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'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ ‖?x' + ?y' + (x - ?x') + (y - ?y')‖ ≤ ‖x' + y'‖ + ‖x - x'‖ + ‖y - y'‖

pp: case x'
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 := ⋯
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 := ⋯
y' : E := ⋯
hxy' : ε / 3 ≤ ‖x' - y'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ E

pp: case y'
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 := ⋯
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 := ⋯
y' : E := ⋯
hxy' : ε / 3 ≤ ‖x' - y'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ E

Tactic:
exact norm_add₃_le

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

Goal state before:
pp: case inr
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
r : ℝ
hr : 0 < r
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y : E⦄, ‖y‖ ≤ 1 → ε / r ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
⊢ ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y : E⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ

Tactic:
refine ⟨δ * r, mul_pos hδ hr, fun x hx y hy hxy => ?_⟩

Goal state after:
case inr
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
r : ℝ
hr : 0 < r
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y : E⦄, ‖y‖ ≤ 1 → ε / r ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
x : E
hx : ‖x‖ ≤ r
y : E
hy : ‖y‖ ≤ r
hxy : ε ≤ ‖x - y‖
⊢ ‖x + y‖ ≤ 2 * r - δ * r
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: x : ℂ
⊢ cos x = 1 ↔ ∃ k, ↑k * (2 * ↑π) = x

Tactic:
by
  rw [← cos_zero, eq_comm, cos_eq_cos_iff]
  simp [mul_assoc, mul_left_comm, eq_comm]

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

Goal state before:
pp: this : ContinuousOn (fun x ↦ sin x / cos x) {x | cos x ≠ 0}
x : ℝ
⊢ sin x / cos x = tan x

Tactic:
rw [tan_eq_sin_div_cos]

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

Goal state before:
pp: case inl
x : ℝ
hx_gt : -(π / 2) < x
hx_lt : ¬π / 2 ≤ x
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : 0 ≤ r
⊢ 1 ≤ 2 * ↑r + 1

Tactic:
simp [h]

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

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < π / 2
⊢ x < tan x

Tactic:
let U := Ico 0 (π / 2)

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
⊢ x < tan x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ arcsin (x / √(1 + x ^ 2)) = arcsin √(1 - (√(1 + x ^ 2))⁻¹ ^ 2)

pp: x : ℝ
h : 0 ≤ x
⊢ 0 ≤ (√(1 + x ^ 2))⁻¹

Tactic:
swap

Goal state after:
x : ℝ
h : 0 ≤ x
⊢ 0 ≤ (√(1 + x ^ 2))⁻¹

x : ℝ
h : 0 ≤ x
⊢ arcsin (x / √(1 + x ^ 2)) = arcsin √(1 - (√(1 + x ^ 2))⁻¹ ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₂
x : ℝ
h : 0 < x
⊢ 0 < arctan x

Tactic:
← arctan_zero

Goal state after:
case hx₂
x : ℝ
h : 0 < x
⊢ arctan 0 < arctan x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inl
x y : ℝ
h : x * y < 1
hy : y ≤ 0
⊢ arctan x + arctan y < π / 2 + 0

Tactic:
← arctan_zero

Goal state after:
case inl
x y : ℝ
h : x * y < 1
hy : y ≤ 0
⊢ arctan x + arctan y < π / 2 + arctan 0
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
⊢ arccos 0 = π / 2

Tactic:
Real.arccos_zero

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
⊢ π / 2 = π / 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
hx : x ≠ 0
hy : y ≠ 0
h : ‖x - y‖ = |‖x‖ - ‖y‖|
⊢ |‖x‖ - ‖y‖| ^ 2 = (‖x‖ - ‖y‖) ^ 2

Tactic:
exact sq_abs (‖x‖ - ‖y‖)

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]
⊢ IsCoveringMapOn (fun x ↦ eval x p) ((fun x ↦ eval x p) '' {k | eval k (derivative p) = 0})ᶜ

Tactic:
refine p.isClosedMap_eval.isCoveringMapOn_of_isLocalHomeomorphOn (fun x hx ↦ ?_)
    fun x hx ↦ ⟨_, ((p.hasStrictDerivAt x).hasStrictFDerivAt_equiv
      fun h ↦ hx ⟨x, h, rfl⟩).mem_toOpenPartialHomeomorph_source, by simp⟩

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

Goal state before:
pp: case inl
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 c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
heq : f z = f c
⊢ dist (f z) (f c) ≤ R₂ * (dist z c / R₁) ^ (n + 1)

pp: case inr
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 c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
⊢ dist (f z) (f c) ≤ R₂ * (dist z c / R₁) ^ (n + 1)

Tactic:
· trans 0 <;> [simp [heq]; positivity]

Goal state after:
case inr
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 c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
⊢ dist (f z) (f c) ≤ R₂ * (dist z c / R₁) ^ (n + 1)
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case neg
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
hf₂ : AnalyticOnNhd 𝕜 f₂ U
x : 𝕜
hx : x ∈ U
h : ¬0 ≤ meromorphicOrderAt f₁ x
⊢ max (-(meromorphicOrderAt (f₁ + f₂) x).untop₀) 0 = max (-(meromorphicOrderAt f₁ x).untop₀) 0

Tactic:
suffices meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x by
        rwa [meromorphicOrderAt_add_eq_left_of_lt (hf₂.meromorphicOn x hx)]
      calc meromorphicOrderAt f₁ x
      _ < 0 := by simpa using h
      _ ≤ meromorphicOrderAt f₂ x := (hf₂ x hx).meromorphicOrderAt_nonneg

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
U : Set 𝕜
h : z₀ ∈ U
⊢ (divisor (fun x ↦ x - z₀) U) z₀ = 1

Tactic:
rw [divisor_apply (show MeromorphicOn (· - z₀) U from fun_sub id <| const z₀) h, ← untop₀_coe 1]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
U : Set 𝕜
h : z₀ ∈ U
⊢ (meromorphicOrderAt (fun x ↦ x - z₀) z₀).untop₀ = (↑1).untop₀
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
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

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
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

Tactic:
swap

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
f : 𝕜 → E
x : 𝕜
hf : ¬MeromorphicAt f x
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

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
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
⊢ MeromorphicNFOn (g • f) U → MeromorphicNFOn f U

Tactic:
intro h z hz

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
f : 𝕜 → E
U : Set 𝕜
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn (g • f) U
z : 𝕜
hz : z ∈ U
⊢ MeromorphicNFAt f z
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
⊢ MeromorphicNFOn f U → MeromorphicNFOn (g • f) U

Tactic:
intro h z hz

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
f : 𝕜 → E
U : Set 𝕜
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn f U
z : 𝕜
hz : z ∈ U
⊢ MeromorphicNFAt (g • f) z
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
c : 𝕜
U : Set 𝕜
⊢ MeromorphicNFOn f (U + {-c}) ↔ MeromorphicNFOn f (U + -{c})

Tactic:
neg_singleton

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

Goal state before:
pp: case h₁
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
U : Set 𝕜
hf : MeromorphicOn f U
hx : x ∈ U
⊢ toMeromorphicNFOn f U =ᶠ[𝓝[≠] x] toMeromorphicNFAt f x

pp: case h₂
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
U : Set 𝕜
hf : MeromorphicOn f U
hx : x ∈ U
⊢ toMeromorphicNFOn f U x = toMeromorphicNFAt f x x

Tactic:
· exact (hf.toMeromorphicNFOn_eq_self_on_nhdsNE hx).trans (hf x hx).eq_nhdsNE_toMeromorphicNFAt

Goal state after:
case h₂
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
U : Set 𝕜
hf : MeromorphicOn f U
hx : x ∈ U
⊢ toMeromorphicNFOn f U x = toMeromorphicNFAt f x x
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
h : HasFiniteSupport d
⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = fun x ↦ ∏ᶠ (u : 𝕜), (x - u) ^ d u

Tactic:
ext x
  rw [finprod_eq_prod_of_mulSupport_subset (s := h.toFinset),
    finprod_eq_prod_of_mulSupport_subset (s := h.toFinset)]
  · simp
  · intro u
    contrapose
    simp_all
  · simp [mulSupport d]

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
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : MeromorphicAt f₁ x
⊢ meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x

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
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : ¬MeromorphicAt f₁ x
⊢ meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x

Tactic:
swap

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
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : ¬MeromorphicAt f₁ x
⊢ meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x

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
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : MeromorphicAt f₁ x
⊢ meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
D : locallyFinsuppWithin U ℤ
hD : D.support.Finite
z : 𝕜
hz : z ∈ U
⊢ (MeromorphicOn.divisor (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) U) z = D z

Tactic:
simp [(meromorphicNFOn D U).meromorphicOn, hz, meromorphicOrderAt_eq D hD]

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
D : locallyFinsuppWithin U ℤ
hD : D.support.Finite
z : 𝕜
hz : z ∉ U
⊢ (MeromorphicOn.divisor (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) U) z = D z

Tactic:
simp [(meromorphicNFOn D U).meromorphicOn, hz, meromorphicOrderAt_eq D hD]

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✝ : ∀ y ∈ Finite.toFinset h✝, ‖(x - y) ^ update d x 0 y‖ ≠ 0
this : (support fun u ↦ ↑(d u) * log ‖x - u‖) ⊆ ↑(Finite.toFinset h✝)
y : 𝕜
hy : y ∈ Finite.toFinset h✝
h : x = y
⊢ ↑(update d x 0 y) * log ‖x - y‖ = ↑(d y) * log ‖x - y‖

Tactic:
simp [h]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
ι : Type u_5
s : Finset ι
f : ι → 𝕜 → 𝕜'
hf : ∀ i ∈ s, MeromorphicAt (f i) x
⊢ meromorphicOrderAt (fun a ↦ ∏ i ∈ s, f i a) x = ∑ i ∈ s, meromorphicOrderAt (f i) x

Tactic:
by
  convert! meromorphicOrderAt_prod hf
  exact (Finset.prod_apply _ s f).symm

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x

pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : ¬MeromorphicAt f x
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x

Tactic:
swap

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : ¬MeromorphicAt f x
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x

case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f⁻¹ x = -meromorphicOrderAt f x
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
⊢ meromorphicOrderAt (f₁ + f₂) x = meromorphicOrderAt f₁ x

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
hf₂ : MeromorphicAt f₂ x
h : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x
hf₁ : ¬MeromorphicAt f₁ x
⊢ meromorphicOrderAt (f₁ + f₂) x = meromorphicOrderAt f₁ x

Tactic:
swap

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
hf₂ : MeromorphicAt f₂ x
h : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x
hf₁ : ¬MeromorphicAt f₁ x
⊢ meromorphicOrderAt (f₁ + f₂) x = meromorphicOrderAt f₁ x

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
⊢ meromorphicOrderAt (f₁ + f₂) x = meromorphicOrderAt f₁ x
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case neg
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)
⊢ ∃ g, CanonicalDecomp f g R

Tactic:
have h₄g {z : ℂ} (hz : z ∈ closedBall 0 R) : meromorphicOrderAt g z ≠ ⊤ := by
    rw [meromorphicOrderAt_toMeromorphicNFOn hφ hz, meromorphicOrderAt_smul _ (h₁f z hz)]
    · simpa [h₂f ⟨z, hz⟩] using canonicalDecomposition_aux₃ hR
    · apply MeromorphicAt.finprod (fun x ↦ (meromorphic_canonicalFactor R x z).zpow _)

Goal state after:
case neg
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 ≠ ⊤
⊢ ∃ g, CanonicalDecomp f g R
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: case mp
g : ℂ →L[ℝ] ℂ
⊢ IsConformalMap g → ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g ∘SL ↑conjCLE) ∧ g ≠ 0

Tactic:
exact fun h => ⟨h.is_complex_or_conj_linear, h.ne_zero⟩

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

Goal state before:
pp: t : |2⁻¹| = 2⁻¹
z : |429895798851992810179897 / 620208537104218167705600 - log 2| ≤ 1 / 17179869184
⊢ |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)

Tactic:
_root_.abs_sub_comm

Goal state after:
t : |2⁻¹| = 2⁻¹
z : |log 2 - 429895798851992810179897 / 620208537104218167705600| ≤ 1 / 17179869184
⊢ |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)
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 3| ≤ (2 / 3) ^ (70 + 1) / 3⁻¹
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)

Tactic:
_root_.abs_sub_comm

Goal state after:
t : |2 / 3| = 2 / 3
z : |log 3 - ∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1)| ≤ (2 / 3) ^ (70 + 1) / 3⁻¹
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)
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 5| ≤ (4 / 5) ^ (130 + 1) / 5⁻¹
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)

Tactic:
_root_.abs_sub_comm

Goal state after:
t : |4 / 5| = 4 / 5
z : |log 5 - ∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1)| ≤ (4 / 5) ^ (130 + 1) / 5⁻¹
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ ‖↑(ε + sSupNormIm f 0) ^ (z - 1)‖ * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)

Tactic:
rw [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _]

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)

Tactic:
rw [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _]

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)

Tactic:
rw [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _]

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
g : G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
⊢ op ((hp.fiberEquivGroup e) (⋯.monodromy 1 e)) = 1

Tactic:
rw [FundamentalGroup.one_def, IsCoveringMap.monodromy_refl]; simp

Goal state after: