mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.RCLike.Inner

Goal state before:
pp: case inl
ι : Type u_1
κ : Type u_2
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
f : κ → ι → 𝕜
hf : ∀ (k : κ), f k ≠ 0
hinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0
h✝ : IsEmpty ι
⊢ LinearIndependent 𝕜 f

pp: case inr
ι : Type u_1
κ : Type u_2
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
f : κ → ι → 𝕜
hf : ∀ (k : κ), f k ≠ 0
hinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0
h✝ : Nonempty ι
⊢ LinearIndependent 𝕜 f

Tactic:
· have : IsEmpty κ := ⟨fun k ↦ hf k <| Subsingleton.elim ..⟩
    exact linearIndependent_empty_type

Goal state after:
case inr
ι : Type u_1
κ : Type u_2
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
f : κ → ι → 𝕜
hf : ∀ (k : κ), f k ≠ 0
hinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0
h✝ : Nonempty ι
⊢ LinearIndependent 𝕜 f
Module: Mathlib.Analysis.Fourier.Convolution

Goal state before:
pp: case h
E : Type u_3
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : InnerProductSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace ℂ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace ℂ F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace ℂ F₃
inst✝² : CompleteSpace F₃
inst✝¹ : CompleteSpace F₁
inst✝ : CompleteSpace F₂
B : F₁ →L[ℂ] F₂ →L[ℂ] F₃
f : 𝓢(E, F₁)
g : 𝓢(E, F₂)
x : E
⊢ BddAbove (Set.range fun x ↦ ‖g x‖)

Tactic:
exact ⟨SchwartzMap.seminorm ℝ 0 0 g, fun x ⟨y, hy⟩ ↦ hy ▸ norm_le_seminorm ℝ g y⟩

Goal state after:
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
this✝ : MemLp (fun x ↦ (1 + ‖x‖ ^ 2) ^ (-s / 2)) 2 volume
this : MemLp (fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (-s / 2))) 2 volume
⊢ 𝓕 f = Lp.toTemperedDistribution (MemLp.toLp (fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (-s / 2))) this • u)

Tactic:
MeasureTheory.Lp.toTemperedDistribution_smul_eq

Goal state after:
case h
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
this✝ : MemLp (fun x ↦ (1 + ‖x‖ ^ 2) ^ (-s / 2)) 2 volume
this : MemLp (fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (-s / 2))) 2 volume
⊢ 𝓕 f = (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (-s / 2))) (Lp.toTemperedDistribution u)

case h.hg₁
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
hs : ↑(Module.finrank ℝ E) < 2 * s
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
u : ↥(Lp F 2 volume)
hu : (smulLeftCLM F fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (𝓕 f) = Lp.toTemperedDistribution u
this✝ : MemLp (fun x ↦ (1 + ‖x‖ ^ 2) ^ (-s / 2)) 2 volume
this : MemLp (fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (-s / 2))) 2 volume
⊢ Function.HasTemperateGrowth fun x ↦ ↑((1 + ‖x‖ ^ 2) ^ (-s / 2))
Module: Mathlib.Analysis.Distribution.Sobolev

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
m x : E
⊢ (1 + ‖x‖ ^ 2) * |(1 + ‖x‖ ^ 2) ^ (-(1 / 2))| ^ 2 = 1

Tactic:
rw [Real.rpow_neg (by positivity), sq_abs, inv_pow]

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : InnerProductSpace ℂ F
inst✝ : CompleteSpace F
s : ℝ
f : 𝓢'(E, F)
hf : MemSobolev s 2 f
m x : E
⊢ (1 + ‖x‖ ^ 2) * (((1 + ‖x‖ ^ 2) ^ (1 / 2)) ^ 2)⁻¹ = 1
Module: Mathlib.Analysis.Fourier.PoissonSummation

Goal state before:
pp: f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖
eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight ↑n) = e
⊢ ∫ (x : ℝ) in 0..1, ∑' (n : ℤ), (e * f.comp (ContinuousMap.addRight ↑n)) x =
    ∑' (n : ℤ), ∫ (x : ℝ) in 0..1, (e * f.comp (ContinuousMap.addRight ↑n)) x

Tactic:
by
      refine (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm ?_).symm
      convert! hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1
      exact funext fun n => neK _ _

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x ↦ |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
  ∀ (x : ℝ),
    ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖ ≤ ‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
⊢ (fun x ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[cocompact ℝ] fun x ↦ |x| ^ (-b)

Tactic:
simp_rw [cocompact_eq_atBot_atTop, isBigO_sup] at hf ⊢

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
  ∀ (x : ℝ),
    ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖ ≤ ‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x ↦ |x| ^ (-b)) ∧ ⇑f =O[atTop] fun x ↦ |x| ^ (-b)
⊢ ((fun x ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[atBot] fun x ↦ |x| ^ (-b)) ∧
    (fun x ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[atTop] fun x ↦ |x| ^ (-b)
Module: Mathlib.RingTheory.RootsOfUnity.Complex

Goal state before:
pp: ζ : ℂˣ
n : ℕ
inst✝ : NeZero n
hζ : ζ ∈ rootsOfUnity n ℂ
⊢ (starRingEnd ℂ) ↑ζ * ↑ζ = 1

Tactic:
conj_mul',

Goal state after:
ζ : ℂˣ
n : ℕ
inst✝ : NeZero n
hζ : ζ ∈ rootsOfUnity n ℂ
⊢ ↑‖↑ζ‖ ^ 2 = 1
Module: Mathlib.RingTheory.RootsOfUnity.Minpoly

Goal state before:
pp: 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
⊢ map (Int.castRingHom ℚ) (P * Q) ∣ map (Int.castRingHom ℚ) (X ^ n - 1)

Tactic:
Polynomial.map_mul

Goal state after:
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
⊢ map (Int.castRingHom ℚ) P * map (Int.castRingHom ℚ) Q ∣ map (Int.castRingHom ℚ) (X ^ n - 1)
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: case inr
n : ℕ
R : Type u_1
inst✝ : Ring R
hn : n > 0
⊢ cyclotomic n ℤ ∣ X ^ n - 1

Tactic:
rw [← prod_cyclotomic_eq_X_pow_sub_one hn]

Goal state after:
case inr
n : ℕ
R : Type u_1
inst✝ : Ring R
hn : n > 0
⊢ cyclotomic n ℤ ∣ ∏ i ∈ n.divisors, cyclotomic i ℤ
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : CommRing K
inst✝ : IsDomain K
ζ : K
n : ℕ
hz : IsPrimitiveRoot ζ n
⊢ cyclotomic n K = cyclotomic' n K

Tactic:
induction n using Nat.strong_induction_on generalizing ζ with | _ k hk

Goal state after:
case h
K : Type u_1
inst✝¹ : CommRing K
inst✝ : IsDomain K
k : ℕ
hk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic m K = cyclotomic' m K
ζ : K
hz : IsPrimitiveRoot ζ k
⊢ cyclotomic k K = cyclotomic' k K
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p n : ℕ
hp : Nat.Prime p
m : ℕ
this :
  cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ i ∈ (p ^ (m + 1)).properDivisors, cyclotomic i R = X ^ p ^ (m + 1) - 1
⊢ (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ i ∈ (p ^ (m + 1)).properDivisors, cyclotomic i R = X ^ p ^ (m + 1) - 1 ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1

Tactic:
Nat.prod_properDivisors_prime_pow hp

Goal state after:
R : Type u_1
inst✝ : CommRing R
p n : ℕ
hp : Nat.Prime p
m : ℕ
this :
  cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ i ∈ (p ^ (m + 1)).properDivisors, cyclotomic i R = X ^ p ^ (m + 1) - 1
⊢ (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1 ↔
    (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: R : Type u_4
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : ℕ
hp : Nat.Prime p
η : R
⊢ η ∈ nthRootsFinset p 1 ↔ IsPrimitiveRoot η p ∨ η = 1

Tactic:
classical
  simp [nthRootsFinset_eq_of_prime hp, mem_primitiveRoots hp.pos, or_comm]

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots

Goal state before:
pp: case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
ha : φ a = 1
⊢ a = 1

Tactic:
specialize hg a

Goal state after:
case h
G : Type u_7
G' : Type u_8
inst✝³ : Group G
inst✝² : IsCyclic G
inst✝¹ : Finite G
inst✝ : CommGroup G'
a : G
inst : Fintype G := Fintype.ofFinite G
ζ : G'
hζ : IsPrimitiveRoot ζ (Nat.card G)
g : G
hg✝ : ∀ (x : G), x ∈ Subgroup.zpowers g
hg : a ∈ Subgroup.zpowers g
hζg : orderOf ζ ∣ orderOf g
φ : G →* G' := monoidHomOfForallMemZpowers hg✝ hζg
hφg : IsPrimitiveRoot (φ g) (Nat.card G)
ha : φ a = 1
⊢ a = 1
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Roots

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : CharZero K
p : ℕ
ζ : K
hp : Nat.Prime p
hζ : IsPrimitiveRoot ζ p
α : Fin p → ℚ
this : Fact (Nat.Prime p)
⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j

Tactic:
let P : ℚ[X] := ∑ i, C (α i) * X ^ i.1

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : CharZero K
p : ℕ
ζ : K
hp : Nat.Prime p
hζ : IsPrimitiveRoot ζ p
α : Fin p → ℚ
this : Fact (Nat.Prime p)
P : ℚ[X] := ∑ i, C (α i) * X ^ ↑i
⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j
Module: Mathlib.RingTheory.Polynomial.Cyclotomic.Expand

Goal state before:
pp: R : Type u_1
p m : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Ring R
inst✝ : CharP R p
hm : ¬p ∣ m
a : ℕ
x✝ : 0 < a + 2
hdiv : p ∣ p ^ a.succ * m
⊢ cyclotomic m R ^ ((p ^ a.succ - p ^ (a.succ - 1)) * p) = cyclotomic m R ^ (p * p ^ (a + 1) - p ^ (a + 2 - 1))

pp: R : Type u_1
p m : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Ring R
inst✝ : CharP R p
hm : ¬p ∣ m
a : ℕ
x✝ : 0 < a + 2
hdiv : p ∣ p ^ a.succ * m
⊢ ¬p ∣ m

Tactic:
· simp only [Nat.succ_sub_succ_eq_sub, Nat.sub_zero]
      rw [Nat.mul_sub_right_distrib, mul_comm, pow_succ]

Goal state after:
R : Type u_1
p m : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : Ring R
inst✝ : CharP R p
hm : ¬p ∣ m
a : ℕ
x✝ : 0 < a + 2
hdiv : p ∣ p ^ a.succ * m
⊢ ¬p ∣ m
Module: Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots

Goal state before:
pp: n : ℕ
inst✝⁵ : NeZero n
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : CommRing L
inst✝² : IsDomain L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
hirr : Irreducible (cyclotomic n K)
this : NeZero ↑n
⊢ Module.finrank K L = φ n

Tactic:
((zeta_spec n K L).powerBasis K).finrank,

Goal state after:
n : ℕ
inst✝⁵ : NeZero n
K : Type u
L : Type v
inst✝⁴ : Field K
inst✝³ : CommRing L
inst✝² : IsDomain L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
hirr : Irreducible (cyclotomic n K)
this : NeZero ↑n
⊢ (IsPrimitiveRoot.powerBasis K ⋯).dim = φ n
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_1.inl
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)
n✝ : ℕ
hn' : n✝ ≠ 0
hn : n✝ ∈ S
b : B
hb : IsPrimitiveRoot b n✝
⊢ ∃ r, IsPrimitiveRoot r n✝

Tactic:
refine ⟨algebraMap B C b, ?_⟩

Goal state after:
case refine_1.inl
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)
n✝ : ℕ
hn' : n✝ ≠ 0
hn : n✝ ∈ S
b : B
hb : IsPrimitiveRoot b n✝
⊢ IsPrimitiveRoot ((algebraMap B C) b) n✝
Module: Mathlib.FieldTheory.Finite.GaloisField

Goal state before:
pp: p : ℕ
h_prime : Fact (Nat.Prime p)
n : ℕ
h : n ≠ 0
this✝ : Fintype (GaloisField p n)
g_poly : (ZMod p)[X] := X ^ p ^ n - X
hp : 1 < p
aux : g_poly ≠ 0
nat_degree_eq : g_poly.natDegree = p ^ n
this : g_poly.rootSet (GaloisField p n) = Set.univ
key : Fintype.card (GaloisField p n) = p ^ n
⊢ Module.finrank (ZMod p) (GaloisField p n) = n

Tactic:
rw [@Module.card_eq_pow_finrank (K := ZMod p), ZMod.card] at key

Goal state after:
p : ℕ
h_prime : Fact (Nat.Prime p)
n : ℕ
h : n ≠ 0
this✝ : Fintype (GaloisField p n)
g_poly : (ZMod p)[X] := X ^ p ^ n - X
hp : 1 < p
aux : g_poly ≠ 0
nat_degree_eq : g_poly.natDegree = p ^ n
this : g_poly.rootSet (GaloisField p n) = Set.univ
key : p ^ Module.finrank (ZMod p) (GaloisField p n) = p ^ n
⊢ Module.finrank (ZMod p) (GaloisField p n) = n
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
j : ℤ
⊢ ↑(toAddCircle ↑j).toCircle = cexp (2 * ↑π * I * ↑j / ↑N)

Tactic:
AddCircle.toCircle,

Goal state after:
N : ℕ
inst✝ : NeZero N
j : ℤ
⊢ ↑(⋯.lift (toAddCircle ↑j)) = cexp (2 * ↑π * I * ↑j / ↑N)
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
χ₀ χ₁ : MulChar R R'
h : (fun χ ↦ (↑χ.toMonoidHom).toFun) χ₀ = (fun χ ↦ (↑χ.toMonoidHom).toFun) χ₁
⊢ χ₀ = χ₁

Tactic:
cases χ₀; cases χ₁; congr; apply MonoidHom.ext (fun _ => congr_fun h _)

Goal state after:
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
⊢ ↑(f ⋯.unit) = ↑(f ⋯.unit) * ↑(f ⋯.unit)

Tactic:
have hm : (hx.mul hy).unit = hx.unit * hy.unit := Units.ext rfl

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 ⋯.unit) = ↑(f ⋯.unit) * ↑(f ⋯.unit)
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ✝ : DirichletCharacter R n
χ χ' : DirichletCharacter R 1
⊢ χ = χ'

Tactic:
simp [level_one]

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p : P
t : ℝ
h_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0
⊢ dist p ((AffineMap.lineMap a b) t) ^ 2 = dist p a ^ 2 + t ^ 2 * dist a b ^ 2

Tactic:
have h := dist_sq_lineMap_lineMap_of_inner_eq_zero (t₁ := 1) (t₂ := t) h_inner

Goal state after:
V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b p : P
t : ℝ
h_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0
h : dist ((AffineMap.lineMap a p) 1) ((AffineMap.lineMap a b) t) ^ 2 = 1 ^ 2 * dist a p ^ 2 + t ^ 2 * dist a b ^ 2
⊢ dist p ((AffineMap.lineMap a b) t) ^ 2 = dist p a ^ 2 + t ^ 2 * dist a b ^ 2
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.rTensor E ((innerₛₗ 𝕜) (e.symm 1)) ∘ₗ adjoint (↑e.symm ∘ₗ mul' 𝕜 A ∘ₗ map ↑e ↑e) =
    (TensorProduct.mk 𝕜 𝕜 E) 1

Tactic:
rw [← adjoint_toSpanSingleton, ← adjoint_rTensor, ← adjoint_comp, ← toLinearMap_symm_lid,
      ← toLinearEquiv_lidIsometry, ← toLinearEquiv_symm, ← adjoint_toLinearMap_eq_symm]
    congr 1; ext; simp

Goal state after:
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
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
⊢ w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ → ⟪w, v⟫ = 0

Tactic:
mem_w_orthogonal

Goal state after:
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
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
⊢ ⟪w, v⟫ = 0
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: case pos
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 ≤ f.closure

Tactic:
refine le_of_le_graph ?_
    rw [← hf.graph_closure_eq_closure_graph]
    exact (graph f).le_topologicalClosure

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

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
p : Submodule 𝕜 E
hp : Dense ↑p
x : F
⊢ x ∈ ((↑A).toPMap p)†.domain ↔ x ∈ ((↑(adjoint A)).toPMap ⊤).domain

Tactic:
simp only [LinearMap.toPMap_domain, Submodule.mem_top, iff_true,
      LinearPMap.mem_adjoint_domain_iff]
    exact ((innerSL 𝕜 x).comp <| A.comp <| Submodule.subtypeL _).cont

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'
⊢ ↑C * eLpNorm (fderiv ℝ v) 1 μ = ↑C * ∫⁻ (x : E), ‖fderiv ℝ v x‖ₑ ∂μ

Tactic:
rw [eLpNorm_one_eq_lintegral_enorm]

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)
⊢ ↑C * ↑(NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯) = ↑(C * (↑p * (↑n - 1) / (↑n - ↑p)).toNNReal)

Tactic:
rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff, eq_comm,
        max_eq_left_iff]

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)
⊢ 0 ≤ ↑p * (↑n - 1) / (↑n - ↑p) ∨ ↑C = 0
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
d : ℝ≥0
h : dimH s < ↑d
⊢ μH[↑d] s = 0

Tactic:
rw [dimH_def] at h

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
d : ℝ≥0
h : ⨆ d, ⨆ (_ : μH[↑d] s = ∞), ↑d < ↑d
⊢ μH[↑d] s = 0