mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Smooth.IntegralClosure

Goal state before:
pp: R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing B
inst✝ : Algebra R B
φ : S[X] →ₐ[R] B
hφ : Function.Surjective ⇑φ
f : S[X]
hf : f.Monic
hf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)
hfx : RingHom.ker φ.toRingHom = Ideal.span {f}
h✝ : Nontrivial B
hfd : f.natDegree ≠ 0
algInst✝¹ : Algebra S B := (φ.comp C).toAlgebra
this : Nontrivial S
y : S[X]
hy : IsIntegral R (φ y)
S' : Type u_2
w✝⁴ : CommRing S'
w✝³ : Algebra S S'
w✝² : Module.Finite S S'
w✝¹ : Module.Free S S'
w✝ : Nontrivial S'
hS' : (Polynomial.map (algebraMap S S') f).Splits
m : Multiset S'
hm : Polynomial.map (algebraMap S S') f = (Multiset.map (fun x ↦ X - C x) m).prod
algInst✝ : Algebra R S' := ((algebraMap S S').comp (algebraMap R S)).toAlgebra
scalarTowerInst✝ : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S'))
hm' : ∀ a ∈ m, IsIntegral R a
hmc : m.card = f.natDegree
⊢ (Multiset.map (fun x ↦ X - C x) m).prod ∣
    (Multiset.map
        (fun i ↦
          (Multiset.map (fun x ↦ X - C x) (m.erase i)).prod * derivative (X - C i) *
              Polynomial.map (algebraMap S S') y -
            (Multiset.map (fun x ↦ X - C x) (m.erase i)).prod * C ((aeval i) y))
        m).sum

Tactic:
refine Multiset.dvd_sum ?_

Goal state after:
R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing B
inst✝ : Algebra R B
φ : S[X] →ₐ[R] B
hφ : Function.Surjective ⇑φ
f : S[X]
hf : f.Monic
hf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)
hfx : RingHom.ker φ.toRingHom = Ideal.span {f}
h✝ : Nontrivial B
hfd : f.natDegree ≠ 0
algInst✝¹ : Algebra S B := (φ.comp C).toAlgebra
this : Nontrivial S
y : S[X]
hy : IsIntegral R (φ y)
S' : Type u_2
w✝⁴ : CommRing S'
w✝³ : Algebra S S'
w✝² : Module.Finite S S'
w✝¹ : Module.Free S S'
w✝ : Nontrivial S'
hS' : (Polynomial.map (algebraMap S S') f).Splits
m : Multiset S'
hm : Polynomial.map (algebraMap S S') f = (Multiset.map (fun x ↦ X - C x) m).prod
algInst✝ : Algebra R S' := ((algebraMap S S').comp (algebraMap R S)).toAlgebra
scalarTowerInst✝ : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S'))
hm' : ∀ a ∈ m, IsIntegral R a
hmc : m.card = f.natDegree
⊢ ∀
    x ∈
      Multiset.map
        (fun i ↦
          (Multiset.map (fun x ↦ X - C x) (m.erase i)).prod * derivative (X - C i) *
              Polynomial.map (algebraMap S S') y -
            (Multiset.map (fun x ↦ X - C x) (m.erase i)).prod * C ((aeval i) y))
        m,
    (Multiset.map (fun x ↦ X - C x) m).prod ∣ x
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
ht : φ.IsIntegralElem t
hp : φ p * t ∈ φ.range
a : R := p.leadingCoeff
R' : Type u_1 := Localization.Away a
S' : Type u_2 := Localization.Away ((algebraMap R S) a)
this✝¹ : Algebra R' S' := (Localization.awayMap (algebraMap R S) a).toAlgebra
this✝ : IsScalarTower R R' S'
ha : IsUnit ((algebraMap R R') a)
H : (aeval ((algebraMap S S') (φ X))).comp (mapRingHom (algebraMap R R')) = (algebraMap S S').comp ↑φ
q : R'[X]
hq : IsIntegral R' ((algebraMap S S') t - (aeval ((algebraMap S S') (φ X))) q)
n : ℕ
q' : R[X]
e : map (algebraMap R R') q' = (fun x ↦ a ^ x) n • q
this : IsIntegral R' ((algebraMap S S') (a ^ n • t - φ q'))
m : ℕ
hm : IsIntegral R (⟨(fun x ↦ a ^ x) m, ⋯⟩ • (algebraMap S S') (a ^ n • t - φ q'))
⊢ ∃ q n, IsIntegral R (a ^ n • t - φ q)

Tactic:
simp only [Algebra.smul_def, Submonoid.smul_def, algebraMap_apply R S S', ← map_mul] at hm

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
ht : φ.IsIntegralElem t
hp : φ p * t ∈ φ.range
a : R := p.leadingCoeff
R' : Type u_1 := Localization.Away a
S' : Type u_2 := Localization.Away ((algebraMap R S) a)
this✝¹ : Algebra R' S' := (Localization.awayMap (algebraMap R S) a).toAlgebra
this✝ : IsScalarTower R R' S'
ha : IsUnit ((algebraMap R R') a)
H : (aeval ((algebraMap S S') (φ X))).comp (mapRingHom (algebraMap R R')) = (algebraMap S S').comp ↑φ
q : R'[X]
hq : IsIntegral R' ((algebraMap S S') t - (aeval ((algebraMap S S') (φ X))) q)
n : ℕ
q' : R[X]
e : map (algebraMap R R') q' = (fun x ↦ a ^ x) n • q
this : IsIntegral R' ((algebraMap S S') (a ^ n • t - φ q'))
m : ℕ
hm : IsIntegral R ((algebraMap S S') ((algebraMap R S) (a ^ m) * ((algebraMap R S) (a ^ n) * t - φ q')))
⊢ ∃ q n, IsIntegral R (a ^ n • t - φ q)
Module: Mathlib.RingTheory.Polynomial.DegreeLT

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
m n : ℕ
Q : ↥R[X]_n
⊢ ↑((addLinearEquiv R m n).symm ((LinearMap.inr R (↥R[X]_m) ↥R[X]_n) Q)) = ↑Q * X ^ m

Tactic:
rw [← (basis ..).sum_repr Q]
  simp [-LinearMap.coe_inr, Finset.sum_mul, addLinearEquiv_symm_apply_inr_basis,
    smul_eq_C_mul, mul_assoc, ← pow_add, add_comm]

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: case h.zero
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
inst✝⁵ : Algebra R✝ S✝
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : WeaklyQuasiFiniteAt R p
x : S
hx : R[x] = ⊤
H : integralClosure R S = ⊥
H₀ : Function.Surjective ⇑(aeval x)
f : R[X]
hf : (aeval x) f = 0
n : ℕ
hfn : (algebraMap R S) (f.coeff n) ∉ p
IH :
  ∀ m < 0,
    ∀ (f : R[X]),
      (aeval x) f = 0 → ∀ (n : ℕ), (algebraMap R S) (f.coeff n) ∉ p → f.natDegree = m → ZariskisMainProperty R p
hm : f.natDegree = 0
⊢ ZariskisMainProperty R p

Tactic:
obtain ⟨r, rfl⟩ := natDegree_eq_zero.mp hm

Goal state after:
case h.zero
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
inst✝⁵ : Algebra R✝ S✝
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : WeaklyQuasiFiniteAt R p
x : S
hx : R[x] = ⊤
H : integralClosure R S = ⊥
H₀ : Function.Surjective ⇑(aeval x)
n : ℕ
IH :
  ∀ m < 0,
    ∀ (f : R[X]),
      (aeval x) f = 0 → ∀ (n : ℕ), (algebraMap R S) (f.coeff n) ∉ p → f.natDegree = m → ZariskisMainProperty R p
r : R
hf : (aeval x) (C r) = 0
hfn : (algebraMap R S) ((C r).coeff n) ∉ p
hm : (C r).natDegree = 0
⊢ ZariskisMainProperty R p
Module: Mathlib.RingTheory.Polynomial.Resultant.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
m n : ℕ
⊢ f.resultant 0 m n = 0 ^ m * f.coeff 0 ^ n

Tactic:
obtain _ | m := m; · simp
  have (i : Fin (m + 1 + n)) : sylvester f 0 (m + 1) n i ⟨0, by lia⟩ = 0 := by
    simp [sylvester, show (0 : Fin (m + 1 + n)) = Fin.castAdd _ 0 from rfl, Fin.addCases_left]
  simpa [resultant] using Matrix.det_eq_zero_of_column_eq_zero ⟨0, by lia⟩ this

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f g : R[X]
m n k : ℕ
hf : f.natDegree ≤ m
⊢ f.resultant g (m + k) n = (-1) ^ (n * k) * g.coeff n ^ k * f.resultant g m n

Tactic:
induction k with

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
g : R[X]
n : ℕ
r : R
hg : g.natDegree ≤ n
⊢ (X - C r).resultant g 1 n = eval r g

Tactic:
nontriviality R
  obtain hg | hg := g.natDegree.eq_zero_or_pos
  · obtain ⟨s, rfl⟩ := natDegree_eq_zero.mp hg
    simp
  conv_lhs => rw [← g.modByMonic_add_div (X - C r)]
  rw [resultant_add_mul_right _ _ _ _ _ _ (natDegree_X_sub_C_le _), modByMonic_X_sub_C_eq_C_eval]
  · simp
  · rw [natDegree_divByMonic g (monic_X_sub_C r), natDegree_sub_C, natDegree_X]
    lia

Goal state after:
Module: Mathlib.AlgebraicGeometry.Normalization

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝¹ : QuasiCompact f
inst✝ : QuasiSeparated f
U : Y.Opens
hU : IsAffineOpen U
⊢ app (fromNormalization f) U =
    CommRingCat.ofHom (algebraMap ↑(Y.presheaf.1 (Opposite.op U)) ↥(integralClosure ↑Γ(Y, U) ↑Γ(X, f ⁻¹ᵁ U))) ≫
      (normalizationObjIso f hU).inv

Tactic:
let := (f.app U).hom.toAlgebra
  have : IsIso (((normalizationOpenCover f).f ⟨U, hU⟩).app (f.fromNormalization ⁻¹ᵁ U)) :=
    Scheme.Hom.isIso_app _ _ (by simp [← fromNormalization_preimage])
  have H : ⊤ = ((normalizationOpenCover f).f ⟨U, hU⟩ ≫ fromNormalization f) ⁻¹ᵁ U := by
    rw [f.ι_fromNormalization]; simp
  rw [← cancel_mono (((normalizationOpenCover f).f ⟨U, hU⟩).app (f.fromNormalization ⁻¹ᵁ U)),
    ← Scheme.Hom.comp_app, Scheme.Hom.congr_app (f.ι_fromNormalization ⟨U, hU⟩) U,
    ← cancel_mono (((normalizationOpenCover f).X ⟨U, hU⟩).presheaf.map (eqToHom H).op)]
  dsimp [normalizationObjIso]
  rw [IsAffineOpen.fromSpec_app_self]
  simp only [app_eq_appLE, Category.assoc, map_appLE, appLE_map]
  simp [Scheme.Hom.appLE, ← ΓSpecIso_inv_naturality]
  rfl

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

Goal state before:
pp: n : ℕ
IH :
  ∀ m < n,
    ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),
      f.Monic →
        g.Monic →
          f.Splits →
            g.Splits →
              g.natDegree ≤ f.natDegree →
                f.natDegree + g.natDegree = m →
                  f.resultant g = (Multiset.map (fun ij ↦ ij.1 - ij.2) (f.roots ×ˢ g.roots)).prod
K : Type u_3
inst✝ : Field K
f g : K[X]
hf : f.Monic
hg : g.Monic
hf' : f.Splits
hg' : g.Splits
hfg : g.natDegree ≤ f.natDegree
hN : f.natDegree + g.natDegree = n
hr : ¬g ∣ f
r : K[X] := C (f %ₘ g).leadingCoeff⁻¹ * (f %ₘ g)
hr₀ : f %ₘ g ≠ 0
⊢ r.natDegree < g.natDegree

Tactic:
simp [r, natDegree_C_mul, hr₀, natDegree_modByMonic_lt _ hg (show g ≠ 1 by aesop)]

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

Goal state before:
pp: R✝ : Type u_1
inst✝² : CommRing R✝
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
φ : R →+* S
hφ : Function.Injective ⇑φ
f : R[X]
IH :
  ∀ (g : S[X]) (r : S),
    map φ f ≠ 0 →
      g ≠ 0 →
        ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =
          r ^ ((map φ f).natDegree * g.natDegree) * (map φ f).resultant g
g : R[X]
r : R
hf : f ≠ 0
hg : g ≠ 0
⊢ map φ f ≠ 0

Tactic:
simpa using (map_injective _ hφ).ne hf

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f g : R[X]
r : R
⊢ (f.scaleRoots r).resultant (g.scaleRoots r) = r ^ (f.natDegree * g.natDegree) * f.resultant g

Tactic:
rw [natDegree_scaleRoots, natDegree_scaleRoots]
  obtain rfl | hf := eq_or_ne f 0; · simp
  obtain rfl | hg := eq_or_ne g 0; · simp
  induction f using induction_of_Splits_of_injective_of_surjective with
  | Splits R f hf' =>
    by_cases hf0 : f.natDegree = 0
    · obtain ⟨a, rfl⟩ := natDegree_eq_zero.mp hf0; simp
    by_cases hg0 : g.natDegree = 0
    · obtain ⟨a, rfl⟩ := natDegree_eq_zero.mp hg0; simp
    obtain rfl | hr := eq_or_ne r 0
    · rw [scaleRoots_zero, scaleRoots_zero, Algebra.smul_def, Algebra.smul_def]
      simp [resultant_C_mul_right, resultant_X_pow_right, *, (Ne.symm hf0)]
    conv_lhs => rw [← natDegree_scaleRoots f r, ← natDegree_scaleRoots g r]
    rw [resultant_eq_prod_eval _ _ _ le_rfl hf',
      resultant_eq_prod_eval _ _ _ le_rfl (hf'.scaleRoots _),
      roots_scaleRoots _ (isUnit_iff_ne_zero.mpr hr)]
    simp only [Multiset.map_map, Function.comp_def, scaleRoots_eval_mul]
    simp only [leadingCoeff_scaleRoots, natDegree_scaleRoots, Multiset.prod_map_mul,
      Multiset.map_const', Multiset.prod_replicate, ← hf'.natDegree_eq_card_roots]
    ring
  | injective R S φ hφ f IH =>
    have := IH (g.map φ) (φ r) (by simpa using (map_injective _ hφ).ne hf)
      (by simpa using (map_injective _ hφ).ne hg)
    apply hφ
    rw [← map_scaleRoots, ← map_scaleRoots] at this
    · simpa [natDegree_map_eq_of_injective hφ] using this
    all_goals simpa [map_eq_zero_iff _ hφ]
  | surjective R S φ hφ f IH =>
    obtain ⟨f', hf', ef⟩ := exists_degree_eq_of_mem_lifts (Polynomial.map_surjective φ hφ f)
    obtain ⟨g', hg', eg⟩ := exists_degree_eq_of_mem_lifts (Polynomial.map_surjective φ hφ g)
    obtain ⟨r, rfl⟩ := hφ r
    have hfl : f.leadingCoeff = φ f'.leadingCoeff := by
      simp_rw [← coeff_natDegree, ← natDegree_eq_natDegree ef, ← hf', coeff_map]
    have hgl : g.leadingCoeff = φ g'.leadingCoeff := by
      simp_rw [← coeff_natDegree, ← natDegree_eq_natDegree eg, ← hg', coeff_map]
    rw [← hf', ← map_scaleRoots _ _ _ (by simpa [← hfl]), ← hg',
      ← map_scaleRoots _ _ _ (by simpa [← hgl]), hf', hg',
      ← natDegree_eq_natDegree ef, ← natDegree_eq_natDegree eg,
      resultant_map_map, IH f' g' r (by aesop) (by aesop)]
    rw [← hf', ← hg', resultant_map_map]
    simp

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

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hg : g.natDegree ≠ 0
this :
  ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),
    g.natDegree ≠ 0 →
      IsDomain R →
        (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =
          g.leadingCoeff ^ (f.natDegree * (g.natDegree - 1)) * f.resultant g
hR : ¬IsDomain R
hf0 : ¬f = 0
hg0 : ¬g = 0
S : Type (max 0 u_1) := MvPolynomial R ℤ
φ : S →+* R := MvPolynomial.eval₂Hom (algebraMap ℤ R) id
hφ : Function.Surjective ⇑φ
f' : S[X]
hf' : map φ f' = f
ef : f'.degree = f.degree
g' : S[X]
hg' : map φ g' = g
eg : g'.degree = g.degree
hfl : f.leadingCoeff = φ f'.leadingCoeff
hgl : g.leadingCoeff = φ g'.leadingCoeff
⊢ (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =
    g.leadingCoeff ^ (f.natDegree * (g.natDegree - 1)) * f.resultant g

Tactic:
rw [← natDegree_eq_natDegree ef, ← natDegree_eq_natDegree eg,
      ← hf', hgl, ← map_scaleRoots _ _ _ (by simpa [← hfl]), ← hg', integralNormalization_map _ _
      (by simpa [← hgl]), resultant_map_map,
      this f' g' (by simpa [natDegree_eq_natDegree eg]) inferInstance]

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hg : g.natDegree ≠ 0
this :
  ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),
    g.natDegree ≠ 0 →
      IsDomain R →
        (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =
          g.leadingCoeff ^ (f.natDegree * (g.natDegree - 1)) * f.resultant g
hR : ¬IsDomain R
hf0 : ¬f = 0
hg0 : ¬g = 0
S : Type (max 0 u_1) := MvPolynomial R ℤ
φ : S →+* R := MvPolynomial.eval₂Hom (algebraMap ℤ R) id
hφ : Function.Surjective ⇑φ
f' : S[X]
hf' : map φ f' = f
ef : f'.degree = f.degree
g' : S[X]
hg' : map φ g' = g
eg : g'.degree = g.degree
hfl : f.leadingCoeff = φ f'.leadingCoeff
hgl : g.leadingCoeff = φ g'.leadingCoeff
⊢ φ (g'.leadingCoeff ^ (f'.natDegree * (g'.natDegree - 1)) * f'.resultant g') =
    φ g'.leadingCoeff ^ (f'.natDegree * (g'.natDegree - 1)) * (map φ f').resultant (map φ g') f'.natDegree g'.natDegree
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
H : m ≠ 0 ∨ n ≠ 0
⊢ ∃ p q, p.degree < ↑n ∧ q.degree < ↑m ∧ f * p + g * q = C (f.resultant g m n)

Tactic:
by
  nontriviality R
  let X := adjSylvester f g ⟨1, by simpa [Polynomial.mem_degreeLT,
    ← Nat.cast_add, Nat.pos_iff_ne_zero, not_and_or, -not_and] using H⟩
  have : ((sylvesterMap f g hf hg X)).1 = _ :=
    congr(($(sylveserMap_comp_adjSylvester f g hf hg) _).1)
  refine ⟨X.2, X.1, by simpa [-SetLike.coe_mem] using X.2.2,
    by simpa [-SetLike.coe_mem] using X.1.2, by simpa [Algebra.smul_def] using this⟩

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
hi hj : 0 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 0 < 5
hj : 1 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 0 < 5
hj : 2 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 0 < 5
hj : 3 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 0 < 5
hj : 4 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 1 < 5
hj : 0 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi hj : 1 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 1 < 5
hj : 2 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 1 < 5
hj : 3 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 1 < 5
hj : 4 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 2 < 5
hj : 0 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 2 < 5
hj : 1 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi hj : 2 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 2 < 5
hj : 3 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 2 < 5
hj : 4 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 3 < 5
hj : 0 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 3 < 5
hj : 1 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 3 < 5
hj : 2 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi hj : 3 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 3 < 5
hj : 4 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 4 < 5
hj : 0 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 4 < 5
hj : 1 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 4 < 5
hj : 2 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi : 4 < 5
hj : 3 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

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
hi hj : 4 < 5
⊢ 2 + 1 = 3

Tactic:
by norm_num

Goal state after:
Module: Mathlib.AlgebraicGeometry.ZariskisMainTheorem

Goal state before:
pp: case e'_2
X S : Scheme
f : X ⟶ S
inst✝¹ : LocallyOfFiniteType f
inst✝ : IsSeparated f
x : ↥X
s : ↥S
h : f x = s
hx : Scheme.Hom.QuasiFiniteAt f x
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : f x ∈ ↑U
V : TopologicalSpace.Opens ↥X
hV : V ∈ X.affineOpens
hxV : x ∈ ↑V
hUV : V ≤ f ⁻¹ᵁ U
this✝⁵ : (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hUV)).FiniteType
algInst✝ : Algebra ↑Γ(S, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hUV)).toAlgebra
algebraizeInst✝ : Algebra.FiniteType ↑Γ(S, U) ↑Γ(X, V)
this✝⁴ : (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal.LiesOver (IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩).asIdeal
this✝³ : Algebra.QuasiFiniteAt (↑Γ(S, U)) (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal
R : Type u
w✝⁷ : CommRing R
w✝⁶ : Algebra (↑Γ(S, U)) R
w✝⁵ : Algebra.Etale (↑Γ(S, U)) R
P : Ideal R
w✝⁴ : P.IsPrime
w✝³ : P.LiesOver (IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩).asIdeal
e : R ⊗[↑Γ(S, U)] ↑Γ(X, V)
w✝² : IsIdempotentElem e
P' : Ideal (R ⊗[↑Γ(S, U)] ↑Γ(X, V))
w✝¹ : P'.IsPrime
w✝ : P'.LiesOver P
hP' : Ideal.comap Algebra.TensorProduct.includeRight.toRingHom P' = (IsAffineOpen.primeIdealOf hV ⟨x, hxV⟩).asIdeal
heP' : e ∉ P'
left✝ : Module.Finite R (Localization.Away e)
this✝² : (algebraMap R (Localization.Away e)).Finite
φ : Γ(S, U) ⟶ CommRingCat.of R := CommRingCat.ofHom (algebraMap (↑Γ(S, U)) R)
hφ : (CommRingCat.Hom.hom φ).Etale
this✝¹ : Etale (Spec.map φ)
e₁ : Spec (CommRingCat.of (R ⊗[↑Γ(S, U)] ↑Γ(X, V))) ≅ pullback (Spec.map (Scheme.Hom.appLE f U V hUV)) (Spec.map φ) :=
  (pullbackSpecIso (↑Γ(S, U)) R ↑Γ(X, V)).symm ≪≫
    pullbackSymmetry (Spec.map (CommRingCat.ofHom (algebraMap (↑Γ(S, U)) R)))
      (Spec.map (CommRingCat.ofHom (algebraMap ↑Γ(S, U) ↑Γ(X, V))))
he₁ :
  e₁.hom ≫ pullback.fst (Spec.map (Scheme.Hom.appLE f U V hUV)) (Spec.map φ) =
    Spec.map (CommRingCat.ofHom Algebra.TensorProduct.includeRight.toRingHom)
g : Spec (CommRingCat.of (R ⊗[↑Γ(S, U)] ↑Γ(X, V))) ⟶ pullback f (Spec.map φ ≫ IsAffineOpen.fromSpec hU) :=
  e₁.hom ≫
    pullback.map (Spec.map (Scheme.Hom.appLE f U V hUV)) (Spec.map φ) f (Spec.map φ ≫ IsAffineOpen.fromSpec hU)
      (IsAffineOpen.fromSpec hV) (𝟙 (Spec (CommRingCat.of R))) (IsAffineOpen.fromSpec hU) ⋯ ⋯
W₁ : (pullback f (Spec.map φ ≫ IsAffineOpen.fromSpec hU)).Opens := g ''ᵁ PrimeSpectrum.basicOpen e
this✝ : IsFinite (W₁.ι ≫ pullback.snd f (Spec.map φ ≫ IsAffineOpen.fromSpec hU))
this : IsFinite W₁.ι
W₂ : (pullback f (Spec.map φ ≫ IsAffineOpen.fromSpec hU)).Opens := { carrier := (↑W₁)ᶜ, is_open' := ⋯ }
⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom φ) { asIdeal := P, isPrime := w✝⁴ } = IsAffineOpen.primeIdealOf hU ⟨f x, hxU⟩

Tactic:
exact PrimeSpectrum.ext (Ideal.over_def _ _).symm

Goal state after: