mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.UniversalFactorizationRing

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq R n
P : Ideal R
inst✝ : P.IsPrime
f : MonicDegreeEq P.ResidueField m
g : MonicDegreeEq P.ResidueField k
H✝ : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g
Hpq : IsCoprime ↑f ↑g
φ : 𝓡' →ₐ[R] P.ResidueField := (homEquiv P.ResidueField m k hn p).symm ⟨(f, g), ⋯⟩
Q : Ideal 𝓡' := RingHom.ker φ.toRingHom
this✝ : Q.IsPrime
this : Q.LiesOver P
φ' : Q.ResidueField →ₐ[R] P.ResidueField := Ideal.ResidueField.liftₐ Q φ ⋯ ⋯
φi : P.ResidueField →ₐ[R] Q.ResidueField := Ideal.ResidueField.mapₐ P Q (Algebra.ofId R 𝓡') ⋯
e : P.ResidueField ≃ₐ[R] Q.ResidueField := AlgEquiv.ofAlgHom φi φ' ⋯ ⋯
H : φi.comp φ = IsScalarTower.toAlgHom R 𝓡' Q.ResidueField
⊢ (↑((homEquiv Q.ResidueField m k hn p) (IsScalarTower.toAlgHom R 𝓡' Q.ResidueField))).1 =
    (factor₁ m k hn p).map (algebraMap 𝓡' Q.ResidueField)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq R n
P : Ideal R
inst✝ : P.IsPrime
f : MonicDegreeEq P.ResidueField m
g : MonicDegreeEq P.ResidueField k
H✝ : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g
Hpq : IsCoprime ↑f ↑g
φ : 𝓡' →ₐ[R] P.ResidueField := (homEquiv P.ResidueField m k hn p).symm ⟨(f, g), ⋯⟩
Q : Ideal 𝓡' := RingHom.ker φ.toRingHom
this✝ : Q.IsPrime
this : Q.LiesOver P
φ' : Q.ResidueField →ₐ[R] P.ResidueField := Ideal.ResidueField.liftₐ Q φ ⋯ ⋯
φi : P.ResidueField →ₐ[R] Q.ResidueField := Ideal.ResidueField.mapₐ P Q (Algebra.ofId R 𝓡') ⋯
e : P.ResidueField ≃ₐ[R] Q.ResidueField := AlgEquiv.ofAlgHom φi φ' ⋯ ⋯
H : φi.comp φ = IsScalarTower.toAlgHom R 𝓡' Q.ResidueField
⊢ (↑((homEquiv Q.ResidueField m k hn p) (IsScalarTower.toAlgHom R 𝓡' Q.ResidueField))).1 =
    (factor₁ m k hn p).map (algebraMap 𝓡' Q.ResidueField)
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: case inr
m n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.natDegree ≤ m
hg : g.natDegree ≤ n
i : Fin (m + n)
j : Fin n
⊢ (if ↑j ≤ ↑i then f.coeff (↑i - ↑j) else 0) = if ↑j ≤ ↑i then if ↑i ≤ ↑j + m then f.coeff (↑i - ↑j) else 0 else 0

Tactic:
]

Goal state after:
case inr
m n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.natDegree ≤ m
hg : g.natDegree ≤ n
i : Fin (m + n)
j : Fin n
⊢ (if ↑j ≤ ↑i then f.coeff (↑i - ↑j) else 0) = if ↑j ≤ ↑i then if ↑i ≤ ↑j + m then f.coeff (↑i - ↑j) else 0 else 0
Module: Mathlib.RingTheory.Etale.QuasiFinite

Goal state before:
pp: R : Type u
S : Type v
T : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra.FiniteType R T
inst✝¹ : Algebra.IsIntegral R S
f : S →ₐ[R] T
g : S
hg : Function.Surjective ⇑(awayMapₐ f g)
p : Ideal R
inst✝ : p.IsPrime
hgp : IsUnit (1 ⊗ₜ[R] g)
this✝¹ : IsClosed (PrimeSpectrum.comap (algebraMap R S) '' PrimeSpectrum.zeroLocus {g})
q : PrimeSpectrum S
hq : q ∈ PrimeSpectrum.zeroLocus {g}
e : PrimeSpectrum.comap (algebraMap R S) q = { asIdeal := p, isPrime := inst✝ }
this✝ : q.asIdeal.LiesOver p
this :
  1 ⊗ₜ[R] g ∉
    RingHom.ker
      (Algebra.TensorProduct.lift
          (Ideal.ResidueField.mapₐ { asIdeal := p, isPrime := inst✝ }.asIdeal (↑⟨q, e⟩).asIdeal (Algebra.ofId R S) ⋯)
          (IsScalarTower.toAlgHom R S (↑⟨q, e⟩).asIdeal.ResidueField) ⋯).toRingHom
⊢ False

Tactic:
]

Goal state after:
R : Type u
S : Type v
T : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra.FiniteType R T
inst✝¹ : Algebra.IsIntegral R S
f : S →ₐ[R] T
g : S
hg : Function.Surjective ⇑(awayMapₐ f g)
p : Ideal R
inst✝ : p.IsPrime
hgp : IsUnit (1 ⊗ₜ[R] g)
this✝¹ : IsClosed (PrimeSpectrum.comap (algebraMap R S) '' PrimeSpectrum.zeroLocus {g})
q : PrimeSpectrum S
hq : q ∈ PrimeSpectrum.zeroLocus {g}
e : PrimeSpectrum.comap (algebraMap R S) q = { asIdeal := p, isPrime := inst✝ }
this✝ : q.asIdeal.LiesOver p
this :
  1 ⊗ₜ[R] g ∉
    RingHom.ker
      (Algebra.TensorProduct.lift
          (Ideal.ResidueField.mapₐ { asIdeal := p, isPrime := inst✝ }.asIdeal (↑⟨q, e⟩).asIdeal (Algebra.ofId R S) ⋯)
          (IsScalarTower.toAlgHom R S (↑⟨q, e⟩).asIdeal.ResidueField) ⋯).toRingHom
⊢ False
Module: Mathlib.AlgebraicGeometry.Normalization

Goal state before:
pp: X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
inst✝⁴ : QuasiCompact f✝
inst✝³ : QuasiSeparated f✝
X S Y : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
inst✝ : Smooth g
x : ↥(pullback (fromNormalization f) g)
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : (pullback.snd (fromNormalization f) g ≫ g) x ∈ ↑U
V : TopologicalSpace.Opens ↥Y
hV : V ∈ Y.affineOpens
hxV : (pullback.snd (fromNormalization f) g) x ∈ ↑V
hVU : V ≤ g ⁻¹ᵁ U
W : (pullback (fromNormalization f) g).Opens := pullback.snd (fromNormalization f) g ⁻¹ᵁ V
this✝² : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
this✝¹ : (CommRingCat.Hom.hom (appLE g U V hVU)).Smooth
algInst✝² : Algebra ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U) := (CommRingCat.Hom.hom (app f U)).toAlgebra
algInst✝¹ : Algebra ↑Γ(S, U) ↑Γ(Y, V) := (CommRingCat.Hom.hom (appLE g U V hVU)).toAlgebra
algInst✝ : Algebra ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) :=
  (CommRingCat.Hom.hom (app (pullback.snd f g) V)).toAlgebra
algebraizeInst✝ : Algebra.Smooth ↑Γ(S, U) ↑Γ(Y, V)
this✝ : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
e₀ : CommRingCat.of
    (↑(Y.presheaf.1 (Opposite.op V)) ⊗[↑(S.presheaf.1 (Opposite.op U))] ↑(X.presheaf.1 (Opposite.op (f ⁻¹ᵁ U)))) ≅
  Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) :=
  ⋯.isoPushout ≪≫ pushout.congrHom ⋯ ⋯ ≪≫ asIso (pushoutSection ⋯ hVU ⋯ ⋯)
e : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↑Γ(X, f ⁻¹ᵁ U) ≃ₐ[↑Γ(Y, V)] ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) :=
  let __RingEquiv := e₀.commRingCatIsoToRingEquiv;
  { toEquiv := __RingEquiv.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
ψ : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↥(integralClosure ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U)) →ₐ[↑Γ(Y, V)]
  ↥(integralClosure ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V)) :=
  (↑e.mapIntegralClosure).comp (toIntegralClosure ↑Γ(S, U) ↑Γ(Y, V) ↑Γ(X, f ⁻¹ᵁ U))
hψ : Function.Bijective ⇑ψ
φ : pushout (app (fromNormalization f) U) (appLE g U V hVU) ⟶
  Γ(normalization (pullback.snd f g), normalizationPullback f g ⁻¹ᵁ W) :=
  pushout.map (app (fromNormalization f) U) (appLE g U V hVU)
      (CommRingCat.ofHom (algebraMap ↑Γ(S, U) ↥(integralClosure ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U)))) (appLE g U V hVU)
      (normalizationObjIso f hU).hom (𝟙 Γ(Y, V)) (𝟙 Γ(S, U)) ⋯ ⋯ ≫
    ⋯.isoPushout.inv ≫
      (RingEquiv.ofBijective ψ.toRingHom hψ).toCommRingCatIso.hom ≫
        (normalizationObjIso (pullback.snd f g) hV).inv ≫ (normalization (pullback.snd f g)).presheaf.map (eqToHom ⋯).op
this : Mono (CommRingCat.ofHom (integralClosure ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V)).val.toRingHom)
H : pullback.snd f g ⁻¹ᵁ V ≤ pullback.fst f g ⁻¹ᵁ f ⁻¹ᵁ U
⊢ appLE (pullback.fst f g ≫ toNormalization f) (fromNormalization f ⁻¹ᵁ U) (pullback.snd f g ⁻¹ᵁ V) ⋯ =
    appLE (pullback.fst f g ≫ toNormalization f) (fromNormalization f ⁻¹ᵁ U) (pullback.snd f g ⁻¹ᵁ V) ⋯

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: m n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.natDegree ≤ m
hg : g.natDegree ≤ n
b₁ : Module.Basis (Fin (m + n)) R (↥R[X]_m × ↥R[X]_n) :=
  ((degreeLT.basis R m).prod (degreeLT.basis R n)).reindex finSumFinEquiv
b₂ : Module.Basis (Fin (m + n)) R ↥R[X]_(m + n) := degreeLT.basis R (m + n)
this : f.sylvesterMap g hf hg ∘ₗ (Matrix.toLin b₂ b₁) (f.sylvester g m n).adjugate = f.resultant g m n • LinearMap.id
⊢ f.sylvesterMap g hf hg ∘ₗ f.adjSylvester g = f.resultant g m n • LinearMap.id

Tactic:
]

Goal state after:
m n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.natDegree ≤ m
hg : g.natDegree ≤ n
b₁ : Module.Basis (Fin (m + n)) R (↥R[X]_m × ↥R[X]_n) :=
  ((degreeLT.basis R m).prod (degreeLT.basis R n)).reindex finSumFinEquiv
b₂ : Module.Basis (Fin (m + n)) R ↥R[X]_(m + n) := degreeLT.basis R (m + n)
this : f.sylvesterMap g hf hg ∘ₗ (Matrix.toLin b₂ b₁) (f.sylvester g m n).adjugate = f.resultant g m n • LinearMap.id
⊢ f.sylvesterMap g hf hg ∘ₗ f.adjSylvester g = f.resultant g m n • LinearMap.id
Module: Mathlib.RingTheory.Polynomial.UniversalFactorizationRing

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq R n
P : Ideal R
inst✝ : P.IsPrime
f : MonicDegreeEq P.ResidueField m
g : MonicDegreeEq P.ResidueField k
H✝ : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g
Hpq : IsCoprime ↑f ↑g
φ : 𝓡' →ₐ[R] P.ResidueField := (homEquiv P.ResidueField m k hn p).symm ⟨(f, g), ⋯⟩
Q : Ideal 𝓡' := RingHom.ker φ.toRingHom
this✝ : Q.IsPrime
this : Q.LiesOver P
φ' : Q.ResidueField →ₐ[R] P.ResidueField := Ideal.ResidueField.liftₐ Q φ ⋯ ⋯
φi : P.ResidueField →ₐ[R] Q.ResidueField := Ideal.ResidueField.mapₐ P Q (Algebra.ofId R 𝓡') ⋯
e : P.ResidueField ≃ₐ[R] Q.ResidueField := AlgEquiv.ofAlgHom φi φ' ⋯ ⋯
H : φi.comp φ = IsScalarTower.toAlgHom R 𝓡' Q.ResidueField
⊢ (↑((homEquiv Q.ResidueField m k hn p) (IsScalarTower.toAlgHom R 𝓡' Q.ResidueField))).2 =
    (factor₂ m k hn p).map (algebraMap 𝓡' Q.ResidueField)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq R n
P : Ideal R
inst✝ : P.IsPrime
f : MonicDegreeEq P.ResidueField m
g : MonicDegreeEq P.ResidueField k
H✝ : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g
Hpq : IsCoprime ↑f ↑g
φ : 𝓡' →ₐ[R] P.ResidueField := (homEquiv P.ResidueField m k hn p).symm ⟨(f, g), ⋯⟩
Q : Ideal 𝓡' := RingHom.ker φ.toRingHom
this✝ : Q.IsPrime
this : Q.LiesOver P
φ' : Q.ResidueField →ₐ[R] P.ResidueField := Ideal.ResidueField.liftₐ Q φ ⋯ ⋯
φi : P.ResidueField →ₐ[R] Q.ResidueField := Ideal.ResidueField.mapₐ P Q (Algebra.ofId R 𝓡') ⋯
e : P.ResidueField ≃ₐ[R] Q.ResidueField := AlgEquiv.ofAlgHom φi φ' ⋯ ⋯
H : φi.comp φ = IsScalarTower.toAlgHom R 𝓡' Q.ResidueField
⊢ (↑((homEquiv Q.ResidueField m k hn p) (IsScalarTower.toAlgHom R 𝓡' Q.ResidueField))).2 =
    (factor₂ m k hn p).map (algebraMap 𝓡' Q.ResidueField)
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: m n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.natDegree ≤ m
hg : g.natDegree ≤ n
b₁ : Module.Basis (Fin (m + n)) R (↥R[X]_m × ↥R[X]_n) :=
  ((degreeLT.basis R m).prod (degreeLT.basis R n)).reindex finSumFinEquiv
b₂ : Module.Basis (Fin (m + n)) R ↥R[X]_(m + n) := degreeLT.basis R (m + n)
this : (Matrix.toLin b₂ b₁) (f.sylvester g m n).adjugate ∘ₗ f.sylvesterMap g hf hg = f.resultant g m n • LinearMap.id
⊢ f.adjSylvester g ∘ₗ f.sylvesterMap g hf hg = f.resultant g m n • LinearMap.id

Tactic:
]

Goal state after:
m n : ℕ
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.natDegree ≤ m
hg : g.natDegree ≤ n
b₁ : Module.Basis (Fin (m + n)) R (↥R[X]_m × ↥R[X]_n) :=
  ((degreeLT.basis R m).prod (degreeLT.basis R n)).reindex finSumFinEquiv
b₂ : Module.Basis (Fin (m + n)) R ↥R[X]_(m + n) := degreeLT.basis R (m + n)
this : (Matrix.toLin b₂ b₁) (f.sylvester g m n).adjugate ∘ₗ f.sylvesterMap g hf hg = f.resultant g m n • LinearMap.id
⊢ f.adjSylvester g ∘ₗ f.sylvesterMap g hf hg = f.resultant g m n • LinearMap.id
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: case neg.refine_2.inl
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
b : R[X]
this : f.coeff f.natDegree ^ (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
e : 0 * f + b * g = 1
⊢ 1 = f.resultant b * f.resultant g

Tactic:
]

Goal state after:
case neg.refine_2.inl
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
b : R[X]
this : f.coeff f.natDegree ^ (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
e : 0 * f + b * g = 1
⊢ 1 = f.resultant b * f.resultant g
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: case neg.refine_2.inr
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
a b : R[X]
e : a * f + b * g = 1
this : f.resultant (C 1) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
hb0 : a ≠ 0
⊢ 1 = f.resultant b * f.resultant g

pp: R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
a b : R[X]
e : a * f + b * g = 1
this : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
hb0 : a ≠ 0
⊢ a.natDegree + f.natDegree ≤ b.natDegree + g.natDegree

Tactic:
]

Goal state after:
case neg.refine_2.inr
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
a b : R[X]
e : a * f + b * g = 1
this : f.resultant (C 1) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
hb0 : a ≠ 0
⊢ 1 = f.resultant b * f.resultant g

R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
a b : R[X]
e : a * f + b * g = 1
this : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
hb0 : a ≠ 0
⊢ a.natDegree + f.natDegree ≤ b.natDegree + g.natDegree
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
a b : R[X]
e : a * f + b * g = 1
this : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
hb0 : a ≠ 0
H : (b * g).natDegree < (a * f).natDegree
⊢ False

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hf : f.Monic
hf0 : ¬f.natDegree = 0
a b : R[X]
e : a * f + b * g = 1
this : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g
hb0 : a ≠ 0
H : (b * g).natDegree < (a * f).natDegree
⊢ False
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: K : Type u_2
inst✝ : Field K
f g : K[X]
hf : f ≠ 0
⊢ f.leadingCoeff⁻¹ * f.coeff f.natDegree = 1

Tactic:
]

Goal state after:
K : Type u_2
inst✝ : Field K
f g : K[X]
hf : f ≠ 0
⊢ f.leadingCoeff⁻¹ * f.coeff f.natDegree = 1
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: case inr
K : Type u_2
inst✝ : Field K
f g : K[X]
hf : f ≠ 0
H : (C f.leadingCoeff⁻¹ * f).Monic
this : IsUnit (f.leadingCoeff⁻¹ ^ g.natDegree) ∧ IsUnit (f.resultant g) ↔ IsCoprime f g
⊢ f.resultant g = 0 ↔ (f ≠ 0 ∨ g ≠ 0) ∧ ¬IsCoprime f g

Tactic:
]

Goal state after:
case inr
K : Type u_2
inst✝ : Field K
f g : K[X]
hf : f ≠ 0
H : (C f.leadingCoeff⁻¹ * f).Monic
this : IsUnit (f.leadingCoeff⁻¹ ^ g.natDegree) ∧ IsUnit (f.resultant g) ↔ IsCoprime f g
⊢ f.resultant g = 0 ↔ (f ≠ 0 ∨ g ≠ 0) ∧ ¬IsCoprime f g
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 1
⊢ f.discr = 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 1
⊢ f.discr = 1
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 2
⊢ f.discr = f.coeff 1 ^ 2 - 4 * f.coeff 0 * f.coeff 2

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 2
⊢ f.discr = f.coeff 1 ^ 2 - 4 * f.coeff 0 * f.coeff 2
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 2
e : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 3 := finCongr ⋯
⊢ ((Matrix.reindex e e) f.sylvesterDeriv).det * (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) =
    f.coeff 1 ^ 2 - 4 * f.coeff 0 * f.coeff 2

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 2
e : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 3 := finCongr ⋯
⊢ ((Matrix.reindex e e) f.sylvesterDeriv).det * (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) =
    f.coeff 1 ^ 2 - 4 * f.coeff 0 * f.coeff 2
Module: Mathlib.RingTheory.Etale.QuasiFinite

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : FiniteType R S
inst✝ : QuasiFiniteAt R q
s₁ : S
hs₁q : s₁ ∉ q
hs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q'.LiesOver p → s₁ ∈ q'
s₂ : ↥(integralClosure R S)
hs₂q : ↑s₂ ∉ q
hs₂ : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom s₂)
s₃ : ↥(integralClosure R S)
n : ℕ
hs₃ :
  (Localization.awayMap (integralClosure R S).val.toRingHom s₂)
      (IsLocalization.mk' (Localization.Away s₂) s₃ ⟨(fun x ↦ s₂ ^ x) n, ⋯⟩) =
    (algebraMap S (Localization.Away ((integralClosure R S).val.toRingHom s₂))) s₁
m : ℕ
hm : ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n)
hm0 : 0 < m
hs₃q : ↑s₃ ∉ q
q' : Ideal ↥(integralClosure R S)
x✝¹ : q'.IsPrime
hq'q : q' ≠ Ideal.under (↥(integralClosure R S)) q
x✝ : q'.LiesOver p
this : Algebra (↥(integralClosure R S)) (Localization.Away ↑s₂) := OreLocalization.instAlgebra
e : Localization.Away s₂ ≃ₐ[↥(integralClosure R S)] Localization.Away ↑s₂ :=
  AlgEquiv.ofBijective (Localization.awayMapₐ (ofId (↥(integralClosure R S)) S) s₂) hs₂
q's : Ideal (Localization.Away s₂) := Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q'
H : ¬⟨↑s₂ ^ m * ↑s₃, ⋯⟩ ∈ q'
⊢ s₂ ∉ q'

Tactic:
]

Goal state after:
R : Type u_2
S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : FiniteType R S
inst✝ : QuasiFiniteAt R q
s₁ : S
hs₁q : s₁ ∉ q
hs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q'.LiesOver p → s₁ ∈ q'
s₂ : ↥(integralClosure R S)
hs₂q : ↑s₂ ∉ q
hs₂ : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom s₂)
s₃ : ↥(integralClosure R S)
n : ℕ
hs₃ :
  (Localization.awayMap (integralClosure R S).val.toRingHom s₂)
      (IsLocalization.mk' (Localization.Away s₂) s₃ ⟨(fun x ↦ s₂ ^ x) n, ⋯⟩) =
    (algebraMap S (Localization.Away ((integralClosure R S).val.toRingHom s₂))) s₁
m : ℕ
hm : ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n)
hm0 : 0 < m
hs₃q : ↑s₃ ∉ q
q' : Ideal ↥(integralClosure R S)
x✝¹ : q'.IsPrime
hq'q : q' ≠ Ideal.under (↥(integralClosure R S)) q
x✝ : q'.LiesOver p
this : Algebra (↥(integralClosure R S)) (Localization.Away ↑s₂) := OreLocalization.instAlgebra
e : Localization.Away s₂ ≃ₐ[↥(integralClosure R S)] Localization.Away ↑s₂ :=
  AlgEquiv.ofBijective (Localization.awayMapₐ (ofId (↥(integralClosure R S)) S) s₂) hs₂
q's : Ideal (Localization.Away s₂) := Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q'
H : ¬⟨↑s₂ ^ m * ↑s₃, ⋯⟩ ∈ q'
⊢ s₂ ∉ q'
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : 0 < f.natDegree
⊢ f.resultant (derivative f) f.natDegree (f.natDegree - 1) =
    (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) * f.leadingCoeff * f.discr

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : 0 < f.natDegree
⊢ f.resultant (derivative f) f.natDegree (f.natDegree - 1) =
    (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) * f.leadingCoeff * f.discr
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : 0 < f.natDegree
⊢ (-1) ^ ((f.natDegree - 1) * f.natDegree) * (f.leadingCoeff * f.sylvesterDeriv.det) =
    (-1) ^ ((f.natDegree - 1) * f.natDegree / 2) * f.leadingCoeff *
      (f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree / 2))

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : 0 < f.natDegree
⊢ (-1) ^ ((f.natDegree - 1) * f.natDegree) * (f.leadingCoeff * f.sylvesterDeriv.det) =
    (-1) ^ ((f.natDegree - 1) * f.natDegree / 2) * f.leadingCoeff *
      (f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree / 2))
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : 0 < f.natDegree
⊢ f.leadingCoeff * f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree) =
    f.leadingCoeff * f.sylvesterDeriv.det * (-1) ^ ((f.natDegree - 1) * f.natDegree)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 3
⊢ 3 - 1 + 3 = 5

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Etale.QuasiFinite

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : FiniteType R S
inst✝ : QuasiFiniteAt R q
s₁ : S
hs₁q : s₁ ∉ q
hs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q'.LiesOver p → s₁ ∈ q'
s₂ : ↥(integralClosure R S)
hs₂q : ↑s₂ ∉ q
hs₂ : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom s₂)
s₃ : ↥(integralClosure R S)
n : ℕ
hs₃ :
  (Localization.awayMap (integralClosure R S).val.toRingHom s₂)
      (IsLocalization.mk' (Localization.Away s₂) s₃ ⟨(fun x ↦ s₂ ^ x) n, ⋯⟩) =
    (algebraMap S (Localization.Away ((integralClosure R S).val.toRingHom s₂))) s₁
m : ℕ
hm : ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n)
hm0 : 0 < m
hs₃q : ↑s₃ ∉ q
q' : Ideal ↥(integralClosure R S)
x✝¹ : q'.IsPrime
x✝ : q'.LiesOver p
this✝² : Algebra (↥(integralClosure R S)) (Localization.Away ↑s₂) := OreLocalization.instAlgebra
e : Localization.Away s₂ ≃ₐ[↥(integralClosure R S)] Localization.Away ↑s₂ :=
  AlgEquiv.ofBijective (Localization.awayMapₐ (ofId (↥(integralClosure R S)) S) s₂) hs₂
q's : Ideal (Localization.Away s₂) := Ideal.map (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q'
H : ¬⟨↑s₂ ^ m * ↑s₃, ⋯⟩ ∈ q'
hq's : Disjoint ↑(Submonoid.powers s₂) ↑q'
this✝¹ : q's.IsPrime
this✝ : q's.LiesOver q'
this : q's.LiesOver p
hq'q :
  Ideal.comap ((↑e.symm).comp (IsScalarTower.toAlgHom (↥(integralClosure R S)) S (Localization.Away ↑s₂))).toRingHom
      q's =
    q
⊢ Ideal.under (↥(integralClosure R S)) q's =
    Ideal.comap (algebraMap (↥(integralClosure R S)) (Localization.Away s₂)) q's

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 3
e : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 5 := finCongr ⋯
⊢ !![f.coeff 0, 0, 1 * f.coeff 1, 0, 0;
          f.coeff 1, f.coeff 0, 2 * f.coeff 2, 1 * f.coeff 1, 0;
          f.coeff 2, f.coeff 1, 3 * f.coeff 3, 2 * f.coeff 2, 1 * f.coeff 1;
          f.coeff 3, f.coeff 2, 0, 3 * f.coeff 3, 2 * f.coeff 2;
          0, 1, 0, 0, 3].det *
      (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) =
    f.coeff 2 ^ 2 * f.coeff 1 ^ 2 - 4 * f.coeff 3 * f.coeff 1 ^ 3 - 4 * f.coeff 2 ^ 3 * f.coeff 0 -
        27 * f.coeff 3 ^ 2 * f.coeff 0 ^ 2 +
      18 * f.coeff 3 * f.coeff 2 * f.coeff 1 * f.coeff 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
hf : f.natDegree = 3
e : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 5 := finCongr ⋯
⊢ !![f.coeff 0, 0, 1 * f.coeff 1, 0, 0;
          f.coeff 1, f.coeff 0, 2 * f.coeff 2, 1 * f.coeff 1, 0;
          f.coeff 2, f.coeff 1, 3 * f.coeff 3, 2 * f.coeff 2, 1 * f.coeff 1;
          f.coeff 3, f.coeff 2, 0, 3 * f.coeff 3, 2 * f.coeff 2;
          0, 1, 0, 0, 3].det *
      (-1) ^ (f.natDegree * (f.natDegree - 1) / 2) =
    f.coeff 2 ^ 2 * f.coeff 1 ^ 2 - 4 * f.coeff 3 * f.coeff 1 ^ 3 - 4 * f.coeff 2 ^ 3 * f.coeff 0 -
        27 * f.coeff 3 ^ 2 * f.coeff 0 ^ 2 +
      18 * f.coeff 3 * f.coeff 2 * f.coeff 1 * f.coeff 0