mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ IsCyclic (ZMod n)ˣ ↔
    (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = x ^ x_1) ∨ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = 2 * x ^ x_1

pp: case neg.inr
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Even n
⊢ IsCyclic (ZMod n)ˣ ↔
    (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = x ^ x_1) ∨ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = 2 * x ^ x_1

Tactic:
·

Goal state after:
case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ IsCyclic (ZMod n)ˣ ↔
    (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = x ^ x_1) ∨ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = 2 * x ^ x_1
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case inl
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Subsingleton E
⊢ (μ.withDensityᵥ f).variation = μ.withDensity fun x ↦ ‖f x‖ₑ

pp: case inr
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
⊢ (μ.withDensityᵥ f).variation = μ.withDensity fun x ↦ ‖f x‖ₑ

Tactic:
·

Goal state after:
case inl
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Subsingleton E
⊢ (μ.withDensityᵥ f).variation = μ.withDensity fun x ↦ ‖f x‖ₑ
Module: Mathlib.RingTheory.PowerSeries.Derivative

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
f g : R⟦X⟧
hg : HasSubst g
n m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0
d : ℕ
hd : d < m
⊢ (coeff d) f • (coeff (n + 1)) (g ^ d) = (coeff d) ↑((trunc (m + 1)) f) • (coeff (n + 1)) (g ^ d)

pp: case inr
R : Type u_1
inst✝ : CommRing R
f g : R⟦X⟧
hg : HasSubst g
n m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0
d : ℕ
hd : m ≤ d
⊢ (coeff d) f • (coeff (n + 1)) (g ^ d) = (coeff d) ↑((trunc (m + 1)) f) • (coeff (n + 1)) (g ^ d)

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝ : CommRing R
f g : R⟦X⟧
hg : HasSubst g
n m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0
d : ℕ
hd : d < m
⊢ (coeff d) f • (coeff (n + 1)) (g ^ d) = (coeff d) ↑((trunc (m + 1)) f) • (coeff (n + 1)) (g ^ d)
Module: Mathlib.RingTheory.PowerSeries.Derivative

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f g : R⟦X⟧
hg : HasSubst g
n m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0
d : ℕ
hd : d < m
⊢ d < m + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ (∃ p m, Nat.Prime p ∧ Odd p ∧ n = p ^ m) ↔ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = x ^ x_1

pp: case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ ¬∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = 2 * x ^ x_1

Tactic:
·

Goal state after:
case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ (∃ p m, Nat.Prime p ∧ Odd p ∧ n = p ^ m) ↔ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = x ^ x_1
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case hf
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
⊢ AEStronglyMeasurable (fun x ↦ ‖f x‖⁻¹ • f x) (μ.withDensity fun x ↦ ‖f x‖ₑ)

pp: case hfC
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
⊢ ∀ᵐ (x : X) ∂μ.withDensity fun x ↦ ‖f x‖ₑ, ‖‖f x‖⁻¹ • f x‖ ≤ 1

Tactic:
·

Goal state after:
case hf
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
⊢ AEStronglyMeasurable (fun x ↦ ‖f x‖⁻¹ • f x) (μ.withDensity fun x ↦ ‖f x‖ₑ)
Module: Mathlib.RingTheory.PowerSeries.Derivative

Goal state before:
pp: case inr
R : Type u_1
inst✝ : CommRing R
f g : R⟦X⟧
hg : HasSubst g
n m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0
d : ℕ
hd : m ≤ d
⊢ (coeff d) f • (coeff (n + 1)) (g ^ d) = (coeff d) ↑((trunc (m + 1)) f) • (coeff (n + 1)) (g ^ d)

Tactic:
·

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
f g : R⟦X⟧
hg : HasSubst g
n m : ℕ
hm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0
d : ℕ
hd : m ≤ d
⊢ (coeff d) f • (coeff (n + 1)) (g ^ d) = (coeff d) ↑((trunc (m + 1)) f) • (coeff (n + 1)) (g ^ d)
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case hfC
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
⊢ ∀ᵐ (x : X) ∂μ.withDensity fun x ↦ ‖f x‖ₑ, ‖‖f x‖⁻¹ • f x‖ ≤ 1

Tactic:
·

Goal state after:
case hfC
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
⊢ ∀ᵐ (x : X) ∂μ.withDensity fun x ↦ ‖f x‖ₑ, ‖‖f x‖⁻¹ • f x‖ ≤ 1
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ ¬∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = 2 * x ^ x_1

Tactic:
·

Goal state after:
case neg.inl
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : ¬n = 2
h4 : ¬n = 4
hn : Odd n
⊢ ¬∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ n = 2 * x ^ x_1
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ IsCyclic (ZMod (2 * n))ˣ ↔
    (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1) ∨
      ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1

pp: case neg.inr.inr
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Even n
⊢ IsCyclic (ZMod (2 * n))ˣ ↔
    (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1) ∨
      ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1

Tactic:
·

Goal state after:
case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ IsCyclic (ZMod (2 * n))ˣ ↔
    (∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1) ∨
      ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ (∃ p m, Nat.Prime p ∧ Odd p ∧ n = p ^ m) ↔ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1

pp: case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ ¬∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1

Tactic:
·

Goal state after:
case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ (∃ p m, Nat.Prime p ∧ Odd p ∧ n = p ^ m) ↔ ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = 2 * x ^ x_1
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ ¬∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1

Tactic:
·

Goal state after:
case neg.inr.inl
n : ℕ
h0 : ¬2 * n = 0
h1 : ¬2 * n = 1
h2 : ¬2 * n = 2
h4 : ¬2 * n = 4
hn✝ : Even (2 * n)
hn : Odd n
⊢ ¬∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * n = x ^ x_1
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ AEMeasurable (fun x ↦ ‖f x‖ₑ) (μ.restrict s)

pp: X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ₑ < ∞

pp: X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, ‖f x‖ₑ.toReal • ‖f x‖⁻¹ • f x ∂μ

Tactic:
·

Goal state after:
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ AEMeasurable (fun x ↦ ‖f x‖ₑ) (μ.restrict s)
Module: Mathlib.RingTheory.ZMod.UnitsCyclic

Goal state before:
pp: case neg.inr.inr.ha
n : ℕ
h0 : ¬2 * (2 * n) = 0
h1 : ¬2 * (2 * n) = 1
h2 : ¬2 * (2 * n) = 2
h4 : ¬2 * (2 * n) = 4
hn✝ : Even (2 * (2 * n))
hn : Even (2 * n)
⊢ ¬IsCyclic (ZMod (2 * (2 * n)))ˣ

pp: case neg.inr.inr.hb
n : ℕ
h0 : ¬2 * (2 * n) = 0
h1 : ¬2 * (2 * n) = 1
h2 : ¬2 * (2 * n) = 2
h4 : ¬2 * (2 * n) = 4
hn✝ : Even (2 * (2 * n))
hn : Even (2 * n)
⊢ ¬((∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * (2 * n) = x ^ x_1) ∨
      ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * (2 * n) = 2 * x ^ x_1)

Tactic:
·

Goal state after:
case neg.inr.inr.ha
n : ℕ
h0 : ¬2 * (2 * n) = 0
h1 : ¬2 * (2 * n) = 1
h2 : ¬2 * (2 * n) = 2
h4 : ¬2 * (2 * n) = 4
hn✝ : Even (2 * (2 * n))
hn : Even (2 * n)
⊢ ¬IsCyclic (ZMod (2 * (2 * n)))ˣ
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ₑ < ∞

pp: X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, ‖f x‖ₑ.toReal • ‖f x‖⁻¹ • f x ∂μ

Tactic:
·

Goal state after:
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
⊢ ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ₑ < ∞
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case e_f.inl
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
x : X
hx : f x = 0
⊢ f x = ‖f x‖ₑ.toReal • ‖f x‖⁻¹ • f x

pp: case e_f.inr
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
x : X
hx : f x ≠ 0
⊢ f x = ‖f x‖ₑ.toReal • ‖f x‖⁻¹ • f x

Tactic:
·

Goal state after:
case e_f.inl
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
s : Set X
hs : MeasurableSet s
x : X
hx : f x = 0
⊢ f x = ‖f x‖ₑ.toReal • ‖f x‖⁻¹ • f x
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case inr.hbc
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) volume (↑⌊b⌋₊) b

pp: case inr.hab
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) volume a ↑⌊b⌋₊

pp: case inr.hac
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) volume a (↑⌊a⌋₊ + 1)

pp: case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ ∀ k ∈ Set.Ico ⌊a⌋₊.succ ⌊b⌋₊, IntervalIntegrable (fun t ↦ deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n) volume ↑k ↑(k + 1)

Tactic:
·

Goal state after:
case inr.hbc
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) volume (↑⌊b⌋₊) b
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case inr
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
this :
  μ.withDensityᵥ f =
    withDensity (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure (fun x ↦ ‖f x‖⁻¹ • f x)
      (ContinuousLinearMap.lsmul ℝ ℝ).flip
⊢ AEMeasurable (fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) μ

pp: case inr
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
this :
  μ.withDensityᵥ f =
    withDensity (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure (fun x ↦ ‖f x‖⁻¹ • f x)
      (ContinuousLinearMap.lsmul ℝ ℝ).flip
⊢ μ.withDensity ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) = μ.withDensity fun x ↦ ‖f x‖ₑ

Tactic:
·

Goal state after:
case inr
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
this :
  μ.withDensityᵥ f =
    withDensity (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure (fun x ↦ ‖f x‖⁻¹ • f x)
      (ContinuousLinearMap.lsmul ℝ ℝ).flip
⊢ AEMeasurable (fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) μ
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case inr.hab
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) volume a ↑⌊b⌋₊

pp: case inr.hac
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) volume a (↑⌊a⌋₊ + 1)

pp: case inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ ∀ k ∈ Set.Ico ⌊a⌋₊.succ ⌊b⌋₊, IntervalIntegrable (fun t ↦ deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n) volume ↑k ↑(k + 1)

Tactic:
·

Goal state after:
case inr.hab
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
a b : ℝ
ha : 0 ≤ a
hab : a ≤ b
hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc a b) volume
aux1 : ↑⌊a⌋₊ ≤ a
aux2 : b ≤ ↑⌊b⌋₊ + 1
hb : ⌊a⌋₊ < ⌊b⌋₊
aux3 : a ≤ ↑⌊a⌋₊ + 1
aux4 : ↑⌊a⌋₊ + 1 ≤ b
aux5 : ↑⌊b⌋₊ ≤ b
aux6 : a ≤ ↑⌊b⌋₊
this :
  ∑ k ∈ Ioc ⌊a⌋₊ (⌊b⌋₊ - 1), (f ↑(k + 1) - f ↑k) * ∑ n ∈ Icc 0 k, c n =
    ∑ k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋₊, ∫ (t : ℝ) in ↑k..↑(k + 1), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n
⊢ IntervalIntegrable (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) volume a ↑⌊b⌋₊
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case inr.e_f.inl
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
this :
  μ.withDensityᵥ f =
    withDensity (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure (fun x ↦ ‖f x‖⁻¹ • f x)
      (ContinuousLinearMap.lsmul ℝ ℝ).flip
x : X
hx : f x = 0
⊢ ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) x = ‖f x‖ₑ

pp: case inr.e_f.inr
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
this :
  μ.withDensityᵥ f =
    withDensity (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure (fun x ↦ ‖f x‖⁻¹ • f x)
      (ContinuousLinearMap.lsmul ℝ ℝ).flip
x : X
hx : f x ≠ 0
⊢ ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) x = ‖f x‖ₑ

Tactic:
·

Goal state after:
case inr.e_f.inl
X : Type u_1
E : Type u_2
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure X
f : X → E
hf : Integrable f μ
hE : Nontrivial E
this✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)
I : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure fun x ↦ ‖f x‖⁻¹ • f x
this :
  μ.withDensityᵥ f =
    withDensity (μ.withDensity fun x ↦ ‖f x‖ₑ).toSignedMeasure (fun x ↦ ‖f x‖⁻¹ • f x)
      (ContinuousLinearMap.lsmul ℝ ℝ).flip
x : X
hx : f x = 0
⊢ ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) x = ‖f x‖ₑ
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ ↑(n ! / (x ! * (n - x)!)) = ↑n ! / (↑x ! * ↑(n - x)!)

pp: case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ x ≤ n

pp: case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ ↑n ! ≠ 0

Tactic:
·

Goal state after:
case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ ↑(n ! / (x ! * (n - x)!)) = ↑n ! / (↑x ! * ↑(n - x)!)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ x ≤ n

pp: case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ ↑n ! ≠ 0

Tactic:
·

Goal state after:
case e_a
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
a b : A
n x : ℕ
hx : x ∈ Finset.range n.succ
⊢ x ≤ n