mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case refine_1
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
U : Set 𝕜
h : MeromorphicOn f U
hx : x ∈ U
this : {x}ᶜ = U \ {x} ∪ Uᶜ
⊢ ∀ᶠ (y : 𝕜) in 𝓝[U \ {x}] x, AnalyticAt 𝕜 f y ∨ y ∉ U

Tactic:
filter_upwards [h.eventually_analyticAt hx] with y hy using Or.inl hy

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
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
H' : ↑I × A → E
x✝¹ : (∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)) ∧ p ∘ H' = ⇑H ∧ ∀ (a : A), H' (0, a) = f a
H'_cont : ∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)
H'_lifts : p ∘ H' = ⇑H
H'_0 : ∀ (a : A), H' (0, a) = f a
x✝ : ↑I × A
t✝ : ↑I
a✝ : A
t : ℝ
a : t ∈ I
⊢ (p ∘ fun x ↦ H' (x, a✝)) ⟨t, a⟩ = (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a✝).2))) ⟨t, a⟩

Tactic:
exact congr_fun H'_lifts _

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y z : X
γ : Path.Homotopic.Quotient x y
γ' : Path.Homotopic.Quotient y z
e : ↑(p ⁻¹' {x})
⊢ cov.monodromy (γ.trans γ') e = cov.monodromy γ' (cov.monodromy γ e)

Tactic:
by
  obtain ⟨γ⟩ := γ; obtain ⟨γ'⟩ := γ'
  exact Subtype.ext (congr($(cov.liftPath_trans e.2.symm ..) 1).trans (Path.target _))

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
ε : ℝ
hε : ε > 0
B : ℝ
z : ℂ
hB : ‖f z‖ ≤ B
hset : z ∈ verticalClosedStrip 0 1
hM0_one : ε + sSupNormIm f 0 < 1
⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) ≤ (ε + sSupNormIm f 0) ^ (-1)

Tactic:
apply Real.rpow_le_rpow_of_exponent_ge (sSupNormIm_eps_pos f hε 0) (le_of_lt hM0_one) _

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
B : ℝ
z : ℂ
hB : ‖f z‖ ≤ B
hset : z ∈ verticalClosedStrip 0 1
hM0_one : ε + sSupNormIm f 0 < 1
⊢ -1 ≤ z.re - 1
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
a✝¹ : ℂ
a✝ : a✝¹ ∈ support fun i ↦ ↑((divisor f B₀R) i) * log ‖canonicalFactor R i w‖
⊢ a✝¹ ∈ ↑t₂

Tactic:
simp_all [S₀R, B₀R]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
a✝¹ : ℂ
a✝ : a✝¹ ∈ support fun i ↦ ↑((divisor f S₀R) i) * log ‖w - i‖
⊢ a✝¹ ∈ ↑t₁

Tactic:
simp_all [S₀R, B₀R]

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
g : Gᵐᵒᵖ
⊢ MulOpposite.unop
        ({ toFun := fun γ ↦ MulOpposite.op ((hp.fiberEquivGroup e) (⋯.monodromy γ e)), map_one' := ⋯, map_mul' := ⋯ }
          γ) =
      MulOpposite.unop g ↔
    MulOpposite.unop g • ↑e = ↑(⋯.monodromy γ e)

Tactic:
iff_comm,

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
g : Gᵐᵒᵖ
⊢ MulOpposite.unop g • ↑e = ↑(⋯.monodromy γ e) ↔
    MulOpposite.unop
        ({ toFun := fun γ ↦ MulOpposite.op ((hp.fiberEquivGroup e) (⋯.monodromy γ e)), map_one' := ⋯, map_mul' := ⋯ }
          γ) =
      MulOpposite.unop g
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
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hd : d < π / 2 / b
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
A : ℝ
hA₀ : A > 0
hA :
  ∀ᶠ (b_1 : ℝ) in atTop,
    ∀ (a_1 : ℂ), |a_1.re| = b_1 → a - b < a_1.im ∧ a_1.im < a + b → ‖f a_1‖ ≤ A * expR (B * expR (c * |a_1.re|))
⊢ ∀ᶠ (R : ℝ) in atTop, ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C

Tactic:
suffices
        Tendsto (fun R => expR (δ * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (𝓝 0) by
      filter_upwards [this.eventually (ge_mem_nhds hC₀), hA] with R hR Hle w hre him
      calc
        ‖g ε w • f w‖ ≤ expR (δ * expR (d * R) + B * expR (c * R) + Real.log A) := ?_
        _ ≤ C := hR
      rw [norm_smul, Real.exp_add, ← hre, Real.exp_add, Real.exp_log hA₀, mul_assoc, mul_comm _ A]
      gcongr
      exacts [hδ <| Ioo_subset_Icc_self him, Hle _ hre him]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hd : d < π / 2 / b
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
A : ℝ
hA₀ : A > 0
hA :
  ∀ᶠ (b_1 : ℝ) in atTop,
    ∀ (a_1 : ℂ), |a_1.re| = b_1 → a - b < a_1.im ∧ a_1.im < a + b → ‖f a_1‖ ≤ A * expR (B * expR (c * |a_1.re|))
⊢ Tendsto (fun R ↦ expR (δ * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R η : ℝ
⊢ circleAverage f c R = (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R (θ + η))

Tactic:
by
  rw [intervalIntegral.integral_comp_add_right (fun θ ↦ f (circleMap c R θ))]
  have t₀ : (fun θ ↦ f (circleMap c R θ)).Periodic (2 * π) :=
    fun x ↦ by simp [periodic_circleMap c R x]
  have := t₀.intervalIntegral_add_eq 0 η
  rw [zero_add, add_comm] at this
  rw [zero_add]
  simp only [circleAverage, mul_inv_rev]
  congr

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

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) ↑x‖ ≤ C

pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) (↑x * I)‖ ≤ C

Tactic:
· rw [comp_apply, ← ofReal_neg]
    exact hre (-x) (neg_nonpos.2 hx)

Goal state after:
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖(f ∘ Neg.neg) (↑x * I)‖ ≤ C
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: case e_6
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ ((starRingEnd ℂ) (g x)).arg + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi

pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (starRingEnd ℂ) (g x) ≠ 0

Tactic:
Complex.arg_conj

Goal state after:
case e_6
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (if (g x).arg = Real.pi then Real.pi else -(g x).arg) + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi

z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (starRingEnd ℂ) (g x) ≠ 0
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: f : ℂ → ℂ
z✝ : ℂ
h₁f : AnalyticAt ℂ f z✝
h₂f : f z✝ ≠ 0
z : ℂ
⊢ 0 ≤ normSq (f z)

Tactic:
exact (f z).normSq_nonneg

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

Goal state before:
pp: case neg
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'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
x₀ : ℝ
hx₀ : 0 ≤ x₀
hne : f ↑x₀ ≠ 0
hlt : ‖0‖ < ‖f ↑x₀‖
⊢ ∀ᶠ (x : ℝ) in atTop ⊓ 𝓟 (Ici 0), ‖f ↑x‖ ≤ ‖f ↑x₀‖

Tactic:
exact (hre.norm.eventually <| ge_mem_nhds hlt).filter_mono inf_le_left

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
⊢ HasDerivAt (fun w ↦ circleAverage (fun ζ ↦ herglotzRieszKernel 0 w ζ • f ζ) 0 R)
    (circleAverage (fun ζ ↦ (2 * ζ / (ζ - w) ^ 2) • f ζ) 0 R) w

Tactic:
have hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ))
      (volume.restrict (uIoc 0 (2 * π))) := (intervalIntegrable_iff.1 hg).aestronglyMeasurable

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ)) (volume.restrict (uIoc 0 (2 * π)))
⊢ HasDerivAt (fun w ↦ circleAverage (fun ζ ↦ herglotzRieszKernel 0 w ζ • f ζ) 0 R)
    (circleAverage (fun ζ ↦ (2 * ζ / (ζ - w) ^ 2) • f ζ) 0 R) w
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
θ : ℝ
⊢ circleMap 0 r θ ≠ w

Tactic:
by_contra h

Goal state after:
w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
θ : ℝ
h : circleMap 0 r θ = w
⊢ False
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
θ : ℝ
⊢ circleMap 0 r θ ≠ ρ

Tactic:
by_contra h

Goal state after:
w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
θ : ℝ
h : circleMap 0 r θ = ρ
⊢ False
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r₀ r : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hw : ‖w‖ < r₀
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ : ℝ
hdR : 0 < ‖circleMap 0 R θ - ρ‖
⊢ |(herglotzRieszKernel 0 w (circleMap 0 r θ)).re| * |log ‖circleMap 0 r θ - ρ‖| ≤
    (R + ‖w‖) / (r₀ - ‖w‖) * (|log (2 * R)| + |log √(r₀ / R)| + |log ‖circleMap 0 R θ - ρ‖|)

Tactic:
have ⟨hrw, hr⟩ : 0 < r₀ - ‖w‖ ∧ 0 < r := by grind

Goal state after:
w ρ : ℂ
R r₀ r : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hw : ‖w‖ < r₀
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ : ℝ
hdR : 0 < ‖circleMap 0 R θ - ρ‖
hrw : 0 < r₀ - ‖w‖
hr : 0 < r
⊢ |(herglotzRieszKernel 0 w (circleMap 0 r θ)).re| * |log ‖circleMap 0 r θ - ρ‖| ≤
    (R + ‖w‖) / (r₀ - ‖w‖) * (|log (2 * R)| + |log √(r₀ / R)| + |log ‖circleMap 0 R θ - ρ‖|)
Module: Mathlib.Analysis.Complex.OpenMapping

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
z : { z // z ≠ 0 }
h : (fun x ↦ x ^ n) 0 = ↑z
⊢ False

Tactic:
exact z.2 (by simpa [zero_pow (NeZero.ne n)] using h.symm)

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

Goal state before:
pp: case inr
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
⊢ ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0

Tactic:
refine ⟨f ∘ (-a + ·), hf_inj.comp (add_right_injective (-a)), ?_, fun z hz ↦ ?_⟩

Goal state after:
case inr.refine_1
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
⊢ ¬Dense ((f ∘ fun x ↦ -a + x) '' U)

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
Module: Mathlib.Order.Monotone.Union

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : StrictMonoOn f s
h₂ : StrictMonoOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
hx✝ : c ∈ s ∪ t
hxc : c ≤ c
hx : c ∈ s
⊢ c ∈ t

pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : StrictMonoOn f s
h₂ : StrictMonoOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
x : α
hx✝ : x ∈ s ∪ t
hxc : c ≤ x
hx : x ∈ s
h'x : c < x
⊢ x ∈ t

Tactic:
· exact ht.1

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : StrictMonoOn f s
h₂ : StrictMonoOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
x : α
hx✝ : x ∈ s ∪ t
hxc : c ≤ x
hx : x ∈ s
h'x : c < x
⊢ x ∈ t
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case pos
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : c ∈ Function.support D
⊢ ∑ᶠ (u : ℂ), ↑(D u) * (log R - log ‖c - u‖) = ∑ᶠ (u : ℂ), ↑(D u) * log (R * ‖c - u‖⁻¹) + ↑(D c) * log R

Tactic:
have {g : ℂ → ℝ} : (fun u ↦ D u * g u).support ⊆ hD.toFinset :=
      fun x ↦ by simp +contextual
    simp only [finsum_eq_sum_of_support_subset _ this,
      Finset.sum_eq_sum_sdiff_singleton_add ((Set.Finite.mem_toFinset hD).mpr h), sub_self,
      norm_zero, log_zero, sub_zero, inv_zero, mul_zero, add_zero, add_left_inj]
    refine Finset.sum_congr rfl fun x hx ↦ ?_
    simp only [Finset.mem_sdiff, Finset.notMem_singleton] at hx
    rw [log_mul hR (inv_ne_zero (norm_ne_zero_iff.mpr (sub_eq_zero.not.2 hx.2.symm))), log_inv]
    ring

Goal state after:
Module: Mathlib.Order.Monotone.Union

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : MonotoneOn f s
h₂ : MonotoneOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
hx✝ : c ∈ s ∪ t
hxc : c ≤ c
hx : c ∈ s
⊢ c ∈ t

pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : MonotoneOn f s
h₂ : MonotoneOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
x : α
hx✝ : x ∈ s ∪ t
hxc : c ≤ x
hx : x ∈ s
h'x : c < x
⊢ x ∈ t

Tactic:
· exact ht.1

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : MonotoneOn f s
h₂ : MonotoneOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
x : α
hx✝ : x ∈ s ∪ t
hxc : c ≤ x
hx : x ∈ s
h'x : c < x
⊢ x ∈ t
Module: Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints

Goal state before:
pp: case mp.inr
g : GL (Fin 2) ℝ
hgc : g ∉ Subgroup.center (GL (Fin 2) ℝ)
hgt : 0 < (↑g).det
this : g.IsElliptic
hg : True
⊢ ∃ z, g • z ≠ z

Tactic:
simp [gl_smul_eq_self_iff_eq_fixedPt hgt this, exists_ne]

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction

Goal state before:
pp: g : GL (Fin 2) ℝ
z : ℍ
⊢ (g • z).re = (num g ↑z / denom g ↑z).re

Tactic:
change (smulAux' g z).re = _
  simp +contextual [smulAux', σ, DFunLike.ite_apply, apply_ite, Complex.div_re]

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Topology

Goal state before:
pp: case neg
w : ℂ
hw : ¬0 < w.im
⊢ ↑ofComplex w = Classical.choice ⋯

Tactic:
· change (Function.invFunOn UpperHalfPlane.coe Set.univ w) = _
    simp only [invFunOn, dite_eq_right_iff, mem_univ, true_and]
    rintro ⟨a, rfl⟩
    exact (a.im_pos.not_ge (by simpa using hw)).elim

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

Goal state before:
pp: K : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
g : GL (Fin 2) K
h_det : det g = 1 ∨ det g = -1
hg10 : ↑g 1 0 = 0
⊢ g.IsParabolic ↔ (∃ x, x ≠ 0 ∧ g = upperRightHom x) ∨ ∃ x, x ≠ 0 ∧ g = -upperRightHom x

Tactic:
rw [isParabolic_iff_of_upperTriangular hg10]

Goal state after:
K : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
g : GL (Fin 2) K
h_det : det g = 1 ∨ det g = -1
hg10 : ↑g 1 0 = 0
⊢ ↑g 0 0 = ↑g 1 1 ∧ ↑g 0 1 ≠ 0 ↔ (∃ x, x ≠ 0 ∧ g = upperRightHom x) ∨ ∃ x, x ≠ 0 ∧ g = -upperRightHom x
Module: Mathlib.RingTheory.Norm.Transitivity

Goal state before:
pp: case inl.inr
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝¹ : Nontrivial R
a✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Infinite ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)

pp: case inl.inl
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝¹ : Nontrivial R
a✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Fintype ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)

Tactic:
· rw [det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis bA), map_one,
      det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis (bS.smulTower bA))]

Goal state after:
case inl.inl
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝¹ : Nontrivial R
a✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
val✝¹ : Fintype ιS
val✝ : Fintype ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
Module: Mathlib.Topology.Category.CompHausLike.Basic

Goal state before:
pp: P : TopCat → Prop
X Y : CompHausLike P
f : X ⟶ Y
bij : Function.Bijective ⇑(ConcreteCategory.hom f)
E : (fun X ↦ ↑X.toTop) X ≃ (fun X ↦ ↑X.toTop) Y := Equiv.ofBijective (⇑(ConcreteCategory.hom f)) bij
S : Set ((fun X ↦ ↑X.toTop) X)
hS : IsClosed S
⊢ IsClosed (⇑E '' S)

Tactic:
exact isClosedMap f S hS

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

Goal state before:
pp: R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y z : P
f : P →ᵃ[R] P'
s : AffineSubspace R P
hf : InjOn ⇑f ↑s
hx : x ∈ s
hy : y ∈ s
hz : z ∈ s
⊢ Wbtw R x y z ∧ f y ≠ f x ∧ f y ≠ f z ↔ Wbtw R x y z ∧ y ≠ x ∧ y ≠ z

Tactic:
hf.ne_iff hy hx,

Goal state after:
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y z : P
f : P →ᵃ[R] P'
s : AffineSubspace R P
hf : InjOn ⇑f ↑s
hx : x ∈ s
hy : y ∈ s
hz : z ∈ s
⊢ Wbtw R x y z ∧ y ≠ x ∧ f y ≠ f z ↔ Wbtw R x y z ∧ y ≠ x ∧ y ≠ z