mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Presentable.CardinalFilteredPresentation

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
h : P.IsCardinalFilteredGenerator κ
X : C
hX : IsCardinalPresentable X κ
J : Type w
w✝¹ : SmallCategory J
w✝ : IsCardinalFiltered J κ
p : P.ColimitOfShape J X
⊢ P.retractClosure X

Tactic:
have := essentiallySmall_of_small_of_locallySmall.{w} J

Goal state after:
case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
κ : Cardinal.{w}
inst✝ : Fact κ.IsRegular
h : P.IsCardinalFilteredGenerator κ
X : C
hX : IsCardinalPresentable X κ
J : Type w
w✝¹ : SmallCategory J
w✝ : IsCardinalFiltered J κ
p : P.ColimitOfShape J X
this : EssentiallySmall.{w, w, w} J
⊢ P.retractClosure X
Module: Mathlib.CategoryTheory.SmallRepresentatives

Goal state before:
pp: κ : Cardinal.{w}
C : Type u
inst✝ : Category.{v, u} C
hC : HasCardinalLT (Arrow C) κ
Ω : Type w := κ.ord.ToType
⊢ Nonempty (Arrow C ↪ Ω)

Tactic:
rw [← Cardinal.lift_mk_le']

Goal state after:
κ : Cardinal.{w}
C : Type u
inst✝ : Category.{v, u} C
hC : HasCardinalLT (Arrow C) κ
Ω : Type w := κ.ord.ToType
⊢ Cardinal.lift.{w, max u v} (Cardinal.mk (Arrow C)) ≤ Cardinal.lift.{max u v, w} (Cardinal.mk Ω)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝³ : LinearOrder ι
f : P.RankFunction ι
inst✝² : P.IsProper
inst✝¹ : SuccOrder ι
inst✝ : NoMaxOrder ι
j : ι
c : f.Cell j
⊢ ↑((ConcreteCategory.hom (c.mapToSucc.app (op ⦋c.dim⦌))) (stdSimplex.objEquiv.symm (SimplexCategory.δ c.index))) =
    (↑c.s).simplex

Tactic:
Cell.mapToSucc

Goal state after:
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝³ : LinearOrder ι
f : P.RankFunction ι
inst✝² : P.IsProper
inst✝¹ : SuccOrder ι
inst✝ : NoMaxOrder ι
j : ι
c : f.Cell j
⊢ ↑((ConcreteCategory.hom ((lift c.map ⋯).app (op ⦋c.dim⦌))) (stdSimplex.objEquiv.symm (SimplexCategory.δ c.index))) =
    (↑c.s).simplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝³ : LinearOrder ι
f : P.RankFunction ι
inst✝² : P.IsProper
inst✝¹ : SuccOrder ι
inst✝ : NoMaxOrder ι
j : ι
⊢ Function.Injective f.mapN

Tactic:
intro s t h

Goal state after:
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝³ : LinearOrder ι
f : P.RankFunction ι
inst✝² : P.IsProper
inst✝¹ : SuccOrder ι
inst✝ : NoMaxOrder ι
j : ι
s t : (range (f.m j)).N
h : f.mapN s = f.mapN t
⊢ s = t
Module: Mathlib.CategoryTheory.Presentable.Presheaf

Goal state before:
pp: A : Type u'
inst✝³ : Category.{v', u'} A
inst✝² : IsLocallyPresentable.{w, v', u'} A
inst✝¹ : HasPullbacks A
C : Type w
inst✝ : SmallCategory C
⊢ ∃ κ, ∃ (x : Fact κ.IsRegular), IsCardinalLocallyPresentable (Cᵒᵖ ⥤ A) κ

Tactic:
obtain ⟨κ, _, _⟩ := IsLocallyPresentable.exists_cardinal.{w} A
    exact ⟨κ, inferInstance, inferInstance⟩

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝⁵ : LinearOrder ι
f : P.RankFunction ι
inst✝⁴ : P.IsProper
inst✝³ : SuccOrder ι
inst✝² : OrderBot ι
inst✝¹ : NoMaxOrder ι
inst✝ : WellFoundedLT ι
m : ι
hm : Order.IsSuccLimit m
⊢ IsLUB (Set.range (⋯.functor ⋙ ⋯.functor).obj) ((Set.principalSegIio m).cocone ⋯.functor).pt

Tactic:
by
    dsimp
    rw [← f.iSup_filtration_iio m hm]
    apply isLUB_iSup

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
inst✝ : P.IsRegular
⊢ WellFounded P.op.AncestralRel

Tactic:
have hP := P.wf
    rw [wellFounded_iff_isEmpty_descending_chain] at hP ⊢
    by_contra!
    obtain ⟨f, hf⟩ := this
    refine hP.false ⟨fun n ↦ ⟨_, (f n).2⟩, fun n ↦ ?_⟩
    simpa [← P.op_ancestralRel_iff] using hf n

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ < min x hd
⊢ ↑i.castSucc < ↑(i.castSucc.succ.castPred ⋯)

Tactic:
by dsimp; grind

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomologyZero

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasCoproducts C
inst✝¹ : Preadditive C
X : SSet
R : C
inst✝ : CategoryWithHomology C
x : X _⦋0⦌
⊢ HomologicalComplex.liftCycles (X.chainComplex R) (X.ιChainComplex x) 0
        liftCycles_ιChainComplex_homologyπ_homology₀Iso_hom._proof_1 ⋯ ≫
      HomologicalComplex.homologyπ (X.chainComplex R) 0 ≫ X.homology₀ε R =
    𝟙 R

Tactic:
simp [homology₀ε]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomologyZero

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasCoproducts C
inst✝² : Preadditive C
X : SSet
R : C
inst✝¹ : CategoryWithHomology C
inst✝ : X.IsConnected
x : X.π₀ := Classical.arbitrary X.π₀
⊢ IsIso (Sigma.desc fun x ↦ 𝟙 R)

Tactic:
refine ⟨Sigma.ι (fun _ ↦ R) x, ?_, by simp⟩

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasCoproducts C
inst✝² : Preadditive C
X : SSet
R : C
inst✝¹ : CategoryWithHomology C
inst✝ : X.IsConnected
x : X.π₀ := Classical.arbitrary X.π₀
⊢ (Sigma.desc fun x ↦ 𝟙 R) ≫ Sigma.ι (fun x ↦ R) x = 𝟙 (∐ fun x ↦ R)
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ : Arrow C₁
inst✝¹ : HasPullbacks C₂
inst✝ : HasPushouts C₃
X₃ : Arrow C₃
⊢ pushout.desc (adj₂.homEquiv.symm (𝟙 ((G.leibnizPullback.obj (op X₁)).obj X₃).left))
        (adj₂.homEquiv.symm
          (pullback.fst ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right)))
        ⋯ ≫
      ((𝟭 (Arrow C₃)).obj X₃).hom =
    ((G.leibnizPullback.obj (op X₁) ⋙ F.leibnizPushout.obj X₁).obj X₃).hom ≫
      adj₂.homEquiv.symm
        (pullback.snd ((G.obj (op (unop (op X₁)).left)).map X₃.hom) ((G.map (unop (op X₁)).hom.op).app X₃.right))

Tactic:
by
    apply pushout.hom_ext
    · simp [← homEquiv_symm_naturality_two, ← homEquiv_symm_naturality_three]
    · simp [← homEquiv_symm_naturality_one, ← homEquiv_symm_naturality_three,
      pullback.condition]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋0⦌
g : X.PtSimplex n x
⊢ ∂Δ[n].ι ≫ yonedaEquiv.symm (opObjEquiv.symm (yonedaEquiv g.map)) =
    const ⟨opObjEquiv.symm x, ⋯⟩ ≫ (Subcomplex.ofSimplex (opObjEquiv.symm x)).ι

Tactic:
obtain _ | n := n
        · ext
        · refine boundary.hom_ext (fun i ↦ ?_)
          simp [stdSimplex.δ_comp_yonedaEquiv_symm, op_δ,
            ← stdSimplex.yonedaEquiv_δ_comp,
            opObjEquiv_symm_yonedaEquiv_const]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋0⦌
f g : X.PtSimplex n x
i : Fin n
h : g.MulStruct RelativeMorphism.const f i
j : Fin (n + 2)
hj : j < i.succ.castSucc
⊢ stdSimplex.δ j ≫ h.map = const x

Tactic:
← Fin.succ_castSucc

Goal state after:
X : SSet
n : ℕ
x : X _⦋0⦌
f g : X.PtSimplex n x
i : Fin n
h : g.MulStruct RelativeMorphism.const f i
j : Fin (n + 2)
hj : j < i.castSucc.succ
⊢ stdSimplex.δ j ≫ h.map = const x
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: n✝ : ℕ
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
H :
  ∀ (x : X.obj (op { obj := ⦋2⦌, property := _proof_14 })) (y : Y.obj (op { obj := ⦋2⦌, property := _proof_14 })),
    f₁ ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op)) x) =
        (ConcreteCategory.hom (Y.map (δ₂ 2 _proof_12 _proof_15).op)) y →
      f₁ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op)) x) =
          (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_12 _proof_15).op)) y →
        f₁ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_12 _proof_15).op)) x) =
          (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_12 _proof_15).op)) y
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
n : ℕ
hn : ⦋n⦌.len ≤ 2
⊢ X.obj (op { obj := ⦋n⦌, property := hn }) ⟶ Y.obj (op { obj := ⦋n⦌, property := hn })

Tactic:
match n with
  | 0 => exact ↾f₀
  | 1 => exact ↾f₁
  | 2 => exact ↾(f₂ f₀ f₁ hδ₁ hδ₀ hY)

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : Field R
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
inst✝¹ : OrderTopology S
ι : Type u_3
inst✝ : Finite ι
v : ι → AbsoluteValue R S
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
⊢ ∀ (i : ι), ∃ a, 1 < (v i) a ∧ ∀ (j : ι), j ≠ i → (v j) a < 1

Tactic:
have := Fintype.ofFinite ι
  let P (ι : Type _) [Fintype ι] : Prop :=
    ∀ v : ι → AbsoluteValue R S, (∀ i, (v i).IsNontrivial) →
      (Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)) → ∀ i, ∃ (a : R), 1 < v i a ∧ ∀ j ≠ i, v j a < 1
  -- Use strong induction on the index.
  revert hv h; refine induction_subsingleton_or_nontrivial (P := P) ι (fun ι _ _ v h hv i ↦ ?_)
    (fun ι _ _ ih v h hv i ↦ ?_) v
  · -- If `ι` is trivial this follows immediately from `(v i).IsNontrivial`.
    let ⟨a, ha⟩ := (h i).exists_abv_gt_one
    exact ⟨a, ha, fun j hij ↦ absurd (Subsingleton.elim i j) hij.symm⟩
  · rcases eq_or_ne (card ι) 2 with (hc | hc)
    · -- If `ι` has two elements this is `exists_one_lt_lt_one_of_not_isEquiv`.
      let ⟨j, hj⟩ := (Nat.card_eq_two_iff' i).1 <| card_eq_nat_card ▸ hc
      let ⟨a, ha⟩ := (v i).exists_one_lt_lt_one_of_not_isEquiv (h i) (h j) (hv hj.1.symm)
      exact ⟨a, ha.1, fun _ h ↦ hj.2 _ h ▸ ha.2⟩
    have hlt : 2 < card ι := Nat.lt_of_le_of_ne (one_lt_card_iff_nontrivial.2 ‹_›) hc.symm
    -- Otherwise, choose another distinguished index `j ≠ i`.
    let ⟨j, hj⟩ := exists_ne i
    -- Apply induction first on the subcollection `v i` for `i ≠ j` to get `a : K`
    let ⟨a, ha⟩ := ih {k : ι // k ≠ j} (card_subtype_lt fun a ↦ a rfl) (restrict _ v)
      (fun i ↦ h _) (hv.comp_of_injective val_injective) ⟨i, hj.symm⟩
    -- Then apply induction next to the subcollection `{v i, v j}` to get `b : K`.
    let ⟨b, hb⟩ := ih {k : ι // k = i ∨ k = j} (by linarith [card_subtype_eq_or_eq_of_ne hj.symm])
      (restrict _ v) (fun _ ↦ h _) (hv.comp_of_injective val_injective) ⟨i, .inl rfl⟩
    rcases eq_or_ne (v j a) 1 with (ha₁ | ha₁)
    · -- If `v j a = 1` then take a large enough value from the sequence `a ^ n * b`.
      let ⟨c, hc⟩ := exists_one_lt_lt_one_pi_of_eq_one ha.1 ha.2 ha₁ hb.1 (hb.2 ⟨j, .inr rfl⟩
        (by grind))
      refine ⟨c, hc.1, fun k hk ↦ ?_⟩
      rcases eq_or_ne k j with (rfl | h); try exact hc.2.2; exact hc.2.1 ⟨k, h⟩ (by grind)
    rcases ha₁.lt_or_gt with (ha_lt | ha_gt)
    · -- If `v j a < 1` then `a` works as the divergent point.
      refine ⟨a, ha.1, fun k hk ↦ ?_⟩
      rcases eq_or_ne k j with (rfl | h); try exact ha_lt; exact ha.2 ⟨k, h⟩ (by grind)
    · -- If `1 < v j a` then take a large enough value from the sequence `b / (1 + a ^ (-n))`.
      let ⟨c, hc⟩ := exists_one_lt_lt_one_pi_of_one_lt ha.1 ha.2 ha_gt hb.1 (hb.2 ⟨j, .inr rfl⟩
        (by grind))
      refine ⟨c, hc.1, fun k hk ↦ ?_⟩
      rcases eq_or_ne k j with (rfl | h); try exact hc.2.2; exact hc.2.1 ⟨k, h⟩ (by grind)

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

Goal state before:
pp: case refine_2.succ.zero.«0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
H :
  ∀ (x : X.obj (op { obj := ⦋2⦌, property := _proof_14 })) (y : Y.obj (op { obj := ⦋2⦌, property := _proof_14 })),
    f₁ ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op)) x) =
        (ConcreteCategory.hom (Y.map (δ₂ 2 _proof_12 _proof_15).op)) y →
      f₁ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op)) x) =
          (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_12 _proof_15).op)) y →
        f₁ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_12 _proof_15).op)) x) =
          (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_12 _proof_15).op)) y
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
hn : 0 + 1 < 2
⊢ naturalityProperty f₀ f₁ hδ₁ hδ₀ hY (σ 2 ((fun i ↦ i) ⟨0, ⋯⟩) ⋯ ⋯)

Tactic:
ext; apply hσ'₀ f₀ f₁ hδ₁ hδ₀ hσ hY

Goal state after:
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
y : 𝕜
hy : y ∈ U
⊢ deriv f y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y

Tactic:
EventuallyEq.deriv_eq (eventually_of_mem (hUo.mem_nhds hy) (fun u hu ↦ (hUf u hu).1)),

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
y : 𝕜
hy : y ∈ U
⊢ deriv (fun u ↦ (u - x) ^ (s + 1) • F u + f x) y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
this : ∀ y ∈ U, deriv f y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 x, (s + 1) • (z - x) ^ s • F z = (z - x) ^ s • g z

Tactic:
refine ⟨fun z ↦ (↑(s + 1) : 𝕜) • F z, hFa.fun_const_smul, ?_, .of_forall fun y ↦ ?_⟩

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
this : ∀ y ∈ U, deriv f y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y
⊢ (fun z ↦ ↑(s + 1) • F z) x ≠ 0

case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
inst✝¹ : CompleteSpace E
inst✝ : CharZero 𝕜
s : ℕ
hrne : s + 1 ≠ 0
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F x
hFne : F x ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ (s + 1) • F z + f x
U : Set 𝕜
hUf : ∀ y ∈ U, f y = (y - x) ^ (s + 1) • F y + f x ∧ AnalyticAt 𝕜 F y
hUo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hUx : x ∈ U
this : ∀ y ∈ U, deriv f y = (y - x) ^ (s + 1) • deriv F y + (s + 1) • (y - x) ^ s • F y
y : 𝕜
⊢ (s + 1) • (y - x) ^ s • F y = (y - x) ^ s • (fun z ↦ ↑(s + 1) • F z) y
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: X : Truncated 2
C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
φ : X ⟶ (truncation 2).obj (nerve C)
s : X.obj (op { obj := ⦋1⦌, property := ⋯ })
⊢ (ConcreteCategory.hom ((homToNerveMk (descOfTruncation φ)).app (op { obj := ⦋1⦌, property := ⋯ }))) s =
    (ConcreteCategory.hom (φ.app (op { obj := ⦋1⦌, property := ⋯ }))) s

Tactic:
by
      obtain ⟨x₀, x₁, f, rfl⟩ := Edge.exists_of_simplex s
      dsimp [nerve.homEquiv]
      simp only [homToNerveMk_app_edge, descOfTruncation_obj_mk,
        descOfTruncation_map_homMk]
      refine ComposableArrows.ext₁ ?_ ?_ rfl
      · dsimp [nerveEquiv, ComposableArrows.right]
        simp only [← f.src_eq, NatTrans.naturality_apply]
        rfl
      · dsimp [nerveEquiv, ComposableArrows.right]
        simp only [← f.tgt_eq, NatTrans.naturality_apply]
        rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Analytic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : E → ℂ
x : E
fa : AnalyticAt ℂ f x
ga : AnalyticAt ℂ g x
m : f x ∈ slitPlane
⊢ AnalyticAt ℂ (fun z ↦ f z ^ g z) x

Tactic:
by
  rw [← analyticWithinAt_univ] at fa ga ⊢
  exact fa.cpow ga m

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ↑n ! ≠ 0

Tactic:
apply (ascPochhammer_eval_eq_zero_iff n _).not.2

Goal state after:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ↑n ! ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval c (ascPochhammer 𝕂 n) ≠ 0

Tactic:
apply (ascPochhammer_eval_eq_zero_iff n _).not.2

Goal state after:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ¬∃ k < n, ↑k = -c
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval b (ascPochhammer 𝕂 n) ≠ 0

Tactic:
apply (ascPochhammer_eval_eq_zero_iff n _).not.2

Goal state after:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ¬∃ k < n, ↑k = -b
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
z₀ : 𝕜
hf : AnalyticAt 𝕜 f (g z₀)
hg : AnalyticAt 𝕜 g z₀
hg_nc : ¬analyticOrderAt (fun x ↦ g x - g z₀) z₀ = ⊤
hf' : ¬analyticOrderAt f (g z₀) = ⊤
r s : ℕ
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F (g z₀)
hFne : F (g z₀) ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 (g z₀), f z = (z - g z₀) ^ r • F z
G : 𝕜 → 𝕜
hGa : AnalyticAt 𝕜 G z₀
hGne : G z₀ ≠ 0
hgG : ∀ᶠ (z : 𝕜) in 𝓝 z₀, g z - g z₀ = (z - z₀) ^ s • G z
z : 𝕜
⊢ (f ∘ g) z = ((fun x ↦ (x - g z₀) ^ r • F x) ∘ g) z →
    g z - g z₀ = (z - z₀) ^ s • G z → (f ∘ g) z = (z - z₀) ^ (r * s) • G z ^ r • F (g z)

Tactic:
hfz

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
z₀ : 𝕜
hf : AnalyticAt 𝕜 f (g z₀)
hg : AnalyticAt 𝕜 g z₀
hg_nc : ¬analyticOrderAt (fun x ↦ g x - g z₀) z₀ = ⊤
hf' : ¬analyticOrderAt f (g z₀) = ⊤
r s : ℕ
F : 𝕜 → E
hFa : AnalyticAt 𝕜 F (g z₀)
hFne : F (g z₀) ≠ 0
hfF : ∀ᶠ (z : 𝕜) in 𝓝 (g z₀), f z = (z - g z₀) ^ r • F z
G : 𝕜 → 𝕜
hGa : AnalyticAt 𝕜 G z₀
hGne : G z₀ ≠ 0
hgG : ∀ᶠ (z : 𝕜) in 𝓝 z₀, g z - g z₀ = (z - z₀) ^ s • G z
z : 𝕜
hfz : (f ∘ g) z = ((fun x ↦ (x - g z₀) ^ r • F x) ∘ g) z
⊢ g z - g z₀ = (z - z₀) ^ s • G z → (f ∘ g) z = (z - z₀) ^ (r * s) • G z ^ r • F (g z)
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ Polynomial.eval a (ascPochhammer 𝕂 n) ≠ 0

Tactic:
apply (ascPochhammer_eval_eq_zero_iff n _).not.2

Goal state after:
case a
𝕂 : Type u_1
inst✝ : RCLike 𝕂
a b c : 𝕂
n : ℕ
habc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c
⊢ ¬∃ k < n, ↑k = -a
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : MeromorphicAt f z₀
⊢ (∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

Tactic:
by
  obtain ⟨n, h⟩ := hf
  rcases h.eventually_eq_zero_or_eventually_ne_zero with h₁ | h₂
  · left
    filter_upwards [nhdsWithin_le_nhds h₁, self_mem_nhdsWithin] with y h₁y h₂y
    rw [Set.mem_compl_iff, Set.mem_singleton_iff, ← sub_eq_zero] at h₂y
    exact smul_eq_zero_iff_right (pow_ne_zero n h₂y) |>.mp h₁y
  · right
    filter_upwards [h₂, self_mem_nhdsWithin] with y h₁y h₂y
    exact (smul_ne_zero_iff.1 h₁y).2

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
h_eq : (fun z ↦ (z - x) ^ m • f z) =ᶠ[𝓝 x] 0
z : 𝕜
⊢ (z - x) ^ m • f z = 0 z → z ∈ {x}ᶜ → 0 = f⁻¹ z

Tactic:
hfz

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
h_eq : (fun z ↦ (z - x) ^ m • f z) =ᶠ[𝓝 x] 0
z : 𝕜
hfz : (z - x) ^ m • f z = 0 z
⊢ z ∈ {x}ᶜ → 0 = f⁻¹ z
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Goal state before:
pp: ⊢ logDeriv sin = cot

Tactic:
ext
  rw [logDeriv, Complex.deriv_sin, Pi.div_apply, Complex.cot]

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ (ascPochhammer ℕ n).smeval r * (X + ↑n).smeval r = (ascPochhammer ℕ n).smeval r * (X + ↑n).smeval r

Tactic:
simp only [smeval_add, smeval_X, ← C_eq_natCast, smeval_C, natCast_zsmul, nsmul_eq_mul,
      Nat.cast_id]

Goal state after: