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

Goal state before:
pp: α β x : ℝ
hx_pos : 0 < x
hx_lt : x < 1
⊢ ENNReal.ofReal (1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)) =
    ENNReal.ofReal (1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1))

Tactic:
]

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

Goal state before:
pp: α β : ℝ
⊢ ∫⁻ (x : ℝ) in Ioo 0 1, ENNReal.ofReal (1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)) ∂volume =
    ∫⁻ (x : ℝ) in Ioo 0 1, ENNReal.ofReal (1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1))

Tactic:
]

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

Goal state before:
pp: α β x : ℝ
hx1 : 0 < x
hx2 : x < 1
hα : 0 < α
hβ : 0 < β
⊢ 0 < 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)

Tactic:
]

Goal state after:
α β x : ℝ
hx1 : 0 < x
hx2 : x < 1
hα : 0 < α
hβ : 0 < β
⊢ 0 < 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)
Module: Mathlib.Probability.Distributions.Bernoulli

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
x : X
p : ↑I
⊢ Ber(x, x, p) = dirac x

Tactic:
by
  simp [bernoulliMeasure_def, ← add_smul]

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : MeasurableSingletonClass X
inst✝ : MeasurableSingletonClass Y
x y : X
f : X → Y
p : ↑I
hf : ∀ (x : X), AEMeasurable f (dirac x)
⊢ map f (toNNReal p • dirac x) + map f (toNNReal (σ p) • dirac y) =
    toNNReal p • dirac (f x) + toNNReal (σ p) • dirac (f y)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝³ : MeasurableSpace X
inst✝² : MeasurableSpace Y
inst✝¹ : MeasurableSingletonClass X
inst✝ : MeasurableSingletonClass Y
x y : X
f : X → Y
p : ↑I
hf : ∀ (x : X), AEMeasurable f (dirac x)
⊢ map f (toNNReal p • dirac x) + map f (toNNReal (σ p) • dirac y) =
    toNNReal p • dirac (f x) + toNNReal (σ p) • dirac (f y)
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ ∫ (a : ℝ) in Ioo 0 1, 1 / beta α β * a ^ (α - 1) * (1 - a) ^ (β - 1) = 1

pp: case hf
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ 0 ≤ᵐ[volume.restrict (Ioo 0 1)] fun x ↦ 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)

pp: case hfm
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ AEStronglyMeasurable (fun x ↦ 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)) (volume.restrict (Ioo 0 1))

Tactic:
]

Goal state after:
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ ∫ (a : ℝ) in Ioo 0 1, 1 / beta α β * a ^ (α - 1) * (1 - a) ^ (β - 1) = 1

case hf
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ 0 ≤ᵐ[volume.restrict (Ioo 0 1)] fun x ↦ 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)

case hfm
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ AEStronglyMeasurable (fun x ↦ 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)) (volume.restrict (Ioo 0 1))
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

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

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝⁴ : MeasurableSpace X
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
x y : X
p : ↑I
f : X → E
⊢ ∫ (x : X), f x ∂toNNReal p • dirac x + ∫ (x : X), f x ∂toNNReal (σ p) • dirac y = ↑p • f x + (1 - ↑p) • f y

pp: case hμ
X : Type u_1
inst✝⁴ : MeasurableSpace X
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
x y : X
p : ↑I
f : X → E
⊢ Integrable f (toNNReal p • dirac x)

pp: case hν
X : Type u_1
inst✝⁴ : MeasurableSpace X
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
x y : X
p : ↑I
f : X → E
⊢ Integrable f (toNNReal (σ p) • dirac y)

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁴ : MeasurableSpace X
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
x y : X
p : ↑I
f : X → E
⊢ ∫ (x : X), f x ∂toNNReal p • dirac x + ∫ (x : X), f x ∂toNNReal (σ p) • dirac y = ↑p • f x + (1 - ↑p) • f y

case hμ
X : Type u_1
inst✝⁴ : MeasurableSpace X
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
x y : X
p : ↑I
f : X → E
⊢ Integrable f (toNNReal p • dirac x)

case hν
X : Type u_1
inst✝⁴ : MeasurableSpace X
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
x y : X
p : ↑I
f : X → E
⊢ Integrable f (toNNReal (σ p) • dirac y)
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ ∫ (t : ℝ) in Ioc 0 1, t ^ (α - 1) * (1 - t) ^ (β - 1) =
    ∫ (x : ℝ) in Ioc 0 1, RCLike.re (↑x ^ (↑α - 1) * (1 - ↑x) ^ (↑β - 1)) ∂volume

pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ Integrable (fun x ↦ ↑x ^ (↑α - 1) * (1 - ↑x) ^ (↑β - 1)) (volume.restrict (Ioc 0 1))

Tactic:
]

Goal state after:
α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ ∫ (t : ℝ) in Ioc 0 1, t ^ (α - 1) * (1 - t) ^ (β - 1) =
    ∫ (x : ℝ) in Ioc 0 1, RCLike.re (↑x ^ (↑α - 1) * (1 - ↑x) ^ (↑β - 1)) ∂volume

α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ Integrable (fun x ↦ ↑x ^ (↑α - 1) * (1 - ↑x) ^ (↑β - 1)) (volume.restrict (Ioc 0 1))
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
x : ℝ
x✝ : x ∈ Ioc 0 1
hx1 : 0 < x
hx₂ : x ≤ 1
⊢ x ^ (α - 1) * (1 - x) ^ (β - 1) = x ^ (α - 1) * (1 - x) ^ (β - 1)

pp: case hx
α β : ℝ
hα : 0 < α
hβ : 0 < β
x : ℝ
x✝ : x ∈ Ioc 0 1
hx1 : 0 < x
hx₂ : x ≤ 1
⊢ 0 ≤ 1 - x

pp: case hx
α β : ℝ
hα : 0 < α
hβ : 0 < β
x : ℝ
x✝ : x ∈ Ioc 0 1
hx1 : 0 < x
hx₂ : x ≤ 1
⊢ 0 ≤ x

Tactic:
]

Goal state after:
case hx
α β : ℝ
hα : 0 < α
hβ : 0 < β
x : ℝ
x✝ : x ∈ Ioc 0 1
hx1 : 0 < x
hx₂ : x ≤ 1
⊢ 0 ≤ 1 - x

case hx
α β : ℝ
hα : 0 < α
hβ : 0 < β
x : ℝ
x✝ : x ∈ Ioc 0 1
hx1 : 0 < x
hx₂ : x ≤ 1
⊢ 0 ≤ x
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝² : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝ : NormedAddCommGroup E
a b : ℕ
hab : a ≤ b
f : ((n : ℕ) → X n) → E
x₀ : (i : ↥(Iic a)) → X ↑i
hf : AEStronglyMeasurable f ((traj κ b ∘ₖ partialTraj κ a b) x₀)
⊢ ∀ᵐ (x : (i : ↥(Iic b)) → X ↑i) ∂(partialTraj κ a b) x₀, AEStronglyMeasurable f ((traj κ b) x)

Tactic:
]

Goal state after:
X : ℕ → Type u_1
inst✝² : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)
E : Type u_2
inst✝ : NormedAddCommGroup E
a b : ℕ
hab : a ≤ b
f : ((n : ℕ) → X n) → E
x₀ : (i : ↥(Iic a)) → X ↑i
hf : AEStronglyMeasurable f ((traj κ b ∘ₖ partialTraj κ a b) x₀)
⊢ ∀ᵐ (x : (i : ↥(Iic b)) → X ↑i) ∂(partialTraj κ a b) x₀, AEStronglyMeasurable f ((traj κ b) x)
Module: Mathlib.Probability.Distributions.Beta

Goal state before:
pp: α β : ℝ
hα : 0 < α
hβ : 0 < β
⊢ Integrable (fun x ↦ ↑x ^ (↑α - 1) * (1 - ↑x) ^ (↑β - 1)) (volume.restrict (Ioc 0 1)) ↔
    Integrable (fun x ↦ ↑x ^ (↑α - 1) * (1 - ↑x) ^ (↑β - 1)) (volume.restrict (Ioc 0 1))

Tactic:
]

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

Goal state before:
pp: case e'_4
α β : ℝ
hα : 0 < α
hβ : 0 < β
x : ℝ
hx : x ∈ Ioo 0 1
⊢ (fun x ↦ 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)) x = 1 / beta α β * x ^ (α - 1) * (1 - x) ^ (β - 1)

Tactic:
]

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 : ℕ
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
mf : AEStronglyMeasurable f ((traj κ a) x₀)
⊢ ∫ (x : (i : ℕ) → X i), f (updateFinset x (Iic a) x₀) ∂(traj κ a) x₀ =
    ∫ (x : (n : ℕ) → X n), f (updateFinset x (Iic a) x₀) ∂(traj κ a) x₀

pp: case hφ
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 : ℕ
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
mf : AEStronglyMeasurable f ((traj κ a) x₀)
⊢ AEMeasurable (fun x ↦ updateFinset x (Iic a) x₀) ((traj κ a) x₀)

pp: case hfm
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 : ℕ
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
mf : AEStronglyMeasurable f ((traj κ a) x₀)
⊢ AEStronglyMeasurable f (Measure.map (fun x ↦ updateFinset x (Iic a) x₀) ((traj κ a) x₀))

Tactic:
]

Goal state after:
case hφ
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 : ℕ
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
mf : AEStronglyMeasurable f ((traj κ a) x₀)
⊢ AEMeasurable (fun x ↦ updateFinset x (Iic a) x₀) ((traj κ a) x₀)

case hfm
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 : ℕ
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
mf : AEStronglyMeasurable f ((traj κ a) x₀)
⊢ AEStronglyMeasurable f (Measure.map (fun x ↦ updateFinset x (Iic a) x₀) ((traj κ a) x₀))
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 : ℕ
x₀ : (i : ↥(Iic a)) → X ↑i
f : ((n : ℕ) → X n) → E
mf : AEStronglyMeasurable f ((traj κ a) x₀)
⊢ (traj κ a) x₀ = (traj κ a) x₀

Tactic:
]

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

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
x₀ : (n : ↥(Iic a)) → X ↑n
hf : (fun x ↦ (frestrictLe b x, x (b + 1))) = (Prod.map id fun x ↦ x (b + 1)) ∘ fun x ↦ (frestrictLe b x, x)
⊢ (partialTraj κ a b) x₀ ⊗ₘ κ b = (partialTraj κ a b) x₀ ⊗ₘ κ b

Tactic:
]

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 : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)
hf' : Integrable (uncurry f) (Measure.map (fun x ↦ (frestrictLe b x, x)) ((traj κ a) x₀))
⊢ ∫ (x : (i : ↥(Iic b)) → X ↑i), ∫ (y : (n : ℕ) → X n), f x y ∂(traj κ b) x ∂(partialTraj κ a b) x₀ =
    ∫ (x : (n : ℕ) → X n), f (frestrictLe b x) x ∂(traj κ a) x₀

Tactic:
]

Goal state after:
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 : ((i : ↥(Iic b)) → X ↑i) → ((n : ℕ) → X n) → E
hf : Integrable (uncurry f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)
hf' : Integrable (uncurry f) (Measure.map (fun x ↦ (frestrictLe b x, x)) ((traj κ a) x₀))
⊢ ∫ (x : (i : ↥(Iic b)) → X ↑i), ∫ (y : (n : ℕ) → X n), f x y ∂(traj κ b) x ∂(partialTraj κ a b) x₀ =
    ∫ (x : (n : ℕ) → X n), f (frestrictLe b x) x ∂(traj κ a) x₀
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
x : ℝ
⊢ π⁻¹ * ↑γ * ((x - x₀) ^ 2 + ↑γ ^ 2)⁻¹ = π⁻¹ * ↑γ⁻¹ * (1 + ((x - x₀) / ↑γ) ^ 2)⁻¹

Tactic:
]

Goal state after:
x₀ : ℝ
γ : ℝ≥0
x : ℝ
⊢ π⁻¹ * ↑γ * ((x - x₀) ^ 2 + ↑γ ^ 2)⁻¹ = π⁻¹ * ↑γ⁻¹ * (1 + ((x - x₀) / ↑γ) ^ 2)⁻¹
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: y z : ℝ
f : ℝ → ℝ≥0∞
hzy : z ≤ y
⊢ (∫⁻ (a : ℝ) in Iio z, f a) + ∫⁻ (a : ℝ) in Icc z y, f a = (∫⁻ (x : ℝ) in Iio z, f x) + ∫⁻ (x : ℝ) in Icc z y, f x

pp: y z : ℝ
f : ℝ → ℝ≥0∞
hzy : z ≤ y
⊢ Disjoint (Iio z) (Icc z y)

Tactic:
]

Goal state after:
y z : ℝ
f : ℝ → ℝ≥0∞
hzy : z ≤ y
⊢ Disjoint (Iio z) (Icc z y)
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: Ω : Type u_2
m : MeasurableSpace Ω
P : Measure Ω
n : ℕ
p : ↑I
X : Ω → ℕ
hX : HasLaw X Bin(n, p) P
⊢ ∀ᵐ (x : Set ℕ) ∂setBer(Iio n, p), x.ncard ≤ n

Tactic:
]

Goal state after:
Ω : Type u_2
m : MeasurableSpace Ω
P : Measure Ω
n : ℕ
p : ↑I
X : Ω → ℕ
hX : HasLaw X Bin(n, p) P
⊢ ∀ᵐ (x : Set ℕ) ∂setBer(Iio n, p), x.ncard ≤ n
Module: Mathlib.Probability.Distributions.Gamma

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

pp: x a r : ℝ
hx : x ≤ 0
⊢ EqOn (gammaPDF a r) (fun x ↦ 0) (Iio x)

Tactic:
]

Goal state after:
x a r : ℝ
hx : x ≤ 0
⊢ ∫⁻ (x : ℝ) in Iio x, 0 = 0

x a r : ℝ
hx : x ≤ 0
⊢ EqOn (gammaPDF a r) (fun x ↦ 0) (Iio x)
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: x a r : ℝ
hx : x ≤ 0
⊢ ENNReal.ofReal 0 = ENNReal.ofReal 0

Tactic:
]

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

Goal state before:
pp: n k : ℕ
p : ↑I
⊢ ↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k) = ↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)

Tactic:
]

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 ((partialTraj κ a b) x₀ ⊗ₘ traj κ b).snd
⊢ Integrable (uncurry fun x ↦ f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)

Tactic:
]

Goal state after:
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 ((partialTraj κ a b) x₀ ⊗ₘ traj κ b).snd
⊢ Integrable (uncurry fun x ↦ f) ((partialTraj κ a b) x₀ ⊗ₘ traj κ b)
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
hγ : γ ≠ 0
x : ℝ
⊢ 0 < π⁻¹ * ↑γ * ((x - x₀) ^ 2 + ↑γ ^ 2)⁻¹

Tactic:
]

Goal state after:
x₀ : ℝ
γ : ℝ≥0
hγ : γ ≠ 0
x : ℝ
⊢ 0 < π⁻¹ * ↑γ * ((x - x₀) ^ 2 + ↑γ ^ 2)⁻¹
Module: Mathlib.Probability.Distributions.Binomial

Goal state before:
pp: n k : ℕ
p : ↑I
⊢ ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) =
    ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k))

Tactic:
]

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

Goal state before:
pp: x a✝¹ r : ℝ
hx : x ≤ 0
a : ℝ
a✝ : a < x
⊢ 0 ≤ 0

Tactic:
]

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(n, p).real (Nat.cast ⁻¹' {↑k}) = ↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)

Tactic:
]

Goal state after:
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(n, p).real (Nat.cast ⁻¹' {↑k}) = ↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: x₀ : ℝ
γ : ℝ≥0
hγ : γ ≠ 0
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ 0 = 0

Tactic:
]

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
⊢ ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) =
    ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k))

Tactic:
]

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
⊢ ↑(n.choose 0) * ↑p ^ 0 * (1 - ↑p) ^ (n - 0) = (1 - ↑p) ^ n

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : MeasurableSpace R
inst✝² : AddMonoidWithOne R
inst✝¹ : MeasurableSingletonClass R
inst✝ : CharZero R
n : ℕ
p : ↑I
⊢ ↑(n.choose 0) * ↑p ^ 0 * (1 - ↑p) ^ (n - 0) = (1 - ↑p) ^ n
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 : (n : ℕ) → X n), A.indicator (fun x_1 ↦ f x_1 x) (frestrictLe b x) ∂(traj κ a) u =
    ∫ (y : (n : ℕ) → X n) in frestrictLe b ⁻¹' A, f (frestrictLe b y) y ∂(traj κ a) u

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
⊢ Integrable (uncurry fun x y ↦ A.indicator (fun x ↦ f x y) x) ((partialTraj κ a b) u ⊗ₘ traj κ b)

Tactic:
]

Goal state after:
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 : (n : ℕ) → X n), A.indicator (fun x_1 ↦ f x_1 x) (frestrictLe b x) ∂(traj κ a) u =
    ∫ (y : (n : ℕ) → X n) in frestrictLe b ⁻¹' A, f (frestrictLe b y) y ∂(traj κ a) u

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
⊢ Integrable (uncurry fun x y ↦ A.indicator (fun x ↦ f x y) x) ((partialTraj κ a b) u ⊗ₘ traj κ b)
Module: Mathlib.Probability.Distributions.Binomial

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

pp: case h₀
n : ℕ
p : ↑I
k : ℕ
⊢ ∀ b ∈ Finset.Iic n, b ≠ k → (ENNReal.ofReal (↑(n.choose b) * ↑p ^ b * (1 - ↑p) ^ (n - b)) • dirac b) {k} = 0

pp: case h₁
n : ℕ
p : ↑I
k : ℕ
⊢ k ∉ Finset.Iic n → (ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac k) {k} = 0

Tactic:
]

Goal state after:
n : ℕ
p : ↑I
k : ℕ
⊢ ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) =
    (ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac k) {k}

case h₀
n : ℕ
p : ↑I
k : ℕ
⊢ ∀ b ∈ Finset.Iic n, b ≠ k → (ENNReal.ofReal (↑(n.choose b) * ↑p ^ b * (1 - ↑p) ^ (n - b)) • dirac b) {k} = 0

case h₁
n : ℕ
p : ↑I
k : ℕ
⊢ k ∉ Finset.Iic n → (ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac k) {k} = 0
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
⊢ ∑ i ≤ n, map Nat.cast (ENNReal.ofReal (↑(n.choose i) * ↑p ^ i * (1 - ↑p) ^ (n - i)) • dirac i) =
    ∑ k ≤ n, ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac ↑k

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : MeasurableSpace R
inst✝¹ : AddMonoidWithOne R
inst✝ : MeasurableSingletonClass R
n : ℕ
p : ↑I
⊢ ∑ i ≤ n, map Nat.cast (ENNReal.ofReal (↑(n.choose i) * ↑p ^ i * (1 - ↑p) ^ (n - i)) • dirac i) =
    ∑ k ≤ n, ENNReal.ofReal (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • dirac ↑k
Module: Mathlib.Probability.Distributions.Exponential

Goal state before:
pp: r x : ℝ
⊢ ENNReal.ofReal (if 0 ≤ x then r ^ 1 / Gamma 1 * x ^ (1 - 1) * rexp (-(r * x)) else 0) =
    ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)

Tactic:
]

Goal state after:
r x : ℝ
⊢ ENNReal.ofReal (if 0 ≤ x then r ^ 1 / Gamma 1 * x ^ (1 - 1) * rexp (-(r * x)) else 0) =
    ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)
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
⊢ 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:
]

Goal state after: