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

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f c
z : ℂ
hz : z ≤ c
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ iteratedDeriv n (fun z ↦ f (-z)) (-c)

Tactic:
simpa using

Goal state after:
f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f c
z : ℂ
hz : z ≤ c
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ (-1) ^ n * iteratedDeriv n f c
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
⊢ circleAverage (herglotzLogIntegrand w ρ) 0 (r n) = log ‖w - ρ‖

Tactic:
unfold herglotzLogIntegrand

Goal state after:
w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
⊢ circleAverage (Complex.re ∘ herglotzRieszKernel 0 w • fun x ↦ log ‖x - ρ‖) 0 (r n) = log ‖w - ρ‖
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
x : ℂ
hx : x ∈ closure (ball 0 (r n))
⊢ ?m.450

Tactic:
simpa using

Goal state after:
w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
x : ℂ
hx : x ∈ closure (ball 0 (r n))
⊢ ?m.450
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case inr.refine_2
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : ¬0 ∉ U
a : ℂ
ha : a ∉ U
f : ℂ → ℂ
hf_inj : Injective f
hf_dense : ¬Dense (f '' (-a +ᵥ U))
hdf : ∀ z ∈ -a +ᵥ U, deriv f z ≠ 0
z : ℂ
hz : z ∈ U
⊢ deriv (f ∘ fun x ↦ -a + x) z ≠ 0

Tactic:
simpa using

Goal state after:
case inr.refine_2
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : ¬0 ∉ U
a : ℂ
ha : a ∉ U
f : ℂ → ℂ
hf_inj : Injective f
hf_dense : ¬Dense (f '' (-a +ᵥ U))
hdf : ∀ z ∈ -a +ᵥ U, deriv f z ≠ 0
z : ℂ
hz : z ∈ U
⊢ ¬deriv f (-a + z) = 0
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
z : ℂ
hz : z ∈ U
hfz : f z = 0
⊢ False

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
z : ℂ
hz : z ∈ U
hfz : f z = 0
⊢ False
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case hf
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
z : ℂ
hz : z ∈ U
⊢ HasStrictDerivAt ?f (2 * f z) (f z)

Tactic:
simpa using

Goal state after:
case hf
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
z : ℂ
hz : z ∈ U
⊢ HasStrictDerivAt ?f (2 * f z) (f z)
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case hf'
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
z : ℂ
hz : z ∈ U
⊢ 2 * f z ≠ 0

Tactic:
simpa using

Goal state after:
case hf'
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
z : ℂ
hz : z ∈ U
⊢ ¬f z = 0
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case refine_1
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
⊢ ¬Dense (f '' U)

Tactic:
simp only [Dense, not_forall, mem_closure_iff_frequently, not_frequently]

Goal state after:
case refine_1
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
⊢ ∃ x, ∀ᶠ (x : ℂ) in 𝓝 x, x ∉ f '' U
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
⊢ (2 * f x)⁻¹ ≠ 0

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
⊢ ¬f x = 0
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case refine_2
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
z : ℂ
hz : z ∈ U
⊢ deriv f z ≠ 0

Tactic:
simpa using

Goal state after:
case refine_2
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
z : ℂ
hz : z ∈ U
⊢ ¬f z = 0
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
⊢ ∃ x ε, 0 < ε ∧ ∀ a ∈ U, ε < dist (f a) x

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
⊢ ∃ x ε, 0 < ε ∧ ∀ a ∈ U, ε < dist (f a) x
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
z : ℂ
hz : z ∈ U
⊢ f z ≠ x

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
z : ℂ
hz : z ∈ U
⊢ ¬f z = x
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ ε < ‖f z - x‖

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ ε < ‖f z - x‖
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ 0 < ‖f z - x‖

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ ¬f z = x
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
w : ℂ
hw : w ∈ U
heq : (fun z ↦ ↑ε / (f z - x)) z = (fun z ↦ ↑ε / (f z - x)) w
⊢ z = w

Tactic:
simpa using

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
w : ℂ
hw : w ∈ U
heq : (fun z ↦ ↑ε / (f z - x)) z = (fun z ↦ ↑ε / (f z - x)) w
⊢ z = w
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
CB : Set ℂ := closedBall c |R|
h₂f✝ : ¬∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
h₂f : ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u = ⊤
this : divisor f CB = 0
z : ℂ
h₁z : MeromorphicNFAt f z
h₂z : z ∈ CB
⊢ f z = 0 z

Tactic:
simpa using

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℂ
hR : R ≠ 0
h₁f : MeromorphicOn f (closedBall c |R|)
CB : Set ℂ := closedBall c |R|
h₂f✝ : ¬∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤
h₂f : ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u = ⊤
this : divisor f CB = 0
z : ℂ
h₁z : MeromorphicNFAt f z
h₂z : z ∈ CB
⊢ f z = 0
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Exp

Goal state before:
pp: τ : ℍ
⊢ ‖cexp (2 * ↑π * Complex.I * ↑τ)‖ < 1

Tactic:
simpa using

Goal state after:
τ : ℍ
⊢ 0 < 2 * π * τ.im
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
this : (∑ᶠ (a : ℂ), ↑((divisor f (closedBall c |r|)) a)) * Real.log (R / r) ≤ Real.log (M / ‖f c‖)
⊢ 0 < Real.log (R / r)

Tactic:
simpa using

Goal state after:
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
this : (∑ᶠ (a : ℂ), ↑((divisor f (closedBall c |r|)) a)) * Real.log (R / r) ≤ Real.log (M / ‖f c‖)
⊢ 0 < Real.log (R / r)
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case h₂f.h.h.inl
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
z : ℂ
hz : z ∈ sphere c |R|
this : ‖f z‖ ≤ M
h : f z = 0
⊢ Real.log ‖f z‖ ≤ Real.log M

Tactic:
simpa using

Goal state after:
case h₂f.h.h.inl
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
z : ℂ
hz : z ∈ sphere c |R|
this : ‖f z‖ ≤ M
h : f z = 0
⊢ 0 ≤ Real.log M
Module: Mathlib.Analysis.Complex.Tietze

Goal state before:
pp: case pos
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
this✝¹ : NormedSpace ℝ E
this✝ : IsScalarTower ℝ 𝕜 E
g : E → E := fun x ↦ ‖x‖⁻¹ • x
this : Continuous ((Metric.closedBall 0 1).piecewise id g)
x : E
hx : x ∈ Metric.closedBall 0 1
⊢ (Metric.closedBall 0 1).piecewise id g x ∈ Metric.closedBall 0 1

Tactic:
simpa using

Goal state after:
case pos
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
this✝¹ : NormedSpace ℝ E
this✝ : IsScalarTower ℝ 𝕜 E
g : E → E := fun x ↦ ‖x‖⁻¹ • x
this : Continuous ((Metric.closedBall 0 1).piecewise id g)
x : E
hx : x ∈ Metric.closedBall 0 1
⊢ ‖x‖ ≤ 1
Module: Mathlib.Analysis.Complex.Tietze

Goal state before:
pp: 𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
this✝ : NormedSpace ℝ E
this : IsScalarTower ℝ 𝕜 E
g : E → E := fun x ↦ ‖x‖⁻¹ • x
x : E
hx : x ∈ frontier (Metric.closedBall 0 1)
⊢ ‖x‖ = 1

Tactic:
simpa using

Goal state after:
𝕜 : Type v
inst✝³ : RCLike 𝕜
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
this✝ : NormedSpace ℝ E
this : IsScalarTower ℝ 𝕜 E
g : E → E := fun x ↦ ‖x‖⁻¹ • x
x : E
hx : x ∈ frontier (Metric.closedBall 0 1)
⊢ ‖x‖ = 1
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
h2 : u ∈ closedBall c |r|
h3 : u ≠ c
⊢ ‖c - u‖ ≠ 0

Tactic:
simpa using

Goal state after:
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
h2 : u ∈ closedBall c |r|
h3 : u ≠ c
⊢ ¬c = u
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
hR : ∀ (r : Rˣ), ∃ k, k ^ Fintype.card n = r
g : GL n R
r : Rˣ
hr : r ^ Fintype.card n = (GeneralLinearGroup.det g)⁻¹
⊢ r ^ Fintype.card n * GeneralLinearGroup.det g = 1

Tactic:
simpa using

Goal state after:
n : Type u_1
R : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
hR : ∀ (r : Rˣ), ∃ k, k ^ Fintype.card n = r
g : GL n R
r : Rˣ
hr : r ^ Fintype.card n = (GeneralLinearGroup.det g)⁻¹
⊢ r ^ Fintype.card n = (GeneralLinearGroup.det g)⁻¹
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case hxy
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
h2 : u ∈ closedBall c |r|
h3 : u ≠ c
this : ‖c - u‖ ≠ 0
⊢ |R / r| ≤ |R * ‖c - u‖⁻¹|

Tactic:
simpa using

Goal state after:
case hxy
c : ℂ
r R M : ℝ
f : ℂ → ℂ
r_pos : 0 < |r|
r_lt_R : |r| < |R|
hM : 1 ≤ M
h₁f : AnalyticOnNhd ℂ f (closedBall c |R|)
h₂f : f c ≠ 0
f_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M
hrR : 1 < |R / r|
jensen :
  circleAverage (fun x ↦ Real.log ‖f x‖) c R =
    ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|)) u) * Real.log (R * ‖c - u‖⁻¹) + Real.log ‖f c‖
integral_bound : circleAverage (fun x ↦ Real.log ‖f x‖) c R ≤ Real.log M
u : ℂ
h1 : u ∈ closedBall c |R|
h2 : u ∈ closedBall c |r|
h3 : u ≠ c
this : ‖c - u‖ ≠ 0
⊢ ‖c - u‖ ≤ |r|
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Nonempty n
h : Function.Surjective ⇑toPGL
r : Rˣ
A : GL n R
hA : GeneralLinearGroup.det A = r
X : SpecialLinearGroup n R
hX : toPGL ↑X = mk A
⊢ ∃ u, toGL X * (GeneralLinearGroup.scalar n) u = A

Tactic:
simpa using

Goal state after:
n : Type u_1
R : Type u_2
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Nonempty n
h : Function.Surjective ⇑toPGL
r : Rˣ
A : GL n R
hA : GeneralLinearGroup.det A = r
X : SpecialLinearGroup n R
hX : toPGL ↑X = mk A
⊢ ∃ u, toGL X * (GeneralLinearGroup.scalar n) u = A
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Nonempty n
h : Function.Surjective ⇑toPGL
r : Rˣ
A : GL n R
hA : GeneralLinearGroup.det A = r
X : SpecialLinearGroup n R
hX : toPGL ↑X = mk A
u : Rˣ
hu : toGL X * (GeneralLinearGroup.scalar n) u = A
⊢ u ^ Fintype.card n = r

Tactic:
simpa using

Goal state after:
n : Type u_1
R : Type u_2
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Nonempty n
h : Function.Surjective ⇑toPGL
r : Rˣ
A : GL n R
hA : GeneralLinearGroup.det A = r
X : SpecialLinearGroup n R
hX : toPGL ↑X = mk A
u : Rˣ
hu : toGL X * (GeneralLinearGroup.scalar n) u = A
⊢ u ^ Fintype.card n = r
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Nonempty n
F : Type u_3
inst✝¹ : Field F
inst✝ : IsAlgClosed F
r : Fˣ
x : F
hx : (X ^ Fintype.card n - C ↑r).IsRoot x
hx' : x ≠ 0
⊢ { val := x, inv := x⁻¹, val_inv := ⋯, inv_val := ⋯ } ^ Fintype.card n = r

Tactic:
simpa using

Goal state after:
n : Type u_1
R : Type u_2
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : Nonempty n
F : Type u_3
inst✝¹ : Field F
inst✝ : IsAlgClosed F
r : Fˣ
x : F
hx : (X ^ Fintype.card n - C ↑r).IsRoot x
hx' : x ≠ 0
⊢ x ^ Fintype.card n = ↑r
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
F : Type u_3
inst✝¹ : Field F
inst✝ : IsAlgClosed F
h : ¬Nonempty n
⊢ IsEmpty n

Tactic:
simpa using

Goal state after:
n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
F : Type u_3
inst✝¹ : Field F
inst✝ : IsAlgClosed F
h : ¬Nonempty n
⊢ IsEmpty n
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: case inr.refine_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
hf3 : -‖f‖ / 3 < ‖f‖ / 3
hc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hc₂ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hd : Disjoint (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
g : Y →ᵇ ℝ
hg₁ : EqOn (⇑g) (Function.const Y (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hg₂ : EqOn (⇑g) (Function.const Y (‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hgf : ∀ (x : Y), g x ∈ Icc (-‖f‖ / 3) (‖f‖ / 3)
y : Y
⊢ ‖g y‖ ≤ ‖f‖ / 3

Tactic:
simpa using

Goal state after:
case inr.refine_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
hf3 : -‖f‖ / 3 < ‖f‖ / 3
hc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hc₂ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hd : Disjoint (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
g : Y →ᵇ ℝ
hg₁ : EqOn (⇑g) (Function.const Y (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hg₂ : EqOn (⇑g) (Function.const Y (‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hgf : ∀ (x : Y), g x ∈ Icc (-‖f‖ / 3) (‖f‖ / 3)
y : Y
⊢ -(‖f‖ / 3) ≤ g y ∧ g y ≤ ‖f‖ / 3
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
hf3 : -‖f‖ / 3 < ‖f‖ / 3
hc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hc₂ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hd : Disjoint (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
g : Y →ᵇ ℝ
hg₁ : EqOn (⇑g) (Function.const Y (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hg₂ : EqOn (⇑g) (Function.const Y (‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hgf : ∀ (x : Y), g x ∈ Icc (-‖f‖ / 3) (‖f‖ / 3)
x : X
⊢ -‖f‖ ≤ f x ∧ f x ≤ ‖f‖

Tactic:
simpa using

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : IsClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
hf3 : -‖f‖ / 3 < ‖f‖ / 3
hc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hc₂ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hd : Disjoint (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
g : Y →ᵇ ℝ
hg₁ : EqOn (⇑g) (Function.const Y (-‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))
hg₂ : EqOn (⇑g) (Function.const Y (‖f‖ / 3)) (⇑e '' ⇑f ⁻¹' Ici (‖f‖ / 3))
hgf : ∀ (x : Y), g x ∈ Icc (-‖f‖ / 3) (‖f‖ / 3)
x : X
⊢ -‖f‖ ≤ f x ∧ f x ≤ ‖f‖
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints

Goal state before:
pp: g : GL (Fin 2) ℝ
z : ℍ
htrace : ↑g 0 0 = -↑g 1 1
hc : ↑g 1 0 = 0
h₀ : ↑g 1 1 = 0
⊢ False

Tactic:
simpa using

Goal state after:
g : GL (Fin 2) ℝ
z : ℍ
htrace : ↑g 0 0 = -↑g 1 1
hc : ↑g 1 0 = 0
h₀ : ↑g 1 1 = 0
⊢ False
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
g : GL (Fin 2) R
⊢ (¬∃ y, (↑g * diagonal fun x ↦ y) = ↑g * m) ∧ m.discr = 0 ↔ (¬∃ y, (diagonal fun x ↦ y) = m) ∧ m.discr = 0

Tactic:
Units.mul_right_inj

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
g : GL (Fin 2) R
⊢ ((↑g)⁻¹ * m * ↑g).IsParabolic ↔ m.IsParabolic

Tactic:
simpa using

Goal state after:
R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
g : GL (Fin 2) R
⊢ ((↑g)⁻¹ * m * ↑g).IsParabolic ↔ m.IsParabolic
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: case right
R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
h : m.IsParabolic
⊢ (-m).discr = 0

Tactic:
simpa using

Goal state after:
case right
R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
h : m.IsParabolic
⊢ m.trace ^ 2 - 4 * m.det = 0
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
h : (-m).IsParabolic
⊢ m.IsParabolic

Tactic:
simpa using

Goal state after:
R : Type u_1
inst✝ : CommRing R
m : Matrix (Fin 2) (Fin 2) R
h : (-m).IsParabolic
⊢ m.IsParabolic
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints

Goal state before:
pp: case inr
g : GL (Fin 2) ℝ
z : ℍ
hpos : 0 < (↑g).det
hell : g.IsElliptic
this : ∀ {g : GL (Fin 2) ℝ}, 0 < (↑g).det → ∀ (hell : g.IsElliptic), 0 < ↑g 1 0 → (g • z = z ↔ z = fixedPt g hell)
hc : ↑g 1 0 < 0
⊢ g • z = z ↔ z = fixedPt g hell

Tactic:
simpa using

Goal state after:
case inr
g : GL (Fin 2) ℝ
z : ℍ
hpos : 0 < (↑g).det
hell : g.IsElliptic
this : ∀ {g : GL (Fin 2) ℝ}, 0 < (↑g).det → ∀ (hell : g.IsElliptic), 0 < ↑g 1 0 → (g • z = z ↔ z = fixedPt g hell)
hc : ↑g 1 0 < 0
⊢ g • z = z ↔ z = fixedPt g hell