mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: 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
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f ((lineMap c z) 0)) =o[𝓝 ((lineMap c z) 0)] fun w ↦ ‖w - (lineMap c z) 0‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
⊢ Tendsto (⇑(lineMap c z)) (𝓝 0) (𝓝 ((lineMap c z) 0))

pp: 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
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f ((lineMap c z) 0)) =o[𝓝 ((lineMap c z) 0)] fun w ↦ ‖w - (lineMap c z) 0‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
⊢ ((fun w ↦ ‖w - (lineMap c z) 0‖ ^ n) ∘ ⇑(lineMap c z)) =O[𝓝 0] fun w ↦ w ^ n

Tactic:
· exact Continuous.tendsto (by fun_prop) 0

Goal state after:
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
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f ((lineMap c z) 0)) =o[𝓝 ((lineMap c z) 0)] fun w ↦ ‖w - (lineMap c z) 0‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
⊢ ((fun w ↦ ‖w - (lineMap c z) 0‖ ^ n) ∘ ⇑(lineMap c z)) =O[𝓝 0] fun w ↦ w ^ n
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
R : ℝ
f : E → F
z : E
hd : DifferentiableOn ℂ f (ball 0 R)
h_maps : MapsTo f (ball 0 R) (closedBall 0 R)
h₀ : f 0 = 0
hz : ‖z‖ < R
⊢ MapsTo f (ball 0 R) (closedBall (f 0) R)

Tactic:
by rwa [h₀]

Goal state after:
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
g : 𝕜 → 𝕜
x : 𝕜
hg : AnalyticAt 𝕜 g x
n : ℤ
q : 𝕜 → E
hq_an : AnalyticAt 𝕜 q (g x)
hq_ne : q (g x) ≠ 0
hf : f =ᶠ[𝓝 (g x)] (fun x_1 ↦ x_1 - g x) ^ n • q
hord : ∀ᶠ (z : 𝕜) in 𝓝 x, g z - g x = 0
h : ¬f (g x) = 0
⊢ MeromorphicNFAt (f ∘ g) x

Tactic:
· refine Or.inr ⟨0, fun _ ↦ f (g x), by fun_prop, h, ?_⟩
      filter_upwards [hord, hg.continuousAt.preimage_mem_nhds (hf.filter_mono (by simp))]
        using by simp_all [sub_eq_zero]

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

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
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₂ : MeromorphicAt f₂ x
h : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x
hf₁ : MeromorphicAt f₁ x
h₁f₂ : meromorphicOrderAt f₂ x = ⊤
⊢ f₁ + f₂ =ᶠ[𝓝[≠] x] f₁

Tactic:
filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₁f₂]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₂ : MeromorphicAt f₂ x
h : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x
hf₁ : MeromorphicAt f₁ x
h₁f₂ : meromorphicOrderAt f₂ x = ⊤
⊢ ∀ (a : 𝕜), f₂ a = 0 → (f₁ + f₂) a = f₁ a
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
x : 𝕜
h : ∀ σ ∈ s, MeromorphicAt (f σ) x
⊢ meromorphicTrailingCoeffAt (fun z ↦ ∏ n ∈ s, f n z) x = ∏ n ∈ s, meromorphicTrailingCoeffAt (f n) x

Tactic:
by
  convert! meromorphicTrailingCoeffAt_prod h
  simp

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = (fun x ↦ x - u₀) ^ d u₀ * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d u₀ 0 u

Tactic:
· have : (fun u ↦ (fun x ↦ x - u) ^ d u).mulSupport ⊆ hd.toFinset := by
      simp [mulSupport]
    rw [finprod_eq_prod_of_mulSupport_subset _ this]
    have : u₀ ∈ hd.toFinset := by simp_all
    rw [← Finset.mul_prod_erase hd.toFinset _ this]
    congr 1
    have : (fun u ↦ (· - u) ^ (update d u₀ 0 u)).mulSupport ⊆ hd.toFinset.erase u₀ := by
      rw [mulSupport]
      intro x hx
      by_cases h₁x : x = u₀ <;> simp_all
    simp_all [finprod_eq_prod_of_mulSupport_subset _ this, Finset.prod_congr rfl]

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
f : 𝕜 → E
x : 𝕜
hf : AnalyticAt 𝕜 f x
⊢ meromorphicOrderAt f x = ENat.map Nat.cast (analyticOrderAt f x)

Tactic:
cases hn : analyticOrderAt f x
  · rw [ENat.map_top, meromorphicOrderAt_eq_top_iff]
    exact (analyticOrderAt_eq_top.mp hn).filter_mono nhdsWithin_le_nhds
  · simp_rw [ENat.map_natCast, meromorphicOrderAt_eq_int_iff hf.meromorphicAt, zpow_natCast]
    rcases hf.analyticOrderAt_eq_natCast.mp hn with ⟨g, h1, h2, h3⟩
    exact ⟨g, h1, h2, h3.filter_mono nhdsWithin_le_nhds⟩

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
⊢ u ∉ ↑h.toFinset → (x - u) ^ update d x 0 u = 1

Tactic:
by_cases h₁ : u = x

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
h₁ : u = x
⊢ u ∉ ↑h.toFinset → (x - u) ^ update d x 0 u = 1

case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
h₁ : ¬u = x
⊢ u ∉ ↑h.toFinset → (x - u) ^ update d x 0 u = 1
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w z : ℂ
hw : w ∈ ball 0 R
hz : z ∈ sphere 0 R
hR : 0 < R
hzw : z - w ≠ 0
⊢ ‖↑R * (z - w)‖ ≠ 0

Tactic:
rw [ne_eq, norm_eq_zero]

Goal state after:
R : ℝ
w z : ℂ
hw : w ∈ ball 0 R
hz : z ∈ sphere 0 R
hR : 0 < R
hzw : z - w ≠ 0
⊢ ¬↑R * (z - w) = 0
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

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
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
u : 𝕜
hx : (fun x ↦ ↑(D u) * log ‖x - u‖) = 0
⊢ D u = 0

pp: case mpr
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
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
u : 𝕜
hx : D u = 0
⊢ (fun x ↦ ↑(D u) * log ‖x - u‖) = 0

Tactic:
· obtain ⟨y, hy⟩ := NormedField.exists_one_lt_norm 𝕜
      have := congrFun hx (y + u)
      simp only [add_sub_cancel_right, Pi.zero_apply, mul_eq_zero, Int.cast_eq_zero, log_eq_zero,
        norm_eq_zero] at this
      rcases this with h | h | h | h
      · assumption
      · simp only [h, norm_zero] at hy
        linarith
      · simp only [h, lt_self_iff_false] at hy
      · simp only [h, lt_neg_self_iff] at hy
        linarith

Goal state after:
case mpr
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
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
u : 𝕜
hx : D u = 0
⊢ (fun x ↦ ↑(D u) * log ‖x - u‖) = 0
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_4
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
homeo : IsLocalHomeomorph p
sep : IsSeparatedMap p
γ₀ γ₁ : C(↑I, X)
γ : γ₀.HomotopyRel γ₁ {0, 1}
Γ : ↑I → C(↑I, E)
Γ_lifts : ∀ (t s : ↑I), p ((Γ t) s) = γ (t, s)
Γ_0 : ∀ (t : ↑I), (Γ t) 0 = (Γ 0) 0
t✝ : ↑I
this : Continuous[_, inst✝¹] fun st ↦ (Γ st.2) st.1
t t' : ↑I
⊢ γ (t, 1) = γ (t', 1)

Tactic:
γ.eq_fst _ (.inr rfl)

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

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
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₂ : MeromorphicAt f₂ x
h : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x
hf₁ : MeromorphicAt f₁ x
h₁f₂ : meromorphicOrderAt f₂ x = ⊤
⊢ f₁ + f₂ =ᶠ[𝓝[≠] x] f₁

Tactic:
filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₁f₂]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₂ : MeromorphicAt f₂ x
h : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x
hf₁ : MeromorphicAt f₁ x
h₁f₂ : meromorphicOrderAt f₂ x = ⊤
⊢ ∀ (a : 𝕜), f₂ a = 0 → (f₁ + f₂) a = f₁ a
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
h₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤
hR : 0 < R
φ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := toMeromorphicNFOn φ (closedBall 0 R)
h₃g : divisor g (ball 0 R) = 0
h₂g : MeromorphicNFOn g (closedBall 0 R)
h₄g : ∀ {z : ℂ}, z ∈ closedBall 0 R → meromorphicOrderAt g z ≠ ⊤
η₀ : (-divisor f (ball 0 R)).support.Finite
⊢ f =ᶠ[codiscreteWithin (closedBall 0 R)] (∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) • g

Tactic:
by
      trans (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) • φ
      · unfold φ
        rw [finprod_eq_prod_of_mulSupport_subset_of_finite _ (by aesop) η₀]
        · filter_upwards [codiscreteWithin_mono (by tauto) η₀.compl_mem_codiscrete,
            self_mem_codiscreteWithin (closedBall 0 R)] with a ha h₂a
          simp only [Pi.smul_apply', Finset.prod_apply, Pi.pow_apply]
          rw [← smul_assoc, ← Finset.prod_smul, Finset.prod_eq_one, one_smul]
          intro x hx
          rw [smul_eq_mul, ← zpow_add', neg_add_cancel, zpow_zero]
          simp_all only [ne_eq, Subtype.forall, mem_closedBall, dist_zero_right,
            locallyFinsuppWithin.support_neg, mem_compl_iff, mem_support, Decidable.not_not,
            Finite.mem_toFinset, neg_add_cancel, not_true_eq_false, neg_eq_zero, and_self, or_self,
            or_false]
          apply canonicalFactor_ne_zero _ (by simp_all) (by grind)
          by_contra h
          simp_all
      · rw [finprod_eq_prod_of_mulSupport_subset_of_finite _ (by aesop) η₀]
        filter_upwards [toMeromorphicNFOn_eqOn_codiscrete hφ] using by simp_all [g]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
x : ℂ
h₁ : DifferentiableAt ℝ f x
h₂ : (fderiv ℝ f x) I = I • (fderiv ℝ f x) 1
⊢ HasDerivAt f (((fderiv ℝ f x).complexOfReal h₂) 1) x

Tactic:
rw [hasDerivAt_iff_hasFDerivAt, toSpanSingleton_apply_map_one]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
x : ℂ
h₁ : DifferentiableAt ℝ f x
h₂ : (fderiv ℝ f x) I = I • (fderiv ℝ f x) 1
⊢ HasFDerivAt f ((fderiv ℝ f x).complexOfReal h₂) x
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
⊢ ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      meromorphicTrailingCoeffAt
        (((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f B₀R) u)) * ∏ᶠ (v : ℂ), (fun x ↦ x - v) ^ (divisor f S₀R) v) •
          h)
        w =
    ((∏ i ∈ t₂, meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ i ∈ t₁, meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      meromorphicTrailingCoeffAt
        (((∏ i ∈ t₂, canonicalFactor R i ^ (-(divisor f B₀R) i)) * ∏ i ∈ t₁, (fun x ↦ x - i) ^ (divisor f S₀R) i) • h) w

Tactic:
finprod_eq_prod_of_mulSupport_subset (s := t₂) _ _,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
⊢ ((∏ i ∈ t₂, meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      meromorphicTrailingCoeffAt
        (((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f B₀R) u)) * ∏ᶠ (v : ℂ), (fun x ↦ x - v) ^ (divisor f S₀R) v) •
          h)
        w =
    ((∏ i ∈ t₂, meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ i ∈ t₁, meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      meromorphicTrailingCoeffAt
        (((∏ i ∈ t₂, canonicalFactor R i ^ (-(divisor f B₀R) i)) * ∏ i ∈ t₁, (fun x ↦ x - i) ^ (divisor f S₀R) i) • h) w

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
⊢ (mulSupport fun i ↦ meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) ⊆ ↑t₂
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ |log 5 - 160943791243 / 100000000000| ≤ 1 / 10 ^ 10

Tactic:
suffices |log 5 - 160943791243 / 100000000000| ≤
      (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹) by
    norm_num1 at *
    assumption
  have t : |4 / 5| = (4 : ℝ) / 5 := by norm_num
  have z := abs_log_sub_add_sum_range_le (x := 4 / 5) (by norm_num) 130
  rw [t, show (1 - (4 : ℝ) / 5) = (1 / 5 : ℝ) by norm_num, one_div (5 : ℝ), log_inv,
    ← sub_eq_add_neg, _root_.abs_sub_comm] at z
  apply le_trans (_root_.abs_sub_le _ _ _) (add_le_add z _)
  norm_num [sum_range_succ]

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

Goal state before:
pp: case η₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
hw : w ∈ closedBall 0 R
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
this : MeromorphicAt h w
⊢ ∀ x ∈ t₁,
    meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ (-divisor f S₀R) x *
        meromorphicTrailingCoeffAt ((fun x_1 ↦ x_1 - x) ^ (divisor f S₀R) x) w =
      1

Tactic:
case η₂ =>
        intro x hx
        rw [MeromorphicAt.meromorphicTrailingCoeffAt_zpow (by fun_prop), ← zpow_add₀,
          locallyFinsuppWithin.coe_neg, Pi.neg_apply, neg_add_cancel, zpow_zero]
        rw [meromorphicTrailingCoeffAt_id_sub_const]
        grind

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ log
      ‖((∏ᶠ (i : ℂ), canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ᶠ (i : ℂ), (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖ =
    log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ i ∈ t₁, (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖

Tactic:
finprod_eq_prod_of_mulSupport_subset (s := t₂) _ _,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ᶠ (i : ℂ), (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖ =
    log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ i ∈ t₁, (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ (mulSupport fun i ↦ canonicalFactor R i w ^ (divisor f B₀R) i) ⊆ ↑t₂
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hd : d < π / 2 / b
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
A : ℝ
hA₀ : A > 0
hA :
  ∀ᶠ (b_1 : ℝ) in atTop,
    ∀ (a_1 : ℂ), |a_1.re| = b_1 → a - b < a_1.im ∧ a_1.im < a + b → ‖f a_1‖ ≤ A * expR (B * expR (c * |a_1.re|))
this : Tendsto (fun R ↦ expR (δ * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (𝓝 0)
R : ℝ
hR : expR (δ * expR (d * R) + B * expR (c * R) + Real.log A) ≤ C
Hle : ∀ (a_1 : ℂ), |a_1.re| = R → a - b < a_1.im ∧ a_1.im < a + b → ‖f a_1‖ ≤ A * expR (B * expR (c * |a_1.re|))
w : ℂ
hre : |w.re| = R
him : w.im ∈ Ioo (a - b) (a + b)
⊢ ‖g ε w • f w‖ ≤ C

Tactic:
calc
        ‖g ε w • f w‖ ≤ expR (δ * expR (d * R) + B * expR (c * R) + Real.log A) := ?_
        _ ≤ C := hR

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hd : d < π / 2 / b
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
A : ℝ
hA₀ : A > 0
hA :
  ∀ᶠ (b_1 : ℝ) in atTop,
    ∀ (a_1 : ℂ), |a_1.re| = b_1 → a - b < a_1.im ∧ a_1.im < a + b → ‖f a_1‖ ≤ A * expR (B * expR (c * |a_1.re|))
this : Tendsto (fun R ↦ expR (δ * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (𝓝 0)
R : ℝ
hR : expR (δ * expR (d * R) + B * expR (c * R) + Real.log A) ≤ C
Hle : ∀ (a_1 : ℂ), |a_1.re| = R → a - b < a_1.im ∧ a_1.im < a + b → ‖f a_1‖ ≤ A * expR (B * expR (c * |a_1.re|))
w : ℂ
hre : |w.re| = R
him : w.im ∈ Ioo (a - b) (a + b)
⊢ ‖g ε w • f w‖ ≤ expR (δ * expR (d * R) + B * expR (c * R) + Real.log A)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: z : ℂ
l u : ℝ
hul : l < u
hz : z ∈ re ⁻¹' Icc l u
⊢ u - l ≠ 0

Tactic:
by linarith

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

Goal state before:
pp: f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
h₁f : ContDiffAt ℝ 2 f x
⊢ (fderiv ℝ (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) x - I • fderiv ℝ (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I) x) I =
    I • (fderiv ℝ (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) x - I • fderiv ℝ (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I) x) 1

Tactic:
fderiv_comp

Goal state after:
f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
h₁f : ContDiffAt ℝ 2 f x
⊢ (fderiv ℝ (⇑ofRealCLM) ((fderiv ℝ f x) 1) ∘SL fderiv ℝ (fun x ↦ (fderiv ℝ f x) 1) x -
        I • fderiv ℝ (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I) x)
      I =
    I •
      (fderiv ℝ (⇑ofRealCLM) ((fderiv ℝ f x) 1) ∘SL fderiv ℝ (fun x ↦ (fderiv ℝ f x) 1) x -
          I • fderiv ℝ (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I) x)
        1

case hg
f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
h₁f : ContDiffAt ℝ 2 f x
⊢ DifferentiableAt ℝ (⇑ofRealCLM) ((fderiv ℝ f x) 1)

case hf
f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
h₁f : ContDiffAt ℝ 2 f x
⊢ DifferentiableAt ℝ (fun x ↦ (fderiv ℝ f x) 1) x