mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.TopAdj

Goal state before:
pp: n m : SimplexCategory
f : n ⟶ m
⊢ ⇑m.toTopHomeo.symm ∘ stdSimplex.map ⇑(ConcreteCategory.hom f) =
    ⇑(TopCat.Hom.hom (SSet.toTop.map (SSet.stdSimplex.map f))) ∘ ⇑n.toTopHomeo.symm

Tactic:
ext x : 1
  exact ConcreteCategory.congr_hom ((forget _).congr_map
    (toTopSimplex.inv.naturality f)) _

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

Goal state before:
pp: X : TopCat
n m : SimplexCategory
f : n ⟶ m
g : C(↑(stdSimplex ℝ (Fin (m.len + 1))), ↑X)
⊢ Continuous (stdSimplex.map ⇑(ConcreteCategory.hom f))

Tactic:
by continuity

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

Goal state before:
pp: 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)
hY : Y.StrictSegal
x : X.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)) (f₂ f₀ f₁ hδ₁ hδ₀ hY x)

Tactic:
simp [← spineEquiv_f₂_arrow_zero f₀ f₁ hδ₁ hδ₀ hY, StrictSegal.spineEquiv,
    SimplexCategory.mkOfSucc_zero_eq_δ]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Connected

Goal state before:
pp: case inr
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
⊢ JoinedIn sᶜ a b

Tactic:
obtain ⟨y, hy⟩ : ∃ y, LinearIndependent ℝ ![x, y] :=
    exists_linearIndependent_pair_of_one_lt_rank h x_ne_zero

Goal state after:
case inr
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
y : E
hy : LinearIndependent ℝ ![x, y]
⊢ JoinedIn sᶜ a b
Module: Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
r : ENNReal
hr_pos : 0 < r
h : AnalyticOnNhd 𝕜 f (Metric.eball x r)
p : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial
hr : r ≤ p.radius
g : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)
hg : HasFPowerSeriesOnBall g p x p.radius
hg' :
  IsPreconnected (Metric.eball x r) →
    ∀ {z₀ : 𝕜}, z₀ ∈ Metric.eball x r → f =ᶠ[nhds z₀] g → Set.EqOn f g (Metric.eball x r)
⊢ HasFPowerSeriesOnBall f p x r

Tactic:
apply (hg.mono hr_pos hr).congr

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
r : ENNReal
hr_pos : 0 < r
h : AnalyticOnNhd 𝕜 f (Metric.eball x r)
p : FormalMultilinearSeries 𝕜 𝕜 𝕜 := ⋯
hr : r ≤ p.radius
g : 𝕜 → 𝕜 := ⋯
hg : HasFPowerSeriesOnBall g p x p.radius
hg' :
  IsPreconnected (Metric.eball x r) →
    ∀ {z₀ : 𝕜}, z₀ ∈ Metric.eball x r → f =ᶠ[nhds z₀] g → Set.EqOn f g (Metric.eball x r)
⊢ Set.EqOn g f (Metric.eball x r)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
f : 𝕜 → 𝕜
x : 𝕜
⊢ 0 < (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial).radius →
    AnalyticOn 𝕜 f
        (Metric.eball x (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial).radius) →
      HasFPowerSeriesOnBall f (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial) x
        (FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial).radius

Tactic:
by
  intro hr hs
  exact hs.hasFPowerSeriesOnSubball hr le_rfl

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
c : 𝕜
⊢ analyticOrderAt (fun x ↦ x - c) c = 1

Tactic:
have := analyticOrderAt_centeredMonomial (n := 1) (z₀ := c)

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
c : 𝕜
this : analyticOrderAt ((fun x ↦ x - c) ^ 1) c = ↑1
⊢ analyticOrderAt (fun x ↦ x - c) c = 1
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case pos
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 : EventuallyConst g (𝓝 z₀)
⊢ analyticOrderAt (f ∘ g) z₀ = analyticOrderAt f (g z₀) * analyticOrderAt (fun x ↦ g x - g z₀) z₀

pp: case neg
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 : ¬EventuallyConst g (𝓝 z₀)
⊢ analyticOrderAt (f ∘ g) z₀ = analyticOrderAt f (g z₀) * analyticOrderAt (fun x ↦ g x - g z₀) z₀

Tactic:
· -- If `g` is eventually constant, both sides are either `⊤` or `0`.
    have := hg_nc.comp f
    rw [eventuallyConst_iff_analyticOrderAt_sub_eq_top] at hg_nc this
    rw [hg_nc]
    by_cases hf' : f (g z₀) = 0
    · simpa [hf', show analyticOrderAt f (g z₀) ≠ 0 by grind [analyticOrderAt_ne_zero]]
    · rw [show analyticOrderAt f (g z₀) = 0 from ?_, zero_mul] <;>
      grind [hf.comp hg, AnalyticAt.analyticOrderAt_eq_zero]

Goal state after:
case neg
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 : ¬EventuallyConst g (𝓝 z₀)
⊢ analyticOrderAt (f ∘ g) z₀ = analyticOrderAt f (g z₀) * analyticOrderAt (fun x ↦ g x - g z₀) z₀
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'
ι : Type u_5
s : Finset ι
F : ι → 𝕜 → 𝕜'
x : 𝕜
h : ∀ σ ∈ s, MeromorphicAt (F σ) x
⊢ MeromorphicAt (fun z ↦ ∏ n ∈ s, F n z) x

Tactic:
by
  convert! prod h (s := s)
  simp

Goal state after:
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
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ Subtype.val '' {u | analyticOrderAt f ↑u = 0 ∨ analyticOrderAt f ↑u = ⊤}))

Tactic:
disjoint_principal_right

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ U, (U \ Subtype.val '' {u | analyticOrderAt f ↑u = 0 ∨ analyticOrderAt f ↑u = ⊤})ᶜ ∈ 𝓝[≠] x
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
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {u | analyticOrderAt f u = 0 ∨ analyticOrderAt f u = ⊤}))

Tactic:
disjoint_principal_right

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ U, (U \ {u | analyticOrderAt f u = 0 ∨ analyticOrderAt f u = ⊤})ᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
x : 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicAt f x
⊢ ∀ᶠ (y : 𝕜) in 𝓝[≠] x, AnalyticAt 𝕜 f y

Tactic:
obtain ⟨n, h⟩ := h
  apply AnalyticAt.eventually_analyticAt at h
  refine (h.filter_mono ?_).mp ?_
  · simp [nhdsWithin]
  · rw [eventually_nhdsWithin_iff]
    apply Filter.Eventually.of_forall
    intro y hy hf
    rw [Set.mem_compl_iff, Set.mem_singleton_iff] at hy
    have := ((analyticAt_id (𝕜 := 𝕜).sub analyticAt_const).pow n).inv
      (pow_ne_zero _ (sub_ne_zero_of_ne hy))
    apply (this.smul hf).congr ∘ (eventually_ne_nhds hy).mono
    intro z hz
    simp [smul_smul, hz, sub_eq_zero]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : 𝕜 → E
x : 𝕜
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
z₀ : 𝕜
h₁ : AnalyticAt 𝕜 g z₀
h₂ : z₀ ∈ {x}ᶜ
⊢ deriv (fun z ↦ (z - x) ^ n • g z) z₀ = (↑n * (z₀ - x) ^ (n - 1)) • g z₀ + (z₀ - x) ^ n • deriv g z₀

Tactic:
rw [deriv_fun_smul (DifferentiableAt.zpow (by fun_prop) (by simp_all [sub_ne_zero_of_ne h₂]))
      (by fun_prop), add_comm, deriv_comp_sub_const (f := (· ^ n))]

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : 𝕜 → E
x : 𝕜
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
z₀ : 𝕜
h₁ : AnalyticAt 𝕜 g z₀
h₂ : z₀ ∈ {x}ᶜ
⊢ deriv (fun x ↦ x ^ n) (z₀ - x) • g z₀ + (z₀ - x) ^ n • deriv g z₀ =
    (↑n * (z₀ - x) ^ (n - 1)) • g z₀ + (z₀ - x) ^ n • deriv g z₀
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
r : R
k : ℕ
ih : (ascPochhammer ℕ k).smeval (-r) = (↑k).negOnePow • (descPochhammer ℤ k).smeval r
h : (X + ↑k).smeval (-r) = -(X + -↑k).smeval r
⊢ -((↑k).negOnePow • (descPochhammer ℤ k).smeval r) * (X + -↑k).smeval r =
    (↑(k + 1)).negOnePow • ((descPochhammer ℤ k).smeval r * (X + -↑k).smeval r)

Tactic:
Int.natCast_add,

Goal state after:
case succ
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
r : R
k : ℕ
ih : (ascPochhammer ℕ k).smeval (-r) = (↑k).negOnePow • (descPochhammer ℤ k).smeval r
h : (X + ↑k).smeval (-r) = -(X + -↑k).smeval r
⊢ -((↑k).negOnePow • (descPochhammer ℤ k).smeval r) * (X + -↑k).smeval r =
    (↑k + ↑1).negOnePow • ((descPochhammer ℤ k).smeval r * (X + -↑k).smeval r)
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 g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₂ : EqOn f g Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {x | f x = g x}))
⊢ f =ᶠ[𝓝[≠] x] g

Tactic:
disjoint_principal_right

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₂ : EqOn f g Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f x = g x})ᶜ ∈ 𝓝[≠] x
⊢ f =ᶠ[𝓝[≠] x] g
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case negSucc
k n : ℕ
⊢ (↑k.factorial *
      match -[n+1] with
      | ofNat n => ↑((n + k - 1).choose k)
      | -[n+1] => ↑(↑k).negOnePow * ↑((n + 1).choose k)) =
    (ascPochhammer ℕ k).smeval -[n+1]

Tactic:
simp only
      rw [mul_comm, mul_assoc, ← Nat.cast_mul, mul_comm _ (k.factorial),
        ← Nat.descFactorial_eq_factorial_mul_choose, ← descPochhammer_smeval_eq_descFactorial,
        ← Int.neg_ofNat_succ, ascPochhammer_smeval_neg_eq_descPochhammer]
      norm_cast

Goal state after:
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 g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {x | f x = g x}))
⊢ f =ᶠ[𝓝[≠] x] g

Tactic:
disjoint_principal_right

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f x = g x})ᶜ ∈ 𝓝[≠] x
⊢ f =ᶠ[𝓝[≠] x] g
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
n k : ℕ
⊢ k.factorial • choose (↑n) k = k.factorial • ↑(n.choose k)

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
n k : ℕ
⊢ (descPochhammer ℤ k).smeval ↑n = k.factorial • ↑(n.choose k)
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 g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h_eq : EqOn f g U
hu : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
⊢ MeromorphicOn g U

Tactic:
refine fun x hx ↦ (hf x hx).congr (EventuallyEq.filter_mono ?_ nhdsWithin_le_nhds)

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h_eq : EqOn f g U
hu : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : 𝕜
hx : x ∈ U
⊢ f =ᶠ[𝓝 x] g
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_2
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
k : ℕ
h_pos : 0 < k
⊢ choose 0 k = 0

Tactic:
by
  rw [← Nat.succ_pred_eq_of_pos h_pos, choose_zero_succ]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case hf
f : ℂ → ℂ
x : ℂ
hf : DifferentiableAt ℂ f x
c : ℂ
⊢ DifferentiableWithinAt ℂ f Set.univ x

Tactic:
rwa [differentiableWithinAt_univ]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x : 𝕜
hx : x ∈ U
⊢ Disjoint (𝓝[≠] x) (𝓟 (U \ {x | AnalyticAt 𝕜 f x}))

Tactic:
disjoint_principal_right

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x : 𝕜
hx : x ∈ U
⊢ (U \ {x | AnalyticAt 𝕜 f x})ᶜ ∈ 𝓝[≠] x
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ n.factorial • choose (-r) n = n.factorial • (↑n).negOnePow • choose (r + ↑n - 1) n

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ (descPochhammer ℤ n).smeval (-r) = n.factorial • (↑n).negOnePow • choose (r + ↑n - 1) n
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ (descPochhammer ℤ n).smeval (-r) = (↑n).negOnePow • n.factorial • choose (r + ↑n - 1) n

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n : ℕ
⊢ (descPochhammer ℤ n).smeval (-r) = (↑n).negOnePow • (descPochhammer ℤ n).smeval (r + ↑n - 1)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • n.factorial • choose r n = n.factorial • (choose r k * choose (r - ↑k) (n - k))

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • (descPochhammer ℤ n).smeval r = n.factorial • (choose r k * choose (r - ↑k) (n - k))
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • (descPochhammer ℤ n).smeval r =
    (n.choose k * k.factorial) • choose r k * (n - k).factorial • choose (r - ↑k) (n - k)

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • (descPochhammer ℤ n).smeval r =
    (n.choose k * k.factorial) • choose r k * (descPochhammer ℤ (n - k)).smeval (r - ↑k)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • (descPochhammer ℤ n).smeval r =
    n.choose k • k.factorial • choose r k * (descPochhammer ℤ (n - k)).smeval (r - ↑k)

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
n k : ℕ
hkn : k ≤ n
⊢ n.choose k • (descPochhammer ℤ n).smeval r =
    n.choose k • (descPochhammer ℤ k).smeval r * (descPochhammer ℤ (n - k)).smeval (r - ↑k)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Ring R
r s : R
h : Commute r s
k : ℕ
ih :
  (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)
⊢ (X - ↑k).smeval (r + s) * (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal (k + 1),
      ↑((k + 1).choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)

Tactic:
sum_antidiagonal_choose_succ_mul
      fun i j => ((descPochhammer ℤ i).smeval r * (descPochhammer ℤ j).smeval s),

Goal state after:
case succ
R : Type u_1
inst✝ : Ring R
r s : R
h : Commute r s
k : ℕ
ih :
  (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)
⊢ (X - ↑k).smeval (r + s) * (descPochhammer ℤ k).smeval (r + s) =
    ∑ ij ∈ antidiagonal k,
        ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ (ij.2 + 1)).smeval s) +
      ∑ ij ∈ antidiagonal k,
        ↑(k.choose ij.2) * ((descPochhammer ℤ (ij.1 + 1)).smeval r * (descPochhammer ℤ ij.2).smeval s)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
⊢ k.factorial • choose (r + s) k = k.factorial • ∑ ij ∈ antidiagonal k, choose r ij.1 * choose s ij.2

Tactic:
← descPochhammer_eq_factorial_smul_choose,

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
⊢ (descPochhammer ℤ k).smeval (r + s) = k.factorial • ∑ ij ∈ antidiagonal k, choose r ij.1 * choose s ij.2
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
k : ℕ
ih : HasFPowerSeriesWithinOnBall (iteratedFDerivWithin 𝕜 k f s) (p.iteratedFDerivSeries k) s x r
⊢ HasFPowerSeriesWithinOnBall (iteratedFDerivWithin 𝕜 (k + 1) f s) (p.iteratedFDerivSeries (k + 1)) s x r

Tactic:
rw [iteratedFDerivWithin_succ_eq_comp_left]

Goal state after:
case succ
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
k : ℕ
ih : HasFPowerSeriesWithinOnBall (iteratedFDerivWithin 𝕜 k f s) (p.iteratedFDerivSeries k) s x r
⊢ HasFPowerSeriesWithinOnBall
    (⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F).symm ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 k f s) s)
    (p.iteratedFDerivSeries (k + 1)) s x r
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := ⋯
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := ⋯
⊢ ∀ y ∈ Metric.eball 0 r, (∑ i ∈ Finset.range (n + 1), (q i) fun x ↦ y) = g (x + y)

pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := ⋯
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := ⋯
⊢ ∀ (m : ℕ), n + 1 ≤ m → q m = 0

Tactic:
· intro y hy
      rw [Finset.sum_eq_single_of_mem n]
      · simp [q, g]
      · simp
      · intro i hi h'i
        simp [q, h'i.symm]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := ⋯
gcont : ContDiff 𝕜 ω g
q : FormalMultilinearSeries 𝕜 E F := ⋯
⊢ ∀ (m : ℕ), n + 1 ≤ m → q m = 0
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: w x : ℂ
hw : w ≠ x
⊢ w - x ≠ 0

Tactic:
simp [sub_eq_zero, hw]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
h : μ s ≠ ∞
this : Fact (μ s < ∞)
⊢ μ s * ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, f x ∂μ

Tactic:
rw [← measure_mul_laverage, restrict_apply_univ]

Goal state after: