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

Goal state before:
pp: n : ℕ
inst✝⁴ : NeZero n
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
hζ : IsPrimitiveRoot ζ n
inst✝ : IsCyclotomicExtension {n} K L
h : 2 < n
hirr : Irreducible (cyclotomic n K)
⊢ (Algebra.norm K) (ζ - 1) = ↑(eval 1 (cyclotomic n ℤ))

Tactic:
by
  have := IsCyclotomicExtension.neZero' n K L
  let E := AlgebraicClosure L
  obtain ⟨z, hz⟩ := IsAlgClosed.exists_root _ (degree_cyclotomic_pos n E (NeZero.pos _)).ne.symm
  apply (algebraMap K E).injective
  let := IsCyclotomicExtension.finiteDimensional {n} K L
  let := IsCyclotomicExtension.isGalois {n} K L
  rw [norm_eq_prod_embeddings]
  conv_lhs =>
    congr
    rfl
    ext
    rw [← neg_sub, map_neg, map_sub, map_one, neg_eq_neg_one_mul]
  rw [prod_mul_distrib, prod_const, Finset.card_univ, AlgHom.card,
    IsCyclotomicExtension.finrank L hirr, (totient_even h).neg_one_pow, one_mul]
  have Hprod : (Finset.univ.prod fun σ : L →ₐ[K] E => 1 - σ ζ) = eval 1 (cyclotomic' n E) := by
    rw [cyclotomic', eval_prod, ← @Finset.prod_attach E E, ← univ_eq_attach]
    refine Fintype.prod_equiv (hζ.embeddingsEquivPrimitiveRoots E hirr) _ _ fun σ => ?_
    simp
  have : NeZero (n : E) := NeZero.of_faithfulSMul K _ n
  rw [Hprod, cyclotomic', ← cyclotomic_eq_prod_X_sub_primitiveRoots (isRoot_cyclotomic_iff.1 hz),
    ← map_cyclotomic_int, _root_.map_intCast, ← Int.cast_one, eval_intCast_map, eq_intCast,
    Int.cast_id]

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

Goal state before:
pp: case algebraMap
S : Set ℕ
A : Type u
B : Type v
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n
x✝ : B
x : A
⊢ ⟨(algebraMap A B) x, ⋯⟩ ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}

Tactic:
| algebraMap x =>

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

Goal state before:
pp: S : Set ℕ
A : Type u
B : Type v
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
h : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n
x : B
hx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}
⊢ ⟨x, hx⟩ ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}

Tactic:
induction hx using adjoin_induction with

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

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

Tactic:
refine le_antisymm (adjoin_mono fun x hx => ?_) (adjoin_le fun x hx => ?_)

Goal state after:
case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : x ∈ (cyclotomic n A).rootSet B
⊢ x ∈ {b | n ≠ 0 ∧ b ^ n = 1}

case refine_2
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : x ∈ {b | n ≠ 0 ∧ b ^ n = 1}
⊢ x ∈ ↑(adjoin A ((cyclotomic n A).rootSet B))
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
inst✝¹ : IsDomain B
ζ : B
n : ℕ
inst✝ : NeZero n
hζ : IsPrimitiveRoot ζ n
x : B
hx : map (algebraMap A B) (cyclotomic n A) ≠ 0 ∧ (aeval x) (cyclotomic n A) = 0
⊢ (aeval x) (cyclotomic n A) = 0

Tactic:
exact hx.2

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

Goal state before:
pp: case refine_1
A : Type u
B : Type v
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Algebra A B
n : ℕ
inst✝¹ : NeZero n
inst✝ : IsDomain B
ζ : B
hζ : IsPrimitiveRoot ζ n
x : B
hx : cyclotomic n B ≠ 0 ∧ (cyclotomic n B).IsRoot x
⊢ (cyclotomic n B).IsRoot x

Tactic:
exact hx.2

Goal state after:
Module: Mathlib.Data.ZMod.Units

Goal state before:
pp: case inl
a : ℤ
⊢ ∃ d, d ∣ 0 ∧ ∃ u, IsUnit u ∧ a = u * ↑d

Tactic:
rcases eq_or_ne a 0 with rfl | ha

Goal state after:
case inl.inl
⊢ ∃ d, d ∣ 0 ∧ ∃ u, IsUnit u ∧ 0 = u * ↑d

case inl.inr
a : ℤ
ha : a ≠ 0
⊢ ∃ d, d ∣ 0 ∧ ∃ u, IsUnit u ∧ a = u * ↑d
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case right.algebraMap
S : Set ℕ
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
i : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})
f : AlgebraicClosure K →ₐ[K] AlgebraicClosure K
y : AlgebraicClosure K
x : K
⊢ f ((algebraMap K (AlgebraicClosure K)) x) ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}

Tactic:
| algebraMap x =>

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_3
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
S : Submonoid R
x : (↥S)ˣ
⊢ (equivToUnitHom.symm f) ↑↑x = ↑(f ↑(S.unitsEquivUnitsType.symm x))

Tactic:
← Submonoid.val_unitsEquivUnitsType_symm_apply_coe S x,

Goal state after:
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_3
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
S : Submonoid R
x : (↥S)ˣ
⊢ (equivToUnitHom.symm f) ↑↑(S.unitsEquivUnitsType.symm x) = ↑(f ↑(S.unitsEquivUnitsType.symm x))
Module: Mathlib.NumberTheory.MulChar.Lemmas

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing R'
inst✝ : Fintype Rˣ
a : Rˣ
χ : MulChar R R'
⊢ (equivToUnitHom χ) (a ^ Fintype.card Rˣ) = 1

Tactic:
← (equivToUnitHom χ).map_one,

Goal state after:
R : Type u_1
R' : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing R'
inst✝ : Fintype Rˣ
a : Rˣ
χ : MulChar R R'
⊢ (equivToUnitHom χ) (a ^ Fintype.card Rˣ) = (equivToUnitHom χ) 1
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
χ : MulChar R R'
⊢ 1 * χ = χ

Tactic:
by
  ext
  simp only [one_mul, Pi.mul_apply, MulChar.coeToFun_mul, MulChar.one_apply_coe]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst✝ : NeZero n
ψ : DirichletCharacter R m
h : (changeLevel ⋯) χ = (changeLevel ⋯) ψ
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
z : ℕ
hz₁ : ↑z = ↑x
hz₂ : ↑z = 1
hz₀ : z.gcd (n * m) = 1
this : ((changeLevel ⋯) χ) ↑z = χ ↑z
⊢ ((changeLevel ⋯) ψ) ↑z = 1

Tactic:
have := changeLevel_eq_cast_of_dvd ψ (m.dvd_mul_left n) (ZMod.unitOfCoprime z hz₀)

Goal state after:
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst✝ : NeZero n
ψ : DirichletCharacter R m
h : (changeLevel ⋯) χ = (changeLevel ⋯) ψ
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
z : ℕ
hz₁ : ↑z = ↑x
hz₂ : ↑z = 1
hz₀ : z.gcd (n * m) = 1
this✝ : ((changeLevel ⋯) χ) ↑z = χ ↑z
this : ((changeLevel ⋯) ψ) ↑(ZMod.unitOfCoprime z hz₀) = ψ (↑(ZMod.unitOfCoprime z hz₀)).cast
⊢ ((changeLevel ⋯) ψ) ↑z = 1
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: 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]

Tactic:
refine adjoin_le fun x ⟨_, hx⟩ ↦ ?_
      obtain ⟨k, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx
      exact Subalgebra.pow_mem _ (self_mem_adjoin_singleton A ζ) _

Goal state after:
Module: Mathlib.NumberTheory.DirichletCharacter.GaussSum

Goal state before:
pp: N : ℕ
inst✝¹ : NeZero N
R : Type u_1
inst✝ : CommRing R
e : AddChar (ZMod N) R
χ : DirichletCharacter R N
d : ℕ
hd : d ∣ N
he : e.mulShift ↑d = 1
u : (ZMod N)ˣ
hu : ↑(ZMod.castHom hd (ZMod d)) ↑u = 1
this : (ZMod.castHom hd (ZMod d)) ↑u = ↑↑(↑u).val
⊢ ↑d ∣ ↑(↑u).val - 1

Tactic:
rwa [MonoidHom.coe_coe, this, ← Int.cast_one, eq_comm,
    ZMod.intCast_eq_intCast_iff_dvd_sub] at hu

Goal state after:
Module: Mathlib.NumberTheory.DirichletCharacter.GaussSum

Goal state before:
pp: N : ℕ
inst✝² : NeZero N
R : Type u_1
inst✝¹ : CommRing R
e : AddChar (ZMod N) R
inst✝ : IsDomain R
χ : DirichletCharacter R N
d : ℕ
hd : d ∣ N
he : e.mulShift ↑d = 1
h_ne : gaussSum χ e ≠ 0
⊢ χ.FactorsThrough d

Tactic:
rw [DirichletCharacter.factorsThrough_iff_ker_unitsMap hd]
  intro _ hu
  simpa [← Units.val_inj, h_ne] using gaussSum_aux_of_mulShift e χ hd he hu

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

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 : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
⊢ T.IsSymmetric → ((∀ (x : F), 0 ≤ re ⟪(↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm) x, x⟫) ↔ ∀ (x : E), 0 ≤ re ⟪T x, x⟫)

Tactic:
coe_comp,

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 : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
⊢ T.IsSymmetric → ((∀ (x : F), 0 ≤ re ⟪(⇑↑f.toLinearEquiv ∘ ⇑T ∘ ⇑↑f.symm) x, x⟫) ↔ ∀ (x : E), 0 ≤ re ⟪T x, x⟫)
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).HolderConjugate p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ (1 + 1 / (↑#ι - 1))

Tactic:
convert lintegral_mul_prod_lintegral_pow_le μ h2 h3 hf

Goal state after:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).HolderConjugate p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)
x✝ : (i : ι) → A i
⊢ ∏ i, (∫⁻ (xᵢ : A i), f (update x✝ i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) =
    f x✝ ^ (1 - (↑#ι - 1) * (1 / (↑#ι - 1))) * ∏ i, (∫⁻ (xᵢ : A i), f (update x✝ i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1))
Module: Mathlib.Analysis.InnerProductSpace.JointEigenspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : DecidableEq (n → 𝕜)
hT : ∀ (i : n), IsSymmetric (T i)
hC : Pairwise (Commute on T)
⊢ DirectSum.IsInternal fun α ↦ ⨅ j, (T j).eigenspace (α j)

Tactic:
OrthogonalFamily.isInternal_iff

Goal state after:
𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : DecidableEq (n → 𝕜)
hT : ∀ (i : n), IsSymmetric (T i)
hC : Pairwise (Commute on T)
⊢ (⨆ α, ⨅ j, (T j).eigenspace (α j))ᗮ = ⊥

𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : DecidableEq (n → 𝕜)
hT : ∀ (i : n), IsSymmetric (T i)
hC : Pairwise (Commute on T)
⊢ OrthogonalFamily 𝕜 (fun i ↦ ↥(⨅ j, (T j).eigenspace (i j))) fun i ↦ (⨅ j, (T j).eigenspace (i j)).subtypeₗᵢ
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
coercive : IsCoercive B
w✝ : ℝ≥0
left✝ : 0 < w✝
antilipschitz : AntilipschitzWith w✝ ⇑(continuousLinearMapOfBilin B)
⊢ IsClosed ↑(↑(continuousLinearMapOfBilin B)).range

Tactic:
exact antilipschitz.isClosed_range B♯.uniformContinuous

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

Goal state before:
pp: R : Type u_1
α : Type u_2
M : Type u_3
inst✝⁴ : DivisionSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_4
inst✝¹ : DivisionSemiring S
inst✝ : Module S M
⊢ birkhoffAverage R = birkhoffAverage S

Tactic:
ext; apply birkhoffAverage_congr_ring

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✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
x : F × E
hx' : x.1 = 0
hx : ∀ a ∈ T.domain, ⟪a, x.2⟫ = 0
⊢ ∀ v ∈ ↑T.domain, ⟪v, x.2⟫ = 0

Tactic:
exact fun a ha ↦ hx a ha

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
⊢ inner 𝕜 ((map f.toLinearMap g.toLinearMap) 0) ((map f.toLinearMap g.toLinearMap) y) = inner 𝕜 0 y

Tactic:
simp [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 : E) (y : F),
    inner 𝕜 ((map f.toLinearMap g.toLinearMap) (x ⊗ₜ[𝕜] y)) ((map f.toLinearMap g.toLinearMap) 0) =
      inner 𝕜 (x ⊗ₜ[𝕜] y) 0

Tactic:
simp [inner_def]

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

Goal state before:
pp: case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
i : ℝ≥0
hi : μH[↑i] s = 0
j : ℝ≥0
hj : μH[↑j] s = ∞
⊢ j ≤ i

Tactic:
by_contra! hij

Goal state after:
case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
i : ℝ≥0
hi : μH[↑i] s = 0
j : ℝ≥0
hj : μH[↑j] s = ∞
hij : i < j
⊢ False
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
s : Set X
h : s.Subsingleton
this✝¹ : MeasurableSpace X := borel X
this✝ : BorelSpace X
⊢ dimH s ≤ 0

Tactic:
apply dimH_le_of_hausdorffMeasure_ne_top

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
s : Set X
h : s.Subsingleton
this✝¹ : MeasurableSpace X := ⋯
this✝ : BorelSpace X
⊢ μH[↑0] s ≠ ∞
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
f : X → Y
s : Set X
hf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t
⊢ dimH (f '' s) ≤ dimH s

Tactic:
have : ∀ x ∈ s, ∃ C : ℝ≥0, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t := by
    simpa only [holderOnWith_one] using hf

Goal state after:
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
f : X → Y
s : Set X
hf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t
this : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t
⊢ dimH (f '' s) ≤ dimH s
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
hcvx : Convex ℝ s
hne : s.Nonempty
this : Nonempty ↑s
φ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩
hs_eq : s = Subtype.val '' Subtype.val ⁻¹' s
⊢ dimH s = ↑(finrank ℝ ↥(vectorSpan ℝ s))

Tactic:
hs_eq,

Goal state after:
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hcvx : Convex ℝ s
hne : s.Nonempty
this : Nonempty ↑s
φ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩
hs_eq : s = Subtype.val '' Subtype.val ⁻¹' s
⊢ dimH (Subtype.val '' Subtype.val ⁻¹' s) = ↑(finrank ℝ ↥(vectorSpan ℝ (Subtype.val '' Subtype.val ⁻¹' s)))
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
K : ℝ≥0
d : ℝ
hf : AntilipschitzWith K f
hd : 0 ≤ d
s : Set Y
⊢ μH[d] (f ⁻¹' s) ≤ ↑K ^ d * μH[d] s

Tactic:
by
  rcases eq_or_ne K 0 with (rfl | h0)
  · rcases eq_empty_or_nonempty (f ⁻¹' s) with (hs | ⟨x, hx⟩)
    · simp only [hs, measure_empty, zero_le]
    have : f ⁻¹' s = {x} := by
      have : Subsingleton X := hf.subsingleton
      have : (f ⁻¹' s).Subsingleton := subsingleton_univ.anti (subset_univ _)
      exact (subsingleton_iff_singleton hx).1 this
    rw [this]
    rcases eq_or_lt_of_le hd with (rfl | h'd)
    · simp only [ENNReal.rpow_zero, one_mul]
      rw [hausdorffMeasure_zero_singleton]
      exact one_le_hausdorffMeasure_zero_of_nonempty ⟨f x, hx⟩
    · have := nullSingletonClass_hausdorff X h'd
      simp only [zero_le, measure_singleton]
  have hKd0 : (K : ℝ≥0∞) ^ d ≠ 0 := by simp [h0]
  have hKd : (K : ℝ≥0∞) ^ d ≠ ∞ := by simp [hd]
  simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hKd0 hKd,
    ← ENNReal.tsum_mul_left]
  refine iSup₂_le fun ε ε0 => ?_
  refine le_iSup₂_of_le (ε / K) (by simp [ε0.ne']) ?_
  refine le_iInf₂ fun t hst => le_iInf fun htε => ?_
  replace hst : f ⁻¹' s ⊆ _ := preimage_mono hst; rw [preimage_iUnion] at hst
  refine iInf₂_le_of_le _ hst (iInf_le_of_le (fun n => ?_) ?_)
  · exact (hf.ediam_preimage_le _).trans (ENNReal.mul_le_of_le_div' <| htε n)
  · refine ENNReal.tsum_le_tsum fun n => iSup_le_iff.2 fun hft => ?_
    simp only [nonempty_of_nonempty_preimage hft, ciSup_pos]
    rw [← ENNReal.mul_rpow_of_nonneg _ _ hd]
    exact ENNReal.rpow_le_rpow (hf.ediam_preimage_le _) hd

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)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
bE' : (𝓝 0).HasBasis (fun x ↦ True) fun n ↦ c ^ n • b n
⊢ ∀ t ∈ 𝓝 0, ∃ i, Set.MapsTo (⇑f) (c ^ i • b i) t

Tactic:
Set.MapsTo

Goal state after:
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)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
bE' : (𝓝 0).HasBasis (fun x ↦ True) fun n ↦ c ^ n • b n
⊢ ∀ t ∈ 𝓝 0, ∃ i, ∀ ⦃x : E⦄, x ∈ c ^ i • b i → f x ∈ t
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
P : Type u_6
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : BorelSpace P
d : ℝ
hd : 0 ≤ d
x : P
c : 𝕜
hc : c ≠ 0
s : Set P
⊢ μH[d] (⇑(AffineMap.homothety x c) '' s) = ‖c‖₊ ^ d • μH[d] s

Tactic:
by
  suffices
    μH[d] (IsometryEquiv.vaddConst x '' (c • ·) '' (IsometryEquiv.vaddConst x).symm '' s) =
      ‖c‖₊ ^ d • μH[d] s by
    simpa only [Set.image_image]
  borelize E
  rw [IsometryEquiv.hausdorffMeasure_image, Set.image_smul, Measure.hausdorffMeasure_smul₀ hd hc,
    IsometryEquiv.hausdorffMeasure_image]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ ⟪o.rightAngleRotation x, y⟫ = (o.areaForm x) y

Tactic:
by
  rw [rightAngleRotation]
  exact o.inner_rightAngleRotationAux₁_left x y

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ (o.kahler x) y = 0 ↔ x = 0 ∨ y = 0

Tactic:
refine ⟨o.eq_zero_or_eq_zero_of_kahler_eq_zero, ?_⟩

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ x = 0 ∨ y = 0 → (o.kahler x) y = 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ x = 0 ∨ y = 0 → (o.kahler x) y = 0

Tactic:
rintro (rfl | rfl)

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
y : E
⊢ (o.kahler 0) y = 0

case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
⊢ (o.kahler x) 0 = 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ x = 0 ∨ y = 0 → ↑⟪x, y⟫ + ↑((o.areaForm x) y) * Complex.I = 0

Tactic:
rintro (rfl | rfl)

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
y : E
⊢ ↑⟪0, y⟫ + ↑((o.areaForm 0) y) * Complex.I = 0

case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
⊢ ↑⟪x, 0⟫ + ↑((o.areaForm x) 0) * Complex.I = 0
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)

Tactic:
rcases eq_or_ne a 0 with rfl | ha

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)

case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)