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

Goal state before:
pp: n : ℕ
a b : ℂ
⊢ ∑ i ∈ Finset.range n, 2 * sin (a / 2) * cos (a * ↑i + b) =
    ∑ x ∈ Finset.range n, (sin (a * (↑(x + 1) - 1 / 2) + b) - sin (a * (↑x - 1 / 2) + b))

Tactic:
by
      congr! 1 with x hx
      rw [sin_sub_sin]
      push_cast
      ring_nf

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

Goal state before:
pp: x : ℝ
hx✝ : cos x = 0
hx : Complex.cos ↑x = 0
⊢ Tendsto (fun x ↦ ‖Complex.tan ↑x‖) (𝓝[≠] x) atTop

Tactic:
refine (Complex.tendsto_norm_tan_of_cos_eq_zero hx).comp ?_

Goal state after:
x : ℝ
hx✝ : cos x = 0
hx : Complex.cos ↑x = 0
⊢ Tendsto Complex.ofReal (𝓝[≠] x) (𝓝[≠] ↑x)
Module: Mathlib.Analysis.Complex.Arg

Goal state before:
pp: case pos
x y : ℂ
hx : x = 0
⊢ x = 0 ∨ y = 0 ∨ x.arg = y.arg ↔ ↑(x / y).arg = 0

Tactic:
simp [hx]

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 : V
hx : x ≠ 0
⊢ angle x (-x) = π

Tactic:
angle_neg_right,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
hx : x ≠ 0
⊢ π - angle x x = π
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
r : ℝ
hr : r < 0
⊢ angle x (-(-r • y)) = angle x (-y)

Tactic:
angle_neg_right,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : r < 0
⊢ π - angle x (-r • y) = angle x (-y)
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|

Tactic:
· rw [Real.sqrt_le_left (by positivity), ← abs_pow, abs_sq]
      calc
        _ = 2 * (1 - θ.cos) := by linear_combination θ.cos_sq_add_sin_sq
        _ ≤ 2 * (1 - (1 - θ ^ 2 / 2)) := by gcongr; exact Real.one_sub_sq_div_two_le_cos
        _ = _ := by ring

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
⊢ cos (angle x y) = 0 ↔ angle x y = π / 2

Tactic:
rw [← cos_pi_div_two]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ cos (angle x y) = cos (π / 2) ↔ angle x y = π / 2
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x = 0
⊢ angle (NormedSpace.normalize x) y = angle x y

Tactic:
simp [hx]

Goal state after:
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)
this : ‖g' z‖ ≤ R₂ / R₁ ^ (n + 1)
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)

Tactic:
have hz' : ‖z - c‖ ≠ 0 := by simpa [sub_eq_zero] using hne

Goal state after:
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)
this : ‖g' z‖ ≤ R₂ / R₁ ^ (n + 1)
hz' : ‖z - c‖ ≠ 0
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
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
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
⊢ (fun x ↦ f ((lineMap c z) x) - f ((lineMap c z) 0)) =o[𝓝 0] fun w ↦ w ^ n

Tactic:
← lineMap_apply_zero (k := ℂ) c z

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f ((lineMap c z) 0)) =o[𝓝 ((lineMap c z) 0)] fun w ↦ ‖w - (lineMap c z) 0‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
⊢ (fun x ↦ f ((lineMap c z) x) - f ((lineMap c z) 0)) =o[𝓝 0] fun w ↦ w ^ n
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
R : ℝ
f : E → F
z : E
hd : DifferentiableOn ℂ f (ball 0 R)
h_maps : MapsTo f (ball 0 R) (closedBall 0 R)
h₀ : f 0 = 0
hz : ‖z‖ < R
⊢ ‖f z‖ ≤ ‖z‖

Tactic:
by
  simpa [h₀] using dist_le_dist_of_mapsTo_ball hd (by rwa [h₀]) (mem_ball_zero_iff.mpr hz)

Goal state after:
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
h₁ : f₁ =ᶠ[codiscreteWithin U] f₂
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∉ U
⊢ (divisor f₁ U) x = (divisor f₂ U) x

Tactic:
simp [hx]

Goal state after:
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
h₁ : f₁ =ᶠ[codiscreteWithin U] f₂
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hf₁ : MeromorphicOn f₁ U
x : 𝕜
hx : x ∉ U
⊢ (divisor f₁ U) x = (divisor f₂ U) x

Tactic:
simp [hx]

Goal state after:
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
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
x : 𝕜
hx : x ∉ U
⊢ 0 x ≤ (divisor f U) x

Tactic:
simp [hx]

Goal state after:
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
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hV : V ⊆ U
x : 𝕜
hx : x ∉ V
⊢ ((divisor f U).restrict hV) x = (divisor f V) x

Tactic:
simp [hx]

Goal state after:
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
⊢ (divisor (f₁ + f₂) U)⁻ x = (divisor f₁ U)⁻ x

Tactic:
simp [hx]

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
g : 𝕜 → 𝕜
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
hprod : MeromorphicNFAt (g • f) x
⊢ MeromorphicNFAt f x

Tactic:
by
    have : f =ᶠ[𝓝 x] g⁻¹ • g • f := by
      filter_upwards [h₁g.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.mpr h₂g)]
      intro y hy
      rw [Set.preimage_compl, Set.mem_compl_iff, Set.mem_preimage,
        Set.mem_singleton_iff] at hy
      simp [hy]
    rw [meromorphicNFAt_congr this]
    exact hprod.smul_analytic (h₁g.inv h₂g) (inv_ne_zero h₂g)

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
h₁f : MeromorphicAt f x
h₃ : meromorphicOrderAt f x = ⊤
⊢ meromorphicTrailingCoeffAt f x = g x

Tactic:
simp only [h₃, WithTop.untop₀_top, zpow_zero, one_smul,
      MeromorphicAt.meromorphicTrailingCoeffAt_of_order_eq_top] at ⊢ h

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
f g : 𝕜 → E
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₁f : MeromorphicAt f x
h₃ : meromorphicOrderAt f x = ⊤
h : f =ᶠ[𝓝[≠] x] fun z ↦ g z
⊢ 0 = g x
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x✝ : 𝕜
f : 𝕜 → 𝕜
h : f =ᶠ[𝓝 x✝] 0
x : 𝕜
hx : f x = 0 x
⊢ f⁻¹ x = 0 x

Tactic:
simp [hx]

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
x : 𝕜
f g : 𝕜 → E
hfg : f =ᶠ[𝓝[≠] x] g
hf : MeromorphicAt f x
⊢ toMeromorphicNFAt f x =ᶠ[𝓝 x] toMeromorphicNFAt g x

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
f g : 𝕜 → E
hfg : f =ᶠ[𝓝[≠] x] g
hf : ¬MeromorphicAt f x
⊢ toMeromorphicNFAt f x =ᶠ[𝓝 x] toMeromorphicNFAt g x

Tactic:
· exact meromorphicNFAt_toMeromorphicNFAt.eventuallyEq_nhdsNE_iff_eventuallyEq_nhds
        meromorphicNFAt_toMeromorphicNFAt |>.mp <| hf.eq_nhdsNE_toMeromorphicNFAt.symm.trans
      <| hfg.trans ((MeromorphicAt.meromorphicAt_congr hfg).mp hf).eq_nhdsNE_toMeromorphicNFAt

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
f g : 𝕜 → E
hfg : f =ᶠ[𝓝[≠] x] g
hf : ¬MeromorphicAt f x
⊢ toMeromorphicNFAt f x =ᶠ[𝓝 x] toMeromorphicNFAt g x
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
U : Set 𝕜
h₁f : MeromorphicNFOn f U
h : AnalyticOnNhd 𝕜 f U
x : 𝕜
hx : x ∉ U
⊢ 0 x ≤ (MeromorphicOn.divisor f U) x

Tactic:
simp [hx]

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
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:
constructor <;> 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

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.FactorizedRational

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

Tactic:
rw [extractFactor z hd]

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : (support d).Finite
z : 𝕜
hz : z ∈ univ
⊢ MeromorphicNFAt ((fun x ↦ x - z) ^ d z * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d z 0 u) z
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

Tactic:
hxy,

Goal state after:
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) y ^ d y = (y - y) ^ update d y 0 y
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case coe.coe
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁶ : NormedRing R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : Module R E
inst✝³ : IsBoundedSMul R E
inst✝² : Module.IsTorsionFree R E
x : 𝕜
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : IsScalarTower 𝕜 R E
f : 𝕜 → R
g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : meromorphicOrderAt f x = ↑m
n : ℤ
h₂g : meromorphicOrderAt g x = ↑n
F : 𝕜 → R
h₁F : AnalyticAt 𝕜 F x
h₂F : F x ≠ 0
h₃F : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ m • F z
⊢ ∃ g_1, AnalyticAt 𝕜 g_1 x ∧ g_1 x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f • g) z = (z - x) ^ (m + n) • g_1 z

Tactic:
obtain ⟨G, h₁G, h₂G, h₃G⟩ := (meromorphicOrderAt_eq_int_iff hg).1 h₂g

Goal state after:
case coe.coe
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁶ : NormedRing R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : Module R E
inst✝³ : IsBoundedSMul R E
inst✝² : Module.IsTorsionFree R E
x : 𝕜
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : IsScalarTower 𝕜 R E
f : 𝕜 → R
g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : meromorphicOrderAt f x = ↑m
n : ℤ
h₂g : meromorphicOrderAt g x = ↑n
F : 𝕜 → R
h₁F : AnalyticAt 𝕜 F x
h₂F : F x ≠ 0
h₃F : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ m • F z
G : 𝕜 → E
h₁G : AnalyticAt 𝕜 G x
h₂G : G x ≠ 0
h₃G : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, g z = (z - x) ^ n • G z
⊢ ∃ g_1, AnalyticAt 𝕜 g_1 x ∧ g_1 x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f • g) z = (z - x) ^ (m + n) • g_1 z
Module: Mathlib.Topology.Homotopy.Lifting

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

Tactic:
intro t t'

Goal state after:
case refine_4
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
homeo : IsLocalHomeomorph p
sep : IsSeparatedMap p
γ₀ γ₁ : C(↑I, X)
γ : γ₀.HomotopyRel γ₁ {0, 1}
Γ : ↑I → C(↑I, E)
Γ_lifts : ∀ (t s : ↑I), p ((Γ t) s) = γ (t, s)
Γ_0 : ∀ (t : ↑I), (Γ t) 0 = (Γ 0) 0
t✝ : ↑I
this : Continuous[_, inst✝¹] fun st ↦ (Γ st.2) st.1
t t' : ↑I
⊢ p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) t) = p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) t')
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] 1 • g
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
h₃f : ¬D.support.Finite
x : 𝕜
hx : f x = (1 • g) x
⊢ log ‖f x‖ = (0 + fun x ↦ log ‖g x‖) x

Tactic:
simp [hx]

Goal state after:
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
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
a : A
p : OpenPartialHomeomorph E X
hep : F a ∈ p.source
homeo : IsLocalHomeomorph ↑p
he : ↑p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → ↑p ∘ ⇑Γ = ⇑(f.comp γ) → ↑p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), ↑p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
this : ∀ (a : A), ↑p (F a) = f a
hfap : f a ∈ p.target
U : Set A
haU : U ∈ 𝓝 a
U_conn : IsPathConnected U
hUp : id U ⊆ ⇑f ⁻¹' p.target
x : A
hxU : x ∈ U
γ : Path a x
hγ : ∀ (t : ↑I), γ t ∈ U
Γ' : Path e₀ ((↑p.symm ∘ ⇑f) a) :=
  { toContinuousMap := Γ (somePath a₀ a).toContinuousMap ⋯, source' := ⋯, target' := ⋯ }
⊢ Set.MapsTo (⇑f) (Set.range ⇑γ) p.target

Tactic:
rintro _ ⟨t, rfl⟩; exact hUp (hγ _)

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

Goal state before:
pp: case mp.inr
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
g : GL n R
hg : g ∈ Subgroup.center (GL n R)
hn : Nonempty n
⊢ g ∈ (scalar n).range

Tactic:
| inr hn =>

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

Goal state before:
pp: f : ℂ →ₗᵢ[ℝ] ℂ
h₂ : ∀ (z : ℂ), (f z).re = z.re
z : ℂ
h₁ : √(normSq (f z)) = √(normSq z)
⊢ (f z).im = z.im ∨ (f z).im = -z.im

Tactic:
rwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z,
    h₂, add_left_cancel_iff, mul_self_eq_mul_self_iff] at h₁

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

Goal state before:
pp: f : ℂ →ₗᵢ[ℝ] ℂ
h₂ : ∀ (z : ℂ), (f z).re = z.re
z : ℂ
⊢ (f z).im = z.im ∨ (f z).im = -z.im

Tactic:
by
  have h₁ := f.norm_map z
  simp only [norm_def] at h₁
  rwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z,
    h₂, add_left_cancel_iff, mul_self_eq_mul_self_iff] at h₁

Goal state after:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
inst✝ : PreconnectedSpace A
f₀' f₁' : C(A, E)
he : ∃ a ∈ S, f₀' a = f₁' a
h₀ : p ∘ ⇑f₀' = ⇑f₀
h₁ : p ∘ ⇑f₁' = ⇑f₁
F_0 : ∀ (a : A), F (0, a) = p (f₀' a)
t : ↑I
a : A
ha : a ∈ S
⊢ ∀ (a_1 a' : ↑I),
    p
        ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t, a).2))) (f₀' (t, a).2) ⋯)
          a_1) =
      p
        ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t, a).2))) (f₀' (t, a).2) ⋯)
          a')

Tactic:
intro t t'

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
inst✝ : PreconnectedSpace A
f₀' f₁' : C(A, E)
he : ∃ a ∈ S, f₀' a = f₁' a
h₀ : p ∘ ⇑f₀' = ⇑f₀
h₁ : p ∘ ⇑f₁' = ⇑f₁
F_0 : ∀ (a : A), F (0, a) = p (f₀' a)
t✝ : ↑I
a : A
ha : a ∈ S
t t' : ↑I
⊢ p ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯) t) =
    p
      ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯)
        t')
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
cov : IsCoveringMap p
inst✝¹ : SimplyConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
⊢ ∃! F, F a₀ = e₀ ∧ p ∘ ⇑F = ⇑f

Tactic:
refine cov.isLocalHomeomorph.existsUnique_continuousMap_lifts f a₀ e₀ he (fun γ γ_0 ↦ ?_)
    fun γ γ' Γ Γ' γ_0 γ'_0 Γ_0 Γ'_0 Γ_lifts Γ'_lifts γγ'1 ↦ ?_

Goal state after:
case refine_1
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
cov : IsCoveringMap p
inst✝¹ : SimplyConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
γ : C(↑I, A)
γ_0 : γ 0 = a₀
⊢ ∃ Γ, Γ 0 = e₀ ∧ p ∘ ⇑Γ = ⇑(f.comp γ)

case refine_2
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
cov : IsCoveringMap p
inst✝¹ : SimplyConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
γ γ' : C(↑I, A)
Γ Γ' : C(↑I, E)
γ_0 : γ 0 = a₀
γ'_0 : γ' 0 = a₀
Γ_0 : Γ 0 = e₀
Γ'_0 : Γ' 0 = e₀
Γ_lifts : p ∘ ⇑Γ = ⇑(f.comp γ)
Γ'_lifts : p ∘ ⇑Γ' = ⇑(f.comp γ')
γγ'1 : γ 1 = γ' 1
⊢ Γ 1 = Γ' 1
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case e'_5
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
x : ℂ
⊢ meromorphicTrailingCoeffAt (canonicalFactor R x) w ^ (divisor f (ball 0 R)) x =
    canonicalFactor R x w ^ (divisor f (ball 0 R)) x

Tactic:
by_cases h₃x : (divisor f (ball 0 R)) x = 0

Goal state after:
case pos
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
x : ℂ
h₃x : (divisor f (ball 0 R)) x = 0
⊢ meromorphicTrailingCoeffAt (canonicalFactor R x) w ^ (divisor f (ball 0 R)) x =
    canonicalFactor R x w ^ (divisor f (ball 0 R)) x

case neg
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
x : ℂ
h₃x : ¬(divisor f (ball 0 R)) x = 0
⊢ meromorphicTrailingCoeffAt (canonicalFactor R x) w ^ (divisor f (ball 0 R)) x =
    canonicalFactor R x w ^ (divisor f (ball 0 R)) x