mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
hpq : p • 1 + q • (mk W') Y = 0
⊢ p = 0 ∧ q = 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x : W'.CoordinateRing
⊢ ∃ p q, p • 1 + q • (mk W') Y = x

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
y p q : R[X]
⊢ (p • 1 + q • (mk W') Y) * (mk W') (C y) = (p * y) • 1 + (q * y) • (mk W') Y

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofiltered I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
s : ↑Γ(c.pt, ⊤)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : CompactSpace ↥c.pt
i : ↥c.pt → I
U : (x : ↥c.pt) → (D.obj (i x)).Opens
hU : ∀ (x : ↥c.pt), IsAffineOpen (U x)
hxU : ∀ (x : ↥c.pt), (c.π.app (i x)) x ∈ U x
t : (x : ↥c.pt) → ↑Γ(D.obj (i x), U x)
ht : ∀ (x : ↥c.pt), (ConcreteCategory.hom (Scheme.Hom.app (c.π.app (i x)) (U x))) (t x) = (s |_ c.π.app (i x) ⁻¹ᵁ U x) ⋯
j : ↥c.pt → ↥c.pt → I
fjx : (x y : ↥c.pt) → j x y ⟶ i x
fjy : (x y : ↥c.pt) → j x y ⟶ i y
hj :
  ∀ (x y : ↥c.pt),
    ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fjx x y)) (U x))) (t x) |_
          (D.map (fjx x y) ⁻¹ᵁ U x ⊓ D.map (fjy x y) ⁻¹ᵁ U y))
        ⋯ =
      ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fjy x y)) (U y))) (t y) |_
          (D.map (fjx x y) ⁻¹ᵁ U x ⊓ D.map (fjy x y) ⁻¹ᵁ U y))
        ⋯
σ : Finset ↥c.pt
hσ : ⋃ x ∈ σ, (c.π.app (i x) ⁻¹ᵁ U x).carrier = ⊤
σi : ↥c.pt → ↥c.pt
hσiσ : ∀ (x : ↥c.pt), σi x ∈ σ
hσi : ∀ (x : ↥c.pt), x ∈ (c.π.app (i (σi x)) ⁻¹ᵁ U (σi x)).carrier
S : Finset I := Finset.image i σ ∪ Finset.image₂ j σ σ
hiS : ∀ {x : ↥c.pt}, x ∈ σ → i x ∈ S
hjS : ∀ {x y : ↥c.pt}, x ∈ σ → y ∈ σ → j x y ∈ S
k : I
fk : {X : I} → X ∈ S → (k ⟶ X)
hk :
  ∀ {X Y : I} (mX : X ∈ S) (mY : Y ∈ S) {f : X ⟶ Y},
    ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈
        Finset.image₂ (fun x y ↦ ⟨j ↑x ↑y, ⟨i ↑x, ⟨⋯, ⟨⋯, fjx ↑x ↑y⟩⟩⟩⟩) σ.attach σ.attach ∪
          Finset.image₂ (fun x y ↦ ⟨j ↑x ↑y, ⟨i ↑y, ⟨⋯, ⟨⋯, fjy ↑x ↑y⟩⟩⟩⟩) σ.attach σ.attach →
      fk mX ≫ f = fk mY
hk₁ : ∀ {x y : ↥c.pt} (hx : x ∈ σ) (hy : y ∈ σ), fk ⋯ ≫ fjx x y = fk ⋯
hk₂ : ∀ {x y : ↥c.pt} (hx : x ∈ σ) (hy : y ∈ σ), fk ⋯ ≫ fjy x y = fk ⋯
k' : I
fk'k : k' ⟶ k
hk' : D.map fk'k ⁻¹ᵁ ⨆ x, ⨆ (hx : x ∈ σ), D.map (fk ⋯) ⁻¹ᵁ U x = ⊤
x : ↥(D.obj k')
x✝ : x ∈ ⊤
⊢ ∃ U_1 f, (Sieve.ofArrows (fun x ↦ D.map (fk'k ≫ fk ⋯) ⁻¹ᵁ U ↑x) fun x ↦ homOfLE ⋯).arrows f ∧ x ∈ U_1

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
⊢ (fun x ↦ x ^ 2) Y - (C (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆) - C (C W'.a₁ * X + C W'.a₃) * Y) =
    W'.polynomial * 1

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
⊢ (p • 1 + q • (mk W') Y) * (mk W') Y =
    (q * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)) • 1 + (p - q * (C W'.a₁ * X + C W'.a₃)) • (mk W') Y

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
S : Type s
A F : Type u
B K : Type v
L : Type w
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Field F
inst✝¹ : Field K
inst✝ : Field L
W' : Affine R
W : Affine F
f : R →+* S
⊢ eval₂ ((AdjoinRoot.of (W'.map f).polynomial).comp (mapRingHom f)) (AdjoinRoot.root (W'.map f).polynomial)
      W'.polynomial =
    0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Affine R
f : R →+* S
x : R[X][Y]
⊢ (map W' f) ((mk W') x) = (mk (W'.map f)) (Polynomial.map (mapRingHom f) x)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Affine R
f : R →+* S
x : R[X]
y : W'.CoordinateRing
⊢ (map W' f) (x • y) = Polynomial.map f x • (map W' f) y

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Affine R
f : R →+* S
hf : Function.Injective ⇑f
y : W'.CoordinateRing
hy : (map W' f) y = 0
⊢ y = 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Affine R
inst✝ : Nontrivial R
x : R
⊢ (C (X - C x)).natDegree < W'.polynomial.natDegree

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Affine R
inst✝ : Nontrivial R
y : R[X]
⊢ (Y - C y).natDegree < W'.polynomial.natDegree

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x y ℓ : R
⊢ C (W'.addPolynomial x y ℓ) - (Y - C (linePolynomial x y ℓ)) * (W'.negPolynomial - C (linePolynomial x y ℓ)) =
    W'.polynomial * 1

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
S : Type s
A F : Type u
B K : Type v
L : Type w
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Field F
inst✝¹ : Field K
inst✝ : Field L
W' : Affine R
W : Affine F
x : R
y : R[X]
h : eval x (eval y W'.polynomial) = 0
⊢ XYIdeal W' x y = Ideal.map (Quotient.mkₐ R (span {W'.polynomial})) (span {C (X - C x), Y - C y})

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x₁ x₂ y₁ ℓ : R
⊢ XYIdeal W' (W'.addX x₁ x₂ ℓ) (C (W'.addY x₁ x₂ y₁ ℓ)) =
    span {(mk W') (W'.negPolynomial - C (linePolynomial x₁ y₁ ℓ))} ⊔ XIdeal W' (W'.addX x₁ x₂ ℓ)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x y ℓ : R
⊢ XYIdeal W' x (C y) = XYIdeal W' x (linePolynomial x y ℓ)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx : x₁ = x₂
⊢ y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))

pp: case neg
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx : ¬x₁ = x₂
⊢ y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))

Tactic:
· have hy : y₁ ≠ W.negY x₂ y₂ := fun h => hxy ⟨hx, h⟩
      rcases hx, Y_eq_of_Y_ne h₁ h₂ hx hy with ⟨rfl, rfl⟩
      simp [linePolynomial]

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx : ¬x₁ = x₂
⊢ y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ XYIdeal W x₂ (C y₂) = XYIdeal W x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
⊢ (Y - C (C y)) * (Y - C (C (W.negY x y))) -
      C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) =
    W.polynomial * 1

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofiltered I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
s : ↑Γ(c.pt, ⊤)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this✝ : CompactSpace ↥c.pt
i : ↥c.pt → I
U : (x : ↥c.pt) → (D.obj (i x)).Opens
hU : ∀ (x : ↥c.pt), IsAffineOpen (U x)
hxU : ∀ (x : ↥c.pt), (c.π.app (i x)) x ∈ U x
t : (x : ↥c.pt) → ↑Γ(D.obj (i x), U x)
ht : ∀ (x : ↥c.pt), (ConcreteCategory.hom (Scheme.Hom.app (c.π.app (i x)) (U x))) (t x) = (s |_ c.π.app (i x) ⁻¹ᵁ U x) ⋯
j : ↥c.pt → ↥c.pt → I
fjx : (x y : ↥c.pt) → j x y ⟶ i x
fjy : (x y : ↥c.pt) → j x y ⟶ i y
hj :
  ∀ (x y : ↥c.pt),
    ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fjx x y)) (U x))) (t x) |_
          (D.map (fjx x y) ⁻¹ᵁ U x ⊓ D.map (fjy x y) ⁻¹ᵁ U y))
        ⋯ =
      ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fjy x y)) (U y))) (t y) |_
          (D.map (fjx x y) ⁻¹ᵁ U x ⊓ D.map (fjy x y) ⁻¹ᵁ U y))
        ⋯
σ : Finset ↥c.pt
hσ : ⋃ x ∈ σ, (c.π.app (i x) ⁻¹ᵁ U x).carrier = ⊤
σi : ↥c.pt → ↥c.pt
hσiσ : ∀ (x : ↥c.pt), σi x ∈ σ
hσi : ∀ (x : ↥c.pt), x ∈ (c.π.app (i (σi x)) ⁻¹ᵁ U (σi x)).carrier
S : Finset I := Finset.image i σ ∪ Finset.image₂ j σ σ
hiS : ∀ {x : ↥c.pt}, x ∈ σ → i x ∈ S
hjS : ∀ {x y : ↥c.pt}, x ∈ σ → y ∈ σ → j x y ∈ S
k : I
fk : {X : I} → X ∈ S → (k ⟶ X)
hk :
  ∀ {X Y : I} (mX : X ∈ S) (mY : Y ∈ S) {f : X ⟶ Y},
    ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈
        Finset.image₂ (fun x y ↦ ⟨j ↑x ↑y, ⟨i ↑x, ⟨⋯, ⟨⋯, fjx ↑x ↑y⟩⟩⟩⟩) σ.attach σ.attach ∪
          Finset.image₂ (fun x y ↦ ⟨j ↑x ↑y, ⟨i ↑y, ⟨⋯, ⟨⋯, fjy ↑x ↑y⟩⟩⟩⟩) σ.attach σ.attach →
      fk mX ≫ f = fk mY
hk₁ : ∀ {x y : ↥c.pt} (hx : x ∈ σ) (hy : y ∈ σ), fk ⋯ ≫ fjx x y = fk ⋯
hk₂ : ∀ {x y : ↥c.pt} (hx : x ∈ σ) (hy : y ∈ σ), fk ⋯ ≫ fjy x y = fk ⋯
k' : I
fk'k : k' ⟶ k
hk' : D.map fk'k ⁻¹ᵁ ⨆ x, ⨆ (hx : x ∈ σ), D.map (fk ⋯) ⁻¹ᵁ U x = ⊤
this :
  ∀ (x : (i_1 : ↥σ) → ((D.obj k').presheaf ⋙ forget CommRingCat).obj (Opposite.op (D.map (fk'k ≫ fk ⋯) ⁻¹ᵁ U ↑i_1))),
    Presieve.Arrows.Compatible ((D.obj k').presheaf ⋙ forget CommRingCat) (fun x ↦ homOfLE ⋯) x →
      ∃! t,
        ∀ (i_1 : ↥σ), (ConcreteCategory.hom (((D.obj k').presheaf ⋙ forget CommRingCat).map (homOfLE ⋯).op)) t = x i_1
x y : ↥σ
V : TopologicalSpace.Opens ↥(D.obj k')
fVx : V ⟶ (fun x ↦ D.map (fk'k ≫ fk ⋯) ⁻¹ᵁ U ↑x) x
fVy : V ⟶ (fun x ↦ D.map (fk'k ≫ fk ⋯) ⁻¹ᵁ U ↑x) y
x✝ : fVx ≫ (fun x ↦ homOfLE ⋯) x = fVy ≫ (fun x ↦ homOfLE ⋯) y
⊢ V ≤ D.map (fk'k ≫ fk ⋯) ⁻¹ᵁ (D.map (fjx ↑x ↑y) ⁻¹ᵁ U ↑x ⊓ D.map (fjy ↑x ↑y) ⁻¹ᵁ U ↑y)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
n : ℕ
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin n)
S : ConstructibleSetData (MvPolynomial (Fin n) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)
⊢ ∃ T,
    comap MvPolynomial.C '' S.toSet = T.toSet ∧
      ∀ C ∈ T,
        C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
          ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
⊢ g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
⊢ g.comp σ = RingHom.id (MvPolynomial (Fin m) R)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' : ∀ (x : MvPolynomial (Fin m) R), g (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
⊢ Ideal.span s₀ ≤ RingHom.ker g

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂) =
    -((X - C x₁) * (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' : ∀ (x : MvPolynomial (Fin m) R), g (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
q : MvPolynomial (Fin m) R
⊢ 0 ∈ Ideal.span s₀

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' : ∀ (x : MvPolynomial (Fin m) R), g (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
q : MvPolynomial (Fin m) R
⊢ (commAlgEquiv R (Fin n) (Fin m)) (MvPolynomial.C q) =
    (commAlgEquiv R (Fin n) (Fin m)) ((commAlgEquiv R (Fin n) (Fin m)).symm (0 + σ q))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' : ∀ (x : MvPolynomial (Fin m) R), g (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
q₁ : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
q₂ : MvPolynomial (Fin m) R
hq₁ : q₁ ∈ Ideal.span s₀
q₃ : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
q₄ : MvPolynomial (Fin m) R
hq₃ : q₃ ∈ Ideal.span s₀
⊢ (commAlgEquiv R (Fin n) (Fin m)).symm (q₁ + σ q₂) + (commAlgEquiv R (Fin n) (Fin m)).symm (q₃ + σ q₄) =
    (commAlgEquiv R (Fin n) (Fin m)).symm (q₁ + q₃ + σ (q₂ + q₄))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ C (W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) =
    -(C (X - C x₁) * C (X - C x₂) * C (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ derivative (W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) =
    -((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) +
        (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' : ∀ (x : MvPolynomial (Fin m) R), g (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
p : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
hp : p ∈ RingHom.ker g
⊢ ∃ q₁ q₂, q₁ ∈ Ideal.span s₀ ∧ p = q₁ + σ q₂

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degrees ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg : g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' : ∀ (x : MvPolynomial (Fin m) R), g (σ x) = x
hg' : Surjective ⇑g
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun x ↦
      match x with
      | { f := fk, n := k, g := gk } =>
        { f := σ fk, n := k + n,
          g := fun s ↦
            Sum.elim (⇑σ ∘ gk) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i)))
              (finSumFinEquiv.symm s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))
H : Ideal.span s₀ ≤ RingHom.ker g
q₁ : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)
q₂ : MvPolynomial (Fin m) R
hq₁ : q₁ ∈ Ideal.span s₀
hp : q₁ + σ q₂ ∈ RingHom.ker g
⊢ q₂ = 0

Tactic:
by

Goal state after: