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

Goal state before:
pp: case neg
F : Type u
K : Type v
inst✝³ : Field F
inst✝² : Field K
W : Affine F
inst✝¹ : DecidableEq F
inst✝ : DecidableEq K
f : F →+* K
x₁ x₂ y₁ y₂ : F
hx : ¬x₁ = x₂
⊢ (W.map f).slope (f x₁) (f x₂) (f y₁) (f y₂) = f (W.slope x₁ x₂ y₁ y₂)

Tactic:
·

Goal state after:
case neg
F : Type u
K : Type v
inst✝³ : Field F
inst✝² : Field K
W : Affine F
inst✝¹ : DecidableEq F
inst✝ : DecidableEq K
f : F →+* K
x₁ x₂ y₁ y₂ : F
hx : ¬x₁ = x₂
⊢ (W.map f).slope (f x₁) (f x₂) (f y₁) (f y₂) = f (W.slope x₁ x₂ y₁ y₂)
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case right
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 := ⋯
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) := ⋯
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)) := ⋯
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) := ⋯
hs : zeroLocus s₀ = Set.range (comap g)
hg'' : ∀ (t : Set (PrimeSpectrum (MvPolynomial (Fin m) R))), comap g '' t = comap σ ⁻¹' t ∩ zeroLocus s₀
hS' : comap g '' S.toSet = S'.toSet
x : BasicConstructibleSetData (MvPolynomial (Fin m) R)
hx : x ∈ S
j : Fin x.n ⊕ Fin n
⊢ (Sum.elim (⇑σ ∘ x.g) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))) j).degrees ≤ d

Tactic:
·

Goal state after:
case right
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 := ⋯
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) := ⋯
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)) := ⋯
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) := ⋯
hs : zeroLocus s₀ = Set.range (comap g)
hg'' : ∀ (t : Set (PrimeSpectrum (MvPolynomial (Fin m) R))), comap g '' t = comap σ ⁻¹' t ∩ zeroLocus s₀
hS' : comap g '' S.toSet = S'.toSet
x : BasicConstructibleSetData (MvPolynomial (Fin m) R)
hx : x ∈ S
j : Fin x.n ⊕ Fin n
⊢ (Sum.elim (⇑σ ∘ x.g) (fun i ↦ MvPolynomial.C (MvPolynomial.X i) - σ (f (MvPolynomial.X i))) j).degrees ≤ d
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap

Goal state before:
pp: case «0»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (s ^ 2 - C W.b₄ * s * u - C W.b₆ * t * u - C W.b₈ * u ^ 2).IsHomogeneous 2

pp: case «1»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (C 2 * t * s + C W.b₂ * s * u + C W.b₄ * t * u + C W.b₆ * u ^ 2).IsHomogeneous 2

pp: case «2»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (t ^ 2 - C 4 * s * u).IsHomogeneous 2

Tactic:
·

Goal state after:
case «0»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (s ^ 2 - C W.b₄ * s * u - C W.b₆ * t * u - C W.b₈ * u ^ 2).IsHomogeneous 2
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap

Goal state before:
pp: case «1»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (C 2 * t * s + C W.b₂ * s * u + C W.b₄ * t * u + C W.b₆ * u ^ 2).IsHomogeneous 2

pp: case «2»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (t ^ 2 - C 4 * s * u).IsHomogeneous 2

Tactic:
·

Goal state after:
case «1»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (C 2 * t * s + C W.b₂ * s * u + C W.b₄ * t * u + C W.b₆ * u ^ 2).IsHomogeneous 2
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap

Goal state before:
pp: case «2»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (t ^ 2 - C 4 * s * u).IsHomogeneous 2

Tactic:
·

Goal state after:
case «2»
R : Type u_1
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (t ^ 2 - C 4 * s * u).IsHomogeneous 2
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)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
hx : x₁ = x₂
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1

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)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
hx : ¬x₁ = x₂
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1

Tactic:
·

Goal state after:
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)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
hx : x₁ = x₂
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommRing R
W : ℤ → R
odd : Function.Odd W
a b c : ℤ
⊢ atom W a b * atom W a c - atom W a a * atom W b c + atom W a c * atom W b a +
        (W a * W 0 * atom W b c + atom W a c * -atom W a b) -
      (W a * W 0 * atom W c b + (W a * W 0 * -atom W c b + atom W a c * atom W b a)) =
    0

Tactic:
atom_same

Goal state after:
case a
R : Type u_1
inst✝ : CommRing R
W : ℤ → R
odd : Function.Odd W
a b c : ℤ
⊢ atom W a b * atom W a c - W a * W 0 * atom W b c + atom W a c * atom W b a +
        (W a * W 0 * atom W b c + atom W a c * -atom W a b) -
      (W a * W 0 * atom W c b + (W a * W 0 * -atom W c b + atom W a c * atom W b a)) =
    0
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 : ℤ
⊢ atom W a a * atom W b c = W a * W 0 * atom W b c

Tactic:
atom_same

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommRing R
W : ℤ → R
odd : Function.Odd W
a b c : ℤ
⊢ atom W a b * atom W c b - atom W a c * atom W b b + atom W a b * atom W b c +
        (W b * W 0 * atom W c a + atom W a b * -atom W b c) -
      (W b * W 0 * atom W c a + (W b * W 0 * -atom W a c + atom W a b * atom W c b)) =
    0

Tactic:
atom_same

Goal state after:
case a
R : Type u_1
inst✝ : CommRing R
W : ℤ → R
odd : Function.Odd W
a b c : ℤ
⊢ atom W a b * atom W c b - atom W a c * (W b * W 0) + atom W a b * atom W b c +
        (W b * W 0 * atom W c a + atom W a b * -atom W b c) -
      (W b * W 0 * atom W c a + (W b * W 0 * -atom W a c + atom W a b * atom W c b)) =
    0
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_1
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (x : ((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).pt),
    ∃ i xi, x = (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) xi

pp: case refine_2
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (i : Iᵒᵖ) (x y : (fun X ↦ X) (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).obj i)),
    (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) x =
        (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) y →
      ∃ k f,
        (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) x =
          (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) y

Tactic:
·

Goal state after:
case refine_1
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (x : ((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).pt),
    ∃ i xi, x = (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) xi
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_1
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (x : ((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).pt),
    ∃ i xi, x = (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) xi

pp: case refine_2
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (i : Iᵒᵖ) (x y : (fun X ↦ X) (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).obj i)),
    (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) x =
        (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) y →
      ∃ k f,
        (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) x =
          (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) y

Tactic:
· exact fun s ↦ ⟨.op _, (exists_appTop_π_eq_of_isLimit D c hc s).choose_spec⟩

Goal state after:
case refine_2
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (i : Iᵒᵖ) (x y : (fun X ↦ X) (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).obj i)),
    (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) x =
        (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) y →
      ∃ k f,
        (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) x =
          (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) y
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_2
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (i : Iᵒᵖ) (x y : (fun X ↦ X) (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).obj i)),
    (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) x =
        (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) y →
      ∃ k f,
        (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) x =
          (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) y

Tactic:
·

Goal state after:
case refine_2
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)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : ReflectsFilteredColimits (forget CommRingCat)
⊢ ∀ (i : Iᵒᵖ) (x y : (fun X ↦ X) (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).obj i)),
    (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) x =
        (ConcreteCategory.hom (((forget CommRingCat).mapCocone (Scheme.Γ.mapCocone c.op)).ι.app i)) y →
      ∃ k f,
        (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) x =
          (ConcreteCategory.hom (((D.op ⋙ Scheme.Γ) ⋙ forget CommRingCat).map f)) y
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: 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
r : ↑Γ(c.pt, ⊤)
hr : IsAffineOpen (c.pt.basicOpen r)
hxr : x ∈ ↑(c.pt.basicOpen r)
hrU : ↑(c.pt.basicOpen r) ⊆ ↑(c.π.app i ⁻¹ᵁ U)
⊢ ∃ i f, IsAffineOpen ((D.obj i).basicOpen f) ∧ (c.π.app i) x ∈ (D.obj i).basicOpen f

Tactic:
obtain ⟨j, r, rfl⟩ := exists_appTop_π_eq_of_isLimit D c hc r

Goal state after:
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)
⊢ ∃ i f, IsAffineOpen ((D.obj i).basicOpen f) ∧ (c.π.app i) x ∈ (D.obj i).basicOpen f
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
b c d : R
n : ℕ
hn : n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n

pp: case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
n : ℕ
hn : ¬n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
b c d : R
n : ℕ
hn : n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
n : ℕ
hn : ¬n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
n : ℕ
hn : ¬n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
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)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
hx : ¬x₁ = x₂
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1

Tactic:
·

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)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
hx : ¬x₁ = x₂
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_1
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
⊢ IsAffineOpen ((D.obj l).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r))

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
⊢ (c.π.app l) x ∈ (D.obj l).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r)

Tactic:
·

Goal state after:
case refine_1
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
⊢ IsAffineOpen ((D.obj l).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r))
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg.zero
R : Type u_1
inst✝ : CommRing R
b c d : R
ih :
  ∀ (n : ℕ),
    preNormEDS b c d (2 * ↑n + 1) =
      (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -
        preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b
⊢ preNormEDS b c d (2 * -↑0 + 1) =
    (preNormEDS b c d (-↑0 + 2) * preNormEDS b c d (-↑0) ^ 3 * if Even (-↑0) then b else 1) -
      preNormEDS b c d (-↑0 - 1) * preNormEDS b c d (-↑0 + 1) ^ 3 * if Even (-↑0) then 1 else b

pp: case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
ih :
  ∀ (n : ℕ),
    preNormEDS b c d (2 * ↑n + 1) =
      (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -
        preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b
m : ℕ
⊢ preNormEDS b c d (2 * -↑(m + 1) + 1) =
    (preNormEDS b c d (-↑(m + 1) + 2) * preNormEDS b c d (-↑(m + 1)) ^ 3 * if Even (-↑(m + 1)) then b else 1) -
      preNormEDS b c d (-↑(m + 1) - 1) * preNormEDS b c d (-↑(m + 1) + 1) ^ 3 * if Even (-↑(m + 1)) then 1 else b

Tactic:
·

Goal state after:
case neg.zero
R : Type u_1
inst✝ : CommRing R
b c d : R
ih :
  ∀ (n : ℕ),
    preNormEDS b c d (2 * ↑n + 1) =
      (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -
        preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b
⊢ preNormEDS b c d (2 * -↑0 + 1) =
    (preNormEDS b c d (-↑0 + 2) * preNormEDS b c d (-↑0) ^ 3 * if Even (-↑0) then b else 1) -
      preNormEDS b c d (-↑0 - 1) * preNormEDS b c d (-↑0 + 1) ^ 3 * if Even (-↑0) then 1 else b
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
⊢ (c.π.app l) x ∈ (D.obj l).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r)

Tactic:
·

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
⊢ (c.π.app l) x ∈ (D.obj l).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r)