mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
⊢ Integrable (cauchyPDFReal x₀ γ) volume

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: x a r : ℝ
ha : 0 < a
hr : 0 < r
hx : 0 < x
⊢ 0 < gammaPDFReal a r x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
hγ : γ ≠ 0
⊢ AEStronglyMeasurable (cauchyPDFReal x₀ γ) volume

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
hγ : γ ≠ 0
⊢ ∫⁻ (x : ℝ), cauchyPDF x₀ γ x = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
⊢ (cauchyMeasure x₀ γ) Set.univ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n k : ℕ
p : ↑I
⊢ Bin(R, n, p).real {↑k} = ↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
k : ℕ
⊢ 0 ≤ ↑(n.choose k)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
k : ℕ
⊢ 0 ≤ ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
x : ℝ
⊢ 0 ≤ gammaPDFReal a r x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
k : ℕ
⊢ 0 ≤ 1 - ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n k : ℕ
p : ↑I
⊢ Bin(R, n, p) {↑k} ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n k : ℕ
p : ↑I
⊢ Bin(R, n, p) {↑k} = ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
⊢ Bin(n, p).real {0} = (1 - ↑p) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n : ℕ
p : ↑I
⊢ Bin(R, n, p).real {0} = (1 - ↑p) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
⊢ Bin(n, p).real {n} = ↑p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n : ℕ
p : ↑I
⊢ Bin(R, n, p).real {↑n} = ↑p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: p : ↑I
⊢ Bin(1, p) = Ber(1, 0, p)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
⊢ Bin(n, p) = ∑ k ≤ n, ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac k

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r x : ℝ
⊢ exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)

Tactic:
exponentialPDF,

Goal state after:
r x : ℝ
⊢ ENNReal.ofReal (exponentialPDFReal r x) = ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r x : ℝ
⊢ exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r x : ℝ
hx : 0 ≤ x
⊢ exponentialPDF r x = ENNReal.ofReal (r * rexp (-(r * x)))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝² : MeasurableSpace R
inst✝¹ : AddMonoidWithOne R
inst✝ : MeasurableSingletonClass R
n : ℕ
p : ↑I
x✝¹ : ℕ
x✝ : x✝¹ ∈ Finset.Iic n
⊢ map Nat.cast (ENNReal.ofReal (↑(n.choose x✝¹) * ↑p ^ x✝¹ * (1 - ↑p) ^ (n - x✝¹)) • dirac x✝¹) =
    ENNReal.ofReal (↑(n.choose x✝¹) * ↑p ^ x✝¹ * (1 - ↑p) ^ (n - x✝¹)) • dirac ↑x✝¹

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r x : ℝ
⊢ HasDerivAt (fun a ↦ -rexp (-(r * a))) (r * rexp (-(r * x))) x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: b x : ℝ
hb : 0 < b
⊢ IntegrableOn (fun x ↦ rexp (-(b * x))) (Ioc 0 x) volume

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝² : MeasurableSpace R
inst✝¹ : AddMonoidWithOne R
inst✝ : MeasurableSingletonClass R
n : ℕ
p : ↑I
⊢ Bin(R, n, p) = ∑ k ≤ n, ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac ↑k

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
n : ℕ
p : ↑I
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : MeasurableSingletonClass R
f : R → E
⊢ Integrable f Bin(R, n, p)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℕ → E
⊢ ∫ (x : ℕ), f x ∂∑ k ≤ n, ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac k =
    ∑ k ≤ n, (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • f k

Tactic:
integral_finsetSum_measure

Goal state after:
n : ℕ
p : ↑I
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℕ → E
⊢ ∑ i ≤ n, ∫ (a : ℕ), f a ∂ENNReal.ofReal (↑(n.choose i) * ↑p ^ i * (1 - ↑p) ^ (n - i)) • dirac i =
    ∑ k ≤ n, (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • f k

n : ℕ
p : ↑I
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℕ → E
⊢ ∀ i ∈ Finset.Iic n, Integrable f (ENNReal.ofReal (↑(n.choose i) * ↑p ^ i * (1 - ↑p) ^ (n - i)) • dirac i)
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝³ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℕ
hab : a ≤ b
u : (i : ↥(Iic a)) → X ↑i
f : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) u ⊗ₘ traj κ b)
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ ∫ (x : (i : ↥(Iic b)) → X ↑i) in A, ∫ (y : (n : ℕ) → X n), f x y ∂(traj κ b) x ∂(partialTraj κ a b) u =
    ∫ (y : (n : ℕ) → X n) in frestrictLe b ⁻¹' A, f (frestrictLe b y) y ∂(traj κ a) u

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : ¬0 ≤ x
a : ℝ
x✝ : a ≤ x
⊢ ¬0 ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : ¬0 ≤ x
a : ℝ
x✝ : a ≤ x
⊢ ENNReal.ofReal (if 0 ≤ a then r * rexp (-(r * a)) else 0) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℕ → E
x✝¹ : ℕ
x✝ : x✝¹ ∈ Finset.Iic n
⊢ ‖f x✝¹‖ₑ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℕ → E
x✝¹ : ℕ
x✝ : x✝¹ ∈ Finset.Iic n
⊢ ENNReal.ofReal (↑(n.choose x✝¹) * ↑p ^ x✝¹ * (1 - ↑p) ^ (n - x✝¹)) ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n : ℕ
p : ↑I
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℕ → E
⊢ ∫ (x : ℕ), f x ∂Bin(n, p) = ∑ k ≤ n, (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • f k

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: R : Type u_1
inst✝⁵ : MeasurableSpace R
inst✝⁴ : AddMonoidWithOne R
n : ℕ
p : ↑I
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass R
f : R → E
⊢ ∫ (x : R), f x ∂Bin(R, n, p) = ∑ k ≤ n, (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • f ↑k

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝³ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
hf : Integrable f ((traj κ a) x₀)
A : Set ((i : ↥(Iic b)) → X ↑i)
hA : MeasurableSet A
⊢ ∫ (x : (i : ↥(Iic b)) → X ↑i) in A, ∫ (y : (n : ℕ) → X n), f y ∂(traj κ b) x ∂(partialTraj κ a b) x₀ =
    ∫ (y : (n : ℕ) → X n) in frestrictLe b ⁻¹' A, f y ∂(traj κ a) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ EqOn (fun x ↦ ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0))
    (fun x ↦ ENNReal.ofReal (r * rexp (-(r * x)))) (Icc 0 x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
⊢ p a = 0 ↔ a ∉ p.support

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
h : p a = 1
a' : α
ha' : a' ∈ p.support
ha : a' ∉ {a}
⊢ 0 < ∑' (b : α), if b = a then 0 else p b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
h : p a = 1
a' : α
ha' : a' ∈ p.support
ha : a' ∉ {a}
this : 0 < ∑' (b : α), if b = a then 0 else p b
⊢ (p a + ∑' (b : α), if b = a then 0 else p b) = (if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
h : p a = 1
a' : α
ha' : a' ∈ p.support
ha : a' ∉ {a}
this : 0 < ∑' (b : α), if b = a then 0 else p b
⊢ ((if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b) =
    (∑' (b : α), if b = a then p b else 0) + ∑' (b : α), if b = a then 0 else p b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
h : p a = 1
a' : α
ha' : a' ∈ p.support
ha : a' ∉ {a}
this : 0 < ∑' (b : α), if b = a then 0 else p b
b : α
⊢ ((if b = a then p b else 0) + if b = a then 0 else p b) = p b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
⊢ p a = 1 ↔ p.support = {a}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
⊢ p a ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
⊢ p.toOuterMeasure.caratheodory = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
s : Finset α
⊢ p.toOuterMeasure ↑s = ∑ x ∈ s, p x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝⁴ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
a b : ℕ
hab : a ≤ b
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
i_f : Integrable f ((traj κ a) x₀)
⊢ Integrable (fun x ↦ ∫ (y : (n : ℕ) → X n), f y ∂(traj κ b) x) (Measure.map (frestrictLe b) ((traj κ a) x₀))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
a : α
⊢ p.toOuterMeasure {a} = p a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
⊢ ∫ (a : ℝ) in uIoc 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0..x, r * rexp (-(r * a))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
s : Set α
⊢ p.toOuterMeasure s = 0 ↔ Disjoint p.support s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
s : Set α
⊢ p.toOuterMeasure s = 1 ↔ p.support ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0
rightSide :
  ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x =
    ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))
x : ℝ
hx : 0 ≤ x
⊢ 0 x ≤ r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
p : PMF α
s : Set α
⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
a : α
h : MeasurableSet {a}
⊢ p.toMeasure {a} = p a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
s : Set α
hs : MeasurableSet s
⊢ p.toMeasure s = 0 ↔ Disjoint p.support s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
s : Set α
hs : MeasurableSet s
⊢ p.toMeasure s = 0 ↔ Disjoint p.support s

Tactic:
by
  rw [p.toMeasure_apply_eq_toOuterMeasure_apply hs, toOuterMeasure_apply_eq_zero_iff]

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
s t : Set α
hs : MeasurableSet s
h : s ∩ p.support ⊆ t
⊢ p.toMeasure s ≤ p.toMeasure t

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
⊢ p.toMeasure.restrict p.support = p.toMeasure

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h : s ∩ p.support = t ∩ p.support
⊢ p.toMeasure s = p.toMeasure t

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
⊢ Function.Injective toMeasure

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
p : PMF α
inst✝ : MeasurableSingletonClass α
s : Set α
⊢ p.toMeasure s = p.toOuterMeasure s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Countable α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
s : Set α
hs : MeasurableSet s
⊢ μ.toPMF.toMeasure s = μ s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : PMF α
⊢ p.toMeasure Set.univ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ∫⁻ (x : ℝ), gammaPDF a r x = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ (gammaMeasure a r) univ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝² : Countable α
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
p : PMF α
x : α
⊢ p.toMeasure.toPMF x = p x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Countable α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
p : PMF α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ p.toMeasure = μ ↔ p = μ.toPMF

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : Countable α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
p : PMF α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ μ.toPMF = p ↔ μ = p.toMeasure

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
this : ∫ (a : ℝ) in uIoc 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0..x, r * rexp (-(r * a))
⊢ -(r * x) ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
x : ℝ
this : IsProbabilityMeasure (gammaMeasure a r)
b : ℝ
⊢ 0 b ≤ gammaPDFReal a r b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
x : ℝ
⊢ ↑(cdf (gammaMeasure a r)) x = ∫ (x : ℝ) in Iic x, gammaPDFReal a r x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
n : ℕ
⊢ 0 ≤ 1 - ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
h1 : p ≠ 0
h2 : p ≠ 1
n : ℕ
⊢ 0 < 1 - ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
h1 : p ≠ 0
h2 : p ≠ 1
n : ℕ
⊢ 0 < ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
hp : p ≠ 0
n : ℕ
⊢ (geometricMeasure p) {n} = ENNReal.ofReal ((1 - ↑p) ^ n * ↑p)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
hp : p ≠ 0
n : ℕ
⊢ (geometricMeasure p).real {n} = (1 - ↑p) ^ n * ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
h1 : p ≠ 0
h2 : p ≠ 1
n : ℕ
⊢ 0 < (geometricMeasure p).real {n}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
hp : p ≠ 0
⊢ 0 ≤ 1 - ↑p

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
hp : p ≠ 0
⊢ 1 - ↑p < 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
hp : p ≠ 0
⊢ HasSum (fun n ↦ (1 - ↑p) ^ n * ↑p) 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
x : ℝ
⊢ ↑(cdf (gammaMeasure a r)) x = (∫⁻ (x : ℝ) in Iic x, gammaPDF a r x).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
⊢ IsProbabilityMeasure (geometricMeasure p)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℕ → E
hp : p ≠ 0
⊢ ∀ (i : ℕ), ENNReal.ofReal ((1 - ↑p) ^ i * ↑p) ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℕ → E
hp : p ≠ 0
⊢ Integrable f (geometricMeasure p) ↔ Summable fun n ↦ (1 - ↑p) ^ n * ↑p * ‖f n‖

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r x : ℝ
hx : x < t
⊢ paretoPDF t r x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r x : ℝ
hx : t ≤ x
⊢ paretoPDF t r x = ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r x : ℝ
hx : x ≤ t
a : ℝ
a✝ : a < x
⊢ ¬t ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝² : NormedAddCommGroup E
f : ℕ → E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hp : p ≠ 0
hf : Integrable f (geometricMeasure p)
⊢ (fun n ↦ ((1 - ↑p) ^ n * ↑p) • f n) = fun n ↦ (ENNReal.ofReal ((1 - ↑p) ^ n * ↑p)).toReal • f n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r x : ℝ
hx : x ≤ t
⊢ ∫⁻ (y : ℝ) in Iio x, paretoPDF t r y = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
this : ∫ (a : ℝ) in uIoc 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0..x, r * rexp (-(r * a))
⊢ Continuous fun a ↦ rexp (-(r * a))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r x : ℝ
ht : 0 < t
hr : 0 < r
hx : t ≤ x
⊢ 0 < x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝² : NormedAddCommGroup E
f : ℕ → E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hp : p ≠ 0
hf : Integrable f (geometricMeasure p)
this : (fun n ↦ ((1 - ↑p) ^ n * ↑p) • f n) = fun n ↦ (ENNReal.ofReal ((1 - ↑p) ^ n * ↑p)).toReal • f n
⊢ ∀ (i : ℕ), ENNReal.ofReal ((1 - ↑p) ^ i * ↑p) ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r x : ℝ
ht : 0 < t
hr : 0 < r
hx : t ≤ x
⊢ 0 < paretoPDFReal t r x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝² : NormedAddCommGroup E
f : ℕ → E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hp : p ≠ 0
hf : Integrable f (geometricMeasure p)
⊢ HasSum (fun n ↦ ((1 - ↑p) ^ n * ↑p) • f n) (∫ (n : ℕ), f n ∂geometricMeasure p)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
hp : p ≠ 0
f : ℕ → E
⊢ ∀ (i : ℕ), ENNReal.ofReal ((1 - ↑p) ^ i * ↑p) ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
hp : p ≠ 0
f : ℕ → E
⊢ ∫ (n : ℕ), f n ∂geometricMeasure p = ∑' (n : ℕ), ((1 - ↑p) ^ n * ↑p) • f n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Pareto

Goal state before:
pp: t r : ℝ
ht : 0 ≤ t
hr : 0 ≤ r
x : ℝ
⊢ 0 ≤ paretoPDFReal t r x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ℝ
hp_pos : 0 < p
hp_le_one : p ≤ 1
⊢ HasSum (fun n ↦ geometricPMFReal p n) 1

Tactic:
by

Goal state after: