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

Goal state before:
pp: case inr
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j ≤ (↑⌈(↑j + 1) / r⌉ - 1) * r
⊢ ⌊↑(⌈(↑j + 1) / r⌉ - 1) * r⌋ = j

Tactic:
Int.floor_eq_iff,

Goal state after:
case inr
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j ≤ (↑⌈(↑j + 1) / r⌉ - 1) * r
⊢ ↑j ≤ ↑(⌈(↑j + 1) / r⌉ - 1) * r ∧ ↑(⌈(↑j + 1) / r⌉ - 1) * r < ↑j + 1
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case inr
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j ≤ (↑⌈(↑j + 1) / r⌉ - 1) * r
⊢ ↑j ≤ (↑⌈(↑j + 1) / r⌉ - 1) * r ∧ ↑⌈(↑j + 1) / r⌉ < (↑j + 1) / r + 1

Tactic:
]

Goal state after:
case inr
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j ≤ (↑⌈(↑j + 1) / r⌉ - 1) * r
⊢ ↑j ≤ (↑⌈(↑j + 1) / r⌉ - 1) * r ∧ ↑⌈(↑j + 1) / r⌉ < (↑j + 1) / r + 1
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case inr
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j < ↑⌊(↑j + 1) / r⌋ * r
⊢ ↑j + 1 - 1 < ↑⌊(↑j + 1) / r⌋ * r ∧ ↑⌊(↑j + 1) / r⌋ * r ≤ ↑j + 1

Tactic:
]

Goal state after:
case inr
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j < ↑⌊(↑j + 1) / r⌋ * r
⊢ ↑j + 1 - 1 < ↑⌊(↑j + 1) / r⌋ * r ∧ ↑⌊(↑j + 1) / r⌋ * r ≤ ↑j + 1
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case inr.left
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j < ↑⌊(↑j + 1) / r⌋ * r
⊢ ↑j < ↑⌊(↑j + 1) / r⌋ * r

Tactic:
]

Goal state after:
case inr.left
r : ℝ
j : ℤ
h : r > 0
h✝ : ↑j < ↑⌊(↑j + 1) / r⌋ * r
⊢ ↑j < ↑⌊(↑j + 1) / r⌋ * r
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
⊢ {x | ∃ k, beattySeq s k = x} = {x | ∃ k, beattySeq s k = x}

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case a.inl
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
hk : k > 0
hjk : beattySeq r k = j
⊢ 1 ≤ ↑k * r

Tactic:
]

Goal state after:
case a.inl
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
hk : k > 0
hjk : beattySeq r k = j
⊢ 1 ≤ ↑k * r
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬π ∣ q
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
⊢ v ((algebraMap K[X] K⟮X⟯) (q % π) + (algebraMap K[X] K⟮X⟯) (π * (q / π))) = 1

Tactic:
]

Goal state after:
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬π ∣ q
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
⊢ v ((algebraMap K[X] K⟮X⟯) (q % π) + (algebraMap K[X] K⟮X⟯) (π * (q / π))) = 1
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case a.inr
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
hk : k > 0
hjk : beattySeq' s k = j
⊢ 1 < ↑k * s

Tactic:
]

Goal state after:
case a.inr
r s : ℝ
hrs : r.HolderConjugate s
j k : ℤ
hk : k > 0
hjk : beattySeq' s k = j
⊢ 1 < ↑k * s
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
s✝ : ℝ
hs : s✝ ≥ 0
x✝ : j ∈ {x | ∃ k, beattySeq s✝ k = x}
k : ℤ
hj : 1 ≤ ↑k * s✝
hk : beattySeq s✝ k = j
⊢ k > 0

Tactic:
]

Goal state after:
r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
s✝ : ℝ
hs : s✝ ≥ 0
x✝ : j ∈ {x | ∃ k, beattySeq s✝ k = x}
k : ℤ
hj : 1 ≤ ↑k * s✝
hk : beattySeq s✝ k = j
⊢ k > 0
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
d : ℕ
hd : d ∣ s.prodPrimes
⊢ s.nu d = s.nu d

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬q % π = 0
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
⊢ v ((algebraMap K[X] K⟮X⟯) (q % π) + (algebraMap K[X] K⟮X⟯) (π * (q / π))) = 1

Tactic:
]

Goal state after:
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬q % π = 0
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
⊢ v ((algebraMap K[X] K⟮X⟯) (q % π) + (algebraMap K[X] K⟮X⟯) (π * (q / π))) = 1
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case hfg
s : BoundingSieve
d : ℕ
hdP : d ∣ s.prodPrimes
hd_ne_one : d ≠ 1
hd_sq : Squarefree d
this : d ≠ 0
p : ℕ
hpd : Nat.Prime p ∧ p ∣ d
⊢ s.nu p < 1

Tactic:
]

Goal state after:
case hfg
s : BoundingSieve
d : ℕ
hdP : d ∣ s.prodPrimes
hd_ne_one : d ≠ 1
hd_sq : Squarefree d
this : d ≠ 0
p : ℕ
hpd : Nat.Prime p ∧ p ∣ d
⊢ s.nu p < 1
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: case refine_1.calc_2
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
span_d : Submodule.restrictScalars R (S ∙ f A.det) ≤ M
this : Nontrivial (R ⧸ p)
i j : Fin n
⊢ -1 i j = -1 i j

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
d : ℕ
⊢ multSum d = s.nu d * s.totalMass + (multSum d - s.nu d * s.totalMass)

Tactic:
]

Goal state after:
s : BoundingSieve
d : ℕ
⊢ multSum d = s.nu d * s.totalMass + (multSum d - s.nu d * s.totalMass)
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬q % π = 0
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
h₁ : v ((algebraMap K[X] K⟮X⟯) (q % π)) = 1
h₂ : v ((algebraMap K[X] K⟮X⟯) (π * (q / π))) < v ((algebraMap K[X] K⟮X⟯) (q % π))
⊢ v ((algebraMap K[X] K⟮X⟯) (q % π) + (algebraMap K[X] K⟮X⟯) (π * (q / π))) = v ((algebraMap K[X] K⟮X⟯) (q % π))

Tactic:
]

Goal state after:
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬q % π = 0
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
h₁ : v ((algebraMap K[X] K⟮X⟯) (q % π)) = 1
h₂ : v ((algebraMap K[X] K⟮X⟯) (π * (q / π))) < v ((algebraMap K[X] K⟮X⟯) (q % π))
⊢ v ((algebraMap K[X] K⟮X⟯) (q % π) + (algebraMap K[X] K⟮X⟯) (π * (q / π))) = v ((algebraMap K[X] K⟮X⟯) (q % π))
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
⊢ (∑ d ∈ s.support, if s.prodPrimes.Coprime d then s.weights d else 0) =
    ∑ d ∈ s.support, s.weights d * if s.prodPrimes.gcd d = 1 then 1 else 0

Tactic:
]

Goal state after:
s : BoundingSieve
⊢ (∑ d ∈ s.support, if s.prodPrimes.Coprime d then s.weights d else 0) =
    ∑ d ∈ s.support, s.weights d * if s.prodPrimes.gcd d = 1 then 1 else 0
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
hb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}
s✝ : ℝ
hs : s✝ ≥ 0
x✝ : j ∈ {x | ∃ k, beattySeq' s✝ k = x}
k : ℤ
hj : 1 < ↑k * s✝
hk : beattySeq' s✝ k = j
⊢ k > 0

Tactic:
]

Goal state after:
r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
hb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}
s✝ : ℝ
hs : s✝ ≥ 0
x✝ : j ∈ {x | ∃ k, beattySeq' s✝ k = x}
k : ℤ
hj : 1 < ↑k * s✝
hk : beattySeq' s✝ k = j
⊢ k > 0
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
hμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d
⊢ (∑ d ∈ s.support, s.weights d * if s.prodPrimes.gcd d = 1 then 1 else 0) ≤
    ∑ n ∈ s.support, s.weights n * ∑ d ∈ (s.prodPrimes.gcd n).divisors, muPlus d

Tactic:
]

Goal state after:
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
hμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d
⊢ (∑ d ∈ s.support, s.weights d * if s.prodPrimes.gcd d = 1 then 1 else 0) ≤
    ∑ n ∈ s.support, s.weights n * ∑ d ∈ (s.prodPrimes.gcd n).divisors, muPlus d
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: case left
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬q % π = 0
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
⊢ v ↑(q % π) = 1

Tactic:
]

Goal state after:
case left
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
p : K[X]
hp : p ≠ 0
π : K[X] := πᵥ
hne : {p | v ↑p < 1 ∧ p ≠ 0}.Nonempty
hπirr : Irreducible π
k : ℕ
q : K[X]
hnq : ¬q % π = 0
heq : p = π ^ k * q
hπ : v ↑π < 1 ∧ ¬π = 0
⊢ v ↑(q % π) = 1
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case e_f.e_s
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
hμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d
n : ℕ
⊢ {d ∈ s.prodPrimes.divisors | d ∣ s.prodPrimes.gcd n} = {a ∈ s.prodPrimes.divisors | a ∣ n}

Tactic:
]

Goal state after:
case e_f.e_s
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
hμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d
n : ℕ
⊢ {d ∈ s.prodPrimes.divisors | d ∣ s.prodPrimes.gcd n} = {a ∈ s.prodPrimes.divisors | a ∣ n}
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: case a
r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
hj : 0 < j
hb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}
hb₂ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq' s k = x} ↔ j ∈ {x | ∃ k, beattySeq' s k = x}
⊢ j ∈ {x | ∃ k, beattySeq r k = x} ∨ j ∉ {x | ∃ k, beattySeq r k = x}

Tactic:
]

Goal state after:
case a
r s : ℝ
hrs : r.HolderConjugate s
j : ℤ
hj : 0 < j
hb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}
hb₂ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq' s k = x} ↔ j ∈ {x | ∃ k, beattySeq' s k = x}
⊢ j ∈ {x | ∃ k, beattySeq r k = x} ∨ j ∉ {x | ∃ k, beattySeq r k = x}
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
⊢ {n | 0 < n} = {n | 0 < n}

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
hμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d
⊢ (∑ y ∈ s.prodPrimes.divisors, ∑ x ∈ s.support, if y ∣ x then s.weights x * muPlus y else 0) =
    ∑ d ∈ s.prodPrimes.divisors, muPlus d * multSum d

Tactic:
]

Goal state after:
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
hμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d
⊢ (∑ y ∈ s.prodPrimes.divisors, ∑ x ∈ s.support, if y ∣ x then s.weights x * muPlus y else 0) =
    ∑ d ∈ s.prodPrimes.divisors, muPlus d * multSum d
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r : ℝ
hr : Irrational r
x✝ k : ℤ
⊢ k > 0 → (⌈↑k * r⌉ - 1 = x✝ ↔ ⌊↑k * r⌋ = x✝)

Tactic:
]

Goal state after:
r : ℝ
hr : Irrational r
x✝ k : ℤ
⊢ k > 0 → (⌈↑k * r⌉ - 1 = x✝ ↔ ⌊↑k * r⌋ = x✝)
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r : ℝ
hr : Irrational r
x✝ k : ℤ
hk : k > 0
⊢ ↑⌊↑k * r⌋ < ↑k * r ∧ ↑k * r ≤ ↑⌊↑k * r⌋ + 1

Tactic:
]

Goal state after:
r : ℝ
hr : Irrational r
x✝ k : ℤ
hk : k > 0
⊢ ↑⌊↑k * r⌋ < ↑k * r ∧ ↑k * r ≤ ↑⌊↑k * r⌋ + 1
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
⊢ ∑ d ∈ s.prodPrimes.divisors, muPlus d * multSum d =
    ∑ x ∈ s.prodPrimes.divisors, (s.totalMass * (muPlus x * s.nu x) + muPlus x * rem x)

Tactic:
]

Goal state after:
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
⊢ ∑ d ∈ s.prodPrimes.divisors, muPlus d * multSum d =
    ∑ x ∈ s.prodPrimes.divisors, (s.totalMass * (muPlus x * s.nu x) + muPlus x * rem x)
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case e_f
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
d : ℕ
⊢ muPlus d * multSum d = s.totalMass * (muPlus d * s.nu d) + muPlus d * (multSum d - s.nu d * s.totalMass)

Tactic:
]

Goal state after:
case e_f
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
d : ℕ
⊢ muPlus d * multSum d = s.totalMass * (muPlus d * s.nu d) + muPlus d * (multSum d - s.nu d * s.totalMass)
Module: Mathlib.NumberTheory.Rayleigh

Goal state before:
pp: r s : ℝ
hrs : r.HolderConjugate s
hr : Irrational r
⊢ {n | 0 < n} = {n | 0 < n}

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
hv : v.IsRankOneDiscrete
h0 : v ↑πᵥ ≠ 0
⊢ ⋯.unit = LinearOrderedCommGroup.Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueGroup

Tactic:
]

Goal state after:
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
hv : v.IsRankOneDiscrete
h0 : v ↑πᵥ ≠ 0
⊢ ⋯.unit = LinearOrderedCommGroup.Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueGroup
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
⊢ s.totalMass * mainSum muPlus + ∑ d ∈ s.prodPrimes.divisors, muPlus d * rem d ≤
    s.totalMass * mainSum muPlus + ∑ d ∈ s.prodPrimes.divisors, |muPlus d| * |rem d|

Tactic:
]

Goal state after:
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
⊢ s.totalMass * mainSum muPlus + ∑ d ∈ s.prodPrimes.divisors, muPlus d * rem d ≤
    s.totalMass * mainSum muPlus + ∑ d ∈ s.prodPrimes.divisors, |muPlus d| * |rem d|
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: case hg
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
hv : v.IsRankOneDiscrete
h0 : v ↑πᵥ ≠ 0
⊢ v ↑πᵥ < 1

Tactic:
]

Goal state after:
case hg
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
hv : v.IsRankOneDiscrete
h0 : v ↑πᵥ ≠ 0
⊢ v ↑πᵥ < 1
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case h
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
d : ℕ
a✝ : d ∈ s.prodPrimes.divisors
⊢ muPlus d * rem d ≤ |muPlus d * rem d|

Tactic:
]

Goal state after:
case h
s : BoundingSieve
muPlus : ℕ → ℝ
h : IsUpperMoebius muPlus
d : ℕ
a✝ : d ∈ s.prodPrimes.divisors
⊢ muPlus d * rem d ≤ |muPlus d * rem d|
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case a
f : ℕ → ℕ → ℕ → ℝ
n d : ℕ
hd : d ∣ n ∧ n ≠ 0
⊢ (∑ d1 ∈ d.divisors, ∑ d2 ∈ d.divisors, if d = d1.lcm d2 then f d1 d2 d else 0) =
    ∑ d1 ∈ n.divisors, ∑ d2 ∈ n.divisors, if d = d1.lcm d2 then f d1 d2 d else 0

Tactic:
]

Goal state after:
case a
f : ℕ → ℕ → ℕ → ℝ
n d : ℕ
hd : d ∣ n ∧ n ≠ 0
⊢ (∑ d1 ∈ d.divisors, ∑ d2 ∈ d.divisors, if d = d1.lcm d2 then f d1 d2 d else 0) =
    ∑ d1 ∈ n.divisors, ∑ d2 ∈ n.divisors, if d = d1.lcm d2 then f d1 d2 d else 0
Module: Mathlib.NumberTheory.RamificationInertia.Basic

Goal state before:
pp: case refine_2
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
span_d : Submodule.restrictScalars R (S ∙ f A.det) ≤ M
x : L
hx : x ∈ span {(algebraMap S L) (f A.det)}
⊢ x ∈ Submodule.span K (⇑(algebraMap S L) '' b)

Tactic:
]

Goal state after:
case refine_2
R : Type u
inst✝¹⁶ : CommRing R
S : Type v
inst✝¹⁵ : CommRing S
inst✝¹⁴ : Algebra R S
p : Ideal R
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : Algebra R K
L : Type u_2
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra S L
inst✝⁹ : IsFractionRing S L
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra K L
inst✝⁵ : Module.Finite R S
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R S L
inst✝² : IsScalarTower R K L
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsTorsionFree R K
hp : p ≠ ⊤
b : Set S
hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map f p) = ⊤
hRL : Function.Injective ⇑(algebraMap R L)
M : Submodule R S := Submodule.span R b
n : ℕ
a : Fin n → S ⧸ M
ha : Submodule.span R (Set.range a) = ⊤
smul_top_eq : p • ⊤ = ⊤
exists_sum : ∀ (x : S ⧸ M), ∃ a', (∀ (i : Fin n), a' i ∈ p) ∧ ∑ i, a' i • a i = x
A' : Fin n → Fin n → R
hA'p : ∀ (i i_1 : Fin n), A' i i_1 ∈ p
hA' : ∀ (i : Fin n), ∑ i_1, A' i i_1 • a i_1 = a i
A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
B : Matrix (Fin n) (Fin n) R := A.adjugate
A_smul : ∀ (i : Fin n), ∑ j, A i j • a j = 0
d_smul : ∀ (i : Fin n), A.det • a i = 0
span_d : Submodule.restrictScalars R (S ∙ f A.det) ≤ M
x : L
hx : x ∈ span {(algebraMap S L) (f A.det)}
⊢ x ∈ Submodule.span K (⇑(algebraMap S L) '' b)
Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
a b c d x y z w : R
⊢ (a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 + (a * z - b * w + c * x + d * y) ^ 2 +
      (a * w + b * z - c * y + d * x) ^ 2 =
    (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)

Tactic:
by ring

Goal state after:
Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: a b c d x y z w : ℕ
⊢ ↑((↑a * ↑x - ↑b * ↑y - ↑c * ↑z - ↑d * ↑w).natAbs ^ 2 + (↑a * ↑y + ↑b * ↑x + ↑c * ↑w - ↑d * ↑z).natAbs ^ 2 +
          (↑a * ↑z - ↑b * ↑w + ↑c * ↑x + ↑d * ↑y).natAbs ^ 2 +
        (↑a * ↑w + ↑b * ↑z - ↑c * ↑y + ↑d * ↑x).natAbs ^ 2) =
    ↑((a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2))

Tactic:
]

Goal state after:
a b c d x y z w : ℕ
⊢ ↑((↑a * ↑x - ↑b * ↑y - ↑c * ↑z - ↑d * ↑w).natAbs ^ 2 + (↑a * ↑y + ↑b * ↑x + ↑c * ↑w - ↑d * ↑z).natAbs ^ 2 +
          (↑a * ↑z - ↑b * ↑w + ↑c * ↑x + ↑d * ↑y).natAbs ^ 2 +
        (↑a * ↑w + ↑b * ↑z - ↑c * ↑y + ↑d * ↑x).natAbs ^ 2) =
    ↑((a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2))
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: case hH.refine_2
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
hv : v.IsRankOneDiscrete
h0 : v ↑πᵥ ≠ 0
γ : Γˣ
x✝ : ∃ a, ¬a = 0 ∧ ∃ x, (MonoidWithZeroHom.ofClass v) a * ↑γ = (MonoidWithZeroHom.ofClass v) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
ka : ℤ
hka : v a = v ↑πᵥ ^ ka
kb : ℤ
hab : v ↑πᵥ ^ ka * ↑γ = v ↑πᵥ ^ kb
hkb : v b = v ↑πᵥ ^ kb
⊢ ∃ k, ⋯.unit ^ k = γ

Tactic:
]

Goal state after:
case hH.refine_2
K : Type u_1
Γ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝¹ : v.IsNontrivial
inst✝ : IsTrivialOn K v
hle : v X ≤ 1
hv : v.IsRankOneDiscrete
h0 : v ↑πᵥ ≠ 0
γ : Γˣ
x✝ : ∃ a, ¬a = 0 ∧ ∃ x, (MonoidWithZeroHom.ofClass v) a * ↑γ = (MonoidWithZeroHom.ofClass v) x
a : K⟮X⟯
ha : ¬a = 0
b : K⟮X⟯
ka : ℤ
hka : v a = v ↑πᵥ ^ ka
kb : ℤ
hab : v ↑πᵥ ^ ka * ↑γ = v ↑πᵥ ^ kb
hkb : v b = v ↑πᵥ ^ kb
⊢ ∃ k, ⋯.unit ^ k = γ
Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 * (x ^ 2 + y ^ 2) = (x - y) ^ 2 + (x + y) ^ 2

Tactic:
]

Goal state after:
m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 * (x ^ 2 + y ^ 2) = (x - y) ^ 2 + (x + y) ^ 2
Module: Mathlib.NumberTheory.SumFourSquares

Goal state before:
pp: m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ (x - y) ^ 2 + (x + y) ^ 2 = (x - y) ^ 2 + (x + y) ^ 2

pp: m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 ∣ x + y

pp: m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 ∣ x - y

Tactic:
]

Goal state after:
m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 ∣ x + y

m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 ∣ x - y
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case neg
weights : ℕ → ℝ
hw : weights 1 = 1
n : ℕ
h✝ : ¬n = 1
⊢ (∑ d ∈ n.divisors, weights d) ^ 2 ≤
    ∑ y ∈ n.divisors, ∑ x ∈ n.divisors, ∑ d2 ∈ n.divisors, if x = y.lcm d2 then weights y * weights d2 else 0

Tactic:
]

Goal state after:
case neg
weights : ℕ → ℝ
hw : weights 1 = 1
n : ℕ
h✝ : ¬n = 1
⊢ (∑ d ∈ n.divisors, weights d) ^ 2 ≤
    ∑ y ∈ n.divisors, ∑ x ∈ n.divisors, ∑ d2 ∈ n.divisors, if x = y.lcm d2 then weights y * weights d2 else 0
Module: Mathlib.NumberTheory.RatFunc.Ostrowski

Goal state before:
pp: K : Type u_1
Γ : Type u_2
inst✝⁴ : Field K
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝² : v.IsNontrivial
inst✝¹ : IsTrivialOn K v
hle : v X ≤ 1
inst✝ : v.IsRankOneDiscrete
⊢ ∀ {x : K⟮X⟯}, v x ≤ 1 ↔ (valuation K⟮X⟯ Pᵥ) x ≤ 1

Tactic:
]

Goal state after:
K : Type u_1
Γ : Type u_2
inst✝⁴ : Field K
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation K⟮X⟯ Γ
inst✝² : v.IsNontrivial
inst✝¹ : IsTrivialOn K v
hle : v X ≤ 1
inst✝ : v.IsRankOneDiscrete
⊢ ∀ {x : K⟮X⟯}, v x ≤ 1 ↔ (valuation K⟮X⟯ Pᵥ) x ≤ 1
Module: Mathlib.NumberTheory.SelbergSieve

Goal state before:
pp: case neg.a
weights : ℕ → ℝ
hw : weights 1 = 1
n : ℕ
h✝ : ¬n = 1
d1 : ℕ
hd1 : d1 ∈ n.divisors
⊢ ∑ i ∈ n.divisors, weights i * weights d1 =
    ∑ y ∈ n.divisors, ∑ x ∈ n.divisors, if x = d1.lcm y then weights d1 * weights y else 0

Tactic:
]

Goal state after:
case neg.a
weights : ℕ → ℝ
hw : weights 1 = 1
n : ℕ
h✝ : ¬n = 1
d1 : ℕ
hd1 : d1 ∈ n.divisors
⊢ ∑ i ∈ n.divisors, weights i * weights d1 =
    ∑ y ∈ n.divisors, ∑ x ∈ n.divisors, if x = d1.lcm y then weights d1 * weights y else 0