mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteIteration

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
i : J
hi : Order.IsSuccLimit i
e : ⋯.functor ⋙ Φ.iterationFunctor J ≅ restrictionLT (Φ.iter i).F ⋯ := ⋯
k : J
hk : k ∈ Set.Iio i
⊢ Arrow.mk ((coconeOfLE (Φ.iter i).F ⋯).ι.app ⟨k, hk⟩ ≫ (eqToIso ⋯).hom) =
    Arrow.mk (((Cocone.precompose e.inv).obj ((Set.principalSegIio i).cocone (Φ.iterationFunctor J))).ι.app ⟨k, hk⟩)

Tactic:
simp [Φ.arrowMk_iterationFunctor_map k i hk.le (Φ.iter i) (by simp), e]

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
W✝ : MorphismProperty C
α : Type u_1
W : α → MorphismProperty C
inst✝¹ : Small.{w, u_1} α
inst✝ : ∀ (a : α), IsSmall.{w, v, u} (W a)
⊢ Function.Surjective fun x ↦
    match x with
    | ⟨i, f⟩ => ⟨↑f, ⋯⟩

Tactic:
rintro ⟨f, hf⟩

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
W✝ : MorphismProperty C
α : Type u_1
W : α → MorphismProperty C
inst✝¹ : Small.{w, u_1} α
inst✝ : ∀ (a : α), IsSmall.{w, v, u} (W a)
f : Arrow C
hf : f ∈ ⋃ i, (W i).toSet
⊢ ∃ a,
    (fun x ↦
          match x with
          | ⟨i, f⟩ => ⟨↑f, ⋯⟩)
        a =
      ⟨f, hf⟩
Module: Mathlib.AlgebraicTopology.RelativeCellComplex.Basic

Goal state before:
pp: case succ
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type w'
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
α : J → Type t
A B : (j : J) → α j → C
basicCell : (j : J) → (i : α j) → A j i ⟶ B j i
X Y : C
f : X ⟶ Y
c : RelativeCellComplex basicCell f
Z : C
φ₁ φ₂ : Y ⟶ Z
h₀ : f ≫ φ₁ = f ≫ φ₂
h : ∀ (γ : c.Cells), γ.ι ≫ φ₁ = γ.ι ≫ φ₂
j : J
hj : ¬IsMax j
hj' : c.incl.app j ≫ φ₁ = c.incl.app j ≫ φ₂
⊢ c.incl.app (Order.succ j) ≫ φ₁ = c.incl.app (Order.succ j) ≫ φ₂

Tactic:
apply (c.attachCells j hj).hom_ext
    · simpa using hj'
    · intro i
      simpa only [Category.assoc, Cells.ι] using h ({ hj := hj, k := i, .. })

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.WellOrderInductionData

Goal state before:
pp: J : Type u
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
F : Jᵒᵖ ⥤ Type v
d : F.WellOrderInductionData
inst✝¹ : OrderBot J
val₀ : F.obj (op ⊥)
inst✝ : WellFoundedLT J
j : J
hj : Order.IsSuccLimit j
e : (i : J) → i < j → d.Extension val₀ i
j✝ j'✝ : { x // x ∈ Set.Iio j }ᵒᵖ
f : j✝ ⟶ j'✝
⊢ (ConcreteCategory.hom (F.map (⋯.functor.map f.unop).op)) (e ↑j✝.1 ⋯).val = (e ↑j'✝.1 ⋯).val

Tactic:
apply compatibility

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.WellOrderInductionData

Goal state before:
pp: case e_x
J : Type u
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
F : Jᵒᵖ ⥤ Type v
d : F.WellOrderInductionData
inst✝¹ : OrderBot J
val₀ : F.obj (op ⊥)
inst✝ : WellFoundedLT J
j : J
hj : Order.IsSuccLimit j
e : (i : J) → i < j → d.Extension val₀ i
i : J
hi : i < j
⊢ (ConcreteCategory.hom (F.map (homOfLE ⋯).op)) (e (Order.succ i) ⋯).val =
    ↑⟨fun x ↦
          match x with
          | op ⟨i, hi⟩ => (e i hi).val,
          ⋯⟩
      (op ⟨i, hi⟩)

Tactic:
apply compatibility

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.WellOrderInductionData

Goal state before:
pp: case e_x
J : Type u
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
F : Jᵒᵖ ⥤ Type v
d : F.WellOrderInductionData
inst✝¹ : OrderBot J
val₀ : F.obj (op ⊥)
inst✝ : WellFoundedLT J
j : J
hj : Order.IsSuccLimit j
e : (i : J) → i < j → d.Extension val₀ i
i : J
hi : Order.IsSuccLimit i
hij : i ≤ j
hij' : i < j
this : (e i hij').val = d.lift i hi ⟨fun x ↦ (ConcreteCategory.hom (F.map (homOfLE ⋯).op)) (e i hij').val, ⋯⟩
l : J
hl : l ∈ Set.Iio i
⊢ (ConcreteCategory.hom (F.map (homOfLE ⋯).op)) (e i hij').val = ↑⟨fun x ↦ (e ↑x.1 ⋯).val, ⋯⟩ (op ⟨l, ⋯⟩)

Tactic:
apply compatibility

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
k : ℕ
⊢ padicValRat p (q ^ k) = ↑k * padicValRat p q

Tactic:
obtain rfl | hq := eq_or_ne q 0
  · cases k <;> simp
  induction k <;>
    simp [*, padicValRat.mul hq (pow_ne_zero _ hq), _root_.pow_succ', add_mul, add_comm]

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

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
z : ℤ
hz : ¬↑z = 0
⊢ ↑(p ^ n) ∣ z ↔ ↑p ^ (-padicValRat p ↑z) ≤ ↑p ^ (-↑n)

Tactic:
zpow_le_zpow_iff_right₀,

Goal state after:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
z : ℤ
hz : ¬↑z = 0
⊢ ↑(p ^ n) ∣ z ↔ -padicValRat p ↑z ≤ -↑n

case neg
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
z : ℤ
hz : ¬↑z = 0
⊢ 1 < ↑p
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : p ≠ 1
a n : ℕ
ha : a ≠ 0
⊢ p ^ n ∣ a ↔ n ≤ padicValNat p a

Tactic:
by
  rw [pow_dvd_iff_le_emultiplicity, ← padicValNat_eq_emultiplicity_of_ne_one hp ha, Nat.cast_le]

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: n : ℕ
hn : n ≠ 0
⊢ ¬log 2 n = padicValNat 2 (n + 1)

Tactic:
log_eq_iff (by simp [hn])

Goal state after:
n : ℕ
hn : n ≠ 0
⊢ ¬(2 ^ padicValNat 2 (n + 1) ≤ n ∧ n < 2 ^ (padicValNat 2 (n + 1) + 1))
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ ↑p ^ (-↑x.valuation) ≤ ↑p ^ (-↑n) ↔ n ≤ x.valuation

Tactic:
zpow_le_zpow_iff_right₀,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ -↑x.valuation ≤ -↑n ↔ n ≤ x.valuation

p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
n : ℕ
⊢ 1 < ↑p
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
⊢ Mono f ↔ c.fst = c.snd

Tactic:
by
  constructor
  · intro hf
    simpa only [← cancel_mono f] using c.condition
  · intro hf
    constructor
    intro Z g g' h
    obtain ⟨φ, rfl, rfl⟩ := PullbackCone.IsLimit.lift' hc g g' h
    rw [hf]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: case inr
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 F₂ : Cᵒᵖ ⥤ Type w
f : F₁ ⟶ F₂
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
this :
  ∀ {F₁ F₂ : Cᵒᵖ ⥤ Type w} (f : F₁ ⟶ F₂) [Mono f],
    (∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))) →
      (∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U) → Presheaf.IsLocallySurjective J f
hF₂ : ¬∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U
U : C
s : ToType (F₂.obj (op U))
f' : pullback f (shrinkYonedaEquiv.symm s) ⟶ shrinkYoneda.{w, v, u}.obj U := pullback.snd f (shrinkYonedaEquiv.symm s)
⊢ Presheaf.imageSieve f s ∈ J U

Tactic:
have hf' (Φ : P.FullSubcategory) :
        Function.Surjective (Φ.obj.presheafFiber.map f') := by
      replace hf := hf Φ
      rw [← CategoryTheory.epi_iff_surjective] at hf ⊢
      exact (MorphismProperty.epimorphisms _).of_isPullback
        ((IsPullback.of_hasPullback f (shrinkYonedaEquiv.{w}.symm s)).map
          Φ.obj.presheafFiber) (.infer_property _)

Goal state after:
case inr
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 F₂ : Cᵒᵖ ⥤ Type w
f : F₁ ⟶ F₂
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
this :
  ∀ {F₁ F₂ : Cᵒᵖ ⥤ Type w} (f : F₁ ⟶ F₂) [Mono f],
    (∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))) →
      (∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U) → Presheaf.IsLocallySurjective J f
hF₂ : ¬∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U
U : C
s : ToType (F₂.obj (op U))
f' : pullback f (shrinkYonedaEquiv.symm s) ⟶ shrinkYoneda.{w, v, u}.obj U := pullback.snd f (shrinkYonedaEquiv.symm s)
hf' : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f'))
⊢ Presheaf.imageSieve f s ∈ J U
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfgne : f.norm ≠ g.norm
⊢ (f + g).norm = max f.norm g.norm

Tactic:
have hfg : ¬f + g ≈ 0 := mt norm_eq_of_add_equiv_zero hfgne

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfgne : f.norm ≠ g.norm
hfg : ¬f + g ≈ 0
⊢ (f + g).norm = max f.norm g.norm
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case hbc.h₁.hbc.h₁
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
⊢ j ≥ N

Tactic:
exact le_of_max_le_left hj

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
h : ¬↑(padicNormE q) = ↑(padicNormE r)
⊢ ↑(padicNormE (q + r)) = max ↑(padicNormE q) ↑(padicNormE r)

Tactic:
have : padicNormE q ≠ padicNormE r := mod_cast h

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ_[p]
h : ¬↑(padicNormE q) = ↑(padicNormE r)
this : padicNormE q ≠ padicNormE r
⊢ ↑(padicNormE (q + r)) = max ↑(padicNormE q) ↑(padicNormE r)
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case mpr
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ
⊢ ‖↑p * ↑x‖ < 1

Tactic:
rw [padicNormE.mul]

Goal state after:
case mpr
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ
⊢ ‖↑p‖ * ‖↑x‖ < 1
Module: Mathlib.AlgebraicTopology.DoldKan.PInfty

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hqn : n ≤ q
⊢ (P (q + 1)).f n = (P q).f n

Tactic:
cases n with
  | zero => simp only [P_f_0_eq]
  | succ n =>
    simp only [P_succ, comp_add, comp_id, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
      add_eq_left]
    exact (HigherFacesVanish.of_P q n).comp_Hσ_eq_zero (Nat.succ_le_iff.mp hqn)

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hqn : n < q
⊢ φ ≫
      ((AlternatingFaceMapComplex.obj X).d (n + 1) n ≫ 0 +
        hσ' q (n + 1) (n + 2) ⋯ ≫ (AlternatingFaceMapComplex.obj X).d (n + 2) (n + 1)) =
    0

Tactic:
comp_zero,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hqn : n < q
⊢ φ ≫ (0 + hσ' q (n + 1) (n + 2) ⋯ ≫ (AlternatingFaceMapComplex.obj X).d (n + 2) (n + 1)) = 0
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ⟦f⟧ ≠ 0
⊢ ↑(↑p ^ (-PadicSeq.valuation f)) = ↑p ^ (-PadicSeq.valuation f)

pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ⟦f⟧ ≠ 0
⊢ ¬f ≈ 0

Tactic:
· rw [Rat.cast_zpow, Rat.cast_natCast]

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ⟦f⟧ ≠ 0
⊢ ¬f ≈ 0
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
n : ℕ
⊢ (x ^ (n + 1)).valuation = ↑(n + 1) * x.valuation

Tactic:
obtain rfl | hx := eq_or_ne x 0
    · simp
    · simp [pow_succ, hx, valuation_mul, valuation_pow, _root_.add_one_mul]

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

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬x = 0
⊢ ↑p ^ (-x.valuation) ≤ ↑p ^ 0 ↔ 0 ≤ x.valuation

Tactic:
zpow_le_zpow_iff_right₀,

Goal state after:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬x = 0
⊢ -x.valuation ≤ 0 ↔ 0 ≤ x.valuation

case neg
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : ¬x = 0
⊢ 1 < ↑p
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: case neg.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ (P ↑j).f (n + 1) ≫ X.δ j.rev.succ ≫ X.σ ⟨↑i, ⋯⟩ ≫ 0 = 0

pp: case neg.succ.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ ⟨↑i, ⋯⟩ ≤ j.rev

Tactic:
comp_zero,

Goal state after:
case neg.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ (P ↑j).f (n + 1) ≫ X.δ j.rev.succ ≫ 0 = 0

case neg.succ.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ ⟨↑i, ⋯⟩ ≤ j.rev
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: case neg.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ (P ↑j).f (n + 1) ≫ X.δ j.rev.succ ≫ 0 = 0

pp: case neg.succ.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ ⟨↑i, ⋯⟩ ≤ j.rev

Tactic:
comp_zero,

Goal state after:
case neg.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ (P ↑j).f (n + 1) ≫ 0 = 0

case neg.succ.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0
i : Fin (n + 1 + 1)
h : ¬n + 1 + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)
j : Fin (n + 1)
hj : True ∧ ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = ⟨↑i, ⋯⟩.castSucc
eq : X.σ j.rev.succ ≫ (P q).f (n + 1 + 1) = 0
⊢ ⟨↑i, ⋯⟩ ≤ j.rev
Module: Mathlib.AlgebraicTopology.DoldKan.Normalized

Goal state before:
pp: case succ
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
n : ℕ
⊢ (NormalizedMooreComplex.objX X (n + 1)).Factors (PInfty.f (n + 1))

Tactic:
PInfty_f,

Goal state after:
case succ
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
n : ℕ
⊢ (NormalizedMooreComplex.objX X (n + 1)).Factors ((P (n + 1)).f (n + 1))
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
i : Fin (n + 1)
⊢ X.σ i ≫ PInfty.f (n + 1) = 0

Tactic:
PInfty_f,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
i : Fin (n + 1)
⊢ X.σ i ≫ (P (n + 1)).f (n + 1) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : HasZeroMorphisms C
Δ : SimplexCategoryᵒᵖ
A B : IndexSet Δ
⊢ s.N (unop B.fst).len ⟶ s.N (unop A.fst).len

Tactic:
by_cases h : B = A

Goal state after:
case pos
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : HasZeroMorphisms C
Δ : SimplexCategoryᵒᵖ
A B : IndexSet Δ
h : B = A
⊢ s.N (unop B.fst).len ⟶ s.N (unop A.fst).len

case neg
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : HasZeroMorphisms C
Δ : SimplexCategoryᵒᵖ
A B : IndexSet Δ
h : ¬B = A
⊢ s.N (unop B.fst).len ⟶ s.N (unop A.fst).len
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
n : ℕ
i : Fin (n + 1)
A : IndexSet (op ⦋n⦌)
⊢ (s.cofan (op ⦋n⦌)).inj A ≫ X.map (SimplexCategory.σ i).op ≫ s.πSummand (IndexSet.id (op ⦋n + 1⦌)) =
    (s.cofan (op ⦋n⦌)).inj A ≫ 0

Tactic:
comp_zero,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
n : ℕ
i : Fin (n + 1)
A : IndexSet (op ⦋n⦌)
⊢ (s.cofan (op ⦋n⦌)).inj A ≫ X.map (SimplexCategory.σ i).op ≫ s.πSummand (IndexSet.id (op ⦋n + 1⦌)) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: case neg
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ PInfty.f n = PInfty.f n' ≫ X.map i.op

Tactic:
Γ₀.Obj.Termwise.mapMono_eq_zero _ i _ hi,

Goal state after:
case neg
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ 0 ≫ PInfty.f n = PInfty.f n' ≫ X.map i.op

C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ ⦋n'⦌ ≠ ⦋n⦌
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X = (Γ₂N₂ToKaroubiIso.app X).inv ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X)

Tactic:
rw [Γ₂N₂.natTrans_app_f_app]
  dsimp only [Karoubi.decompId_i_toKaroubi, Karoubi.decompId_p_toKaroubi, Functor.comp_map,
    NatTrans.comp_app]
  rw [N₂.map_id, Γ₂.map_id, Iso.app_inv]
  dsimp only [toKaroubi]
  erw [id_comp]
  rw [comp_id, Iso.inv_hom_id_app_assoc]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
      PInfty.f n ≫ (Γ₂N₂.natTrans.app P).f.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)

Tactic:
PInfty_on_Γ₀_splitting_summand_eq_self_assoc,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌))
⊢ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
      (Γ₂N₂.natTrans.app P).f.app (op ⦋n⦌) =
    PInfty.f n ≫ P.p.app (op ⦋n⦌)
Module: Mathlib.CategoryTheory.SingleObj

Goal state before:
pp: M : Type u
N : Type v
inst✝¹ : Monoid M
inst✝ : Monoid N
e : M ≃* N
⊢ 𝟭 (SingleObj M) = e.toMonoidHom.toFunctor ⋙ e.symm.toMonoidHom.toFunctor

Tactic:
rw [← MonoidHom.comp_toFunctor, ← MonoidHom.id_toFunctor]
    congr 1
    simp

Goal state after:
Module: Mathlib.CategoryTheory.SingleObj

Goal state before:
pp: M : Type u
N : Type v
inst✝¹ : Monoid M
inst✝ : Monoid N
e : M ≃* N
⊢ e.symm.toMonoidHom.toFunctor ⋙ e.toMonoidHom.toFunctor = 𝟭 (SingleObj N)

Tactic:
rw [← MonoidHom.comp_toFunctor, ← MonoidHom.id_toFunctor]
    congr 1
    simp

Goal state after:
Module: Mathlib.Topology.Homotopy.Contractible

Goal state before:
pp: case mpr.refine_2
Y : Type u_1
inst✝ : TopologicalSpace Y
p : Y
h : (ContinuousMap.id Y).Homotopic (const Y p)
⊢ ((const Y ()).comp (const Unit p)).Homotopic (ContinuousMap.id Unit)

Tactic:
· convert! Homotopic.refl (ContinuousMap.id Unit)

Goal state after:
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X✝
inst✝² : TopologicalSpace Y
X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : SimplyConnectedSpace X
x y : X
⊢ Nonempty (Unique (Path.Homotopic.Quotient x y))

Tactic:
rw [simply_connected_iff_unique_homotopic] at *; tauto

Goal state after:
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic

Goal state before:
pp: case pos.right
x : ↑I × ↑I
h✝ : ↑x.2 ≤ 1 / 2
⊢ ↑x.1 * 2 * ↑x.2 ≤ 1

Tactic:
rw [mul_assoc]

Goal state after:
case pos.right
x : ↑I × ↑I
h✝ : ↑x.2 ≤ 1 / 2
⊢ ↑x.1 * (2 * ↑x.2) ≤ 1