mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Uniform

Goal state before:
pp: E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
x' : E := ‖x‖⁻¹ • x
y' : E := ‖y‖⁻¹ • y
hxy' : ε / 3 ≤ ‖x' - y'‖
⊢ 2 - δ + δ' + δ' ≤ 2 - δ'

Tactic:
by
      suffices δ' ≤ δ / 3 by linarith
      exact min_le_of_right_le <| min_le_right _ _

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)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ ContinuousOn f (Icc x w)

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
⊢ StrictMonoOn (deriv f) (Ioo x w)

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
⊢ ∀ w_1 ∈ Ioo x w, deriv f w_1 ≠ 0

Tactic:
· exact hf.mono (Icc_subset_Icc le_rfl hwy.le)

Goal state after:
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
⊢ StrictMonoOn (deriv f) (Ioo x w)

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
⊢ ∀ w_1 ∈ Ioo x w, deriv f w_1 ≠ 0
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : StrictConvexSpace ℝ F
f : E → F
s : Set E
z w : E
hd : DiffContOnCl ℂ f s
hz : IsMaxOn (norm ∘ f) s z
hsub : ball z (dist w z) ⊆ s
H₁ : ‖f w‖ = ‖f z‖
H₂ : ‖f w + f z‖ = ‖f z + f z‖
⊢ ‖f w + f z‖ = ‖f w‖ + ‖f z‖

Tactic:
by simp only [H₂, SameRay.rfl.norm_add, H₁]

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

Goal state before:
pp: E : Type u
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : StrictConvexSpace ℝ F
inst✝ : ProperSpace E
f : E → F
r b : ℝ
h_an : DifferentiableOn ℂ f (ball 0 b)
hr_nn : 0 ≤ r
hr_lt : r < b
hr : ∀ z ∈ ball 0 b, ∃ w ∈ closedBall 0 r, ‖f z‖ ≤ ‖f w‖
x : E
hx_mem : x ∈ closedBall 0 r
hx_max : IsMaxOn (fun x ↦ ‖f x‖) (closedBall 0 r) x
⊢ EqOn f (Function.const E (f 0)) (ball 0 b)

Tactic:
suffices Set.EqOn f (Function.const E (f x)) (ball 0 b) by
    rwa [this (mem_ball_self (hr_nn.trans_lt hr_lt))]

Goal state after:
E : Type u
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : StrictConvexSpace ℝ F
inst✝ : ProperSpace E
f : E → F
r b : ℝ
h_an : DifferentiableOn ℂ f (ball 0 b)
hr_nn : 0 ≤ r
hr_lt : r < b
hr : ∀ z ∈ ball 0 b, ∃ w ∈ closedBall 0 r, ‖f z‖ ≤ ‖f w‖
x : E
hx_mem : x ∈ closedBall 0 r
hx_max : IsMaxOn (fun x ↦ ‖f x‖) (closedBall 0 r) x
⊢ EqOn f (Function.const E (f x)) (ball 0 b)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: z w : ℂ
⊢ cos z = w ↔ cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0

Tactic:
rw [← sub_eq_zero]
  simpa [field, cos, exp_neg] using Eq.congr (by ring) rfl

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ frontier U ∨ z ∈ U
⊢ ‖f z‖ ≤ C

Tactic:
rcases hz with hz | hz

Goal state after:
case inl
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ frontier U
⊢ ‖f z‖ ≤ C

case inr
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
U : Set E
hU : Bornology.IsBounded U
hd : DiffContOnCl ℂ f U
C : ℝ
hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C
z : E
hz : z ∈ U
⊢ ‖f z‖ ≤ C
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ cos (arctan x) ^ 2 = 1 / (1 + x ^ 2)

Tactic:
by
  rw [one_div, ← inv_one_add_tan_sq (cos_arctan_pos x).ne', tan_arctan]

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

Goal state before:
pp: case inl
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy2' : y < π / 2
hy4 : 0 < cos y
⊢ y ^ 2 + 1 < (1 + tan y ^ 2)⁻¹⁻¹

Tactic:
inv_inv,

Goal state after:
case inl
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy2' : y < π / 2
hy4 : 0 < cos y
⊢ y ^ 2 + 1 < 1 + tan y ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
⊢ sin (arctan x) < 0 ↔ x < 0

Tactic:
by
  simpa using sin_arctan_strictMono.lt_iff_lt (b := 0)

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

Goal state before:
pp: case refine_1
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (2 / π) ^ 2 * |θ ^ 2| ≤ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2

Tactic:
abs_sq

Goal state after:
case refine_1
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (2 / π) ^ 2 * θ ^ 2 ≤ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: case refine_2
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2 ≤ |θ ^ 2|

Tactic:
abs_sq

Goal state after:
case refine_2
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2 ≤ θ ^ 2
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : angle x y = π
⊢ ⟪x, y⟫ = -(‖x‖ * ‖y‖)

Tactic:
simp [← cos_angle_mul_norm_mul_norm, h]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : angle x y = 0
⊢ ⟪x, y⟫ = ‖x‖ * ‖y‖

Tactic:
simp [← cos_angle_mul_norm_mul_norm, h]

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

Goal state before:
pp: f : ℂ → ℂ
M R : ℝ
z : ℂ
hM : 0 < M
hf : DifferentiableOn ℂ f (ball 0 R)
hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}
hR : 0 < R
hz : z ∈ ball 0 R
hf₂ : f 0 = 0
w : ℂ := f z / (2 * ↑M - f z)
hzR : ‖z‖ < R
hwR : ‖f z / (2 * ↑M - f z)‖ ≤ ‖z‖ / R
h : 2 * ↑M = f z
⊢ False

Tactic:
simpa [← h, hM] using hf₁ hz

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
p : 𝕜[X]
x : 𝕜
hx : x ∈ ((fun x ↦ eval x p) '' {k | eval k (derivative p) = 0})ᶜ
⊢ ((fun x ↦ eval x p) ⁻¹' {x}).Finite

Tactic:
obtain rfl | ne := eq_or_ne p (C x)

Goal state after:
case inl
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
x : 𝕜
hx : x ∈ ((fun x_1 ↦ eval x_1 (C x)) '' {k | eval k (derivative (C x)) = 0})ᶜ
⊢ ((fun x_1 ↦ eval x_1 (C x)) ⁻¹' {x}).Finite

case inr
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
p : 𝕜[X]
x : 𝕜
hx : x ∈ ((fun x ↦ eval x p) '' {k | eval k (derivative p) = 0})ᶜ
ne : p ≠ C x
⊢ ((fun x ↦ eval x p) ⁻¹' {x}).Finite
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: f : ℂ → ℂ
c z : ℂ
R₂ : ℝ
n : ℕ
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n
R₁ : ℝ
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hd : DifferentiableOn ℂ f (closedBall c R₁)
h_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)
hne : z ≠ c
g : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f c)
g' : ℂ → ℂ := update g c ((𝓝[≠] c).limUnder g)
hdg' : DifferentiableOn ℂ g' (closedBall c R₁)
w : ℂ
hw : w ∈ sphere c R₁
hwc : w ≠ c
⊢ ‖f w - f c‖ ≤ R₂

Tactic:
simpa [dist_eq_norm] using h_maps (sphere_subset_closedBall hw)

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

Goal state before:
pp: 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, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
⊢ U ∈ 𝓝[≠] x

Tactic:
apply mem_nhdsWithin.mpr

Goal state after:
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, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
⊢ ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ U
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
g : 𝕜 → E
hf : MeromorphicNFAt f x
hg : MeromorphicNFAt g x
⊢ f =ᶠ[𝓝[≠] x] g ↔ f =ᶠ[𝓝 x] g

Tactic:
constructor
  · intro h
    have t₀ := meromorphicOrderAt_congr h
    by_cases cs : meromorphicOrderAt f x = 0
    · rw [cs] at t₀
      have Z := (hf.meromorphicOrderAt_nonneg_iff_analyticAt.1 (le_of_eq cs.symm)).continuousAt
      have W := (hg.meromorphicOrderAt_nonneg_iff_analyticAt.1 (le_of_eq t₀)).continuousAt
      exact (Z.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE W).1 h
    · apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE h
      let h₁f := cs
      rw [hf.meromorphicOrderAt_eq_zero_iff] at h₁f
      let h₁g := cs
      rw [t₀, hg.meromorphicOrderAt_eq_zero_iff] at h₁g
      simp only [not_not] at *
      rw [h₁f, h₁g]
  · exact (Filter.EventuallyEq.filter_mono · nhdsWithin_le_nhds)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : 𝕜 → E
⊢ (divisor (f ∘ fun x ↦ x - c) U) (z + c) = (divisor f (U + {-c})) z

Tactic:
← divisor_comp_add_const_eq_divisor

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : 𝕜 → E
⊢ (divisor (f ∘ fun x ↦ x - c) U) (z + c) = (divisor (f ∘ fun x ↦ x + -c) U) (z - -c)
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
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 = ((y - x) ^ (meromorphicOrderAt f x).untop₀)⁻¹ • f y

Tactic:
h₁y,

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 = ((y - x) ^ (meromorphicOrderAt f x).untop₀)⁻¹ • (y - x) ^ (meromorphicOrderAt f x).untop₀ • g y
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case neg
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
ta : ↑I × A
_ : ¬ta.1 ≤ t n
hv : f ta ∉ (q e).target
⊢ p (g ta) = f ta

Tactic:
· exact congr($g_lifts ta)

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 : HasFiniteSupport d
y : 𝕜
⊢ y ∈ Finite.toFinset h → ‖(x - y) ^ update d x 0 y‖ ≠ 0

Tactic:
_

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
y : 𝕜
a✝ : y ∈ Finite.toFinset h
⊢ ‖(x - y) ^ update d x 0 y‖ ≠ 0
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 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
⊢ (fun x ↦ ↑(D u) * log ‖x - u‖) = 0 ↔ D u = 0

Tactic:
constructor <;> intro hx

Goal state after:
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

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.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
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₂f₁ : meromorphicOrderAt f₁ x = ⊤
⊢ f₂ =ᶠ[𝓝[≠] x] f₁ + f₂

Tactic:
filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₂f₁]

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
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₂f₁ : meromorphicOrderAt f₁ x = ⊤
⊢ ∀ (a : 𝕜), f₁ a = 0 → f₂ a = (f₁ + f₂) a
Module: Mathlib.Analysis.Meromorphic.Order

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
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
meromorphicOrderAt_add : MeromorphicAt f₁ x → MeromorphicAt f₂ x → min ↑n₁ ↑n₂ ≤ meromorphicOrderAt (f₁ + f₂) x
g₁ : 𝕜 → E
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n₁ • g₁ z
g₂ : 𝕜 → E
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n₂ • g₂ z
n : ℤ := min n₁ n₂
g : 𝕜 → E := (fun z ↦ (z - x) ^ (n₁ - n)) • g₁ + (fun z ↦ (z - x) ^ (n₂ - n)) • g₂
h₁g : AnalyticAt 𝕜 g x
this : f₁ + f₂ =ᶠ[𝓝[≠] x] (fun x_1 ↦ x_1 - x) ^ n • g
⊢ min ↑n₁ ↑n₂ ≤ meromorphicOrderAt (f₁ + f₂) x

Tactic:
have t₀ : MeromorphicAt ((· - x) ^ n) x := by fun_prop

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
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
meromorphicOrderAt_add : MeromorphicAt f₁ x → MeromorphicAt f₂ x → min ↑n₁ ↑n₂ ≤ meromorphicOrderAt (f₁ + f₂) x
g₁ : 𝕜 → E
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n₁ • g₁ z
g₂ : 𝕜 → E
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n₂ • g₂ z
n : ℤ := min n₁ n₂
g : 𝕜 → E := (fun z ↦ (z - x) ^ (n₁ - n)) • g₁ + (fun z ↦ (z - x) ^ (n₂ - n)) • g₂
h₁g : AnalyticAt 𝕜 g x
this : f₁ + f₂ =ᶠ[𝓝[≠] x] (fun x_1 ↦ x_1 - x) ^ n • g
t₀ : MeromorphicAt ((fun x_1 ↦ x_1 - x) ^ n) x
⊢ min ↑n₁ ↑n₂ ≤ meromorphicOrderAt (f₁ + f₂) x
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
⊢ (divisor g (closedBall 0 R)) x = (divisor f (sphere 0 R)) x

Tactic:
rcases lt_trichotomy ‖x‖ R with h|h|h

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : ‖x‖ < R
⊢ (divisor g (closedBall 0 R)) x = (divisor f (sphere 0 R)) x

case inr.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : ‖x‖ = R
⊢ (divisor g (closedBall 0 R)) x = (divisor f (sphere 0 R)) x

case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : R < ‖x‖
⊢ (divisor g (closedBall 0 R)) x = (divisor f (sphere 0 R)) x
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : ‖x‖ < R
this✝ : x ∉ sphere 0 R
this : meromorphicOrderAt g x = 0
⊢ (divisor g (closedBall 0 R)) x = (divisor f (sphere 0 R)) x

Tactic:
divisor_apply D.meromorphicNFOn.meromorphicOn (mem_closedBall_zero_iff.mpr h.le)

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : ‖x‖ < R
this✝ : x ∉ sphere 0 R
this : meromorphicOrderAt g x = 0
⊢ (meromorphicOrderAt g x).untop₀ = (divisor f (sphere 0 R)) x
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_1.refl
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
hs : t n ∈ Set.Icc 0 (t (n + 1)) ∩ frontier {a | a ≤ t n}
⊢ Γ (t n) ∈ q.source

Tactic:
rw [q.mem_source, pΓtn]

Goal state after:
case succ.refine_1.refl
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
hs : t n ∈ Set.Icc 0 (t (n + 1)) ∩ frontier {a | a ≤ t n}
⊢ γ (t n) ∈ q.baseSet
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
s : Set ℂ
h₁ : DifferentiableWithinAt ℝ f s x
h₂ : (fderivWithin ℝ f s x) I = I • (fderivWithin ℝ f s x) 1
⊢ HasDerivWithinAt f (((fderivWithin ℝ f s x).complexOfReal h₂) 1) s x

Tactic:
by
  rw [hasDerivWithinAt_iff_hasFDerivWithinAt, toSpanSingleton_apply_map_one]
  exact h₁.hasFDerivWithinAt.complexOfReal h₂

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
a✝ : ℂ
⊢ (a✝ ∈ support fun i ↦ ↑((divisor f B₀R) i) * log ‖canonicalFactor R i w‖) → a✝ ∈ ↑t₂

Tactic:
_

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
a✝¹ : ℂ
a✝ : a✝¹ ∈ support fun i ↦ ↑((divisor f B₀R) i) * log ‖canonicalFactor R i w‖
⊢ a✝¹ ∈ ↑t₂
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
a✝ : ℂ
⊢ (a✝ ∈ support fun i ↦ ↑((divisor f S₀R) i) * log ‖w - i‖) → a✝ ∈ ↑t₁

Tactic:
_

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
a✝¹ : ℂ
a✝ : a✝¹ ∈ support fun i ↦ ↑((divisor f S₀R) i) * log ‖w - i‖
⊢ a✝¹ ∈ ↑t₁