mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
this : ∀ {y : ℝ}, 0 < y → y ≤ 3 * π / 2 → cos y < 1 / √(y ^ 2 + 1)
h✝ : x < 0
⊢ 0 < -x

Tactic:
linarith

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

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

Tactic:
· rw [← lt_div_iff₀ hy, ← inv_eq_one_div] at h
    replace h : arctan x < arctan y⁻¹ := tanOrderIso.symm.strictMono h
    rwa [arctan_inv_of_pos hy, lt_tsub_iff_right] at h

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

Goal state before:
pp: x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
this : ∀ {y : ℝ}, 0 < y → y ≤ 3 * π / 2 → cos y < 1 / √(y ^ 2 + 1)
h✝ : x < 0
⊢ -x ≤ 3 * π / 2

Tactic:
linarith

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

Goal state before:
pp: 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
⊢ -(π / 2) < y

Tactic:
linarith

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

Goal state before:
pp: 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
⊢ -(π / 2) ≤ y

Tactic:
linarith

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

Goal state before:
pp: x : ℝ
h : 1 < x
⊢ 0 < x

Tactic:
linarith

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

Goal state before:
pp: x : ℝ
h : x < -1
⊢ x < 0

Tactic:
linarith

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
⊢ cos y ^ 2 < 1 / √(y ^ 2 + 1) ^ 2

Tactic:
sq_sqrt hy3.le,

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
⊢ cos y ^ 2 < 1 / (y ^ 2 + 1)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E →ₗᵢ[ℝ] F
u v : E
⊢ angle (f u) (f v) = angle u v

Tactic:
rw [angle, angle, f.inner_map_map, f.norm_map, f.norm_map]

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
hx : x ≠ 0
hy : y ≠ 0
h : ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = (‖x‖ + ‖y‖) ^ 2
hxy₁ : 0 ≤ ‖x - y‖
hxy₂ : 0 ≤ ‖x‖ + ‖y‖
⊢ ⟪x, y⟫ = (‖x‖ ^ 2 + ‖y‖ ^ 2 - (‖x‖ + ‖y‖) ^ 2) / 2

Tactic:
linarith

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
hx : x ≠ 0
hy : y ≠ 0
h : ‖x - y‖ = |‖x‖ - ‖y‖|
h1 : ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = (‖x‖ - ‖y‖) ^ 2
⊢ ⟪x, y⟫ = ((‖x‖ + ‖y‖) ^ 2 - ‖x‖ ^ 2 - ‖y‖ ^ 2) / 2

Tactic:
linarith

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

Goal state before:
pp: case mp
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ‖x‖ ^ 2 + 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2
⊢ ⟪x, y⟫ = 0

Tactic:
linarith

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

Goal state before:
pp: case mpr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
⊢ ‖x‖ ^ 2 + 2 * ⟪x, y⟫ + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫ + ‖y‖ ^ 2

Tactic:
linarith

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
n : ℤ
hn : ↑n ≠ 0
this : ∀ (x : 𝕜), x ^ n = 0 ↔ x = 0
x✝ : 𝕜
⊢ x✝ ∈ (fun x ↦ x ^ n) ⁻¹' {0}ᶜ ↔ x✝ ∈ {0}ᶜ

Tactic:
simp [this]

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

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

Tactic:
rw [accPt_iff_frequently_nhdsNE, not_frequently] at hacc

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
hacc : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, x ∉ U
⊢ {x | deriv f x = deriv g x} ∪ Uᶜ ∈ 𝓝[≠] x
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
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
⊢ ∀ (a : 𝕜),
    meromorphicOrderAt f a = 0 ∨ meromorphicOrderAt f a = ⊤ →
      a ∈ U → MeromorphicOn f U → a ∈ U → meromorphicOrderAt f a = 0 ∨ meromorphicOrderAt f a = ⊤

Tactic:
tauto

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
hf₁ : MeromorphicOn f₁ U
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
a : 𝕜
ha : a ∈ U → f₁ a = f₂ a
⊢ f₁ a = f₂ a

Tactic:
tauto

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
this : U ∈ 𝓝[≠] x
a : 𝕜
h₁a : a ∈ U
h₂a : a ∈ U → ¬¬f₁ a = f₂ a
⊢ f₁ a = f₂ a

Tactic:
tauto

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
e : E
x : 𝕜
⊢ MeromorphicOn (fun x ↦ e) U → x ∈ U → ¬e = 0 ∨ e = 0

Tactic:
tauto

Goal state after:
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
h : AnalyticAt 𝕜 f x
h₂f : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = 0
⊢ MeromorphicNFAt f x

Tactic:
tauto

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

Goal state before:
pp: case mpr.inl
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 : MeromorphicNFAt f x
h : f x ≠ 0
h₁ : f =ᶠ[𝓝 x] 0
this : f x = 0 x
⊢ meromorphicOrderAt f x = 0

Tactic:
tauto

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 : MeromorphicNFAt f x
h : f x ≠ 0
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
hContra : ¬n = 0
this : f x = 0
⊢ False

Tactic:
tauto

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
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 = 1 • g y

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
⊢ y - x ≠ 0 ∨
    -(meromorphicOrderAt f x).untop₀ + (meromorphicOrderAt f x).untop₀ ≠ 0 ∨
      -(meromorphicOrderAt f x).untop₀ = 0 ∧ (meromorphicOrderAt f x).untop₀ = 0

Tactic:
one_smul

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 = g y

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
⊢ y - x ≠ 0 ∨
    -(meromorphicOrderAt f x).untop₀ + (meromorphicOrderAt f x).untop₀ ≠ 0 ∨
      -(meromorphicOrderAt f x).untop₀ = 0 ∧ (meromorphicOrderAt f x).untop₀ = 0
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x e : 𝕜
⊢ meromorphicTrailingCoeffAt (fun x ↦ e) x = e

Tactic:
by
  by_cases he : e = 0
  · rw [he]
    apply MeromorphicAt.meromorphicTrailingCoeffAt_of_order_eq_top
    rw [meromorphicOrderAt_eq_top_iff]
    simp
  · exact analyticAt_const.meromorphicTrailingCoeffAt_of_ne_zero he

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : DecidableEq 𝕜
x y : 𝕜
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x = if x = y then 1 else x - y

Tactic:
by_cases h : x = y

Goal state after:
case pos
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : DecidableEq 𝕜
x y : 𝕜
h : x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x = if x = y then 1 else x - y

case neg
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : DecidableEq 𝕜
x y : 𝕜
h : ¬x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x = if x = y then 1 else x - y
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x z : 𝕜
hz : z = x
h₀f : MeromorphicNFAt f x
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
this : meromorphicOrderAt f x = ↑0
h₃f : ¬meromorphicOrderAt f x = 0
hn : n = 0
⊢ False

Tactic:
tauto

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

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
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₂ : MeromorphicAt f₂ x
hf₁ : MeromorphicAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
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
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h : n₁ < n₂
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
τ₀ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f₁ + f₂) z = (z - x) ^ n₁ • (g₁ + (z - x) ^ (n₂ - n₁) • g₂) z
τ₁ : AnalyticAt 𝕜 (fun z ↦ g₁ z + (z - x) ^ (n₂ - n₁) • g₂ z) x
τ₂ : g₁ x + (x - x) ^ (n₂ - n₁) • g₂ x ≠ 0
⊢ g₁ x + 0 ^ (n₂ - n₁) • g₂ x = g₁ x

Tactic:
add_eq_left,

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
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₂ : MeromorphicAt f₂ x
hf₁ : MeromorphicAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
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
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h : n₁ < n₂
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
τ₀ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f₁ + f₂) z = (z - x) ^ n₁ • (g₁ + (z - x) ^ (n₂ - n₁) • g₂) z
τ₁ : AnalyticAt 𝕜 (fun z ↦ g₁ z + (z - x) ^ (n₂ - n₁) • g₂ z) x
τ₂ : g₁ x + (x - x) ^ (n₂ - n₁) • g₂ x ≠ 0
⊢ 0 ^ (n₂ - n₁) • g₂ x = 0
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

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
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₂ : MeromorphicAt f₂ x
hf₁ : MeromorphicAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
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
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h : n₁ < n₂
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
τ₀ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f₁ + f₂) z = (z - x) ^ n₁ • (g₁ + (z - x) ^ (n₂ - n₁) • g₂) z
τ₁ : AnalyticAt 𝕜 (fun z ↦ g₁ z + (z - x) ^ (n₂ - n₁) • g₂ z) x
τ₂ : g₁ x + (x - x) ^ (n₂ - n₁) • g₂ x ≠ 0
⊢ 0 = 0 ∨ g₂ x = 0

Tactic:
tauto

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

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
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h₁f₁ : ¬↑n₁ = ⊤
g₁ : 𝕜 → E
h₂ : g₁ x + meromorphicTrailingCoeffAt f₂ x ≠ 0
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n₁ • g₁ z
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
h₁ : n₁ = n₂
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
⊢ meromorphicTrailingCoeffAt (f₁ + f₂) x = g₁ x + meromorphicTrailingCoeffAt f₂ x

Tactic:
h₁g₂.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g₂ h₃g₂

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
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h₁f₁ : ¬↑n₁ = ⊤
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
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
h₁ : n₁ = n₂
g₂ : 𝕜 → E
h₂ : g₁ x + g₂ x ≠ 0
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n₂ • g₂ z
⊢ meromorphicTrailingCoeffAt (f₁ + f₂) x = g₁ x + g₂ x
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
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
h₁f : MeromorphicOn f U
z : 𝕜
hz : z ∈ U
⊢ E

Tactic:
exact toMeromorphicNFAt f z z

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(support d).Finite
⊢ MeromorphicNFOn (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) univ

Tactic:
· rw [← mulSupport d] at hd
    rw [finprod_of_infinite_mulSupport hd]
    exact AnalyticOnNhd.meromorphicNFOn analyticOnNhd_const

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
⊢ (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)

Tactic:
simp [Function.FactorizedRational.mulSupport]

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
a✝ : y ∈ Finite.toFinset h
⊢ ‖(x - y) ^ update d x 0 y‖ ≠ 0

Tactic:
by_cases h : x = y

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h✝ : HasFiniteSupport d
y : 𝕜
a✝ : y ∈ Finite.toFinset h✝
h : x = y
⊢ ‖(x - y) ^ update d x 0 y‖ ≠ 0

case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h✝ : HasFiniteSupport d
y : 𝕜
a✝ : y ∈ Finite.toFinset h✝
h : ¬x = y
⊢ ‖(x - y) ^ update d x 0 y‖ ≠ 0
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 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
⊢ ↑(update d x 0 y) * log ‖x - y‖ = ↑(d y) * log ‖x - y‖

Tactic:
by_cases h : x = y

Goal state after:
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‖

case neg
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‖
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 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
⊢ y ≠ x

Tactic:
tauto

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
h₁f : MeromorphicOn f U
h₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤
h₃f : (divisor f U).support.Finite
φ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u
hφ : MeromorphicOn φ U
g : 𝕜 → E := toMeromorphicNFOn (φ⁻¹ • f) U
hg : MeromorphicNFOn g U
a : 𝕜
h₂a : (divisor f U) a = 0 a
h₃a : MeromorphicNFAt (φ⁻¹ • f) a
h₄a : a ∈ U
⊢ f a = 1 • f a

pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
h₁f : MeromorphicOn f U
h₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤
h₃f : (divisor f U).support.Finite
φ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u
hφ : MeromorphicOn φ U
g : 𝕜 → E := toMeromorphicNFOn (φ⁻¹ • f) U
hg : MeromorphicNFOn g U
a : 𝕜
h₂a : (divisor f U) a = 0 a
h₃a : MeromorphicNFAt (φ⁻¹ • f) a
h₄a : a ∈ U
⊢ (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u) a ≠ 0

Tactic:
one_smul

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
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
a : 𝕜
h₂a : (divisor f U) a = 0 a
h₃a : MeromorphicNFAt (φ⁻¹ • f) a
h₄a : a ∈ U
⊢ f a = f a

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
a : 𝕜
h₂a : (divisor f U) a = 0 a
h₃a : MeromorphicNFAt (φ⁻¹ • f) a
h₄a : a ∈ U
⊢ (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u) a ≠ 0
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case mp.inr.inl
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
y : 𝕜
h : y = 0
hy : 1 < 0
⊢ D u = 0

Tactic:
linarith

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

Goal state before:
pp: case mp.inr.inr.inr
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
y : 𝕜
h : ‖y‖ = -1
hy : 1 < 0
⊢ D u = 0

Tactic:
linarith

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f 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
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
h₃f : ¬D.support.Finite
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] (∑ᶠ (u : 𝕜), fun x ↦ ↑(D u) * log ‖x - u‖) + fun x ↦ log ‖g x‖

Tactic:
finsum_of_infinite_support (by simpa [t₁] using h₃f)

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f 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
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
h₃f : ¬D.support.Finite
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] 0 + fun x ↦ log ‖g 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
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₂f₁ : ¬meromorphicOrderAt f₁ x = ⊤
h₂f₂ : meromorphicOrderAt f₂ x = ⊤
⊢ min (meromorphicOrderAt f₁ x) (meromorphicOrderAt f₂ x) ≤ meromorphicOrderAt (f₁ + f₂) x

Tactic:
simp only [h₂f₂, le_top, inf_of_le_left]

Goal state after:
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 = ⊤
h₂f₂ : meromorphicOrderAt f₂ x = ⊤
⊢ meromorphicOrderAt f₁ x ≤ meromorphicOrderAt (f₁ + f₂) x