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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C✝ : ℝ
f : ℂ → E
z : ℂ
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)
hB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))
hza : a < z.im
hzb : z.im < b
C : ℝ
hle_a : ∀ (z : ℂ), z.im = a → ‖f z‖ ≤ C
hle_b : ∀ (z : ℂ), z.im = b → ‖f z‖ ≤ C
hC₀ : 0 < C
⊢ a = (a + b) / 2 - (b - a) / 2

Tactic:
by

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
hB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B
z : ℂ
hset : z ∈ verticalClosedStrip 0 1
⊢ ‖f z‖ ∈ norm ∘ f '' verticalClosedStrip 0 1

Tactic:
by

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✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
inst✝¹ : SimplyConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
s : Set X
cov : IsCoveringMapOn p s
a₀ : A
f : C(A, ↑s)
hs : ∀ (a : A), ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp f) a ∈ s
F : C(A, ↑(p ⁻¹' s))
hF : s.restrictPreimage p ∘ ⇑F = ⇑f
he : p ↑(F a₀) = ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp f) a₀
hF_unique : ∀ (y : C(A, ↑(p ⁻¹' s))), (fun F_1 ↦ F_1 a₀ = F a₀ ∧ s.restrictPreimage p ∘ ⇑F_1 = ⇑f) y → y = F
F' : C(A, E)
hF'₁ : F' a₀ = ↑(F a₀)
hF'₂ : ∀ (x : A), p (F' x) = ↑(f x)
a : A
⊢ F' a ∈ p ⁻¹' s

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C✝ : ℝ
f : ℂ → E
z : ℂ
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)
hB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))
hza : a < z.im
hzb : z.im < b
C : ℝ
hle_a : ∀ (z : ℂ), z.im = a → ‖f z‖ ≤ C
hle_b : ∀ (z : ℂ), z.im = b → ‖f z‖ ≤ C
hC₀ : 0 < C
⊢ b = (a + b) / 2 + (b - a) / 2

Tactic:
by

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✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
inst✝¹ : SimplyConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
s : Set X
cov : IsCoveringMapOn p s
a₀ : A
f : C(A, ↑s)
hs : ∀ (a : A), ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp f) a ∈ s
F : C(A, ↑(p ⁻¹' s))
hF : s.restrictPreimage p ∘ ⇑F = ⇑f
he : p ↑(F a₀) = ({ toFun := Subtype.val, continuous_toFun := ⋯ }.comp f) a₀
hF_unique : ∀ (y : C(A, ↑(p ⁻¹' s))), (fun F_1 ↦ F_1 a₀ = F a₀ ∧ s.restrictPreimage p ∘ ⇑F_1 = ⇑f) y → y = F
F' : C(A, E)
hF'₁ : F' a₀ = ↑(F a₀)
hF'₂ : ∀ (x : A), p (F' x) = ↑(f x)
⊢ Continuous[inst✝², instTopologicalSpaceSubtype] fun a ↦ ⟨F' a, ⋯⟩

Tactic:
by

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
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))
hB :
  ∃ c < π / (a + b - (a - b)),
    ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
hab : a - b < a + b
⊢ 0 < b

Tactic:
by

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

Goal state before:
pp: case neg
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 : ¬1 ≤ ε + sSupNormIm f 0
⊢ ‖↑(ε + sSupNormIm f 0) ^ (z - 1)‖ ≤ max 1 ((ε + sSupNormIm f 0) ^ (-1))

Tactic:
rw [not_le] at hM0_one; apply le_trans _ (le_max_right _ _)
      simp only [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε 0), sub_re,
        one_re]
      apply Real.rpow_le_rpow_of_exponent_ge (sSupNormIm_eps_pos f hε 0) (le_of_lt hM0_one) _
      simp only [neg_le_sub_iff_le_add, le_add_iff_nonneg_left, hset.1]

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✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
inst✝¹ : SimplyConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
s : Set X
cov : IsCoveringMapOn p s
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
hs : ∀ (a : A), f a ∈ s
⊢ ∃! F, F a₀ = e₀ ∧ p ∘ ⇑F = ⇑f

Tactic:
by

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
η₀ : ∀ x ∈ t₁, ‖w - x‖ ^ (-divisor f S₀R) x ≠ 0
η₁ : ∀ x ∈ t₂, ‖canonicalFactor R x w‖ ^ (divisor f B₀R) x ≠ 0
⊢ meromorphicOrderAt f w ≠ ⊤

Tactic:
by

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
g : G
x y : X
γ : Path.Homotopic.Quotient x y
e : ↑(p ⁻¹' {x})
Γ : Path.Homotopic.Quotient ↑e ↑(⋯.monodromy γ e) := ⋯.liftPathQuotient γ e
g' : C(E, E) := { toFun := fun x ↦ g • x, continuous_toFun := ⋯ }
p' : C(E, X) := { toFun := p, continuous_toFun := ⋯ }
⊢ p'.comp g' = p'

Tactic:
by

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
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
⊢ BddAbove (norm ∘ F f ε '' verticalClosedStrip 0 1)

Tactic:
by

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
⊢ log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ i ∈ t₁, (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖ =
    ∑ i ∈ t₂, log (‖canonicalFactor R i w‖ ^ (divisor f B₀R) i) + ∑ i ∈ t₁, log (‖w - i‖ ^ (-divisor f S₀R) i) +
      log ‖meromorphicTrailingCoeffAt f w‖

Tactic:
by

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
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, (c < d ∧ 0 < d) ∧ d < π / 2 / b

Tactic:
by

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
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz1 : z.re = 1
⊢ z ∈ verticalClosedStrip 0 1

Tactic:
simp only [verticalClosedStrip, mem_preimage, zero_le_one, hz1, right_mem_Icc]

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
g : G
x : X
γ : FundamentalGroup X x
⊢ Commute ((⋯.monodromyPerm x) γ) ((hp.toPermFiber x) g)

Tactic:
by

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
z : ℂ
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ re ⁻¹' {0, 1}
⊢ ‖F f ε z‖ ≤ 1

Tactic:
by

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 y : X
γ γ' : Path.Homotopic.Quotient x y
e : ↑(p ⁻¹' {x})
eq : ⋯.monodromy γ e = ⋯.monodromy γ' e
⊢ ⋯.monodromy γ = ⋯.monodromy γ'

Tactic:
by

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
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)))
this : ∀ᶠ (ε : ℝ) in 𝓝[<] 0, ‖g ε z • f z‖ ≤ C
⊢ ‖f z‖ ≤ C

Tactic:
by

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})
⊢ (⋯.monodromyPerm x).ker = (FundamentalGroup.mapOfEq { toFun := p, continuous_toFun := ⋯ } ⋯).range

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
inst✝ : SimplyConnectedSpace E
⊢ Injective ⇑(⋯.monodromyPerm x)

Tactic:
by

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✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
g : G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
⊢ op ((hp.fiberEquivGroup e) (⋯.monodromy 1 e)) = 1

Tactic:
by

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✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
g : G
x : X
e : ↑(p ⁻¹' {x})
γ✝ γ γ' : FundamentalGroup X x
⊢ op ((hp.fiberEquivGroup e) (⋯.monodromy (γ * γ') e)) =
    op ((hp.fiberEquivGroup e) (⋯.monodromy γ e)) * op ((hp.fiberEquivGroup e) (⋯.monodromy γ' e))

Tactic:
by

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
⊢ (hp.fundamentalGroupToMulOpposite e) γ = g ↔ MulOpposite.unop g • ↑e = ↑(⋯.monodromy γ e)

Tactic:
by

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
⊢ MulOpposite.unop ((hp.fundamentalGroupToMulOpposite e) γ) • ↑e = ↑(⋯.monodromy γ e)

Tactic:
by

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
h : (hp.fundamentalGroupToMulOpposite e) γ = 1
⊢ ↑(⋯.monodromy γ e) = ↑e

Tactic:
by

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
h : ⋯.monodromy γ e = e
⊢ MulOpposite.unop ((hp.fundamentalGroupToMulOpposite e) γ) • ↑e = MulOpposite.unop 1 • ↑e

Tactic:
by

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})
⊢ (hp.fundamentalGroupToMulOpposite e).ker = (⋯.monodromyPerm x).ker

Tactic:
by

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
⊢ ∑ i ∈ t₂, log (‖canonicalFactor R i w‖ ^ (divisor f B₀R) i) + ∑ i ∈ t₁, log (‖w - i‖ ^ (-divisor f S₀R) i) +
      log ‖meromorphicTrailingCoeffAt f w‖ =
    ∑ᶠ (i : ℂ), ↑((divisor f B₀R) i) * log ‖canonicalFactor R i w‖ - ∑ᶠ (i : ℂ), ↑((divisor f S₀R) i) * log ‖w - i‖ +
      log ‖meromorphicTrailingCoeffAt f w‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
g : Gᵐᵒᵖ
⊢ MulOpposite.unop g • ↑e ∈ p ⁻¹' {x}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
g : Gᵐᵒᵖ
e' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩
he' : e' = ⟨MulOpposite.unop g • ↑e, ⋯⟩
⊢ Continuous[_, inst✝⁴] ⇑(PathConnectedSpace.somePath ↑e ↑e')

Tactic:
by

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
⊢ log ‖h w‖ =
    ∑ᶠ (i : ℂ), ↑((divisor f (ball 0 R)) i) * log ‖canonicalFactor R i w‖ -
        ∑ᶠ (i : ℂ), ↑((divisor f (sphere 0 R)) i) * log ‖w - i‖ +
      log ‖meromorphicTrailingCoeffAt f w‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
g : Gᵐᵒᵖ
e' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩
he' : e' = ⟨MulOpposite.unop g • ↑e, ⋯⟩
⊢ (PathConnectedSpace.somePath ↑e ↑e') 0 = ↑e

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
g : Gᵐᵒᵖ
e' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩
he' : e' = ⟨MulOpposite.unop g • ↑e, ⋯⟩
⊢ (PathConnectedSpace.somePath ↑e ↑e') 1 = ↑e'

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
g : Gᵐᵒᵖ
e' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩
he' : e' = ⟨MulOpposite.unop g • ↑e, ⋯⟩
Γ : Path ↑e ↑e' := { toFun := ⇑(PathConnectedSpace.somePath ↑e ↑e'), continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⊢ x = p ↑e

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ IsOpen {z | z.re < x}

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ IsOpen {z | x < z.re}

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ IsOpen {z | z.im < x}

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ IsOpen {z | x < z.im}

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r : ℝ
hz : z ∈ ball c r
⊢ 0 ≤ √(((↑z.re + ↑c.im * I).re - c.re) ^ 2 + ((↑z.re + ↑c.im * I).im - c.im) ^ 2)

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r : ℝ
hz : z ∈ ball c r
⊢ 0 ≤ (z.re - c.re) ^ 2 + (z.im - c.im) ^ 2

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r : ℝ
hz : z ∈ ball c r
⊢ 0 ≤ ((↑z.re + ↑c.im * I).re - c.re) ^ 2 + ((↑z.re + ↑c.im * I).im - c.im) ^ 2

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r : ℝ
hz : z ∈ ball c r
⊢ ↑z.re + ↑c.im * I ∈ ball c r

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
g : Gᵐᵒᵖ
e' : ↑(p ⁻¹' {x}) := ⟨MulOpposite.unop g • ↑e, ⋯⟩
he' : e' = ⟨MulOpposite.unop g • ↑e, ⋯⟩
Γ : Path ↑e ↑e' := { toFun := ⇑(PathConnectedSpace.somePath ↑e ↑e'), continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⊢ x = p ↑e'

Tactic:
by

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 : 0 < ε
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalClosedStrip 0 1
⊢ ‖F f ε z‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
h0 : 0 < sSupNormIm f 0
h1 : 0 < sSupNormIm f 1
⊢ interpStrip f z = ↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r x : ℝ
r₁ : ℝ := r - dist z c
s : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)
hx : x ∈ s
⊢ s ×ℂ {z.im} ⊆ ball z r₁

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r x : ℝ
r₁ : ℝ := r - dist z c
s : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)
hx : x ∈ s
s_ball₁ : s ×ℂ {z.im} ⊆ ball z r₁
this : s ×ℂ {z.im} ⊆ ball c r
⊢ ↑x + ↑z.im * I ∈ s ×ℂ {z.im}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : PathConnectedSpace E
⊢ Surjective ⇑(hp.fundamentalGroupToMulOpposite e)

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r x : ℝ
r₁ : ℝ := r - dist z c
s : Set ℝ := Ioo (z.re - r₁) (z.re + r₁)
hx : x ∈ s
s_ball₁ : s ×ℂ {z.im} ⊆ ball z r₁
⊢ r₁ + dist z c ≤ r

Tactic:
by

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:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
inst✝ : SimplyConnectedSpace E
⊢ Injective ⇑(hp.fundamentalGroupToMulOpposite e)

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r y : ℝ
z_in_ball : z ∈ closedBall c r
y_in_I : y ∈ Ι c.im z.im
⊢ 0 ≤ √(((↑z.re + ↑y * I).re - c.re) ^ 2 + ((↑z.re + ↑y * I).im - c.im) ^ 2)

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r y : ℝ
z_in_ball : z ∈ closedBall c r
y_in_I : y ∈ Ι c.im z.im
⊢ 0 ≤ (z.re - c.re) ^ 2 + (z.im - c.im) ^ 2

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
hz : z ∈ verticalStrip 0 1
⊢ interpStrip f z = ↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r y : ℝ
z_in_ball : z ∈ closedBall c r
y_in_I : y ∈ Ι c.im z.im
⊢ 0 ≤ ((↑z.re + ↑y * I).re - c.re) ^ 2 + ((↑z.re + ↑y * I).im - c.im) ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.CanonicalTensor

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
ι : Type u_2
inst✝ : Fintype ι
v : OrthonormalBasis ι ℝ E
w : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
⊢ ∑ m, w m ⊗ₜ[ℝ] w m = ∑ m, ∑ n, ⟪w m, w n⟫_ℝ • w m ⊗ₜ[ℝ] w n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.CanonicalTensor

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
ι : Type u_2
inst✝ : Fintype ι
v : OrthonormalBasis ι ℝ E
w : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
⊢ ∑ m, ∑ n, ⟪w m, w n⟫_ℝ • w m ⊗ₜ[ℝ] w n = ∑ m, ∑ n, (∑ i, ⟪w m, v i⟫_ℝ * ⟪v i, w n⟫_ℝ) • w m ⊗ₜ[ℝ] w n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.CanonicalTensor

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
ι : Type u_2
inst✝ : Fintype ι
v : OrthonormalBasis ι ℝ E
w : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
⊢ ∑ m, ∑ n, (∑ i, ⟪w m, v i⟫_ℝ * ⟪v i, w n⟫_ℝ) • w m ⊗ₜ[ℝ] w n =
    ∑ m, ∑ n, (∑ i, ⟪w m, v i⟫_ℝ * ⟪w n, v i⟫_ℝ) • w m ⊗ₜ[ℝ] w n

Tactic:
by

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

Goal state before:
pp: c z : ℂ
r y : ℝ
z_in_ball : z ∈ closedBall c r
y_in_I : y ∈ Ι c.im z.im
this : (y - c.im) ^ 2 ≤ (z.im - c.im) ^ 2
⊢ ((↑z.re + ↑y * I).re - c.re) ^ 2 + ((↑z.re + ↑y * I).im - c.im) ^ 2 ≤ (z.re - c.re) ^ 2 + (z.im - c.im) ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.CanonicalTensor

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
ι : Type u_2
inst✝ : Fintype ι
v : OrthonormalBasis ι ℝ E
w : OrthonormalBasis (Fin (Module.finrank ℝ E)) ℝ E := stdOrthonormalBasis ℝ E
⊢ ∑ m, ∑ n, (∑ i, ⟪w m, v i⟫_ℝ * ⟪w n, v i⟫_ℝ) • w m ⊗ₜ[ℝ] w n =
    ∑ i, (∑ m, ⟪w m, v i⟫_ℝ • w m) ⊗ₜ[ℝ] ∑ n, ⟪w n, v i⟫_ℝ • w n

Tactic:
by

Goal state after: