mathlib-initiative/mathlib-tactics huggingface dataset:
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:
ext x : 2

Goal state after:
case hC
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
x : R
⊢ ((g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))).comp MvPolynomial.C)
      x =
    ((↑f).comp MvPolynomial.C) x

case hX
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
x : Fin n
m✝ : Fin m →₀ ℕ
⊢ MvPolynomial.coeff m✝
      ((g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))))
        (MvPolynomial.X x)) =
    MvPolynomial.coeff m✝ (↑f (MvPolynomial.X x))
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))
⊢ zeroLocus ↑(Ideal.span s₀) = zeroLocus ↑(RingHom.ker g)

Tactic:
congr! 2

Goal state after:
case e'_3
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
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

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.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ 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.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

Tactic:
· by_cases hx : x₁ = x₂
      · subst hx
        contradiction
      · rwa [negAddY, ← neg_sub, mul_neg, hx₂, slope_of_X_ne hx,
          div_mul_cancel₀ _ <| sub_ne_zero_of_ne hx, neg_sub, sub_add_cancel]

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.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).negPolynomial = ((W'⁄A).map ↑f).negPolynomial

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).negPolynomial = (W'⁄B).negPolynomial
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x y : A
⊢ (W'⁄B).negY (↑f x) (↑f y) = ((W'⁄A).map ↑f).negY (↑f x) (↑f y)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x y : A
⊢ (W'⁄B).negY (↑f x) (↑f y) = (W'⁄B).negY (↑f x) (↑f y)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x y ℓ : A
⊢ (W'⁄B).addPolynomial (↑f x) (↑f y) (↑f ℓ) = ((W'⁄A).map ↑f).addPolynomial (↑f x) (↑f y) (↑f ℓ)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x y ℓ : A
⊢ (W'⁄B).addPolynomial (↑f x) (↑f y) (↑f ℓ) = (W'⁄B).addPolynomial (↑f x) (↑f y) (↑f ℓ)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ x₂ ℓ : A
⊢ (W'⁄B).addX (↑f x₁) (↑f x₂) (↑f ℓ) = ((W'⁄A).map ↑f).addX (↑f x₁) (↑f x₂) (↑f ℓ)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ x₂ ℓ : A
⊢ (W'⁄B).addX (↑f x₁) (↑f x₂) (↑f ℓ) = (W'⁄B).addX (↑f x₁) (↑f x₂) (↑f ℓ)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ y₁ x₂ ℓ : A
⊢ (W'⁄B).negAddY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ) = ((W'⁄A).map ↑f).negAddY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ y₁ x₂ ℓ : A
⊢ (W'⁄B).negAddY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ) = (W'⁄B).negAddY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ y₁ x₂ ℓ : A
⊢ (W'⁄B).addY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ) = ((W'⁄A).map ↑f).addY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Affine R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
x₁ y₁ x₂ ℓ : A
⊢ (W'⁄B).addY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ) = (W'⁄B).addY (↑f x₁) (↑f x₂) (↑f y₁) (↑f ℓ)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: R : Type r
S : Type s
F : Type u
K : Type v
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Field F
inst✝⁹ : Field K
W' : Affine R
inst✝⁸ : Algebra R S
inst✝⁷ : DecidableEq F
inst✝⁶ : DecidableEq K
inst✝⁵ : Algebra R F
inst✝⁴ : Algebra S F
inst✝³ : IsScalarTower R S F
inst✝² : Algebra R K
inst✝¹ : Algebra S K
inst✝ : IsScalarTower R S K
f : F →ₐ[S] K
x₁ x₂ y₁ y₂ : F
⊢ (W'⁄K).slope (↑f x₁) (↑f x₂) (↑f y₁) (↑f y₂) = ((W'⁄F).map ↑f).slope (↑f x₁) (↑f x₂) (↑f y₁) (↑f y₂)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
F : Type u
K : Type v
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : Field F
inst✝⁹ : Field K
W' : Affine R
inst✝⁸ : Algebra R S
inst✝⁷ : DecidableEq F
inst✝⁶ : DecidableEq K
inst✝⁵ : Algebra R F
inst✝⁴ : Algebra S F
inst✝³ : IsScalarTower R S F
inst✝² : Algebra R K
inst✝¹ : Algebra S K
inst✝ : IsScalarTower R S K
f : F →ₐ[S] K
x₁ x₂ y₁ y₂ : F
⊢ (W'⁄K).slope (↑f x₁) (↑f x₂) (↑f y₁) (↑f y₂) = (W'⁄K).slope (↑f x₁) (↑f x₂) (↑f y₁) (↑f y₂)
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
W : ℤ → R
odd : Function.Odd W
a b c d : ℤ
⊢ atomRel W (-a) b c d = atomRel W a b c d

Tactic:
simp_rw [atomRel, atom_neg_left odd]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_2
I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
inst✝ : c.pt.IsQuasiAffine
x : ↥c.pt
i : I
U : TopologicalSpace.Opens ↥(D.obj i)
hU : U ∈ (D.obj i).affineOpens
hxU : (c.π.app i) x ∈ ↑U
j : I
r : ↑Γ(D.obj j, ⊤)
hr : IsAffineOpen (c.pt.basicOpen ((ConcreteCategory.hom (Hom.appTop (c.π.app j))) r))
hxr : x ∈ ↑(c.pt.basicOpen ((ConcreteCategory.hom (Hom.appTop (c.π.app j))) r))
hrU : ↑(c.pt.basicOpen ((ConcreteCategory.hom (Hom.appTop (c.π.app j))) r)) ⊆ ↑(c.π.app i ⁻¹ᵁ U)
k : I
fki : k ⟶ i
fkj : k ⟶ j
l : I
flk : l ⟶ k
hl :
  D.map flk ⁻¹ᵁ (D.obj k).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map fkj))) r) ≤ D.map flk ⁻¹ᵁ D.map fki ⁻¹ᵁ U
⊢ x ∈
    (((Functor.const I).obj c.pt).obj l).basicOpen
      ((ConcreteCategory.hom (Hom.appTop (c.π.app l))) ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r))

Tactic:
← elementwise_of% Scheme.Hom.comp_appTop,

Goal state after:
case refine_2
I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
inst✝ : c.pt.IsQuasiAffine
x : ↥c.pt
i : I
U : TopologicalSpace.Opens ↥(D.obj i)
hU : U ∈ (D.obj i).affineOpens
hxU : (c.π.app i) x ∈ ↑U
j : I
r : ↑Γ(D.obj j, ⊤)
hr : IsAffineOpen (c.pt.basicOpen ((ConcreteCategory.hom (Hom.appTop (c.π.app j))) r))
hxr : x ∈ ↑(c.pt.basicOpen ((ConcreteCategory.hom (Hom.appTop (c.π.app j))) r))
hrU : ↑(c.pt.basicOpen ((ConcreteCategory.hom (Hom.appTop (c.π.app j))) r)) ⊆ ↑(c.π.app i ⁻¹ᵁ U)
k : I
fki : k ⟶ i
fkj : k ⟶ j
l : I
flk : l ⟶ k
hl :
  D.map flk ⁻¹ᵁ (D.obj k).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map fkj))) r) ≤ D.map flk ⁻¹ᵁ D.map fki ⁻¹ᵁ U
⊢ x ∈
    (((Functor.const I).obj c.pt).obj l).basicOpen
      ((ConcreteCategory.hom (Hom.appTop (c.π.app l ≫ D.map (flk ≫ fkj)))) r)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
P : W.Point
⊢ toClass P = 0 ↔ P = 0

Tactic:
by
  constructor
  · intro hP
    rcases P with (_ | ⟨_, _, h, _⟩)
    · rfl
    · rcases (ClassGroup.mk_eq_one_of_coe_ideal <| by rfl).mp hP with ⟨p, h0, hp⟩
      apply (p.natDegree_norm_ne_one _).elim
      rw [← finrank_quotient_span_eq_natDegree_norm (CoordinateRing.basis W) h0,
        ← (quotientEquivAlgOfEq F hp).toLinearEquiv.finrank_eq,
        (CoordinateRing.quotientXYIdealEquiv h).toLinearEquiv.finrank_eq, Module.finrank_self]
  · exact congr_arg toClass

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).ψ₂ = ((W⁄A).map ↑f).ψ₂

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).ψ₂ = (W⁄B).ψ₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).Ψ₂Sq = ((W⁄A).map ↑f).Ψ₂Sq

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).Ψ₂Sq = (W⁄B).Ψ₂Sq
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).Ψ₃ = ((W⁄A).map ↑f).Ψ₃

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).Ψ₃ = (W⁄B).Ψ₃
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).preΨ₄ = ((W⁄A).map ↑f).preΨ₄

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W⁄B).preΨ₄ = (W⁄B).preΨ₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℕ
⊢ (W⁄B).preΨ' n = ((W⁄A).map ↑f).preΨ' n

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℕ
⊢ (W⁄B).preΨ' n = (W⁄B).preΨ' n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).preΨ n = ((W⁄A).map ↑f).preΨ n

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).preΨ n = (W⁄B).preΨ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).ΨSq n = ((W⁄A).map ↑f).ΨSq n

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).ΨSq n = (W⁄B).ΨSq n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).Ψ n = ((W⁄A).map ↑f).Ψ n

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).Ψ n = (W⁄B).Ψ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).Φ n = ((W⁄A).map ↑f).Φ n

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
W : WeierstrassCurve R
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
n : ℤ
⊢ (W⁄B).Φ n = (W⁄B).Φ n