mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) (Set.Icc a ↑⌊b⌋₊) volume

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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) (Set.Icc a ↑⌊b⌋₊) volume
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:
· rw [isCyclic_units_two_mul_iff_of_odd _ hn, isCyclic_units_iff_of_odd hn, or_iff_right]
    · congr! with p m
      rw [Nat.mul_left_cancel_iff zero_lt_two, and_iff_right_of_imp]
      rintro rfl
      contrapose! h2
      cases Nat.lt_one_iff.mp h2
      rw [pow_zero, mul_one]
    · rintro ⟨p, m, -, odd, -, eq⟩
      have := eq ▸ odd.pow
      simp [← Nat.not_even_iff_odd] at this

Goal state after:
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
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)
⊢ ¬(n = 0 ∨ n = 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)
⊢ ¬(n = 0 ∨ n = 1)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
⊢ (algebraMap ℚ A) (1 / ↑0!) = 1

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
⊢ (algebraMap ℚ A) (1 / ↑0!) = 1
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (algebraMap ℚ A) (1 / ↑(n + 1)!) * (↑n + 1) = (algebraMap ℚ A) (1 / ↑n !)

Tactic:
]

Goal state after:
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (algebraMap ℚ A) (1 / ↑(n + 1)!) * (↑n + 1) = (algebraMap ℚ A) (1 / ↑n !)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ↑n + 1 = ↑n + 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
key : ↑n + 1 = (algebraMap ℚ A) (↑n + 1)
⊢ (algebraMap ℚ A) (1 / ((↑n + 1) * ↑n !) * (↑n + 1)) = (algebraMap ℚ A) (1 / ↑n !)

Tactic:
]

Goal state after:
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
key : ↑n + 1 = (algebraMap ℚ A) (↑n + 1)
⊢ (algebraMap ℚ A) (1 / ((↑n + 1) * ↑n !) * (↑n + 1)) = (algebraMap ℚ A) (1 / ↑n !)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case zero
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) f
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)

Tactic:
]

Goal state after:
case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) f
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) f
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)

Tactic:
]

Goal state after:
case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) f
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) (exp A)
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)

Tactic:
]

Goal state after:
case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) (exp A)
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) (exp A)
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
⊢ (coeff n) (exp A) = (coeff n) (exp A)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Exp

Goal state before:
pp: case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) (exp A)
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
h : n.succ • (coeff (n + 1)) f = n.succ • (coeff (n + 1)) (exp A)
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)

Tactic:
]

Goal state after:
case succ
A : Type u_3
inst✝² : CommRing A
inst✝¹ : Algebra ℚ A
inst✝ : IsAddTorsionFree A
f : A⟦X⟧
hd : (d⁄dX A) f = f
hc : constantCoeff f = 1
n : ℕ
ih : (coeff n) f = (coeff n) (exp A)
eq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) (exp A)
eq2 : (coeff (n + 1)) (exp A) * (↑n + 1) = (coeff n) (exp A)
h : n.succ • (coeff (n + 1)) f = n.succ • (coeff (n + 1)) (exp A)
⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
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

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‖ ≠ 0

Tactic:
]

Goal state after:
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‖ ≠ 0
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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) (Set.Icc (↑⌊b⌋₊) b) volume

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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ x ∈ Icc 0 ⌊x⌋₊, c x) (Set.Icc (↑⌊b⌋₊) b) volume
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
⊢ μ.withDensity ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) = 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
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

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
⊢ μ.withDensity ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖‖f x‖⁻¹ • f x‖ₑ) = μ.withDensity fun x ↦ ‖f x‖ₑ

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.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
⊢ (algebraMap ℚ A) (1 / ↑x ! * (1 / ↑(n - x)!)) = (algebraMap ℚ A) (↑(n.choose x) * (1 / ↑n !))

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
⊢ (algebraMap ℚ A) (1 / ↑x ! * (1 / ↑(n - x)!)) = (algebraMap ℚ A) (↑(n.choose x) * (1 / ↑n !))
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
⊢ 1 / (↑x ! * ↑(n - x)!) = ↑(n.choose x) / ↑n !

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
⊢ 1 / (↑x ! * ↑(n - x)!) = ↑(n.choose x) / ↑n !
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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) (Set.Icc a ↑⌊b⌋₊) volume

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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) (Set.Icc a ↑⌊b⌋₊) volume
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)!)

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

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
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 - x)! ∣ n !

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 - x)! ∣ n !
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
h : ↑n ! = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
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
h'x : ‖f x‖ ≠ 0
⊢ ‖f x‖ₑ = ‖f x‖ₑ

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: n : ℕ
⊢ ∑ i ∈ range (n + 1), (monomial (n - i)) (_root_.bernoulli i * ↑(n.choose i)) =
    ∑ j ∈ range (n + 1), (monomial (n - j)) (_root_.bernoulli (n - (n - j)) * ↑(n.choose (n - j)))

Tactic:
]

Goal state after:
n : ℕ
⊢ ∑ i ∈ range (n + 1), (monomial (n - i)) (_root_.bernoulli i * ↑(n.choose i)) =
    ∑ j ∈ range (n + 1), (monomial (n - j)) (_root_.bernoulli (n - (n - j)) * ↑(n.choose (n - j)))
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: n x : ℕ
hx : x ≤ n
⊢ (monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x)) =
    (monomial (n - x)) (_root_.bernoulli (n - (n - x)) * ↑(n.choose (n - x)))

Tactic:
]

Goal state after:
n x : ℕ
hx : x ≤ n
⊢ (monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x)) =
    (monomial (n - x)) (_root_.bernoulli (n - (n - x)) * ↑(n.choose (n - x)))
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: n x : ℕ
hx : x ≤ n
⊢ (monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x)) = (monomial (n - x)) (_root_.bernoulli x * ↑(n.choose x))

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) (Set.Icc a (↑⌊a⌋₊ + 1)) volume

Tactic:
]

Goal state after:
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
⊢ IntegrableOn (fun x ↦ deriv f x * ∑ n ∈ Icc 0 ⌊x⌋₊, c n) (Set.Icc a (↑⌊a⌋₊ + 1)) volume
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: n : ℕ
⊢ _root_.bernoulli n = _root_.bernoulli n

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: ⊢ _root_.bernoulli 0 * ↑(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * ↑(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +
        _root_.bernoulli 2 * ↑(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +
      _root_.bernoulli 3 * ↑(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =
    3 / 64

Tactic:
]

Goal state after:
⊢ _root_.bernoulli 0 * ↑(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * ↑(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +
        _root_.bernoulli 2 * ↑(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +
      _root_.bernoulli 3 * ↑(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =
    3 / 64
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case e'_2.h.e'_4
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
n m : ℕ
h : n ≤ m
hf_diff : ∀ t ∈ Set.Icc ↑n ↑m, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc ↑n ↑m) volume
⊢ n = n

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: ⊢ 1 * ↑(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * ↑(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +
        1 / 6 * ↑(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +
      0 * ↑(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =
    3 / 64

Tactic:
]

Goal state after:
⊢ 1 * ↑(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * ↑(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +
        1 / 6 * ↑(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +
      0 * ↑(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =
    3 / 64
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case e'_2.h.e'_5
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
n m : ℕ
h : n ≤ m
hf_diff : ∀ t ∈ Set.Icc ↑n ↑m, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc ↑n ↑m) volume
⊢ m = m

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case e'_3.e'_5.h
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
n m : ℕ
h : n ≤ m
hf_diff : ∀ t ∈ Set.Icc ↑n ↑m, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc ↑n ↑m) volume
⊢ m = m

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.AbelSummation

Goal state before:
pp: case e'_3.e'_6.h
𝕜 : Type u_1
inst✝ : RCLike 𝕜
c : ℕ → 𝕜
f : ℝ → 𝕜
n m : ℕ
h : n ≤ m
hf_diff : ∀ t ∈ Set.Icc ↑n ↑m, DifferentiableAt ℝ f t
hf_int : IntegrableOn (deriv f) (Set.Icc ↑n ↑m) volume
⊢ n = n

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: k : ℕ
⊢ ∑ x ∈ range (k + 1), (monomial (k - x)) (_root_.bernoulli x * ↑((k + 1).choose x) * ↑(k + 1 - x)) =
    ∑ i ∈ range (k + 1), (↑k + 1) * (monomial (k - i)) (_root_.bernoulli i * ↑(k.choose i))

Tactic:
]

Goal state after:
k : ℕ
⊢ ∑ x ∈ range (k + 1), (monomial (k - x)) (_root_.bernoulli x * ↑((k + 1).choose x) * ↑(k + 1 - x)) =
    ∑ i ∈ range (k + 1), (↑k + 1) * (monomial (k - i)) (_root_.bernoulli i * ↑(k.choose i))
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: k m : ℕ
x✝ : m ∈ range (k + 1)
⊢ (monomial (k - m)) (_root_.bernoulli m * ↑((k + 1).choose m * (k + 1 - m))) =
    (monomial (k - m)) (_root_.bernoulli m * ↑(k.choose m * (k + 1)))

Tactic:
]

Goal state after:
k m : ℕ
x✝ : m ∈ range (k + 1)
⊢ (monomial (k - m)) (_root_.bernoulli m * ↑((k + 1).choose m * (k + 1 - m))) =
    (monomial (k - m)) (_root_.bernoulli m * ↑(k.choose m * (k + 1)))
Module: Mathlib.NumberTheory.BernoulliPolynomials

Goal state before:
pp: case e_6.e_a
k m : ℕ
x✝ : m ∈ range (k + 1)
⊢ k.choose m * (k + 1) = k.choose m * (k + 1)

Tactic:
]

Goal state after: