOnePowerBasis ℚ hζ).dim
⊢ IsIntegral ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: case refine_2
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
j : Fin (subOnePowerBasis ℚ hζ).dim
⊢ minpoly ℚ (IsPrimitiveRoot.powerBasis ℚ hζ).gen =
    Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen)

pp: case refine_3
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ IsIntegral ℤ (subOnePowerBasis ℚ hζ).gen

pp: case refine_4
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ minpoly ℚ (subOnePowerBasis ℚ hζ).gen = Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (subOnePowerBasis ℚ hζ).gen)

Tactic:
·

Goal state after:
case refine_2
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
j : Fin (subOnePowerBasis ℚ hζ).dim
⊢ minpoly ℚ (IsPrimitiveRoot.powerBasis ℚ hζ).gen =
    Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen)
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: case refine_3
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ IsIntegral ℤ (subOnePowerBasis ℚ hζ).gen

pp: case refine_4
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ minpoly ℚ (subOnePowerBasis ℚ hζ).gen = Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (subOnePowerBasis ℚ hζ).gen)

Tactic:
·

Goal state after:
case refine_3
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ IsIntegral ℤ (subOnePowerBasis ℚ hζ).gen
Module: Mathlib.NumberTheory.Cyclotomic.Discriminant

Goal state before:
pp: case refine_4
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ minpoly ℚ (subOnePowerBasis ℚ hζ).gen = Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (subOnePowerBasis ℚ hζ).gen)

Tactic:
·

Goal state after:
case refine_4
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
ζ : K
ce : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
this : NumberField K
H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen
H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen
i : Fin (subOnePowerBasis ℚ hζ).dim
j : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim
⊢ minpoly ℚ (subOnePowerBasis ℚ hζ).gen = Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (subOnePowerBasis ℚ hζ).gen)
Module: Mathlib.NumberTheory.Padics.RingHoms

Goal state before:
pp: case mp
p : ℕ
hp_prime : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ (∀ (n : ℕ), (toZModPow n) x = (toZModPow n) y) → x = y

pp: case mpr
p : ℕ
hp_prime : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ x = y → ∀ (n : ℕ), (toZModPow n) x = (toZModPow n) y

Tactic:
·

Goal state after:
case mp
p : ℕ
hp_prime : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ (∀ (n : ℕ), (toZModPow n) x = (toZModPow n) y) → x = y
Module: Mathlib.NumberTheory.Padics.RingHoms

Goal state before:
pp: case mpr
p : ℕ
hp_prime : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ x = y → ∀ (n : ℕ), (toZModPow n) x = (toZModPow n) y

Tactic:
·

Goal state after:
case mpr
p : ℕ
hp_prime : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ x = y → ∀ (n : ℕ), (toZModPow n) x = (toZModPow n) y
Module: Mathlib.NumberTheory.Padics.RingHoms

Goal state before:
pp: case mp
p : ℕ
hp_prime : Fact (Nat.Prime p)
R : Type u_1
inst✝ : NonAssocSemiring R
g g' : R →+* ℤ_[p]
⊢ (∀ (n : ℕ), (toZModPow n).comp g = (toZModPow n).comp g') → g = g'

pp: case mpr
p : ℕ
hp_prime : Fact (Nat.Prime p)
R : Type u_1
inst✝ : NonAssocSemiring R
g g' : R →+* ℤ_[p]
⊢ g = g' → ∀ (n : ℕ), (toZModPow n).comp g = (toZModPow n).comp g'

Tactic:
·

Goal state after:
case mp
p : ℕ
hp_prime : Fact (Nat.Prime p)
R : Type u_1
inst✝ : NonAssocSemiring R
g g' : R →+* ℤ_[p]
⊢ (∀ (n : ℕ), (toZModPow n).comp g = (toZModPow n).comp g') → g = g'
Module: Mathlib.NumberTheory.Padics.RingHoms

Goal state before:
pp: case mpr
p : ℕ
hp_prime : Fact (Nat.Prime p)
R : Type u_1
inst✝ : NonAssocSemiring R
g g' : R →+* ℤ_[p]
⊢ g = g' → ∀ (n : ℕ), (toZModPow n).comp g = (toZModPow n).comp g'

Tactic:
·

Goal state after:
case mpr
p : ℕ
hp_prime : Fact (Nat.Prime p)
R : Type u_1
inst✝ : NonAssocSemiring R
g g' : R →+* ℤ_[p]
⊢ g = g' → ∀ (n : ℕ), (toZModPow n).comp g = (toZModPow n).comp g'