mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x c : 𝕜
f : 𝕜 → E
h : MeromorphicNFAt (f ∘ fun x ↦ x + c) x
⊢ x = x + c - c

Tactic:
ring

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 : MeromorphicAt f x
m : ℕ
hm : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
z : 𝕜
hfz : f z = 0
⊢ (z - x) ^ Exists.choose hf • 0 = 0

Tactic:
smul_zero

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
hf : MeromorphicAt f x
m : ℕ
hm : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
z : 𝕜
hfz : f z = 0
⊢ 0 = 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

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
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
h₂f : meromorphicOrderAt f x = ⊤
⊢ MeromorphicNFAt (toMeromorphicNFAt f x) x

Tactic:
have : toMeromorphicNFAt f x =ᶠ[𝓝 x] 0 := by
        apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE
        · exact hf.eq_nhdsNE_toMeromorphicNFAt.symm.trans (meromorphicOrderAt_eq_top_iff.1 h₂f)
        · simp [h₂f, toMeromorphicNFAt, hf]
      apply AnalyticAt.meromorphicNFAt
      rw [analyticAt_congr this]
      exact analyticAt_const

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

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 : toMeromorphicNFAt f x = f
⊢ MeromorphicNFAt f x

Tactic:
rw [hf.symm]
    exact meromorphicNFAt_toMeromorphicNFAt

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x

Tactic:
by
  by_cases h₁ : ¬ MeromorphicAt f x
  · aesop
  rw [not_not] at h₁
  by_cases h₂ : meromorphicOrderAt f x = ⊤
  · simp_all [← meromorphicOrderAt_neg]
  obtain ⟨g, h₁g, h₂g, h₃g⟩ := (meromorphicOrderAt_ne_top_iff h₁).1 h₂
  rw [h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g h₃g]
  rw [AnalyticAt.meromorphicTrailingCoeffAt_of_eq_nhdsNE (g := -g)]
  · simp
  · fun_prop
  · filter_upwards [h₃g] with a ha
    simp [ha, ← meromorphicOrderAt_neg]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
⊢ (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) = support d

Tactic:
ext u
  constructor <;> intro h
  · simp_all only [mem_mulSupport, ne_eq, mem_support]
    by_contra hCon
    simp_all
  · simp_all only [mem_mulSupport, ne_eq, ne_iff]
    use u
    simp_all [zero_zpow_eq_one₀]

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
hxy : x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y

pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
hxy : ¬x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y

Tactic:
· rw [hxy, meromorphicTrailingCoeffAt_id_sub_const]
    simp_all

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
hxy : ¬x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case coe
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hf : MeromorphicAt f x
n : ℕ
hn : meromorphicOrderAt (f ^ n) x = ↑n * meromorphicOrderAt f x
a✝ : ℤ
⊢ ↑n * a✝ + a✝ = (↑n + 1) * a✝

Tactic:
ring

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
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ -((divisor f (ball 0 R)) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯) =
    (-divisor f (ball 0 R)) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ x ∈ ball 0 R

Tactic:
locallyFinsuppWithin.coe_neg,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ -((divisor f (ball 0 R)) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯) =
    (-⇑(divisor f (ball 0 R))) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ x ∈ ball 0 R
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case refine_3
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
hg : MeromorphicNFOn g U
⊢ f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u) • g

Tactic:
· -- f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), fun z ↦ (z - u) ^ (divisor f U) u) * g
    filter_upwards [(divisor f U).eq_zero_codiscreteWithin,
      (hφ.inv.smul h₁f).meromorphicNFAt_mem_codiscreteWithin,
      self_mem_codiscreteWithin U] with a h₂a h₃a h₄a
    unfold g
    simp only [Pi.smul_apply', toMeromorphicNFOn_eq_toMeromorphicNFAt (hφ.inv.smul h₁f) h₄a,
      toMeromorphicNFAt_eq_self.2 h₃a, Pi.inv_apply]
    rw [← smul_assoc, smul_eq_mul, mul_inv_cancel₀ _, one_smul]
    rwa [← ((meromorphicNFOn_univ (divisor f U)) trivial).meromorphicOrderAt_eq_zero_iff,
      meromorphicOrderAt_eq, h₂a, Pi.zero_apply, WithTop.coe_zero]

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
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
z : ℂ
hz : z ∈ ball 0 R
this : (divisor g (ball 0 R)) z = 0
⊢ meromorphicOrderAt g z = 0

Tactic:
divisor_apply (fun x hx ↦ (h₂g (ball_subset_closedBall hx)).meromorphicAt) hz

Goal state after:
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
z : ℂ
hz : z ∈ ball 0 R
this : (meromorphicOrderAt g z).untop₀ = 0
⊢ meromorphicOrderAt g z = 0
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ℓ : ℂ →ₗ[ℝ] E
h : ℓ I = I • ℓ 1
a b : ℂ
t₀ : (↑a.im • I) • ↑b.re = ↑(a.im * b.re) • I
t₁ : (↑a.im • I) • ↑b.im • I = ↑(-a.im * b.im) • 1
t₂ : ∀ {r : ℝ}, ℓ ↑r = r • ℓ 1
⊢ ↑a.im * ↑b.re * I = ↑a.im * I * ↑b.re

Tactic:
ring

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

Goal state before:
pp: t : |2 / 3| = 2 / 3
z : |∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1) + log (1 - 2 / 3)| ≤ |2 / 3| ^ (70 + 1) / (1 - |2 / 3|)
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)

Tactic:
t,

Goal state after:
t : |2 / 3| = 2 / 3
z : |∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1) + log (1 - 2 / 3)| ≤ (2 / 3) ^ (70 + 1) / (1 - 2 / 3)
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)
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
⊢ ((∏ 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 =
    ((∏ i ∈ t₂, meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ i ∈ t₁, meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      ((∏ n ∈ t₂, meromorphicTrailingCoeffAt (canonicalFactor R n ^ (-(divisor f B₀R) n)) w) *
          ∏ n ∈ t₁, meromorphicTrailingCoeffAt ((fun x ↦ x - n) ^ (divisor f S₀R) n) w) •
        h w

Tactic:
rw [MeromorphicAt.meromorphicTrailingCoeffAt_smul (by fun_prop)
        (D.analyticOnNhd w hw).meromorphicAt,
        MeromorphicAt.meromorphicTrailingCoeffAt_mul (by fun_prop) (by fun_prop),
        meromorphicTrailingCoeffAt_prod (by fun_prop),
        meromorphicTrailingCoeffAt_prod (by fun_prop),
        (D.analyticOnNhd w hw).meromorphicTrailingCoeffAt_of_ne_zero (D.ne_zero w hw)]

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

Goal state before:
pp: t : |4 / 5| = 4 / 5
z : |∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1) + log (1 - 4 / 5)| ≤ |4 / 5| ^ (130 + 1) / (1 - |4 / 5|)
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)

Tactic:
t,

Goal state after:
t : |4 / 5| = 4 / 5
z : |∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1) + log (1 - 4 / 5)| ≤ (4 / 5) ^ (130 + 1) / (1 - 4 / 5)
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)
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
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ ((-divisor f S₀R) x + (divisor f S₀R) x) = 1

pp: case ha
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 : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ≠ 0

Tactic:
locallyFinsuppWithin.coe_neg,

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
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ ((-⇑(divisor f S₀R)) x + (divisor f S₀R) x) = 1

case ha
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 : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ≠ 0
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y : X
γ : Path.Homotopic.Quotient x y
ex : ↑(p ⁻¹' {x})
ey : ↑(p ⁻¹' {y})
Γ : Path.Homotopic.Quotient ↑ex ↑ey
eq : Γ.map { toFun := p, continuous_toFun := ⋯ } = γ.cast ⋯ ⋯
e_1✝ : ↑(p ⁻¹' {{ toFun := p, continuous_toFun := ⋯ } ↑ey}) = ↑(p ⁻¹' {y})
e_7✝ : { toFun := p, continuous_toFun := ⋯ } ↑ex = x
e_8✝ : { toFun := p, continuous_toFun := ⋯ } ↑ey = y
⊢ Γ.map { toFun := p, continuous_toFun := ⋯ } ≍ γ

Tactic:
rw [eq]

Goal state after:
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y : X
γ : Path.Homotopic.Quotient x y
ex : ↑(p ⁻¹' {x})
ey : ↑(p ⁻¹' {y})
Γ : Path.Homotopic.Quotient ↑ex ↑ey
eq : Γ.map { toFun := p, continuous_toFun := ⋯ } = γ.cast ⋯ ⋯
e_1✝ : ↑(p ⁻¹' {{ toFun := p, continuous_toFun := ⋯ } ↑ey}) = ↑(p ⁻¹' {y})
e_7✝ : { toFun := p, continuous_toFun := ⋯ } ↑ex = x
e_8✝ : { toFun := p, continuous_toFun := ⋯ } ↑ey = y
⊢ γ.cast ⋯ ⋯ ≍ γ
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C✝ : ℝ
f : ℂ → E
z : ℂ
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)
hB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))
hza : a < z.im
hzb : z.im < b
C : ℝ
hle_a : ∀ (z : ℂ), z.im = a → ‖f z‖ ≤ C
hle_b : ∀ (z : ℂ), z.im = b → ‖f z‖ ≤ C
hC₀ : 0 < C
⊢ a = (a + b) / 2 - (b - a) / 2

Tactic:
ring

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C✝ : ℝ
f : ℂ → E
z : ℂ
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)
hB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))
hza : a < z.im
hzb : z.im < b
C : ℝ
hle_a : ∀ (z : ℂ), z.im = a → ‖f z‖ ≤ C
hle_b : ∀ (z : ℂ), z.im = b → ‖f z‖ ≤ C
hC₀ : 0 < C
⊢ b = (a + b) / 2 + (b - a) / 2

Tactic:
ring

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

Goal state before:
pp: case hf.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      ‖↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z‖
⊢ Tendsto (fun a ↦ a + sSupNormIm f 0) (𝓝[>] 0) (𝓝 (sSupNormIm f 0))

Tactic:
nth_rw 2 [← zero_add (sSupNormIm f 0)]

Goal state after:
case hf.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
this :
  ∀ x ∈ Ioi 0,
    (x + sSupNormIm f 0) ^ (1 - z.re) * (x + sSupNormIm f 1) ^ z.re =
      ‖↑(x + sSupNormIm f 0) ^ (1 - z) * ↑(x + sSupNormIm f 1) ^ z‖
⊢ Tendsto (fun a ↦ a + sSupNormIm f 0) (𝓝[>] 0) (𝓝 (0 + sSupNormIm f 0))
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
⊢ ‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re

Tactic:
have : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re := by
    by_cases! h : sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0
    · rw [interpStrip_eq_of_zero f z h, norm_zero, mul_nonneg_iff]
      left
      exact ⟨Real.rpow_nonneg (sSupNormIm_nonneg f _) _,
        Real.rpow_nonneg (sSupNormIm_nonneg f _) _ ⟩
    · rcases h with ⟨h0, h1⟩
      rw [ne_comm] at h0 h1
      simp_rw [interpStrip_eq_of_pos f _ (lt_of_le_of_ne (sSupNormIm_nonneg f 0) h0)
        (lt_of_le_of_ne (sSupNormIm_nonneg f 1) h1)]
      simp only [norm_mul]
      rw [norm_cpow_eq_rpow_re_of_pos ((Ne.le_iff_lt h0).mp (sSupNormIm_nonneg f _)) _]
      rw [norm_cpow_eq_rpow_re_of_pos ((Ne.le_iff_lt h1).mp (sSupNormIm_nonneg f _)) _]
      simp only [sub_re, one_re, le_refl]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ ‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ 0 ≤ b ^ z.re

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ^ (1 - z.re) ≤ a ^ (1 - z.re)

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 1 ^ z.re ≤ b ^ z.re

Tactic:
· apply (Real.rpow_nonneg _ _)
    specialize hb 1
    simp only [mem_preimage, one_re, mem_singleton_iff, forall_true_left] at hb
    exact (norm_nonneg _).trans hb

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ^ (1 - z.re) ≤ a ^ (1 - z.re)

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 1 ^ z.re ≤ b ^ z.re
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

Goal state before:
pp: E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f : E → F
x : E
l : F →L[ℝ] G
h : ContDiffAt ℝ 2 f x
⊢ Δ (⇑l ∘ f) x = (⇑l ∘ Δ f) x

Tactic:
by
  simp [laplacian_eq_iteratedFDeriv_stdOrthonormalBasis, l.iteratedFDeriv_comp_left h]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
⊢ (volume.real (Ι a b))⁻¹ • ∫ (x : ℝ) in Ι a b, f x = (volume.real (Ι b a))⁻¹ • ∫ (x : ℝ) in Ι b a, f x

Tactic:
uIoc_comm

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
⊢ (volume.real (Ι b a))⁻¹ • ∫ (x : ℝ) in Ι b a, f x = (volume.real (Ι b a))⁻¹ • ∫ (x : ℝ) in Ι b a, f x
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: case e_a.e_f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R θ : ℝ
⊢ ↑R * cexp (↑θ * I) + c = c + ↑R * cexp (↑θ * I)

Tactic:
ring

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
⊢ circleAverage f c R = f c

Tactic:
have h : CircleIntegrable f c R :=
    (hf.continuousOn.mono sphere_subset_closedBall).circleIntegrable'
  rw [SeparatingDual.eq_iff_forall_dual_eq (R := ℝ)]
  intro g
  rw [← g.circleAverage_comp_comm h]
  obtain ⟨e, h₁e, h₂e⟩ := (isCompact_closedBall c |R|).exists_thickening_subset_open
    (isOpen_setOfPred_harmonicAt (g ∘ f)) (hf.comp_CLM g)
  rw [thickening_closedBall h₁e (abs_nonneg R)] at h₂e
  obtain ⟨F, h₁F, h₂F⟩ := InnerProductSpace.HarmonicOnNhd.exists_analyticOnNhd_ball_re_eq h₂e
  have h₃F : DifferentiableOn ℂ F (closure (ball c |R|)) := by
    intro x hx
    apply (h₁F x _).differentiableWithinAt
    grind [mem_ball, mem_closedBall.1 (closure_ball_subset_closedBall hx)]
  have h₄F : Set.EqOn (Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|) :=
    fun x hx ↦ h₂F (sphere_subset_ball (lt_add_of_pos_left |R| h₁e) hx)
  rw [← circleAverage_congr_sphere h₄F, Complex.reCLM.circleAverage_comp_comm,
    h₃F.diffContOnCl.circleAverage]
  · apply h₂F
    simp [mem_ball, dist_self, add_pos_of_pos_of_nonneg h₁e (abs_nonneg R)]
  · apply (h₁F.continuousOn.mono (fun _ _ ↦ by simp_all [dist_eq_norm])).circleIntegrable'

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

Goal state before:
pp: a b : ℂ
⊢ (a.re + b.re) * (a.re - b.re) + (a.im + b.im) * (a.im - b.im) =
    a.re * a.re + a.im * a.im - (b.re * b.re + b.im * b.im)

Tactic:
ring

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

Goal state before:
pp: φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
h_subst : (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)
⊢ R * R + r * r - 2 * (R * Real.cos θ * (r * Real.cos φ) + R * Real.sin θ * (r * Real.sin φ)) =
    R ^ 2 + r ^ 2 - 2 * R * r * (Real.cos θ * Real.cos φ + Real.sin θ * Real.sin φ)

Tactic:
ring

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

Goal state before:
pp: θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
key : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r
⊢ R ^ 2 + r ^ 2 + 2 * (R * r) = (R + r) * (R + r)

Tactic:
ring

Goal state after: