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

Goal state before:
pp: case inr.refine_1
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
⊢ (slope f x '' {y | y ∈ S ∧ y < x}).Nonempty

pp: case inr.refine_2
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
⊢ ∀ b ∈ slope f x '' {y | y ∈ S ∧ y < x}, b ≤ slope f x y

Tactic:
· have hxs' := hxs
    rw [mem_interior_iff_mem_nhds, mem_nhds_iff_exists_Ioo_subset] at hxs'
    obtain ⟨a, b, hxab, habs⟩ := hxs'
    rw [image_nonempty]
    obtain ⟨z, hxz, hzb⟩ := exists_between hxab.1
    exact ⟨z, habs ⟨hxz, hzb.trans hxab.2⟩, hzb⟩

Goal state after:
case inr.refine_2
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
⊢ ∀ b ∈ slope f x '' {y | y ∈ S ∧ y < x}, b ≤ slope f x y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
⊢ ∀ ⦃b : ℝ⦄, b ∈ S → x ≤ b → derivWithin f S x ≤ derivWithin f S b

Tactic:
y

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
y : ℝ
⊢ y ∈ S → x ≤ y → derivWithin f S x ≤ derivWithin f S y
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ ‖cos ↑θ - 1 + sin ↑θ * I‖ ∈ Icc (2 / π * |θ|) |θ|

pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ θ ∈ Ioc (-π) (-π + 2 * π)

Tactic:
· norm_cast
    rw [norm_add_mul_I]
    refine ⟨Real.le_sqrt_of_sq_le ?_, ?_⟩
    · rw [mul_pow, ← abs_pow, abs_sq]
      calc
        _ = 2 * (1 - (1 - 2 / π ^ 2 * θ ^ 2)) := by ring
        _ ≤ 2 * (1 - θ.cos) := by
            gcongr; exact Real.cos_le_one_sub_mul_cos_sq <| abs_le.2 <| Ioc_subset_Icc_self hθ
        _ = _ := by linear_combination -θ.cos_sq_add_sin_sq
    · 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:
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ θ ∈ Ioc (-π) (-π + 2 * π)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
x : ℝ
hx : x ∈ S
⊢ ∀ ⦃b : ℝ⦄, b ∈ S → x ≤ b → deriv f x ≤ deriv f b

Tactic:
y

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
x : ℝ
hx : x ∈ S
y : ℝ
⊢ y ∈ S → x ≤ y → deriv f x ≤ deriv f y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
hxy : x < y
this : 0 ≤ slope f x y
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}

Tactic:
by
      simp only [slope_def_field, div_nonneg_iff, sub_nonneg, tsub_le_iff_right, zero_add,
        not_le.mpr hxy, and_false, or_false] at this
      exact this.1

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

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
⊢ ∀ ⦃b : ℝ⦄, b ∈ S → x < b → derivWithin f S x < derivWithin f S b

Tactic:
y

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
y : ℝ
⊢ y ∈ S → x < y → derivWithin f S x < derivWithin f S y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
x : ℝ
hx : x ∈ S
⊢ ∀ ⦃b : ℝ⦄, b ∈ S → x < b → deriv f x < deriv f b

Tactic:
y

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : StrictConvexOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x
x : ℝ
hx : x ∈ S
y : ℝ
⊢ y ∈ S → x < y → deriv f x < deriv f y
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_denom : 2 * ↑M - f z ≠ 0
⊢ ‖2 * ↑M * w / (1 + w)‖ ≤ 2 * M * ‖w‖ / (1 - ‖w‖)

Tactic:
by
      simp only [norm_div, norm_mul, norm_ofNat, norm_real, Real.norm_eq_abs, abs_of_pos hM]
      gcongr
      · linarith [hwR.trans_lt ((div_lt_one₀ hR).mpr hzR)]
      · simpa using norm_sub_le_norm_add 1 w

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

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : ConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
⊢ ∀ ⦃b : ℝ⦄, b ∈ S → x ≤ b → derivWithin f S b ≤ derivWithin f S x

Tactic:
y

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : ConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
y : ℝ
⊢ y ∈ S → x ≤ y → derivWithin f S y ≤ derivWithin f S x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
hfc : StrictConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
⊢ ∀ ⦃b : ℝ⦄, b ∈ S → x < b → derivWithin f S b < derivWithin f S x

Tactic:
y

Goal state after:
S : Set ℝ
f : ℝ → ℝ
hfc : StrictConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
y : ℝ
⊢ y ∈ S → x < y → derivWithin f S y < derivWithin f S x
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: f : ℂ → ℂ
c z : ℂ
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
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)

Tactic:
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
  wlog hd' : DifferentiableOn ℂ f (closedBall c R₁) ∧
    MapsTo f (closedBall c R₁) (closedBall (f c) R₂) generalizing R₁
  · suffices ∀ᶠ r in 𝓝[<] R₁, ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / r) ^ (n + 1) by
      refine ge_of_tendsto ?_ this
      refine ContinuousAt.continuousWithinAt ?_
      fun_prop (disch := positivity)
    rw [mem_ball_iff_norm] at hz
    filter_upwards [Ioo_mem_nhdsLT hz] with r ⟨hzr, hrR₁⟩
    apply this
    · exact hd.mono <| by gcongr
    · exact h_maps.mono_left <| by gcongr
    · rwa [mem_ball_iff_norm]
    · exact (norm_nonneg _).trans_lt hzr
    · exact ⟨hd.mono <| closedBall_subset_ball hrR₁, h_maps.mono_left <|
        closedBall_subset_ball hrR₁⟩
  -- Cleanup, discard the case `z = c`.
  clear hd h_maps
  rcases hd' with ⟨hd, h_maps⟩
  rcases eq_or_ne z c with rfl | hne
  · simp
  -- Consider the function given by `g w := ((w - c) ^ (n + 1))⁻¹ * (f w - f c)`.
  -- It is differentiable away from `c` and satisfies `g w = o((w - c)⁻¹)`,
  -- thus it can be extended to a function g'` differentiable on the whole closed ball
  -- with center c` and radius `R₁`.
  set g : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f c)
  set g' := update g c (limUnder (𝓝[≠] c) g)
  have hdg' : DifferentiableOn ℂ g' (closedBall c R₁) := by
    refine .mono ?_ (subset_insert_sdiff_singleton c _)
    apply differentiableOn_update_limUnder_insert_of_isLittleO
    · exact sdiff_mem_nhdsWithin_compl (closedBall_mem_nhds _ hR₁) _
    · refine .mul ?_ (hd.mono sdiff_subset |>.sub_const _)
      fun_prop (disch := simp +contextual [sub_eq_zero])
    · refine Asymptotics.isBigO_refl (fun w ↦ ((w - c) ^ (n + 1))⁻¹) _ |>.mul_isLittleO hn
        |>.mono (nhdsWithin_le_nhds (s := {c}ᶜ)) |>.congr' ?_ ?_
      · simp [g]
      · refine eventually_mem_nhdsWithin.mono fun w hw ↦ ?_
        rw [mem_compl_singleton_iff, ← sub_ne_zero] at hw
        simp [pow_succ, field]
  -- Finally, we apply the maximum modulus principle to this function.
  -- On the sphere `dist w c = R₁`, its norm is bounded by `R₂ / R₁ ^ (n + 1)`,
  -- thus it's bounded by the same constant on the whole closed ball,
  -- in particular, at `w = z`.
  suffices ‖g' z‖ ≤ R₂ / R₁ ^ (n + 1) by
    have hz' : ‖z - c‖ ≠ 0 := by simpa [sub_eq_zero] using hne
    simpa [g', hne, g, div_pow, mul_comm, field] using this
  refine norm_le_of_forall_mem_frontier_norm_le isBounded_ball (hdg'.diffContOnCl_ball subset_rfl)
    ?_ ?_
  · grw [frontier_ball_subset_sphere]
    intro w hw
    have hwc := ne_of_mem_sphere hw hR₁.ne'
    have hfw : ‖f w - f c‖ ≤ R₂ := by
      simpa [dist_eq_norm] using h_maps (sphere_subset_closedBall hw)
    rw [mem_sphere_iff_norm] at hw
    simpa [g', hwc, g, hw, field]
  · exact subset_closure hz

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✝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
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : a' = a
⊢ (t0, a') ∈ ⇑f ⁻¹' (q e).target

Tactic:
rw [← g_lifts, hpq e, ha]

Goal state after:
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
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : a' = a
⊢ (t0, a) ∈ ↑(q e) ∘ g ⁻¹' (q e).target
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
h₁ : f₁ =ᶠ[codiscreteWithin U] f₂
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
⊢ (divisor f₁ U) x = (divisor f₂ U) x

Tactic:
by_cases hx : x ∈ U

Goal state after:
case pos
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

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
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case pos
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
⊢ (divisor f₁ U) x = (divisor f₂ U) x

Tactic:
by_cases hx : x ∈ U

Goal state after:
case pos
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

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
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
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
x : 𝕜
⊢ 0 x ≤ (divisor f U) x

Tactic:
by_cases hx : x ∈ U

Goal state after:
case pos
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

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
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
hf₂ : AnalyticOnNhd 𝕜 f₂ U
x : 𝕜
⊢ (divisor (f₁ + f₂) U)⁻ x = (divisor f₁ U)⁻ x

Tactic:
by_cases hx : x ∈ U

Goal state after:
case pos
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

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
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
z₀ x : 𝕜
hx : x ≠ z₀
hu : x ∈ U
⊢ (meromorphicOrderAt (fun x ↦ x - z₀) x).untop₀ = (↑0).untop₀

Tactic:
congr

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
z₀ x : 𝕜
hx : x ≠ z₀
hu : x ∈ U
⊢ meromorphicOrderAt (fun x ↦ x - z₀) x = ↑0
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
U : Set 𝕜
h : z₀ ∈ U
⊢ (meromorphicOrderAt (fun x ↦ x - z₀) z₀).untop₀ = (↑1).untop₀

Tactic:
congr

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
U : Set 𝕜
h : z₀ ∈ U
⊢ meromorphicOrderAt (fun x ↦ x - z₀) z₀ = ↑1
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
h₁ : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ E

Tactic:
exact 0

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
f g : 𝕜 → E
x : 𝕜
h₁ : ¬MeromorphicAt f x
⊢ E

Tactic:
exact 0

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
h₄f : ¬∀ ⦃x_1 : ι⦄, x_1 ∈ s → ¬f x_1 x = 0
⊢ MeromorphicNFAt (∏ i ∈ s, f i) x

Tactic:
push Not at h₄f

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
h₄f : ∃ x_1 ∈ s, f x_1 x = 0
⊢ MeromorphicNFAt (∏ i ∈ s, f i) 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
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
hf : MeromorphicAt f x
h₁f : ¬meromorphicOrderAt f x = ↑0
⊢ E

Tactic:
exact 0

Goal state after:
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
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
hf : ¬MeromorphicAt f x
⊢ 𝕜 → E

Tactic:
exact 0

Goal state after:
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
h₁f : MeromorphicNFOn f U
h : AnalyticOnNhd 𝕜 f U
x : 𝕜
⊢ 0 x ≤ (MeromorphicOn.divisor f U) x

Tactic:
by_cases hx : x ∈ U

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 : 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

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
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
ho : meromorphicOrderAt f x < 0
hf : MeromorphicAt f x
m : ℤ
hm : ↑m = meromorphicOrderAt f x
⊢ m < 0

Tactic:
simpa [← hm] using ho

Goal state after:
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
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
h₁f : ¬MeromorphicOn f U
⊢ 𝕜 → E

Tactic:
exact 0

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 : 𝕜 → 𝕜
f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₁f₁ : ¬meromorphicOrderAt f₁ x = ⊤
h₁f₂ : ¬meromorphicOrderAt f₂ x = ⊤
⊢ meromorphicTrailingCoeffAt (f₁ • f₂) x = meromorphicTrailingCoeffAt f₁ x • meromorphicTrailingCoeffAt f₂ x

Tactic:
obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (meromorphicOrderAt_ne_top_iff hf₁).1 h₁f₁

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 : 𝕜 → 𝕜
f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₁f₁ : ¬meromorphicOrderAt f₁ x = ⊤
h₁f₂ : ¬meromorphicOrderAt f₂ x = ⊤
g₁ : 𝕜 → 𝕜
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : f₁ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₁ x).untop₀ • g₁ z
⊢ meromorphicTrailingCoeffAt (f₁ • f₂) x = meromorphicTrailingCoeffAt f₁ x • meromorphicTrailingCoeffAt f₂ x
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
h : MeromorphicNFOn f U
x : 𝕜
⊢ toMeromorphicNFOn f U x = f x

Tactic:
by_cases hx : x ∈ U

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 : 𝕜 → E
U : Set 𝕜
h : MeromorphicNFOn f U
x : 𝕜
hx : x ∈ U
⊢ toMeromorphicNFOn f U x = f x

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
h : MeromorphicNFOn f U
x : 𝕜
hx : x ∉ U
⊢ toMeromorphicNFOn f U x = f x
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
h : MeromorphicNFOn f U
⊢ toMeromorphicNFOn f U = f

Tactic:
· ext x
    by_cases hx : x ∈ U
    · simp only [toMeromorphicNFOn, h.meromorphicOn, ↓reduceDIte, hx]
      rw [toMeromorphicNFAt_eq_self.2 (h hx)]
    · simp [toMeromorphicNFOn, h.meromorphicOn, hx]

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
f : 𝕜 → 𝕜
h₁ : MeromorphicAt f x
h₂ : ¬meromorphicOrderAt f x = ⊤
⊢ meromorphicTrailingCoeffAt (f ^ n) x = meromorphicTrailingCoeffAt f x ^ n

Tactic:
· obtain ⟨g, h₁g, h₂g, h₃g⟩ := (meromorphicOrderAt_ne_top_iff h₁).1 h₂
    rw [h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE
        (n := (meromorphicOrderAt f x).untop₀) h₂g h₃g,
      (h₁g.zpow h₂g (n := n)).meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE
        (n := (meromorphicOrderAt (f ^ n) x).untop₀)
        (by simp_all [zpow_ne_zero])]
    · simp only [Pi.pow_apply]
    · filter_upwards [h₃g] with a ha
      simp_all [mul_zpow, ← zpow_mul, meromorphicOrderAt_zpow h₁, mul_comm]

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
f : 𝕜 → E
x : 𝕜
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
h : ¬meromorphicOrderAt f (g x) = ⊤
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
F : 𝕜 → E
h₁F : AnalyticAt 𝕜 F (g x)
h₂F : F (g x) ≠ 0
h₃F : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ (meromorphicOrderAt f (g x)).untop₀ • F z
h₁ : meromorphicTrailingCoeffAt (f ∘ g) x = meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g) x
⊢ meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r) x • (F ∘ g) x =
    meromorphicTrailingCoeffAt (fun x_1 ↦ g x_1 - g x) x ^ r • meromorphicTrailingCoeffAt f (g x)

Tactic:
h₁F.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂F h₃F

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
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
h : ¬meromorphicOrderAt f (g x) = ⊤
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
F : 𝕜 → E
h₁F : AnalyticAt 𝕜 F (g x)
h₂F : F (g x) ≠ 0
h₃F : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ (meromorphicOrderAt f (g x)).untop₀ • F z
h₁ : meromorphicTrailingCoeffAt (f ∘ g) x = meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g) x
⊢ meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r) x • (F ∘ g) x =
    meromorphicTrailingCoeffAt (fun x_1 ↦ g x_1 - g x) x ^ r • F (g x)