mathlib-initiative/mathlib-tactics huggingface dataset:
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) ⋯
x y : ↥c.pt
j : I
fjx : j ⟶ i x
fjy : j ⟶ i y
⊢ (ConcreteCategory.hom (Scheme.Hom.app (c.π.app j) (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y)))
      (((ConcreteCategory.hom (Scheme.Hom.app (D.map fjx) (U x))) (t x) |_ (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y)) ⋯ -
        ((ConcreteCategory.hom (Scheme.Hom.app (D.map fjy) (U y))) (t y) |_ (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y))
          ⋯) =
    0

Tactic:
dsimp +instances [TopCat.Presheaf.restrictOpen, TopCat.Presheaf.restrict]
      simp only [map_sub, sub_eq_zero, ← ConcreteCategory.comp_apply,
        Scheme.Hom.app_eq_appLE, Scheme.Hom.appLE_map, Scheme.Hom.appLE_comp_appLE,
        Cone.w]
      simp_rw [Scheme.Hom.appLE, ConcreteCategory.comp_apply, ht, TopCat.Presheaf.restrictOpen,
        TopCat.Presheaf.restrict, ← ConcreteCategory.comp_apply, ← Functor.map_comp]
      rfl

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:
rw [map, AdjoinRoot.lift_mk, ← eval₂_map]

Goal state after:
R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Affine R
f : R →+* S
x : R[X][Y]
⊢ eval₂ (AdjoinRoot.of (W'.map f).polynomial) (AdjoinRoot.root (W'.map f).polynomial)
      (Polynomial.map (mapRingHom f) x) =
    (mk (W'.map f)) (Polynomial.map (mapRingHom f) 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
⊢ ∃ T,
    comap ↑f '' S.toSet = T.toSet ∧
      ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d

Tactic:
let s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
    .range fun i ↦ .C (.X i) - σ (f (.X i))

Goal state after:
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))
⊢ ∃ T,
    comap ↑f '' S.toSet = T.toSet ∧
      ∀ C ∈ T, C.n ≤ numBound k m n d ∧ ∀ (i : Fin C.n) (j : Fin n), degreeOf j (C.g i) ≤ degBound k m n d
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
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
⊢ (c.π.app k ≫ D.map fkj) ⁻¹ᵁ (D.obj j).basicOpen r ≤ (c.π.app k ≫ D.map fki) ⁻¹ᵁ U

Tactic:
c.w,

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)
k : I
fki : k ⟶ i
fkj : k ⟶ j
⊢ c.π.app j ⁻¹ᵁ (D.obj j).basicOpen r ≤ (c.π.app k ≫ D.map fki) ⁻¹ᵁ U
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
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
⊢ c.π.app j ⁻¹ᵁ (D.obj j).basicOpen r ≤ (c.π.app k ≫ D.map fki) ⁻¹ᵁ U

Tactic:
c.w,

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)
k : I
fki : k ⟶ i
fkj : k ⟶ j
⊢ c.π.app j ⁻¹ᵁ (D.obj j).basicOpen r ≤ c.π.app i ⁻¹ᵁ U
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 y : F
h : W.Nonsingular x y
⊢ ↑(XYIdeal W x (C (W.negY x y)) * XYIdeal W x (C y)) * (↑(XIdeal W x))⁻¹ = 1

Tactic:
XYIdeal_neg_mul h,

Goal state after:
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 y : F
h : W.Nonsingular x y
⊢ ↑(XIdeal W x) * (↑(XIdeal W x))⁻¹ = 1
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
⊢ ∃ i, (D.obj i).IsQuasiAffine

Tactic:
by
  classical
  have (x : c.pt) : ∃ (i : I) (f : Γ(D.obj i, ⊤)),
      IsAffineOpen (Scheme.basicOpen _ f) ∧ c.π.app i x ∈ (Scheme.basicOpen _ f) := by
    obtain ⟨i⟩ := IsCofiltered.nonempty (C := I)
    obtain ⟨_, ⟨U, hU, rfl⟩, hxU, -⟩ := (D.obj i).isBasis_affineOpens.exists_subset_of_mem_open
      (Set.mem_univ (c.π.app i x)) isOpen_univ
    obtain ⟨_, ⟨_, ⟨r, hr, rfl⟩, rfl⟩, hxr, hrU⟩ :=
      (IsQuasiAffine.isBasis_basicOpen c.pt).exists_subset_of_mem_open hxU (c.π.app i ⁻¹ᵁ U).isOpen
    obtain ⟨j, r, rfl⟩ := exists_appTop_π_eq_of_isLimit D c hc r
    obtain ⟨k, fki, fkj, -⟩ := IsCofilteredOrEmpty.cone_objs i j
    obtain ⟨l, flk, hl⟩ := exists_map_preimage_le_map_preimage D c hc (isCompact_basicOpen _
      isCompact_univ ((D.map fkj).appTop r)) (V := D.map fki ⁻¹ᵁ U) (by
        rwa [← preimage_basicOpen_top, ← Hom.comp_preimage, ← Hom.comp_preimage,
          c.w, c.w, preimage_basicOpen_top])
    refine ⟨l, (D.map (flk ≫ fkj)).appTop r, ?_, ?_⟩
    · convert!
      (hU.preimage (D.map (flk ≫ fki))).basicOpen
        ((D.obj _).presheaf.map (homOfLE le_top).op ((D.map (flk ≫ fkj)).appTop r)) using 1
      rwa [Scheme.basicOpen_res, eq_comm, inf_eq_right, Functor.map_comp,
        elementwise_of% Scheme.Hom.comp_appTop, ← Scheme.preimage_basicOpen_top, Functor.map_comp,
        Scheme.Hom.comp_preimage]
    · change x ∈ c.π.app l ⁻¹ᵁ (D.obj l).basicOpen _
      rwa [Scheme.preimage_basicOpen_top, ← elementwise_of% Scheme.Hom.comp_appTop, Cone.w]
  choose i f hf hi using this
  obtain ⟨σ, hσ⟩ := CompactSpace.elim_nhds_subcover
    (fun x ↦ (((c.π.app (i x)) ⁻¹ᵁ (D.obj (i x)).basicOpen (f x)).1))
    (fun x ↦ ((c.π.app (i x)) ⁻¹ᵁ (D.obj (i x)).basicOpen (f x)).2.mem_nhds (by exact hi x))
  choose σi hσiσ hσi using fun x ↦ Set.mem_iUnion₂.mp (hσ.ge (Set.mem_univ x))
  obtain ⟨j, fj⟩ := IsCofiltered.inf_objs_exists (σ.image i)
  replace fj := fun i h ↦ (@fj i h).some
  obtain ⟨k, fkj, hk⟩ := exists_map_eq_top D c hc
    (⨆ k, D.map (fj _ (Finset.mem_image_of_mem i (hσiσ k))) ⁻¹ᵁ (D.obj (i _)).basicOpen (f _)) (by
      refine top_le_iff.mp fun x _ ↦ TopologicalSpace.Opens.mem_iSup.mpr ⟨x, ?_⟩
      change (c.π.app j ≫ D.map _).base x ∈ (D.obj (i (σi x))).basicOpen (f (σi x))
      rw [Cone.w]
      exact hσi _)
  refine ⟨k, .of_forall_exists_mem_basicOpen _ fun x ↦ ?_⟩
  obtain ⟨y, hy⟩ := TopologicalSpace.Opens.mem_iSup.mp (hk.ge (Set.mem_univ x))
  use (D.map fkj).appTop ((D.map (fj _ (Finset.mem_image_of_mem i (hσiσ y)))).appTop (f _))
  rw [← Scheme.preimage_basicOpen_top, ← Scheme.preimage_basicOpen_top]
  exact ⟨((hf _).preimage _).preimage _, hy⟩

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
W : ℤ → R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
b c d : R
k : ℤ
P : ℕ → Sort u_4
zero : P 0
one : P 1
even : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))
odd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)
n : ℕ
⊢ (n : ℕ) → ((k : ℕ) → k < 2 * n → P k) → P (2 * n)

Tactic:
rintro (_ | _) h

Goal state after:
case zero
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
W : ℤ → R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
b c d : R
k : ℤ
P : ℕ → Sort u_4
zero : P 0
one : P 1
even : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))
odd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)
n : ℕ
h : (k : ℕ) → k < 2 * 0 → P k
⊢ P (2 * 0)

case succ
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
W : ℤ → R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
b c d : R
k : ℤ
P : ℕ → Sort u_4
zero : P 0
one : P 1
even : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))
odd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)
n n✝ : ℕ
h : (k : ℕ) → k < 2 * (n✝ + 1) → P k
⊢ P (2 * (n✝ + 1))
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
W : ℤ → R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
b c d : R
k : ℤ
P : ℕ → Sort u_4
zero : P 0
one : P 1
even : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))
odd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)
n : ℕ
⊢ (n : ℕ) → ((k : ℕ) → k < 2 * n + 1 → P k) → P (2 * n + 1)

Tactic:
rintro (_ | _) h

Goal state after:
case zero
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
W : ℤ → R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
b c d : R
k : ℤ
P : ℕ → Sort u_4
zero : P 0
one : P 1
even : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))
odd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)
n : ℕ
h : (k : ℕ) → k < 2 * 0 + 1 → P k
⊢ P (2 * 0 + 1)

case succ
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
W : ℤ → R
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : RingHomClass F R S
f : F
b c d : R
k : ℤ
P : ℕ → Sort u_4
zero : P 0
one : P 1
even : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) → P k) → P (2 * (m + 1))
odd : (m : ℕ) → ((k : ℕ) → k < 2 * (m + 1) + 1 → P k) → P (2 * (m + 1) + 1)
n n✝ : ℕ
h : (k : ℕ) → k < 2 * (n✝ + 1) + 1 → P k
⊢ P (2 * (n✝ + 1) + 1)
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)
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
⊢ ⨆ i_1, (c.π.app i ≫ D.map (fi i_1)) ⁻¹ᵁ V i_1 = ⊤

Tactic:
c.w,

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)
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
⊢ ⨆ i_1, c.π.app (j i_1) ⁻¹ᵁ V i_1 = ⊤
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x y : R
h : W'.Nonsingular x y
⊢ W'.nonsingularPointEquiv (Point.some x y h) = some ⟨(x, y), h⟩

Tactic:
by
  rfl

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

Goal state before:
pp: R : Type r
inst✝² : CommRing R
W' : Affine R
inst✝¹ : Nontrivial R
inst✝ : WeierstrassCurve.IsElliptic W'
x y : R
h : W'.Equation x y
⊢ W'.pointEquiv (Point.mk h) = some ⟨(x, y), h⟩

Tactic:
by
  rfl

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

Goal state before:
pp: case zero.zero
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
⊢ zero + zero = 0 ↔ zero = -zero

pp: case zero.some
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ zero + some x₂ y₂ h✝ = 0 ↔ zero = -some x₂ y₂ h✝

pp: case some.zero
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝ : W.Nonsingular x₁ y₁
⊢ some x₁ y₁ h✝ + zero = 0 ↔ some x₁ y₁ h✝ = -zero

pp: case some.some
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0 ↔ some x₁ y₁ h✝¹ = -some x₂ y₂ h✝

Tactic:
any_goals rfl

Goal state after:
case zero.some
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ zero + some x₂ y₂ h✝ = 0 ↔ zero = -some x₂ y₂ h✝

case some.some
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0 ↔ some x₁ y₁ h✝¹ = -some x₂ y₂ h✝
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: case some.some.mp
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0 → x₁ = x₂ ∧ y₁ = W.negY x₂ y₂

pp: case some.some.mpr
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ x₁ = x₂ ∧ y₁ = W.negY x₂ y₂ → some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0

Tactic:
· contrapose
      exact fun hxy => by simpa only [add_some hxy] using some_ne_zero _

Goal state after:
case some.some.mpr
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ x₁ = x₂ ∧ y₁ = W.negY x₂ y₂ → some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharNeTwoNF
⊢ W.b₆ = 4 * W.a₆

Tactic:
by
  rw [b₆, a₃_of_isCharNeTwoNF]
  ring1

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

Goal state before:
pp: case h.a₆
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).a₆ = E'.a₆

Tactic:
simp_rw [variableChange_a₆, inv_one, Units.val_one, heq]
    ring1

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJNeZeroNF
⊢ W.b₄ = 0

Tactic:
by
  rw [b₄, a₃_of_isCharTwoJNeZeroNF, a₄_of_isCharTwoJNeZeroNF]
  ring1

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

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP R 2
⊢ W.b₂ = 1

Tactic:
b₂_of_isCharTwoJNeZeroNF

Goal state after:
R : Type u_1
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP R 2
⊢ 1 + 4 * W.a₂ = 1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (W.ΨSq n).coeff (n.natAbs ^ 2 - 1) = ↑n ^ 2

Tactic:
induction n using Int.negInduction with
  | nat n => exact_mod_cast (W.natDegree_coeff_ΨSq_ofNat n).right
  | neg ih => rw [ΨSq_neg, Int.natAbs_neg, ← Int.cast_pow, neg_sq, Int.cast_pow, ih]

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

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
inst✝² : IsSepClosed F
E E' : WeierstrassCurve F
inst✝¹ : E.IsElliptic
inst✝ : E'.IsElliptic
heq : E.j = E'.j
⊢ ∃ C, C • E = E'

Tactic:
by
  obtain ⟨p, _⟩ := CharP.exists F
  by_cases hchar2 : p = 2
  · subst hchar2
    exact exists_variableChange_of_char_two _ _ heq
  by_cases hchar3 : p = 3
  · subst hchar3
    exact exists_variableChange_of_char_three _ _ heq
  exact exists_variableChange_of_char_ne_two_or_three _ _ hchar2 hchar3 heq

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

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Jacobian R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
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).polynomialZ = (MvPolynomial.map ↑f) (W'⁄A).polynomialZ

Tactic:
by
  rw [← map_polynomialZ, map_baseChange]

Goal state after:
Module: Mathlib.RingTheory.Valuation.ExtendToLocalization

Goal state before:
pp: case e'_3
A : Type u_1
inst✝⁴ : CommRing A
Γ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation A Γ
S : Submonoid A
hS : S ≤ v.supp.primeCompl
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
f : S.LocalizationMap B := IsLocalization.toLocalizationMap S B
h : ∀ (s : ↥S), IsUnit (↑v.toMonoidWithZeroHom ↑s) := fun s ↦ isUnit_iff_ne_zero.mpr (hS s.property)
a b : A
s : ↥S
⊢ f.mk' a s * (algebraMap A B) ↑s + f.mk' b s * (algebraMap A B) ↑s = (algebraMap A B) a + (algebraMap A B) b

Tactic:
← IsLocalization.toLocalizationMap_apply S B,

Goal state after:
case e'_3
A : Type u_1
inst✝⁴ : CommRing A
Γ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation A Γ
S : Submonoid A
hS : S ≤ v.supp.primeCompl
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
f : S.LocalizationMap B := IsLocalization.toLocalizationMap S B
h : ∀ (s : ↥S), IsUnit (↑v.toMonoidWithZeroHom ↑s) := fun s ↦ isUnit_iff_ne_zero.mpr (hS s.property)
a b : A
s : ↥S
⊢ f.mk' a s * (IsLocalization.toLocalizationMap S B) ↑s + f.mk' b s * (IsLocalization.toLocalizationMap S B) ↑s =
    (algebraMap A B) a + (algebraMap A B) b
Module: Mathlib.RingTheory.Valuation.ExtendToLocalization

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
Γ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation A Γ
S : Submonoid A
hS : S ≤ v.supp.primeCompl
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
x : A
y : ↥S
⊢ ↑v.toMonoidWithZeroHom x * ↑((IsUnit.liftRight ((↑v.toMonoidWithZeroHom).domRestrict S) ⋯) y)⁻¹ = v x * (v ↑y)⁻¹

Tactic:
simp [IsUnit.coe_liftRight]

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W'.addX P Q * (P z * Q z) ^ 2 = (P y * Q z ^ 3 - Q y * P z ^ 3) ^ 2

Tactic:
rw [addX_eq' hP hQ, addZ_of_X_eq hx]
  ring1

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
r : R
s : ↥(posSubmonoid R)
⊢ ValueGroupWithZero.mk r s = (valuation R) r / (valuation R) ↑s

Tactic:
by
  rw [eq_div_iff (valuation_posSubmonoid_ne_zero _)]
  simp [valuation, mk_eq_mk]

Goal state after: