mathlib-initiative/mathlib-tactics huggingface dataset:
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 ∣ (∑ i ∈ range (p ^ multiplicity p (n + 1)), x ^ i * y ^ (p ^ multiplicity p (n + 1) - 1 - i)) * (x - y)

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 ∣ (∑ i ∈ range (p ^ multiplicity p (n + 1)), x ^ i * y ^ (p ^ multiplicity p (n + 1) - 1 - i)) * (x - y)
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.NumberTheory.ModularForms.LFunction

Goal state before:
pp: 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)
n : ℕ
p : ℂ
⊢ ↑2 ^ s * ↑↑n ^ s / ↑(h Γ) ^ s = ↑2 ^ s * ↑↑n ^ s / ↑(h Γ) ^ s

Tactic:
]

Goal state after:
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
c : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * (p * c)
⊢ n + 1 = p ^ multiplicity p (n + 1) * (p * c)

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
c : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * (p * c)
⊢ n + 1 = p ^ multiplicity p (n + 1) * (p * c)
Module: Mathlib.NumberTheory.Multiplicity

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

Tactic:
]

Goal state after:
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - -y
hx : ¬↑p ∣ x
n : ℕ
hn : Odd n
⊢ emultiplicity (↑p) (x ^ n + y ^ n) = emultiplicity (↑p) (x + y) + emultiplicity p n
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)
n : ℕ
p : ℂ
this : (2 * ↑n / ↑(h Γ)) ^ s = 2 ^ s * ↑n ^ s / ↑(h Γ) ^ s
⊢ p / ↑n ^ s = (↑π ^ s)⁻¹ * Gamma s * p / (2 ^ s * ↑n ^ s / ↑(h Γ) ^ s) * (2 ^ s * ↑π ^ s / Gamma s * (↑(h Γ) ^ s)⁻¹)

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)
n : ℕ
p : ℂ
this : (2 * ↑n / ↑(h Γ)) ^ s = 2 ^ s * ↑n ^ s / ↑(h Γ) ^ s
⊢ p / ↑n ^ s = (↑π ^ s)⁻¹ * Gamma s * p / (2 ^ s * ↑n ^ s / ↑(h Γ) ^ s) * (2 ^ s * ↑π ^ s / Gamma s * (↑(h Γ) ^ s)⁻¹)
Module: Mathlib.NumberTheory.Multiplicity

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

Tactic:
]

Goal state after:
p : ℕ
hp : Nat.Prime p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - -y
hx : ¬↑p ∣ x
n : ℕ
hn : Odd n
⊢ emultiplicity (↑p) (x ^ n - (-y) ^ n) = emultiplicity (↑p) (x - -y) + emultiplicity p n
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

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.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

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.RingTheory.DedekindDomain.FiniteAdeleRing

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
k : K
n d : R
hd : d ∈ nonZeroDivisors R
hd' : d ≠ 0
this : {v | (valuation K v) ((algebraMap R K) d) < 1}.Finite
v : HeightOneSpectrum R
hv : v ∈ Support R k
hk : (valuation K v) k * v.intValuation d = v.intValuation n
⊢ v.intValuation d < 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
k : K
n d : R
hd : d ∈ nonZeroDivisors R
hd' : d ≠ 0
this : {v | (valuation K v) ((algebraMap R K) d) < 1}.Finite
v : HeightOneSpectrum R
hv : v ∈ Support R k
hk : (valuation K v) k * v.intValuation d = v.intValuation n
⊢ v.intValuation d < 1
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

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.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

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.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

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.RingTheory.DedekindDomain.FiniteAdeleRing

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
k : K
n d : R
hd : d ∈ nonZeroDivisors R
hd' : d ≠ 0
this✝ : {v | (valuation K v) ((algebraMap R K) d) < 1}.Finite
v : HeightOneSpectrum R
hv : v ∈ Support R k
hk : (valuation K v) k * v.intValuation d = v.intValuation n
this : 1 ≤ v.intValuation d
⊢ 1 < v.intValuation d * (valuation K v) k

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
k : K
n d : R
hd : d ∈ nonZeroDivisors R
hd' : d ≠ 0
this✝ : {v | (valuation K v) ((algebraMap R K) d) < 1}.Finite
v : HeightOneSpectrum R
hv : v ∈ Support R k
hk : (valuation K v) k * v.intValuation d = v.intValuation n
this : 1 ≤ v.intValuation d
⊢ 1 < v.intValuation d * (valuation K v) k
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x + y
hx : ¬p ∣ x
n : ℕ
hn : Odd n
⊢ emultiplicity ↑p ↑(x ^ n + y ^ n) = emultiplicity p (x + y) + emultiplicity p n

Tactic:
]

Goal state after:
p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x + y
hx : ¬p ∣ x
n : ℕ
hn : Odd n
⊢ emultiplicity ↑p ↑(x ^ n + y ^ n) = emultiplicity p (x + y) + emultiplicity p n
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x + y
hx : ¬p ∣ x
n : ℕ
hn : Odd n
⊢ emultiplicity ↑p ↑(x ^ n + y ^ n) = emultiplicity ↑p ↑(x + y) + emultiplicity p n

Tactic:
]

Goal state after:
p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : p ∣ x + y
hx : ¬p ∣ x
n : ℕ
hn : Odd n
⊢ emultiplicity ↑p ↑(x ^ n + y ^ n) = emultiplicity ↑p ↑(x + y) + emultiplicity p n
Module: Mathlib.NumberTheory.ModularForms.LFunction

Goal state before:
pp: case refine_1
Γ : Subgroup (GL (Fin 2) ℝ)
inst✝² : Γ.IsArithmetic
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
f : F
s : ℂ
inst✝ : CuspFormClass F Γ k
hk : 0 < k
hs : ↑k / 2 + 1 < s.re
⊢ Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s

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

Tactic:
· simp [Λ_eq_mellin, (CuspFormClass.zero_at_infty f).valueAtInfty_eq_zero]

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

Goal state before:
pp: p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : ↑p ∣ ↑(x + y)
hx : ¬↑p ∣ ↑x
n : ℕ
hn : Odd n
⊢ emultiplicity ↑p ↑(x ^ n + y ^ n) = emultiplicity ↑p ↑(x + y) + emultiplicity p n

Tactic:
]

Goal state after:
p : ℕ
hp : Prime p
hp1 : Odd p
x y : ℕ
hxy : ↑p ∣ ↑(x + y)
hx : ¬↑p ∣ ↑x
n : ℕ
hn : Odd n
⊢ emultiplicity ↑p ↑(x ^ n + y ^ n) = emultiplicity ↑p ↑(x + y) + emultiplicity p n
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i ∈ range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
this : x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)
⊢ (∏ i ∈ range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x ^ 2 ^ d + y ^ 2 ^ d) * (x - y) =
    (∏ x_1 ∈ range d, (x ^ 2 ^ x_1 + y ^ 2 ^ x_1)) * (x ^ 2 ^ d + y ^ 2 ^ d) * (x - y)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
a : 𝔸ᶠ[R, K]
⊢ ((∀ (i : HeightOneSpectrum R), IsUnit (a i)) ∧
      ∀ᶠ (i : HeightOneSpectrum R) in Filter.cofinite, ∃ (h : a i ∈ adicCompletionIntegers K i), IsUnit ⟨a i, h⟩) ↔
    (∀ (v : HeightOneSpectrum R), a v ≠ 0) ∧ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, Valued.v (a v) = 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
a : 𝔸ᶠ[R, K]
⊢ ((∀ (i : HeightOneSpectrum R), IsUnit (a i)) ∧
      ∀ᶠ (i : HeightOneSpectrum R) in Filter.cofinite, ∃ (h : a i ∈ adicCompletionIntegers K i), IsUnit ⟨a i, h⟩) ↔
    (∀ (v : HeightOneSpectrum R), a v ≠ 0) ∧ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, Valued.v (a v) = 1
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i ∈ range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
⊢ x ^ 2 ^ (d + 1) - y ^ 2 ^ (d + 1) = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)

Tactic:
]

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
x y : R
d : ℕ
hd : x ^ 2 ^ d - y ^ 2 ^ d = (∏ i ∈ range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)
⊢ x ^ 2 ^ (d + 1) - y ^ 2 ^ (d + 1) = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)
Module: Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
a : 𝔸ᶠ[R, K]
ha₀ : ∀ (v : HeightOneSpectrum R), a v ≠ 0
ha : {v | Valued.v (a v) ≠ 1}.Finite
⊢ (∀ (v : HeightOneSpectrum R), a v ≠ 0) ∧ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, Valued.v (a v) = 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
a : 𝔸ᶠ[R, K]
ha₀ : ∀ (v : HeightOneSpectrum R), a v ≠ 0
ha : {v | Valued.v (a v) ≠ 1}.Finite
⊢ (∀ (v : HeightOneSpectrum R), a v ≠ 0) ∧ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, Valued.v (a v) = 1
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: w✝ : ℤ
⊢ 1 % 4 = 1

Tactic:
]

Goal state after:
w✝ : ℤ
⊢ 1 % 4 = 1
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
S : Set ι
hS : S ∈ cofinite
p : ι → Prop
⊢ IsOpen[topologicalSpace R A (𝓟 S)] {f | ∀ (i : ι), p i → ↑f i ∈ A i}

Tactic:
]

Goal state after:
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
S : Set ι
hS : S ∈ cofinite
p : ι → Prop
⊢ IsOpen[topologicalSpace R A (𝓟 S)] {f | ∀ (i : ι), p i → ↑f i ∈ A i}
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
⊢ ¬2 ∣ x

Tactic:
]

Goal state after:
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
⊢ ¬2 ∣ x
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case refine_1
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ Even (x ^ 2 ^ i + y ^ 2 ^ i)

Tactic:
]

Goal state after:
case refine_1
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
⊢ Even (x ^ 2 ^ i + y ^ 2 ^ i)
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
i : ℕ
this : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
⊢ ¬(1 + 1) % 4 = 0

Tactic:
]

Goal state after:
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
i : ℕ
this : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
⊢ ¬(1 + 1) % 4 = 0
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case refine_2.succ
x✝ y : ℤ
hx✝ : ¬2 ∣ x✝
hxy : 4 ∣ x✝ - y
hx_odd : Odd x✝
hxy_even : Even (x✝ - y)
hy_odd : Odd y
i : ℕ
x : ℤ
hx : Odd x
⊢ (x ^ 2 ^ i) ^ 2 % 4 = 1

Tactic:
Int.sq_mod_four_eq_one_of_odd hx.pow

Goal state after:
case refine_2.succ
x✝ y : ℤ
hx✝ : ¬2 ∣ x✝
hxy : 4 ∣ x✝ - y
hx_odd : Odd x✝
hxy_even : Even (x✝ - y)
hy_odd : Odd y
i : ℕ
x : ℤ
hx : Odd x
⊢ 1 = 1
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case refine_2.succ
x✝ y : ℤ
hx✝ : ¬2 ∣ x✝
hxy : 4 ∣ x✝ - y
hx_odd : Odd x✝
hxy_even : Even (x✝ - y)
hy_odd : Odd y
i : ℕ
x : ℤ
hx : Odd x
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
S : Set ι
hS : cofinite ≤ 𝓟 S
⊢ IsOpen[topologicalSpace R A cofinite] {x | ∀ᶠ (i : ι) in 𝓟 S, x i ∈ A i}

Tactic:
]

Goal state after:
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
S : Set ι
hS : cofinite ≤ 𝓟 S
⊢ IsOpen[topologicalSpace R A cofinite] {x | ∀ᶠ (i : ι) in 𝓟 S, x i ∈ A i}
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
⊢ ¬2 ∣ x

Tactic:
]

Goal state after:
x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
⊢ ¬2 ∣ x
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
⊢ IsOpen[topologicalSpace R A cofinite] {f | ∀ (i : ι), ↑f i ∈ A i}

Tactic:
]

Goal state after:
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
⊢ IsOpen[topologicalSpace R A cofinite] {f | ∀ (i : ι), ↑f i ∈ A i}
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
S : Set ι
hS : cofinite ≤ 𝓟 S
x : Πʳ (i : ι), [R i, A i]_[𝓟 S]
⊢ 𝓝 (inclusion R A hS x) = 𝓝 (inclusion R A hS x)

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝ i] (A i)
x : (i : ι) → ↑(A i)
⊢ 𝓝 (structureMap R A cofinite x) = 𝓝 (structureMap R A cofinite x)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)

pp: case succ.hp
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ Prime 2

pp: case succ.hxy
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)

pp: case succ.hx
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)

pp: case succ.hn
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ ↑k

Tactic:
]

Goal state after:
case succ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1)) = emultiplicity 2 (x - y) + emultiplicity 2 ↑(n + 1)

case succ.hp
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ Prime 2

case succ.hxy
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 (n + 1)

case succ.hx
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)

case succ.hn
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ ¬2 ∣ ↑k
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
hpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
⊢ emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1)) = emultiplicity 2 (x - y) + ↑(multiplicity 2 (n + 1))

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: case succ.hn
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
hpn : 2 ∣ k
⊢ 2 ^ multiplicity 2 (n + 1) * 2 ∣ n + 1

Tactic:
]

Goal state after:
case succ.hn
x y : ℤ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
n : ℕ
h : FiniteMultiplicity 2 (n + 1)
k : ℕ
hk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k
hpn : 2 ∣ k
⊢ 2 ^ multiplicity 2 (n + 1) * 2 ∣ n + 1
Module: Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace

Goal state before:
pp: ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝¹ i] (A i)
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hAcompact : ∀ᶠ (i : ι) in cofinite, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]
S : Set ι := {i | IsCompact (A i) ∧ x i ∈ A i}
hS : cofinite ≤ 𝓟 S
hSx : ∀ i ∈ S, x i ∈ A i
hSA : ∀ i ∈ S, IsCompact (A i)
this : WeaklyLocallyCompactSpace Πʳ (i : ι), [R i, A i]_[𝓟 S]
x' : Πʳ (i : ι), [R i, A i]_[𝓟 S]
hxx' : inclusion R A hS x' = x
⊢ ∃ s, IsCompact s ∧ s ∈ Filter.map (inclusion R A hS) (𝓝 x')

Tactic:
]

Goal state after:
ι : Type u_1
R : ι → Type u_2
A : (i : ι) → Set (R i)
inst✝¹ : (i : ι) → TopologicalSpace (R i)
hAopen : ∀ (i : ι), IsOpen[inst✝¹ i] (A i)
inst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)
hAcompact : ∀ᶠ (i : ι) in cofinite, IsCompact (A i)
x : Πʳ (i : ι), [R i, A i]
S : Set ι := {i | IsCompact (A i) ∧ x i ∈ A i}
hS : cofinite ≤ 𝓟 S
hSx : ∀ i ∈ S, x i ∈ A i
hSA : ∀ i ∈ S, IsCompact (A i)
this : WeaklyLocallyCompactSpace Πʳ (i : ι), [R i, A i]_[𝓟 S]
x' : Πʳ (i : ι), [R i, A i]_[𝓟 S]
hxx' : inclusion R A hS x' = x
⊢ ∃ s, IsCompact s ∧ s ∈ Filter.map (inclusion R A hS) (𝓝 x')
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
n : ℕ
hxy : 2 ∣ x - y
hx : Odd x
hn : Even n
⊢ Odd y

Tactic:
]

Goal state after:
x y : ℤ
n : ℕ
hxy : 2 ∣ x - y
hx : Odd x
hn : Even n
⊢ Odd y
Module: Mathlib.NumberTheory.Multiplicity

Goal state before:
pp: x y : ℤ
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
hy : Odd y
d : ℕ
⊢ (1 - 1) % 4 = 0

pp: x y : ℤ
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
hy : Odd y
d : ℕ
⊢ Odd x

Tactic:
]

Goal state after:
x y : ℤ
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
hy : Odd y
d : ℕ
⊢ (1 - 1) % 4 = 0

x y : ℤ
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
hy : Odd y
d : ℕ
⊢ Odd x