mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.TemperedDistribution

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℂ F
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : 𝓢(E, F)
⊢ Δ ((toTemperedDistributionCLM E F volume) f) = (toTemperedDistributionCLM E F volume) (Δ f)

Tactic:
ext u
  simp [SchwartzMap.integral_smul_laplacian_right_eq_left]

Goal state after:
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
E : ι → Type u_4
inst✝³ : Fintype ι
inst✝² : RCLike 𝕜
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (E i)
w : ι → ℝ
f g₁ g₂ : (i : ι) → E i
⊢ ⟪f, g₁ - g₂⟫_[𝕜, w] = ⟪f, g₁⟫_[𝕜, w] - ⟪f, g₂⟫_[𝕜, w]

Tactic:
simp_rw [sub_eq_add_neg, wInner_add_right, wInner_neg_right]

Goal state after:
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
w : ι → ℝ
f g : ι → 𝕜
hw : 0 ≤ w
i : ι
x✝ : i ∈ univ
⊢ ‖w i • ⟪f i, g i⟫_𝕜‖ = w i • ⟪(fun i ↦ ‖f i‖) i, (fun i ↦ ‖g i‖) i⟫_ℝ

Tactic:
by
    simp [Algebra.smul_def, norm_mul, abs_of_nonneg (hw i)]

Goal state after:
Module: Mathlib.Data.PNat.Prime

Goal state before:
pp: m n : ℕ+
⊢ m.gcd n = m ↔ ↑m ∣ ↑n

Tactic:
← Nat.gcd_eq_left_iff_dvd,

Goal state after:
m n : ℕ+
⊢ m.gcd n = m ↔ (↑m).gcd ↑n = ↑m
Module: Mathlib.NumberTheory.ArithmeticFunction.Moebius

Goal state before:
pp: p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hn : IsPrimePow (p ^ k)
hn' : ¬Nat.Prime (p ^ k)
⊢ μ (p ^ k) = 0

Tactic:
rw [moebius_apply_prime_pow hp hk.ne', if_neg]

Goal state after:
case hnc
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hn : IsPrimePow (p ^ k)
hn' : ¬Nat.Prime (p ^ k)
⊢ ¬k = 1
Module: Mathlib.RingTheory.RootsOfUnity.Complex

Goal state before:
pp: q : ℚ
h : Odd q.num
⊢ IsPrimitiveRoot (cexp (↑π * I * ↑q)) (2 * q.den)

Tactic:
convert! isPrimitiveRoot_exp_rat (q / 2) using 1

Goal state after:
case e'_3
q : ℚ
h : Odd q.num
⊢ cexp (↑π * I * ↑q) = cexp (2 * ↑π * I * ↑(q / 2))

case e'_4
q : ℚ
h : Odd q.num
⊢ 2 * q.den = (q / 2).den
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
n : ℕ
ζ : M
hζ : IsPrimitiveRoot ζ n
i : ℕ
hi : i ∈ ↑(range n)
j : ℕ
hj : j ∈ ↑(range n)
e : (fun x ↦ ζ ^ x) i = (fun x ↦ ζ ^ x) j
⊢ i = j

Tactic:
rw [Finset.coe_range, Set.mem_Iio] at hi hj

Goal state after:
M : Type u_1
inst✝ : CommMonoid M
n : ℕ
ζ : M
hζ : IsPrimitiveRoot ζ n
i : ℕ
hi : i < n
j : ℕ
hj : j < n
e : (fun x ↦ ζ ^ x) i = (fun x ↦ ζ ^ x) j
⊢ i = j
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
hv : R + 1 < ‖v‖
⊢ R + ‖i w‖ ≤ ‖v‖

Tactic:
refine le_trans ?_ hv.le

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
v : V
hv : R + 1 < ‖v‖
⊢ R + ‖i w‖ ≤ R + 1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case right
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝³ : CommMonoid M
inst✝² : CommMonoid N
inst✝¹ : DivisionCommMonoid G
k l : ℕ
inst✝ : CommRing R
ζ : Rˣ
h✝ h : IsPrimitiveRoot ζ k
i : ℤ
⊢ (((Int.castAddHom (ZMod k)).liftOfRightInverse ZMod.cast ⋯)
        ⟨{ toFun := fun i ↦ Additive.ofMul ⟨(fun x ↦ ζ ^ x) i, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }, ⋯⟩)
      ((Int.castAddHom (ZMod k)) i) =
    ⟨(fun x ↦ ζ ^ x) i, ⋯⟩

Tactic:
AddMonoidHom.liftOfRightInverse_comp_apply

Goal state after:
case right
M : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝³ : CommMonoid M
inst✝² : CommMonoid N
inst✝¹ : DivisionCommMonoid G
k l : ℕ
inst✝ : CommRing R
ζ : Rˣ
h✝ h : IsPrimitiveRoot ζ k
i : ℤ
⊢ ↑⟨{ toFun := fun i ↦ Additive.ofMul ⟨(fun x ↦ ζ ^ x) i, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }, ⋯⟩ i =
    ⟨(fun x ↦ ζ ^ x) i, ⋯⟩
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
⊢ cyclotomic n R = (X ^ n - 1) /ₘ ∏ i ∈ n.properDivisors, cyclotomic i R

Tactic:
nontriviality R
  rw [← prod_cyclotomic_eq_X_pow_sub_one hpos, ← Nat.cons_self_properDivisors hpos.ne',
    Finset.prod_cons]
  have prod_monic : (∏ i ∈ Nat.properDivisors n, cyclotomic i R).Monic := by
    apply monic_prod_of_monic
    intro i _
    exact cyclotomic.monic i R
  rw [(div_modByMonic_unique (cyclotomic n R) 0 prod_monic _).1]
  simp only [degree_zero, zero_add]
  constructor
  · rw [mul_comm]
  rw [bot_lt_iff_ne_bot]
  intro h
  exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)

Goal state after:
Module: Mathlib.Analysis.Fourier.RiemannLebesgueLemma

Goal state before:
pp: E : Type u_1
V : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℂ E
f : V → E
inst✝⁹ : AddCommGroup V
inst✝⁸ : TopologicalSpace V
inst✝⁷ : IsTopologicalAddGroup V
inst✝⁶ : T2Space V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : Module ℝ V
inst✝² : ContinuousSMul ℝ V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
⊢ Tendsto (fun w ↦ ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (cocompact (StrongDual ℝ V)) (𝓝 0)

Tactic:
let V' := EuclideanSpace ℝ (Fin (finrank ℝ V))

Goal state after:
E : Type u_1
V : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℂ E
f : V → E
inst✝⁹ : AddCommGroup V
inst✝⁸ : TopologicalSpace V
inst✝⁷ : IsTopologicalAddGroup V
inst✝⁶ : T2Space V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : Module ℝ V
inst✝² : ContinuousSMul ℝ V
inst✝¹ : FiniteDimensional ℝ V
μ : Measure V
inst✝ : μ.IsAddHaarMeasure
V' : Type := EuclideanSpace ℝ (Fin (finrank ℝ V))
⊢ Tendsto (fun w ↦ ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (cocompact (StrongDual ℝ V)) (𝓝 0)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
ζ : R
n : ℕ
x y : R
inst✝ : IsDomain R
hpos : 0 < n
h : IsPrimitiveRoot ζ n
K : Type u_1 := ⋯
⊢ (algebraMap R K) (x ^ n - y ^ n) = (algebraMap R K) (∏ ζ ∈ nthRootsFinset n 1, (x - ζ * y))

Tactic:
map_sub,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
ζ : R
n : ℕ
x y : R
inst✝ : IsDomain R
hpos : 0 < n
h : IsPrimitiveRoot ζ n
K : Type u_1 := FractionRing R
⊢ (algebraMap R K) (x ^ n) - (algebraMap R K) (y ^ n) = (algebraMap R K) (∏ ζ ∈ nthRootsFinset n 1, (x - ζ * y))
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
ζ : R
n : ℕ
x y : R
inst✝ : IsDomain R
hpos : 0 < n
h : IsPrimitiveRoot ζ n
K : Type u_1 := FractionRing R
⊢ (algebraMap R K) x ^ n - (algebraMap R K) y ^ n = ∏ x_1 ∈ nthRootsFinset n 1, (algebraMap R K) (x - x_1 * y)

Tactic:
map_sub,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
ζ : R
n : ℕ
x y : R
inst✝ : IsDomain R
hpos : 0 < n
h : IsPrimitiveRoot ζ n
K : Type u_1 := FractionRing R
⊢ (algebraMap R K) x ^ n - (algebraMap R K) y ^ n =
    ∏ x_1 ∈ nthRootsFinset n 1, ((algebraMap R K) x - (algebraMap R K) (x_1 * y))
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
ζ : R
n : ℕ
x y : R
inst✝ : IsDomain R
hpos : 0 < n
h : IsPrimitiveRoot ζ n
K : Type u_1 := FractionRing R
⊢ (algebraMap R K) x ^ n - (algebraMap R K) y ^ n =
    ∏ x_1 ∈ nthRootsFinset n 1, ((algebraMap R K) x - (algebraMap R K) (x_1 * y))

Tactic:
map_mul

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
ζ : R
n : ℕ
x y : R
inst✝ : IsDomain R
hpos : 0 < n
h : IsPrimitiveRoot ζ n
K : Type u_1 := FractionRing R
⊢ (algebraMap R K) x ^ n - (algebraMap R K) y ^ n =
    ∏ x_1 ∈ nthRootsFinset n 1, ((algebraMap R K) x - (algebraMap R K) x_1 * (algebraMap R K) y)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Eval

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝ : Ring R
n : ℕ
h✝ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n
hn' : n > 0
hn : 1 < n
h : eval 1 (cyclotomic n ℤ) = -1
⊢ eval 1 (cyclotomic n R) = 1

pp: R : Type u_1
inst✝ : Ring R
n : ℕ
h : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n
hn' : n > 0
hn : 1 < n
⊢ eval 1 (cyclotomic n ℤ) = 1 ∨ eval 1 (cyclotomic n ℤ) = -1

Tactic:
· exfalso
    linarith [cyclotomic_nonneg n (le_refl (1 : ℤ))]

Goal state after:
R : Type u_1
inst✝ : Ring R
n : ℕ
h : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n
hn' : n > 0
hn : 1 < n
⊢ eval 1 (cyclotomic n ℤ) = 1 ∨ eval 1 (cyclotomic n ℤ) = -1
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_2
S T : Set ℕ
A : Type u
B : Type v
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
C : Type w
inst✝³ : CommRing C
inst✝² : Algebra A C
inst✝¹ : Algebra B C
inst✝ : IsScalarTower A B C
hS : IsCyclotomicExtension S A B
hT : IsCyclotomicExtension T B C
h : Function.Injective ⇑(algebraMap B C)
x : C
b : B
f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C
⊢ (algebraMap B C) b ∈ adjoin A {b | ∃ n ∈ S ∪ T, n ≠ 0 ∧ b ^ n = 1}

Tactic:
have hb : f b ∈ (adjoin A {b : B | ∃ a : ℕ, a ∈ S ∧ a ≠ 0 ∧ b ^ a = 1}).map f :=
      ⟨b, ((isCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩

Goal state after:
case refine_2
S T : Set ℕ
A : Type u
B : Type v
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
C : Type w
inst✝³ : CommRing C
inst✝² : Algebra A C
inst✝¹ : Algebra B C
inst✝ : IsScalarTower A B C
hS : IsCyclotomicExtension S A B
hT : IsCyclotomicExtension T B C
h : Function.Injective ⇑(algebraMap B C)
x : C
b : B
f : B →ₐ[A] C := IsScalarTower.toAlgHom A B C
hb : f b ∈ Subalgebra.map f (adjoin A {b | ∃ a ∈ S, a ≠ 0 ∧ b ^ a = 1})
⊢ (algebraMap B C) b ∈ adjoin A {b | ∃ n ∈ S ∪ T, n ≠ 0 ∧ b ^ n = 1}
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: case f
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)
this✝¹ : NeZero ↑n
E : Type v := AlgebraicClosure L
z : E
hz : (cyclotomic n E).IsRoot z
this✝ : FiniteDimensional K L := finiteDimensional {n} K L
this : IsGalois K L := isGalois {n} K L
x✝ : L →ₐ[K] E
| -x✝ (1 - ζ)

Tactic:
map_sub,

Goal state after:
case f
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)
this✝¹ : NeZero ↑n
E : Type v := AlgebraicClosure L
z : E
hz : (cyclotomic n E).IsRoot z
this✝ : FiniteDimensional K L := finiteDimensional {n} K L
this : IsGalois K L := isGalois {n} K L
x✝ : L →ₐ[K] E
| -(x✝ 1 - x✝ ζ)
Module: Mathlib.Data.ZMod.Units

Goal state before:
pp: ⊢ 0 = 1 * 0

Tactic:
mul_zero

Goal state after:
⊢ 0 = 0
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: case refine_2.e_a
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✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) (p ^ (k - s + 1))
this✝¹ : FiniteDimensional K L
this✝ : IsGalois K L
H : (Algebra.norm K) η₁ = ↑(p ^ (k - s + 1)).minFac
this : p ^ ((k - s).succ - 1) * Module.finrank (↥K⟮η⟯) L = p ^ (k.succ - 1)
Hex : k.succ - 1 = (k - s).succ - 1 + s
⊢ Module.finrank (↥K⟮η⟯) L = p ^ s

Tactic:
rw [Hex, pow_add] at this

Goal state after:
case refine_2.e_a
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✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯
hη : IsPrimitiveRoot (η₁ + 1) (p ^ (k - s + 1))
this✝¹ : FiniteDimensional K L
this✝ : IsGalois K L
H : (Algebra.norm K) η₁ = ↑(p ^ (k - s + 1)).minFac
this : p ^ ((k - s).succ - 1) * Module.finrank (↥K⟮η⟯) L = p ^ ((k - s).succ - 1) * p ^ s
Hex : k.succ - 1 = (k - s).succ - 1 + s
⊢ Module.finrank (↥K⟮η⟯) L = p ^ s
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: 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})
⊢ Normal K L

Tactic:
i.transfer_normal,

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})
⊢ Normal K ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case right.mul
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 y : AlgebraicClosure K
hx : x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
hy : y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihx : f x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihy : f y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
⊢ f (x * y) ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}

Tactic:
map_mul

Goal state after:
case right.mul
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 y : AlgebraicClosure K
hx : x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
hy : y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihx : f x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
ihy : f y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
⊢ f x * f y ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
Module: Mathlib.FieldTheory.Finite.GaloisField

Goal state before:
pp: p✝ : ℕ
inst✝⁴ : Fact (Nat.Prime p✝)
n : ℕ
K : Type u_1
K' : Type u_2
inst✝³ : Field K
inst✝² : Field K'
inst✝¹ : Fintype K
inst✝ : Fintype K'
hKK' : Fintype.card K = Fintype.card K'
p : ℕ
_char_p_K : CharP K p
p' : ℕ
_char_p'_K' : CharP K' p'
⊢ K ≃+* K'

Tactic:
choose n hp hK using FiniteField.card K p

Goal state after:
p✝ : ℕ
inst✝⁴ : Fact (Nat.Prime p✝)
n✝ : ℕ
K : Type u_1
K' : Type u_2
inst✝³ : Field K
inst✝² : Field K'
inst✝¹ : Fintype K
inst✝ : Fintype K'
hKK' : Fintype.card K = Fintype.card K'
p : ℕ
_char_p_K : CharP K p
p' : ℕ
_char_p'_K' : CharP K' p'
n : ℕ+
hp : Nat.Prime p
hK : Fintype.card K = p ^ ↑n
⊢ K ≃+* K'
Module: Mathlib.NumberTheory.LegendreSymbol.ZModChar

Goal state before:
pp: n : ℤ
help : ∀ (m : ℤ), 0 ≤ m → m < 4 → χ₄ ↑m = if m % 2 = 0 then 0 else if m = 1 then 1 else -1
⊢ χ₄ ↑n = if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1

Tactic:
rw [← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 4), ← ZMod.intCast_mod n 4]

Goal state after:
n : ℤ
help : ∀ (m : ℤ), 0 ≤ m → m < 4 → χ₄ ↑m = if m % 2 = 0 then 0 else if m = 1 then 1 else -1
⊢ χ₄ ↑(n % ↑4) = if n % 4 % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
x y : R
hx : IsUnit x
hy : IsUnit y
hm : ⋯.unit = hx.unit * hy.unit
⊢ ↑(f (hx.unit * hy.unit)) = ↑(f ⋯.unit) * ↑(f ⋯.unit)

Tactic:
map_mul

Goal state after:
case pos
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
x y : R
hx : IsUnit x
hy : IsUnit y
hm : ⋯.unit = hx.unit * hy.unit
⊢ ↑(f hx.unit * f hy.unit) = ↑(f ⋯.unit) * ↑(f ⋯.unit)
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:
Subtype.ext_iff,

Goal state after:
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
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:
push_cast at h

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommMonoid R
R' : Type u_2
inst✝³ : CommMonoidWithZero R'
S : Type u_3
inst✝² : SetLike S R
inst✝¹ : SubmonoidClass S R
T : S
R'' : Type u_4
inst✝ : CommMonoidWithZero R''
x : (↥T)ˣ
⊢ (domRestrict T 1) ↑x = 1 ↑x

Tactic:
rw [domRestrict_apply, if_pos x.isUnit, MulChar.one_apply x.isUnit.coe, 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
d : ℕ
inst✝ : NeZero n
this✝ : NeZero d
hd : d ∣ n
χ₀ : DirichletCharacter R d
hχ₀ : χ = (changeLevel hd) χ₀
this : NeZero (d * χ.conductor * n)
⊢ (changeLevel ⋯) χ₀ = (changeLevel ⋯) χ.primitiveCharacter

Tactic:
changeLevel_trans _ hd (n.dvd_mul_left (d * χ.conductor)),

Goal state after:
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
d : ℕ
inst✝ : NeZero n
this✝ : NeZero d
hd : d ∣ n
χ₀ : DirichletCharacter R d
hχ₀ : χ = (changeLevel hd) χ₀
this : NeZero (d * χ.conductor * n)
⊢ (changeLevel ⋯) ((changeLevel hd) χ₀) = (changeLevel ⋯) χ.primitiveCharacter
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝³ : Field R
inst✝² : Fintype R
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
ψ : AddChar R R'
hψ : ψ ≠ 1
⊢ gaussSum 1 ψ = -1

Tactic:
by
  classical
  simp only [gaussSum, ← add_eq_zero_iff_eq_neg]
  calc ∑ a, (1 : MulChar R R') a * ψ a + 1
  _ = ∑ a ∈ {0}ᶜ, (1 : MulChar R R') a * ψ a + 1 := by
    simp [← ({0} : Finset R).sum_compl_add_sum]
  _ = ∑ a ∈ {0}ᶜ, ψ a + ψ 0 := by
    congr! <;> aesop (add simp MulChar.one_apply)
  _ = 0 := by
    rw [← AddChar.sum_eq_zero_of_ne_one hψ, ← Finset.sum_compl_add_sum (s := {0})]
    simp

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

Goal state before:
pp: S : Type u_2
inst✝ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
x : ZMod m
hψ : ψ (-1) = -1
⊢ ψ (-1 * x) = -ψ x

Tactic:
map_mul

Goal state after:
S : Type u_2
inst✝ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
x : ZMod m
hψ : ψ (-1) = -1
⊢ ψ (-1) * ψ x = -ψ x
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: S : Type u_2
inst✝ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
x : ZMod m
hψ : ψ (-1) = 1
⊢ ψ (-1 * x) = ψ x

Tactic:
map_mul

Goal state after:
S : Type u_2
inst✝ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
x : ZMod m
hψ : ψ (-1) = 1
⊢ ψ (-1) * ψ x = ψ x
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: 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'
hψ : ψ.IsPrimitive
⊢ (∑ x, if x = 1 then χ x * ↑(Fintype.card R) else χ x * 0) = ↑(Fintype.card R)

Tactic:
mul_zero

Goal state after:
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'
hψ : ψ.IsPrimitive
⊢ (∑ x, if x = 1 then χ x * ↑(Fintype.card R) else 0) = ↑(Fintype.card R)
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
p q : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
hq : q.IsSymmetricProjection
x✝ : p ≤ q
a : E
ha : a ∈ p.range
h1 : (q - p).IsSymmetric
hh : ∀ {T : E →ₗ[𝕜] E}, T.IsSymmetricProjection → re ⟪T a, a⟫ = ‖T a‖ ^ 2
h2 : 0 ≤ re (⟪q a, a⟫ - ⟪p a, a⟫)
⊢ a ∈ q.range

Tactic:
map_sub,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
p q : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
hq : q.IsSymmetricProjection
x✝ : p ≤ q
a : E
ha : a ∈ p.range
h1 : (q - p).IsSymmetric
hh : ∀ {T : E →ₗ[𝕜] E}, T.IsSymmetricProjection → re ⟪T a, a⟫ = ‖T a‖ ^ 2
h2 : 0 ≤ re ⟪q a, a⟫ - re ⟪p a, a⟫
⊢ a ∈ q.range
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_4
inst✝ : Finite n
M : Matrix n n 𝕜
this : Fintype n
⊢ (∃ m u, toEuclideanLin.symm ↑(∑ i, ((rankOne 𝕜) (u i)) (u i)) = M) ↔ ∃ m v, M = ∑ i, vecMulVec (v i) (star (v i))

Tactic:
ContinuousLinearMap.toLinearMap_sum,

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_4
inst✝ : Finite n
M : Matrix n n 𝕜
this : Fintype n
⊢ (∃ m u, toEuclideanLin.symm (∑ d, ↑(((rankOne 𝕜) (u d)) (u d))) = M) ↔ ∃ m v, M = ∑ i, vecMulVec (v i) (star (v i))