mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case pos
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
a : K
ha : (MonoidWithZeroHom.ofClass v) a = ↑↑z
hz1 : z ≤ 1
z0' : 0 < ↑z
a0 : 0 < v a
U : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v w ≤ ↑↑z} else {w | v w = v y}
this : (∀ (i : K), IsOpen (U i)) → ↑𝒪[K] ⊆ ⋃ i, U i → ∃ t, ↑𝒪[K] ⊆ ⋃ i ∈ t, U i
w : K
hw : v w ≤ ↑↑z
b : K
z0 : 0 < (MonoidWithZeroHom.ofClass v) b
hb : (MonoidWithZeroHom.ofClass v) b = ↑↑z
⊢ ¬v b = 0

Tactic:
]

Goal state after:
case pos
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
a : K
ha : (MonoidWithZeroHom.ofClass v) a = ↑↑z
hz1 : z ≤ 1
z0' : 0 < ↑z
a0 : 0 < v a
U : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v w ≤ ↑↑z} else {w | v w = v y}
this : (∀ (i : K), IsOpen (U i)) → ↑𝒪[K] ⊆ ⋃ i, U i → ∃ t, ↑𝒪[K] ⊆ ⋃ i ∈ t, U i
w : K
hw : v w ≤ ↑↑z
b : K
z0 : 0 < (MonoidWithZeroHom.ofClass v) b
hb : (MonoidWithZeroHom.ofClass v) b = ↑↑z
⊢ ¬v b = 0
Module: Mathlib.NumberTheory.LocalField.Basic

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : ValuativeRel K
inst✝¹ : TopologicalSpace K
inst✝ : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup
s : Set K
hs : s ∈ nhds 0
hs' : IsCompact s
r : K
hr : r ≠ 0
hr1 : (valuation K) r < 1
H : {x | (valuation K).restrict x ≤ (valuation K).restrict r} ⊆ s
x : K
⊢ (valuation K) x ≤ (valuation K) γ ↔
    Valued.v.restrict r * Valued.v.restrict x ≤ (valuation K).restrict r * Valued.v.restrict γ

Tactic:
]

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : ValuativeRel K
inst✝¹ : TopologicalSpace K
inst✝ : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup
s : Set K
hs : s ∈ nhds 0
hs' : IsCompact s
r : K
hr : r ≠ 0
hr1 : (valuation K) r < 1
H : {x | (valuation K).restrict x ≤ (valuation K).restrict r} ⊆ s
x : K
⊢ (valuation K) x ≤ (valuation K) γ ↔
    Valued.v.restrict r * Valued.v.restrict x ≤ (valuation K).restrict r * Valued.v.restrict γ
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: case e_a
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
t : ℝ
ht : ↑t ≠ 0
⊢ S t * (↑t ^ (-↑s) / ↑t) - l * ↑t ^ (-↑s) = S t * (↑t ^ (-↑s) / ↑t) - l * ↑t ^ (-↑s)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case ind.isFalse.isFalse.isFalse.isTrue
a : ℕ
IH :
  ∀ m < a,
    ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0},
      b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)
b : ℕ
flip : Bool
ha0 : a > 0
hb2 : b % 2 = 1
hb1 : b > 1
ha4 : ¬a % 4 = 0
ha2 : ¬a % 2 = 0
ha1 : ¬a = 1
hba : b % a = 0
⊢ a ≠ 1

Tactic:
]

Goal state after:
case ind.isFalse.isFalse.isFalse.isTrue
a : ℕ
IH :
  ∀ m < a,
    ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0},
      b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)
b : ℕ
flip : Bool
ha0 : a > 0
hb2 : b % 2 = 1
hb1 : b > 1
ha4 : ¬a % 4 = 0
ha2 : ¬a % 2 = 0
ha1 : ¬a = 1
hba : b % a = 0
⊢ a ≠ 1
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case ind.isFalse.isFalse.isFalse.isFalse
a : ℕ
IH :
  ∀ m < a,
    ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0},
      b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)
b : ℕ
flip : Bool
ha0 : a > 0
hb2 : b % 2 = 1
hb1 : b > 1
ha4 : ¬a % 4 = 0
ha2 : ¬a % 2 = 0
ha1 : ¬a = 1
hba : ¬b % a = 0
⊢ (if (decide (a % 4 = 3 ∧ b % 4 = 3) ^^ flip) = true then -J(↑(b % a) | a) else J(↑(b % a) | a)) =
    if flip = true then -J(↑a | b) else J(↑a | b)

Tactic:
]

Goal state after:
case ind.isFalse.isFalse.isFalse.isFalse
a : ℕ
IH :
  ∀ m < a,
    ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0},
      b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)
b : ℕ
flip : Bool
ha0 : a > 0
hb2 : b % 2 = 1
hb1 : b > 1
ha4 : ¬a % 4 = 0
ha2 : ¬a % 2 = 0
ha1 : ¬a = 1
hba : ¬b % a = 0
⊢ (if (decide (a % 4 = 3 ∧ b % 4 = 3) ^^ flip) = true then -J(↑(b % a) | a) else J(↑(b % a) | a)) =
    if flip = true then -J(↑a | b) else J(↑a | b)
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case neg
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
a : K
ha : (MonoidWithZeroHom.ofClass v) a = ↑↑z
hz1 : z ≤ 1
z0' : 0 < ↑z
z0 : 0 < ↑↑z
a0 : 0 < v a
U : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v w ≤ ↑↑z} else {w | v w = v y}
this : (∀ (i : K), IsOpen (U i)) → ↑𝒪[K] ⊆ ⋃ i, U i → ∃ t, ↑𝒪[K] ⊆ ⋃ i ∈ t, U i
w : K
hw : ↑↑z < v w
⊢ ¬v w = v 0

Tactic:
]

Goal state after:
case neg
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
a : K
ha : (MonoidWithZeroHom.ofClass v) a = ↑↑z
hz1 : z ≤ 1
z0' : 0 < ↑z
z0 : 0 < ↑↑z
a0 : 0 < v a
U : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v w ≤ ↑↑z} else {w | v w = v y}
this : (∀ (i : K), IsOpen (U i)) → ↑𝒪[K] ⊆ ⋃ i, U i → ∃ t, ↑𝒪[K] ⊆ ⋃ i ∈ t, U i
w : K
hw : ↑↑z < v w
⊢ ¬v w = v 0
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case ind.isFalse.isFalse.isFalse.isFalse
a : ℕ
IH :
  ∀ m < a,
    ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0},
      b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)
b : ℕ
flip : Bool
ha0 : a > 0
hb2 : b % 2 = 1
hb1 : b > 1
ha4 : ¬a % 4 = 0
ha2 : ¬a % 2 = 0
ha1 : ¬a = 1
hba : ¬b % a = 0
⊢ (if (decide (a % 4 = 3 ∧ b % 4 = 3) ^^ flip) = true then -J(↑b | a) else J(↑b | a)) =
    if flip = true then -if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)
    else if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)

Tactic:
]

Goal state after:
case ind.isFalse.isFalse.isFalse.isFalse
a : ℕ
IH :
  ∀ m < a,
    ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0},
      b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)
b : ℕ
flip : Bool
ha0 : a > 0
hb2 : b % 2 = 1
hb1 : b > 1
ha4 : ¬a % 4 = 0
ha2 : ¬a % 2 = 0
ha1 : ¬a = 1
hba : ¬b % a = 0
⊢ (if (decide (a % 4 = 3 ∧ b % 4 = 3) ^^ flip) = true then -J(↑b | a) else J(↑b | a)) =
    if flip = true then -if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)
    else if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ (s - 1) * s * ‖∫ (t : ℝ) in Set.Ioi 1, (S t - l * ↑t) * ↑t ^ (-↑s - 1)‖ ≤
    (s - 1) * s * ∫ (t : ℝ) in Set.Ioi 1, ‖(S t - l * ↑t) * ↑t ^ (-↑s - 1)‖

Tactic:
]

Goal state after:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ (s - 1) * s * ‖∫ (t : ℝ) in Set.Ioi 1, (S t - l * ↑t) * ↑t ^ (-↑s - 1)‖ ≤
    (s - 1) * s * ∫ (t : ℝ) in Set.Ioi 1, ‖(S t - l * ↑t) * ↑t ^ (-↑s - 1)‖
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case pos
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : b = 0
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case neg
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : b % 2 = 0
ha2 : ¬a % 2 = 0
⊢ J(a | b) = J(a | b / 2)

Tactic:
]

Goal state after:
case neg
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : b % 2 = 0
ha2 : ¬a % 2 = 0
⊢ J(a | b) = J(a | b / 2)
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case neg
a : ℤ
ha2 : ¬a % 2 = 0
b : ℕ
IH : ∀ m < 2 * b, J(a | m) = fastJacobiSym a m
hb0 : ¬2 * b = 0
hb2 : 2 * b % 2 = 0
⊢ J(a | b) = J(a | b)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case pos
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : b = 1
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case pos
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : ¬b = 1
hab : a % ↑b = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: case e_a
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
t : ℝ
ht : 0 ≤ t
⊢ ‖S t - l * ↑t‖ * t ^ (-s - 1) = ‖S t - l * ↑t‖ * t ^ (-s - 1)

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: case neg
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : ¬b = 1
hab : ¬a % ↑b = 0
⊢ J(a % ↑b | b) = J(a % ↑b | b)

pp: case neg.hb2
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : ¬b = 1
hab : ¬a % ↑b = 0
⊢ b % 2 = 1

pp: case neg.hb1
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : ¬b = 1
hab : ¬a % ↑b = 0
⊢ b > 1

Tactic:
]

Goal state after:
case neg.hb2
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : ¬b = 1
hab : ¬a % ↑b = 0
⊢ b % 2 = 1

case neg.hb1
a : ℤ
b : ℕ
IH : ∀ m < b, J(a | m) = fastJacobiSym a m
hb0 : ¬b = 0
hb2 : ¬b % 2 = 0
hb1 : ¬b = 1
hab : ¬a % ↑b = 0
⊢ b > 1
Module: Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol

Goal state before:
pp: p : ℕ
x✝ : Fact (Nat.Prime p)
a : ℤ
⊢ J(a | p) = J(a | p)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : ⟨x, hx1✝⟩ ≠ 0
⊢ ∃ b, ⟨x, hx1✝⟩ * b = 1

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : ⟨x, hx1✝⟩ ≠ 0
⊢ ∃ b, ⟨x, hx1✝⟩ * b = 1
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
a c : A
⊢ a * (x * c) = a * (x * c)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
b a : A
⊢ a * (x * b) = a * (x * b)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case neg
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc✝ : IsCompact ↑𝒪[K]
z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
a : K
ha : (MonoidWithZeroHom.ofClass v) a = ↑↑z
hz1 : z ≤ 1
z0' : 0 < ↑z
z0 : 0 < ↑↑z
a0 : 0 < v a
U : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v w ≤ ↑↑z} else {w | v w = v y}
t : Finset K
ht : ↑𝒪[K] ⊆ ⋃ i ∈ t, U i
i : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
c : K
hc : v c = ↑↑i
hzi : z ≤ i
hi1 : i ≤ 1
j : K
hj : j ∈ t
hcj : ¬v j ≤ ↑↑z
hj' : v c = v j
⊢ Units.mk0 ⟨v j, ⋯⟩ ⋯ = i

Tactic:
]

Goal state after:
case neg
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc✝ : IsCompact ↑𝒪[K]
z : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
a : K
ha : (MonoidWithZeroHom.ofClass v) a = ↑↑z
hz1 : z ≤ 1
z0' : 0 < ↑z
z0 : 0 < ↑↑z
a0 : 0 < v a
U : K → Set K := fun y ↦ if v y ≤ ↑↑z then {w | v w ≤ ↑↑z} else {w | v w = v y}
t : Finset K
ht : ↑𝒪[K] ⊆ ⋃ i ∈ t, U i
i : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ
c : K
hc : v c = ↑↑i
hzi : z ≤ i
hi1 : i ≤ 1
j : K
hj : j ∈ t
hcj : ¬v j ≤ ↑↑z
hj' : v c = v j
⊢ Units.mk0 ⟨v j, ⋯⟩ ⋯ = i
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ a * y = a * y

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ x * y * a * y = x * y * a * y

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (x * a) * y = y * (x * a) * y

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (a * x * y) = y * (a * x * y)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
⊢ MulArchimedean (↥(MonoidHom.mrange v))ˣ

Tactic:
]

Goal state after:
K : Type u_1
Γ₀ : Type u_2
inst✝² : Field K
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
hc : IsCompact ↑𝒪[K]
⊢ MulArchimedean (↥(MonoidHom.mrange v))ˣ
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (a * (x * y)) = y * (a * (x * y))

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * a = y * a

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.SimpleRing.Field

Goal state before:
pp: A : Type u_1
inst✝ : CommRing A
x✝ : IsSimpleRing A
⊢ IsField ↥(Subring.center A)

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CommRing A
x✝ : IsSimpleRing A
⊢ IsField ↥(Subring.center A)
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case refine_1
K : Type u_1
Γ₀ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : Valued K Γ₀
inst✝ : v.RankOne
h : Finite 𝓀[K] ∧ IsComplete Set.univ
this : IsDiscreteValuationRing ↥𝒪[K]
⊢ Finite 𝓀[K]

Tactic:
]

Goal state after:
case refine_1
K : Type u_1
Γ₀ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : Valued K Γ₀
inst✝ : v.RankOne
h : Finite 𝓀[K] ∧ IsComplete Set.univ
this : IsDiscreteValuationRing ↥𝒪[K]
⊢ Finite 𝓀[K]
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case refine_2
K : Type u_1
Γ₀ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : Valued K Γ₀
inst✝ : v.RankOne
x✝ : CompleteSpace ↥𝒪[K] ∧ IsDiscreteValuationRing ↥𝒪[K] ∧ Finite 𝓀[K]
left✝¹ : CompleteSpace ↥𝒪[K]
left✝ : IsDiscreteValuationRing ↥𝒪[K]
h : TotallyBounded Set.univ
⊢ IsCompact Set.univ

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
Γ₀ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : Valued K Γ₀
inst✝ : v.RankOne
x✝ : CompleteSpace ↥𝒪[K] ∧ IsDiscreteValuationRing ↥𝒪[K] ∧ Finite 𝓀[K]
left✝¹ : CompleteSpace ↥𝒪[K]
left✝ : IsDiscreteValuationRing ↥𝒪[K]
h : TotallyBounded Set.univ
⊢ IsCompact Set.univ
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: p : ℕ
h : Prime (mersenne p)
⊢ Prime p

Tactic:
apply Nat.prime_of_pow_sub_one_prime _ h |>.2

Goal state after:
p : ℕ
h : Prime (mersenne p)
⊢ p ≠ 1
Module: Mathlib.Topology.Algebra.Valued.LocallyCompact

Goal state before:
pp: case refine_2
K : Type u_1
Γ₀ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : Valued K Γ₀
inst✝ : v.RankOne
x✝ : CompleteSpace ↥𝒪[K] ∧ IsDiscreteValuationRing ↥𝒪[K] ∧ Finite 𝓀[K]
left✝¹ : CompleteSpace ↥𝒪[K]
left✝ : IsDiscreteValuationRing ↥𝒪[K]
h : TotallyBounded Set.univ
⊢ TotallyBounded Set.univ ∧ IsComplete Set.univ

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
Γ₀ : Type u_2
inst✝³ : Field K
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : Valued K Γ₀
inst✝ : v.RankOne
x✝ : CompleteSpace ↥𝒪[K] ∧ IsDiscreteValuationRing ↥𝒪[K] ∧ Finite 𝓀[K]
left✝¹ : CompleteSpace ↥𝒪[K]
left✝ : IsDiscreteValuationRing ↥𝒪[K]
h : TotallyBounded Set.univ
⊢ TotallyBounded Set.univ ∧ IsComplete Set.univ
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: k : ℕ
⊢ 2 ^ k = 2 ^ k

pp: k : ℕ
⊢ 1 ≤ 2 ^ k

Tactic:
]

Goal state after:
k : ℕ
⊢ 1 ≤ 2 ^ k
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: case succ
n n✝ : ℕ
hmn✝ : 2 ≤ n✝
a✝ : mersenne n✝ % 4 = 3
⊢ (2 * mersenne n✝ + 1) % 4 = 3

Tactic:
]

Goal state after:
case succ
n n✝ : ℕ
hmn✝ : 2 ≤ n✝
a✝ : mersenne n✝ % 4 = 3
⊢ (2 * mersenne n✝ + 1) % 4 = 3
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: case succ
k j : ℕ
hmn✝ : 1 ≤ j
a✝ : mersenne (2 * j + 1) % 3 = 1
⊢ (2 * (2 * mersenne (2 * j + 1) + 1) + 1) % 3 = 1

Tactic:
]

Goal state after:
case succ
k j : ℕ
hmn✝ : 1 ≤ j
a✝ : mersenne (2 * j + 1) % 3 = 1
⊢ (2 * (2 * mersenne (2 * j + 1) + 1) + 1) % 3 = 1
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: case succ
n n✝ : ℕ
hmn✝ : 3 ≤ n✝
a✝ : mersenne n✝ % 8 = 7
⊢ (2 * mersenne n✝ + 1) % 8 = 7

Tactic:
]

Goal state after:
case succ
n n✝ : ℕ
hmn✝ : 3 ≤ n✝
a✝ : mersenne n✝ % 8 = 7
⊢ (2 * mersenne n✝ + 1) % 8 = 7
Module: Mathlib.NumberTheory.LSeries.SumCoeff

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ (s - 1) * s * Cs + (s - 1) * s * ∫ (t : ℝ) in Set.Ioi T, ‖S t - l * ↑t‖ * t ^ (-s - 1) ≤
    (s - 1) * s * C + (s - 1) * s * ∫ (t : ℝ) in Set.Ioi T, ‖S t - l * ↑t‖ * t ^ (-s - 1)

Tactic:
]

Goal state after:
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : 1 ≤ T
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
⊢ (s - 1) * s * Cs + (s - 1) * s * ∫ (t : ℝ) in Set.Ioi T, ‖S t - l * ↑t‖ * t ^ (-s - 1) ≤
    (s - 1) * s * C + (s - 1) * s * ∫ (t : ℝ) in Set.Ioi T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime (mersenne p))
hp : 3 ≤ p
this : mersenne p % 8 = 7
⊢ (if mersenne p % 2 = 0 then 0 else if mersenne p % 8 = 1 ∨ mersenne p % 8 = 7 then 1 else -1) = 1

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime (mersenne p))
hp : 3 ≤ p
this : mersenne p % 8 = 7
⊢ (if mersenne p % 2 = 0 then 0 else if mersenne p % 8 = 1 ∨ mersenne p % 8 = 7 then 1 else -1) = 1
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime (mersenne p))
hp : 3 ≤ p
odd : Odd p
⊢ -legendreSym 3 (↑(mersenne p) % ↑3) = -1

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime (mersenne p))
hp : 3 ≤ p
odd : Odd p
⊢ -legendreSym 3 (↑(mersenne p) % ↑3) = -1
Module: Mathlib.NumberTheory.LucasLehmer

Goal state before:
pp: p : ℕ
hp : p ≠ 0
i : ℕ
⊢ sMod p i % (2 ^ p - 1) < 2 ^ p - 1

Tactic:
]

Goal state after:
p : ℕ
hp : p ≠ 0
i : ℕ
⊢ sMod p i % (2 ^ p - 1) < 2 ^ p - 1
Module: Mathlib.NumberTheory.MahlerMeasure

Goal state before:
pp: p : ℤ[X]
hp : p ≠ 0
⊢ 1 ≤ ‖↑p.leadingCoeff‖

Tactic:
]

Goal state after:
p : ℤ[X]
hp : p ≠ 0
⊢ 1 ≤ ‖↑p.leadingCoeff‖