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

Goal state before:
pp: S : Set ℕ
K : Type w
L : Type z
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : IsCyclotomicExtension S K L
M : Type u_1
inst✝² : Field M
inst✝¹ : Algebra K M
inst✝ : IsSepClosed M
this : Algebra.IsSeparable K L
i : L →ₐ[K] M := IsSepClosed.lift
htop : IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1} = ⊤
y : L
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : y ^ n = 1
⊢ i y ^ n = 1

Tactic:
by simpa using congrArg i h2

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
z : ↥(rootsOfUnity n ℂ)
⊢ ↑((rootsOfUnitytoCircle n).toHomUnits z) ^ n = 1

Tactic:
MonoidHom.coe_toHomUnits,

Goal state after:
n : ℕ
inst✝ : NeZero n
z : ↥(rootsOfUnity n ℂ)
⊢ (rootsOfUnitytoCircle n) z ^ n = 1
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case h
n : ℕ
inst✝¹⁰ : NeZero n
S T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑n
x✝ : ↥(nonZeroDivisors (CyclotomicRing n A K))
x : CyclotomicRing n A K
hx : x ∈ nonZeroDivisors (CyclotomicRing n A K)
⊢ ↑⟨x, hx⟩ ∈ nonZeroDivisors (CyclotomicRing n A K)

Tactic:
exact hx

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
χ₁ χ₂ : MulChar R R'
a : Rˣ
⊢ ↑((equivToUnitHom (χ₁ * χ₂)) a) = ↑((equivToUnitHom χ₁) a) * ↑((equivToUnitHom χ₂) a)

Tactic:
simp_rw [coe_equivToUnitHom, coeToFun_mul, Pi.mul_apply]

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Lemmas

Goal state before:
pp: case h.refine_2
F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
R : Type u_2
inst✝ : CommRing R
n : ℕ
h✝ : n ∣ Fintype.card F - 1
ζ : R
hζ : IsPrimitiveRoot ζ n
hn₀ : 0 < n
e : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card Fˣ) R) := equiv_rootsOfUnity F R
ζ' : Rˣ := ⋯.unit
h' : ζ' ^ Fintype.card Fˣ = 1
m : ℕ
hm : m < n
hm₀ : 0 < m
h : ↑ζ' ^ m = 1
⊢ False

Tactic:
exact ((Nat.le_of_dvd hm₀ <| hζ.dvd_of_pow_eq_one _ h).trans_lt hm).false

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Lemmas

Goal state before:
pp: F : Type u_1
inst✝⁴ : Field F
inst✝³ : Finite F
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
χ : MulChar F R
n : ℕ
inst✝ : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
ha : a ≠ 0
ζ : Rˣ
hζ₁ : ζ ∈ rootsOfUnity n R
hζ₂ : ↑ζ = χ a
hζ' : ↑ζ ^ n = 1
k : ℕ
hk₁ : k < n
hk₂ : μ ^ k = ↑ζ
⊢ ∃ k < n, χ a = μ ^ k

Tactic:
exact ⟨k, hk₁, (hζ₂ ▸ hk₂).symm⟩

Goal state after:
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: 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✝

Tactic:
aesop (add simp MulChar.one_apply)

Goal state after:
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: 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

Tactic:
aesop (add simp MulChar.one_apply)

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

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
⊢ HasFDerivAt (fun x ↦ ‖x‖ ^ p) 0 0

Tactic:
have h2p : 0 < p - 1 := sub_pos.mpr hp

Goal state after:
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
h2p : 0 < p - 1
⊢ HasFDerivAt (fun x ↦ ‖x‖ ^ p) 0 0
Module: Mathlib.Analysis.InnerProductSpace.NormPow

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
h2p : 0 < p - 1
⊢ (fun x ↦ ‖x‖ * ‖x‖ ^ (p - 1)) =o[𝓝 0] fun x ↦ ‖x‖ * 1

Tactic:
refine (isBigO_refl _ _).mul_isLittleO <| (isLittleO_const_iff <| by simp).mpr ?_

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
h2p : 0 < p - 1
⊢ Filter.Tendsto (fun x ↦ ‖x‖ ^ (p - 1)) (𝓝 0) (𝓝 0)
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
inst✝² : Fintype R
R' : Type v
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ.IsQuadratic
ψ : AddChar R R'
p n : ℕ
fp : Fact (Nat.Prime p)
hch : CharP R' p
hp : IsUnit ↑p
hp' : p ≠ 2
hg : gaussSum χ ψ ^ 2 = χ (-1) * ↑(Fintype.card R)
this : gaussSum χ ψ ≠ 0
⊢ gaussSum χ ψ ^ (2 * (p ^ n / 2) + 1) = gaussSum χ ψ ^ p ^ n

Tactic:
Nat.two_mul_div_two_add_one_of_odd (fp.1.eq_two_or_odd'.resolve_left hp').pow

Goal state after:
R : Type u
inst✝³ : CommRing R
inst✝² : Fintype R
R' : Type v
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ.IsQuadratic
ψ : AddChar R R'
p n : ℕ
fp : Fact (Nat.Prime p)
hch : CharP R' p
hp : IsUnit ↑p
hp' : p ≠ 2
hg : gaussSum χ ψ ^ 2 = χ (-1) * ↑(Fintype.card R)
this : gaussSum χ ψ ≠ 0
⊢ gaussSum χ ψ ^ p ^ n = gaussSum χ ψ ^ p ^ n
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' : Nat.Prime (ringChar F')
hc' : Fintype.card F' = ringChar F' ^ ↑n'
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' hch₁
FF' : Type u_2 := CyclotomicField (↑ψ.n) F'
hchar : ringChar F' = ringChar FF'
this : Fact (Nat.Prime (ringChar F'))
⊢ ((χ.ringHomComp (algebraMap F' FF')) (-1) * ↑(Fintype.card F)) ^ (ringChar FF' ^ ↑n' / 2) =
    (χ.ringHomComp (algebraMap F' FF')) (↑(ringChar FF') ^ ↑n')

Tactic:
have := Fact.mk (hchar.subst hp')

Goal state after:
F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' : Nat.Prime (ringChar F')
hc' : Fintype.card F' = ringChar F' ^ ↑n'
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' hch₁
FF' : Type u_2 := CyclotomicField (↑ψ.n) F'
hchar : ringChar F' = ringChar FF'
this✝ : Fact (Nat.Prime (ringChar F'))
this : Fact (Nat.Prime (ringChar FF'))
⊢ ((χ.ringHomComp (algebraMap F' FF')) (-1) * ↑(Fintype.card F)) ^ (ringChar FF' ^ ↑n' / 2) =
    (χ.ringHomComp (algebraMap F' FF')) (↑(ringChar FF') ^ ↑n')
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case hxy.e'_6.a.e_s
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
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t ↦ (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t ↦ f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
j : ι
hj : j ∈ s
⊢ {i, j} = {j} ∪ {i}

Tactic:
Finset.union_comm

Goal state after:
case hxy.e'_6.a.e_s
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
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t ↦ (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t ↦ f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
j : ι
hj : j ∈ s
⊢ {i, j} = {i} ∪ {j}
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsPositive
hT' : T ≠ 0
α : 𝕜
h : (α • T).IsPositive
⊢ ?m.48

Tactic:
by simpa only [hT.1 _] using hT.ne_zero_iff.mp hT'

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_4
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
⊢ (A.IsHermitian ∧ ∀ (x : EuclideanSpace 𝕜 n), 0 ≤ re ⟪(Matrix.toEuclideanLin A) x, x⟫) ↔ A.PosSemidef

Tactic:
inner_re_symm,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_4
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
⊢ (A.IsHermitian ∧ ∀ (x : EuclideanSpace 𝕜 n), 0 ≤ re ⟪x, (Matrix.toEuclideanLin A) x⟫) ↔ A.PosSemidef
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: ι✝ : Type u_1
A : ι✝ → Type u_2
inst✝⁸ : (i : ι✝) → MeasurableSpace (A i)
μ✝ : (i : ι✝) → Measure (A i)
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
p : ℝ
ι : Type := Fin (finrank ℝ E)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
Fintype.card_fin (finrank ℝ E)

Goal state after:
ι✝ : Type u_1
A : ι✝ → Type u_2
inst✝⁸ : (i : ι✝) → MeasurableSpace (A i)
μ✝ : (i : ι✝) → Measure (A i)
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
p : ℝ
ι : Type := Fin (finrank ℝ E)
⊢ finrank ℝ E = finrank ℝ E
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
inst✝¹ : U.HasOrthogonalProjection
inst✝ : V.HasOrthogonalProjection
⊢ ↑U.starProjection ≤ ↑V.starProjection ↔ U ≤ V

Tactic:
isSymmetricProjection_starProjection _
      |>.le_iff_range_le_range <| isSymmetricProjection_starProjection _,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
inst✝¹ : U.HasOrthogonalProjection
inst✝ : V.HasOrthogonalProjection
⊢ (↑U.starProjection).range ≤ (↑V.starProjection).range ↔ U ≤ V
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 g : E →ₗ.[R] F
hg : g.IsClosable
h : f ≤ g
⊢ f.graph.topologicalClosure ≤ g.closure.graph

Tactic:
← hg.graph_closure_eq_closure_graph

Goal state after:
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 g : E →ₗ.[R] F
hg : g.IsClosable
h : f ≤ g
⊢ f.graph.topologicalClosure ≤ g.graph.topologicalClosure
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
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
hcf : f.closure.toFun.ker = ⊥
⊢ f.inverse.closure.graph = f.closure.inverse.graph

Tactic:
rw [closure_inverse_graph hf hf' hcf,
    ((inverse_isClosable_iff hf hf').mpr hcf).graph_closure_eq_closure_graph]

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
⊢ (↑n')⁻¹ - (1 - (↑p)⁻¹) = ↑p⁻¹ - (↑n)⁻¹

Tactic:
← hn.coe.one_sub_inv,

Goal state after:
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
⊢ 1 - (↑↑n)⁻¹ - (1 - (↑p)⁻¹) = ↑p⁻¹ - (↑n)⁻¹
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:
exact top_unique (hsd' ▸ hausdorffMeasure_mono hdd'.le _)

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

Goal state before:
pp: V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
⊢ μHE[finrank ℝ V] = volume

Tactic:
← (stdOrthonormalBasis ℝ V).measurePreserving_repr_symm.map_eq,

Goal state after:
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
⊢ μHE[finrank ℝ V] = map (⇑(stdOrthonormalBasis ℝ V).repr.symm) volume
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
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 ↥f.range
hrank : finrank 𝕜 U = finrank 𝕜 ↥f.range
h : Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range)
h1 : bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis * (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix ⇑bu.toBasis = 1
h2 : (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix ⇑bu.toBasis * bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis = 1
this : ‖(bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis * h.some.toBasis.toMatrix ⇑bv.toBasis).det‖ = 1
⊢ ‖(bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis * h.some.toBasis.toMatrix ⇑bv.toBasis).det‖ *
      ‖((toMatrix bu.toBasis bv.toBasis) f.rangeRestrict).det‖ =
    ‖((toMatrix bu.toBasis bv.toBasis) f.rangeRestrict).det‖

Tactic:
by
    rw [this, one_mul]

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

Goal state before:
pp: case inl
ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
h✝ : IsEmpty ι
⊢ dimH (Metric.ball x r) = ↑(Fintype.card ι)

Tactic:
dimH_subsingleton,

Goal state after:
case inl
ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
h✝ : IsEmpty ι
⊢ 0 = ↑(Fintype.card ι)

case inl
ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
h✝ : IsEmpty ι
⊢ (Metric.ball x r).Subsingleton
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℝ
hd : ↑(finrank ℝ E) < d
⊢ μH[d] = 0

Tactic:
lift d to ℝ≥0 using (Nat.cast_nonneg _).trans hd.le
  rw [← measure_univ_eq_zero]
  apply hausdorffMeasure_of_dimH_lt
  rw [dimH_univ_eq_finrank]
  exact mod_cast hd

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

Goal state before:
pp: case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : (g ∘ₗ f).ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet

pp: case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : ¬(g ∘ₗ f).ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet

Tactic:
· obtain ⟨bw⟩ := ((g ∘ₗ f).normDet_ne_zero_tfae.out 1 3).mp hgf
      let bw' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 (g.domRestrict f.range).range :=
        bw.map (LinearIsometryEquiv.ofEq _ _ (by simp [LinearMap.range_comp]))
      rw [(g ∘ₗ f).normDet_eq_norm_det_toMatrix_rangeRestrict bu bw,
        f.normDet_eq_norm_det_toMatrix_rangeRestrict bu bv,
        (g.domRestrict f.range).normDet_eq_norm_det_toMatrix_rangeRestrict bv bw']
      rw [← norm_mul, ← Matrix.det_mul, ← LinearMap.toMatrix_comp]
      rfl

Goal state after:
case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : ¬(g ∘ₗ f).ker = ⊥
⊢ (g ∘ₗ f).normDet = (g.domRestrict f.range).normDet * f.normDet
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε

Tactic:
have hnorm := (tendsto_iff_norm_sub_tendsto_zero.mp h).mul_const C

Goal state after:
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
hε : ε > 0
hnorm : Tendsto (fun x ↦ ‖F x - f‖ * C) l (𝓝 (0 * C))
⊢ ∀ᶠ (n : ι) in l, ∀ x ∈ s, dist (f x) ((F n) x) < ε
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case pos
U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h : f.ker = ⊥
hrank : finrank ℝ ↥f.range = finrank ℝ U
bv : OrthonormalBasis (Fin (finrank ℝ U)) ℝ ↥f.range
g : U ≃ₗᵢ[ℝ] ↥f.range := (stdOrthonormalBasis ℝ U).equiv bv (Equiv.refl (Fin (finrank ℝ U)))
⊢ μH[↑(finrank ℝ U)]
      (⇑(f.range.subtypeₗᵢ.comp g.toLinearIsometry) '' ⇑(g.symm.toLinearIsometry.toLinearMap ∘ₗ f.rangeRestrict) '' s) =
    ENNReal.ofReal f.normDet * μH[↑(finrank ℝ U)] s

Tactic:
(LinearIsometry.isometry _).hausdorffMeasure_image (by simp),

Goal state after:
case pos
U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h : f.ker = ⊥
hrank : finrank ℝ ↥f.range = finrank ℝ U
bv : OrthonormalBasis (Fin (finrank ℝ U)) ℝ ↥f.range
g : U ≃ₗᵢ[ℝ] ↥f.range := (stdOrthonormalBasis ℝ U).equiv bv (Equiv.refl (Fin (finrank ℝ U)))
⊢ μH[↑(finrank ℝ U)] (⇑(g.symm.toLinearIsometry.toLinearMap ∘ₗ f.rangeRestrict) '' s) =
    ENNReal.ofReal f.normDet * μH[↑(finrank ℝ U)] s
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
x : H
⊢ x ∈ Sᗮ.mulI ↔ x ∈ S.symplComp

Tactic:
mem_symplComp_iff,

Goal state after:
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
x : H
⊢ x ∈ Sᗮ.mulI ↔ ∀ y ∈ S, ⟪y, x⟫.im = 0
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S T : ClosedSubmodule ℝ H
⊢ (mapEquiv (scalarSMulCLE H UnitI)) (S ⊔ T) = S.mulI ⊔ T.mulI

Tactic:
← mapEquiv_sup_eq

Goal state after:
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S T : ClosedSubmodule ℝ H
⊢ (mapEquiv (scalarSMulCLE H UnitI)) (S ⊔ T) = (mapEquiv (scalarSMulCLE H UnitI)) (S ⊔ T)
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
x✝² x✝¹ x✝ : H₀ K
⊢ (Finsupp.sum (x✝² + x✝¹) fun x z ↦
      match x with
      | (y, u) =>
        Finsupp.sum x✝ fun x w ↦
          match x with
          | (x, v) => star z * w * ⟪(K x y) u, v⟫_𝕜) =
    (Finsupp.sum x✝² fun x z ↦
        match x with
        | (y, u) =>
          Finsupp.sum x✝ fun x w ↦
            match x with
            | (x, v) => star z * w * ⟪(K x y) u, v⟫_𝕜) +
      Finsupp.sum x✝¹ fun x z ↦
        match x with
        | (y, u) =>
          Finsupp.sum x✝ fun x w ↦
            match x with
            | (x, v) => star z * w * ⟪(K x y) u, v⟫_𝕜

Tactic:
rw [Finsupp.sum_add_index'] <;> simp [← Finsupp.sum_add, add_mul]

Goal state after: