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

Goal state before:
pp: case refine_2
C₀ : Type u₀
C : Type u
inst✝² : Category.{v₀, u₀} C₀
inst✝¹ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
inst✝ : IsDenseSubsite J₀ J F
X : C
data : F.OneHypercoverDenseData J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this✝ : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ → ⦃p : W ⟶ F.obj X₀⦄ → (Sieve.pullback f data.toOneHypercover.sieve₀).arrows p → Sieve W :=
  fun W p hp ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun U π hπ ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f data.toOneHypercover.sieve₀).arrows p
⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ W⦄ (H : (Sieve.coverByImage F W).arrows f),
    Sieve.pullback (Nonempty.some H).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some H).map ≫ p))) ∈ J Y

Tactic:
intro U

Goal state after:
case refine_2
C₀ : Type u₀
C : Type u
inst✝² : Category.{v₀, u₀} C₀
inst✝¹ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
inst✝ : IsDenseSubsite J₀ J F
X : C
data : F.OneHypercoverDenseData J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this✝ : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ → ⦃p : W ⟶ F.obj X₀⦄ → (Sieve.pullback f data.toOneHypercover.sieve₀).arrows p → Sieve W :=
  fun W p hp ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun U π hπ ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f data.toOneHypercover.sieve₀).arrows p
U : C
⊢ ∀ ⦃f : U ⟶ W⦄ (H : (Sieve.coverByImage F W).arrows f),
    Sieve.pullback (Nonempty.some H).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some H).map ≫ p))) ∈ J U
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec (CommRingCat.of ↑R)).Modules
N : ModuleCat ↑R
e : (tilde.functor R).obj N ≅ M
⊢ SheafOfModules.isQuasicoherent (Spec R).ringCatSheaf ((tilde.functor R).obj N)

Tactic:
dsimp; infer_instance

Goal state after:
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
data : (X : C) → F.OneHypercoverDenseData J₀ J X
G : Cᵒᵖ ⥤ A
⊢ Presheaf.IsSheaf J G ↔
    Presheaf.IsSheaf J₀ (F.op ⋙ G) ∧ ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork G))

Tactic:
by
  refine ⟨fun hG ↦ ⟨op_comp_isSheaf F J₀ J ⟨_, hG⟩,
    fun X ↦ ⟨(data X).toOneHypercover.isLimitMultifork ⟨G, hG⟩⟩⟩, fun ⟨hG₀, hG⟩ ↦ ?_⟩
  rw [Presheaf.isSheaf_iff_multifork]
  replace hG := fun X ↦ (hG X).some
  exact fun X S ↦ ⟨isSheaf_iff.isLimit hG₀ hG S⟩

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: case h.e_a.e_a.e_f
σ : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : SetLike σ A
inst✝⁸ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝⁷ : GradedRing 𝒜
O : Type u_3
inst✝⁶ : CommRing O
inst✝⁵ : IsDomain O
inst✝⁴ : ValuationRing O
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra O K
inst✝¹ : IsFractionRing O K
φ₀ : ↥(𝒜 0) →+* O
ι : Type u_5
inst✝ : Finite ι
x : ι → A
h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤
j : ι
φ : Away 𝒜 (x j) →+* K
hcomm : (algebraMap O K).comp φ₀ = φ.comp (fromZeroRingHom 𝒜 (Submonoid.powers (x j)))
d : ι → ℕ
hdi : ∀ (i : ι), 0 < d i
hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)
val✝ : Fintype ι
ψ : ι → ValueGroup O K := fun i ↦ (valuation O K) (φ (Away.isLocalizationElem ⋯ ⋯) ^ ∏ k ∈ Finset.univ.erase i, d k)
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := (Finset.image ψ Finset.univ).max' ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : 0 < Kmax
this✝ : Algebra (Away 𝒜 (x j)) (Away 𝒜 (x j * x i₀)) := (awayMap 𝒜 ⋯ ⋯).toAlgebra
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := IsLocalization.Away.lift (Away.isLocalizationElem ⋯ ⋯) hunit
hφ'1 : ∀ (s : Away 𝒜 (x j)), φ' ((awayMap 𝒜 ⋯ ⋯) s) = φ s
sx : Away 𝒜 (x i₀)
a : ℕ
ai : ι → ℕ
hai✝ : ∑ i, ai i • d i = a • d i₀
hai : ∑ x, ai x * d x = a * d i₀
H : (∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1)) ∈ 𝒜 ((a * d i₀) • d j)
⊢ (fun x_1 ↦ x x_1 ^ (ai x_1 * (d j * ∏ i, d i))) = fun x_1 ↦ x x_1 ^ (d j * ((∏ i, d i) * ai x_1))

Tactic:
ext i

Goal state after:
case h.e_a.e_a.e_f
σ : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : SetLike σ A
inst✝⁸ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝⁷ : GradedRing 𝒜
O : Type u_3
inst✝⁶ : CommRing O
inst✝⁵ : IsDomain O
inst✝⁴ : ValuationRing O
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra O K
inst✝¹ : IsFractionRing O K
φ₀ : ↥(𝒜 0) →+* O
ι : Type u_5
inst✝ : Finite ι
x : ι → A
h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤
j : ι
φ : Away 𝒜 (x j) →+* K
hcomm : (algebraMap O K).comp φ₀ = φ.comp (fromZeroRingHom 𝒜 (Submonoid.powers (x j)))
d : ι → ℕ
hdi : ∀ (i : ι), 0 < d i
hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)
val✝ : Fintype ι
ψ : ι → ValueGroup O K := fun i ↦ (valuation O K) (φ (Away.isLocalizationElem ⋯ ⋯) ^ ∏ k ∈ Finset.univ.erase i, d k)
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := (Finset.image ψ Finset.univ).max' ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : 0 < Kmax
this✝ : Algebra (Away 𝒜 (x j)) (Away 𝒜 (x j * x i₀)) := (awayMap 𝒜 ⋯ ⋯).toAlgebra
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := IsLocalization.Away.lift (Away.isLocalizationElem ⋯ ⋯) hunit
hφ'1 : ∀ (s : Away 𝒜 (x j)), φ' ((awayMap 𝒜 ⋯ ⋯) s) = φ s
sx : Away 𝒜 (x i₀)
a : ℕ
ai : ι → ℕ
hai✝ : ∑ i, ai i • d i = a • d i₀
hai : ∑ x, ai x * d x = a * d i₀
H : (∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1)) ∈ 𝒜 ((a * d i₀) • d j)
i : ι
⊢ x i ^ (ai i * (d j * ∏ i, d i)) = x i ^ (d j * ((∏ i, d i) * ai i))
Module: Mathlib.AlgebraicGeometry.Sites.QuasiCompact

Goal state before:
pp: S✝ : Scheme
P : MorphismProperty Scheme
S : Scheme
𝒰 : Cover (propQCPrecoverage P) S
⊢ QuasiCompactCover 𝒰.forgetQc.toPreZeroHypercover

Tactic:
dsimp; infer_instance

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: σ : Type u_1
A : Type u_2
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) A
O : Type u_2
commRing✝ : CommRing O
domain✝ : IsDomain O
valuationRing✝ : ValuationRing O
K : Type u_2
field✝ : Field K
algebra✝ : Algebra O K
isFractionRing✝ : IsFractionRing O K
i₁ : Spec (CommRingCat.of K) ⟶ Proj 𝒜
i₂ : Spec (CommRingCat.of O) ⟶ Spec (CommRingCat.of ↥(𝒜 0))
w : CommSq i₁ (Spec.map (CommRingCat.ofHom (algebraMap O K))) (toSpecZero 𝒜) i₂
x : Finset A
hx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤
hx' : ∀ i ∈ x, ∃ n, n ≠ 0 ∧ i ∈ 𝒜 n
⊢ ⋯.HasLift

Tactic:
choose d hd hxd using hx'

Goal state after:
σ : Type u_1
A : Type u_2
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) A
O : Type u_2
commRing✝ : CommRing O
domain✝ : IsDomain O
valuationRing✝ : ValuationRing O
K : Type u_2
field✝ : Field K
algebra✝ : Algebra O K
isFractionRing✝ : IsFractionRing O K
i₁ : Spec (CommRingCat.of K) ⟶ Proj 𝒜
i₂ : Spec (CommRingCat.of O) ⟶ Spec (CommRingCat.of ↥(𝒜 0))
w : CommSq i₁ (Spec.map (CommRingCat.ofHom (algebraMap O K))) (toSpecZero 𝒜) i₂
x : Finset A
hx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤
d : (i : A) → i ∈ x → ℕ
hd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0
hxd : ∀ (i : A) (a : i ∈ x), i ∈ 𝒜 (d i a)
⊢ ⋯.HasLift
Module: Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact

Goal state before:
pp: P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ QuasiCompactCover
    (Precoverage.ZeroHypercover.pullback₂ (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i)
        (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover)).toPreZeroHypercover

Tactic:
dsimp; infer_instance

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 : C
⊢ presheafMap data G₀ (𝟙 X) = 𝟙 (presheafObj data G₀ X)

Tactic:
ext i

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 : C
i : (data X).I₀
⊢ presheafMap data G₀ (𝟙 X) ≫ presheafObjπ data G₀ X i = 𝟙 (presheafObj data G₀ X) ≫ presheafObjπ data G₀ X i
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 Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ presheafMap data G₀ (f ≫ g) = presheafMap data G₀ g ≫ presheafMap data G₀ f

Tactic:
ext i

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 Z : C
f : X ⟶ Y
g : Y ⟶ Z
i : (data X).I₀
⊢ presheafMap data G₀ (f ≫ g) ≫ presheafObjπ data G₀ X i =
    (presheafMap data G₀ g ≫ presheafMap data G₀ f) ≫ presheafObjπ data G₀ X i
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 Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ presheafMap data G₀ (f ≫ g) = presheafMap data G₀ g ≫ presheafMap data G₀ f

Tactic:
by
  ext i
  rw [assoc, presheafMap_π, presheafMap_π, presheafMap_restriction, assoc]

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

Goal state before:
pp: ⊢ fppfPrecoverage ≤ fpqcPrecoverage

Tactic:
rw [fpqcPrecoverage, propQCPrecoverage, le_inf_iff]

Goal state after:
⊢ fppfPrecoverage ≤ qcPrecoverage ∧ fppfPrecoverage ≤ precoverage @Flat
Module: Mathlib.CategoryTheory.Sites.Subcanonical

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X Y : C
f : X ⟶ Y
⊢ J.uliftYonedaEquiv ((uliftYoneda.{v', v, u} J).map f) = { down := f }

Tactic:
by
  rw [uliftYonedaEquiv_apply]
  simp [uliftYoneda]

Goal state after:
Module: Mathlib.Order.Interval.Set.Limit

Goal state before:
pp: case pos
J : Type u
inst✝ : LinearOrder J
j : J
m : ↑(Ici j)
hm : Order.IsSuccLimit m
b : J
hb : b < ↑m
hb' : j ≤ b
⊢ ∃ x, ∃ (_ : b < x), x < ↑m

Tactic:
have := hm.2 ⟨b, hb'⟩
      rw [not_covBy_iff (by exact hb)] at this
      obtain ⟨⟨x, h₁⟩, h₂, h₃⟩ := this
      refine ⟨x, h₂, h₃⟩

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Preorder.HasIterationOfShape

Goal state before:
pp: J : Type w
inst✝⁶ : LinearOrder J
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasIterationOfShape J C
inst✝³ : SuccOrder J
inst✝² : WellFoundedLT J
α : Type u_1
inst✝¹ : PartialOrder α
f : α ≤i J
inst✝ : Nonempty α
⊢ HasColimitsOfShape α C

Tactic:
by
  by_cases hf : Function.Surjective f
  · exact hasColimitsOfShape_of_equivalence
      (OrderIso.ofRelIsoLT (RelIso.ofSurjective f.toRelEmbedding hf)).equivalence.symm
  · let s := f.toPrincipalSeg hf
    obtain ⟨i, hi₀⟩ : ∃ i, i = s.top := ⟨_, rfl⟩
    induction i using SuccOrder.limitRecOn with
    | isMin i hi =>
      subst hi₀
      exact (hi.not_lt (s.lt_top (Classical.arbitrary _))).elim
    | succ i hi _ =>
      obtain ⟨a, rfl⟩ := (s.mem_range_iff_rel (b := i)).2 (by
        simpa only [← hi₀] using Order.lt_succ_of_not_isMax hi)
      have : OrderTop α :=
        { top := a
          le_top b := by
            rw [← s.le_iff_le]
            exact Order.le_of_lt_succ (by simpa only [hi₀] using s.lt_top b) }
      infer_instance
    | isSuccLimit i hi =>
      subst hi₀
      exact hasColimitsOfShape_of_isSuccLimit' C s hi

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
J : Type w
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
inst✝¹ : OrderBot J
inst✝ : WellFoundedLT J
A B : C
f : A ⟶ B
hf : W.transfiniteCompositionsOfShape J f
⊢ transfiniteCompositions.{w, v, u} W f

Tactic:
transfiniteCompositions_iff

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
J : Type w
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
inst✝¹ : OrderBot J
inst✝ : WellFoundedLT J
A B : C
f : A ⟶ B
hf : W.transfiniteCompositionsOfShape J f
⊢ ∃ J x x_1 x_2, ∃ (x_3 : WellFoundedLT J), W.transfiniteCompositionsOfShape J f
Module: Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W₁ W₂ : MorphismProperty C
h : W₁ ≤ W₂
X Y : C
f : X ⟶ Y
hf : transfiniteCompositions.{w, v, u} W₁ f
⊢ transfiniteCompositions.{w, v, u} W₂ f

Tactic:
transfiniteCompositions_iff

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
W₁ W₂ : MorphismProperty C
h : W₁ ≤ W₂
X Y : C
f : X ⟶ Y
hf : ∃ J x x_1 x_2, ∃ (x_3 : WellFoundedLT J), W₁.transfiniteCompositionsOfShape J f
⊢ transfiniteCompositions.{w, v, u} W₂ f
Module: Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
inst✝ : W.IsStableUnderTransfiniteComposition
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : transfiniteCompositions.{w, v, u} W f
⊢ W f

Tactic:
transfiniteCompositions_iff

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
inst✝ : W.IsStableUnderTransfiniteComposition
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : ∃ J x x_1 x_2, ∃ (x_3 : WellFoundedLT J), W.transfiniteCompositionsOfShape J f
⊢ W f
Module: Mathlib.CategoryTheory.SmallObject.Iteration.Basic

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : Type w
Φ : SuccStruct C
inst✝⁴ : LinearOrder J
inst✝³ : SuccOrder J
inst✝² : OrderBot J
inst✝¹ : HasIterationOfShape J C
inst✝ : WellFoundedLT J
j₁ j₂ : J
iter₁ : Φ.Iteration j₁
iter₂ : Φ.Iteration j₂
k₁ k₂ : J
h : k₁ ≤ k₂
h₁ : k₂ ≤ j₁
h₂ : k₂ ≤ j₂
⊢ iter₁.F.map (homOfLE h) = eqToHom ⋯ ≫ iter₂.F.map (homOfLE h) ≫ eqToHom ⋯

Tactic:
by
  have := (Arrow.mk_eq_mk_iff _ _).1 (congr_arrowMap iter₁ iter₂ h h₁ h₂)
  tauto

Goal state after:
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
i₁ i₂ : J
h₁₂ : i₁ ≤ i₂
j : J
iter : Φ.Iteration j
hj : i₂ ≤ j
⊢ Arrow.mk ((Φ.iterationFunctor J).map (homOfLE h₁₂)) = Arrow.mk (iter.F.map (homOfLE h₁₂))

Tactic:
dsimp [iterationFunctor]
  rw [Iteration.arrow_mk_mapObj]
  exact Arrow.ext (Iteration.congr_obj _ _ _ _ _)
    (Iteration.congr_obj _ _ _ _ _) (Iteration.congr_map _ _ _ _ _)

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

Goal state before:
pp: case succ.h₀
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.F.map (homOfLE ⋯) ≫ c.incl.app (Order.succ j) ≫ φ₁ = c.F.map (homOfLE ⋯) ≫ c.incl.app (Order.succ j) ≫ φ₂

Tactic:
simpa using hj'

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
F : J ⥤ C
c : Cocone F
hc : IsColimit c
X Y : C
p : X ⟶ Y
f : F.obj ⊥ ⟶ X
g : c.pt ⟶ Y
inst✝² : F.IsWellOrderContinuous
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hF : ∀ (j : J), ¬IsMax j → HasLiftingPropertyFixedBot (F.map (homOfLE ⋯)) p (c.ι.app (Order.succ j) ≫ g)
sq : CommSq f (c.ι.app ⊥) p g
s : ↑(sqFunctor c p f g).sections
hs : (↑s (Opposite.op ⊥)).f' = f
t : Cocone F := { pt := X, ι := { app := fun j ↦ (↑s (Opposite.op j)).f', naturality := ⋯ } }
l : c.pt ⟶ t.pt := hc.desc t
hl : ∀ (j : J), c.ι.app j ≫ l = (↑s (Opposite.op j)).f'
j : J
⊢ c.ι.app j ≫ l ≫ p = c.ι.app j ≫ g

Tactic:
reassoc_of% (hl j),

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
F : J ⥤ C
c : Cocone F
hc : IsColimit c
X Y : C
p : X ⟶ Y
f : F.obj ⊥ ⟶ X
g : c.pt ⟶ Y
inst✝² : F.IsWellOrderContinuous
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hF : ∀ (j : J), ¬IsMax j → HasLiftingPropertyFixedBot (F.map (homOfLE ⋯)) p (c.ι.app (Order.succ j) ≫ g)
sq : CommSq f (c.ι.app ⊥) p g
s : ↑(sqFunctor c p f g).sections
hs : (↑s (Opposite.op ⊥)).f' = f
t : Cocone F := { pt := X, ι := { app := fun j ↦ (↑s (Opposite.op j)).f', naturality := ⋯ } }
l : c.pt ⟶ t.pt := hc.desc t
hl : ∀ (j : J), c.ι.app j ≫ l = (↑s (Opposite.op j)).f'
j : J
⊢ (↑s (Opposite.op j)).f' ≫ p = c.ι.app j ≫ g
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : transfiniteCompositions.{w, v, u} W f
⊢ W.rlp.llp f

Tactic:
transfiniteCompositions_iff

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : ∃ J x x_1 x_2, ∃ (x_3 : WellFoundedLT J), W.transfiniteCompositionsOfShape J f
⊢ W.rlp.llp f
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
hq : ¬q = 0
⊢ padicValRat p 1 = 0

Tactic:
padicValRat.one

Goal state after:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
hq : ¬q = 0
⊢ 0 = 0
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n₁ n₂ d₁ d₂ : ℤ
hn₁ : n₁ ≠ 0
hn₂ : n₂ ≠ 0
hd₁ : d₁ ≠ 0
hd₂ : d₂ ≠ 0
hf1 : FiniteMultiplicity (↑p) (n₁ * d₂)
hf2 : FiniteMultiplicity (↑p) (n₂ * d₁)
| padicValRat p (n₁ /. d₁) ≤ padicValRat p (n₂ /. d₂) ↔ ∀ (n : ℕ), ↑p ^ n ∣ n₁ * d₂ → ↑p ^ n ∣ n₂ * d₁

Tactic:
lhs
    rw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn₁ hd₁) rfl,
      padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn₂ hd₂) rfl, sub_le_iff_le_add', ←
      add_sub_assoc, le_sub_iff_add_le]
    norm_cast
    rw [← multiplicity_mul (Nat.prime_iff_prime_int.1 hp.1) hf1, add_comm,
        ← multiplicity_mul (Nat.prime_iff_prime_int.1 hp.1) hf2,
        hf1.multiplicity_le_multiplicity_iff hf2]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n₁ n₂ d₁ d₂ : ℤ
hn₁ : n₁ ≠ 0
hn₂ : n₂ ≠ 0
hd₁ : d₁ ≠ 0
hd₂ : d₂ ≠ 0
hf1 : FiniteMultiplicity (↑p) (n₁ * d₂)
hf2 : FiniteMultiplicity (↑p) (n₂ * d₁)
| ∀ (n : ℕ), ↑p ^ n ∣ n₁ * d₂ → ↑p ^ n ∣ n₂ * d₁
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
h : padicValRat p q ≤ padicValRat p r
hq : ¬q = 0
hr : ¬r = 0
hqn : q.num ≠ 0
hqd : ↑q.den ≠ 0
hrn : r.num ≠ 0
hrd : ↑r.den ≠ 0
hqreq : q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)
hqrd : q.num * ↑r.den + ↑q.den * r.num ≠ 0
⊢ emultiplicity (↑p) (q.num * (↑q.den * ↑r.den)) ≤ emultiplicity (↑p) ((q.num * ↑r.den + ↑q.den * r.num) * ↑q.den)

Tactic:
mul_left_comm,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
h : padicValRat p q ≤ padicValRat p r
hq : ¬q = 0
hr : ¬r = 0
hqn : q.num ≠ 0
hqd : ↑q.den ≠ 0
hrn : r.num ≠ 0
hrd : ↑r.den ≠ 0
hqreq : q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)
hqrd : q.num * ↑r.den + ↑q.den * r.num ≠ 0
⊢ emultiplicity (↑p) (↑q.den * (q.num * ↑r.den)) ≤ emultiplicity (↑p) ((q.num * ↑r.den + ↑q.den * r.num) * ↑q.den)
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:
pow_dvd_iff_le_emultiplicity,

Goal state after:
p : ℕ
hp : p ≠ 1
a n : ℕ
ha : a ≠ 0
⊢ ↑n ≤ emultiplicity p a ↔ n ≤ padicValNat p a
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case succ.succ.succ
n p : ℕ
⊢ padicValNat (p + 1 + 1) (n + 1) ≤ log (p + 1 + 1) (n + 1)

Tactic:
exact Nat.le_log_of_pow_le p.one_lt_succ_succ (le_of_dvd n.succ_pos pow_padicValNat_dvd)

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : n ≠ 0
⊢ (p - 1) * padicValNat p n ! < n

Tactic:
rw [sub_one_mul_padicValNat_factorial n]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : n ≠ 0
⊢ n - (p.digits n).sum < n
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
this : ↑p ∣ r.num ∧ ↑p ∣ ↑r.den
⊢ p ∣ r.num.natAbs ∧ p ∣ r.den

Tactic:
← Int.natCast_dvd,

Goal state after:
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖
key : ‖↑r.num‖ < 1
this : ↑p ∣ r.num ∧ ↑p ∣ ↑r.den
⊢ ↑p ∣ r.num ∧ p ∣ r.den
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.num ≤ (Int.padicValuation p) ↑x.den ↔ (Int.padicValuation p) ↑x.den = 1

Tactic:
· simp only [h.ne, iff_false, not_le]
      rcases (Int.padicValuation_le_one p x.num).eq_or_lt with h' | h'
      · simp [h, h']
      · rw [Int.padicValuation_lt_one_iff] at h h'
        exfalso
        rw [Int.natCast_dvd_natCast] at h
        rw [Int.natCast_dvd] at h'
        exact Nat.not_coprime_of_dvd_of_dvd (Nat.Prime.one_lt Fact.out) h h' x.reduced.symm

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

Goal state before:
pp: case coe
p : ℕ
hp : Fact (Nat.Prime p)
x : Multiplicative ℤ
⊢ ∃ a, (padicValuation p) a = ↑x

Tactic:
induction x with

Goal state after:
case coe.ofAdd
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ
⊢ ∃ a, (padicValuation p) a = ↑(Multiplicative.ofAdd x)