mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B : ℝ≥0
hB_pos : 0 < B
hB_vol : volume A ≤ ↑B
δ : ℝ
hδ1 : δ > 0
hδ2 : ∀ ⦃a b : V⦄, dist a b < δ → dist (f a) (f b) < ε / ↑B
w : V
hw_bd : 1 / 2 + 1 / (2 * δ) ≤ ‖w‖
hw_ne : w ≠ 0
hw'_nm : ‖i w‖ = 1 / (2 * ‖w‖)
this : ‖1 / 2‖ = 2⁻¹
⊢ ∫ (v : V) in A, ‖f v - f (v + i w)‖ < 2 * ε

Tactic:
have bdA : ∀ v : V, v ∈ A → ‖‖f v - f (v + i w)‖‖ ≤ ε / B := by
    simp_rw [norm_norm]
    simp_rw [dist_eq_norm] at hδ2
    refine fun x _ => (hδ2 ?_).le
    rw [sub_add_cancel_left, norm_neg, hw'_nm, ← div_div, div_lt_iff₀ (norm_pos_iff.mpr hw_ne), ←
      div_lt_iff₀' hδ1, div_div]
    exact (lt_add_of_pos_left _ one_half_pos).trans_le hw_bd

Goal state after:
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B : ℝ≥0
hB_pos : 0 < B
hB_vol : volume A ≤ ↑B
δ : ℝ
hδ1 : δ > 0
hδ2 : ∀ ⦃a b : V⦄, dist a b < δ → dist (f a) (f b) < ε / ↑B
w : V
hw_bd : 1 / 2 + 1 / (2 * δ) ≤ ‖w‖
hw_ne : w ≠ 0
hw'_nm : ‖i w‖ = 1 / (2 * ‖w‖)
this : ‖1 / 2‖ = 2⁻¹
bdA : ∀ v ∈ A, ‖‖f v - f (v + i w)‖‖ ≤ ε / ↑B
⊢ ∫ (v : V) in A, ‖f v - f (v + i w)‖ < 2 * ε
Module: Mathlib.RingTheory.RootsOfUnity.Minpoly

Goal state before:
pp: case neg
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
prod : map (Int.castRingHom (ZMod p)) P * map (Int.castRingHom (ZMod p)) Q ∣ X ^ n - 1
R : (ZMod p)[X]
hR : map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) = map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R
⊢ False

Tactic:
hR,

Goal state after:
case neg
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
R : (ZMod p)[X]
prod : map (Int.castRingHom (ZMod p)) P * (map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R) ∣ X ^ n - 1
hR : map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) = map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) * R
⊢ False
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hpos : n > 0
h :
  ∀ (n : ℕ),
    0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)
⊢ ∀ (n : ℕ), 0 < n → ¬X ^ n - 1 = 0

Tactic:
intro n hn
    apply Monic.ne_zero
    apply monic_X_pow_sub_C _ (ne_of_gt hn)

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

Goal state before:
pp: n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : ∏ i ∈ n.divisors, cyclotomic i K = X ^ n - 1
⊢ False

Tactic:
rw [← Finset.prod_sdiff hni, Finset.prod_pair key.ne, hk, hj] at this

Goal state after:
n : ℕ
K : Type u_2
inst✝¹ : Field K
μ : K
inst✝ : NeZero ↑n
hnpos : 0 < n
hμn : orderOf μ ∣ n
hnμ : orderOf μ ≠ n
ho : 0 < orderOf μ
i : ℕ
hio : i ∣ orderOf μ
key : i < n
key' : i ∣ n
hni : {i, n} ⊆ n.divisors
k : K[X]
hk : cyclotomic i K = (X - C μ) * k
j : K[X]
hj : cyclotomic n K = (X - C μ) * j
this : (∏ x ∈ n.divisors \ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1
⊢ False
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
⊢ -((cyclotomic n R).coeff 0 * 1) = -(cyclotomic n R).coeff 0

Tactic:
mul_one

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
⊢ -(cyclotomic n R).coeff 0 = -(cyclotomic n R).coeff 0
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
hpos : 0 < n
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)
hpow : ↑a ^ n - 1 = 0
⊢ ↑(ZMod.unitOfCoprime a ⋯ ^ n) = 1

Tactic:
simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]

Goal state after:
Module: Mathlib.RingTheory.Adjoin.PowerBasis

Goal state before:
pp: case hsp
K : Type u_1
S : Type u_2
inst✝² : Field K
inst✝¹ : CommRing S
inst✝ : Algebra K S
x : S
hx : IsIntegral K x
hST : Function.Injective ⇑(algebraMap (↥K[x]) S)
hx' : IsIntegral K ⟨x, ⋯⟩
⊢ ⊤ ≤ Submodule.span K (Set.range fun i ↦ ⟨x, ⋯⟩ ^ ↑i)

Tactic:
rintro ⟨y, hy⟩ _

Goal state after:
case hsp
K : Type u_1
S : Type u_2
inst✝² : Field K
inst✝¹ : CommRing S
inst✝ : Algebra K S
x : S
hx : IsIntegral K x
hST : Function.Injective ⇑(algebraMap (↥K[x]) S)
hx' : IsIntegral K ⟨x, ⋯⟩
y : S
hy : y ∈ K[x]
a✝ : ⟨y, hy⟩ ∈ ⊤
⊢ ⟨y, hy⟩ ∈ Submodule.span K (Set.range fun i ↦ ⟨x, ⋯⟩ ^ ↑i)
Module: Mathlib.RingTheory.Adjoin.PowerBasis

Goal state before:
pp: case pos
S : Type u_2
inst✝⁶ : CommRing S
R : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R S
A : Type u_4
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
B : PowerBasis S A
hB : IsIntegral R B.gen
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n : ℕ
i : Fin B.dim
a✝ : Nontrivial S
Q : R[X] := X ^ n %ₘ minpoly R B.gen
this : B.gen ^ n = (aeval B.gen) Q
hQ : ¬Q = 0
hlt : Q.natDegree < B.dim
j : ℕ
hj : j < B.dim
hij : ⟨j, hj⟩ = i
⊢ IsIntegral R ((algebraMap R S) (Q.coeff j) * 1)

Tactic:
mul_one

Goal state after:
case pos
S : Type u_2
inst✝⁶ : CommRing S
R : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R S
A : Type u_4
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
B : PowerBasis S A
hB : IsIntegral R B.gen
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n : ℕ
i : Fin B.dim
a✝ : Nontrivial S
Q : R[X] := X ^ n %ₘ minpoly R B.gen
this : B.gen ^ n = (aeval B.gen) Q
hQ : ¬Q = 0
hlt : Q.natDegree < B.dim
j : ℕ
hj : j < B.dim
hij : ⟨j, hj⟩ = i
⊢ IsIntegral R ((algebraMap R S) (Q.coeff j))
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: case inl
n : ℕ
inst✝³ : NeZero n
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
ζ : K
l : ℕ
hζ : IsPrimitiveRoot ζ l
hl✝ : l ≠ 0
hl : NeZero l
hroot : IsPrimitiveRoot (zeta n ℚ K) n
r : ℕ
hr : GCDMonoid.lcm l n = n * 1
ineq : φ n * φ r ≤ φ (GCDMonoid.lcm l n)
rpos : 0 < r
key : r = 1
⊢ l ∣ 2 * n

Tactic:
mul_one

Goal state after:
case inl
n : ℕ
inst✝³ : NeZero n
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
ζ : K
l : ℕ
hζ : IsPrimitiveRoot ζ l
hl✝ : l ≠ 0
hl : NeZero l
hroot : IsPrimitiveRoot (zeta n ℚ K) n
r : ℕ
hr : GCDMonoid.lcm l n = n
ineq : φ n * φ r ≤ φ (GCDMonoid.lcm l n)
rpos : 0 < r
key : r = 1
⊢ l ∣ 2 * n
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: n : ℕ
inst✝³ : NeZero n
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
hno : Odd n
ζ x : K
hζ : IsPrimitiveRoot ζ n
hx : IsOfFinOrder x
⊢ 2 * n = orderOf (-1 * ζ)

Tactic:
(Commute.all _ _).orderOf_mul_eq_mul_orderOf_of_coprime

Goal state after:
n : ℕ
inst✝³ : NeZero n
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
hno : Odd n
ζ x : K
hζ : IsPrimitiveRoot ζ n
hx : IsOfFinOrder x
⊢ 2 * n = orderOf (-1) * orderOf ζ

n : ℕ
inst✝³ : NeZero n
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
hno : Odd n
ζ x : K
hζ : IsPrimitiveRoot ζ n
hx : IsOfFinOrder x
⊢ (orderOf (-1)).Coprime (orderOf ζ)
Module: Mathlib.NumberTheory.LegendreSymbol.AddCharacter

Goal state before:
pp: C : Type v
inst✝¹ : CommMonoid C
n : ℕ
inst✝ : NeZero n
ψ : AddChar (ZMod n) C
h₁ : ψ 1 = 1
a : ZMod n
⊢ a = ↑a.val * 1

Tactic:
mul_one

Goal state after:
C : Type v
inst✝¹ : CommMonoid C
n : ℕ
inst✝ : NeZero n
ψ : AddChar (ZMod n) C
h₁ : ψ 1 = 1
a : ZMod n
⊢ a = ↑a.val
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
a : ZMod n
ha : a ≠ 0
hf : ψ.mulShift a = 1
h : ψ (a * 1) = 1 1
⊢ False

Tactic:
mul_one

Goal state after:
C : Type v
inst✝ : CommMonoid C
n : ℕ
ψ : AddChar (ZMod n) C
hψ : ∀ (a : ZMod n), ψ a = 1 → a = 0
a : ZMod n
ha : a ≠ 0
hf : ψ.mulShift a = 1
h : ψ a = 1 1
⊢ False
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case right.mem
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 x : AlgebraicClosure K
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : x ^ n = 1
⊢ f x ∈ {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}

Tactic:
use n, hn, h1

Goal state after:
case right
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 x : AlgebraicClosure K
n : ℕ
hn : n ∈ S
h1 : n ≠ 0
h2 : x ^ n = 1
⊢ f x ^ n = 1
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝ : ↥(rootsOfUnity n ℂ)
w : ℂˣ
hw : w ∈ rootsOfUnity n ℂ
⊢ ∃ a, (⇑(rootsOfUnityCircleEquiv n) ∘ ⇑(ZMod.rootsOfUnityAddChar n)) a = ⟨w, hw⟩

Tactic:
by
  obtain ⟨j, hj1, hj2⟩ := (Complex.mem_rootsOfUnity n w).mp hw
  exact ⟨j, by simp [Units.ext_iff, Subtype.ext_iff, ← hj2, ZMod.toCircle_natCast, mul_div_assoc]⟩

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

Goal state before:
pp: n : ℕ
hn : n % 2 = 1
⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n % 4 / 2) * 1

Tactic:
mul_one

Goal state after:
n : ℕ
hn : n % 2 = 1
⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n % 4 / 2)
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
inst✝⁸ : NeZero n
S✝ T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSepClosed K
inst✝ : CharZero K
S : Set ℕ
⊢ IsCyclotomicExtension S K K

Tactic:
IsCyclotomicExtension.eq_self_sdiff_zero

Goal state after:
n : ℕ
inst✝⁸ : NeZero n
S✝ T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra A B
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSepClosed K
inst✝ : CharZero K
S : Set ℕ
⊢ IsCyclotomicExtension (S \ {0}) K K
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: K : Type w
L : Type z
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
n₁ n₂ : ℕ
F₁ F₂ : IntermediateField K L
h₁ : IsCyclotomicExtension {n₁} K ↥F₁
h₂ : IsCyclotomicExtension {n₂} K ↥F₂
inst✝¹ : NeZero n₁
inst✝ : NeZero n₂
this : FiniteDimensional K ↥F₁
⊢ IsCyclotomicExtension {n₁.lcm n₂} K ↥(F₁ ⊔ F₂)

Tactic:
have := IsCyclotomicExtension.lcm_sup n₁ n₂ F₁.toSubalgebra F₂.toSubalgebra

Goal state after:
K : Type w
L : Type z
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
n₁ n₂ : ℕ
F₁ F₂ : IntermediateField K L
h₁ : IsCyclotomicExtension {n₁} K ↥F₁
h₂ : IsCyclotomicExtension {n₂} K ↥F₂
inst✝¹ : NeZero n₁
inst✝ : NeZero n₂
this✝ : FiniteDimensional K ↥F₁
this : IsCyclotomicExtension {n₁.lcm n₂} K ↥(F₁.toSubalgebra ⊔ F₂.toSubalgebra)
⊢ IsCyclotomicExtension {n₁.lcm n₂} K ↥(F₁ ⊔ F₂)
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: case inr
R : Type u
inst✝³ : Field R
inst✝² : Fintype R
R' : Type v
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ ≠ 1
ψ : AddChar R R'
b : R
hb : b ≠ 0
x : R
⊢ χ x * ψ (b * x - b * 1) = χ x * ψ (b * x - b)

Tactic:
mul_one

Goal state after:
case inr
R : Type u
inst✝³ : Field R
inst✝² : Fintype R
R' : Type v
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ ≠ 1
ψ : AddChar R R'
b : R
hb : b ≠ 0
x : R
⊢ χ x * ψ (b * x - b) = χ x * ψ (b * x - b)
Module: Mathlib.Analysis.InnerProductSpace.Coalgebra

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : FiniteDimensional 𝕜 E
A : Type u_3
inst✝³ : Ring A
inst✝² : Module 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
e : E ≃ₗ[𝕜] A
⊢ LinearMap.lTensor E ((innerₛₗ 𝕜) (e.symm 1)) ∘ₗ adjoint (↑e.symm ∘ₗ mul' 𝕜 A ∘ₗ map ↑e ↑e) =
    (TensorProduct.mk 𝕜 E 𝕜).flip 1

Tactic:
rw [← adjoint_toSpanSingleton, ← adjoint_lTensor, ← adjoint_comp, ← toLinearMap_symm_rid,
      ← comm_trans_lid, ← toLinearEquiv_commIsometry, ← toLinearEquiv_lidIsometry,
      ← toLinearEquiv_trans, ← toLinearEquiv_symm, ← adjoint_toLinearMap_eq_symm]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : FiniteDimensional 𝕜 E
A : Type u_3
inst✝³ : Ring A
inst✝² : Module 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
e : E ≃ₗ[𝕜] A
⊢ adjoint ((↑e.symm ∘ₗ mul' 𝕜 A ∘ₗ map ↑e ↑e) ∘ₗ LinearMap.lTensor E (toSpanSingleton 𝕜 E (e.symm 1))) =
    adjoint ↑((commIsometry 𝕜 E 𝕜).trans (lidIsometry 𝕜 E)).toLinearEquiv
Module: Mathlib.Analysis.InnerProductSpace.ExteriorPower

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : LinearOrder ι
b : OrthonormalBasis ι ℝ E
x y : ↥(⋀[ℝ]^n E)
| (innerProductForm
      (∑ i, ((Module.Basis.exteriorPower n b.toBasis).repr x) i • (Module.Basis.exteriorPower n b.toBasis) i))
    y

Tactic:
← (b.toBasis.exteriorPower n).sum_repr y

Goal state after:
n : ℕ
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : LinearOrder ι
b : OrthonormalBasis ι ℝ E
x y : ↥(⋀[ℝ]^n E)
| (innerProductForm
      (∑ i, ((Module.Basis.exteriorPower n b.toBasis).repr x) i • (Module.Basis.exteriorPower n b.toBasis) i))
    (∑ i, ((Module.Basis.exteriorPower n b.toBasis).repr y) i • (Module.Basis.exteriorPower n b.toBasis) i)
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
⊢ C * c ^ p = c * ‖↑e.symm‖₊ ^ p

Tactic:
rw [h0C, inv_mul_cancel_right₀ (NNReal.rpow_pos hc).ne']

Goal state after:
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
⊢ f.graph.topologicalClosure = f.closure.graph

Tactic:
rw [closure_def hf]
  exact hf.choose_spec

Goal state after:
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.closure.inverse.graph = f.inverse.graph.topologicalClosure

Tactic:
rw [inverse_graph hf, inverse_graph hcf, ← hf'.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
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 = ⊥
⊢ Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph.topologicalClosure =
    (Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph).topologicalClosure
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
hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E
hT : (adjoint T ∘ₗ T).IsSymmetric
this : T.singularValues.support.attachFin hS = {i | ↑(hT.eigenvalues ⋯ i) = 0}ᶜ
⊢ finrank 𝕜 E - finrank 𝕜 ↥(End.eigenspace (adjoint T ∘ₗ T) 0) = finrank 𝕜 ↥T.range

Tactic:
Module.End.eigenspace_zero,

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
hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E
hT : (adjoint T ∘ₗ T).IsSymmetric
this : T.singularValues.support.attachFin hS = {i | ↑(hT.eigenvalues ⋯ i) = 0}ᶜ
⊢ finrank 𝕜 E - finrank 𝕜 ↥(adjoint T ∘ₗ T).ker = finrank 𝕜 ↥T.range
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:
Finsupp.mem_support_iff,

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 : T.singularValues a ≠ 0
⊢ T.singularValues b ≠ 0
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

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
x x' : E
y y' : F
⊢ nndist (x ⊗ₜ[𝕜] y) (x' ⊗ₜ[𝕜] y') ≤ ‖x‖₊ * ‖y‖₊ + ‖x'‖₊ * ‖y'‖₊

Tactic:
grw [nndist_eq_nnnorm, nnnorm_sub_le]; simp

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

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
f : E →L[𝕜] E
g : F →L[𝕜] F
n : ℕ
⊢ mapL f g ^ n = mapL (f ^ n) (g ^ n)

Tactic:
simp [← ContinuousLinearMap.coe_inj]

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

Goal state before:
pp: case inr
X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℕ
h✝ : d₁ < d₂
s : Set X
h : μH[↑d₁] s = ⊤
⊢ volume.addHaarScalarFactor μH[↑d₁] • ⊤ = ⊤

Tactic:
ENNReal.smul_top

Goal state after:
case inr
X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℕ
h✝ : d₁ < d₂
s : Set X
h : μH[↑d₁] s = ⊤
⊢ (if volume.addHaarScalarFactor μH[↑d₁] = 0 then 0 else ⊤) = ⊤
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
h : s.Nonempty
⊢ 1 ≤ μH[0] s

Tactic:
by
  rcases h with ⟨x, hx⟩
  calc
    (1 : ℝ≥0∞) = μH[0] ({x} : Set X) := (hausdorffMeasure_zero_singleton x).symm
    _ ≤ μH[0] s := measure_mono (singleton_subset_iff.2 hx)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℝ
hd : 0 ≤ d
r : 𝕜
hr : r ≠ 0
s : Set E
this : ∀ {r : 𝕜} (s : Set E), μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s
⊢ μH[d] s = μH[d] (r⁻¹ • r • s)

Tactic:
rw [inv_smul_smul₀ hr]

Goal state after:
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
⊢ ‖↑(Units.mk0 c hc)⁻¹‖₊ ^ d • μH[d] s = ‖c‖₊⁻¹ ^ d • μH[d] s

Tactic:
Units.val_inv_eq_inv_val,

Goal state after:
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
⊢ ‖(↑(Units.mk0 c hc))⁻¹‖₊ ^ d • μH[d] s = ‖c‖₊⁻¹ ^ d • μH[d] s
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j

Tactic:
LDL.lowerInv,

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j