mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2
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
cov : IsCoveringMap p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
le : (FundamentalGroup.map f a₀).range ≤ (mapOfEq { toFun := p, continuous_toFun := ⋯ } he).range
γ γ' : C(↑I, A)
Γ Γ' : C(↑I, E)
γ_0 : γ 0 = a₀
γ'_0 : γ' 0 = a₀
Γ_0 : Γ 0 = e₀
Γ'_0 : Γ' 0 = e₀
Γ_lifts : p ∘ ⇑Γ = ⇑(f.comp γ)
Γ'_lifts : p ∘ ⇑Γ' = ⇑(f.comp γ')
γγ'1 : γ 1 = γ' 1
pγ : Path a₀ (γ 1) := ⋯
pγ' : Path a₀ (γ 1) := ⋯
⊢ cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯).symm)
      (cov.monodromy (Path.Homotopic.Quotient.mk (pγ.map ⋯)) ⟨e₀, he⟩) =
    cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯).symm)
      (cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯)) ⟨e₀, he⟩)

Tactic:
[

Goal state after:
case refine_2
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
cov : IsCoveringMap p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
le : (FundamentalGroup.map f a₀).range ≤ (mapOfEq { toFun := p, continuous_toFun := ⋯ } he).range
γ γ' : C(↑I, A)
Γ Γ' : C(↑I, E)
γ_0 : γ 0 = a₀
γ'_0 : γ' 0 = a₀
Γ_0 : Γ 0 = e₀
Γ'_0 : Γ' 0 = e₀
Γ_lifts : p ∘ ⇑Γ = ⇑(f.comp γ)
Γ'_lifts : p ∘ ⇑Γ' = ⇑(f.comp γ')
γγ'1 : γ 1 = γ' 1
pγ : Path a₀ (γ 1) := ⋯
pγ' : Path a₀ (γ 1) := ⋯
⊢ cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯).symm)
      (cov.monodromy (Path.Homotopic.Quotient.mk (pγ.map ⋯)) ⟨e₀, he⟩) =
    cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯).symm)
      (cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯)) ⟨e₀, he⟩)
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
⊢ 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:
[

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

Goal state before:
pp: case inl
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)
hz0 : z.re = 0
⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1

pp: case inr
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}
⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1

Tactic:
· simp only [hz0, zero_sub, Real.rpow_neg_one, neg_zero, Real.rpow_zero, mul_one,
      inv_mul_le_iff₀ (sSupNormIm_eps_pos f hε 0)]
    rw [← hz0]
    apply le_of_lt (norm_lt_sSupNormIm_eps f ε hε _ _ hB)
    simp only [verticalClosedStrip, mem_preimage, zero_le_one, left_mem_Icc, hz0]

Goal state after:
case inr
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}
⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1
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
g : G
⊢ ↑(⋯.monodromy γ (((hp.toPermFiber x) g) e)) = ↑(⋯.monodromy γ' (((hp.toPermFiber x) g) e))

Tactic:
[

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 y : X
γ γ' : Path.Homotopic.Quotient x y
e : ↑(p ⁻¹' {x})
eq : ⋯.monodromy γ e = ⋯.monodromy γ' e
g : G
⊢ ↑(⋯.monodromy γ (((hp.toPermFiber x) g) e)) = ↑(⋯.monodromy γ' (((hp.toPermFiber x) g) e))
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
⊢ (hp.fiberEquivGroup e) (⋯.monodromy γ (⋯.monodromy γ' e)) • ↑e =
    ((hp.fiberEquivGroup e) (⋯.monodromy γ' e) * (hp.fiberEquivGroup e) (⋯.monodromy γ e)) • ↑e

Tactic:
[

Goal state after:
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
⊢ (hp.fiberEquivGroup e) (⋯.monodromy γ (⋯.monodromy γ' e)) • ↑e =
    ((hp.fiberEquivGroup e) (⋯.monodromy γ' e) * (hp.fiberEquivGroup e) (⋯.monodromy γ e)) • ↑e
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case pos
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h : sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0
⊢ DiffContOnCl ℂ (fun z ↦ interpStrip (fun a ↦ f a) z) fun a ↦ verticalStrip 0 1 a

Tactic:
[

Goal state after:
case pos
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
h : sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0
⊢ DiffContOnCl ℂ (fun z ↦ interpStrip (fun a ↦ f a) z) fun a ↦ verticalStrip 0 1 a
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:
[

Goal state after:
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
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 ↦ interpStrip (fun a ↦ f a) z) z

Tactic:
[

Goal state after:
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 ↦ interpStrip (fun a ↦ f a) z) z
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSupNormIm (scale f l u) 0 = sSupNormIm f l

Tactic:
[

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSupNormIm (scale f l u) 0 = sSupNormIm f l
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
this : scale f l u '' re ⁻¹' {0} = f '' re ⁻¹' {l}
⊢ sSup (norm '' scale f l u '' re ⁻¹' {0}) = sSup (norm '' f '' re ⁻¹' {l})

Tactic:
this

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
this : scale f l u '' re ⁻¹' {0} = f '' re ⁻¹' {l}
⊢ sSup (norm '' f '' re ⁻¹' {l}) = sSup (norm '' f '' re ⁻¹' {l})
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
⊢ ∃ R, |z.re| < R ∧ ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C

Tactic:
refine ((eventually_gt_atTop _).and ?_).exists

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
⊢ ∀ᶠ (R : ℝ) in atTop, ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSupNormIm (scale f l u) 1 = sSupNormIm f u

Tactic:
[

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSupNormIm (scale f l u) 1 = sSupNormIm f u
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
this : scale f l u '' re ⁻¹' {1} = f '' re ⁻¹' {u}
⊢ sSup (norm '' scale f l u '' re ⁻¹' {1}) = sSup (norm '' f '' re ⁻¹' {u})

Tactic:
this

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
this : scale f l u '' re ⁻¹' {1} = f '' re ⁻¹' {u}
⊢ sSup (norm '' f '' re ⁻¹' {u}) = sSup (norm '' f '' re ⁻¹' {u})
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
z : ℂ
⊢ (if sSupNormIm (scale f l u) 0 = 0 ∨ sSupNormIm (scale f l u) 1 = 0 then 0
    else
      ↑(sSupNormIm (scale f l u) 0) ^ (1 - (z - ↑l) / (↑u - ↑l)) *
        ↑(sSupNormIm (scale f l u) 1) ^ ((z - ↑l) / (↑u - ↑l))) =
    if sSupNormIm f l = 0 ∨ sSupNormIm f u = 0 then 0
    else ↑(sSupNormIm f l) ^ (1 - (z - ↑l) / (↑u - ↑l)) * ↑(sSupNormIm f u) ^ ((z - ↑l) / (↑u - ↑l))

Tactic:
[

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
z : ℂ
⊢ (if sSupNormIm (scale f l u) 0 = 0 ∨ sSupNormIm (scale f l u) 1 = 0 then 0
    else
      ↑(sSupNormIm (scale f l u) 0) ^ (1 - (z - ↑l) / (↑u - ↑l)) *
        ↑(sSupNormIm (scale f l u) 1) ^ ((z - ↑l) / (↑u - ↑l))) =
    if sSupNormIm f l = 0 ∨ sSupNormIm f u = 0 then 0
    else ↑(sSupNormIm f l) ^ (1 - (z - ↑l) / (↑u - ↑l)) * ↑(sSupNormIm f u) ^ ((z - ↑l) / (↑u - ↑l))
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case convert_8
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ContinuousWithinAt (fun ε ↦ ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (Ioi 0) 0

Tactic:
[

Goal state after:
case convert_8
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℂ → E
inst✝ : NormedSpace ℂ E
z : ℂ
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
hz : z ∈ verticalStrip 0 1
⊢ ContinuousWithinAt (fun ε ↦ ‖(↑ε + ↑(sSupNormIm f 0)) ^ (1 - z) * (↑ε + ↑(sSupNormIm f 1)) ^ z‖) (Ioi 0) 0
Module: Mathlib.Analysis.Distribution.DerivNotation

Goal state before:
pp: case succ
V : Type u_5
E : Type u_6
inst✝³ : LineDeriv V E E
inst✝² : AddCommGroup V
inst✝¹ : AddCommGroup E
inst✝ : LineDerivAdd V E E
x y : E
n : ℕ
IH : ∀ (m : Fin n → V), ∂^{m} (x + y) = ∂^{m} x + ∂^{m} y
m : Fin (n + 1) → V
⊢ ∂^{m} (x + y) = ∂^{m} x + ∂^{m} y

Tactic:
[

Goal state after:
case succ
V : Type u_5
E : Type u_6
inst✝³ : LineDeriv V E E
inst✝² : AddCommGroup V
inst✝¹ : AddCommGroup E
inst✝ : LineDerivAdd V E E
x y : E
n : ℕ
IH : ∀ (m : Fin n → V), ∂^{m} (x + y) = ∂^{m} x + ∂^{m} y
m : Fin (n + 1) → V
⊢ ∂^{m} (x + y) = ∂^{m} x + ∂^{m} y
Module: Mathlib.Analysis.Distribution.DerivNotation

Goal state before:
pp: case succ
R : Type u_4
V : Type u_5
E : Type u_6
inst✝² : LineDeriv V E E
inst✝¹ : SMul R E
inst✝ : LineDerivSMul R V E E
r : R
x : E
n : ℕ
IH : ∀ (m : Fin n → V), ∂^{m} (r • x) = r • ∂^{m} x
m : Fin (n + 1) → V
⊢ ∂^{m} (r • x) = r • ∂^{m} x

Tactic:
[

Goal state after:
case succ
R : Type u_4
V : Type u_5
E : Type u_6
inst✝² : LineDeriv V E E
inst✝¹ : SMul R E
inst✝ : LineDerivSMul R V E E
r : R
x : E
n : ℕ
IH : ∀ (m : Fin n → V), ∂^{m} (r • x) = r • ∂^{m} x
m : Fin (n + 1) → V
⊢ ∂^{m} (r • x) = r • ∂^{m} x
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
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
⊢ ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re

Tactic:
[

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
h0 : 0 ≠ sSupNormIm f 0
h1 : 0 ≠ sSupNormIm f 1
⊢ ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: case hbc.ha₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
⊢ ↑w.re + ↑c.im * I ∈ ball c r

Tactic:
· exact re_add_im_mul_mem_ball w_mem

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

Goal state before:
pp: z : ℂ
l u : ℝ
hul : l < u
hz : z ∈ re ⁻¹' Icc l u
⊢ l * (u - l) / normSq ↑(u - l) ≤ z.re * (u - l) / normSq ↑(u - l) ∧
    z.re * (u - l) / normSq ↑(u - l) ≤ 1 + l * (u - l) / normSq ↑(u - l)

Tactic:
[

Goal state after:
z : ℂ
l u : ℝ
hul : l < u
hz : z ∈ re ⁻¹' Icc l u
⊢ l * (u - l) / normSq ↑(u - l) ≤ z.re * (u - l) / normSq ↑(u - l) ∧
    z.re * (u - l) / normSq ↑(u - l) ≤ 1 + l * (u - l) / normSq ↑(u - l)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case right
z : ℂ
l u : ℝ
hul : l < u
hz : z ∈ re ⁻¹' Icc l u
⊢ z.re ≤ u

Tactic:
exact hz.2

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic

Goal state before:
pp: case left
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
h₁ : HarmonicAt f₁ x
h₂ : HarmonicAt f₂ x
⊢ ContDiffAt ℝ 2 (f₁ - f₂) x

pp: case right
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
h₁ : HarmonicAt f₁ x
h₂ : HarmonicAt f₂ x
⊢ Δ (f₁ - f₂) =ᶠ[𝓝 x] 0

Tactic:
· exact h₁.1.sub h₂.1

Goal state after:
case right
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
h₁ : HarmonicAt f₁ x
h₂ : HarmonicAt f₂ x
⊢ Δ (f₁ - f₂) =ᶠ[𝓝 x] 0
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
⊢ DifferentiableAt ℂ (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) x

Tactic:
this

Goal state after:
f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
this :
  (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =
    (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I
⊢ DifferentiableAt ℂ ((⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I) x
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 : ℝ
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c (-R) θ) = (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)

Tactic:
[

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c (-R) θ) = (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
hR : R < 0
⊢ circleAverage f c (-R) = circleAverage f c R

Tactic:
circleAverage_neg_radius

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
hR : R < 0
⊢ circleAverage f c R = circleAverage f c R
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ Complex.reCLM (F c) = g (f c)

pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ CircleIntegrable F c R

Tactic:
· apply h₂F
    simp [mem_ball, dist_self, add_pos_of_pos_of_nonneg h₁e (abs_nonneg R)]

Goal state after:
F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ CircleIntegrable F c R
Module: Mathlib.Analysis.Complex.IntegerCompl

Goal state before:
pp: x : ℂ
hx : x ∈ ℂ_ℤ
n : ℤ
⊢ x ^ 2 ≠ ↑n ^ 2

Tactic:
have := not_exists.mp hx n

Goal state after:
x : ℂ
hx : x ∈ ℂ_ℤ
n : ℤ
this : ¬↑n = x
⊢ x ^ 2 ≠ ↑n ^ 2
Module: Mathlib.Analysis.Complex.IsIntegral

Goal state before:
pp: ⊢ eval₂ (algebraMap ℤ ℂ) I (X ^ 2) + eval₂ (algebraMap ℤ ℂ) I (C 1) = 0

Tactic:
eval₂_X_pow,

Goal state after:
⊢ I ^ 2 + eval₂ (algebraMap ℤ ℂ) I (C 1) = 0
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
x✝¹ : ℂ
x✝ : x✝¹ ∈ sphere c |R|
⊢ (poissonKernel c w • f) x✝¹ = (re ∘ herglotzRieszKernel c w • f) x✝¹

Tactic:
[

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
x✝¹ : ℂ
x✝ : x✝¹ ∈ sphere c |R|
⊢ (poissonKernel c w • f) x✝¹ = (re ∘ herglotzRieszKernel c w • f) x✝¹
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
x✝¹ : ℂ
x✝ : x✝¹ ∈ sphere c |R|
⊢ (poissonKernel c w • f) x✝¹ = (re ∘ herglotzRieszKernel c w • f) x✝¹

Tactic:
[

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
x✝¹ : ℂ
x✝ : x✝¹ ∈ sphere c |R|
⊢ (poissonKernel c w • f) x✝¹ = (re ∘ herglotzRieszKernel c w • f) x✝¹
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: case inr
⊢ Filter.Tendsto (fun x ↦ x * log x) (𝓝 0) (𝓝 0)

Tactic:
[

Goal state after:
case inr
⊢ Filter.Tendsto (fun x ↦ x * log x) (𝓝 0) (𝓝 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
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
⊢ HarmonicAt (fun x ↦ Real.log ‖f x‖) z

Tactic:
this

Goal state after:
f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
⊢ HarmonicAt (2⁻¹ • Real.log ∘ ⇑normSq ∘ f) z
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: ⊢ ¬ContinuousAt (fun x ↦ (log x)⁻¹) 1

Tactic:
by
  suffices Tendsto (fun x ↦ (log x)⁻¹) (nhdsWithin 1 {1}ᶜ) (cobounded ℝ) from
    not_continuousAt_of_tendsto this nhdsWithin_le_nhds (disjoint_nhds_cobounded _)
  exact tendsto_inv₀_nhdsNE_zero.comp <| log_one ▸
    HasDerivAt.tendsto_nhdsNE (by simpa using hasDerivAt_log one_ne_zero) one_ne_zero

Goal state after: