mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
z : ℤ
hp : p ≠ 1
hz : z ≠ 0
⊢ padicValRat p ↑z = ↑(multiplicity (↑p) z)

Tactic:
rw [of_int, padicValInt.of_ne_one_ne_zero hp hz]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
G : C
inst✝⁶ : Abelian C
hG : IsSeparator G
X : C
inst✝⁵ : IsGrothendieckAbelian.{w, v, u} C
A : C
f : A ⟶ X
inst✝⁴ : Mono f
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
hj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤
t : Subobject X := ⋯
this : IsIso t.arrow
⊢ Arrow.mk ((functor hG (Subobject.mk f) J).map (homOfLE ⋯)) ≅ Arrow.mk f

Tactic:
refine Arrow.isoMk ((Subobject.isoOfEq _ _ (transfiniteIterate_bot _ _) ≪≫
    Subobject.underlyingIso f)) (asIso t.arrow) ?_

Goal state after:
C : Type u
inst✝⁷ : Category.{v, u} C
G : C
inst✝⁶ : Abelian C
hG : IsSeparator G
X : C
inst✝⁵ : IsGrothendieckAbelian.{w, v, u} C
A : C
f : A ⟶ X
inst✝⁴ : Mono f
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
hj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤
t : Subobject X := transfiniteIterate (largerSubobject hG) j (Subobject.mk f)
this : IsIso t.arrow
⊢ ((transfiniteIterate (largerSubobject hG) ⊥ (Subobject.mk f)).isoOfEq (Subobject.mk f) ⋯ ≪≫
          Subobject.underlyingIso f).hom ≫
      (Arrow.mk f).hom =
    (Arrow.mk ((functor hG (Subobject.mk f) J).map (homOfLE ⋯))).hom ≫ (asIso t.arrow).hom
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case cons
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
⊢ padicValRat p (F j) < padicValRat p (∑ i ∈ Finset.cons s S' Hnot, F i)

Tactic:
| cons s S' Hnot Hne Hind =>

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < ↑k
⊢ ∃ k, ↑p ^ (-↑k) < ε

Tactic:
use k

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < ↑k
⊢ ↑p ^ (-↑k) < ε
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
ε : ℚ
hε : 0 < ε
k : ℕ
hk : ↑p ^ (-↑k) < ↑ε
⊢ ∃ k, ↑p ^ (-↑k) < ε

Tactic:
use k

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℚ
hε : 0 < ε
k : ℕ
hk : ↑p ^ (-↑k) < ↑ε
⊢ ↑p ^ (-↑k) < ε
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
⊢ 1 ≤ ‖z‖

Tactic:
have := mul_le_mul_of_nonneg_left (norm_le_one w) (norm_nonneg z)

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ_[p]
h : IsUnit z
w : ℤ_[p]
eq : 1 = z * w
this : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1
⊢ 1 ≤ ‖z‖
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ ↑p ∈ nonunits ℤ_[p]

Tactic:
by
  have : (p : ℝ)⁻¹ < 1 := inv_lt_one_of_one_lt₀ <| mod_cast hp.out.one_lt
  rwa [← norm_p, ← mem_nonunits] at this

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.ElladicCohomology

Goal state before:
pp: X : Scheme
⊢ IsGrothendieckAbelian.{u + 1, u + 1, u + 2} (Sheaf (ProEt.topology X) Ab)

Tactic:
have : EssentiallySmall.{u + 1} X.ProEt := inferInstance
  exact Sheaf.isGrothendieckAbelian_of_essentiallySmall (ProEt.topology X) Ab.{u + 1}

Goal state after:
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: case mp
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
h : c.fst = c.snd
φ : X ⟶ c.pt
hφ₁ : φ ≫ c.fst = 𝟙 X
hφ₂ : φ ≫ c.snd = 𝟙 X
⊢ IsIso c.fst

Tactic:
refine ⟨φ, PullbackCone.IsLimit.hom_ext hc ?_ ?_, hφ₁⟩

Goal state after:
case mp.refine_1
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
h : c.fst = c.snd
φ : X ⟶ c.pt
hφ₁ : φ ≫ c.fst = 𝟙 X
hφ₂ : φ ≫ c.snd = 𝟙 X
⊢ (c.fst ≫ φ) ≫ c.fst = 𝟙 c.pt ≫ c.fst

case mp.refine_2
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
h : c.fst = c.snd
φ : X ⟶ c.pt
hφ₁ : φ ≫ c.fst = 𝟙 X
hφ₂ : φ ≫ c.snd = 𝟙 X
⊢ (c.fst ≫ φ) ≫ c.snd = 𝟙 c.pt ≫ c.snd
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
V : C
g : unop (op V) ⟶ U
this :
  Presheaf.imageSieve f
      ((ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.obj U).map g.op)) (shrinkYonedaObjObjEquiv.symm (𝟙 U))) ∈
    J (unop (op V))
⊢ Presheaf.imageSieve f (shrinkYonedaObjObjEquiv.symm g) ∈ J V

Tactic:
have hg := shrinkYoneda_obj_map_shrinkYonedaObjObjEquiv_symm g.op (𝟙 _)

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
V : C
g : unop (op V) ⟶ U
this :
  Presheaf.imageSieve f
      ((ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.obj U).map g.op)) (shrinkYonedaObjObjEquiv.symm (𝟙 U))) ∈
    J (unop (op V))
hg :
  (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.obj (unop (op U))).map g.op))
      (shrinkYonedaObjObjEquiv.symm (𝟙 (unop (op U)))) =
    shrinkYonedaObjObjEquiv.symm (g.op.unop ≫ 𝟙 (unop (op U)))
⊢ Presheaf.imageSieve f (shrinkYonedaObjObjEquiv.symm g) ∈ J V
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.toPresheafFiber V v (shrinkYoneda.{w, v, u}.obj U))) (shrinkYonedaObjObjEquiv.symm t) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u

Tactic:
rw [← hx₁]
      refine Eq.trans (congr_arg _ ht)
        (Φ.obj.toPresheafFiber_naturality_apply f _ v y).symm

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
F G : Cᵒᵖ ⥤ Type v₁
f : F ⟶ G
hf : (monomorphisms C).presheaf f
⊢ ∀ {X : C} {a b : yoneda.obj X ⟶ F}, a ≫ f = b ≫ f → a = b

Tactic:
intro X a b h

Goal state after:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
F G : Cᵒᵖ ⥤ Type v₁
f : F ⟶ G
hf : (monomorphisms C).presheaf f
X : C
a b : yoneda.obj X ⟶ F
h : a ≫ f = b ≫ f
⊢ a = b
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
F G : Cᵒᵖ ⥤ Type v₁
f : F ⟶ G
hf : (monomorphisms C).presheaf f
X : C
a b : yoneda.obj X ⟶ F
h : a ≫ f = b ≫ f
this : ⋯.lift a (𝟙 X) ⋯ = ⋯.lift b (𝟙 X) ⋯
⊢ a = b

Tactic:
by
    simpa using yoneda.congr_map this =≫ (hf.rep.fst (a ≫ f))

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.Compatibility

Goal state before:
pp: A : Type u_1
A' : Type u_2
B : Type u_3
B' : Type u_4
inst✝³ : Category.{v_1, u_1} A
inst✝² : Category.{v_2, u_2} A'
inst✝¹ : Category.{v_3, u_3} B
inst✝ : Category.{v_4, u_4} B'
eA : A ≌ A'
eB : B ≌ B'
e' : A' ≌ B'
F : A ⥤ B'
hF : eA.functor ⋙ e'.functor ≅ F
G : B ⥤ A
hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor
ε : eA.functor ≅ F ⋙ e'.inverse
hε : υ hF = ε
⊢ (equivalence hF hG).unitIso = equivalenceUnitIso hG ε

Tactic:
ext1; apply NatTrans.ext; ext X
  dsimp [equivalence]
  simp only [assoc, comp_id, equivalenceUnitIso_hom_app, equivalence₂_inverse, Functor.comp_obj,
    id_comp, equivalence₂UnitIso_eq eB hF, equivalence₂UnitIso_hom_app,
    ← eA.inverse.map_comp_assoc, assoc, ← hε, υ_hom_app]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.Projections

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
⊢ P q + Q q = 𝟙 K[X]

Tactic:
by
  rw [Q]
  abel

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'
i : ℕ
hi : i ≥ N
⊢ ‖↑(f - const norm q) i‖ < ε

Tactic:
have h := hN i hi

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
q : ℚ_[p]
hq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε
ε : ℝ
hε : ε > 0
ε' : ℚ
hε' : 0 < ε' ∧ ↑ε' < ε
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'
i : ℕ
hi : i ≥ N
h : padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'
⊢ ‖↑(f - const norm q) i‖ < ε
Module: Mathlib.AlgebraicTopology.DoldKan.Decomposition

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
Z Z' : C
f : MorphComponents X n Z
h : Z ⟶ Z'
⊢ PInfty.f (n + 1) ≫ { a := f.a ≫ h, b := fun i ↦ f.b i ≫ h }.a +
      ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ { a := f.a ≫ h, b := fun i ↦ f.b i ≫ h }.b i.rev =
    (PInfty.f (n + 1) ≫ f.a + ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ f.b i.rev) ≫ h

Tactic:
simp only [add_comp, sum_comp, assoc]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.Faces

Goal state before:
pp: case neg.inr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
ham'' : a = m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.δ j ≫ X.σ (⟨a, ⋯⟩.castLT ⋯)

Tactic:
X.δ_comp_δ_self'_assoc

Goal state after:
case neg.inr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
ham'' : a = m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ j.succ ≫ X.δ j ≫ X.σ (⟨a, ⋯⟩.castLT ⋯)

case neg.inr.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
ham'' : a = m
⊢ ⟨a + 1, ⋯⟩ = j.castSucc
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: case mpr
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
a✝ : Mono A.e
⊢ (unop Δ).len ≤ (unop A.fst).len

Tactic:
exact len_le_of_mono A.e

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Δ' : SimplexCategory
θ : ⦋0⦌ ⟶ Δ'
hθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)
⊢ X.map θ.op ≫ PInfty.f 0 = 0

pp: case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Δ' : SimplexCategory
n✝ : ℕ
θ : ⦋n✝ + 1⦌ ⟶ Δ'
hθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)
⊢ X.map θ.op ≫ PInfty.f (n✝ + 1) = 0

Tactic:
· exfalso
    apply hθ
    intro x y h
    fin_cases x
    fin_cases y
    rfl

Goal state after:
case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Δ' : SimplexCategory
n✝ : ℕ
θ : ⦋n✝ + 1⦌ ⟶ Δ'
hθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)
⊢ X.map θ.op ≫ PInfty.f (n✝ + 1) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ K.d (n + 1) n =
    ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫
      (Γ₀.obj K).map (SimplexCategory.δ 0).op ≫ (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌))

Tactic:
Γ₀.Obj.mapMono_on_summand_id_assoc,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ K.d (n + 1) n =
    Γ₀.Obj.Termwise.mapMono K (SimplexCategory.δ 0) ≫
      ((Γ₀.splitting K).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
        (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌))
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
i : Fin (n + 2)
hi : i ≠ 0
⊢ (-1) ^ ↑i •
      ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫
        (Γ₀.obj K).map (SimplexCategory.δ i).op ≫ (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌)) =
    0

Tactic:
Γ₀.Obj.mapMono_on_summand_id_assoc,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
i : Fin (n + 2)
hi : i ≠ 0
⊢ (-1) ^ ↑i •
      Γ₀.Obj.Termwise.mapMono K (SimplexCategory.δ i) ≫
        ((Γ₀.splitting K).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
          (Γ₀.splitting K).πSummand (Splitting.IndexSet.id (op ⦋n⦌)) =
    0
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (N₂Γ₂ToKaroubiIso.hom.app X).f.f n = PInfty.f n

Tactic:
dsimp [N₂Γ₂ToKaroubiIso]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (((PInfty.f n ≫
            (Γ₀.splitting X).desc (op ⦋n⦌) fun A ↦ 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op ⦋n⦌)).inj A) ≫
          PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op ⦋n⦌))) ≫
        PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op ⦋n⦌))) ≫
      PInfty.f n =
    PInfty.f n
Module: Mathlib.AlgebraicTopology.DoldKan.GammaCompN

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (N₂Γ₂ToKaroubiIso.inv.app X).f.f n = PInfty.f n

Tactic:
dsimp [N₂Γ₂ToKaroubiIso]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (PInfty.f n ≫
      (PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op ⦋n⦌))) ≫
        (PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op ⦋n⦌))) ≫
          PInfty.f n ≫
            (Γ₀.splitting X).desc (op ⦋n⦌) fun A ↦ 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op ⦋n⦌)).inj A) =
    PInfty.f n
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
G : C₁ ⥤ C₂
F : C₂ ⥤ C₁
inst✝¹ : F.Full
inst✝ : F.Faithful
adj : G ⊣ F
W : MorphismProperty C₁
hW : W.IsInvertedBy G
hW' : W.functorCategory C₁ adj.unit
⊢ G.IsLocalization W

Tactic:
by
  let Φ : W.Localization ⥤ C₂ := Localization.lift _ hW W.Q
  let e : W.Q ⋙ Φ ≅ G := by apply Localization.fac
  have : IsIso (Functor.whiskerRight adj.unit W.Q) := by
    rw [NatTrans.isIso_iff_isIso_app]
    intro X
    exact Localization.inverts W.Q W _ (hW' X)
  exact Functor.IsLocalization.of_equivalence_target W.Q W _
    (Equivalence.mk Φ (F ⋙ W.Q)
      (Localization.liftNatIso W.Q W W.Q (G ⋙ F ⋙ W.Q) _ _
        (W.Q.leftUnitor.symm ≪≫ asIso (Functor.whiskerRight adj.unit W.Q) ≪≫
        Functor.associator _ _ _))
      (Functor.associator _ _ _ ≪≫ Functor.isoWhiskerLeft _ e ≪≫ asIso adj.counit)) e

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
Hcof : Type u_1 := (weakEquivalences (HoCat C)).Localization
Lcofπ : HoCat C ⥤ Hcof := (weakEquivalences (HoCat C)).Q
Lcof : CofibrantObject C ⥤ Hcof := toHoCat ⋙ Lcofπ
H : Type u_1 := (weakEquivalences C).Localization
L : C ⥤ H := (weakEquivalences C).Q
F : Hcof ⥤ H := (localizerMorphism C).localizedFunctor Lcof L
eF : ι ⋙ L ≅ Lcof ⋙ F := CatCommSq.iso (localizerMorphism C).functor Lcof L F
eF' : HoCat.toLocalization L ≅ Lcofπ ⋙ F :=
  Quotient.natIsoLift (homRel C) (HoCat.toHoCatCompToLocalizationIso L ≪≫ eF ≪≫ toHoCat.associator Lcofπ F)
G : H ⥤ Hcof := (HoCat.localizerMorphismResolution C).localizedFunctor L Lcofπ
eG : HoCat.resolution ⋙ Lcofπ ≅ L ⋙ G := CatCommSq.iso (HoCat.localizerMorphismResolution C).functor L Lcofπ G
this✝ : Localization.Lifting L (weakEquivalences C) (HoCat.resolution ⋙ HoCat.toLocalization L) (G ⋙ F)
this : Localization.Lifting Lcof (weakEquivalences (CofibrantObject C)) (ι ⋙ HoCat.resolution ⋙ Lcofπ) (F ⋙ G)
⊢ (localizerMorphism C).IsLocalizedEquivalence

Tactic:
let E : Hcof ≌ H := CategoryTheory.Equivalence.mk F G
    (Localization.liftNatIso Lcof (weakEquivalences _) Lcof (ι ⋙ HoCat.resolution ⋙ Lcofπ) _ _
      ((asIso (whiskerRight HoCat.ιCompResolutionNatTrans Lcofπ)).symm ≪≫
          associator _ _ _))
    (Localization.liftNatIso L (weakEquivalences _)
      (HoCat.resolution ⋙ HoCat.toLocalization L) L _ _
      (asIso (HoCat.resolutionCompToLocalizationNatTrans L)))

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
Hcof : Type u_1 := (weakEquivalences (HoCat C)).Localization
Lcofπ : HoCat C ⥤ Hcof := (weakEquivalences (HoCat C)).Q
Lcof : CofibrantObject C ⥤ Hcof := toHoCat ⋙ Lcofπ
H : Type u_1 := (weakEquivalences C).Localization
L : C ⥤ H := (weakEquivalences C).Q
F : Hcof ⥤ H := (localizerMorphism C).localizedFunctor Lcof L
eF : ι ⋙ L ≅ Lcof ⋙ F := CatCommSq.iso (localizerMorphism C).functor Lcof L F
eF' : HoCat.toLocalization L ≅ Lcofπ ⋙ F :=
  Quotient.natIsoLift (homRel C) (HoCat.toHoCatCompToLocalizationIso L ≪≫ eF ≪≫ toHoCat.associator Lcofπ F)
G : H ⥤ Hcof := (HoCat.localizerMorphismResolution C).localizedFunctor L Lcofπ
eG : HoCat.resolution ⋙ Lcofπ ≅ L ⋙ G := CatCommSq.iso (HoCat.localizerMorphismResolution C).functor L Lcofπ G
this✝ : Localization.Lifting L (weakEquivalences C) (HoCat.resolution ⋙ HoCat.toLocalization L) (G ⋙ F)
this : Localization.Lifting Lcof (weakEquivalences (CofibrantObject C)) (ι ⋙ HoCat.resolution ⋙ Lcofπ) (F ⋙ G)
E : Hcof ≌ H :=
  CategoryTheory.Equivalence.mk F G
    (Localization.liftNatIso Lcof (weakEquivalences (CofibrantObject C)) Lcof (ι ⋙ HoCat.resolution ⋙ Lcofπ) (𝟭 Hcof)
      (F ⋙ G) ((asIso (whiskerRight HoCat.ιCompResolutionNatTrans Lcofπ)).symm ≪≫ ι.associator HoCat.resolution Lcofπ))
    (Localization.liftNatIso L (weakEquivalences C) (HoCat.resolution ⋙ HoCat.toLocalization L) L (G ⋙ F) (𝟭 H)
      (asIso (HoCat.resolutionCompToLocalizationNatTrans L)))
⊢ (localizerMorphism C).IsLocalizedEquivalence
Module: Mathlib.CategoryTheory.Limits.Set

Goal state before:
pp: case hinj
J : Type w
inst✝¹ : Category.{w', w} J
X : Type u
inst✝ : IsFilteredOrEmpty J
F : J ⥤ Set X
i j : J
x : X
hx : x ∈ F.obj i
hy : x ∈ F.obj j
h :
  (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app i)) ⟨x, hx⟩ =
    (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app j)) ⟨x, hy⟩
⊢ ∃ k f g,
    (ConcreteCategory.hom ((F ⋙ functorToTypes).map f)) ⟨x, hx⟩ =
      (ConcreteCategory.hom ((F ⋙ functorToTypes).map g)) ⟨x, hy⟩

Tactic:
exact ⟨IsFiltered.max i j, IsFiltered.leftToMax i j, IsFiltered.rightToMax i j, rfl⟩

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h : IsPushout t l r b
x : (span t l).obj none
⊢ (ConcreteCategory.hom r) ((ConcreteCategory.hom t) x) = (ConcreteCategory.hom (h.cocone.ι.app none)) x

Tactic:
by simp; rfl

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
⊢ t i d ≫ r i d = l i d ≫ b i d

Tactic:
by
  ext c : 1
  simp [← cancel_mono (Subcomplex.ι _)]

Goal state after:
Module: Mathlib.CategoryTheory.Category.ReflQuiv

Goal state before:
pp: V : Type u_1
inst✝ : ReflQuiver V
x y : V
e : x ⟶ y
⊢ morphismPropertyHomMk V (homMk e)

Tactic:
by
  dsimp only [morphismPropertyHomMk]
  rw [MorphismProperty.ofHoms_iff]
  exact ⟨⟨x, y, e⟩, rfl⟩

Goal state after:
Module: Mathlib.CategoryTheory.Category.ReflQuiv

Goal state before:
pp: V : Type u_1
inst✝¹ : ReflQuiver V
D : Type u_2
inst✝ : Category.{v_1, u_2} D
F : V ⥤rq D
⊢ ∀ (x y : Paths V) (f₁ f₂ : x ⟶ y),
    FreeReflRel V x y f₁ f₂ → (Paths.lift F.toPrefunctor).map f₁ = (Paths.lift F.toPrefunctor).map f₂

Tactic:
rintro _ _ _ _ ⟨h⟩
    simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat

Goal state before:
pp: V : Truncated 2
x₀ x₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e : Edge x₀ x₁
y₀ y₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e' : Edge y₀ y₁
h : e.edge = e'.edge
⊢ (ConcreteCategory.hom (V.map (δ₂ 1 Edge._proof_1 Edge._proof_3).op)) e.edge = y₀

Tactic:
← e'.src_eq,

Goal state after:
V : Truncated 2
x₀ x₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e : Edge x₀ x₁
y₀ y₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e' : Edge y₀ y₁
h : e.edge = e'.edge
⊢ (ConcreteCategory.hom (V.map (δ₂ 1 Edge._proof_1 Edge._proof_3).op)) e.edge =
    (ConcreteCategory.hom (V.map (δ₂ 1 Edge._proof_1 Edge._proof_3).op)) e'.edge
Module: Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal

Goal state before:
pp: X : SSet
sx : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
⊢ (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) lift._proof_1).hom ≫ (inclusion 2).op.map (Hom.tr (⦋1⦌.const ⦋0⦌ 0) ⋯ ⋯).op =
    (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) lift._proof_3).hom

Tactic:
by simp; rfl

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat

Goal state before:
pp: V : Truncated 2
x y : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })
e : Edge x y
⊢ morphismPropertyHomMk V (homMk e)

Tactic:
by
  dsimp only [morphismPropertyHomMk]
  rw [MorphismProperty.ofHoms_iff]
  exact ⟨⟨x, y, e⟩, rfl⟩

Goal state after: