mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: case mp
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
f : E →ₗ.[R] F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
hf : f.toFun.ker = ⊥
hf' : f.IsClosable
⊢ f.inverse.IsClosable → f.closure.toFun.ker = ⊥

Tactic:
intro ⟨f', h⟩
    rw [LinearMap.ker_eq_bot']
    intro ⟨x, hx⟩ hx'
    simp only [Submodule.mk_eq_zero]
    rw [toFun_eq_coe, eq_comm, image_iff] at hx'
    have : (0, x) ∈ graph f' := by
      rw [← h, inverse_graph hf]
      rw [← hf'.graph_closure_eq_closure_graph, ← SetLike.mem_coe,
        Submodule.topologicalClosure_coe] at hx'
      apply image_closure_subset_closure_image continuous_swap
      simp only [Set.mem_image, Prod.exists, Prod.swap_prod_mk, Prod.mk.injEq]
      exact ⟨x, 0, hx', rfl, rfl⟩
    exact graph_fst_eq_zero_snd f' this rfl

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
this : ↑p * (↑n - 1) - (↑n - ↑p) = ↑n * (↑p - 1)
⊢ ((↑p * (↑n - 1) / (↑n - ↑p) - 1) * (↑p / (↑p - 1)))⁻¹ = ↑p⁻¹ - (↑n)⁻¹

Tactic:
simp [field, this]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →ₗᵢ[𝕜] G
g : F →ₗᵢ[𝕜] H
x y : E ⊗[𝕜] F
⊢ ∀ (x y : E ⊗[𝕜] F),
    (∀ (x_1 : E) (y : F),
        inner 𝕜 ((map f.toLinearMap g.toLinearMap) (x_1 ⊗ₜ[𝕜] y)) ((map f.toLinearMap g.toLinearMap) x) =
          inner 𝕜 (x_1 ⊗ₜ[𝕜] y) x) →
      (∀ (x : E) (y_1 : F),
          inner 𝕜 ((map f.toLinearMap g.toLinearMap) (x ⊗ₜ[𝕜] y_1)) ((map f.toLinearMap g.toLinearMap) y) =
            inner 𝕜 (x ⊗ₜ[𝕜] y_1) y) →
        ∀ (x_1 : E) (y_1 : F),
          inner 𝕜 ((map f.toLinearMap g.toLinearMap) (x_1 ⊗ₜ[𝕜] y_1)) ((map f.toLinearMap g.toLinearMap) (x + y)) =
            inner 𝕜 (x_1 ⊗ₜ[𝕜] y_1) (x + y)

Tactic:
by simp_all [inner_def]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →ₗᵢ[𝕜] G
g : F →ₗᵢ[𝕜] H
x y : E ⊗[𝕜] F
⊢ ∀ (x y_1 : E ⊗[𝕜] F),
    inner 𝕜 ((map f.toLinearMap g.toLinearMap) x) ((map f.toLinearMap g.toLinearMap) y) = inner 𝕜 x y →
      inner 𝕜 ((map f.toLinearMap g.toLinearMap) y_1) ((map f.toLinearMap g.toLinearMap) y) = inner 𝕜 y_1 y →
        inner 𝕜 ((map f.toLinearMap g.toLinearMap) (x + y_1)) ((map f.toLinearMap g.toLinearMap) y) =
          inner 𝕜 (x + y_1) y

Tactic:
by simp_all [inner_def]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹⁰ : EMetricSpace X
inst✝⁹ : MeasurableSpace X
inst✝⁸ : BorelSpace X
inst✝⁷ : EMetricSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ ↑(volume.addHaarScalarFactor μH[↑(finrank ℝ E)]) ≠ 0

Tactic:
simpa using addHaarScalarFactor_volume_hausdorffMeasure_ne_zero (Module.finrank ℝ E)

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
d d' : ℝ≥0
hsd' : μH[↑d'] s = ∞
hdd' : ↑d < ↑d'
⊢ μH[↑d] s = ∞

Tactic:
ENNReal.coe_lt_coe,

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
d d' : ℝ≥0
hsd' : μH[↑d'] s = ∞
hdd' : d < d'
⊢ μH[↑d] s = ∞
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
d : ℝ≥0
h : ⨆ d, ⨆ (_ : μH[↑d] s = ∞), ↑d < ↑d
d' : ℝ≥0
hsd' : ⨆ d, ⨆ (_ : μH[↑d] s = ∞), ↑d < ↑d'
hd'd : ↑d' < ↑d
⊢ μH[↑d] s = 0

Tactic:
ENNReal.coe_lt_coe,

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
d : ℝ≥0
h : ⨆ d, ⨆ (_ : μH[↑d] s = ∞), ↑d < ↑d
d' : ℝ≥0
hsd' : ⨆ d, ⨆ (_ : μH[↑d] s = ∞), ↑d < ↑d'
hd'd : d' < d
⊢ μH[↑d] s = 0
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: case e_f
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
u : Set ↥(mk' (↑(x +ᵥ p)) s.directionᗮ) := Subtype.val ⁻¹' (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
hu : MeasurableSet u
hinter : t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ) = Subtype.val '' u
⊢ volume (Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑((IsometryEquiv.vaddConst p) x)) s.directionᗮ))

Tactic:
have hxp : (x +ᵥ p).val ∈ mk' (x +ᵥ p).val s.directionᗮ := by simp

Goal state after:
case e_f
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
u : Set ↥(mk' (↑(x +ᵥ p)) s.directionᗮ) := Subtype.val ⁻¹' (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
hu : MeasurableSet u
hinter : t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ) = Subtype.val '' u
hxp : ↑(x +ᵥ p) ∈ mk' (↑(x +ᵥ p)) s.directionᗮ
⊢ volume (Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑((IsometryEquiv.vaddConst p) x)) s.directionᗮ))
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto (fun r ↦ (pre m r) s) (𝓝[>] 0) (𝓝 ((mkMetric' m) s))

Tactic:
← tendsto_comp_coe_Ioi_atBot

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s : Set X
⊢ Tendsto (fun x ↦ (pre m ↑x) s) atBot (𝓝 ((mkMetric' m) s))
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
r : ℝ≥0∞
hr0 : r ∈ Ioi 0
hr : Ico 0 r ⊆ {x | (fun x ↦ m₁ x ≤ (c • m₂) x) x}
s : Set X
r' : ℝ≥0∞
hr' : r' ∈ Ioo 0 r
⊢ (boundedBy (extend fun s x ↦ m₁ (ediam s))) s ≤ c * (boundedBy (extend fun s x ↦ m₂ (ediam s))) s

Tactic:
rw [← smul_eq_mul, ← smul_apply, smul_boundedBy hc]

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
r : ℝ≥0∞
hr0 : r ∈ Ioi 0
hr : Ico 0 r ⊆ {x | (fun x ↦ m₁ x ≤ (c • m₂) x) x}
s : Set X
r' : ℝ≥0∞
hr' : r' ∈ Ioo 0 r
⊢ (boundedBy (extend fun s x ↦ m₁ (ediam s))) s ≤ (boundedBy (c • extend fun s x ↦ m₂ (ediam s))) s
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case neg
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
x✝ : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
htr : ¬∀ (n : ℕ), ediam (t n) ≤ r
⊢ ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)) =
    ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n))

Tactic:
rw [iInf_eq_if, if_neg htr]
    push Not at htr; rcases htr with ⟨n, hn⟩
    refine ENNReal.tsum_eq_top_of_eq_top ⟨n, ?_⟩
    rw [iSup_eq_if, if_pos, iInf_eq_if, if_neg]
    · exact hn.not_ge
    rcases ediam_pos_iff.1 hn.pos with ⟨x, hx, -⟩
    exact ⟨x, hx⟩

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
hI : ¬I = 0
hI' : I * I = -1
I_smul : ∀ (v : E), ‖I • v‖ = ‖v‖
h₁ : ‖I • y - x‖ = ‖I • x + y‖
⊢ 4⁻¹ *
      (↑‖x + y‖ * ↑‖x + y‖ - ↑‖x - y‖ * ↑‖x - y‖ + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ -
        -I * ↑‖I • y - x‖ * ↑‖I • y - x‖) =
    4⁻¹ *
      (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Tactic:
have h₂ : ‖(I : 𝕜) • y + x‖ = ‖(I : 𝕜) • x - y‖ := by
    convert (I_smul ((I : 𝕜) • y + x)).symm
    linear_combination (norm := module) -hI' • y

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
hI : ¬I = 0
hI' : I * I = -1
I_smul : ∀ (v : E), ‖I • v‖ = ‖v‖
h₁ : ‖I • y - x‖ = ‖I • x + y‖
h₂ : ‖I • y + x‖ = ‖I • x - y‖
⊢ 4⁻¹ *
      (↑‖x + y‖ * ↑‖x + y‖ - ↑‖x - y‖ * ↑‖x - y‖ + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ -
        -I * ↑‖I • y - x‖ * ↑‖I • y - x‖) =
    4⁻¹ *
      (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
f : OfKernel K
h : (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f = 0
⊢ f = 0

Tactic:
refine UniformSpace.Completion.denseRange_coe.eq_zero_of_inner_right 𝕜 fun ff ↦ ?_

Goal state after:
𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
f : OfKernel K
h : (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f = 0
ff : H₀ K
⊢ ⟪↑ff, f⟫_𝕜 = 0
Module: Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E →WOT[𝕜] F
⊢ (∀ (x : α), ContinuousAt f x) ↔ ∀ (b : α) (b_1 : E) (a : F), ContinuousAt (fun x ↦ ⟪a, (f x) b_1⟫_𝕜) b

Tactic:
continuousAt_iff

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
⊢ ContinuousAt (⇑f) 0

Tactic:
obtain ⟨c, hc0, hc1⟩ := NormedField.exists_norm_lt_one 𝕜
  have c_ne := norm_pos_iff.mp hc0
  -- We use a fast decreasing balanced basis for 0 : E, and reformulate continuity in terms of
  -- this basis
  rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩
  simp only [_root_.id] at bE
  have bE' : (𝓝 (0 : E)).HasBasis (fun _ ↦ True) (fun n : ℕ ↦ (c ^ n) • b n) := by
    refine bE.1.to_hasBasis' ?_ ?_
    · intro n _
      use n
      exact ⟨trivial, (bE1 n).2 _ (by grw [norm_pow, hc1, one_pow])⟩
    · intro n _
      simpa using smul_mem_nhds_smul₀ (pow_ne_zero n c_ne) (bE1 n).1
  simp_rw [ContinuousAt, map_zero, bE'.tendsto_left_iff, true_and, Set.MapsTo]
  -- Assume that f is not continuous at 0
  by_contra! h
  rcases h with ⟨V, hV, h⟩
  -- There exists `u : ℕ → E` such that for all `n : ℕ` we have `u n ∈ c ^ n • b n` and
  -- `f (u n) ∉ V`, with `V` some neighborhood of `0` in `F`.
  choose! u hu hu' using h
  -- The sequence `fun n ↦ c ^ (-n) • u n` converges to `0`
  have h_tendsto : Tendsto (fun n : ℕ => (c ^ n)⁻¹ • u n) atTop (𝓝 (0 : E)) := by
    apply bE.tendsto
    intro n
    simpa only [Set.mem_smul_set_iff_inv_smul_mem₀ (pow_ne_zero n c_ne)] using hu n
  -- The range of `fun n ↦ c ^ (-n) • u n` is von Neumann bounded
  have h_bounded : IsVonNBounded 𝕜 (Set.range fun n : ℕ => (c ^ n)⁻¹ • u n) :=
    h_tendsto.isVonNBounded_range 𝕜
  -- Hence, by assumption, the range of `fun n ↦ (σ c) ^ (-n) • f (u n)` is von Neumann bounded
  specialize hf _ h_bounded
  simp only [← Set.range_comp', LinearMap.map_smulₛₗ, map_inv₀, map_pow] at hf
  -- Since `fun n ↦ (σ c) ^ n` tends to zero, this implies that `f ∘ u` converges to zero.
  have : Tendsto (f ∘ u) atTop (𝓝 0) := by
    have : Tendsto (fun n ↦ σ c ^ n) atTop (𝓝 0) :=
      tendsto_pow_atTop_nhds_zero_of_norm_lt_one (by simpa using hc1)
    have := hf.smul_tendsto_zero (.of_forall fun n ↦ Set.mem_range_self n) this
    exact this.congr fun n ↦ by simp [c_ne]
  -- But this is a contradiction.
  refine frequently_false (atTop : Filter ℕ) <| Eventually.frequently ?_
  filter_upwards [this.eventually_mem hV] using hu'

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E →ₗ[𝕜] F
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he' : Continuous fun x ↦ (toWeakSpace 𝕜 F) (e ((toWeakSpace 𝕜 E).symm x))
⊢ ⇑e '' closure s ⊆ closure (⇑e '' s)

Tactic:
by
    have h_convex : Convex ℝ (e '' s) := hs.linear_image (F := F) e
    rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜]
    simpa only [Set.image_image, ← hs.toWeakSpace_closure 𝕜, LinearEquiv.symm_apply_apply]
      using he'.continuousOn.image_closure (s := toWeakSpace 𝕜 E '' s)

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ ↑(-σ) • x ^ σ • f x = f x

Tactic:
← Complex.ofReal_cpow hx.le

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑(x ^ (-σ)) • x ^ σ • f x = f x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case zero
s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
s : ℂ
hsre : -↑0 < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ Gamma s

Tactic:
replace hsre : 0 < s.re := by simpa using hsre
    have : IsOpen {s : ℂ | 0 < s.re} := continuous_re.isOpen_preimage _ isOpen_Ioi
    apply (hasDerivAt_GammaIntegral (by simpa using hsre)).differentiableAt.congr_of_eventuallyEq
    filter_upwards [this.mem_nhds hsre] with a using Gamma_eq_integral

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
c u : ℝ
hc : 0 < c
hu : 0 < u
A : ENNReal.ofReal (1 / c) ≠ 0
B : ENNReal.ofReal (1 / c) ≠ ∞
⊢ MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))

Tactic:
rw [← memLp_norm_rpow_iff _ A B, ENNReal.toReal_ofReal (one_div_nonneg.mpr hc.le),
      ENNReal.div_self A B, memLp_one_iff_integrable]

Goal state after:
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
c u : ℝ
hc : 0 < c
hu : 0 < u
A : ENNReal.ofReal (1 / c) ≠ 0
B : ENNReal.ofReal (1 / c) ≠ ∞
⊢ Integrable (fun x ↦ ‖f c u x‖ ^ (1 / c)) (volume.restrict (Ioi 0))

s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
c u : ℝ
hc : 0 < c
hu : 0 < u
A : ENNReal.ofReal (1 / c) ≠ 0
B : ENNReal.ofReal (1 / c) ≠ ∞
⊢ AEStronglyMeasurable (f c u) (volume.restrict (Ioi 0))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)

Tactic:
convert!
    integral_mul_deriv_eq_deriv_mul der1 (fun x _ => antideriv_sin_comp_const_mul hz x) _ _ using 1

Goal state after:
case e'_2
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ∫ (x : ℝ) in 0..π / 2, ↑(sin x) * ↑(cos x) ^ (n - 1) * Complex.sin (2 * z * ↑x)

case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    ↑(sin (π / 2)) * ↑(cos (π / 2)) ^ (n - 1) * (-Complex.cos (2 * z * ↑(π / 2)) / (2 * z)) -
        ↑(sin 0) * ↑(cos 0) ^ (n - 1) * (-Complex.cos (2 * z * ↑0) / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2,
        (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z))

case convert_1
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ IntervalIntegrable (fun x ↦ ↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) volume 0 (π / 2)

case convert_2
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ IntervalIntegrable (fun x ↦ Complex.sin (2 * z * ↑x)) volume 0 (π / 2)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
convert! hm.sub (tendsto_log_nat_add_one_sub_log.const_mul x) using 2

Goal state after:
case e'_3
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
x✝ : ℕ
⊢ logGammaSeq x x✝ = logGammaSeq x x✝ + x * (log (↑x✝ + 1) - log ↑x✝) - x * (log (↑x✝ + 1) - log ↑x✝)

case e'_5
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 ↦
        (fun n ↦ logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b ↦ log (x - 1)) x_1) ∘
      fun a ↦ a + 1) =
    fun n ↦ logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ 0 ∈ closure (interior (Icc 0 (π / 2)))

Tactic:
rw [interior_Icc, closure_Ioo pi_div_two_pos.ne, left_mem_Icc]

Goal state after:
f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ 0 ≤ π / 2