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

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
heq :
  ∀ (j : A.𝒰D.I₀),
    (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.a =
      (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.b
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
w v : 𝒰Df.I₀
⊢ v ∈ Finset.univ

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C✝ : Type u_1
inst✝⁶ : Category.{v_1, u_1} C✝
D✝ : Type u_2
inst✝⁵ : Category.{v_2, u_2} D✝
C : Type u_3
D : Type u_4
inst✝⁴ : Category.{v_3, u_3} C
inst✝³ : Category.{v_4, u_4} D
X Y : C
f : X ⟶ Y
inst✝² : EffectiveEpi f
F : C ⥤ D
inst✝¹ : F.PreservesEffectiveEpis
inst✝ : PreservesLimitsOfShape WalkingCospan F
c : PullbackCone f f
hc : IsLimit c
⊢ F.map c.fst ≫ F.map f = F.map c.snd ≫ F.map f

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
heq :
  ∀ (j : A.𝒰D.I₀),
    (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.a =
      (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.b
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
w v : 𝒰Df.I₀
⊢ hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑w) = hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑v)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C✝ : Type u_1
inst✝⁶ : Category.{v_1, u_1} C✝
D✝ : Type u_2
inst✝⁵ : Category.{v_2, u_2} D✝
C : Type u_3
D : Type u_4
inst✝⁴ : Category.{v_3, u_3} C
inst✝³ : Category.{v_4, u_4} D
X Y : C
f : X ⟶ Y
inst✝² : EffectiveEpi f
F : C ⥤ D
inst✝¹ : F.PreservesEffectiveEpis
inst✝ : PreservesLimitsOfShape WalkingCospan F
c : PullbackCone f f
hc : IsLimit c
⊢ ∀ (j : WalkingCospan),
    ((Cone.postcompose
                  (cospanIsoMk (Iso.refl ((cospan f f ⋙ F).obj WalkingCospan.one))
                      (Iso.refl ((cospan f f ⋙ F).obj WalkingCospan.left))
                      (Iso.refl ((cospan f f ⋙ F).obj WalkingCospan.right)) ⋯ ⋯).hom).obj
              (F.mapCone c)).π.app
        j =
      (Iso.refl
            ((Cone.postcompose
                    (cospanIsoMk (Iso.refl ((cospan f f ⋙ F).obj WalkingCospan.one))
                        (Iso.refl ((cospan f f ⋙ F).obj WalkingCospan.left))
                        (Iso.refl ((cospan f f ⋙ F).obj WalkingCospan.right)) ⋯ ⋯).hom).obj
                (F.mapCone c)).pt).hom ≫
        (PullbackCone.mk (F.map c.fst) (F.map c.snd) ⋯).π.app j

Tactic:
by

Goal state after:
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.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 j : 𝒱.I₀
hij : i ⟶ j
⊢ d.transitionMap hij ≫ (d.isColimit j).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).mapCocone s) =
    (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

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C✝ : Type u_1
inst✝⁶ : Category.{v_1, u_1} C✝
D✝ : Type u_2
inst✝⁵ : Category.{v_2, u_2} D✝
C : Type u_3
D : Type u_4
inst✝⁴ : Category.{v_3, u_3} C
inst✝³ : Category.{v_4, u_4} D
X Y : C
f : X ⟶ Y
inst✝² : EffectiveEpi f
F : C ⥤ D
inst✝¹ : F.PreservesEffectiveEpis
inst✝ : PreservesLimitsOfShape WalkingCospan F
c : PullbackCone f f
hc : IsLimit c
⊢ IsColimit (Cofork.ofπ (F.map f) ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : F.PreservesEffectiveEpis
inst✝ : G.PreservesEffectiveEpis
X✝ Y✝ : C
x✝¹ : X✝ ⟶ Y✝
x✝ : EffectiveEpi x✝¹
⊢ EffectiveEpi ((F ⋙ G).map x✝¹)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
inst✝² : F.PreservesFiniteEffectiveEpiFamilies
inst✝¹ : G.PreservesFiniteEffectiveEpiFamilies
α✝ : Type
inst✝ : Finite α✝
B✝ : C
x✝² : α✝ → C
x✝¹ : (a : α✝) → x✝² a ⟶ B✝
x✝ : EffectiveEpiFamily x✝² x✝¹
⊢ EffectiveEpiFamily (fun a ↦ (F ⋙ G).obj (x✝² a)) fun a ↦ (F ⋙ G).map (x✝¹ a)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : F.PreservesEffectiveEpiFamilies
inst✝ : G.PreservesEffectiveEpiFamilies
α✝ : Type u
B✝ : C
x✝² : α✝ → C
x✝¹ : (a : α✝) → x✝² a ⟶ B✝
x✝ : EffectiveEpiFamily x✝² x✝¹
⊢ EffectiveEpiFamily (fun a ↦ (F ⋙ G).obj (x✝² a)) fun a ↦ (F ⋙ G).map (x✝¹ a)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
F : C ⥤ D
inst✝¹ : F.ReflectsEffectiveEpiFamilies
α✝ : Type
inst✝ : Finite α✝
B✝ : C
x✝¹ : α✝ → C
x✝ : (a : α✝) → x✝¹ a ⟶ B✝
h : EffectiveEpiFamily (fun a ↦ F.obj (x✝¹ a)) fun a ↦ F.map (x✝ a)
⊢ EffectiveEpiFamily x✝¹ x✝

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝ : F.ReflectsFiniteEffectiveEpiFamilies
X✝ Y✝ : C
x✝ : X✝ ⟶ Y✝
h : EffectiveEpi (F.map x✝)
⊢ EffectiveEpi x✝

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case pos
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
l : Fin n
m : ℕ
hlj : l = j
⊢ (c.val j %ₘ c.val i).coeff m ∈ coeffSubmodule R₀ c ^ (c.val j).degree.succ

Tactic:
convert!
          coeff_modByMonic_mem_pow_natDegree_mul _ _ _ (fun _ ↦ coeff_mem_coeffSubmodule)
            one_mem_coeffSubmodule _ (fun _ ↦ coeff_mem_coeffSubmodule) one_mem_coeffSubmodule _

Goal state after:
case e'_4
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
l : Fin n
m : ℕ
hlj : l = j
⊢ coeffSubmodule R₀ c ^ (c.val j).degree.succ = coeffSubmodule R₀ c ^ (c.val j).natDegree * coeffSubmodule R₀ c
Module: Mathlib.CategoryTheory.EffectiveEpi.Preserves

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝ : F.IsEquivalence
α : Type u_3
B : C
X : α → C
π : (a : α) → X a ⟶ B
x✝ : EffectiveEpiFamily (fun a ↦ F.obj (X a)) fun a ↦ F.map (π a)
⊢ EffectiveEpiFamily X π

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this✝ : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
heq :
  ∀ (j : A.𝒰D.I₀),
    (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.a =
      (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.b
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
this : ∀ (w v : 𝒰Df.I₀), hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑w) = hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑v)
u : (Precoverage.ZeroHypercover.pullback₁ (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) 𝒰Df).I₀
⊢ k (Cover.idx A.𝒰D ↑u) ∈ O

Tactic:
by

Goal state after:
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

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝⁴ : IsAffine X
inst✝³ : IsAffine Y
π : X ⟶ Y
inst✝² : Surjective π
inst✝¹ : Flat π
f : X ⟶ S
hf : pullback.fst π π ≫ f = pullback.snd π π ≫ f
inst✝ : IsAffine S
⊢ EffectiveEpi (AffineScheme.ofHom π)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝⁴ : IsAffine X
inst✝³ : IsAffine Y
π : X ⟶ Y
inst✝² : Surjective π
inst✝¹ : Flat π
f : X ⟶ S
hf : pullback.fst π π ≫ f = pullback.snd π π ≫ f
inst✝ : IsAffine S
⊢ ∃ u, π ≫ u = f

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
⊢ ∃ u, π.base ≫ u = f.base

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this✝ : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
heq :
  ∀ (j : A.𝒰D.I₀),
    (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.a =
      (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.b
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
this : ∀ (w v : 𝒰Df.I₀), hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑w) = hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑v)
u : (Precoverage.ZeroHypercover.pullback₁ (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) 𝒰Df).I₀
⊢ D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o)) ≫ 𝟙 (D.obj A.i') = D.map (hl1 ⋯) ≫ D.map (hki' (Cover.idx A.𝒰D ↑u))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
b : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace
hfac : π.base ≫ b = f.base
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯
i : 𝒰.I₀
this : IsAffine (𝒰.X i)
⊢ Set.range ⇑(TopCat.Hom.hom (𝒰.f i).base) ⊆ ↑(↑i).1

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
b : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace
hfac : π.base ≫ b = f.base
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯
i : 𝒰.I₀
this : IsAffine (𝒰.X i)
⊢ pullback π (𝒰.f i) ⟶ ↑↑(↑i).2

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
b : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace
hfac : π.base ≫ b = f.base
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯
i : 𝒰.I₀
this : IsAffine (𝒰.X i)
f' : pullback π (𝒰.f i) ⟶ ↑↑(↑i).2 := IsOpenImmersion.lift (Opens.ι ↑(↑i).2) (fst π (𝒰.f i) ≫ f) ⋯
⊢ fst (snd π (𝒰.f i)) (snd π (𝒰.f i)) ≫ fst π (𝒰.f i) =
    map (snd π (𝒰.f i)) (snd π (𝒰.f i)) π π (fst π (𝒰.f i)) (fst π (𝒰.f i)) (𝒰.f i) ⋯ ⋯ ≫ fst π π

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this✝ : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
heq :
  ∀ (j : A.𝒰D.I₀),
    (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.a =
      (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.b
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
this : ∀ (w v : 𝒰Df.I₀), hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑w) = hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑v)
u : (Precoverage.ZeroHypercover.pullback₁ (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) 𝒰Df).I₀
F : pullback (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) ⟶
  pullback (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) :=
  pullback.map (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) (D.map (hki' (Cover.idx A.𝒰D ↑u)))
    (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) (D.map (hl1 ⋯)) (𝟙 (𝒰Df.X u)) (𝟙 (D.obj A.i')) ⋯ ⋯
⊢ k (Cover.idx A.𝒰D ↑u) ∈ O

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
b : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace
hfac : π.base ≫ b = f.base
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯
i : 𝒰.I₀
this : IsAffine (𝒰.X i)
f' : pullback π (𝒰.f i) ⟶ ↑↑(↑i).2 := IsOpenImmersion.lift (Opens.ι ↑(↑i).2) (fst π (𝒰.f i) ≫ f) ⋯
h1 :
  fst (snd π (𝒰.f i)) (snd π (𝒰.f i)) ≫ fst π (𝒰.f i) =
    map (snd π (𝒰.f i)) (snd π (𝒰.f i)) π π (fst π (𝒰.f i)) (fst π (𝒰.f i)) (𝒰.f i) ⋯ ⋯ ≫ fst π π
⊢ snd (snd π (𝒰.f i)) (snd π (𝒰.f i)) ≫ fst π (𝒰.f i) =
    map (snd π (𝒰.f i)) (snd π (𝒰.f i)) π π (fst π (𝒰.f i)) (fst π (𝒰.f i)) (𝒰.f i) ⋯ ⋯ ≫ snd π π

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this✝ : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
heq :
  ∀ (j : A.𝒰D.I₀),
    (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.a =
      (Precoverage.ZeroHypercover.pullback₁ (D.map (hki' j)) A.𝒰D).f j ≫ D.map (hki' j ≫ A.hii') ≫ A.b
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
this : ∀ (w v : 𝒰Df.I₀), hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑w) = hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑v)
u : (Precoverage.ZeroHypercover.pullback₁ (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) 𝒰Df).I₀
F : pullback (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) ⟶
  pullback (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) :=
  pullback.map (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) (D.map (hki' (Cover.idx A.𝒰D ↑u)))
    (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) (D.map (hl1 ⋯)) (𝟙 (𝒰Df.X u)) (𝟙 (D.obj A.i')) ⋯ ⋯
⊢ F ≫ pullback.fst (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) =
    pullback.fst (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) ≫ D.map (hl1 ⋯)

Tactic:
by

Goal state after: