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

Goal state before:
pp: case refine_2
P : MorphismProperty Scheme
S : Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝⁶ : W.Over S
inst✝⁵ : X.Over S
inst✝⁴ : Cover.Over S 𝒰
inst✝³ : Hom.IsOver f S
Q : MorphismProperty Scheme
inst✝² : Q.HasOfPostcompProperty Q
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j :
  { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left,
      f := fun x ↦ (pullback.snd (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left }.I₀
⊢ P ((Over.forget S).map (pullback.snd (Hom.asOverProp (𝒰.f j) S) (Hom.asOverProp f S)).toCommaMorphism)

Tactic:
MorphismProperty.Comma.toCommaMorphism_eq_hom,

Goal state after:
case refine_2
P : MorphismProperty Scheme
S : Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover (precoverage P) X
f : W ⟶ X
inst✝⁶ : W.Over S
inst✝⁵ : X.Over S
inst✝⁴ : Cover.Over S 𝒰
inst✝³ : Hom.IsOver f S
Q : MorphismProperty Scheme
inst✝² : Q.HasOfPostcompProperty Q
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j :
  { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left,
      f := fun x ↦ (pullback.snd (Hom.asOverProp (𝒰.f x) S) (Hom.asOverProp f S)).left }.I₀
⊢ P
    ((Over.forget S).map
      (MorphismProperty.Comma.Hom.hom (pullback.snd (Hom.asOverProp (𝒰.f j) S) (Hom.asOverProp f S))))
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case hP₃
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
⊢ Statement R₀ R n c

Tactic:
f

Goal state after:
case hP₃
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
⊢ ∃ T,
    comap Polynomial.C '' (zeroLocus (Set.range c.val) \ zeroLocus {f}) = T.toSet ∧
      ∀ C ∈ T, C.n ≤ c.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ coeffSubmodule R₀ c ^ c.powBound
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_1.refine_1
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝ : (i : ι) → TopologicalSpace (X i)
U : Set S
x✝ : IsCompactOpenCovered f U
s : Set ι
hs : s.Finite
V : (i : ι) → i ∈ s → Opens (X i)
hc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier
hU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
i : ι
⊢ IsOpen (Sigma.mk i ⁻¹' s.sigma fun i ↦ if h : i ∈ s then ↑(V i h) else ∅)

Tactic:
by_cases h : i ∈ s

Goal state after:
case pos
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝ : (i : ι) → TopologicalSpace (X i)
U : Set S
x✝ : IsCompactOpenCovered f U
s : Set ι
hs : s.Finite
V : (i : ι) → i ∈ s → Opens (X i)
hc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier
hU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
i : ι
h : i ∈ s
⊢ IsOpen (Sigma.mk i ⁻¹' s.sigma fun i ↦ if h : i ∈ s then ↑(V i h) else ∅)

case neg
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝ : (i : ι) → TopologicalSpace (X i)
U : Set S
x✝ : IsCompactOpenCovered f U
s : Set ι
hs : s.Finite
V : (i : ι) → i ∈ s → Opens (X i)
hc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier
hU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
i : ι
h : i ∉ s
⊢ IsOpen (Sigma.mk i ⁻¹' s.sigma fun i ↦ if h : i ∈ s then ↑(V i h) else ∅)
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:
simp [𝒱, pullback.condition, glued]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Cover.QuasiCompact

Goal state before:
pp: case h
P : MorphismProperty Scheme
S : Scheme
𝒰 : Scheme.Cover (Scheme.precoverage P) S
inst✝² : P.RespectsLeft IsOpenImmersion
inst✝¹ : CompactSpace ↥S
inst✝ : QuasiCompactCover 𝒰.toPreZeroHypercover
n : ℕ
f : Fin n → 𝒰.I₀
V : (i : Fin n) → (𝒰.X (f i)).Opens
hV : ∀ (i : Fin n), IsAffineOpen (V i)
idx : ↥S → Fin n
x : (x : ↥S) → ↥(𝒰.X (f (idx x)))
hmem : ∀ (x_1 : ↥S), x x_1 ∈ V (idx x_1)
hx : ∀ (x_1 : ↥S), (𝒰.f (f (idx x_1))) (x x_1) = x_1
s : ↥S
⊢ (⋯.fromSpec ≫ 𝒰.f (f { down := idx s }.down)) (⋯.isoSpec.hom ⟨x s, ⋯⟩) = s

Tactic:
rw [← Scheme.Hom.comp_apply, ← IsAffineOpen.isoSpec_inv_ι, Category.assoc,
            Iso.hom_inv_id_assoc]

Goal state after:
case h
P : MorphismProperty Scheme
S : Scheme
𝒰 : Scheme.Cover (Scheme.precoverage P) S
inst✝² : P.RespectsLeft IsOpenImmersion
inst✝¹ : CompactSpace ↥S
inst✝ : QuasiCompactCover 𝒰.toPreZeroHypercover
n : ℕ
f : Fin n → 𝒰.I₀
V : (i : Fin n) → (𝒰.X (f i)).Opens
hV : ∀ (i : Fin n), IsAffineOpen (V i)
idx : ↥S → Fin n
x : (x : ↥S) → ↥(𝒰.X (f (idx x)))
hmem : ∀ (x_1 : ↥S), x x_1 ∈ V (idx x_1)
hx : ∀ (x_1 : ↥S), (𝒰.f (f (idx x_1))) (x x_1) = x_1
s : ↥S
⊢ ((V { down := idx s }.down).ι ≫ 𝒰.f (f { down := idx s }.down)) ⟨x s, ⋯⟩ = s
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝² : Flat f
inst✝¹ : Surjective f
inst✝ : QuasiCompact f
⊢ EffectiveEpi f.base

Tactic:
by
  rw [TopCat.effectiveEpi_iff_isQuotientMap]
  exact Flat.isQuotientMap_of_surjective _

Goal state after:
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.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 := d.relativeGluingData.cover
s : Cocone D
i : 𝒱.I₀
this :
  ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)).left ≫
      pullback.snd s.pt.hom (𝒰.f i) =
    (d.cocone i).pt.hom
⊢ ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)).left ≫
      pullback.snd s.pt.hom (𝒰.f i) ≫ 𝒰.f i =
    d.relativeGluingData.cover.f i ≫ Over.hom (MorphismProperty.Over.mk ⊤ d.relativeGluingData.toBase ⋯).toComma

Tactic:
rw [reassoc_of% this]

Goal state after:
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.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 := d.relativeGluingData.cover
s : Cocone D
i : 𝒱.I₀
this :
  ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)).left ≫
      pullback.snd s.pt.hom (𝒰.f i) =
    (d.cocone i).pt.hom
⊢ (d.cocone i).pt.hom ≫ 𝒰.f i =
    d.relativeGluingData.cover.f i ≫ Over.hom (MorphismProperty.Over.mk ⊤ d.relativeGluingData.toBase ⋯).toComma
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: k₁ k₂ : ℕ
D₁ D₂ : ℕ → ℕ
hk : k₁ ≤ k₂
n : ℕ
hD : ∀ i < n + 1, D₁ i ≤ D₂ i
⊢ degBound k₁ D₁ (n + 1) ≤ degBound k₂ D₂ (n + 1)

Tactic:
rw [degBound_succ, degBound_succ]

Goal state after:
k₁ k₂ : ℕ
D₁ D₂ : ℕ → ℕ
hk : k₁ ≤ k₂
n : ℕ
hD : ∀ i < n + 1, D₁ i ≤ D₂ i
⊢ numBound k₁ D₁ (n + 1) ^ numBound k₁ D₁ (n + 1) * degBound k₁ D₁ n ≤
    numBound k₂ D₂ (n + 1) ^ numBound k₂ D₂ (n + 1) * degBound k₂ D₂ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
A : Type v
inst✝ : CommRing A
φ : R →+* A
hφ : Function.Injective ⇑φ
x✝¹ x✝ : VariableChange R
h : (fun C ↦ C.map φ) x✝¹ = (fun C ↦ C.map φ) x✝
⊢ x✝¹ = x✝

Tactic:
rcases mk.inj h with ⟨h, _, _, _⟩

Goal state after:
R : Type u
inst✝¹ : CommRing R
A : Type v
inst✝ : CommRing A
φ : R →+* A
hφ : Function.Injective ⇑φ
x✝¹ x✝ : VariableChange R
h✝ : (fun C ↦ C.map φ) x✝¹ = (fun C ↦ C.map φ) x✝
h : (Units.map ↑φ) x✝¹.u = (Units.map ↑φ) x✝.u
left✝¹ : φ x✝¹.r = φ x✝.r
left✝ : φ x✝¹.s = φ x✝.s
right✝ : φ x✝¹.t = φ x✝.t
⊢ x✝¹ = x✝
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
hpq : p • 1 + q • (mk W') Y = 0
h : ![p, q] 0 • 1 + ∑ i, ![p, q] i.succ • (CoordinateRing.basis W') i.succ = 0 → ∀ (i : Fin 2), ![p, q] i = 0
⊢ p = 0 ∧ q = 0

Tactic:
Fin.sum_univ_one,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
hpq : p • 1 + q • (mk W') Y = 0
h : ![p, q] 0 • 1 + ![p, q] (Fin.succ 0) • (CoordinateRing.basis W') (Fin.succ 0) = 0 → ∀ (i : Fin 2), ![p, q] i = 0
⊢ p = 0 ∧ q = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x : W'.CoordinateRing
h :
  (CoordinateRing.basis W').equivFun x 0 • (CoordinateRing.basis W') 0 +
      ∑ i, (CoordinateRing.basis W').equivFun x i.succ • (CoordinateRing.basis W') i.succ =
    x
⊢ ∃ p q, p • 1 + q • (mk W') Y = x

Tactic:
Fin.sum_univ_one,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Affine R
x : W'.CoordinateRing
h :
  (CoordinateRing.basis W').equivFun x 0 • (CoordinateRing.basis W') 0 +
      (CoordinateRing.basis W').equivFun x (Fin.succ 0) • (CoordinateRing.basis W') (Fin.succ 0) =
    x
⊢ ∃ p q, p • 1 + q • (mk W') Y = 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
⊢ g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))) = ↑f

Tactic:
ext x : 2 <;> simp [g]

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

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
⊢ span {(mk W) (Y - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))} *
      (span
            {(mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))),
              (mk W) (Y - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))) *
                  XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) -
                (mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))) *
                  XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
          span {XClass W x₁} * span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
        span {XClass W x₂} * span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))}) =
    YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
      (span {(mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))} ⊔
        span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))})

Tactic:
← sub_mul,

Goal state after:
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
⊢ span {(mk W) (Y - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))} *
      (span
            {(mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))),
              ((mk W) (Y - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))) -
                  (mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))) *
                XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
          span {XClass W x₁} * span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
        span {XClass W x₂} * span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))}) =
    YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
      (span {(mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))} ⊔
        span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))})
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
⊢ preNormEDS' b c d 0 = 0

Tactic:
by
  rw [preNormEDS']

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
⊢ preNormEDS' b c d 1 = 1

Tactic:
by
  rw [preNormEDS']

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
⊢ preNormEDS' b c d 2 = 1

Tactic:
by
  rw [preNormEDS']

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
⊢ preNormEDS' b c d 3 = c

Tactic:
by
  rw [preNormEDS']

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
⊢ preNormEDS' b c d 4 = d

Tactic:
by
  rw [preNormEDS']

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
ih :
  ∀ (n : ℕ),
    preNormEDS b c d (2 * ↑n) =
      preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -
        preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2
m : ℕ
⊢ -preNormEDS b c d (2 * ↑m) =
    (-preNormEDS b c d (↑m + 1)) ^ 2 * -preNormEDS b c d ↑m * -preNormEDS b c d (↑m - 2) -
      -preNormEDS b c d (↑m + 2) * -preNormEDS b c d ↑m * (-preNormEDS b c d (↑m - 1)) ^ 2

Tactic:
ih

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
ih :
  ∀ (n : ℕ),
    preNormEDS b c d (2 * ↑n) =
      preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -
        preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2
m : ℕ
⊢ -(preNormEDS b c d (↑m - 1) ^ 2 * preNormEDS b c d ↑m * preNormEDS b c d (↑m + 2) -
        preNormEDS b c d (↑m - 2) * preNormEDS b c d ↑m * preNormEDS b c d (↑m + 1) ^ 2) =
    (-preNormEDS b c d (↑m + 1)) ^ 2 * -preNormEDS b c d ↑m * -preNormEDS b c d (↑m - 2) -
      -preNormEDS b c d (↑m + 2) * -preNormEDS b c d ↑m * (-preNormEDS b c d (↑m - 1)) ^ 2
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
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:
show 2 * -(m + 1 : ℤ) + 1 = -(2 * m + 1) by rfl,

Goal state after:
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)) =
    (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
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
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) =
    (-preNormEDS b c d (↑m - 1) * (-preNormEDS b c d (↑m + 1)) ^ 3 * if Even ↑m then 1 else b) -
      -preNormEDS b c d (↑m + 2) * (-preNormEDS b c d ↑m) ^ 3 * if Even ↑m then b else 1

Tactic:
ih

Goal state after:
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 (↑m + 2) * preNormEDS b c d ↑m ^ 3 * if Even ↑m then b else 1) -
        preNormEDS b c d (↑m - 1) * preNormEDS b c d (↑m + 1) ^ 3 * if Even ↑m then 1 else b) =
    (-preNormEDS b c d (↑m - 1) * (-preNormEDS b c d (↑m + 1)) ^ 3 * if Even ↑m then 1 else b) -
      -preNormEDS b c d (↑m + 2) * (-preNormEDS b c d ↑m) ^ 3 * if Even ↑m then b else 1
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih :
  ∀ (n : ℕ),
    complEDS b c d k (2 * ↑n + 1) =
      complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -
        complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k - 1)
m : ℕ
⊢ complEDS b c d k (2 * -(↑m + 1) + 1) =
    complEDS b c d k (-(↑m + 1)) ^ 2 * normEDS b c d ((-(↑m + 1) + 1) * k + 1) *
        normEDS b c d ((-(↑m + 1) + 1) * k - 1) -
      complEDS b c d k (-(↑m + 1) + 1) ^ 2 * normEDS b c d (-(↑m + 1) * k + 1) * normEDS b c d (-(↑m + 1) * k - 1)

Tactic:
show 2 * -(m + 1 : ℤ) + 1 = -(2 * m + 1) by rfl,

Goal state after:
case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih :
  ∀ (n : ℕ),
    complEDS b c d k (2 * ↑n + 1) =
      complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -
        complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k - 1)
m : ℕ
⊢ complEDS b c d k (-(2 * ↑m + 1)) =
    complEDS b c d k (-(↑m + 1)) ^ 2 * normEDS b c d ((-(↑m + 1) + 1) * k + 1) *
        normEDS b c d ((-(↑m + 1) + 1) * k - 1) -
      complEDS b c d k (-(↑m + 1) + 1) ^ 2 * normEDS b c d (-(↑m + 1) * k + 1) * normEDS b c d (-(↑m + 1) * k - 1)
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih :
  ∀ (n : ℕ),
    complEDS b c d k (2 * ↑n + 1) =
      complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -
        complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k - 1)
m : ℕ
⊢ -complEDS b c d k (2 * ↑m + 1) =
    (-complEDS b c d k (↑m + 1)) ^ 2 * -normEDS b c d (↑m * k - 1) * -normEDS b c d (↑m * k + 1) -
      (-complEDS b c d k ↑m) ^ 2 * -normEDS b c d ((↑m + 1) * k - 1) * -normEDS b c d ((↑m + 1) * k + 1)

Tactic:
ih

Goal state after:
case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih :
  ∀ (n : ℕ),
    complEDS b c d k (2 * ↑n + 1) =
      complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -
        complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k - 1)
m : ℕ
⊢ -(complEDS b c d k ↑m ^ 2 * normEDS b c d ((↑m + 1) * k + 1) * normEDS b c d ((↑m + 1) * k - 1) -
        complEDS b c d k (↑m + 1) ^ 2 * normEDS b c d (↑m * k + 1) * normEDS b c d (↑m * k - 1)) =
    (-complEDS b c d k (↑m + 1)) ^ 2 * -normEDS b c d (↑m * k - 1) * -normEDS b c d (↑m * k + 1) -
      (-complEDS b c d k ↑m) ^ 2 * -normEDS b c d ((↑m + 1) * k - 1) * -normEDS b c d ((↑m + 1) * k + 1)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Affine R
inst✝ : IsDomain R
p q : R[X]
⊢ ((Algebra.norm R[X]) (p • 1 + q • (mk W') Y)).degree ≠ 1

Tactic:
degree_norm_smul_basis

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Affine R
inst✝ : IsDomain R
p q : R[X]
⊢ max (2 • p.degree) (2 • q.degree + 3) ≠ 1
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case inr
I : Type u
inst✝⁷ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁶ : IsCofiltered I
inst✝⁵ : ∀ (i : I), IsAffine (D.obj i)
R : CommRingCat
t : D ⟶ (Functor.const I).obj (Spec R)
inst✝⁴ : IsAffine (Spec R)
S : CommRingCat
inst✝³ : IsAffine (Spec S)
a : c.pt ⟶ Spec S
φ : R ⟶ S
inst✝² : LocallyOfFinitePresentation (Spec.map φ)
ha : c.π ≫ t = (Functor.const I).map (a ≫ Spec.map φ)
this :
  ∀ (D : I ⥤ Scheme) (c : Cone D) (hc : IsLimit c) [∀ (i : I), IsAffine (D.obj i)]
    (t : D ⟶ (Functor.const I).obj (Spec R)) (a : c.pt ⟶ Spec S),
    c.π ≫ t = (Functor.const I).map (a ≫ Spec.map φ) →
      (∃ D', D = D' ⋙ Scheme.Spec) → ∃ i g, c.π.app i ≫ g = a ∧ g ≫ Spec.map φ = t.app i
hD : ¬∃ D', D = D' ⋙ Scheme.Spec
e : D ⟶ D ⋙ Γ.rightOp ⋙ Scheme.Spec := D.whiskerLeft ΓSpec.adjunction.unit
inst✝¹ : ∀ (i : I), IsIso (e.app i)
inst✝ : IsIso e
inst : ∀ (i : I), IsAffine ((D ⋙ Γ.rightOp ⋙ Scheme.Spec).obj i)
⊢ ∃ i g, c.π.app i ≫ g = a ∧ g ≫ Spec.map φ = t.app i

Tactic:
obtain ⟨i, g, hg, hg'⟩ := this _ _ ((IsLimit.postcomposeHomEquiv (asIso e) c).symm hc)
      (inv e ≫ t) a (by simpa using! ha) ⟨D ⋙ Scheme.Γ.rightOp, rfl⟩

Goal state after:
case inr
I : Type u
inst✝⁷ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁶ : IsCofiltered I
inst✝⁵ : ∀ (i : I), IsAffine (D.obj i)
R : CommRingCat
t : D ⟶ (Functor.const I).obj (Spec R)
inst✝⁴ : IsAffine (Spec R)
S : CommRingCat
inst✝³ : IsAffine (Spec S)
a : c.pt ⟶ Spec S
φ : R ⟶ S
inst✝² : LocallyOfFinitePresentation (Spec.map φ)
ha : c.π ≫ t = (Functor.const I).map (a ≫ Spec.map φ)
this :
  ∀ (D : I ⥤ Scheme) (c : Cone D) (hc : IsLimit c) [∀ (i : I), IsAffine (D.obj i)]
    (t : D ⟶ (Functor.const I).obj (Spec R)) (a : c.pt ⟶ Spec S),
    c.π ≫ t = (Functor.const I).map (a ≫ Spec.map φ) →
      (∃ D', D = D' ⋙ Scheme.Spec) → ∃ i g, c.π.app i ≫ g = a ∧ g ≫ Spec.map φ = t.app i
hD : ¬∃ D', D = D' ⋙ Scheme.Spec
e : D ⟶ D ⋙ Γ.rightOp ⋙ Scheme.Spec := D.whiskerLeft ΓSpec.adjunction.unit
inst✝¹ : ∀ (i : I), IsIso (e.app i)
inst✝ : IsIso e
inst : ∀ (i : I), IsAffine ((D ⋙ Γ.rightOp ⋙ Scheme.Spec).obj i)
i : I
g : (D ⋙ Γ.rightOp ⋙ Scheme.Spec).obj i ⟶ Spec S
hg : ((Cone.postcompose (asIso e).hom).obj c).π.app i ≫ g = a
hg' : g ≫ Spec.map φ = (inv e ≫ t).app i
⊢ ∃ i g, c.π.app i ≫ g = a ∧ g ≫ Spec.map φ = t.app i