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

Goal state before:
pp: x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : arctan x < π / 2
⊢ False

Tactic:
rw [h, ← one_mul (π / 2), mul_div_assoc, mul_lt_mul_iff_left₀ (by positivity)] at ub

Goal state after:
x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : 2 * ↑k + 1 < 1
⊢ False
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ HasDerivWithinAt f (sSup (slope f x '' {y | y ∈ S ∧ y < x})) (Iio x) x

Tactic:
[

Goal state after:
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hfc : ConvexOn ℝ S f
hxs : x ∈ interior S
a b : ℝ
hxab : x ∈ Ioo a b
habs : Ioo a b ⊆ S
⊢ HasDerivWithinAt f (sSup (slope f x '' {y | y ∈ S ∧ y < x})) (Iio x) x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr
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
⊢ derivWithin f (Ioi x) x ≤ derivWithin f (Ioi y) y

Tactic:
[

Goal state after:
case inr
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
⊢ derivWithin f (Ioi x) x ≤ derivWithin f (Ioi y) y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr
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
⊢ derivWithin f (Iio x) x ≤ derivWithin f (Iio y) y

Tactic:
[

Goal state after:
case inr
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
⊢ derivWithin f (Iio x) x ≤ derivWithin f (Iio y) y
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℝ
⊢ angle (cexp (↑x * I)) (cexp (↑y * I)) = |toIocMod Real.two_pi_pos (-π) (x - y)|

Tactic:
[

Goal state after:
x y : ℝ
⊢ angle (cexp (↑x * I)) (cexp (↑y * I)) = |toIocMod Real.two_pi_pos (-π) (x - y)|
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : ConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Ioi x) x
⊢ ∀ᶠ (c : ℝ) in 𝓝[>] x, slope f x c ≤ slope f x y

Tactic:
[

Goal state after:
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : ConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Ioi x) x
⊢ ∀ᶠ (c : ℝ) in 𝓝[>] x, slope f x c ≤ slope f x y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : ConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Iio y) y
⊢ ∀ᶠ (c : ℝ) in 𝓝[<] y, slope f x y ≤ slope f y c

Tactic:
[

Goal state after:
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : ConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Iio y) y
⊢ ∀ᶠ (c : ℝ) in 𝓝[<] y, slope f x y ≤ slope f y c
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:
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

Goal state after:
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 : f x ≤ f y ∧ x ≤ y
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Iio y) y
u : ℝ
hxu : x < u
huy : u < y
hu : u ∈ S
this : (f x - f y) / (x - y) < (f u - f y) / (u - y)
⊢ slope f x y < f'

Tactic:
[

Goal state after:
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Iio y) y
u : ℝ
hxu : x < u
huy : u < y
hu : u ∈ S
this : (f x - f y) / (x - y) < (f u - f y) / (u - y)
⊢ slope f x y < f'
Module: Mathlib.Analysis.Complex.CoveringMap

Goal state before:
pp: case refine_2
z x✝ : ℂ
⊢ ⟨cexp z, ⋯⟩ = ⟨cexp x✝, ⋯⟩ ↔ x✝ + -z ∈ AddSubgroup.zmultiples (2 * ↑Real.pi * I)

Tactic:
[

Goal state after:
case refine_2
z x✝ : ℂ
⊢ ⟨cexp z, ⋯⟩ = ⟨cexp x✝, ⋯⟩ ↔ x✝ + -z ∈ AddSubgroup.zmultiples (2 * ↑Real.pi * I)
Module: Mathlib.Analysis.Complex.CoveringMap

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

Tactic:
simp at hx

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Action

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
c : ↑(sphere 0 1)
x : ↑(sphere 0 r)
⊢ 1 * ‖↑x‖ = r

Tactic:
mem_sphere_zero_iff_norm.1 x.coe_prop,

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
c : ↑(sphere 0 1)
x : ↑(sphere 0 r)
⊢ 1 * r = r
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 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
hf : MeromorphicAt f x
hg : MeromorphicAt g x
⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = g z) ↔ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, (f - g) z = 0

Tactic:
[

Goal state after:
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
hf : MeromorphicAt f x
hg : MeromorphicAt g x
⊢ (∃ᶠ (z : 𝕜) in 𝓝[≠] x, f z = g z) ↔ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, (f - g) z = 0
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
this :
  ∀ (n : ℕ),
    ∃ 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
hN :
  ∃ g',
    ContinuousOn g' (Set.Icc 0 (t n_max) ×ˢ N) ∧
      p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n_max, g' (t', a) = g (t', a)
⊢ ∃ N ∈ 𝓝 a,
    ∃ g',
      ContinuousOn g' (Set.univ ×ˢ N) ∧
        p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ (t : ↑I), g' (t, a) = g (t, a)

Tactic:
[

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
this :
  ∀ (n : ℕ),
    ∃ 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
hN :
  ∃ g',
    ContinuousOn g' (Set.Icc 0 (t n_max) ×ˢ N) ∧
      p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n_max, g' (t', a) = g (t', a)
⊢ ∃ N ∈ 𝓝 a,
    ∃ g',
      ContinuousOn g' (Set.univ ×ˢ N) ∧
        p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ (t : ↑I), g' (t, a) = g (t, a)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case e'_2
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
this :
  ∀ (n : ℕ),
    ∃ 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
hN :
  ∃ g',
    ContinuousOn g' (Set.Icc 0 1 ×ˢ N) ∧
      p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ 1, g' (t', a) = g (t', a)
x✝ : ↑I × A → E
a✝ : ↑I
⊢ x✝ (a✝, a) = g (a✝, a) ↔ a✝ ≤ 1 → x✝ (a✝, a) = g (a✝, a)

Tactic:
rw [imp_iff_right]

Goal state after:
case e'_2
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
this :
  ∀ (n : ℕ),
    ∃ 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
hN :
  ∃ g',
    ContinuousOn g' (Set.Icc 0 1 ×ˢ N) ∧
      p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ 1, g' (t', a) = g (t', a)
x✝ : ↑I × A → E
a✝ : ↑I
⊢ a✝ ≤ 1
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
hf₁ : MeromorphicOn f₁ U
h₁ : f₁ =ᶠ[codiscreteWithin U] f₂
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∈ U
⊢ f₁ =ᶠ[𝓝[≠] x] f₂

Tactic:
[

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
⊢ f₁ =ᶠ[𝓝[≠] x] f₂
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
hx : x ∈ U
⊢ f₁ =ᶠ[𝓝[≠] x] f₂

Tactic:
[

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
⊢ f₁ =ᶠ[𝓝[≠] x] f₂
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
⊢ 0 ≤ meromorphicOrderAt f x ↔ AnalyticAt 𝕜 f x

Tactic:
constructor <;> intro h₂f

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
x : 𝕜
hf : MeromorphicNFAt f x
h₂f : 0 ≤ meromorphicOrderAt f x
⊢ AnalyticAt 𝕜 f x

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
x : 𝕜
hf : MeromorphicNFAt f x
h₂f : AnalyticAt 𝕜 f x
⊢ 0 ≤ meromorphicOrderAt f x
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 = ((fun x_1 ↦ x_1 - x) ^ n • g) x
⊢ False

Tactic:
simp only [Pi.smul_apply', Pi.pow_apply, sub_self, zero_zpow n hContra, zero_smul] at this

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
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
Module: Mathlib.Analysis.Meromorphic.Divisor

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

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
z : 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : 𝕜 → E
⊢ (divisor (f ∘ fun x ↦ x - c) U) (z + c) = (divisor (f ∘ fun x ↦ x + -c) U) (z - -c)
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 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
⊢ E

Tactic:
by_cases h₂ : meromorphicOrderAt f x = ⊤
    · exact 0
    · exact ((meromorphicOrderAt_ne_top_iff h₁).1 h₂).choose x

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
h₁f : ∀ (i : ι), MeromorphicNFAt (f i) x
h₂f : {σ | f σ x = 0}.Subsingleton
h₃f : Function.HasFiniteMulSupport f
⊢ MeromorphicNFAt (∏ᶠ (i : ι), f i) x

Tactic:
[

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
h₁f : ∀ (i : ι), MeromorphicNFAt (f i) x
h₂f : {σ | f σ x = 0}.Subsingleton
h₃f : Function.HasFiniteMulSupport f
⊢ MeromorphicNFAt (∏ᶠ (i : ι), 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
hg : AnalyticAt 𝕜 g x
n : ℤ
q : 𝕜 → E
hq_an : AnalyticAt 𝕜 q (g x)
hq_ne : q (g x) ≠ 0
hf : f =ᶠ[𝓝 (g x)] (fun x_1 ↦ x_1 - g x) ^ n • q
m : ℕ
hm : ↑m = analyticOrderAt (fun x_1 ↦ g x_1 - g x) x
p : 𝕜 → 𝕜
h₁p : AnalyticAt 𝕜 p x
h₂p : p x ≠ 0
h₃p : ∀ᶠ (z : 𝕜) in 𝓝 x, g z - g x = (z - x) ^ m • p z
⊢ MeromorphicNFAt (f ∘ g) x

Tactic:
refine Or.inr ⟨n * m, fun z ↦ (p z) ^ n • q (g z), (h₁p.zpow h₂p).smul (by fun_prop), ?_⟩

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
x : 𝕜
hg : AnalyticAt 𝕜 g x
n : ℤ
q : 𝕜 → E
hq_an : AnalyticAt 𝕜 q (g x)
hq_ne : q (g x) ≠ 0
hf : f =ᶠ[𝓝 (g x)] (fun x_1 ↦ x_1 - g x) ^ n • q
m : ℕ
hm : ↑m = analyticOrderAt (fun x_1 ↦ g x_1 - g x) x
p : 𝕜 → 𝕜
h₁p : AnalyticAt 𝕜 p x
h₂p : p x ≠ 0
h₃p : ∀ᶠ (z : 𝕜) in 𝓝 x, g z - g x = (z - x) ^ m • p z
⊢ (fun z ↦ p z ^ n • q (g z)) x ≠ 0 ∧ f ∘ g =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ (n * ↑m) • fun z ↦ p z ^ n • q (g z)
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 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

Tactic:
simp_all only [sub_self, ite_true]

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) y = 1
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
x c : 𝕜
f : 𝕜 → E
⊢ MeromorphicNFAt (f ∘ fun x ↦ x - c) x ↔ MeromorphicNFAt f (x - c)

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x c : 𝕜
f : 𝕜 → E
⊢ MeromorphicNFAt (f ∘ fun x ↦ x - c) x ↔ MeromorphicNFAt f (x - c)
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 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 = ↑n
h₃f : meromorphicOrderAt f x = 0
hn : n = 0
⊢ Function.update f x (Classical.choose ⋯ x) x = 0 ^ n • g x

Tactic:
[

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
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 = ↑n
h₃f : meromorphicOrderAt f x = 0
hn : n = 0
⊢ Function.update f x (Classical.choose ⋯ x) x = 0 ^ n • g x
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2.refine_5
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
⊢ ContinuousOn (fun ta ↦ if f ta ∈ (q e).target then ↑(q e).symm (f ta) else g ta)
    (Set.Icc 0 (t n.succ) ×ˢ (v ∩ (N ∩ (g' ∘ fun y ↦ (t n, y)) ⁻¹' (q e).source)) ∩
      closure[instTopologicalSpaceProd] {a | ¬a.1 ≤ t n})

Tactic:
[

Goal state after:
case refine_2.refine_5
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
⊢ ContinuousOn (fun ta ↦ if f ta ∈ (q e).target then ↑(q e).symm (f ta) else g ta)
    (Set.Icc 0 (t n.succ) ×ˢ (v ∩ (N ∩ (g' ∘ fun y ↦ (t n, y)) ⁻¹' (q e).source)) ∩
      closure[instTopologicalSpaceProd] {a | ¬a.1 ≤ t n})
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
U : Set 𝕜
f : 𝕜 → E
hf : MeromorphicOn f U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
⊢ ∀ᶠ (x : 𝕜) in codiscreteWithin U, f x ≠ 0

Tactic:
[

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
hf : MeromorphicOn f U
h'f : ∀ x ∈ U, meromorphicOrderAt f x ≠ ⊤
⊢ ∀ᶠ (x : 𝕜) in codiscreteWithin U, f x ≠ 0
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 ∘ fun x ↦ x - c) U ↔ MeromorphicNFOn f (U - {c})

Tactic:
[

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
c : 𝕜
U : Set 𝕜
⊢ MeromorphicNFOn (f ∘ fun x ↦ x - c) U ↔ MeromorphicNFOn f (U - {c})
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case coe
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 : 0 < meromorphicOrderAt f x
hf : MeromorphicAt f x
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
n : ℕ
h'o : meromorphicOrderAt f x = ↑↑n
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ ↑n • g z
this : 0 = (x - x) ^ n • g x
⊢ Tendsto f (𝓝[≠] x) (𝓝 0)

Tactic:
this

Goal state after:
case coe
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 : 0 < meromorphicOrderAt f x
hf : MeromorphicAt f x
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
n : ℕ
h'o : meromorphicOrderAt f x = ↑↑n
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ ↑n • g z
this : 0 = (x - x) ^ n • g x
⊢ Tendsto f (𝓝[≠] x) (𝓝 ((x - x) ^ n • g x))
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : d u₀ = 0
⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = (fun x ↦ x - u₀) ^ d u₀ * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d u₀ 0 u

pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = (fun x ↦ x - u₀) ^ d u₀ * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d u₀ 0 u

Tactic:
· simp [← eq_update_self_iff.2 h₁d, h₁d]

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
⊢ ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u = (fun x ↦ x - u₀) ^ d u₀ * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d u₀ 0 u