mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: 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)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
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
hunion : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
Us : (i : ↑s) → Set (Opens (X ↑i))
UsB : ∀ (i : ↑s), Us i ⊆ B ↑i
hUsf : ∀ (i : ↑s), (Us i).Finite
hUs : ∀ (i : ↑s), V ↑i ⋯ = sSup (Us i)
σ : Type (max u_2 u_3) := (i : ↑s) × ↑(Us i)
this✝ : Finite ↑s
this : ∀ (i : ↑s), Finite ↑(Us i)
⊢ ⋃ i, f ((fun i ↦ ↑i.fst) i) '' ↑((fun i ↦ ↑i.snd) i) = ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h)

Tactic:
]

Goal state after:
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)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
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
hunion : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
Us : (i : ↑s) → Set (Opens (X ↑i))
UsB : ∀ (i : ↑s), Us i ⊆ B ↑i
hUsf : ∀ (i : ↑s), (Us i).Finite
hUs : ∀ (i : ↑s), V ↑i ⋯ = sSup (Us i)
σ : Type (max u_2 u_3) := (i : ↑s) × ↑(Us i)
this✝ : Finite ↑s
this : ∀ (i : ↑s), Finite ↑(Us i)
⊢ ⋃ i, f ((fun i ↦ ↑i.fst) i) '' ↑((fun i ↦ ↑i.snd) i) = ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h)
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
a : J
⊢ (MorphismProperty.Over.map ⊤ ⋯).map
        ((D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
                (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
                    (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm).hom.app
          a) ≫
      (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans 𝒰 hij) ⋯ trivial).counit.app
          ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).obj a) ≫
        (d.cocone j).ι.app a =
    (d.trans hij).app a ≫ (d.cocone j).ι.app a

Tactic:
]

Goal state after:
P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
a : J
⊢ (MorphismProperty.Over.map ⊤ ⋯).map
        ((D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
                (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
                    (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm).hom.app
          a) ≫
      (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans 𝒰 hij) ⋯ trivial).counit.app
          ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).obj a) ≫
        (d.cocone j).ι.app a =
    (d.trans hij).app a ≫ (d.cocone j).ι.app a
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: 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)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
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
hunion : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
Us : (i : ↑s) → Set (Opens (X ↑i))
UsB : ∀ (i : ↑s), Us i ⊆ B ↑i
hUsf : ∀ (i : ↑s), (Us i).Finite
hUs : ∀ (i : ↑s), V ↑i ⋯ = sSup (Us i)
σ : Type (max u_2 u_3) := (i : ↑s) × ↑(Us i)
this✝ : Finite ↑s
this : ∀ (i : ↑s), Finite ↑(Us i)
x : S
x✝ : ∃ i, ∃ (i_1 : i ∈ s), x ∈ f i '' ↑(V i i_1)
i : ι
hi : i ∈ s
h : X i
hmem : ∃ i_1, h ∈ ⋃ (_ : i_1 ∈ Us ⟨i, hi⟩), ↑i_1
heq : f i h = x
⊢ ∃ i, x ∈ (fun a ↦ f (↑i.fst) a) '' ↑↑i.snd

Tactic:
]

Goal state after:
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)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
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
hunion : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V i h) = U
Us : (i : ↑s) → Set (Opens (X ↑i))
UsB : ∀ (i : ↑s), Us i ⊆ B ↑i
hUsf : ∀ (i : ↑s), (Us i).Finite
hUs : ∀ (i : ↑s), V ↑i ⋯ = sSup (Us i)
σ : Type (max u_2 u_3) := (i : ↑s) × ↑(Us i)
this✝ : Finite ↑s
this : ∀ (i : ↑s), Finite ↑(Us i)
x : S
x✝ : ∃ i, ∃ (i_1 : i ∈ s), x ∈ f i '' ↑(V i i_1)
i : ι
hi : i ∈ s
h : X i
hmem : ∃ i_1, h ∈ ⋃ (_ : i_1 ∈ Us ⟨i, hi⟩), ↑i_1
heq : f i h = x
⊢ ∃ i, x ∈ (fun a ↦ f (↑i.fst) a) '' ↑↑i.snd
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : Finite ι
inst✝ : TopologicalSpace S
hf : ∀ (i : ι), IsSpectralMap (f i)
U : Set S
hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)
hU : IsOpen U
hc : IsCompact U
x : S
hx : x ∈ U
⊢ x ∈ ⋃ i, ⋃ (h : i ∈ Set.univ), f i '' ↑((fun i x ↦ { carrier := f i ⁻¹' U, is_open' := ⋯ }) i h)

Tactic:
obtain ⟨i, y, rfl⟩ := hs x hx

Goal state after:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : Finite ι
inst✝ : TopologicalSpace S
hf : ∀ (i : ι), IsSpectralMap (f i)
U : Set S
hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)
hU : IsOpen U
hc : IsCompact U
i : ι
y : X i
hx : f i y ∈ U
⊢ f i y ∈ ⋃ i, ⋃ (h : i ∈ Set.univ), f i '' ↑((fun i x ↦ { carrier := f i ⁻¹' U, is_open' := ⋯ }) i h)
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : TopologicalSpace S
inst✝ : ∀ (i : ι), PrespectralSpace (X i)
hfc : ∀ (i : ι), Continuous (f i)
h : ∀ (i : ι), IsOpenMap (f i)
U : Set S
hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)
hU : IsOpen U
hc : IsCompact U
⊢ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U

Tactic:
]

Goal state after:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝² : (i : ι) → TopologicalSpace (X i)
inst✝¹ : TopologicalSpace S
inst✝ : ∀ (i : ι), PrespectralSpace (X i)
hfc : ∀ (i : ι), Continuous (f i)
h : ∀ (i : ι), IsOpenMap (f i)
U : Set S
hs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)
hU : IsOpen U
hc : IsCompact U
⊢ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝³ : (i : ι) → TopologicalSpace (X i)
inst✝² : ∀ (i : ι), PrespectralSpace (X i)
inst✝¹ : TopologicalSpace S
σ : Type u_4
Y : σ → Type u_5
inst✝ : (i : σ) → TopologicalSpace (Y i)
g : (i : σ) → Y i → S
a : σ → ι
t : (i : σ) → Y i → X (a i)
ht : ∀ (i : σ), Continuous (t i)
hge : ∀ (i : σ), g i = f (a i) ∘ t i
hf : ∀ (i : ι), Continuous (f i)
U : Set S
ho : IsOpen U
hU : ∃ V, IsCompact V.carrier ∧ (fun p ↦ g p.fst p.snd) '' V.carrier = U
⊢ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U

Tactic:
]

Goal state after:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝³ : (i : ι) → TopologicalSpace (X i)
inst✝² : ∀ (i : ι), PrespectralSpace (X i)
inst✝¹ : TopologicalSpace S
σ : Type u_4
Y : σ → Type u_5
inst✝ : (i : σ) → TopologicalSpace (Y i)
g : (i : σ) → Y i → S
a : σ → ι
t : (i : σ) → Y i → X (a i)
ht : ∀ (i : σ), Continuous (t i)
hge : ∀ (i : σ), g i = f (a i) ∘ t i
hf : ∀ (i : ι), Continuous (f i)
U : Set S
ho : IsOpen U
hU : ∃ V, IsCompact V.carrier ∧ (fun p ↦ g p.fst p.snd) '' V.carrier = U
⊢ ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
a : J
⊢ ((pullback.map (D.obj a).hom (𝒰.f i) (D.obj a).hom (Cover.trans 𝒰 hij ≫ 𝒰.f j) (𝟙 (D.obj a).left) (𝟙 (𝒰.X i)) (𝟙 S) ⋯
            ⋯ ≫
          (pullbackLeftPullbackSndIso (D.obj a).hom (𝒰.f j) (Cover.trans 𝒰 hij)).inv) ≫
        pullback.fst (pullback.snd (D.obj a).hom (𝒰.f j)) (Cover.trans 𝒰 hij)) ≫
      (Comma.Hom.hom ((d.cocone j).ι.app a)).left =
    pullback.map (D.obj a).hom (𝒰.f i) (D.obj a).hom (𝒰.f j) (𝟙 (D.obj a).left) (Cover.trans 𝒰 hij) (𝟙 S) ⋯ ⋯ ≫
      (Comma.Hom.hom ((d.cocone j).ι.app a)).left

Tactic:
]

Goal state after:
P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
a : J
⊢ ((pullback.map (D.obj a).hom (𝒰.f i) (D.obj a).hom (Cover.trans 𝒰 hij ≫ 𝒰.f j) (𝟙 (D.obj a).left) (𝟙 (𝒰.X i)) (𝟙 S) ⋯
            ⋯ ≫
          (pullbackLeftPullbackSndIso (D.obj a).hom (𝒰.f j) (Cover.trans 𝒰 hij)).inv) ≫
        pullback.fst (pullback.snd (D.obj a).hom (𝒰.f j)) (Cover.trans 𝒰 hij)) ≫
      (Comma.Hom.hom ((d.cocone j).ι.app a)).left =
    pullback.map (D.obj a).hom (𝒰.f i) (D.obj a).hom (𝒰.f j) (𝟙 (D.obj a).left) (Cover.trans 𝒰 hij) (𝟙 S) ⋯ ⋯ ≫
      (Comma.Hom.hom ((d.cocone j).ι.app a)).left
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝³ : (i : ι) → TopologicalSpace (X i)
inst✝² : ∀ (i : ι), PrespectralSpace (X i)
inst✝¹ : TopologicalSpace S
σ : Type u_4
Y : σ → Type u_5
inst✝ : (i : σ) → TopologicalSpace (Y i)
g : (i : σ) → Y i → S
a : σ → ι
t : (i : σ) → Y i → X (a i)
ht : ∀ (i : σ), Continuous (t i)
hge : ∀ (i : σ), g i = f (a i) ∘ t i
hf✝ : ∀ (i : ι), Continuous (f i)
U : Set S
ho✝ : IsOpen U
p : (i : σ) × Y i → (i : ι) × X i := Sigma.map a t
hcomp : (fun x ↦ f x.fst x.snd) ∘ p = fun x ↦ g x.fst x.snd
hp : Continuous p
hf : Continuous fun p ↦ f p.fst p.snd
V : Opens ((i : σ) × Y i)
hV : IsCompact V.carrier
heq : (fun p ↦ g p.fst p.snd) '' V.carrier = U
K : Set ((i : ι) × X i)
hK : IsCompact K
ho : IsOpen K
hVK : p '' V.carrier ⊆ K
hKU : K ⊆ (fun p ↦ f p.fst p.snd) ⁻¹' U
⊢ (fun x ↦ f x.fst x.snd) '' p '' V.carrier ⊆ (fun p ↦ f p.fst p.snd) '' { carrier := K, is_open' := ho }.carrier

Tactic:
]

Goal state after:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝³ : (i : ι) → TopologicalSpace (X i)
inst✝² : ∀ (i : ι), PrespectralSpace (X i)
inst✝¹ : TopologicalSpace S
σ : Type u_4
Y : σ → Type u_5
inst✝ : (i : σ) → TopologicalSpace (Y i)
g : (i : σ) → Y i → S
a : σ → ι
t : (i : σ) → Y i → X (a i)
ht : ∀ (i : σ), Continuous (t i)
hge : ∀ (i : σ), g i = f (a i) ∘ t i
hf✝ : ∀ (i : ι), Continuous (f i)
U : Set S
ho✝ : IsOpen U
p : (i : σ) × Y i → (i : ι) × X i := Sigma.map a t
hcomp : (fun x ↦ f x.fst x.snd) ∘ p = fun x ↦ g x.fst x.snd
hp : Continuous p
hf : Continuous fun p ↦ f p.fst p.snd
V : Opens ((i : σ) × Y i)
hV : IsCompact V.carrier
heq : (fun p ↦ g p.fst p.snd) '' V.carrier = U
K : Set ((i : ι) × X i)
hK : IsCompact K
ho : IsOpen K
hVK : p '' V.carrier ⊆ K
hKU : K ⊆ (fun p ↦ f p.fst p.snd) ⁻¹' U
⊢ (fun x ↦ f x.fst x.snd) '' p '' V.carrier ⊆ (fun p ↦ f p.fst p.snd) '' { carrier := K, is_open' := ho }.carrier
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case ht
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]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : c.val i = 1
⊢ 0 < ∑ i, (c.val i).degree.succ

Tactic:
]

Goal state after:
case ht
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]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : c.val i = 1
⊢ 0 < ∑ i, (c.val i).degree.succ
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case neg
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]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : ¬c.val i = 1
⊢ coeffSubmodule R₀ c ^ ((c.val j).degree.succ * c'.degBound ^ c'.degBound) ≤
    coeffSubmodule R₀ c ^ c.degBound ^ c.degBound

Tactic:
]

Goal state after:
case neg
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]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : ¬c.val i = 1
⊢ coeffSubmodule R₀ c ^ ((c.val j).degree.succ * c'.degBound ^ c'.degBound) ≤
    coeffSubmodule R₀ c ^ c.degBound ^ c.degBound
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: 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.71} 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.71} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
⊢ ∀ (i : 𝒰.toPreZeroHypercover.1), P (pullbackHom 𝒰 d.relativeGluingData.toBase i)

Tactic:
]

Goal state after:
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.71} 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.71} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
⊢ ∀ (i : 𝒰.toPreZeroHypercover.1), P (pullbackHom 𝒰 d.relativeGluingData.toBase i)
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: 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.71} 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.71} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
i : 𝒰.toPreZeroHypercover.1
⊢ P (d.relativeGluingData.natTrans.app i)

Tactic:
]

Goal state after:
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.71} 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.71} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
i : 𝒰.toPreZeroHypercover.1
⊢ P (d.relativeGluingData.natTrans.app i)