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

Goal state before:
pp: case inl
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
x : E
ha' : ‖x‖ ≤ a
⊢ ‖x‖ ≤ a ∨ ∃ i, t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

pp: case inr
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
x : E
ha' : a < ‖x‖
⊢ ‖x‖ ≤ a ∨ ∃ i, t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
x : E
ha' : ‖x‖ ≤ a
⊢ ‖x‖ ≤ a ∨ ∃ i, t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: case inr
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
x : E
ha' : a < ‖x‖
⊢ ‖x‖ ≤ a ∨ ∃ i, t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)

Tactic:
·

Goal state after:
case inr
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
x : E
ha' : a < ‖x‖
⊢ ‖x‖ ≤ a ∨ ∃ i, t i < ‖x‖ ∧ ‖x‖ ≤ t (i + 1)
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case inl
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
⊢ iIndepFun (∅.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P

pp: case inr
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
⊢ iIndepFun (s.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P

Tactic:
·

Goal state after:
case inl
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
⊢ iIndepFun (∅.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case inr
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
⊢ iIndepFun (s.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P

Tactic:
·

Goal state after:
case inr
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
⊢ iIndepFun (s.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case inr
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
⊢ iIndepFun (s.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P

Tactic:
let g (x : s) : Fin (s.max' hs + 1) := ⟨x.1, Nat.lt_add_one_of_le (s.le_max' x.1 x.2)⟩

Goal state after:
case inr
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
g : ↥s → Fin (s.max' hs + 1) := fun x ↦ ⟨↑x, ⋯⟩
⊢ iIndepFun (s.restrict fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case inr.refine_1
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
g : ↥s → Fin (s.max' hs + 1) := fun x ↦ ⟨↑x, ⋯⟩
⊢ Function.Injective g

pp: case inr.refine_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
g : ↥s → Fin (s.max' hs + 1) := fun x ↦ ⟨↑x, ⋯⟩
⊢ Monotone fun m ↦ t ↑m

Tactic:
·

Goal state after:
case inr.refine_1
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
g : ↥s → Fin (s.max' hs + 1) := fun x ↦ ⟨↑x, ⋯⟩
⊢ Function.Injective g
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case inr.refine_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
g : ↥s → Fin (s.max' hs + 1) := fun x ↦ ⟨↑x, ⋯⟩
⊢ Monotone fun m ↦ t ↑m

Tactic:
·

Goal state after:
case inr.refine_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
s : Finset ℕ
hs : s.Nonempty
g : ↥s → Fin (s.max' hs + 1) := fun x ↦ ⟨↑x, ⋯⟩
⊢ Monotone fun m ↦ t ↑m
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case e'_6.e'_5.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.succ = min n (↑i + 1)

pp: case e'_6.e'_6.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.castSucc = min n ↑i

pp: case convert_1
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ Monotone t'

pp: case convert_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ EventuallyConst t' atTop

Tactic:
·

Goal state after:
case e'_6.e'_5.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.succ = min n (↑i + 1)
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case e'_6.e'_6.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.castSucc = min n ↑i

pp: case convert_1
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ Monotone t'

pp: case convert_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ EventuallyConst t' atTop

Tactic:
·

Goal state after:
case e'_6.e'_6.e'_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
i : Fin n
ω : Ω
⊢ ↑i.castSucc = min n ↑i
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case convert_1
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ Monotone t'

pp: case convert_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ EventuallyConst t' atTop

Tactic:
·

Goal state after:
case convert_1
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ Monotone t'
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: case convert_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ EventuallyConst t' atTop

Tactic:
·

Goal state after:
case convert_2
T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
⊢ EventuallyConst t' atTop
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : ∀ (t : T), AEMeasurable (X t) P
I J : Finset T
hJI : J ⊆ I
⊢ (fun I ↦ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P) J =
    Measure.map (Finset.restrict₂ hJI ∘ fun ω ↦ I.restrict fun x ↦ X x ω) P

pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : ∀ (t : T), AEMeasurable (X t) P
I J : Finset T
hJI : J ⊆ I
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P

Tactic:
·

Goal state after:
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : ∀ (t : T), AEMeasurable (X t) P
I J : Finset T
hJI : J ⊆ I
⊢ (fun I ↦ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P) J =
    Measure.map (Finset.restrict₂ hJI ∘ fun ω ↦ I.restrict fun x ↦ X x ω) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : ∀ (t : T), AEMeasurable (X t) P
I J : Finset T
hJI : J ⊆ I
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P

Tactic:
·

Goal state after:
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : ∀ (t : T), AEMeasurable (X t) P
I J : Finset T
hJI : J ⊆ I
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: case refine_1
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P

pp: case refine_2
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h :
  ∀ (I : Finset T), Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P
⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P

Tactic:
·

Goal state after:
case refine_1
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: case refine_2
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h :
  ∀ (I : Finset T), Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P
⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P

Tactic:
·

Goal state after:
case refine_2
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h :
  ∀ (I : Finset T), Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P
⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: case refine_1
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P

pp: case refine_2
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ Y x ω) P

pp: case refine_3
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P

pp: case refine_4
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : ∀ (I : Finset T), IdentDistrib (fun ω ↦ I.restrict fun x ↦ X x ω) (fun ω ↦ I.restrict fun x ↦ Y x ω) P P
⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P

Tactic:
·

Goal state after:
case refine_1
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: case refine_2
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ Y x ω) P

pp: case refine_3
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P

pp: case refine_4
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : ∀ (I : Finset T), IdentDistrib (fun ω ↦ I.restrict fun x ↦ X x ω) (fun ω ↦ I.restrict fun x ↦ Y x ω) P P
⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P

Tactic:
·

Goal state after:
case refine_2
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ Y x ω) P
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: case refine_3
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P

pp: case refine_4
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : ∀ (I : Finset T), IdentDistrib (fun ω ↦ I.restrict fun x ↦ X x ω) (fun ω ↦ I.restrict fun x ↦ Y x ω) P P
⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map (fun ω x ↦ Y x ω) P

Tactic:
·

Goal state after:
case refine_3
T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
inst✝ : IsFiniteMeasure P
hX : AEMeasurable (fun ω x ↦ X x ω) P
hY : AEMeasurable (fun ω x ↦ Y x ω) P
h : IdentDistrib (fun ω x ↦ X x ω) (fun ω x ↦ Y x ω) P P
I : Finset T
⊢ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P = Measure.map (fun ω ↦ I.restrict fun x ↦ Y x ω) P