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

Goal state before:
pp: case h
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1
x : E
hx : ‖x‖ = 1
y : E
hy : ‖y‖ = 1
hne : x ≠ y
a b : ℝ
hab : a + b = 1
hlt : ‖a • x + b • y‖ < 1
⊢ (AffineMap.lineMap x y) b ∈ interior (closedBall 0 1)

Tactic:
AffineMap.lineMap_apply_module,

Goal state after:
case h
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1
x : E
hx : ‖x‖ = 1
y : E
hy : ‖y‖ = 1
hne : x ≠ y
a b : ℝ
hab : a + b = 1
hlt : ‖a • x + b • y‖ < 1
⊢ (1 - b) • x + b • y ∈ interior (closedBall 0 1)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
f_deriv_integral_ac : AbsolutelyContinuousOnInterval (fun x ↦ ∫ (v : ℝ) in a..x, deriv f v) a b
g : ℝ → ℝ := fun x ↦ f x - ∫ (t : ℝ) in a..x, deriv f t
g_ac : AbsolutelyContinuousOnInterval g a b
g_ae_deriv_zero : ∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt g 0 x
C : ℝ
hC : ∀ x ∈ uIcc a b, g x = C
⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a

Tactic:
have : f a = g a := by simp [g]

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
f_deriv_integral_ac : AbsolutelyContinuousOnInterval (fun x ↦ ∫ (v : ℝ) in a..x, deriv f v) a b
g : ℝ → ℝ := fun x ↦ f x - ∫ (t : ℝ) in a..x, deriv f t
g_ac : AbsolutelyContinuousOnInterval g a b
g_ae_deriv_zero : ∀ᵐ (x : ℝ), x ∈ uIcc a b → HasDerivAt g 0 x
C : ℝ
hC : ∀ x ∈ uIcc a b, g x = C
this : f a = g a
⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
hx : 0 < ‖x‖
hy : 0 < ‖y‖
hne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y
hxy : 0 < ‖x‖ + ‖y‖
⊢ ‖x‖ / (‖x‖ + ‖y‖) + ‖y‖ / (‖x‖ + ‖y‖) = 1

Tactic:
by rw [← add_div, div_self hxy.ne']

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

Goal state before:
pp: E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
x' : E := ‖x‖⁻¹ • x
y' : E := ‖y‖⁻¹ • y
hxy' : ε / 3 ≤ ‖x' - y'‖
⊢ δ' ≤ δ / 3

Tactic:
exact min_le_of_right_le <| min_le_right _ _

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

Goal state before:
pp: case succ
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
⊢ 0 ≤ (∏ x ∈ Finset.range ((1 + 1) * n), (m - ↑x)) * (m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1))

Tactic:
mul_assoc

Goal state after:
case succ
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
⊢ 0 ≤ (∏ x ∈ Finset.range ((1 + 1) * n), (m - ↑x)) * ((m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1)))
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
⊢ ‖f w‖ = ‖f z‖

Tactic:
refine (isMaxOn_iff.1 hz _ hw).antisymm (not_lt.1 ?_)

Goal state after:
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
⊢ ¬(norm ∘ f) w < (norm ∘ f) z
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
⊢ ‖1 - z‖ * (F + 1) < ε / 4

Tactic:
by rwa [norm_sub_rev, lt_div_iff₀ (by positivity)] at zd

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

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a

Tactic:
have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
    (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
A : DifferentiableOn ℝ f (Ioo x y)
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : f w - f x < deriv f a * (w - x)
hb : f y - f w < deriv f b * (y - w)
⊢ deriv f a * (w - x) < deriv f b * (w - x)

Tactic:
apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : f w - f x < deriv f a * (w - x)
hb : f y - f w < deriv f b * (y - w)
⊢ deriv f a < deriv f b

x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : f w - f x < deriv f a * (w - x)
hb : f y - f w < deriv f b * (y - w)
⊢ 0 ≤ deriv f b
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
    (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
A : DifferentiableOn ℝ f (Ioo x y)
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: x y : ℂ
h :
  ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), y * I ≠ (2 * ↑l + 1) * ↑π / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, y * I = (2 * ↑l + 1) * ↑π / 2
⊢ (tan x + tanh y * I) / (1 - tan x * (tanh y * I)) = (tan x + tanh y * I) / (1 - tan x * tanh y * I)

Tactic:
mul_assoc

Goal state after:
x y : ℂ
h :
  ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), y * I ≠ (2 * ↑l + 1) * ↑π / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, y * I = (2 * ↑l + 1) * ↑π / 2
⊢ (tan x + tanh y * I) / (1 - tan x * (tanh y * I)) = (tan x + tanh y * I) / (1 - tan x * (tanh y * I))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ ContinuousOn tan (Ioo (-(π / 2)) (π / 2))

Tactic:
refine ContinuousOn.mono continuousOn_tan fun x => ?_
  simp only [and_imp, mem_Ioo, mem_ofPred_eq, Ne]
  rw [cos_eq_zero_iff]
  rintro hx_gt hx_lt ⟨r, hxr_eq⟩
  rcases le_or_gt 0 r with h | h
  · rw [lt_iff_not_ge] at hx_lt
    refine hx_lt ?_
    rw [hxr_eq, ← one_mul (π / 2), mul_div_assoc, mul_le_mul_iff_left₀ (half_pos pi_pos)]
    simp [h]
  · rw [lt_iff_not_ge] at hx_gt
    refine hx_gt ?_
    rw [hxr_eq, ← one_mul (π / 2), mul_div_assoc, neg_mul_eq_neg_mul,
      mul_le_mul_iff_left₀ (half_pos pi_pos)]
    have hr_le : r ≤ -1 := by rwa [Int.lt_iff_add_one_le, ← le_neg_iff_add_nonpos_right] at h
    rw [← le_sub_iff_add_le, mul_comm, ← le_div_iff₀]
    · norm_num
      assumption_mod_cast
    · exact zero_lt_two

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‖ * ‖y‖ ≠ 0
⊢ angle x y = 0

Tactic:
rw [angle, h, div_self h₁, Real.arccos_one]

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

Goal state before:
pp: case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
⊢ ContinuousOn (U.piecewise (fun x ↦ cexp (f x / ↑n)) fun x ↦ g x ^ (1 / ↑n)) U

Tactic:
rw [continuousOn_iff_continuous_domRestrict, domRestrict_piecewise,
      ← continuousOn_iff_continuous_domRestrict]

Goal state after:
case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
⊢ ContinuousOn (fun x ↦ cexp (f x / ↑n)) U
Module: Mathlib.Analysis.Meromorphic.Divisor

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
U : Set 𝕜
hf₁ : MeromorphicOn f₁ U
hf₂ : MeromorphicOn f₂ U
h : (divisor f₁ U)⁻ ≤ (divisor f₂ U)⁻
⊢ (divisor f₁ U)⁻ ⊔ (divisor f₂ U)⁻ ≤ (divisor f₁ U)⁻ + (divisor f₂ U)⁻

Tactic:
simp_all [negPart_nonneg]

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

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
f₁ f₂ : 𝕜 → E
U : Set 𝕜
hf₁ : MeromorphicOn f₁ U
hf₂ : MeromorphicOn f₂ U
h : ¬(divisor f₁ U)⁻ ≤ (divisor f₂ U)⁻
⊢ (divisor f₁ U)⁻ ⊔ (divisor f₂ U)⁻ ≤ (divisor f₁ U)⁻ + (divisor f₂ U)⁻

Tactic:
simp_all [negPart_nonneg]

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

Goal state before:
pp: case empty
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ ∅, MeromorphicOn (f i) U
h₂f : ∀ i ∈ ∅, ∀ z ∈ U, meromorphicOrderAt (f i) z ≠ ⊤
⊢ divisor (∏ i ∈ ∅, f i) U = ∑ i ∈ ∅, divisor (f i) U

Tactic:
| empty =>

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
h₂ : ¬meromorphicOrderAt f x = ⊤
⊢ E

Tactic:
· exact ((meromorphicOrderAt_ne_top_iff h₁).1 h₂).choose x

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 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
⊢ meromorphicTrailingCoeffAt f x = g x

Tactic:
by
  have h₁f : MeromorphicAt f x := by
    rw [MeromorphicAt.meromorphicAt_congr h]
    fun_prop
  by_cases h₃ : meromorphicOrderAt f x = ⊤
  · simp only [h₃, WithTop.untop₀_top, zpow_zero, one_smul,
      MeromorphicAt.meromorphicTrailingCoeffAt_of_order_eq_top] at ⊢ h
    apply EventuallyEq.eq_of_nhds (f := 0)
    rw [← ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE (by fun_prop) (by fun_prop)]
    apply (h.symm.trans (meromorphicOrderAt_eq_top_iff.1 h₃)).symm
  · unfold meromorphicTrailingCoeffAt
    simp only [h₁f, reduceDIte, h₃, ne_eq]
    obtain ⟨h'₁, h'₂, h'₃⟩ := ((meromorphicOrderAt_ne_top_iff h₁f).1 h₃).choose_spec
    apply Filter.EventuallyEq.eq_of_nhds
    rw [← h'₁.continuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE h₁g.continuousAt]
    filter_upwards [h, h'₃, self_mem_nhdsWithin] with y h₁y h₂y h₃y
    rw [← sub_eq_zero]
    rwa [h₂y, ← sub_eq_zero, ← smul_sub, smul_eq_zero_iff_right] at h₁y
    simp_all [zpow_ne_zero, sub_ne_zero]

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 : ¬MeromorphicAt f x
⊢ toMeromorphicNFAt f x = 0

Tactic:
simp [toMeromorphicNFAt, hf]

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

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
h : HasFiniteSupport d
x : 𝕜
⊢ (Function.mulSupport fun u ↦ (x - u) ^ d u) ⊆ ↑(Finite.toFinset h)

Tactic:
intro u
    contrapose
    simp_all

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

Goal state before:
pp: case empty
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
x : 𝕜
h : ∀ σ ∈ ∅, MeromorphicAt (f σ) x
⊢ meromorphicTrailingCoeffAt (∏ n ∈ ∅, f n) x = ∏ n ∈ ∅, meromorphicTrailingCoeffAt (f n) x

Tactic:
| empty =>

Goal state after:
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
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : MeromorphicAt f₂ x
⊢ MeromorphicAt f₁ x

Tactic:
exact hf₁.congr hf₁₂.symm

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
hf : Meromorphic f
⊢ (∃ u, meromorphicOrderAt f u ≠ ⊤) ↔ ∀ (u : 𝕜), meromorphicOrderAt f u ≠ ⊤

Tactic:
by
  simpa using (meromorphicOn_univ.2 hf).exists_meromorphicOrderAt_ne_top_iff_forall isConnected_univ

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

Goal state before:
pp: case pos.inr
R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
hz : z ∈ ball 0 R
h₂z : z ≠ w
⊢ (meromorphicOrderAt (canonicalFactor R w) z).untop₀ =
    (-locallyFinsuppWithin.restrict (locallyFinsuppWithin.single w 1) ⋯) z

Tactic:
have : meromorphicOrderAt (canonicalFactor R w) z = 0 := by
        rw [(meromorphicNFOn_canonicalFactor hw (Set.mem_univ z)).meromorphicOrderAt_eq_zero_iff]
        exact canonicalFactor_ne_zero hw (ball_subset_closedBall hz) h₂z

Goal state after:
case pos.inr
R : ℝ
w : ℂ
hw : w ∈ ball 0 R
z : ℂ
hz : z ∈ ball 0 R
h₂z : z ≠ w
this : meromorphicOrderAt (canonicalFactor R w) z = 0
⊢ (meromorphicOrderAt (canonicalFactor R w) z).untop₀ =
    (-locallyFinsuppWithin.restrict (locallyFinsuppWithin.single w 1) ⋯) z
Module: Mathlib.Analysis.Meromorphic.Order

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

Tactic:
| empty =>

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

Goal state before:
pp: c z : ℂ
r x : ℝ
hx : x ∈ Ioo (z.re - (r - dist z c)) (z.re + (r - dist z c))
⊢ ↑x + ↑z.im * I ∈ ball c r

Tactic:
set r₁ := r - dist z c

Goal state after:
c z : ℂ
r x : ℝ
r₁ : ℝ := r - dist z c
hx : x ∈ Ioo (z.re - r₁) (z.re + r₁)
⊢ ↑x + ↑z.im * I ∈ ball c r
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case pos.inr
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
hz : z ∈ verticalStrip 0 1
h1 : sSupNormIm f 1 = 0
⊢ ¬0 = z

Tactic:
simp only [Complex.ext_iff, zero_re, ne_of_lt hz.1, false_and, not_false_eq_true]

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

Goal state before:
pp: case neg
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
z : ℂ
⊢ DifferentiableAt ℂ (fun z ↦ ↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z) z

Tactic:
refine DifferentiableAt.mul ?_ ?_

Goal state after:
case neg.refine_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
z : ℂ
⊢ DifferentiableAt ℂ (fun z ↦ ↑(sSupNormIm f 0) ^ (1 - z)) z

case neg.refine_2
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
z : ℂ
⊢ DifferentiableAt ℂ (HPow.hPow ↑(sSupNormIm f 1)) z
Module: Mathlib.Analysis.Distribution.DerivNotation

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
R : Type u_4
V : Type u_5
E : Type u_6
F : Type u_7
V₁ : Type u_8
V₂ : Type u_9
V₃ : Type u_10
inst✝¹⁴ : LineDeriv E V₁ V₂
inst✝¹³ : LineDeriv E V₂ V₃
inst✝¹² : AddCommGroup V₁
inst✝¹¹ : AddCommGroup V₂
inst✝¹⁰ : AddCommGroup V₃
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module R E
inst✝⁶ : Module R V₂
inst✝⁵ : Module R V₃
inst✝⁴ : LineDerivAdd E V₂ V₃
inst✝³ : LineDerivAdd E V₁ V₂
inst✝² : LineDerivSMul R E V₂ V₃
inst✝¹ : LineDerivLeftSMul R E V₁ V₂
inst✝ : LineDerivLeftSMul R E V₂ V₃
f : V₁
⊢ ∀ (m₁ m₂ n : E), ∂_{m₁ + m₂} (∂_{n} f) = ∂_{m₁} (∂_{n} f) + ∂_{m₂} (∂_{n} f)

Tactic:
simp [lineDerivOp_left_add]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
⊢ ‖f z‖ ≤ ‖interpStrip f z‖

Tactic:
apply le_on_closure (fun w hw ↦ norm_le_interpStrip_of_mem_verticalStrip_zero f w hd hB hw)
    (Continuous.comp_continuousOn' continuous_norm hd.2)
    (Continuous.comp_continuousOn' continuous_norm (diffContOnCl_interpStrip f).2)

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
⊢ z ∈ closure (verticalStrip 0 1)
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: case calc_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
h : IsConservativeOn f (ball c r)
w : ℂ
I₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₂ : E := ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
⊢ I₁ + I • I₂ - (↑w.re - ↑z.re + (↑w.im - ↑z.im) * I) • f z =
    I₁ - (↑w.re - ↑z.re) • f z + I • (I₂ - (↑w.im - ↑z.im) • f z)

Tactic:
rw [add_smul, smul_sub, smul_smul, mul_comm I]

Goal state after:
case calc_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
h : IsConservativeOn f (ball c r)
w : ℂ
I₁ : E := ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)
I₂ : E := ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)
⊢ I₁ + I • I₂ - ((↑w.re - ↑z.re) • f z + ((↑w.im - ↑z.im) * I) • f z) =
    I₁ - (↑w.re - ↑z.re) • f z + (I • I₂ - ((↑w.im - ↑z.im) * I) • f z)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
inst✝ : CompleteSpace E
⊢ circleAverage f c 0 = f c

Tactic:
rw [circleAverage]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
inst✝ : CompleteSpace E
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c 0 θ) = f c
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
⊢ ∃ F, AnalyticOnNhd ℂ F univ ∧ (fun z ↦ (F z).re) = f

Tactic:
have h₃F : Differentiable ℝ F := h₂F.restrictScalars (𝕜 := ℝ)

Goal state after:
f : ℂ → ℝ
hf : HarmonicOnNhd f univ
g : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
hg : Differentiable ℂ g
F : ℂ → ℂ
hF : F 0 = ↑(f 0) ∧ ∀ x ∈ univ, HasDerivAt F (g x) x
h₁F : ∀ (z₁ : ℂ), HasDerivAt F (g z₁) z₁
h₂F : Differentiable ℂ F
h₃F : Differentiable ℝ F
⊢ ∃ F, AnalyticOnNhd ℂ F univ ∧ (fun z ↦ (F z).re) = f
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
inst✝ : CompleteSpace E
a : E
hf : ∀ x ∈ sphere c |R|, f x = a
⊢ circleAverage f c R = a

Tactic:
rw [circleAverage]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
inst✝ : CompleteSpace E
a : E
hf : ∀ x ∈ sphere c |R|, f x = a
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) = a
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : ¬x = 0
this : ∀ᶠ (y : ℝ) in 𝓝 x, deriv (fun x ↦ x * log x) y = log y + 1
⊢ deriv (deriv fun x ↦ x * log x) x = x⁻¹

Tactic:
by
      refine (Filter.EventuallyEq.deriv_eq this).trans ?_
      rw [deriv_add_const, deriv_log x]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C

Tactic:
rcases le_or_gt ‖f x₀‖ C with h | h

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : ‖f ↑x₀‖ ≤ C
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C

case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : C < ‖f ↑x₀‖
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
this✝ : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (ball w d) (thickening d (ball c R)ᶜ)
x : ℂ
hx : x ∈ ball w d
θ : ℝ
this : ∀ (x_1 : ℂ), R ≤ dist x_1 c → d ≤ dist x x_1
⊢ R ≤ dist (circleMap c R θ) c

Tactic:
by simp [dist_eq_norm, le_abs_self]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case empty
α : Type u_1
f : α → ℝ
⊢ log⁺ (∏ t ∈ ∅, f t) ≤ ∑ t ∈ ∅, log⁺ (f t)

Tactic:
| empty =>

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
hab : a ≤ b
x : ℝ
hx : x ∈ [[0, b - a]]
⊢ 0 ≤ x

Tactic:
uIcc_of_le (sub_nonneg.2 hab)

Goal state after:
a b : ℝ
n : ℕ
hab : a ≤ b
x : ℝ
hx : x ∈ Set.Icc 0 (b - a)
⊢ 0 ≤ x
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
⊢ log 1 = 0

pp: a : ℂ
h : ‖a‖ < 1
⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖1 - x * a‖) (closedBall 0 |1|)

Tactic:
log_one

Goal state after:
a : ℂ
h : ‖a‖ < 1
⊢ 0 = 0

a : ℂ
h : ‖a‖ < 1
⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖1 - x * a‖) (closedBall 0 |1|)