mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case refine_2
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : ↑k + 1 < s.re
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ ↑k

Tactic:
·

Goal state after:
case refine_2
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : ↑k + 1 < s.re
⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ ↑k
Module: Mathlib.NumberTheory.NumberField.Completion.InfinitePlace

Goal state before:
pp: case hp
K : Type u_1
inst✝⁶ : Field K
L : Type u_2
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝³ : Algebra v.Completion w.Completion
inst✝² : IsScalarTower K v.Completion w.Completion
inst✝¹ : ContinuousSMul v.Completion w.Completion
inst✝ : ComplexEmbedding.LiesOver (conjugate w.embedding) v.embedding
⊢ IsClosed
    {x | ((conjugate (extensionEmbedding w)).comp (algebraMap v.Completion w.Completion)) x = (extensionEmbedding v) x}

pp: case ih
K : Type u_1
inst✝⁶ : Field K
L : Type u_2
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝³ : Algebra v.Completion w.Completion
inst✝² : IsScalarTower K v.Completion w.Completion
inst✝¹ : ContinuousSMul v.Completion w.Completion
inst✝ : ComplexEmbedding.LiesOver (conjugate w.embedding) v.embedding
a✝ : WithAbs ↑v
⊢ ((conjugate (extensionEmbedding w)).comp (algebraMap v.Completion w.Completion)) { toCompletion := ↑a✝ } =
    (extensionEmbedding v) { toCompletion := ↑a✝ }

Tactic:
·

Goal state after:
case hp
K : Type u_1
inst✝⁶ : Field K
L : Type u_2
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝³ : Algebra v.Completion w.Completion
inst✝² : IsScalarTower K v.Completion w.Completion
inst✝¹ : ContinuousSMul v.Completion w.Completion
inst✝ : ComplexEmbedding.LiesOver (conjugate w.embedding) v.embedding
⊢ IsClosed
    {x | ((conjugate (extensionEmbedding w)).comp (algebraMap v.Completion w.Completion)) x = (extensionEmbedding v) x}
Module: Mathlib.NumberTheory.NumberField.Completion.InfinitePlace

Goal state before:
pp: case ih
K : Type u_1
inst✝⁶ : Field K
L : Type u_2
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝³ : Algebra v.Completion w.Completion
inst✝² : IsScalarTower K v.Completion w.Completion
inst✝¹ : ContinuousSMul v.Completion w.Completion
inst✝ : ComplexEmbedding.LiesOver (conjugate w.embedding) v.embedding
a✝ : WithAbs ↑v
⊢ ((conjugate (extensionEmbedding w)).comp (algebraMap v.Completion w.Completion)) { toCompletion := ↑a✝ } =
    (extensionEmbedding v) { toCompletion := ↑a✝ }

Tactic:
·

Goal state after:
case ih
K : Type u_1
inst✝⁶ : Field K
L : Type u_2
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
w : InfinitePlace L
v : InfinitePlace K
inst✝³ : Algebra v.Completion w.Completion
inst✝² : IsScalarTower K v.Completion w.Completion
inst✝¹ : ContinuousSMul v.Completion w.Completion
inst✝ : ComplexEmbedding.LiesOver (conjugate w.embedding) v.embedding
a✝ : WithAbs ↑v
⊢ ((conjugate (extensionEmbedding w)).comp (algebraMap v.Completion w.Completion)) { toCompletion := ↑a✝ } =
    (extensionEmbedding v) { toCompletion := ↑a✝ }
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case refine_1
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
⊢ ↑p ^ 1 ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)

pp: case refine_2
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
⊢ ¬↑p ^ (1 + 1) ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
⊢ ↑p ^ 1 ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case e'_5
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : 0 < s.re
hΛ : HasSum (fun n ↦ ↑π ^ (-s) * Gamma s * (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑n / h Γ) ^ s) (Λ hk f s)
⊢ (fun i ↦ (PowerSeries.coeff i) (qExpansion (h Γ) ⇑f) / ↑i ^ s) = fun i ↦
    ↑π ^ (-s) * Gamma s * (PowerSeries.coeff i) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑i / h Γ) ^ s *
      (2 / s.Gammaℂ * ↑(h Γ) ^ (-s))

pp: case e'_6
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : 0 < s.re
hΛ : HasSum (fun n ↦ ↑π ^ (-s) * Gamma s * (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑n / h Γ) ^ s) (Λ hk f s)
⊢ ↑(h Γ) ^ (-s) * L hk f s = Λ hk f s * (2 / s.Gammaℂ * ↑(h Γ) ^ (-s))

Tactic:
·

Goal state after:
case e'_5
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : 0 < s.re
hΛ : HasSum (fun n ↦ ↑π ^ (-s) * Gamma s * (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑n / h Γ) ^ s) (Λ hk f s)
⊢ (fun i ↦ (PowerSeries.coeff i) (qExpansion (h Γ) ⇑f) / ↑i ^ s) = fun i ↦
    ↑π ^ (-s) * Gamma s * (PowerSeries.coeff i) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑i / h Γ) ^ s *
      (2 / s.Gammaℂ * ↑(h Γ) ^ (-s))
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hxy
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ↑p ∣ x ^ p ^ a - y ^ p ^ a

pp: case succ.hx
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ¬↑p ∣ x ^ p ^ a

Tactic:
·

Goal state after:
case succ.hxy
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ↑p ∣ x ^ p ^ a - y ^ p ^ a
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hx
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ¬↑p ∣ x ^ p ^ a

Tactic:
·

Goal state after:
case succ.hx
R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
h_ind : emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a
⊢ ¬↑p ∣ x ^ p ^ a
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
x y : R
p : ℕ
inst✝ : IsDomain R
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
a : ℕ
⊢ emultiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = emultiplicity (↑p) (x - y) + ↑a

Tactic:
by
  induction a with
  | zero => rw [Nat.cast_zero, add_zero, pow_zero, pow_one, pow_one]
  | succ a h_ind =>
    rw [Nat.cast_add, Nat.cast_one, ← add_assoc, ← h_ind, pow_succ, pow_mul, pow_mul]
    apply emultiplicity_pow_prime_sub_pow_prime hp hp1
    · rw [← geom_sum₂_mul]
      exact dvd_mul_of_dvd_right hxy _
    · exact fun h => hx (hp.dvd_of_dvd_pow h)

Goal state after:
Module: Mathlib.Topology.Algebra.RestrictedProduct.Basic

Goal state before:
pp: case inl
ι : Type u_1
S : ι → Type u_3
G : ι → Type u_4
inst✝⁴ : (i : ι) → SetLike (S i) (G i)
A : (i : ι) → S i
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → MulZeroClass (G i)
inst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)
inst✝ : ∀ (i : ι), MulMemClass (S i) (G i)
i : ι
r : G i
x : Πʳ (i : ι), [G i, ↑(A i)]
⊢ (single A i (x i * r)) i = (x * single A i r) i

pp: case inr
ι : Type u_1
S : ι → Type u_3
G : ι → Type u_4
inst✝⁴ : (i : ι) → SetLike (S i) (G i)
A : (i : ι) → S i
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → MulZeroClass (G i)
inst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)
inst✝ : ∀ (i : ι), MulMemClass (S i) (G i)
i : ι
r : G i
x : Πʳ (i : ι), [G i, ↑(A i)]
j : ι
hne : i ≠ j
⊢ (single A i (x i * r)) j = (x * single A i r) j

Tactic:
·

Goal state after:
case inl
ι : Type u_1
S : ι → Type u_3
G : ι → Type u_4
inst✝⁴ : (i : ι) → SetLike (S i) (G i)
A : (i : ι) → S i
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → MulZeroClass (G i)
inst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)
inst✝ : ∀ (i : ι), MulMemClass (S i) (G i)
i : ι
r : G i
x : Πʳ (i : ι), [G i, ↑(A i)]
⊢ (single A i (x i * r)) i = (x * single A i r) i
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case zero
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
⊢ emultiplicity (↑p) (x ^ 0 - y ^ 0) = emultiplicity (↑p) (x - y) + emultiplicity p 0

pp: case succ
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
⊢ emultiplicity (↑p) (x ^ (n + 1) - y ^ (n + 1)) = emultiplicity (↑p) (x - y) + emultiplicity p (n + 1)

Tactic:
·

Goal state after:
case zero
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
⊢ emultiplicity (↑p) (x ^ 0 - y ^ 0) = emultiplicity (↑p) (x - y) + emultiplicity p 0
Module: Mathlib.Topology.Algebra.RestrictedProduct.Basic

Goal state before:
pp: case inl
ι : Type u_1
S : ι → Type u_3
G : ι → Type u_4
inst✝⁴ : (i : ι) → SetLike (S i) (G i)
A : (i : ι) → S i
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → MulZeroClass (G i)
inst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)
inst✝ : ∀ (i : ι), MulMemClass (S i) (G i)
i : ι
r : G i
x : Πʳ (i : ι), [G i, ↑(A i)]
⊢ (single A i (r * x i)) i = (single A i r * x) i

pp: case inr
ι : Type u_1
S : ι → Type u_3
G : ι → Type u_4
inst✝⁴ : (i : ι) → SetLike (S i) (G i)
A : (i : ι) → S i
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → MulZeroClass (G i)
inst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)
inst✝ : ∀ (i : ι), MulMemClass (S i) (G i)
i : ι
r : G i
x : Πʳ (i : ι), [G i, ↑(A i)]
j : ι
hne : i ≠ j
⊢ (single A i (r * x i)) j = (single A i r * x) j

Tactic:
·

Goal state after:
case inl
ι : Type u_1
S : ι → Type u_3
G : ι → Type u_4
inst✝⁴ : (i : ι) → SetLike (S i) (G i)
A : (i : ι) → S i
inst✝³ : DecidableEq ι
inst✝² : (i : ι) → MulZeroClass (G i)
inst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)
inst✝ : ∀ (i : ι), MulMemClass (S i) (G i)
i : ι
r : G i
x : Πʳ (i : ι), [G i, ↑(A i)]
⊢ (single A i (r * x i)) i = (single A i r * x) i
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: case pos
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
i : ι
his : i ∈ S
⊢ IsCompact (A i)

pp: case neg
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
i : ι
his : i ∉ S
⊢ IsCompact (K i)

Tactic:
·

Goal state after:
case pos
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
i : ι
his : i ∈ S
⊢ IsCompact (A i)
Module: Mathlib.Topology.Algebra.RestrictedProduct.Units

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
inst✝² : (i : ι) → Monoid (R i)
S : ι → Type u_3
inst✝¹ : (i : ι) → SetLike (S i) (R i)
inst✝ : ∀ (i : ι), SubmonoidClass (S i) (R i)
B : (i : ι) → S i
𝓕 : Filter ι
x : Πʳ (i : ι), [R i, ↑(B i)]_[𝓕]
hx : ∀ (i : ι), IsUnit (x i)
hxr : ∀ᶠ (i : ι) in 𝓕, ∃ (h : x i ∈ B i), IsUnit ⟨x i, h⟩
⊢ ∀ᶠ (i : ι) in 𝓕, ⋯.unit.inv ∈ ↑(B i)

Tactic:
filter_upwards [hxr] with i ⟨h, hu⟩
    have hu : (hx i).unit.1 * hu.unit.inv = 1 := Subtype.val_inj.2 hu.mul_val_inv
    simp [← Units.eq_inv_of_mul_eq_one_left hu]

Goal state after:
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: case neg
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
i : ι
his : i ∉ S
⊢ IsCompact (K i)

Tactic:
·

Goal state after:
case neg
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
i : ι
his : i ∉ S
⊢ IsCompact (K i)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hxy
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ↑p ∣ x ^ p ^ multiplicity p (n + 1) - y ^ p ^ multiplicity p (n + 1)

pp: case succ.hx
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ x ^ p ^ multiplicity p (n + 1)

pp: case succ.hn
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ ↑k

Tactic:
·

Goal state after:
case succ.hxy
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ↑p ∣ x ^ p ^ multiplicity p (n + 1) - y ^ p ^ multiplicity p (n + 1)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hx
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ x ^ p ^ multiplicity p (n + 1)

pp: case succ.hn
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ ↑k

Tactic:
·

Goal state after:
case succ.hx
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ x ^ p ^ multiplicity p (n + 1)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hn
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ ↑k

Tactic:
·

Goal state after:
case succ.hn
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬↑p ∣ ↑k
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: case pos
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
y : Πʳ (i : ι), [R i, A i]_[𝓟 S]
H : y ∈ DFunLike.coe ⁻¹' U
i : ι
x✝ : i ∈ univ
hi : i ∈ S
⊢ y i ∈ A i

pp: case neg
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
y : Πʳ (i : ι), [R i, A i]_[𝓟 S]
H : y ∈ DFunLike.coe ⁻¹' U
i : ι
x✝ : i ∈ univ
hi : i ∉ S
⊢ y i ∈ K i

Tactic:
·

Goal state after:
case pos
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
y : Πʳ (i : ι), [R i, A i]_[𝓟 S]
H : y ∈ DFunLike.coe ⁻¹' U
i : ι
x✝ : i ∈ univ
hi : i ∈ S
⊢ y i ∈ A i
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: case neg
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
y : Πʳ (i : ι), [R i, A i]_[𝓟 S]
H : y ∈ DFunLike.coe ⁻¹' U
i : ι
x✝ : i ∈ univ
hi : i ∉ S
⊢ y i ∈ K i

Tactic:
·

Goal state after:
case neg
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
y : Πʳ (i : ι), [R i, A i]_[𝓟 S]
H : y ∈ DFunLike.coe ⁻¹' U
i : ι
x✝ : i ∈ univ
hi : i ∉ S
⊢ y i ∈ K i
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case inl
p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x - y
hx : ¬p ∣ x
n : ℕ
hyx : y ≤ x
⊢ emultiplicity p (x ^ n - y ^ n) = emultiplicity p (x - y) + emultiplicity p n

pp: case inr
p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x - y
hx : ¬p ∣ x
n : ℕ
hyx : x ≤ y
⊢ emultiplicity p (x ^ n - y ^ n) = emultiplicity p (x - y) + emultiplicity p n

Tactic:
·

Goal state after:
case inl
p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x - y
hx : ¬p ∣ x
n : ℕ
hyx : y ≤ x
⊢ emultiplicity p (x ^ n - y ^ n) = emultiplicity p (x - y) + emultiplicity p n
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: case refine_1
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
QU : DFunLike.coe ⁻¹' U ⊆ DFunLike.coe ⁻¹' Q
⊢ IsCompact (DFunLike.coe ⁻¹' Q)

pp: case refine_2
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
QU : DFunLike.coe ⁻¹' U ⊆ DFunLike.coe ⁻¹' Q
⊢ DFunLike.coe ⁻¹' U ∈ 𝓝 x

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
S : Set ι
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hS : Sᶜ.Finite
hAcompact : ∀ i ∈ S, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
K : (i : ι) → Set (R i)
K_compact : ∀ (i : ι), IsCompact (K i)
hK : ∀ (i : ι), K i ∈ 𝓝 (x i)
Q : Set ((i : ι) → R i) := univ.pi fun i ↦ if i ∈ S then A i else K i
Q_def : Q = univ.pi fun i ↦ if i ∈ S then A i else K i
Q_compact : IsCompact Q
U : Set ((i : ι) → R i) := Sᶜ.pi K
U_nhds : U ∈ 𝓝 ⇑x
QU : DFunLike.coe ⁻¹' U ⊆ DFunLike.coe ⁻¹' Q
⊢ IsCompact (DFunLike.coe ⁻¹' Q)
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case e'_6
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : 0 < s.re
hΛ : HasSum (fun n ↦ ↑π ^ (-s) * Gamma s * (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑n / h Γ) ^ s) (Λ hk f s)
⊢ ↑(h Γ) ^ (-s) * L hk f s = Λ hk f s * (2 / s.Gammaℂ * ↑(h Γ) ^ (-s))

Tactic:
·

Goal state after:
case e'_6
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
hk : 0 < k
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : ModularFormClass F Γ k
hs : 0 < s.re
hΛ : HasSum (fun n ↦ ↑π ^ (-s) * Gamma s * (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f) / ↑(2 * ↑n / h Γ) ^ s) (Λ hk f s)
⊢ ↑(h Γ) ^ (-s) * L hk f s = Λ hk f s * (2 / s.Gammaℂ * ↑(h Γ) ^ (-s))