mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
⊢ μ s < ∞ → ∫ (x : α) in s, (f - g) x ∂μ = 0

Tactic:
hμs

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, (f - g) x ∂μ = 0
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, (f - g) x ∂μ = 0

Tactic:
rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
htf_zero : f =ᵐ[μ.restrict tᶜ] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
⊢ ((μ.restrict t).trim hm) s < ∞ → IntegrableOn f s ((μ.restrict t).trim hm)

Tactic:
hμs

Goal state after:
case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
htf_zero : f =ᵐ[μ.restrict tᶜ] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ((μ.restrict t).trim hm) s < ∞
⊢ IntegrableOn f s ((μ.restrict t).trim hm)
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
htf_zero : f =ᵐ[μ.restrict tᶜ] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
⊢ ((μ.restrict t).trim hm) s < ∞ → ∫ (x : α) in s, f x ∂(μ.restrict t).trim hm = 0

Tactic:
hμs

Goal state after:
case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero✝ : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
htf_zero : f =ᵐ[μ.restrict tᶜ] 0
hf_meas_m : StronglyMeasurable f
s : Set α
hs : MeasurableSet s
hμs : ((μ.restrict t).trim hm) s < ∞
⊢ ∫ (x : α) in s, f x ∂(μ.restrict t).trim hm = 0
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
β : Type u_3
inst✝² : TopologicalSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ = 0
⊢ f =ᵐ[μ] 0

Tactic:
suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from
    hf.ae_eq_zero_of_forall_setIntegral_eq_zero (fun s hs _ ↦ this s hs)

Goal state after:
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
β : Type u_3
inst✝² : TopologicalSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ = 0
⊢ ∀ (s : Set β), MeasurableSet s → ∫ (x : β) in s, f x ∂μ = 0
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
⊢ ∫ (x : E), lineDeriv ℝ f x v * g x ∂μ = ∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ

Tactic:
suffices S1 : ∀ (t : ℝ), ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ =
                            ∫ x, (t⁻¹ • (g (x + t • (-v)) - g x)) * f x ∂μ by
    simp only [S1] at A; exact tendsto_nhds_unique A B

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
⊢ ∀ (t : ℝ), ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ = ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed[inst✝⁴] s
t : ℕ → Set β := fun n ↦ closure[inst✝⁴] (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n ↦ ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in s, f x ∂μ))

Tactic:
rw [B]

Goal state after:
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed[inst✝⁴] s
t : ℕ → Set β := fun n ↦ closure[inst✝⁴] (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n ↦ ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in ⋃ n, t n, f x ∂μ))
Module: Mathlib.Order.SuccPred.IntervalSucc

Goal state before:
pp: case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
m n : α
hmn : m ≤ n
k : α
hmk : m ≤ k
ihk : ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k)
⊢ ⋃ i ∈ Ico m (succ k), Ioc (f i) (f (succ i)) = Ioc (f m) (f (succ k))

Tactic:
← Ioc_union_Ioc_eq_Ioc (hf hmk) (hf <| le_succ _),

Goal state after:
case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
m n : α
hmn : m ≤ n
k : α
hmk : m ≤ k
ihk : ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k)
⊢ ⋃ i ∈ Ico m (succ k), Ioc (f i) (f (succ i)) = Ioc (f m) (f k) ∪ Ioc (f k) (f (succ k))
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
⊢ ∫ (x : E), g x • lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∑ i ∈ s, ∫ (a_1 : E), g a_1 • (a i • fun x ↦ lineDeriv ℝ f x (v i)) a_1 ∂μ

Tactic:
suffices S1 : ∫ x, lineDeriv ℝ f x (∑ i ∈ s, a i • v i) * g x ∂μ
      = ∑ i ∈ s, a i * ∫ x, lineDeriv ℝ f x (v i) * g x ∂μ by
    dsimp only [smul_eq_mul, Pi.smul_apply]
    simp_rw [← mul_assoc, mul_comm _ (a _), mul_assoc, integral_const_mul, mul_comm (g _), S1]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
⊢ ∫ (x : E), lineDeriv ℝ f x (∑ i ∈ s, a i • v i) * g x ∂μ = ∑ i ∈ s, a i * ∫ (x : E), lineDeriv ℝ f x (v i) * g x ∂μ
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a

Tactic:
by_cases! h : ∀ w ∈ Ioo x y, deriv f w ≠ 0

Goal state after:
case pos
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a

case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b

Tactic:
apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
      · refine hf.mono (Icc_subset_Icc hxw.le le_rfl)
      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
      · intro z hz
        rw [← hw]
        apply ne_of_gt
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1

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

Goal state before:
pp: case inr
x y : ℂ
x_eq : ∃ k, x = (2 * ↑k + 1) * ↑π / 2
minus_y_eq : ∃ l, y = (2 * ↑l + 1) * ↑π / 2
⊢ ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), -y ≠ (2 * ↑l + 1) * ↑π / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, -y = (2 * ↑l + 1) * ↑π / 2

Tactic:
refine .inr ⟨x_eq, ?_⟩
      rcases minus_y_eq with ⟨l, rfl⟩
      use -l - 1
      push_cast
      ring

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
by_cases! h : ∀ w ∈ Ioo x y, deriv f w ≠ 0

Goal state after:
case pos
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: this : -(π / 2) < π / 2
⊢ Tendsto (fun x ↦ tan ↑x) atBot atBot

Tactic:
rw [tendsto_comp_coe_Ioo_atBot this]; exact tendsto_tan_neg_pi_div_two

Goal state after:
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
R₁ R₂ : ℝ
f : E → F
c z : E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c

Tactic:
refine dist_le_mul_div_pow_of_mapsTo_ball_of_isLittleO (n := 0) hd h_maps ?_ hz |>.trans_eq ?_

Goal state after:
case refine_1
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
R₁ R₂ : ℝ
f : E → F
c z : E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
⊢ (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ 0

case refine_2
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
R₁ R₂ : ℝ
f : E → F
c z : E
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hz : z ∈ ball c R₁
⊢ R₂ * (dist z c / R₁) ^ (0 + 1) = R₂ / R₁ * dist z c
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h✝ : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
hacc : AccPt x (𝓟 U)
h : f =ᶠ[𝓝[≠] x] g
⊢ {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x

Tactic:
filter_upwards [h.nhdsNE_deriv] using by simp +contextual

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
hacc : ¬AccPt x (𝓟 U)
⊢ {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x

Tactic:
accPt_iff_frequently_nhdsNE,

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
hacc : ¬∃ᶠ (y : 𝕜) in 𝓝[≠] x, y ∈ U
⊢ {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicOn f₁ U
h₁ : {x | f₁ x = f₂ x} ∈ codiscreteWithin U
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∈ U
⊢ f₁ =ᶠ[𝓝[≠] x] f₂

Tactic:
mem_codiscreteWithin,

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicOn f₁ U
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {x | f₁ x = f₂ x}))
⊢ f₁ =ᶠ[𝓝[≠] x] f₂
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
h₁ : {x | f₁ x = f₂ x} ∈ codiscreteWithin U
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hf₁ : MeromorphicOn f₁ U
x : 𝕜
hx : x ∈ U
⊢ f₁ =ᶠ[𝓝[≠] x] f₂

Tactic:
mem_codiscreteWithin,

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hf₁ : MeromorphicOn f₁ U
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {x | f₁ x = f₂ x}))
⊢ f₁ =ᶠ[𝓝[≠] x] f₂
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
g : 𝕜 → E
hf : MeromorphicNFAt f x
hg : MeromorphicNFAt g x
h : f =ᶠ[𝓝[≠] x] g
t₀ : meromorphicOrderAt f x = meromorphicOrderAt g x
cs : ¬meromorphicOrderAt f x = 0
h₁f : f x = 0
h₁g : g x = 0
⊢ f x = g x

Tactic:
rw [h₁f, h₁g]

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

Goal state before:
pp: case neg.h₁g
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
this : ∀ {μ : ι}, μ ∈ s.erase τ → f μ x ≠ 0
⊢ AnalyticAt 𝕜 (∏ x ∈ s.erase τ, f x) x

Tactic:
apply Finset.analyticAt_prod _ (fun μ hμ ↦ h₃f (Finset.mem_of_mem_erase hμ) (by aesop))

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

Goal state before:
pp: case pos.hl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ 0 =ᶠ[𝓝[≠] x] f

Tactic:
filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₂] with y hy

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
y : 𝕜
hy : f y = 0
⊢ 0 y = f y
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: case neg.hl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
⊢ g =ᶠ[𝓝[≠] x] (fun x_1 ↦ x_1 - x) ^ (-(meromorphicOrderAt f x).untop₀) • f

Tactic:
filter_upwards [h₃g, self_mem_nhdsWithin] with y h₁y h₂y

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
y : 𝕜
h₁y : f y = (y - x) ^ (meromorphicOrderAt f x).untop₀ • g y
h₂y : y ∈ {x}ᶜ
⊢ g y = ((fun x_1 ↦ x_1 - x) ^ (-(meromorphicOrderAt f x).untop₀) • f) y
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : ¬MeromorphicAt f x
⊢ MeromorphicNFAt 0 x

Tactic:
exact analyticAt_const.meromorphicNFAt

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : MeromorphicOn f U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
⊢ {x | f x ≠ 0} ∈ codiscreteWithin U

Tactic:
mem_codiscreteWithin,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : MeromorphicOn f U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
⊢ ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {x | f x ≠ 0}))
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
h : toMeromorphicNFOn f U = f
⊢ MeromorphicNFOn (toMeromorphicNFOn f U) U

Tactic:
apply meromorphicNFOn_toMeromorphicNFOn

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

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho✝ : 0 ≤ meromorphicOrderAt f x
ho : 0 < meromorphicOrderAt f x
⊢ ∃ c, Tendsto f (𝓝[≠] x) (𝓝 c)

Tactic:
exact ⟨0, tendsto_zero_of_meromorphicOrderAt_pos ho⟩

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : d x = 0
h₁ : ¬HasFiniteMulSupport fun u ↦ (fun x ↦ x - u) ^ d u
⊢ (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) x ≠ 0

Tactic:
· simp [finprod_of_infinite_mulSupport h₁]

Goal state after:
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case pos.inl
R : ℝ
z : ℂ
hz hw : z ∈ ball 0 R
⊢ (meromorphicOrderAt (canonicalFactor R z) z).untop₀ =
    (-locallyFinsuppWithin.restrict (locallyFinsuppWithin.single z 1) ⋯) z

Tactic:
rw [meromorphicOrderAt_canonicalFactor hz]
      simp_all [Function.locallyFinsuppWithin.restrict_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
h₁f : MeromorphicOn f U
h₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤
h₃f : (divisor f U).support.Finite
φ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u
hφ : MeromorphicOn φ U
g : 𝕜 → E := toMeromorphicNFOn (φ⁻¹ • f) U
⊢ MeromorphicNFOn g U

Tactic:
apply meromorphicNFOn_toMeromorphicNFOn

Goal state after: