mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology

Goal state before:
pp: case mp.transitive
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : (extensiveCoverage C).Saturate X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → (extensiveCoverage C).Saturate Y (Sieve.pullback f S)
a : ∃ α, ∃ (_ : Finite α), ∃ Y π, Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), R.arrows (π a)
b :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∃ α, ∃ (_ : Finite α), ∃ Y_1 π, Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
⊢ ∃ α, ∃ (_ : Finite α), ∃ Y π, Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
obtain ⟨α, w, Y₁, π, h, h'⟩ := a

Goal state after:
case mp.transitive
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : (extensiveCoverage C).Saturate X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → (extensiveCoverage C).Saturate Y (Sieve.pullback f S)
b :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∃ α, ∃ (_ : Finite α), ∃ Y_1 π, Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Finite α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), R.arrows (π a)
⊢ ∃ α, ∃ (_ : Finite α), ∃ Y π, Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ =
    D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯
⊢ mor w φ q a₁ fac₁ = mor w φ q a₂ fac₂

Tactic:
← cancel_mono (D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q),

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  (pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯) ≫
      D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q ⋯ ⋯ =
    (D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯) ≫
      D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q ⋯ ⋯
⊢ mor w φ q a₁ fac₁ = mor w φ q a₂ fac₂
Module: Mathlib.CategoryTheory.Sites.MayerVietorisSquare

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : HasWeakSheafify J (Type v)
sq : Square C
inst✝ : Mono sq.f₁₃
H : ∀ (F : Sheaf J (Type v)), (sq.op.map F.obj).IsPullback
⊢ (sq.map (yoneda ⋙ presheafToSheaf J (Type v))).IsPushout

Tactic:
rw [Square.isPushout_iff_op_map_yoneda_isPullback]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : HasWeakSheafify J (Type v)
sq : Square C
inst✝ : Mono sq.f₁₃
H : ∀ (F : Sheaf J (Type v)), (sq.op.map F.obj).IsPullback
⊢ ∀ (X : Sheaf J (Type v)), ((sq.map (yoneda ⋙ presheafToSheaf J (Type v))).op.map (yoneda.obj X)).IsPullback
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: case hunique
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
y₁ y₂ : (presheafHom F G).obj (op X)
hy₁ : ∀ (Y : C) (g : Y ⟶ X) (hg : S.arrows g), y₁.app (op (Over.mk g)) = (x g hg).app (op (Over.mk (𝟙 Y)))
hy₂ : ∀ (Y : C) (g : Y ⟶ X) (hg : S.arrows g), y₂.app (op (Over.mk g)) = (x g hg).app (op (Over.mk (𝟙 Y)))
Y : Over X
⊢ y₁.app (op Y) = y₂.app (op Y)

Tactic:
apply (hG Y.hom).hom_ext

Goal state after:
case hunique
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
y₁ y₂ : (presheafHom F G).obj (op X)
hy₁ : ∀ (Y : C) (g : Y ⟶ X) (hg : S.arrows g), y₁.app (op (Over.mk g)) = (x g hg).app (op (Over.mk (𝟙 Y)))
hy₂ : ∀ (Y : C) (g : Y ⟶ X) (hg : S.arrows g), y₂.app (op (Over.mk g)) = (x g hg).app (op (Over.mk (𝟙 Y)))
Y : Over X
⊢ ∀ (j : (Sieve.pullback Y.hom S).arrows.categoryᵒᵖ),
    y₁.app (op Y) ≫ (G.mapCone (Sieve.pullback Y.hom S).arrows.cocone.op).π.app j =
      y₂.app (op Y) ≫ (G.mapCone (Sieve.pullback Y.hom S).arrows.cocone.op).π.app j
Module: Mathlib.CategoryTheory.Sites.Descent.DescentData

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
S' : C
p : S' ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S'
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j ≫ p
β : ι' → ι
p'' : (j : ι') → X' j ⟶ X (β j)
w' : ∀ (j : ι'), p'' j ≫ f (β j) = f' j ≫ p
D : F.DescentData f
Y : C
q : Y ⟶ S'
j₁ j₂ : ι'
f₁ : Y ⟶ X' j₁
f₂ : Y ⟶ X' j₂
hf₁ : f₁ ≫ f' j₁ = q
hf₂ : f₂ ≫ f' j₂ = q
⊢ f₂ ≫ p' j₂ ≫ f (α j₂) = q ≫ p

Tactic:
by rw [w, reassoc_of% hf₂]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: case refine_1
C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
X : C
S : Sieve X
hS : S ∈ J X
⊢ Presieve.IsSeparatedFor F S.arrows

Tactic:
intro x t₁ t₂ ht₁ ht₂
    let 𝒮 (Z : C) : Set (F.obj (.op Z)) :=
      .range (fun (g : { g : Z ⟶ X | S.arrows g }) ↦ x _ g.2) ∪
      .range (fun (g : Z ⟶ X) ↦ F.map g.op t₁) ∪ .range (fun (g : Z ⟶ X) ↦ F.map g.op t₂)
    let Q : Subfunctor F := K.subsheafify 𝒮
    have (Z : C) : _root_.Small.{max u v} (Q.toFunctor.obj (Opposite.op Z)) :=
      small_subsheafify_of_small H _ inferInstance _
    have hQ : Presieve.IsSheaf J Q.toFunctor :=
      h.isSheaf_of_forall_aux _ fun X R hR ↦ isSheafFor_subsheafify _ hR (H _ hR)
    let x' : S.arrows.FamilyOfElements Q.toFunctor :=
      fun Z g hg ↦ ⟨x g hg, .base <| .inl <| .inl ⟨⟨g, hg⟩, rfl⟩⟩
    have ht₁' : t₁ ∈ Q.obj (.op X) := .base (.inl <| .inr ⟨𝟙 _, by simp⟩)
    have ht₂' : t₂ ∈ Q.obj (.op X) := .base (.inr ⟨𝟙 _, by simp⟩)
    have : (⟨t₁, ht₁'⟩ : Q.obj _) = ⟨t₂, ht₂'⟩ :=
      (hQ _ hS).isSeparatedFor x' ⟨_, ht₁'⟩ ⟨_, ht₂'⟩ (.of_mono Q.ι ht₁) (.of_mono Q.ι ht₂)
    simp_all

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
K : Precoverage C
J : GrothendieckTopology C
H : K.Generates J
F : Cᵒᵖ ⥤ A
⊢ Presheaf.IsSheaf J F ↔
    ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, ∀ (M : A), Presieve.IsSheafFor (F ⋙ coyoneda.obj (Opposite.op M)) R

Tactic:
by
  grind [Presheaf.IsSheaf, H.isSheaf_type_iff]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Descent.DescentData

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
S : C
R : Sieve S
S₀ : C
M N : ↑(F.obj { as := op S₀ })
a : S ⟶ S₀
h : Presieve.IsSheafFor ((Over.map a).op ⋙ F.presheafHom M N) ((Sieve.overEquiv (Over.mk (𝟙 S))).symm R).arrows
⊢ Presieve.IsSheafFor (F.presheafHom M N) ((Sieve.overEquiv (Over.mk a)).symm R).arrows

Tactic:
Presieve.isSheafFor_over_map_op_comp_iff (X' := Over.mk a)
      (e := Over.isoMk (Iso.refl _))

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat
S : C
R : Sieve S
S₀ : C
M N : ↑(F.obj { as := op S₀ })
a : S ⟶ S₀
h :
  Presieve.IsSheafFor (F.presheafHom M N)
    (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv
        (Sieve.functorPushforward (Over.map a) ((Sieve.overEquiv (Over.mk (𝟙 S))).symm R))).arrows
⊢ Presieve.IsSheafFor (F.presheafHom M N) ((Sieve.overEquiv (Over.mk a)).symm R).arrows
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : HasShift D ℤ
F : C ⥤ D
inst✝⁹ : F.CommShift ℤ
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasZeroObject D
inst✝³ : Preadditive D
inst✝² : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝¹ : Pretriangulated D
inst✝ : F.op.IsTriangulated
T : Triangle C
dT : T ∈ Pretriangulated.distinguishedTriangles
⊢ F.mapTriangle.obj T ∈ Pretriangulated.distinguishedTriangles

Tactic:
have := distinguished_iff_of_iso ((triangleOpEquivalence D).unitIso.app
      (Opposite.op (F.mapTriangle.obj T))).unop
    rw [Functor.id_obj, Opposite.unop_op (F.mapTriangle.obj T)] at this
    rw [← this, Functor.comp_obj, ← mem_distTriang_op_iff, ← Functor.op_obj, ← Functor.comp_obj,
      distinguished_iff_of_iso ((mapTriangleOpCompTriangleOpEquivalenceFunctor F).app
      (Opposite.op T))]
    exact F.op.map_distinguished _ (op_distinguished _ dT)

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Heart

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
t : TStructure C
H : Type u'
inst✝² : Category.{v', u'} H
inst✝¹ : Preadditive H
inst✝ : t.Heart H
X : H
⊢ t.heart (t.ιHeart.obj X)

Tactic:
rw [← t.essImage_ιHeart H]
  exact t.ιHeart.obj_mem_essImage X

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.WeakKernels

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
K : C
i : K ⟶ X✝
p : Y✝ ⟶ (shiftFunctor C 1).obj K
h : Triangle.mk i f p ∈ distinguishedTriangles
⊢ Nonempty (WeakLimitCone (parallelPair f 0))

Tactic:
exact ⟨_, isWeakLimitKernelForkOfDistTriangle _ h⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: case refine_3
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝¹ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this✝ : Mono ((adj.homEquiv Z T.obj₃) h)
this : IsIso ((adj.homEquiv Z T.obj₃) h)
⊢ g =
    g ≫
      adj.unit.app ((shiftFunctor C 1).obj (G.obj T.obj₁)) ≫
        G.map ((Functor.commShiftIso F 1).hom.app (G.obj T.obj₁)) ≫
          G.map ((shiftFunctor D 1).map (adj.counit.app T.obj₁)) ≫ (Functor.commShiftIso G 1).hom.app T.obj₁

Tactic:
Functor.commShiftIso_hom_naturality,

Goal state after:
case refine_3
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝¹ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this✝ : Mono ((adj.homEquiv Z T.obj₃) h)
this : IsIso ((adj.homEquiv Z T.obj₃) h)
⊢ g =
    g ≫
      adj.unit.app ((shiftFunctor C 1).obj (G.obj T.obj₁)) ≫
        G.map ((Functor.commShiftIso F 1).hom.app (G.obj T.obj₁)) ≫
          (Functor.commShiftIso G 1).hom.app (F.obj (G.obj T.obj₁)) ≫
            (shiftFunctor C 1).map (G.map (adj.counit.app T.obj₁))
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
hs : ∀ a ∈ s, ∃ b ∈ t, r a b
ht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton
a : α
h : a ∈ s
⊢ (bipartiteAbove r t a).Nonempty

Tactic:
by
        rw [← coe_nonempty, coe_bipartiteAbove]
        exact hs _ h

Goal state after:
Module: Mathlib.Combinatorics.Enumerative.DoubleCounting

Goal state before:
pp: case e'_3.a
α : Type u_2
β : Type u_3
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
B : α → Finset β
s : Finset α
x✝ : β
a✝ : x✝ ∈ s.biUnion B
⊢ {j | j ∈ s ∧ x✝ ∈ B j} = bipartiteBelow (fun j x ↦ x ∈ B j) s x✝

Tactic:
grind [bipartiteBelow]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d k : ℕ
x : Fin n → ℕ
hx : x ∈ ↑(sphere n d k)
⊢ x ∈ (fun x ↦ toLp 2 (Nat.cast ∘ x)) ⁻¹' Metric.sphere 0 √↑k

Tactic:
by rw [Set.mem_preimage, mem_sphere_zero_iff_norm, norm_of_mem_sphere hx]

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: n d✝ k N : ℕ
x : Fin n → ℕ
d : ℕ
⊢ ∑ i, 0 i * d ^ ↑i = 0

Tactic:
by simp_rw [Pi.zero_apply, zero_mul, sum_const_zero]

Goal state after:
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: case refine_2
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.card = n
h : s.prod = t.prod
⊢ ∀ ⦃x : β⦄, x ∈ map f t → x ∈ B

Tactic:
simpa using fun a h ↦ hf.mapsTo (htA h)

Goal state after:
Module: Mathlib.Combinatorics.Additive.AP.Three.Defs

Goal state before:
pp: M N : ℕ
⊢ addRothNumber (range (M + N)) ≤ addRothNumber (range M) + addRothNumber (range N)

Tactic:
rw [range_add_eq_union, ← addRothNumber_map_add_left (range N) M]

Goal state after:
M N : ℕ
⊢ addRothNumber (range M ∪ map (addLeftEmbedding M) (range N)) ≤
    addRothNumber (range M) + addRothNumber (map (addLeftEmbedding M) (range N))
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: ⊢ log 2 * 2 ≤ √(log 8)

Tactic:
rw [show (8 : ℝ) = 2 ^ 3 by norm_num1, Real.log_pow, Nat.cast_ofNat]

Goal state after:
⊢ log 2 * 2 ≤ √(3 * log 2)
Module: Mathlib.Combinatorics.SimpleGraph.Maps

Goal state before:
pp: V : Type u_1
W : Type u_2
f : V ↪ W
G : SimpleGraph V
⊢ (SimpleGraph.map (⇑f) G).edgeSet = ⇑f.sym2Map '' G.edgeSet

Tactic:
by
  ext v
  induction v
  rw [mem_edgeSet, map_adj, Set.mem_image]
  constructor
  · intro ⟨a, b, hadj, ha, hb⟩
    use s(a, b), hadj
    rw [Embedding.sym2Map_apply, Sym2.map_mk, ha, hb]
  · intro ⟨e, hadj, he⟩
    induction e
    rw [Embedding.sym2Map_apply, Sym2.map_mk, Sym2.eq_iff] at he
    exact he.elim (fun ⟨h, h'⟩ ↦ ⟨_, _, hadj, h, h'⟩) (fun ⟨h', h⟩ ↦ ⟨_, _, hadj.symm, h, h'⟩)

Goal state after:
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: case pos
α : Type u_1
inst✝² : Lattice α
inst✝¹ : OrderBot α
a : α
inst✝ : Decidable (a = ⊥)
b : α
hb✝ : b ∈ ⊤.parts
h✝ : a = ⊥
hb : b ∈ ((Finpartition.empty α).copy ⋯).parts
⊢ b ∈ {a}

pp: case neg
α : Type u_1
inst✝² : Lattice α
inst✝¹ : OrderBot α
a : α
inst✝ : Decidable (a = ⊥)
b : α
hb✝ : b ∈ ⊤.parts
h✝ : ¬a = ⊥
hb : b ∈ (indiscrete h✝).parts
⊢ b ∈ {a}

Tactic:
· simp only [copy_parts, empty_parts, notMem_empty] at hb

Goal state after:
case neg
α : Type u_1
inst✝² : Lattice α
inst✝¹ : OrderBot α
a : α
inst✝ : Decidable (a = ⊥)
b : α
hb✝ : b ∈ ⊤.parts
h✝ : ¬a = ⊥
hb : b ∈ (indiscrete h✝).parts
⊢ b ∈ {a}
Module: Mathlib.Order.Partition.Equipartition

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
z : #({x ∈ P.parts | #x = #s / #P.parts + 1}) + #P.parts * (#s / #P.parts) = #s
⊢ #({p ∈ P.parts | #p = #s / #P.parts + 1}) = #s % #P.parts

Tactic:
← add_left_inj,

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
z : #({x ∈ P.parts | #x = #s / #P.parts + 1}) + #P.parts * (#s / #P.parts) = #s
⊢ #({p ∈ P.parts | #p = #s / #P.parts + 1}) + ?a = #s % #P.parts + ?a

case a
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
z : #({x ∈ P.parts | #x = #s / #P.parts + 1}) + #P.parts * (#s / #P.parts) = #s
⊢ ℕ
Module: Mathlib.Order.Partition.Finpartition

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
r : Set α
hrs : r ⊆ ↑s
hr : Set.BijOn P.part r ↑P.parts
⊢ ∃ r ⊆ s, Set.BijOn P.part ↑r ↑P.parts

Tactic:
lift r to Finset α using s.finite_toSet.subset hrs

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
r : Finset α
hrs : ↑r ⊆ ↑s
hr : Set.BijOn P.part ↑r ↑P.parts
⊢ ∃ r ⊆ s, Set.BijOn P.part ↑r ↑P.parts
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Bound

Goal state before:
pp: ι : Type u_2
𝕜 : Type u_3
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s t : Finset ι
x : 𝕜
hst : s ⊆ t
f : ι → 𝕜
d : 𝕜
hx : 0 ≤ x
hs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|
ht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2
hscard : 0 < ↑(#s)
htcard : 0 < ↑(#t)
h₁ : x ^ 2 ≤ ((∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)) ^ 2
⊢ ((∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)) ^ 2 ≤ ((∑ x ∈ s, f x - ∑ x ∈ s, (∑ j ∈ t, f j) / ↑(#t)) / ↑(#s)) ^ 2

Tactic:
sum_const,

Goal state after:
ι : Type u_2
𝕜 : Type u_3
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s t : Finset ι
x : 𝕜
hst : s ⊆ t
f : ι → 𝕜
d : 𝕜
hx : 0 ≤ x
hs : x ≤ |(∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)|
ht : d ≤ ((∑ i ∈ t, f i) / ↑(#t)) ^ 2
hscard : 0 < ↑(#s)
htcard : 0 < ↑(#t)
h₁ : x ^ 2 ≤ ((∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)) ^ 2
⊢ ((∑ i ∈ s, f i) / ↑(#s) - (∑ i ∈ t, f i) / ↑(#t)) ^ 2 ≤ ((∑ x ∈ s, f x - #s • ((∑ j ∈ t, f j) / ↑(#t))) / ↑(#s)) ^ 2
Module: Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
n : ℤ
inst✝ : t.IsGE X n
i : EInt
⊢ t.IsGE ((t.eTruncLT.obj i).obj X) n

Tactic:
by
  induction i using WithBotTop.rec with
  | bot => exact isGE_of_isZero _ (by simp) _
  | coe _ => dsimp; infer_instance
  | top => dsimp; infer_instance

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : DecidableEq α
A : Finset α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
hε : 0 < ε
⊢ ⊥.nonUniforms G ε = ∅

Tactic:
rw [eq_empty_iff_forall_notMem]
  rintro ⟨u, v⟩
  simp only [mk_mem_nonUniforms, parts_bot, mem_map, not_and,
    Classical.not_not, exists_imp]; dsimp
  rintro x ⟨_, rfl⟩ y ⟨_, rfl⟩ _
  rwa [SimpleGraph.isUniform_singleton]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Increment

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ¬P.IsUniform G ε
⊢ stepBound #P.parts ≤ #P.parts * 16 ^ #P.parts

Tactic:
stepBound

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ¬P.IsUniform G ε
⊢ #P.parts * 4 ^ #P.parts ≤ #P.parts * 16 ^ #P.parts
Module: Mathlib.Combinatorics.SimpleGraph.Regularity.Increment

Goal state before:
pp: α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#(P.nonUniforms G ε)) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25)) =
    ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      ((∑ x ∈ P.parts.offDiag, if G.IsUniform ε x.1 x.2 then 0 else ε ^ 4 / 3) - ∑ x ∈ P.parts.offDiag, ε ^ 5 / 25)

Tactic:
sum_const,

Goal state after:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
hP : P.IsEquipartition
hP₇ : 7 ≤ #P.parts
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPG : ↑(#P.parts.offDiag) * ε < ↑(#(P.nonUniforms G ε))
hε₀ : 0 ≤ ε
hε₁ : ε ≤ 1
⊢ ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      (↑(#(P.nonUniforms G ε)) * (ε ^ 4 / 3) - ↑(#P.parts.offDiag) * (ε ^ 5 / 25)) =
    ∑ x ∈ P.parts.offDiag, ↑(G.edgeDensity x.1 x.2) ^ 2 +
      ((∑ x ∈ P.parts.offDiag, if G.IsUniform ε x.1 x.2 then 0 else ε ^ 4 / 3) - #P.parts.offDiag • (ε ^ 5 / 25))