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

Goal state before:
pp: case pos
n : ℕ
inst✝⁵ : NeZero n
K : Type u
L : Type v
inst✝⁴ : CommRing L
ζ : L
inst✝³ : Field K
inst✝² : Algebra K L
hζ : IsPrimitiveRoot ζ n
inst✝¹ : IsDomain L
inst✝ : IsCyclotomicExtension {n} K L
hn : n ≠ 2
hirr : Irreducible (cyclotomic n K)
this : NeZero ↑n
h1 : n = 1
⊢ (Algebra.norm K) ζ = 1

Tactic:
rw [h1, one_right_iff] at hζ

Goal state after:
case pos
n : ℕ
inst✝⁵ : NeZero n
K : Type u
L : Type v
inst✝⁴ : CommRing L
ζ : L
inst✝³ : Field K
inst✝² : Algebra K L
hζ : ζ = 1
inst✝¹ : IsDomain L
inst✝ : IsCyclotomicExtension {n} K L
hn : n ≠ 2
hirr : Irreducible (cyclotomic n K)
this : NeZero ↑n
h1 : n = 1
⊢ (Algebra.norm K) ζ = 1
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_2
n : ℕ
inst✝³ : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
⊢ ⊤ ≤ adjoin A {b | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}

Tactic:
← ((iff_adjoin_eq_top S A B).1 H).2

Goal state after:
case refine_2
n : ℕ
inst✝³ : NeZero n
S : Set ℕ
A : Type u
B : Type v
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
h : ∃ s ∈ S, s ≠ 0 ∧ n ∣ s
H : IsCyclotomicExtension S A B
⊢ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} ≤ adjoin A {b | ∃ n_1 ∈ S ∪ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case convert_6
n : ℕ
q : ℝ
hn' : 2 ≤ n
hq' : 1 < q
hn : 0 < n
hq : 0 < q
hfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖
ζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)
hζ : IsPrimitiveRoot ζ n
hex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖
this : ¬eval (↑q) (cyclotomic n ℂ) = 0
⊢ ∃ a, ∃ (_ : a ∈ primitiveRoots n ℂ), True ∧ ↑(q - 1).toNNReal < ‖↑q - a‖

Tactic:
simpa [hq'.le, Real.coe_toNNReal', max_eq_left, sub_nonneg] using hex

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: q : ℝ
hq' : 1 < q
⊢ (q - 1) ^ φ 1 ≤ eval q (cyclotomic 1 ℝ)

Tactic:
by simp only [totient_one, pow_one, cyclotomic_one, eval_sub, eval_X, eval_one, le_refl]

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

Goal state before:
pp: p : ℕ
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hpri : Fact (Nat.Prime p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (p ^ (k + 1)) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hirr₁ : Irreducible (cyclotomic (p ^ (k - s + 1)) K)
η : L := ζ ^ p ^ s - 1
η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
this : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) (p ^ (k - s + 1))
⊢ (Algebra.norm K) η = ↑p ^ p ^ s

Tactic:
have := IsCyclotomicExtension.finiteDimensional {p ^ (k + 1)} K L

Goal state after:
p : ℕ
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hpri : Fact (Nat.Prime p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (p ^ (k + 1)) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hirr₁ : Irreducible (cyclotomic (p ^ (k - s + 1)) K)
η : L := ζ ^ p ^ s - 1
η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
this✝ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) (p ^ (k - s + 1))
this : FiniteDimensional K L
⊢ (Algebra.norm K) η = ↑p ^ p ^ s
Module: Mathlib.FieldTheory.Finite.Trace

Goal state before:
pp: F : Type u_1
inst✝² : Field F
inst✝¹ : Finite F
inst✝ : Algebra (ZMod (ringChar F)) F
a : F
ha : a ≠ 0
this : Fact (Nat.Prime (ringChar F))
htr : (∀ (y : F), ((Algebra.traceForm (ZMod (ringChar F)) F) a) y = 0) → a = 0
⊢ ∃ b, (Algebra.trace (ZMod (ringChar F)) F) (a * b) ≠ 0

Tactic:
Algebra.traceForm_apply

Goal state after:
F : Type u_1
inst✝² : Field F
inst✝¹ : Finite F
inst✝ : Algebra (ZMod (ringChar F)) F
a : F
ha : a ≠ 0
this : Fact (Nat.Prime (ringChar F))
htr : (∀ (y : F), (Algebra.trace (ZMod (ringChar F)) F) (a * y) = 0) → a = 0
⊢ ∃ b, (Algebra.trace (ZMod (ringChar F)) F) (a * b) ≠ 0
Module: Mathlib.NumberTheory.LegendreSymbol.AddCharacter

Goal state before:
pp: C : Type v
inst✝ : CommMonoid C
n : ℕ
ψ : AddChar (ZMod n) C
hψ : ∀ (a : ZMod n), ψ a = 1 → a = 0
⊢ ψ.IsPrimitive

Tactic:
by
  intro a ha hf
  have h : mulShift ψ a 1 = (1 : AddChar (ZMod n) C) (1 : ZMod n) :=
    congr_fun (congr_arg (↑) hf) 1
  rw [mulShift_apply, mul_one] at h; norm_cast at h
  exact ha (hψ a h)

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.AddCharacter

Goal state before:
pp: case pos
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : Fintype R
inst✝² : DecidableEq R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
b : R
hψ : ψ.IsPrimitive
h : b = 0
⊢ ∑ x, ψ (x * b) = ↑(Fintype.card R)

pp: case neg
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : Fintype R
inst✝² : DecidableEq R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
b : R
hψ : ψ.IsPrimitive
h : ¬b = 0
⊢ ∑ x, ψ (x * b) = ↑0

Tactic:
· -- case `b = 0`
    simp only [h, mul_zero, map_zero_eq_one, Finset.sum_const, Nat.smul_one_eq_cast]
    rfl

Goal state after:
case neg
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : Fintype R
inst✝² : DecidableEq R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
b : R
hψ : ψ.IsPrimitive
h : ¬b = 0
⊢ ∑ x, ψ (x * b) = ↑0
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_4
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 : IsDomain A
inst✝¹ : IsFractionRing A K
inst✝ : NeZero ↑n
x : CyclotomicField n K
hx : x ∈ adjoin A {b | b ^ n = 1}
y z : CyclotomicField n K
x✝¹ : y ∈ adjoin A {b | b ^ n = 1}
x✝ : z ∈ adjoin A {b | b ^ n = 1}
hy : ⟨y, x✝¹⟩ ∈ adjoin A {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}
hz : ⟨z, x✝⟩ ∈ adjoin A {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}
⊢ ⟨y * z, ⋯⟩ ∈ adjoin A {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}

Tactic:
· exact Subalgebra.mul_mem _ hy hz

Goal state after:
Module: Mathlib.Data.Int.Range

Goal state before:
pp: m n r : ℤ
⊢ (∃ a, m + ↑a < n ∧ m + ↑a = r) ↔ m ≤ r ∧ r < n

Tactic:
exact ⟨fun ⟨a, ha⟩ => ha.2 ▸ ⟨le_add_of_nonneg_right (Int.natCast_nonneg _), ha.1⟩,
    fun h => ⟨toNat (r - m), by simp [toNat_of_nonneg (sub_nonneg.2 h.1), h.2] ⟩⟩

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.ZModChar

Goal state before:
pp: a : ℤ
⊢ χ₈' ↑a = χ₄ (↑a).cast * χ₈ ↑a

Tactic:
exact χ₈'_eq_χ₄_mul_χ₈ a

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'
f : Rˣ →* R'ˣ
⊢ ∀ (x y : R),
    (if hx : IsUnit (x * y) then ↑(f hx.unit) else 0) =
      (if hx : IsUnit x then ↑(f hx.unit) else 0) * if hx : IsUnit y then ↑(f hx.unit) else 0

Tactic:
intro x y
      by_cases hx : IsUnit x
      · simp only [hx, IsUnit.mul_iff, true_and, dif_pos]
        by_cases hy : IsUnit y
        · simp only [hy, dif_pos]
          have hm : (hx.mul hy).unit = hx.unit * hy.unit := Units.ext rfl
          rw [hm, map_mul]
          norm_cast
        · simp only [hy, not_false_iff, dif_neg, mul_zero]
      · simp only [hx, IsUnit.mul_iff, false_and, not_false_iff, dif_neg, zero_mul]

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

Goal state before:
pp: M : Type u_1
inst✝³ : CommMonoid M
inst✝² : Fintype M
inst✝¹ : DecidableEq M
R : Type u_2
inst✝ : CommMonoidWithZero R
ζ : Rˣ
hζ : ζ ∈ rootsOfUnity (Fintype.card Mˣ) R
g : Mˣ
hg : ∀ (x : Mˣ), x ∈ Subgroup.zpowers g
⊢ MulChar M R

Tactic:
by
  have : orderOf ζ ∣ Fintype.card Mˣ :=
    orderOf_dvd_iff_pow_eq_one.mpr <| (mem_rootsOfUnity _ ζ).mp hζ
  refine ofUnitHom <| monoidHomOfForallMemZpowers hg <| this.trans <| dvd_of_eq ?_
  rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card]

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 χ₃ : MulChar R R'
a : Rˣ
⊢ (χ₁ * χ₂ * χ₃) ↑a = (χ₁ * (χ₂ * χ₃)) ↑a

Tactic:
simp only [mul_assoc, Pi.mul_apply, MulChar.coeToFun_mul]

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
d : ℕ
inst✝ : NeZero n
hd : d ∈ χ.conductorSet
⊢ d ≠ 0

Tactic:
by
    contrapose hd
    exact hd ▸ zero_ne_mem_conductorSet χ

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.unitsMap hd) u = 1
⊢ e.mulShift (↑u - 1) = 1

Tactic:
rsuffices ⟨a, ha⟩ : (d : ℤ) ∣ (u.val.val - 1 : ℤ)

Goal state after:
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.unitsMap hd) u = 1
a : ℤ
ha : ↑(↑u).val - 1 = ↑d * a
⊢ e.mulShift (↑u - 1) = 1

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.unitsMap hd) u = 1
⊢ ↑d ∣ ↑(↑u).val - 1
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
⊢ HasStrictFDerivAt (fun x ↦ ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) x ↔
    HasStrictFDerivAt (fun x ↦ (‖x‖ ^ 2) ^ (p / 2)) ((p / 2 * (‖x‖ ^ 2) ^ (p / 2 - 1)) • 2 • (innerSL ℝ) x) x

Tactic:
simp_rw [← Real.rpow_natCast_mul (norm_nonneg _), ← Nat.cast_smul_eq_nsmul ℝ, smul_smul]

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
⊢ HasStrictFDerivAt (fun x ↦ ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) x ↔
    HasStrictFDerivAt (fun x ↦ ‖x‖ ^ (↑2 * (p / 2))) ((p / 2 * ‖x‖ ^ (↑2 * (p / 2 - 1)) * ↑2) • (innerSL ℝ) x) x
Module: Mathlib.Data.Finset.Grade

Goal state before:
pp: α : Type u_1
s t : Multiset α
h : s ⋖ t
⊢ s.card ⋖ t.card

Tactic:
obtain ⟨a, rfl⟩ := h.exists_multiset_cons; rw [card_cons]; exact covBy_succ _

Goal state after:
Module: Mathlib.Analysis.Hofer

Goal state before:
pp: X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this✝ : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n ↦ Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
v : ℕ → ℝ := fun n ↦ (ϕ ∘ u) (n + 1)
this : Tendsto v atTop atTop
⊢ Tendsto (ϕ ∘ u) atTop atTop

Tactic:
tendsto_add_atTop_iff_nat

Goal state after:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this✝ : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n ↦ Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
y : X
limy : Tendsto u atTop (𝓝 y)
v : ℕ → ℝ := fun n ↦ (ϕ ∘ u) (n + 1)
this : Tendsto (ϕ ∘ u) atTop atTop
⊢ Tendsto (ϕ ∘ u) atTop atTop
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case h₂
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
p : ℝ
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
hp : (↑#ι - 1) * p ≤ 1
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
s : Finset ι
i : ι
hi : i ∉ s
⊢ 1 ≤ #sᶜ

Tactic:
have hi' : sᶜ.Nonempty := ⟨i, by rwa [Finset.mem_compl]⟩

Goal state after:
case h₂
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
p : ℝ
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
hp : (↑#ι - 1) * p ≤ 1
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
s : Finset ι
i : ι
hi : i ∉ s
hi' : sᶜ.Nonempty
⊢ 1 ≤ #sᶜ
Module: Mathlib.Analysis.InnerProductSpace.JointEigenspace

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
h₁ : i.1 ≠ j.1
⊢ v ∈ (eigenspace A j.2 ⊓ eigenspace B j.1)ᗮ

Tactic:
intro w ⟨hw1, hw2⟩

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
h₁ : i.1 ≠ j.1
w : E
hw1 : w ∈ ↑(eigenspace A j.2)
hw2 : w ∈ ↑(eigenspace B j.1)
⊢ inner 𝕜 w v = 0
Module: Mathlib.Analysis.InnerProductSpace.JointEigenspace

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
h₂ : i.2 ≠ j.2
⊢ v ∈ (eigenspace A j.2 ⊓ eigenspace B j.1)ᗮ

Tactic:
intro w ⟨hw1, hw2⟩

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
h₂ : i.2 ≠ j.2
w : E
hw1 : w ∈ ↑(eigenspace A j.2)
hw2 : w ∈ ↑(eigenspace B j.1)
⊢ inner 𝕜 w v = 0
Module: Mathlib.Analysis.InnerProductSpace.JointEigenspace

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
h₁ : i.1 ≠ j.1
w : E
hw1 : w ∈ ↑(eigenspace A j.2)
hw2 : w ∈ ↑(eigenspace B j.1)
⊢ inner 𝕜 w v = 0

Tactic:
exact hB.orthogonalFamily_eigenspaces.pairwise h₁ hv2 w hw2

Goal state after:
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
x : E
⊢ (((rankOne 𝕜) x) x).IsPositive

Tactic:
ContinuousLinearMap.isPositive_iff,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ (↑(((rankOne 𝕜) x) x)).IsSymmetric ∧ ∀ (x_1 : E), 0 ≤ ⟪(((rankOne 𝕜) x) x) x_1, x_1⟫
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace F
f : E →ₗ.[R] F
hf : f.toFun.ker = ⊥
⊢ f.inverse.IsClosed ↔ f.IsClosed

Tactic:
rw [IsClosed, inverse_graph hf]
  exact (ContinuousLinearEquiv.prodComm R E F).isClosed_image

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

Goal state before:
pp: 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
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝
hp : (↑#ι).HolderConjugate p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this :
  ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑(c ^ p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p
⊢ ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * (↑c * ∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p

Tactic:
rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ← mul_assoc, ← ENNReal.coe_rpow_of_ne_zero hc.ne']

Goal state after:
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
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝
hp : (↑#ι).HolderConjugate p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this :
  ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑(c ^ p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p
⊢ ↑c * ∫⁻ (x : E), ‖u x‖ₑ ^ p ∂Measure.map (⇑e.symm) volume ≤
    ↑C * ↑(c ^ p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂Measure.map (⇑e.symm) volume) ^ p
Module: Mathlib.Dynamics.BirkhoffSum.Average

Goal state before:
pp: R : Type u_1
α : Type u_2
M : Type u_3
inst✝² : DivisionSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : α → α
g : α → M
h : g ∘ f = g
n : ℕ
hn : ↑n ≠ 0
x : α
this : (↑n)⁻¹ • n • g x = g x
⊢ birkhoffAverage R f g n x = g x

Tactic:
by simpa [birkhoffAverage, birkhoffSum_of_comp_eq h]

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✝² : DivisionSemiring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : α → α
g : α → M
⊢ birkhoffAverage R f (-g) = -birkhoffAverage R f g

Tactic:
by
  funext _ x
  simp [birkhoffAverage, birkhoffSum]

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
⊢ _root_.IsClosed
    (⇑(WithLp.linearEquiv 2 𝕜 (F × E)).symm ⁻¹'
      ↑(Submodule.map (↑((LinearEquiv.skewSwap 𝕜 F E).symm.trans (WithLp.linearEquiv 2 𝕜 (F × E)).symm)) T.graph)ᗮ)

Tactic:
exact (Submodule.isClosed_orthogonal _).preimage (WithLp.prod_continuous_toLp _ _ _)

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'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') / (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p)

Tactic:
ENNReal.div_le_iff h5u h6u

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
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'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)