mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝³ : Category.{v₀, u₀} C₀
inst✝² : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : IsDenseSubsite J₀ J F
X✝ : C
data✝ : F.OneHypercoverDenseData J₀ J X✝
X : C
data : F.OneHypercoverDenseData J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
⊢ ∀ {Y Z : C₀} {f_1 : Y ⟶ X₀},
    Nonempty (data.SieveStruct f f_1) → ∀ (g : Z ⟶ Y), Nonempty (data.SieveStruct f (g ≫ f_1))

Tactic:
rintro Y₀ Z₀ g ⟨h⟩ p
    exact ⟨{ i₀ := h.i₀, q := F.map p ≫ h.q, fac := by rw [assoc, h.fac, map_comp_assoc]}⟩

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

Goal state before:
pp: S : Scheme
X Y : Scheme
f : X ⟶ Y
hf : IsIso f
⊢ qcCoverFamily.property (PreZeroHypercover.singleton f)

Tactic:
qcCoverFamily_property,

Goal state after:
S : Scheme
X Y : Scheme
f : X ⟶ Y
hf : IsIso f
⊢ Y.quasiCompactCover (PreZeroHypercover.singleton f)
Module: Mathlib.AlgebraicGeometry.Sites.QuasiCompact

Goal state before:
pp: S✝ : Scheme
P : MorphismProperty Scheme
S : Scheme
𝒰 : Cover (precoverage P) S
inst✝ : QuasiCompactCover 𝒰.toPreZeroHypercover
⊢ (𝒰.ulift.sum (QuasiCompactCover.ulift 𝒰.toPreZeroHypercover)).presieve₀ ∈ (precoverage P).coverings S

Tactic:
by
    rw [presieve₀_mem_precoverage_iff]
    refine ⟨fun x ↦ ⟨.inl x, 𝒰.covers _⟩, fun i ↦ ?_⟩
    induction i <;> exact 𝒰.map_prop _

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
f : X ⟶ Y
i₁ i₂ : (data X).I₀
j : (data X).I₁ (i₁, i₂).1 (i₁, i₂).2
⊢ restriction data G₀ ((data X).f ((data X).multicospanShape.fst ⟨(i₁, i₂), j⟩) ≫ f) ≫
      ((data X).multicospanIndex G₀.obj).fst ⟨(i₁, i₂), j⟩ =
    restriction data G₀ ((data X).f ((data X).multicospanShape.snd ⟨(i₁, i₂), j⟩) ≫ f) ≫
      ((data X).multicospanIndex G₀.obj).snd ⟨(i₁, i₂), j⟩

Tactic:
obtain ⟨a, h₁, h₂⟩ : ∃ a, a = F.map ((data X).p₁ j) ≫ (data X).f i₁ ≫ f ∧
        a = F.map ((data X).p₂ j) ≫ (data X).f i₂ ≫ f := ⟨_, rfl, (data X).w_assoc j _⟩

Goal state after:
C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
f : X ⟶ Y
i₁ i₂ : (data X).I₀
j : (data X).I₁ (i₁, i₂).1 (i₁, i₂).2
a : F.obj ((data X).Y j) ⟶ Y
h₁ : a = F.map ((data X).p₁ j) ≫ (data X).f i₁ ≫ f
h₂ : a = F.map ((data X).p₂ j) ≫ (data X).f i₂ ≫ f
⊢ restriction data G₀ ((data X).f ((data X).multicospanShape.fst ⟨(i₁, i₂), j⟩) ≫ f) ≫
      ((data X).multicospanIndex G₀.obj).fst ⟨(i₁, i₂), j⟩ =
    restriction data G₀ ((data X).f ((data X).multicospanShape.snd ⟨(i₁, i₂), j⟩) ≫ f) ≫
      ((data X).multicospanIndex G₀.obj).snd ⟨(i₁, i₂), j⟩
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ (restriction data G₀ ((data X).f (Sieve.ofArrows.i hd) ≫ g) ≫
        IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ Sieve.ofArrows.h hd)) ≫
      G₀.obj.map a.op =
    (presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ IsDenseSubsite.mapPreimage J F G₀ (Sieve.ofArrows.h hc)) ≫
      G₀.obj.map a.op

Tactic:
rw [assoc, assoc,
    IsDenseSubsite.mapPreimage_map_of_fac J F G₀ _ _ x₂ (by simpa using fac₂.symm),
    IsDenseSubsite.mapPreimage_map_of_fac J F G₀ _ _ x₁ fac₁.symm]

Goal state after:
C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ restriction data G₀ ((data X).f (Sieve.ofArrows.i hd) ≫ g) ≫ G₀.obj.map x₂.op =
    presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ G₀.obj.map x₁.op
Module: Mathlib.CategoryTheory.Sites.SubcanonicalOver

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X : C
⊢ (J.over X).Subcanonical

Tactic:
refine .of_isSheaf_yoneda_obj _ fun E Z R hR t ht ↦ ?_

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X : C
E Z : Over X
R : Sieve Z
hR : R ∈ (J.over X) Z
t : Presieve.FamilyOfElements (CategoryTheory.yoneda.obj E) R.arrows
ht : t.Compatible
⊢ ∃! t_1, t.IsAmalgamation t_1
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : IsGrothendieckAbelian.{w, v, u} C
X : C
J : Type w
inst✝¹ : SmallCategory J
Y : J ⥤ C
c : Cocone Y
hc : IsColimit c
z : X ⟶ c.pt
inst✝ : IsFiltered J
⊢ Epi (f z)

Tactic:
have := isIso_f hc z
  infer_instance

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
p : X ⟶ Y
⊢ HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C

Tactic:
have := locallySmall I κ
  have := isSmall I κ
  have := hasCoproducts I κ
  exact hasColimitsOfShape_of_equivalence
    (Discrete.equivalence (equivShrink.{w} _)).symm

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
⊢ I.rlp.llp (ιObj I κ f)

Tactic:
apply I.transfiniteCompositionsOfShape_pushouts_coproducts_le_llp_rlp κ.ord.ToType
  apply transfiniteCompositionsOfShape_ιObj

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
l : (↑⟨Arrow.mk i, hi⟩).right ⟶ (relativeCellComplexιObj I κ f).F.obj (Order.succ j)
hl₁ : I.homFamily ⟨Arrow.mk i, hi⟩ ≫ l = t ≫ (relativeCellComplexιObj I κ f).F.map (homOfLE ⋯)
hl₂ : l ≫ (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫ πObj I κ f = b
⊢ sq.HasLift

Tactic:
dsimp at hl₁

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
l : (↑⟨Arrow.mk i, hi⟩).right ⟶ (relativeCellComplexιObj I κ f).F.obj (Order.succ j)
hl₁ : i ≫ l = t ≫ (relativeCellComplexιObj I κ f).F.map (homOfLE ⋯)
hl₂ : l ≫ (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫ πObj I κ f = b
⊢ sq.HasLift
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: case e'_5
C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
k : ↑(Set.Iic j)
hk : ¬IsMax k
⊢ transfiniteIterate (largerSubobject hG) (↑(Order.succ k)) A₀ =
    largerSubobject hG (transfiniteIterate (largerSubobject hG) (↑k) A₀)

pp: case e'_6
C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
j : J
k : ↑(Set.Iic j)
hk : ¬IsMax k
e_5✝ :
  Subobject.underlying.obj (transfiniteIterate (largerSubobject hG) (↑(Order.succ k)) A₀) =
    Subobject.underlying.obj (largerSubobject hG (transfiniteIterate (largerSubobject hG) (↑k) A₀))
⊢ transfiniteIterate (largerSubobject hG) (↑(Order.succ k)) A₀ =
    largerSubobject hG (transfiniteIterate (largerSubobject hG) (↑k) A₀)

Tactic:
all_goals
      rw [Set.Iic.succ_eq_of_not_isMax hk,
        transfiniteIterate_succ _ _ _ (Set.not_isMax_coe _ hk)]

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

Goal state before:
pp: p : ℕ
hp : 1 < p
⊢ (↑p)⁻¹ < 1

Tactic:
inv_lt_one_iff₀

Goal state after:
p : ℕ
hp : 1 < p
⊢ ↑p ≤ 0 ∨ 1 < ↑p
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x✝ : ℚ
hx : x✝ = 0
⊢ padicNorm p x✝ = 0

Tactic:
by simp [hx]

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hval : padicValRat p q ≠ padicValRat p r
h1 : min (padicValRat p q) (padicValRat p r) ≤ padicValRat p (q + r)
h2 : min (padicValRat p (q + r)) (padicValRat p (-r)) ≤ padicValRat p (q + r + -r)
⊢ padicValRat p (q + r) = min (padicValRat p q) (padicValRat p r)

Tactic:
have h3 := min_le_padicValRat_add (p := p) (ne_of_eq_of_ne (add_neg_cancel_right r q) hr)

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hval : padicValRat p q ≠ padicValRat p r
h1 : min (padicValRat p q) (padicValRat p r) ≤ padicValRat p (q + r)
h2 : min (padicValRat p (q + r)) (padicValRat p (-r)) ≤ padicValRat p (q + r + -r)
h3 : min (padicValRat p (r + q)) (padicValRat p (-q)) ≤ padicValRat p (r + q + -q)
⊢ padicValRat p (q + r) = min (padicValRat p q) (padicValRat p r)
Module: Mathlib.NumberTheory.Padics.PadicIntegers

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

Tactic:
apply lt_of_lt_of_le hk

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

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

Tactic:
apply le_of_lt

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

Goal state before:
pp: case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : ℚ
h : (Int.padicValuation p) ↑x.den < 1
⊢ (Int.padicValuation p) ↑x.den < (Int.padicValuation p) x.num

Tactic:
rcases (Int.padicValuation_le_one p x.num).eq_or_lt with h' | h'

Goal state after:
case inr.inl
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : ℚ
h : (Int.padicValuation p) ↑x.den < 1
h' : (Int.padicValuation p) x.num = 1
⊢ (Int.padicValuation p) ↑x.den < (Int.padicValuation p) x.num

case inr.inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : ℚ
h : (Int.padicValuation p) ↑x.den < 1
h' : (Int.padicValuation p) x.num < 1
⊢ (Int.padicValuation p) ↑x.den < (Int.padicValuation p) x.num
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this✝² : ∃ ε > 0, ∃ N1, ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p (↑f j - ↑f k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this✝¹ : padicNorm p (↑f n - ↑f m) < ε
this✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)
this : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))
hne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)
hnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (-↑f m)) (padicNorm p (↑f n))
⊢ False

Tactic:
padicNorm.neg,

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this✝² : ∃ ε > 0, ∃ N1, ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p (↑f j - ↑f k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this✝¹ : padicNorm p (↑f n - ↑f m) < ε
this✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)
this : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))
hne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)
hnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (↑f m)) (padicNorm p (↑f n))
⊢ False
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 ↔ IsIso c.fst

Tactic:
by
  rw [mono_iff_fst_eq_snd hc]
  constructor
  · intro h
    obtain ⟨φ, hφ₁, hφ₂⟩ := PullbackCone.IsLimit.lift' hc (𝟙 X) (𝟙 X) (by simp)
    refine ⟨φ, PullbackCone.IsLimit.hom_ext hc ?_ ?_, hφ₁⟩
    · simp only [assoc, hφ₁, id_comp, comp_id]
    · simp only [assoc, hφ₂, id_comp, comp_id, h]
  · intro
    obtain ⟨φ, hφ₁, hφ₂⟩ := PullbackCone.IsLimit.lift' hc (𝟙 X) (𝟙 X) (by simp)
    have : IsSplitEpi φ := IsSplitEpi.mk ⟨SplitEpi.mk c.fst (by
      rw [← cancel_mono c.fst, assoc, id_comp, hφ₁, comp_id])⟩
    rw [← cancel_epi φ, hφ₁, hφ₂]

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

Goal state before:
pp: case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
hf : IsPullback (𝟙 X) (𝟙 X) f f
⊢ Mono f

Tactic:
exact (mono_iff_fst_eq_snd hf.isLimit).2 rfl

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  max (padicNorm p (↑f i)) (padicNorm p (-↑g i)) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False

Tactic:
padicNorm.neg,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
  padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h :
  padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠
    padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' :
  max (padicNorm p (↑f i)) (padicNorm p (↑g i)) <
    padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -
      padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : f ≈ 0
⊢ (f + g).norm ≤ max f.norm g.norm

Tactic:
by
      have hfg' : f + g ≈ g := by
        change LimZero (f - 0) at hf
        change LimZero (f + g - g); · simpa only [sub_zero, add_sub_cancel_right] using hf
      have hcfg : (f + g).norm = g.norm := norm_equiv hfg'
      have hcl : f.norm = 0 := (norm_zero_iff f).2 hf
      have : max f.norm g.norm = g.norm := by rw [hcl]; exact max_eq_right (norm_nonneg _)
      rw [this, hcfg]

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

Goal state before:
pp: case ha
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 0 ≤ 1 * ε

Tactic:
apply le_of_lt

Goal state after:
case ha
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 0 < 1 * ε
Module: Mathlib.NumberTheory.Padics.PadicNumbers

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

Tactic:
exact mod_cast padicNormE.nonarchimedean' _ _

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 a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
⊢ ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f

Tactic:
intro a b c d e f h1 h2 h3

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a✝ q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a✝ + q
hnaq_shift : ∀ (d : ℕ), n + d = a✝ + d + q
a b c d e f : Y ⟶ X _⦋n + 1⦌
h1 : b = f
h2 : d + e = 0
h3 : c + a = 0
⊢ a + b + (c + d + e) = f
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 a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
⊢ ⟨i, ⋯⟩ ≤ ⟨a, ⋯⟩.castSucc

Tactic:
Fin.le_iff_val_le_val

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n a q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnaq : n = a + q
hnaq_shift : ∀ (d : ℕ), n + d = a + d + q
simplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f
i : ℕ
hi : i < a + 1
a✝ : ⟨i, hi⟩ ∈ Finset.univ
⊢ ↑⟨i, ⋯⟩ ≤ ↑⟨a, ⋯⟩.castSucc
Module: Mathlib.AlgebraicTopology.DoldKan.Faces

Goal state before:
pp: case neg.inl.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
h✝ : a < m
⊢ ⟨a + 1, ⋯⟩ ≤ j.castSucc

Tactic:
Fin.le_iff_val_le_val

Goal state after:
case neg.inl.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
h✝ : a < m
⊢ ↑⟨a + 1, ⋯⟩ ≤ ↑j.castSucc
Module: Mathlib.AlgebraicTopology.DoldKan.Decomposition

Goal state before:
pp: case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X X' : SimplicialObject C
n : ℕ
Z : C
f : MorphComponents X n Z
g : X' ⟶ X
⊢ ∑ x, (P ↑x).f (n + 1) ≫ X'.δ x.rev.succ ≫ g.app (op ⦋n⦌) ≫ f.b x.rev =
    g.app (op ⦋n + 1⦌) ≫ ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ f.b i.rev

Tactic:
· simp only [comp_sum, P_f_naturality_assoc, SimplicialObject.δ_naturality_assoc]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
A✝ B✝ : Karoubi (SimplicialObject C)
f : A✝ ⟶ B✝
⊢ ∀ [IsIso (N₂.map f)], IsIso f

Tactic:
intro
    -- The following functor `F` reflects isomorphisms because it is
    -- a composition of four functors which reflect isomorphisms.
    -- Then, it suffices to show that `F.map f` is an isomorphism.
    let F₁ := karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C
    let F₂ : SimplicialObject (Karoubi C) ⥤ _ := N₁
    let F₃ := (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor
    let F₄ := Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse
      (ComplexShape.down ℕ)
    let F := F₁ ⋙ F₂ ⋙ F₃ ⋙ F₄
    have : IsIso (F.map f) := by
      simp only [F, F₁]
      rw [← compatibility_N₂_N₁_karoubi, Functor.comp_map]
      apply Functor.map_isIso
    exact isIso_of_reflects_iso f F

Goal state after: