mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: case h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ (∑ j_1, (if h : i = b then eqToHom ⋯ else 0) ≫ if h : b = j then eqToHom ⋯ else 0) = 0

Tactic:
Fintype.univ_ofSubsingleton,

Goal state after:
case h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ (∑ j_1 ∈ {PUnit.unit}, (if h : i = b then eqToHom ⋯ else 0) ≫ if h : b = j then eqToHom ⋯ else 0) = 0
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: case h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ ((if h : i = b then eqToHom ⋯ else 0) ≫ if h : b = j then eqToHom ⋯ else 0) = 0

Tactic:
dif_neg hb.symm,

Goal state after:
case h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
M : Mat_ C
i j b : M.ι
a✝ : b ∈ Finset.univ
hb : b ≠ i
⊢ (0 ≫ if h : b = j then eqToHom ⋯ else 0) = 0
Module: Mathlib.CategoryTheory.Preadditive.Mat

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : F.Additive
M N : Mat_ C
f : M ⟶ N
i : N.ι
⊢ (F.map f ≫ (additiveObjIsoBiproduct F N).hom) ≫ biproduct.π (fun i ↦ F.obj ((embedding C).obj (N.X i))) i =
    ((additiveObjIsoBiproduct F M).hom ≫ biproduct.matrix fun i j ↦ F.map ((embedding C).map (f i j))) ≫
      biproduct.π (fun i ↦ F.obj ((embedding C).obj (N.X i))) i

Tactic:
simp only [Category.assoc, additiveObjIsoBiproduct_hom_π, isoBiproductEmbedding_hom,
    biproduct.lift_π, biproduct.matrix_π,
    ← cancel_epi (additiveObjIsoBiproduct F M).inv, Iso.inv_hom_id_assoc]

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Preadditive C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : Preadditive D
inst✝¹ : HasFiniteBiproducts D
F : Mat_ C ⥤ D
inst✝ : F.Additive
M N : Mat_ C
f : M ⟶ N
i : N.ι
⊢ ((additiveObjIsoBiproduct F M).inv ≫ F.map f ≫ F.map fun j x ↦ if h : j = i then eqToHom ⋯ else 0) =
    biproduct.desc fun j ↦ F.map ((embedding C).map (f j i))
Module: Mathlib.CategoryTheory.Presentable.CardinalDirectedPoset

Goal state before:
pp: κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
S : Subtype (J.PropSetWithTop κ')
x✝ : ↑↑S
⊢ x✝ ≤ ⟨⊤, ⋯⟩

Tactic:
rw [Subtype.mk_le_mk]

Goal state after:
κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
S : Subtype (J.PropSetWithTop κ')
x✝ : ↑↑S
⊢ ↑x✝ ≤ ⊤
Module: Mathlib.CategoryTheory.Presentable.CardinalDirectedPoset

Goal state before:
pp: κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
K : Type u
α : K → Subtype (J.PropSetWithTop κ')
hK : HasCardinalLT K κ'
hκ' : Cardinal.aleph0 ≤ κ'
k : K
⊢ α k ≤ ⟨(⋃ k, ↑(α k)) ∪ {⊤}, ⋯⟩

Tactic:
rw [Subtype.mk_le_mk]

Goal state after:
κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
K : Type u
α : K → Subtype (J.PropSetWithTop κ')
hK : HasCardinalLT K κ'
hκ' : Cardinal.aleph0 ≤ κ'
k : K
⊢ ↑(α k) ⊆ (⋃ k, ↑(α k)) ∪ {⊤}
Module: Mathlib.CategoryTheory.Presentable.OrthogonalReflection

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
Z : C
inst✝³ : HasCoproduct D₁.obj₁
inst✝² : HasCoproduct D₁.obj₂
inst✝¹ : HasPushouts C
inst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)
X Y : C
f : X ⟶ Y
hf : W f
g : X ⟶ Z
⊢ f ≫ D₁.ιRight f hf g ≫ pushout.inl (D₁.t W Z) (D₁.l W Z) ≫ fromStep W Z = g ≫ toSucc W Z

Tactic:
by
    simp [← D₁.ιLeft_comp_t_assoc, pushout.condition_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves

Goal state before:
pp: case refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Precoherent C
W X : C
α : Type
inst✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : Limits.IsColimit (Sieve.generate (Presieve.ofArrows Y π)).arrows.cocone
x : Presieve.FamilyOfElements (yoneda.obj W) (Presieve.ofArrows Y π)
hx : x.Compatible
x_ext : Presieve.FamilyOfElements (yoneda.obj W) (Sieve.generate (Presieve.ofArrows Y π)).arrows := x.sieveExtend
hx_ext : x.sieveExtend.Compatible
S : Sieve X := Sieve.generate (Presieve.ofArrows Y π)
t : (yoneda.obj W).obj (Opposite.op X)
t_amalg : x_ext.IsAmalgamation t
t_uniq : ∀ (y : (yoneda.obj W).obj (Opposite.op X)), (fun t ↦ x_ext.IsAmalgamation t) y → y = t
⊢ (fun t ↦ x.IsAmalgamation t) t

pp: case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Precoherent C
W X : C
α : Type
inst✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : Limits.IsColimit (Sieve.generate (Presieve.ofArrows Y π)).arrows.cocone
x : Presieve.FamilyOfElements (yoneda.obj W) (Presieve.ofArrows Y π)
hx : x.Compatible
x_ext : Presieve.FamilyOfElements (yoneda.obj W) (Sieve.generate (Presieve.ofArrows Y π)).arrows := x.sieveExtend
hx_ext : x.sieveExtend.Compatible
S : Sieve X := Sieve.generate (Presieve.ofArrows Y π)
t : (yoneda.obj W).obj (Opposite.op X)
t_amalg : x_ext.IsAmalgamation t
t_uniq : ∀ (y : (yoneda.obj W).obj (Opposite.op X)), (fun t ↦ x_ext.IsAmalgamation t) y → y = t
⊢ ∀ (y : (yoneda.obj W).obj (Opposite.op X)), (fun t ↦ x.IsAmalgamation t) y → y = t

Tactic:
· convert!
    Presieve.isAmalgamation_restrict (Sieve.le_generate (Presieve.ofArrows Y π)) _ _ t_amalg
    exact (Presieve.restrict_extend hx).symm

Goal state after:
case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Precoherent C
W X : C
α : Type
inst✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
H : EffectiveEpiFamily Y π
h_colim : Limits.IsColimit (Sieve.generate (Presieve.ofArrows Y π)).arrows.cocone
x : Presieve.FamilyOfElements (yoneda.obj W) (Presieve.ofArrows Y π)
hx : x.Compatible
x_ext : Presieve.FamilyOfElements (yoneda.obj W) (Sieve.generate (Presieve.ofArrows Y π)).arrows := x.sieveExtend
hx_ext : x.sieveExtend.Compatible
S : Sieve X := Sieve.generate (Presieve.ofArrows Y π)
t : (yoneda.obj W).obj (Opposite.op X)
t_amalg : x_ext.IsAmalgamation t
t_uniq : ∀ (y : (yoneda.obj W).obj (Opposite.op X)), (fun t ↦ x_ext.IsAmalgamation t) y → y = t
⊢ ∀ (y : (yoneda.obj W).obj (Opposite.op X)), (fun t ↦ x.IsAmalgamation t) y → y = t
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularTopology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preregular C
X Y Y' : C
π : Y ⟶ X
inst✝¹ : EffectiveEpi π
π' : Y' ⟶ Y
inst✝ : EffectiveEpi π'
⊢ EffectiveEpi (π' ≫ π)

Tactic:
effectiveEpi_iff_effectiveEpiFamily,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preregular C
X Y Y' : C
π : Y ⟶ X
inst✝¹ : EffectiveEpi π
π' : Y' ⟶ Y
inst✝ : EffectiveEpi π'
⊢ EffectiveEpiFamily (fun x ↦ Y') fun x ↦ π' ≫ π
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves

Goal state before:
pp: case refine_1.refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : FinitaryPreExtensive C
inst✝ : FinitaryExtensive C
F : Cᵒᵖ ⥤ Type w
hF : ∀ {X : C}, ∀ R ∈ (extensiveCoverage C).coverings X, IsSheafFor F R
n : ℕ
K : Discrete (Fin n) ⥤ Cᵒᵖ
Z : Fin n → C := fun i ↦ unop (K.obj { as := i })
this✝ : (ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (∐ Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun i ↦ op (Z i) := Discrete.natIsoFunctor
⊢ (ofArrows Z fun i ↦ Sigma.ι Z i) ∈ (extensiveCoverage C).coverings (∐ Z)

Tactic:
· exact ⟨Fin n, inferInstance, Z, (fun i ↦ Sigma.ι Z i), rfl, instIsIsoDescι⟩

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
this : Precoherent C
⊢ S ∈ (coherentTopology C) X ↔ ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)

Tactic:
rw [← coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily S]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
FD : D → D → Type u_3
CD : D → Type w
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁴ : ConcreteCategory D FD
inst✝³ : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ D
f : F ⟶ G
inst✝² : PreservesFiniteProducts F
inst✝¹ : PreservesFiniteProducts G
inst✝ : PreservesFiniteProducts (forget D)
⊢ (∀ (X : C), Function.Surjective ⇑(ConcreteCategory.hom (f.app (op X)))) →
    Presheaf.IsLocallySurjective (extensiveTopology C) f

Tactic:
· exact fun a ↦
      Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _)

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : ToType (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
⊢ ∃ Y π, EffectiveEpi π ∧ (Presheaf.imageSieve f y).arrows π

Tactic:
let x : (a : α) → (F.obj ⟨Z a⟩) := fun a ↦ (h' a).choose

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preregular C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
U✝ : C
y : ToType (G.obj (op U✝))
α : Type
w✝ : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U✝
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) := fun a ↦ Exists.choose ⋯
⊢ ∃ Y π, EffectiveEpi π ∧ (Presheaf.imageSieve f y).arrows π
Module: Mathlib.CategoryTheory.Sites.Hypercover.Homotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
f g : E.Hom F
i✝ j✝ : (cylinder f g).I₀
k : (cylinder f g).I₁ i✝ j✝
⊢ pullback.snd
        (pullback.map (cylinderf f g i✝.snd) (cylinderf f g j✝.snd) (E.f i✝.fst) (E.f j✝.fst)
          (pullback.fst (pullback.lift (f.h₀ i✝.fst) (g.h₀ i✝.fst) ⋯) (F.toPullback i✝.snd))
          (pullback.fst (pullback.lift (f.h₀ j✝.fst) (g.h₀ j✝.fst) ⋯) (F.toPullback j✝.snd)) (𝟙 S) ⋯ ⋯)
        (pullback.lift (E.p₁ k.down) (E.p₂ k.down) ⋯) ≫
      E.p₁ k.down =
    (cylinder f g).p₁ k ≫ pullback.fst (pullback.lift (f.h₀ i✝.fst) (g.h₀ i✝.fst) ⋯) (F.toPullback i✝.snd)

Tactic:
by
    have : E.p₁ k.down = pullback.lift _ _ (E.w k.down) ≫ pullback.fst _ _ := by simp
    nth_rw 2 [this]
    rw [← pullback.condition_assoc]
    simp

Goal state after:
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
Y₁ Y₂ Z : Over (X i)
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₁ : Y₁ ⟶ Over.mk (𝟙 (X i))
f₂ : Y₂ ⟶ Over.mk (𝟙 (X i))
h₁ : (sieve f f' i).arrows f₁
h₂ : (sieve f f' i).arrows f₂
fac : g₁ ≫ f₁ = g₂ ≫ f₂
⊢ (ConcreteCategory.hom ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g₁.op)) (familyOfElements w φ i f₁ h₁) =
    (ConcreteCategory.hom ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g₂.op)) (familyOfElements w φ i f₂ h₂)

Tactic:
obtain rfl : f₁ = Over.homMk Y₁.hom := by ext; simpa using Over.w f₁

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
Y₁ Y₂ Z : Over (X i)
g₁ : Z ⟶ Y₁
g₂ : Z ⟶ Y₂
f₂ : Y₂ ⟶ Over.mk (𝟙 (X i))
h₂ : (sieve f f' i).arrows f₂
h₁ : (sieve f f' i).arrows (Over.homMk Y₁.hom ⋯)
fac : g₁ ≫ Over.homMk Y₁.hom ⋯ = g₂ ≫ f₂
⊢ (ConcreteCategory.hom ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g₁.op))
      (familyOfElements w φ i (Over.homMk Y₁.hom ⋯) h₁) =
    (ConcreteCategory.hom ((F.presheafHom (D₁.obj i) (D₂.obj i)).map g₂.op)) (familyOfElements w φ i f₂ h₂)
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
D : F.DescentData f
Y' Y : C
g : Y' ⟶ Y
q : Y ⟶ S'
q' : Y' ⟶ S'
hq : g ≫ q = q'
j₁ j₂ : ι'
f₁ : Y ⟶ X' j₁
f₂ : Y ⟶ X' j₂
hf₁ : f₁ ≫ f' j₁ = q
hf₂ : f₂ ≫ f' j₂ = q
gf₁ : Y' ⟶ X' j₁
gf₂ : Y' ⟶ X' j₂
hgf₁ : g ≫ f₁ = gf₁
hgf₂ : g ≫ f₂ = gf₂
⊢ pullHom (pullFunctorObjHom w D (f₁ ≫ f' j₁) f₁ f₂ ⋯ ⋯) g gf₁ gf₂ ⋯ ⋯ = pullFunctorObjHom w D (gf₁ ≫ f' j₁) gf₁ gf₂ ⋯ ⋯

Tactic:
rw [pullFunctorObjHom_eq _ _ _ _ _ (q' ≫ p) (gf₁ ≫ p' j₁) (gf₂ ≫ p' j₂),
      pullFunctorObjHom_eq _ _ _ _ _ (q ≫ p) (f₁ ≫ p' j₁) (f₂ ≫ p' j₂)]
    rw [← D.pullHom_hom g (q ≫ p) (q' ≫ p) (by rw [reassoc_of% hq])
      (f₁ ≫ p' j₁) (f₂ ≫ p' j₂) (by rw [Category.assoc, w, reassoc_of% hf₁])
      (by rw [Category.assoc, w, reassoc_of% hf₂]) (gf₁ ≫ p' j₁) (gf₂ ≫ p' j₂)
      (by cat_disch) (by cat_disch)]
    dsimp [pullHom]
    simp only [Functor.map_comp, Category.assoc]
    rw [F.mapComp'₀₁₃_inv_comp_mapComp'₀₂₃_hom_app_assoc _ _ _ _ _ _ _ _ (by cat_disch),
      mapComp'₀₂₃_inv_comp_mapComp'₀₁₃_hom_app _ _ _ _ _ _ _ _ _ (by cat_disch)]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: 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
⊢ ∀ ⦃X_1 Y : (Over (unop (op X)))ᵒᵖ⦄ (f : X_1 ⟶ Y),
    ((Over.forget (unop (op X))).op ⋙ F).map f ≫ app hG x hx (unop Y).hom =
      app hG x hx (unop X_1).hom ≫ ((Over.forget (unop (op X))).op ⋙ G).map f

Tactic:
rintro ⟨Y₁ : Over X⟩ ⟨Y₂ : Over X⟩ ⟨φ : Y₂ ⟶ Y₁⟩

Goal state after:
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 : Over X
φ : Y₂ ⟶ Y₁
⊢ ((Over.forget (unop (op X))).op ⋙ F).map (op φ) ≫ app hG x hx (unop (op Y₂)).hom =
    app hG x hx (unop (op Y₁)).hom ≫ ((Over.forget (unop (op X))).op ⋙ G).map (op φ)
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)))
⊢ y₁ = y₂

Tactic:
apply NatTrans.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)))
⊢ y₁.app = y₂.app
Module: Mathlib.CategoryTheory.Subfunctor.Finite

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
ι : Type w'
X : ι → Cᵒᵖ
x : (i : ι) → F.obj (X i)
h : G.IsGeneratedBy x
i : ι
⊢ x i ∈ G.obj (X i)

Tactic:
rw [← ofSection_le_iff]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
G : Subfunctor F
ι : Type w'
X : ι → Cᵒᵖ
x : (i : ι) → F.obj (X i)
h : G.IsGeneratedBy x
i : ι
⊢ ofSection (x i) ≤ G
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
M N : ↑(F.obj { as := op S })
φ : Subtype (Presieve.Arrows.Compatible (F.presheafHom M N) fun i ↦ Over.homMk (f i) ⋯)
Y : C
q : Y ⟶ S
i₁ i₂ : ι
f₁ : Y ⟶ X i₁
f₂ : Y ⟶ X i₂
hf₁ : f₁ ≫ f i₁ = q
hf₂ : f₂ ≫ f i₂ = q
⊢ (F.map f₁.op.toLoc).toFunctor.map (↑φ i₁) ≫ ((F.toDescentData f).obj N).hom q f₁ f₂ ⋯ ⋯ =
    ((F.toDescentData f).obj M).hom q f₁ f₂ ⋯ ⋯ ≫ (F.map f₂.op.toLoc).toFunctor.map (↑φ i₂)

Tactic:
have := φ.property i₁ i₂ (Over.mk q) (Over.homMk f₁) (Over.homMk f₂) (by cat_disch)
        simp_all [map_eq_pullHom]

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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b c : EInt
hab : a ≤ b
hbc : b ≤ c
X : C
⊢ (t.eTruncGE.obj a).map ((t.eTruncLT.map (homOfLE hbc)).app X) =
    (t.eTruncGE.obj a).map ((t.eTruncLTLTIsoLT c b hbc).inv.app X) ≫
      (t.eTruncLTGEIsoGELT a b).inv.app ((t.eTruncLT.obj c).obj X) ≫
        (t.eTruncLTι b).app ((t.eTruncGE.obj a).obj ((t.eTruncLT.obj c).obj X))

Tactic:
rw [← cancel_epi ((t.eTruncGE.obj a).map ((t.eTruncLTLTIsoLT c b hbc).hom.app X)),
      ← Functor.map_comp_assoc, Iso.hom_inv_id_app, Functor.map_id, Category.id_comp,
      ← cancel_epi ((t.eTruncLTGEIsoGELT a b).hom.app ((t.eTruncLT.obj c).obj X)),
      Iso.hom_inv_id_app_assoc, eTruncLTLTIsoLT_hom, eTruncLTLTToLT_app,
      ← Functor.map_comp, eTruncLT_obj_map_eTruncLTι_app_eTruncLT_map_app]

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b c : EInt
hab : a ≤ b
hbc : b ≤ c
X : C
⊢ (t.eTruncLTGEIsoGELT a b).hom.app ((t.eTruncLT.obj c).obj X) ≫
      (t.eTruncGE.obj a).map ((t.eTruncLTι b).app ((t.eTruncLT.obj c).obj X)) =
    (t.eTruncLTι b).app ((t.eTruncGE.obj a).obj ((t.eTruncLT.obj c).obj X))
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
A : Set α
B : Set β
f : α → β
hf : IsMulFreimanIso 2 A B f
a b c d : α
ha : a ∈ A
hb : b ∈ A
hc : c ∈ A
hd : d ∈ A
⊢ {f a, f b}.prod = {f c, f d}.prod ↔ {a, b}.prod = {c, d}.prod

Tactic:
refine hf.map_prod_eq_map_prod ?_ ?_ (card_pair _ _) (card_pair _ _) <;> simp [ha, hb, hc, hd]

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

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
A : Set α
B : Set β
n : ℕ
inst✝¹ : EquivLike F α β
inst✝ : MulEquivClass F α β
f : F
hfAB : BijOn (⇑f) A B
s t : Multiset α
x✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
x✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
x✝¹ : s.card = n
x✝ : t.card = n
⊢ (map (⇑f) s).prod = (map (⇑f) t).prod ↔ s.prod = t.prod

Tactic:
← map_multiset_prod,

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
A : Set α
B : Set β
n : ℕ
inst✝¹ : EquivLike F α β
inst✝ : MulEquivClass F α β
f : F
hfAB : BijOn (⇑f) A B
s t : Multiset α
x✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
x✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
x✝¹ : s.card = n
x✝ : t.card = n
⊢ f s.prod = (map (⇑f) t).prod ↔ s.prod = t.prod
Module: Mathlib.Combinatorics.Additive.FreimanHom

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
A : Set α
B : Set β
n : ℕ
inst✝¹ : EquivLike F α β
inst✝ : MulEquivClass F α β
f : F
hfAB : BijOn (⇑f) A B
s t : Multiset α
x✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
x✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
x✝¹ : s.card = n
x✝ : t.card = n
⊢ f s.prod = (map (⇑f) t).prod ↔ s.prod = t.prod

Tactic:
← map_multiset_prod,

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : CommMonoid α
inst✝² : CommMonoid β
A : Set α
B : Set β
n : ℕ
inst✝¹ : EquivLike F α β
inst✝ : MulEquivClass F α β
f : F
hfAB : BijOn (⇑f) A B
s t : Multiset α
x✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
x✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
x✝¹ : s.card = n
x✝ : t.card = n
⊢ f s.prod = f t.prod ↔ s.prod = t.prod
Module: Mathlib.Combinatorics.Additive.RuzsaCovering

Goal state before:
pp: case pos
G : Type u_1
inst✝¹ : Group G
K : ℝ
inst✝ : DecidableEq G
A B : Finset G
hB : B.Nonempty
hK : ↑(#(A * B)) ≤ K * ↑(#B)
this : (F : Set G) → Decidable (F.PairwiseDisjoint fun x ↦ x • B)
C : Finset (Finset G) := {F ∈ A.powerset | (↑F).PairwiseDisjoint fun x ↦ x • B}
F : Finset G
hFmax : Maximal (fun x ↦ x ⊆ A ∧ (↑x).PairwiseDisjoint fun x ↦ x • B) F
hFA : F ⊆ A
hF : (↑F).PairwiseDisjoint fun x ↦ x • B
a : G
ha : a ∈ A
hau : a ∉ F
H : ∀ b ∈ F, Disjoint (a • B) (b • B)
⊢ (a ∈ A ∧ F ⊆ A) ∧ (↑(insert a F)).PairwiseDisjoint fun x ↦ x • B

Tactic:
coe_insert

Goal state after:
case pos
G : Type u_1
inst✝¹ : Group G
K : ℝ
inst✝ : DecidableEq G
A B : Finset G
hB : B.Nonempty
hK : ↑(#(A * B)) ≤ K * ↑(#B)
this : (F : Set G) → Decidable (F.PairwiseDisjoint fun x ↦ x • B)
C : Finset (Finset G) := {F ∈ A.powerset | (↑F).PairwiseDisjoint fun x ↦ x • B}
F : Finset G
hFmax : Maximal (fun x ↦ x ⊆ A ∧ (↑x).PairwiseDisjoint fun x ↦ x • B) F
hFA : F ⊆ A
hF : (↑F).PairwiseDisjoint fun x ↦ x • B
a : G
ha : a ∈ A
hau : a ∉ F
H : ∀ b ∈ F, Disjoint (a • B) (b • B)
⊢ (a ∈ A ∧ F ⊆ A) ∧ (insert a ↑F).PairwiseDisjoint fun x ↦ x • B
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: N : ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this : (2 * dValue N - 1) ^ n ≤ N
⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ (↑N ^ (↑(nValue N))⁻¹) ^ ↑(n - 2) / rexp 1 ^ ↑(n - 2) / ↑n

Tactic:
← rpow_mul hN₀.le,

Goal state after:
N : ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this : (2 * dValue N - 1) ^ n ≤ N
⊢ ↑N * rexp (-4 * √(log ↑N)) ≤ ↑N ^ ((↑(nValue N))⁻¹ * ↑(n - 2)) / rexp 1 ^ ↑(n - 2) / ↑n
Module: Mathlib.Combinatorics.Additive.AP.Three.Behrend

Goal state before:
pp: case h₁
N : ℕ
hN : 4096 ≤ N
n : ℕ := ⋯
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this✝ : (2 * dValue N - 1) ^ n ≤ N
this : rexp (-4 * √(log ↑N)) = rexp (-2 * √(log ↑N)) * rexp (-2 * √(log ↑N))
⊢ rexp (-2 * √(log ↑N)) ≤ ↑N ^ (-(2 / ↑n))

pp: case h₂
N : ℕ
hN : 4096 ≤ N
n : ℕ := ⋯
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this✝ : (2 * dValue N - 1) ^ n ≤ N
this : rexp (-4 * √(log ↑N)) = rexp (-2 * √(log ↑N)) * rexp (-2 * √(log ↑N))
⊢ rexp (-2 * √(log ↑N)) ≤ rexp (2 - ↑n) / ↑n

Tactic:
· rw [← le_log_iff_exp_le (rpow_pos_of_pos hN₀ _), log_rpow hN₀, ← le_div_iff₀, mul_div_assoc,
      div_sqrt, neg_mul, neg_le_neg_iff, div_mul_eq_mul_div, div_le_iff₀ hn]
    · gcongr
      apply le_ceil
    refine Real.sqrt_pos.2 (log_pos ?_)
    rw [one_lt_cast]
    exact hN.trans_lt' (by norm_num1)

Goal state after:
case h₂
N : ℕ
hN : 4096 ≤ N
n : ℕ := ⋯
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this✝ : (2 * dValue N - 1) ^ n ≤ N
this : rexp (-4 * √(log ↑N)) = rexp (-2 * √(log ↑N)) * rexp (-2 * √(log ↑N))
⊢ rexp (-2 * √(log ↑N)) ≤ rexp (2 - ↑n) / ↑n