mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
inst✝⁴ : NeZero n
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C : Subalgebra A B
ζ : B
hζ : IsPrimitiveRoot ζ n
⊢ adjoin A {b | n ≠ 0 ∧ b ^ n = 1} = A[ζ]

Tactic:
apply le_antisymm

Goal state after:
case a
n : ℕ
inst✝⁴ : NeZero n
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C : Subalgebra A B
ζ : B
hζ : IsPrimitiveRoot ζ n
⊢ adjoin A {b | n ≠ 0 ∧ b ^ n = 1} ≤ A[ζ]

case a
n : ℕ
inst✝⁴ : NeZero n
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C : Subalgebra A B
ζ : B
hζ : IsPrimitiveRoot ζ n
⊢ A[ζ] ≤ adjoin A {b | n ≠ 0 ∧ b ^ n = 1}
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : Fintype R
χ : MulChar R ℂ
ψ : AddChar R ℂ
⊢ ∑ x, ψ⁻¹ x * χ⁻¹ x = gaussSum χ⁻¹ ψ⁻¹

Tactic:
simp [mul_comm, gaussSum]

Goal state after:
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝³ : Field R
inst✝² : Fintype R
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
hψ : ψ ≠ 1
⊢ ∑ a ∈ {0}ᶜ, 1 a * ψ a + 1 = ∑ a ∈ {0}ᶜ, ψ a + ψ 0

Tactic:
congr!

Goal state after:
case e'_5.a
R : Type u_1
R' : Type u_2
inst✝³ : Field R
inst✝² : Fintype R
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
hψ : ψ ≠ 1
x✝ : R
a✝ : x✝ ∈ {0}ᶜ
⊢ 1 x✝ * ψ x✝ = ψ x✝

case e'_6
R : Type u_1
R' : Type u_2
inst✝³ : Field R
inst✝² : Fintype R
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
hψ : ψ ≠ 1
⊢ 1 = ψ 0
Module: Mathlib.Data.Finset.Interval

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Preorder β
f : Finset α → β
⊢ StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f s < f (cons a s ha)

Tactic:
simp [strictMono_iff_forall_covBy, covBy_iff_exists_cons]

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

Goal state before:
pp: ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ∫⋯∫⁻_insert i s, f ^ (1 - (↑(#s + 1) - 1) * p) * ∏ i ∈ insert i s, (∫⋯∫⁻_{i}, f ∂μ) ^ p ∂μ =
    ∫⋯∫⁻_insert i s, f ^ (1 - ↑(#s) * p) * ∏ i ∈ insert i s, (∫⋯∫⁻_{i}, f ∂μ) ^ p ∂μ

Tactic:
congr!

Goal state after:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ↑(#s + 1) - 1 = ↑(#s)
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ Measurable fun x ↦ f x ^ (1 - ↑(#s) * p) * ∏ x_1 ∈ insert i s, (∫⋯∫⁻_{x_1}, f ∂μ) x ^ p

Tactic:
refine (hf.pow_const _).mul <| Finset.measurable_fun_prod _ ?_

Goal state after:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ∀ i_1 ∈ insert i s, Measurable fun x ↦ (∫⋯∫⁻_{i_1}, f ∂μ) x ^ p
Module: Mathlib.Analysis.InnerProductSpace.ExteriorPower

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
ι : Type u_2
inst✝ : LinearOrder ι
v : ι → E
hv : Orthonormal ℝ v
s t : ↑(Set.powersetCard ι n)
⊢ (innerProductForm (ιMulti_family ℝ n v s)) (ιMulti_family ℝ n v t) = if s = t then 1 else 0

Tactic:
simp only [ιMulti_family]

Goal state after:
n : ℕ
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
ι : Type u_2
inst✝ : LinearOrder ι
v : ι → E
hv : Orthonormal ℝ v
s t : ↑(Set.powersetCard ι n)
⊢ (innerProductForm ((ιMulti ℝ n) (v ∘ ⇑(Set.powersetCard.ofFinEmbEquiv.symm s))))
      ((ιMulti ℝ n) (v ∘ ⇑(Set.powersetCard.ofFinEmbEquiv.symm t))) =
    if s = t then 1 else 0
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:

Tactic:
‖deriv (u ∘ update x i) y‖ₑ = ‖fderiv ℝ u (update x i y) (deriv (update x i) y)‖ₑ := by
        rw [fderiv_comp_deriv _ (hu.differentiable one_ne_zero).differentiableAt
          (hasDerivAt_update x i y).differentiableAt]
    _ ≤ ‖fderiv ℝ u (update x i y)‖ₑ * ‖deriv (update x i) y‖ₑ := ContinuousLinearMap.le_opENorm _ _
    _ ≤ ‖fderiv ℝ u (update x i y)‖ₑ := by simp [deriv_update, Pi.enorm_single]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
hT : ∀ (i : n), IsSymmetric (T i)
f g : n → 𝕜
hfg : f ≠ g
Ef : ↥(⨅ j, (T j).eigenspace (f j))
Eg : ↥(⨅ j, (T j).eigenspace (g j))
a : n
ha : f a ≠ g a
⊢ inner 𝕜 (((fun γ ↦ (⨅ j, (T j).eigenspace (γ j)).subtypeₗᵢ) f) Ef)
      (((fun γ ↦ (⨅ j, (T j).eigenspace (γ j)).subtypeₗᵢ) g) Eg) =
    0

Tactic:
have H := orthogonalFamily_eigenspaces (hT a) ha

Goal state after:
𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
hT : ∀ (i : n), IsSymmetric (T i)
f g : n → 𝕜
hfg : f ≠ g
Ef : ↥(⨅ j, (T j).eigenspace (f j))
Eg : ↥(⨅ j, (T j).eigenspace (g j))
a : n
ha : f a ≠ g a
H :
  (fun i j ↦
      ∀ (v : (fun μ ↦ ↥((T a).eigenspace μ)) i) (w : (fun μ ↦ ↥((T a).eigenspace μ)) j),
        inner 𝕜 (((fun μ ↦ ((T a).eigenspace μ).subtypeₗᵢ) i) v) (((fun μ ↦ ((T a).eigenspace μ).subtypeₗᵢ) j) w) = 0)
    (f a) (g a)
⊢ inner 𝕜 (((fun γ ↦ (⨅ j, (T j).eigenspace (γ j)).subtypeₗᵢ) f) Ef)
      (((fun γ ↦ (⨅ j, (T j).eigenspace (γ j)).subtypeₗᵢ) g) Eg) =
    0
Module: Mathlib.Analysis.InnerProductSpace.LaxMilgram

Goal state before:
pp: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ C * ‖w‖ * ‖w‖ = 0

Tactic:
apply le_antisymm

Goal state after:
case a
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ C * ‖w‖ * ‖w‖ ≤ 0

case a
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ 0 ≤ C * ‖w‖ * ‖w‖
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: 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
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f : E →ₗ.[R] F
hf : f.IsClosable
x✝¹ x✝ : E →ₗ.[R] F
hy₁ : f.graph.topologicalClosure = x✝¹.graph
hy₂ : f.graph.topologicalClosure = x✝.graph
⊢ x✝¹.graph = x✝.graph

Tactic:
rw [← hy₁, ← hy₂]

Goal state after:
Module: Mathlib.Dynamics.BirkhoffSum.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
E : Type u_3
inst✝³ : PseudoEMetricSpace X
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : X → X
g : X → E
hf : LipschitzWith 1 f
hg : UniformContinuous g
⊢ UniformEquicontinuous (birkhoffAverage 𝕜 f g)

Tactic:
refine Metric.uniformity_basis_dist_le.uniformEquicontinuous_iff_right.2 fun ε hε ↦ ?_
  rcases (uniformity_basis_edist_le.uniformContinuous_iff Metric.uniformity_basis_dist_le).1 hg ε hε
    with ⟨δ, hδ₀, hδε⟩
  refine mem_uniformity_edist.2 ⟨δ, hδ₀, fun {x y} h n ↦ ?_⟩
  calc
    dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y)
      ≤ (∑ k ∈ Finset.range n, dist (g (f^[k] x)) (g (f^[k] y))) / n :=
      dist_birkhoffAverage_birkhoffAverage_le ..
    _ ≤ (∑ _k ∈ Finset.range n, ε) / n := by
      gcongr
      refine hδε _ _ ?_
      simpa using (hf.iterate _).edist_le_mul_of_le h.le
    _ = n * ε / n := by simp
    _ ≤ ε := by
      rcases eq_or_ne n 0 with hn | hn <;> simp [hn, hε.le, mul_div_cancel_left₀]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
a✝ b✝ : F
hx : a✝ ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}
hy : b✝ ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}
⊢ a✝ + b✝ ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}

Tactic:
Set.mem_ofPred_eq,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
a✝ b✝ : F
hx : Continuous ⇑((innerₛₗ 𝕜) a✝ ∘ₗ T.toFun)
hy : Continuous ⇑((innerₛₗ 𝕜) b✝ ∘ₗ T.toFun)
⊢ Continuous ⇑((innerₛₗ 𝕜) (a✝ + b✝) ∘ₗ T.toFun)
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
⊢ 0 ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}

Tactic:
Set.mem_ofPred_eq,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
⊢ Continuous ⇑((innerₛₗ 𝕜) 0 ∘ₗ T.toFun)
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
a : 𝕜
x : F
hx : x ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}
⊢ a • x ∈ {y | Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)}

Tactic:
Set.mem_ofPred_eq,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
a : 𝕜
x : F
hx : Continuous ⇑((innerₛₗ 𝕜) x ∘ₗ T.toFun)
⊢ Continuous ⇑((innerₛₗ 𝕜) (a • x) ∘ₗ T.toFun)
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
a b : ℕ
hl : b ≤ a
ha : a ∈ ↑T.singularValues.support
⊢ b ∈ ↑T.singularValues.support

Tactic:
Finset.mem_coe,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
a b : ℕ
hl : b ≤ a
ha : a ∈ T.singularValues.support
⊢ b ∈ T.singularValues.support
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
n : ℕ
⊢ T.singularValues n = 0 ↔ finrank 𝕜 ↥T.range ≤ n

Tactic:
rw [← Finsupp.notMem_support_iff, support_singularValues, Finset.mem_range, not_lt]

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

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := EuclideanSpace ℝ (Fin (finrank ℝ F))
e : F ≃L[ℝ] F' := toEuclidean
C₁ : ℝ≥0 := ‖↑e.symm‖₊
C : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst μ ↑p
C₂ : ℝ≥0 := ‖↑e‖₊
v : E → F' := ⇑e ∘ u
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
h4v : ∀ (x : E), ‖fderiv ℝ v x‖ ≤ ↑C₂ * ‖fderiv ℝ u x‖
⊢ ↑C₁ * ↑C * (↑‖↑e‖₊ * eLpNorm (fderiv ℝ u) (↑p) μ) =
    ↑(SNormLESNormFDerivOfEqConst F μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
C₁,

Goal state after:
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := EuclideanSpace ℝ (Fin (finrank ℝ F))
e : F ≃L[ℝ] F' := toEuclidean
C₁ : ℝ≥0 := ‖↑e.symm‖₊
C : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst μ ↑p
C₂ : ℝ≥0 := ‖↑e‖₊
v : E → F' := ⇑e ∘ u
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
h4v : ∀ (x : E), ‖fderiv ℝ v x‖ ≤ ↑C₂ * ‖fderiv ℝ u x‖
⊢ ↑‖↑e.symm‖₊ * ↑C * (↑‖↑e‖₊ * eLpNorm (fderiv ℝ u) (↑p) μ) =
    ↑(SNormLESNormFDerivOfEqConst F μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
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
⊢ dimH s = ⨅ d, ⨅ (_ : μH[↑d] s = 0), ↑d

Tactic:
apply le_antisymm

Goal state after:
case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
⊢ dimH s ≤ ⨅ d, ⨅ (_ : μH[↑d] s = 0), ↑d

case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
⊢ ⨅ d, ⨅ (_ : μH[↑d] s = 0), ↑d ≤ dimH s
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: 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
⊢ MeasurableSet (⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t)

Tactic:
measurability

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

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
r : ℝ≥0
f : X → Y
hr : 0 < r
s : Set X
C : X → ℝ≥0
t : X → Set X
htn : ∀ x ∈ s, t x ∈ 𝓝[s] x
hC : ∀ x ∈ s, HolderOnWith (C x) r f (t x)
u : Set X
hus : u ⊆ s
huc : u.Countable
huU : ⋃ i ∈ u, s ∩ t i = s
⊢ ⨆ i ∈ u, dimH (f '' (s ∩ t i)) ≤ (⨆ i ∈ u, dimH (s ∩ t i)) / ↑r

Tactic:
simp only [ENNReal.iSup_div]

Goal state after:
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
r : ℝ≥0
f : X → Y
hr : 0 < r
s : Set X
C : X → ℝ≥0
t : X → Set X
htn : ∀ x ∈ s, t x ∈ 𝓝[s] x
hC : ∀ x ∈ s, HolderOnWith (C x) r f (t x)
u : Set X
hus : u ⊆ s
huc : u.Countable
huU : ⋃ i ∈ u, s ∩ t i = s
⊢ ⨆ i ∈ u, dimH (f '' (s ∩ t i)) ≤ ⨆ i ∈ u, dimH (s ∩ t i) / ↑r
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
⊢ ⨆ r, ⨆ (_ : r > 0), ⊤ = ⊤

Tactic:
eq_top_iff

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
⊢ ⊤ ≤ ⨆ r, ⨆ (_ : r > 0), ⊤
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
⊢ dimH s < ∞

Tactic:
calc
  dimH s ≤ dimH (univ : Set E) := dimH_mono (subset_univ s)
  _ = finrank ℝ E := dimH_univ_eq_finrank E
  _ < ⊤ := by simp

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

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := ⋯
h : ¬(↑f).ker = ⊥
⊢ ↑((↑f).normDet ^ 2) = 0

Tactic:
simp [show f.normDet = 0 from (f.normDet_eq_zero_tfae.out 1 0).mp h]

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

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := ⋯
h : ¬(↑f).ker = ⊥
⊢ ↑((↑f).normDet ^ 2) = 0

pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := ⋯
h : ¬(↑f).ker = ⊥
⊢ 0 = (ContinuousLinearMap.adjoint f ∘SL f).det

Tactic:
· simp [show f.normDet = 0 from (f.normDet_eq_zero_tfae.out 1 0).mp h]

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := ⋯
h : ¬(↑f).ker = ⊥
⊢ 0 = (ContinuousLinearMap.adjoint f ∘SL f).det
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : ¬f.ker = ⊥
⊢ ↑(f.normDet ^ 2) = 0

Tactic:
simp [show f.normDet = 0 from (f.normDet_eq_zero_tfae.out 1 0).mp h]

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

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : ¬f.ker = ⊥
⊢ ↑(f.normDet ^ 2) = 0

pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : ¬f.ker = ⊥
⊢ 0 = (Matrix.gram 𝕜 fun x ↦ f.rangeRestrict (b x)).det

Tactic:
· simp [show f.normDet = 0 from (f.normDet_eq_zero_tfae.out 1 0).mp h]

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : ¬f.ker = ⊥
⊢ 0 = (Matrix.gram 𝕜 fun x ↦ f.rangeRestrict (b x)).det
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
x y z : E
h1 :
  ‖x + y + z + (x - z)‖ * ‖x + y + z + (x - z)‖ + ‖x + y + z - (x - z)‖ * ‖x + y + z - (x - z)‖ =
    2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖)
h2 :
  ‖x + y - z + (x + z)‖ * ‖x + y - z + (x + z)‖ + ‖x + y - z - (x + z)‖ * ‖x + y - z - (x + z)‖ =
    2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖)
h3 : ‖y + z + z‖ * ‖y + z + z‖ + ‖y + z - z‖ * ‖y + z - z‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖)
h4 : ‖y - z + z‖ * ‖y - z + z‖ + ‖y - z - z‖ * ‖y - z - z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
h5 :
  ‖I • (x + y) + z + (I • x - z)‖ * ‖I • (x + y) + z + (I • x - z)‖ +
      ‖I • (x + y) + z - (I • x - z)‖ * ‖I • (x + y) + z - (I • x - z)‖ =
    2 * (‖I • (x + y) + z‖ * ‖I • (x + y) + z‖ + ‖I • x - z‖ * ‖I • x - z‖)
⊢ 4⁻¹ *
      (𝓚 ‖x + y + z‖ * 𝓚 ‖x + y + z‖ - 𝓚 ‖x + y - z‖ * 𝓚 ‖x + y - z‖ + I * 𝓚 ‖I • (x + y) + z‖ * 𝓚 ‖I • (x + y) + z‖ -
        I * 𝓚 ‖I • (x + y) - z‖ * 𝓚 ‖I • (x + y) - z‖) =
    4⁻¹ *
        (𝓚 ‖x + z‖ * 𝓚 ‖x + z‖ - 𝓚 ‖x - z‖ * 𝓚 ‖x - z‖ + I * 𝓚 ‖I • x + z‖ * 𝓚 ‖I • x + z‖ -
          I * 𝓚 ‖I • x - z‖ * 𝓚 ‖I • x - z‖) +
      4⁻¹ *
        (𝓚 ‖y + z‖ * 𝓚 ‖y + z‖ - 𝓚 ‖y - z‖ * 𝓚 ‖y - z‖ + I * 𝓚 ‖I • y + z‖ * 𝓚 ‖I • y + z‖ -
          I * 𝓚 ‖I • y - z‖ * 𝓚 ‖I • y - z‖)

Tactic:
have h6 := parallelogram_identity ((I : 𝕜) • (x + y) - z) ((I : 𝕜) • x + z)

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
x y z : E
h1 :
  ‖x + y + z + (x - z)‖ * ‖x + y + z + (x - z)‖ + ‖x + y + z - (x - z)‖ * ‖x + y + z - (x - z)‖ =
    2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖)
h2 :
  ‖x + y - z + (x + z)‖ * ‖x + y - z + (x + z)‖ + ‖x + y - z - (x + z)‖ * ‖x + y - z - (x + z)‖ =
    2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖)
h3 : ‖y + z + z‖ * ‖y + z + z‖ + ‖y + z - z‖ * ‖y + z - z‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖)
h4 : ‖y - z + z‖ * ‖y - z + z‖ + ‖y - z - z‖ * ‖y - z - z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
h5 :
  ‖I • (x + y) + z + (I • x - z)‖ * ‖I • (x + y) + z + (I • x - z)‖ +
      ‖I • (x + y) + z - (I • x - z)‖ * ‖I • (x + y) + z - (I • x - z)‖ =
    2 * (‖I • (x + y) + z‖ * ‖I • (x + y) + z‖ + ‖I • x - z‖ * ‖I • x - z‖)
h6 :
  ‖I • (x + y) - z + (I • x + z)‖ * ‖I • (x + y) - z + (I • x + z)‖ +
      ‖I • (x + y) - z - (I • x + z)‖ * ‖I • (x + y) - z - (I • x + z)‖ =
    2 * (‖I • (x + y) - z‖ * ‖I • (x + y) - z‖ + ‖I • x + z‖ * ‖I • x + z‖)
⊢ 4⁻¹ *
      (𝓚 ‖x + y + z‖ * 𝓚 ‖x + y + z‖ - 𝓚 ‖x + y - z‖ * 𝓚 ‖x + y - z‖ + I * 𝓚 ‖I • (x + y) + z‖ * 𝓚 ‖I • (x + y) + z‖ -
        I * 𝓚 ‖I • (x + y) - z‖ * 𝓚 ‖I • (x + y) - z‖) =
    4⁻¹ *
        (𝓚 ‖x + z‖ * 𝓚 ‖x + z‖ - 𝓚 ‖x - z‖ * 𝓚 ‖x - z‖ + I * 𝓚 ‖I • x + z‖ * 𝓚 ‖I • x + z‖ -
          I * 𝓚 ‖I • x - z‖ * 𝓚 ‖I • x - z‖) +
      4⁻¹ *
        (𝓚 ‖y + z‖ * 𝓚 ‖y + z‖ - 𝓚 ‖y - z‖ * 𝓚 ‖y - z‖ + I * 𝓚 ‖I • y + z‖ * 𝓚 ‖I • y + z‖ -
          I * 𝓚 ‖I • y - z‖ * 𝓚 ‖I • y - z‖)