mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case refine_1.refine_3
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.129} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.129} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : (a : J) → (D.obj a).left.OpenCover := fun a ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).hom 𝒰
a : J
⊢ ∀ (i : (𝒱 a).I₀),
    (((d.cocone i).ι.app a).left ≫ colimit.ι d.relativeGluingData.functor i) ≫
        Over.hom (((Functor.const J).obj d.glued).obj a).toComma =
      (𝒱 a).f i ≫ Over.hom (D.obj a).toComma

Tactic:
· intro
      simp [𝒱, pullback.condition, glued]

Goal state after:
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case h₀
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.252} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.252} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := ⋯
s : Cocone D
i j : 𝒱.I₀
hij : i ⟶ j
j✝ : J
⊢ (((MorphismProperty.Over.map ⊤ ⋯).mapCocone (d.cocone i)).ι.app j✝ ≫
          d.transitionMap hij ≫ (d.isColimit j).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).mapCocone s)).left ≫
      pullback.fst s.pt.hom (𝒰.f j) =
    (((MorphismProperty.Over.map ⊤ ⋯).mapCocone (d.cocone i)).ι.app j✝ ≫
          (MorphismProperty.Over.map ⊤ ⋯).map
              ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)) ≫
            (Over.pullbackMapHomPullback (Cover.trans 𝒰 hij) ⋯ trivial (𝒰.f j) (𝒰.f i) ⋯).app s.pt).left ≫
      pullback.fst s.pt.hom (𝒰.f j)

Tactic:
simp [cocone_ι_transitionMap_assoc, ← Functor.map_comp_assoc]

Goal state after:
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case h₁
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.252} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.252} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := ⋯
s : Cocone D
i j : 𝒱.I₀
hij : i ⟶ j
j✝ : J
⊢ (((MorphismProperty.Over.map ⊤ ⋯).mapCocone (d.cocone i)).ι.app j✝ ≫
          d.transitionMap hij ≫ (d.isColimit j).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).mapCocone s)).left ≫
      pullback.snd s.pt.hom (𝒰.f j) =
    (((MorphismProperty.Over.map ⊤ ⋯).mapCocone (d.cocone i)).ι.app j✝ ≫
          (MorphismProperty.Over.map ⊤ ⋯).map
              ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)) ≫
            (Over.pullbackMapHomPullback (Cover.trans 𝒰 hij) ⋯ trivial (𝒰.f j) (𝒰.f i) ⋯).app s.pt).left ≫
      pullback.snd s.pt.hom (𝒰.f j)

Tactic:
simp [cocone_ι_transitionMap_assoc, ← Functor.map_comp_assoc]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
inst✝ : CompactSpace ↥(D.obj i)
s : ↑Γ(D.obj i, ⊤)
hs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0
x : ↥(D.obj i)
U : TopologicalSpace.Opens ↥(D.obj i)
hU : IsAffineOpen U
hxU : x ∈ ↑U
this : ∀ (j : Over i), IsAffine ((opensDiagram D i U).obj j)
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hj :
  (ConcreteCategory.hom (Scheme.Hom.appTop ((opensDiagram D i U).map f)))
      ((ConcreteCategory.hom (Scheme.Hom.appTop ((opensDiagramι D i U).app (Over.mk (𝟙 i))))) s) =
    0
⊢ ∃ U, ∃ (_ : IsAffineOpen U) (_ : x ∈ U), ∃ j f, (ConcreteCategory.hom (Scheme.Hom.app (D.map f) U)) ((s |_ U) ⋯) = 0

Tactic:
refine ⟨U, hU, hxU, j.left, j.hom, ?_⟩

Goal state after:
I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
inst✝ : CompactSpace ↥(D.obj i)
s : ↑Γ(D.obj i, ⊤)
hs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0
x : ↥(D.obj i)
U : TopologicalSpace.Opens ↥(D.obj i)
hU : IsAffineOpen U
hxU : x ∈ ↑U
this : ∀ (j : Over i), IsAffine ((opensDiagram D i U).obj j)
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hj :
  (ConcreteCategory.hom (Scheme.Hom.appTop ((opensDiagram D i U).map f)))
      ((ConcreteCategory.hom (Scheme.Hom.appTop ((opensDiagramι D i U).app (Over.mk (𝟙 i))))) s) =
    0
⊢ (ConcreteCategory.hom (Scheme.Hom.app (D.map j.hom) U)) ((s |_ U) ⋯) = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : Affine R
x y : R
⊢ (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).a₆ = 0 ∧
      (2 * y + W.a₁ * x + W.a₃ ≠ 0 ∨ -(W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄)) ≠ 0) ↔
    (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).a₆ = 0 ∧
      (↑{ u := 1, r := x, s := 0, t := y }.u⁻¹ ^ 3 *
            (W.a₃ + { u := 1, r := x, s := 0, t := y }.r * W.a₁ + 2 * { u := 1, r := x, s := 0, t := y }.t) ≠
          0 ∨
        (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).a₄ ≠ 0)

Tactic:
variableChange_a₄,

Goal state after:
R : Type r
inst✝ : CommRing R
W : Affine R
x y : R
⊢ (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).a₆ = 0 ∧
      (2 * y + W.a₁ * x + W.a₃ ≠ 0 ∨ -(W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄)) ≠ 0) ↔
    (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).a₆ = 0 ∧
      (↑{ u := 1, r := x, s := 0, t := y }.u⁻¹ ^ 3 *
            (W.a₃ + { u := 1, r := x, s := 0, t := y }.r * W.a₁ + 2 * { u := 1, r := x, s := 0, t := y }.t) ≠
          0 ∨
        ↑{ u := 1, r := x, s := 0, t := y }.u⁻¹ ^ 4 *
            (W.a₄ - { u := 1, r := x, s := 0, t := y }.s * W.a₃ + 2 * { u := 1, r := x, s := 0, t := y }.r * W.a₂ -
                  ({ u := 1, r := x, s := 0, t := y }.t +
                      { u := 1, r := x, s := 0, t := y }.r * { u := 1, r := x, s := 0, t := y }.s) *
                    W.a₁ +
                3 * { u := 1, r := x, s := 0, t := y }.r ^ 2 -
              2 * { u := 1, r := x, s := 0, t := y }.s * { u := 1, r := x, s := 0, t := y }.t) ≠
          0)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
C : VariableChange R
inst✝ : W.IsElliptic
⊢ (C.u⁻¹ ^ 12 * W.Δ')⁻¹ = C.u ^ 12 * W.Δ'⁻¹

Tactic:
mul_inv,

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
C : VariableChange R
inst✝ : W.IsElliptic
⊢ (C.u⁻¹ ^ 12)⁻¹ * W.Δ'⁻¹ = C.u ^ 12 * W.Δ'⁻¹
Module: Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
C✝ : VariableChange R
A : Type v
inst✝ : CommRing A
φ : R →+* A
C C' : VariableChange R
⊢ (C * C').map φ = C.map φ * C'.map φ

Tactic:
simp only [mul_def, map]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
C✝ : VariableChange R
A : Type v
inst✝ : CommRing A
φ : R →+* A
C C' : VariableChange R
⊢ { u := (Units.map ↑φ) (C.u * C'.u), r := φ (C.r * ↑C'.u ^ 2 + C'.r), s := φ (↑C'.u * C.s + C'.s),
      t := φ (C.t * ↑C'.u ^ 3 + C.r * C'.s * ↑C'.u ^ 2 + C'.t) } =
    { u := (Units.map ↑φ) C.u * (Units.map ↑φ) C'.u, r := φ C.r * ↑((Units.map ↑φ) C'.u) ^ 2 + φ C'.r,
      s := ↑((Units.map ↑φ) C'.u) * φ C.s + φ C'.s,
      t := φ C.t * ↑((Units.map ↑φ) C'.u) ^ 3 + φ C.r * φ C'.s * ↑((Units.map ↑φ) C'.u) ^ 2 + φ C'.t }
Module: Mathlib.LinearAlgebra.FreeModule.Norm

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : IsPrincipalIdealRing R
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra R S
inst✝ : Fintype ι
b : Basis ι R S
f : S
hf : f ≠ 0
hI : ¬span {f} = ⊥
b' : Basis ι R S := ringBasis b (span {f}) hI
⊢ Associated ((Algebra.norm R) f) (∏ i, smithCoeffs b (span {f}) ⋯ i)

Tactic:
rw [← Matrix.det_diagonal, ← LinearMap.det_toLin b']
  let e :=
    (b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans
      ((LinearEquiv.coord S S f hf).restrictScalars R)
  refine (LinearMap.associated_det_of_eq_comp e _ _ ?_).symm
  dsimp only [e, LinearEquiv.trans_apply]
  simp_rw [← LinearEquiv.coe_toLinearMap, ← LinearMap.comp_apply, ← LinearMap.ext_iff]
  refine b'.ext fun i => ?_
  simp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,
    Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul,
    zero_smul, Finset.sum_ite_eq', mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]
  change _ = f * _
  rw [mul_comm, ← smul_eq_mul, LinearEquiv.restrictScalars_apply, LinearEquiv.coord_apply_smul]
  grind [Ideal.selfBasis_def]

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

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

Tactic:
simp_rw [atomRel, atom_neg_right]

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

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

Tactic:
simpa only [preNormEDS_ofNat] using preNormEDS'_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 ∈ c.π.app l ⁻¹ᵁ (D.obj l).basicOpen ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r)

Tactic:
Scheme.preimage_basicOpen_top,

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))) ((ConcreteCategory.hom (Hom.appTop (D.map (flk ≫ fkj)))) r))
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case intro
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)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
k : I
fkj : k ⟶ i
e : D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤
⊢ ∃ i V, IsOpenCover V ∧ ∀ (j : J), IsAffineOpen (V j) ∧ U j = c.π.app i ⁻¹ᵁ V j

Tactic:
refine ⟨k, fun x ↦ D.map (fkj ≫ fi x) ⁻¹ᵁ V _, ?_, fun k ↦ ⟨(hV k).preimage _, ?_⟩⟩

Goal state after:
case intro.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)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
k : I
fkj : k ⟶ i
e : D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤
⊢ IsOpenCover fun x ↦ D.map (fkj ≫ fi x) ⁻¹ᵁ V x

case intro.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)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
k✝ : I
fkj : k✝ ⟶ i
e : D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤
k : J
⊢ U k = c.π.app k✝ ⁻¹ᵁ (fun x ↦ D.map (fkj ≫ fi x) ⁻¹ᵁ V x) k
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case inr
I : Type u
inst✝⁵ : Category.{u, u} I
X : Scheme
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : IsAffine X
inst✝² : ∀ (i : I), IsAffine (D.obj i)
a : c.pt ⟶ X
R : CommRingCat
t : D ⟶ (Functor.const I).obj (Spec R)
f : X ⟶ Spec R
inst✝¹ : LocallyOfFinitePresentation f
inst✝ : IsAffine (Spec R)
ha : c.π ≫ t = (Functor.const I).map (a ≫ f)
this :
  ∀ {X : Scheme} [IsAffine X] (a : c.pt ⟶ X) (f : X ⟶ Spec R) [LocallyOfFinitePresentation f],
    c.π ≫ t = (Functor.const I).map (a ≫ f) → (∃ S, X = Spec S) → ∃ i g, c.π.app i ≫ g = a ∧ g ≫ f = t.app i
hX : ¬∃ S, X = Spec S
⊢ ∃ i g, c.π.app i ≫ g = a ∧ g ≫ f = t.app i

Tactic:
obtain ⟨i, f, hf⟩ := this (a ≫ X.isoSpec.hom) (X.isoSpec.inv ≫ f)
      (by simp [ha, -Functor.map_comp]) ⟨_, rfl⟩
    exact ⟨i, f ≫ X.isoSpec.inv, by simpa [← Iso.comp_inv_eq] using! hf⟩

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
W' : Affine R
P Q : W'.Point
⊢ P.sym2x Q = Q.sym2x P

Tactic:
by
  cases P <;> cases Q <;> simp [← zero_def, mul_comm, add_comm]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: F : Type u_2
inst✝² : Field F
W : WeierstrassCurve F
inst✝¹ : W.IsElliptic
inst✝ : W.IsShortNF
h : ↑W.Δ' ≠ 0
⊢ W.j = 6912 * W.a₄ ^ 3 / (4 * W.a₄ ^ 3 + 27 * W.a₆ ^ 2)

Tactic:
rw [coe_Δ', Δ_of_isShortNF] at h

Goal state after:
F : Type u_2
inst✝² : Field F
W : WeierstrassCurve F
inst✝¹ : W.IsElliptic
inst✝ : W.IsShortNF
h : -16 * (4 * W.a₄ ^ 3 + 27 * W.a₆ ^ 2) ≠ 0
⊢ W.j = 6912 * W.a₄ ^ 3 / (4 * W.a₄ ^ 3 + 27 * W.a₆ ^ 2)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₄
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₄ = E'.a₄

Tactic:
variableChange_a₄,

Goal state after:
case h.a₄
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ↑1⁻¹ ^ 4 * (E.a₄ - s * E.a₃ + 2 * 0 * E.a₂ - (0 + 0 * s) * E.a₁ + 3 * 0 ^ 2 - 2 * s * 0) = E'.a₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: F : Type u_2
inst✝³ : Field F
W : WeierstrassCurve F
inst✝² : W.IsElliptic
inst✝¹ : W.IsCharThreeJNeZeroNF
inst✝ : CharP F 3
⊢ -W.a₂ ^ 3 / W.a₆ ≠ 0

Tactic:
div_ne_zero_iff

Goal state after:
F : Type u_2
inst✝³ : Field F
W : WeierstrassCurve F
inst✝² : W.IsElliptic
inst✝¹ : W.IsCharThreeJNeZeroNF
inst✝ : CharP F 3
⊢ -W.a₂ ^ 3 ≠ 0 ∧ W.a₆ ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
⊢ E.a₃ / E'.a₃ ≠ 0

Tactic:
div_ne_zero_iff

Goal state after:
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
⊢ E.a₃ ≠ 0 ∧ E'.a₃ ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: F : Type u_2
inst✝³ : Field F
W : WeierstrassCurve F
inst✝² : W.IsElliptic
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP F 2
⊢ 1 / W.a₆ ≠ 0

Tactic:
div_ne_zero_iff

Goal state after:
F : Type u_2
inst✝³ : Field F
W : WeierstrassCurve F
inst✝² : W.IsElliptic
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP F 2
⊢ 1 ≠ 0 ∧ W.a₆ ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
⊢ E.a₂ / E'.a₂ ≠ 0

Tactic:
div_ne_zero_iff

Goal state after:
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
⊢ E.a₂ ≠ 0 ∧ E'.a₂ ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₄ = E'.a₄

Tactic:
variableChange_a₄,

Goal state after:
case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
hu0 : u ≠ 0
⊢ ↑(Units.mk0 u hu0)⁻¹ ^ 4 * (E.a₄ - 0 * E.a₃ + 2 * 0 * E.a₂ - (0 + 0 * 0) * E.a₁ + 3 * 0 ^ 2 - 2 * 0 * 0) = E'.a₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
⊢ E.a₄ / E'.a₄ ≠ 0

Tactic:
div_ne_zero_iff

Goal state after:
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
⊢ E.a₄ ≠ 0 ∧ E'.a₄ ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := r, s := 0, t := 0 } • E).a₄ = E'.a₄

Tactic:
variableChange_a₄,

Goal state after:
case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ↑(Units.mk0 u hu0)⁻¹ ^ 4 * (E.a₄ - 0 * E.a₃ + 2 * r * E.a₂ - (0 + r * 0) * E.a₁ + 3 * r ^ 2 - 2 * 0 * 0) = E'.a₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : E.a₄ = 0
ha₆ : E.a₆ ≠ 0
ha₄' : E'.a₄ = 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 6 = E.a₆ / E'.a₆
⊢ E.a₆ / E'.a₆ ≠ 0

Tactic:
div_ne_zero_iff

Goal state after:
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : E.a₄ = 0
ha₆ : E.a₆ ≠ 0
ha₄' : E'.a₄ = 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 6 = E.a₆ / E'.a₆
⊢ E.a₆ ≠ 0 ∧ E'.a₆ ≠ 0