state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rfl | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case succ
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' ⟨this, _⟩ | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case succ
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | ext ω | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case succ.h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [lowerCrossingTime, this, hitting, Set.mem_Iic] | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case succ.h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | split_ifs with h₁ h₂ h₂ | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [hf' ω] | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' False.elim (h₂ _) | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_all only [Set.mem_Iic, not_true_eq_false] | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' False.elim (h₁ _) | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_all only [Set.mem_Iic] | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
upperCrossingTime 0 (b - a) (fun n ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rfl | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n := by
have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤... | Mathlib.Probability.Martingale.Upcrossing.670_0.80Cpy4Qgm9i1y9y | theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
⊢ upcrossingsBefore 0 (b - a) (fun n ω => (f n ω - a)⁺) N ω = upcrossingsBefore a b f N ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [upcrossingsBefore, (crossing_pos_eq hab).1] | theorem upcrossingsBefore_pos_eq (hab : a < b) :
upcrossingsBefore 0 (b - a) (fun n ω => (f n ω - a)⁺) N ω = upcrossingsBefore a b f N ω := by
| Mathlib.Probability.Martingale.Upcrossing.719_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_pos_eq (hab : a < b) :
upcrossingsBefore 0 (b - a) (fun n ω => (f n ω - a)⁺) N ω = upcrossingsBefore a b f N ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' le_trans (le_of_eq _)
(integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_const _ _ _)).pos
(fun ω => LatticeOrderedGroup.pos_nonneg _)
(fun ω => LatticeOrderedGroup.pos_nonneg _) (sub_pos.2 hab)) | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by
| Mathlib.Probability.Martingale.Upcrossing.724_0.80Cpy4Qgm9i1y9y | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ =
(b - a - 0) * ∫ (x : Ω), ↑(upcrossingsBefore 0 (b - a) (f - fun x x => a)... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [sub_zero, ← upcrossingsBefore_pos_eq hab] | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by
refine' le_trans (le_of_eq _)
(integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_co... | Mathlib.Probability.Martingale.Upcrossing.724_0.80Cpy4Qgm9i1y9y | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore 0 (b - a) (fun n ω => (f n ω - a)⁺) N x) ∂μ =
(b - a) * ∫ (x : Ω), ↑(upcrossingsBefore 0 ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rfl | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] := by
refine' le_trans (le_of_eq _)
(integral_mul_upcrossingsBefore_le_integral (hf.sub_martingale (martingale_co... | Mathlib.Probability.Martingale.Upcrossing.724_0.80Cpy4Qgm9i1y9y | theorem mul_integral_upcrossingsBefore_le_integral_pos_part_aux [IsFiniteMeasure μ]
(hf : Submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
N : ℕ
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | by_cases hab : a < b | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib.Probability.Martingale.Upcrossing.735_0.80Cpy4Qgm9i1y9y | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
N : ℕ
hab : a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact mul_integral_upcrossingsBefore_le_integral_pos_part_aux hf hab | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib.Probability.Martingale.Upcrossing.735_0.80Cpy4Qgm9i1y9y | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
N : ℕ
hab : ¬a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [not_lt, ← sub_nonpos] at hab | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib.Probability.Martingale.Upcrossing.735_0.80Cpy4Qgm9i1y9y | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
N : ℕ
hab : b - a ≤ 0
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (integral_nonneg fun ω => Nat.cast_nonneg _))
(integral_nonneg fun ω => LatticeOrderedGroup.pos_nonneg _) | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib.Probability.Martingale.Upcrossing.735_0.80Cpy4Qgm9i1y9y | /-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossingsBefore a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossingsBefore a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. -/
theorem ... | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
⊢ upcrossingsBefore a b f N ω = ∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | by_cases hN : N = 0 | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
| Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : N = 0
⊢ upcrossingsBefore a b f N ω = ∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp [hN] | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
⊢ upcrossingsBefore a b f N ω = ∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω hab))] | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
| Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
⊢ upcrossingsBefore a b f N ω =
∑ i in Finset.Ico (Nat.succ 0) (Nat.succ (upcrossingsBefore a b f N ω)),
Set.indicator {n | upperCrossingTime a b f N n ω ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | have h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1),
{n : ℕ | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1 := by
rintro k hk
rw [Finset.mem_Ico] at hk
rw [Set.indicator_of_mem]
· rfl
· exact upperCrossingTime_lt_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab
(Nat.lt_... | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
⊢ ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1 | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rintro k hk | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
k : ℕ
hk : k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1)
⊢ Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1 | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Finset.mem_Ico] at hk | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
k : ℕ
hk : 1 ≤ k ∧ k < upcrossingsBefore a b f N ω + 1
⊢ Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1 | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Set.indicator_of_mem] | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
k : ℕ
hk : 1 ≤ k ∧ k < upcrossingsBefore a b f N ω + 1
⊢ OfNat.ofNat 1 k = 1 | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rfl | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
k : ℕ
hk : 1 ≤ k ∧ k < upcrossingsBefore a b f N ω + 1
⊢ k ∈ {n | upperCrossingTime a b f N n ω < N} | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact upperCrossingTime_lt_of_le_upcrossingsBefore (zero_lt_iff.2 hN) hab
(Nat.lt_succ_iff.1 hk.2) | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
⊢ upcrossingsBefore a b f N ω =
∑ i in Fi... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | have h₂ : ∀ k ∈ Finset.Ico (upcrossingsBefore a b f N ω + 1) (N + 1),
{n : ℕ | upperCrossingTime a b f N n ω < N}.indicator 1 k = 0 := by
rintro k hk
rw [Finset.mem_Ico, Nat.succ_le_iff] at hk
rw [Set.indicator_of_not_mem]
simp only [Set.mem_setOf_eq, not_lt]
exact (upperCrossingTime_eq_of_upc... | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
⊢ ∀ k ∈ Finset.Ico (upcrossingsBefore a b f N ω + 1) (... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rintro k hk | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
k : ℕ
hk : k ∈ Finset.Ico (upcrossingsBefore a b f N ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Finset.mem_Ico, Nat.succ_le_iff] at hk | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
k : ℕ
hk : upcrossingsBefore a b f N ω < k ∧ k < N + 1... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Set.indicator_of_not_mem] | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
k : ℕ
hk : upcrossingsBefore a b f N ω < k ∧ k ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [Set.mem_setOf_eq, not_lt] | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
k : ℕ
hk : upcrossingsBefore a b f N ω < k ∧ k ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact (upperCrossingTime_eq_of_upcrossingsBefore_lt hab hk.1).symm.le | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 k = 1
h₂ :
∀ k ∈ Finset.Ico (upcrossingsBefore a ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Finset.sum_congr rfl h₁, Finset.sum_congr rfl h₂, Finset.sum_const, Finset.sum_const,
smul_eq_mul, mul_one, smul_eq_mul, mul_zero, Nat.card_Ico, Nat.add_succ_sub_one,
add_zero, add_zero] | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
by_cases hN : N = 0
· simp [hN]
rw [← Finset.sum_Ico_consecutive _ (Nat.succ_le_succ zero_le')
(Nat.succ_le_succ (upcrossingsBefore_le f ω... | Mathlib.Probability.Martingale.Upcrossing.775_0.80Cpy4Qgm9i1y9y | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hab : a < b
⊢ Measurable (upcrossingsBefore a b f N) | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | have : upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
ext ω
exact upcrossingsBefore_eq_sum hab | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) := by
| Mathlib.Probability.Martingale.Upcrossing.801_0.80Cpy4Qgm9i1y9y | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hab : a < b
⊢ upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | ext ω | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) := by
have : upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
| Mathlib.Probability.Martingale.Upcrossing.801_0.80Cpy4Qgm9i1y9y | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) | Mathlib_Probability_Martingale_Upcrossing |
case h
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hab : a < b
ω : Ω
⊢ upcrossingsBefore a b f N ω = ∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact upcrossingsBefore_eq_sum hab | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) := by
have : upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
ext ω
| Mathlib.Probability.Martingale.Upcrossing.801_0.80Cpy4Qgm9i1y9y | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hab : a < b
this :
upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i
⊢ Measurable (upcrossingsBefore a... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [this] | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) := by
have : upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
ext ω
exact upcrossingsBefore_eq_sum hab
... | Mathlib.Probability.Martingale.Upcrossing.801_0.80Cpy4Qgm9i1y9y | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
hf : Adapted ℱ f
hab : a < b
this :
upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), Set.indicator {n | upperCrossingTime a b f N n ω < N} 1 i
⊢ Measurable fun ω => ∑ i in Fins... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact Finset.measurable_sum _ fun i _ => Measurable.indicator measurable_const <|
ℱ.le N _ (hf.isStoppingTime_upperCrossingTime.measurableSet_lt_of_pred N) | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) := by
have : upcrossingsBefore a b f N = fun ω =>
∑ i in Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i := by
ext ω
exact upcrossingsBefore_eq_sum hab
... | Mathlib.Probability.Martingale.Upcrossing.801_0.80Cpy4Qgm9i1y9y | theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
Measurable (upcrossingsBefore a b f N) | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Adapted ℱ f
hab : a < b
⊢ ∀ᵐ (ω : Ω) ∂μ, ‖↑(upcrossingsBefore a b f N ω)‖ ≤ ↑N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' eventually_of_forall fun ω => _ | theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ :=
haveI : ∀ᵐ ω ∂μ, ‖(upcrossingsBefore a b f N ω : ℝ)‖ ≤ N := by
| Mathlib.Probability.Martingale.Upcrossing.812_0.80Cpy4Qgm9i1y9y | theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Adapted ℱ f
hab : a < b
ω : Ω
⊢ ‖↑(upcrossingsBefore a b f N ω)‖ ≤ ↑N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Real.norm_eq_abs, Nat.abs_cast, Nat.cast_le] | theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ :=
haveI : ∀ᵐ ω ∂μ, ‖(upcrossingsBefore a b f N ω : ℝ)‖ ≤ N := by
refine' eventually_of_forall fun ω => _
| Mathlib.Probability.Martingale.Upcrossing.812_0.80Cpy4Qgm9i1y9y | theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Adapted ℱ f
hab : a < b
ω : Ω
⊢ upcrossingsBefore a b f N ω ≤ N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' upcrossingsBefore_le _ _ hab | theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ :=
haveI : ∀ᵐ ω ∂μ, ‖(upcrossingsBefore a b f N ω : ℝ)‖ ≤ N := by
refine' eventually_of_forall fun ω => _
rw [Real.norm_eq_abs, Nat.abs_cast, Nat.ca... | Mathlib.Probability.Martingale.Upcrossing.812_0.80Cpy4Qgm9i1y9y | theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
⊢ upcrossings a b f ω < ⊤ ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩
exact lt_of_le_of_lt hk ENNReal.coe_lt_top | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
| Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
⊢ upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | constructor | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
| Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
⊢ upcrossings a b f ω < ⊤ → ∃ k, upcrossings a b f ω ≤ ↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | intro h | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
h : upcrossings a b f ω < ⊤
⊢ ∃ k, upcrossings a b f ω ≤ ↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
| Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
r : ℝ≥0
hr : ↑r = upcrossings a b f ω
upcrossings_lt_top_iff✝ upcrossings_lt_top_iff : ↑r < ⊤ ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k
h✝ h : ↑r < ⊤
⊢ ∃ k, ↑r ≤ ↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact ⟨r, le_rfl⟩ | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
| Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mpr
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
⊢ (∃ k, upcrossings a b f ω ≤ ↑k) → upcrossings a b f ω < ⊤ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rintro ⟨k, hk⟩ | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mpr.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
k : ℝ≥0
hk : upcrossings a b f ω ≤ ↑k
⊢ upcrossings a b f ω < ⊤ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact lt_of_le_of_lt hk ENNReal.coe_lt_top | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
⊢ upcrossings a b f ω < ⊤ ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [this, upcrossings, iSup_le_iff] | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
⊢ (∃ k, ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k) ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | constructor | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
⊢ (∃ k, ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k) → ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rintro ⟨k, hk⟩ | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mpr
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
⊢ (∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k) → ∃ k, ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rintro ⟨k, hk⟩ | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
k : ℝ≥0
hk : ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k
⊢ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | obtain ⟨m, hm⟩ := exists_nat_ge k | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp.intro.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m✝ : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
k : ℝ≥0
hk : ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k
m : ℕ
hm : k ≤ ↑m
⊢ ∃ k, ∀ (N : ℕ), upcrossingsBefore a... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' ⟨m, fun N => Nat.cast_le.1 ((hk N).trans _)⟩ | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mp.intro.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m✝ : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
k : ℝ≥0
hk : ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k
m : ℕ
hm : k ≤ ↑m
N : ℕ
⊢ ↑k ≤ ↑m | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rwa [← ENNReal.coe_nat, ENNReal.coe_le_coe] | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mpr.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
k : ℕ
hk : ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k
⊢ ∃ k, ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' ⟨k, fun N => _⟩ | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
case mpr.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
k : ℕ
hk : ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k
N : ℕ
⊢ ↑(upcrossingsBefore a b f N ω) ≤ ↑↑k | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [ENNReal.coe_nat, Nat.cast_le, hk N] | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k := by
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k := by
constructor
· intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩
· rintro ⟨k, hk⟩... | Mathlib.Probability.Martingale.Upcrossing.834_0.80Cpy4Qgm9i1y9y | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | by_cases hab : a < b | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [upcrossings] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), ⨆ N, ↑(upcrossingsBefore a b f N ω) ∂μ ≤
⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | have : ∀ N, ∫⁻ ω, ENNReal.ofReal ((f N ω - a)⁺) ∂μ = ENNReal.ofReal (∫ ω, (f N ω - a)⁺ ∂μ) := by
intro N
rw [ofReal_integral_eq_lintegral_ofReal]
· exact (hf.sub_martingale (martingale_const _ _ _)).pos.integrable _
· exact eventually_of_forall fun ω => LatticeOrderedGroup.pos_nonneg _ | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
⊢ ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ) | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | intro N | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
N : ℕ
⊢ ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ) | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [ofReal_integral_eq_lintegral_ofReal] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case hfi
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
N : ℕ
⊢ Integrable fun ω => (f N ω - a)⁺ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact (hf.sub_martingale (martingale_const _ _ _)).pos.integrable _ | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case f_nn
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
N : ℕ
⊢ 0 ≤ᵐ[μ] fun ω => (f N ω - a)⁺ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact eventually_of_forall fun ω => LatticeOrderedGroup.pos_nonneg _ | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
⊢ ENNReal.... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [lintegral_iSup'] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
⊢ ENNReal.... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [this, ENNReal.mul_iSup, iSup_le_iff] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
⊢ ∀ (i : ℕ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | intro N | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
N : ℕ
⊢ E... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [(by simp :
∫⁻ ω, upcrossingsBefore a b f N ω ∂μ = ∫⁻ ω, ↑(upcrossingsBefore a b f N ω : ℝ≥0) ∂μ),
lintegral_coe_eq_integral, ← ENNReal.ofReal_mul (sub_pos.2 hab).le] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
N : ℕ
⊢ ∫⁻ (ω : Ω)... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
N : ℕ
⊢ E... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp_rw [NNReal.coe_nat_cast] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
N : ℕ
⊢ E... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact (ENNReal.ofReal_le_ofReal
(hf.mul_integral_upcrossingsBefore_le_integral_pos_part a b N)).trans
(le_iSup (α := ℝ≥0∞) _ N) | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos.hfi
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
N : ℕ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [NNReal.coe_nat_cast, hf.adapted.integrable_upcrossingsBefore hab] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos.hf
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
⊢ ∀ (n ... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact fun n => measurable_from_top.comp_aemeasurable
(hf.adapted.measurable_upcrossingsBefore hab).aemeasurable | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos.h_mono
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
⊢ ∀... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' eventually_of_forall fun ω N M hNM => _ | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos.h_mono
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [Nat.cast_le] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case pos.h_mono
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N✝ n m : ℕ
ω✝ : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
ω... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact upcrossingsBefore_mono hab hNM ω | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : ¬a < b
⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [not_lt, ← sub_nonpos] at hab | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : b - a ≤ 0
⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [ENNReal.ofReal_of_nonpos hab, zero_mul] | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
case neg
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a✝ b✝ : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : b - a ≤ 0
⊢ 0 ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact zero_le _ | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib.Probability.Martingale.Upcrossing.852_0.80Cpy4Qgm9i1y9y | /-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/
theorem Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [IsFiniteMeasure μ] (a b : ℝ)
(hf : Submartingale f ℱ μ) : ENNReal.ofReal (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ENNReal.ofReal ((f N... | Mathlib_Probability_Martingale_Upcrossing |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
⊢ ∀ (a b c : F ⟶ G), a + b + c = a + (b + c) | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ b✝ c✝ : F ⟶ G
⊢ a✝ + b✝ + c✝ = a✝ + (b✝ + c✝) | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | ext | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
case w.h
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ b✝ c✝ : F ⟶ G
x✝ : C
⊢ (a✝ + b✝ + c✝).app x✝ = (a✝ + (b✝ + c✝)).app x✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | apply add_assoc | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
⊢ ∀ (a : F ⟶ G), 0 + a = a | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ : F ⟶ G
⊢ 0 + a✝ = a✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | ext | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
case w.h
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ : F ⟶ G
x✝ : C
⊢ (0 + a✝).app x✝ = a✝.app x✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | apply zero_add | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
⊢ ∀ (a : F ⟶ G), a + 0 = a | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ : F ⟶ G
⊢ a✝ + 0 = a✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | ext | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
case w.h
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ : F ⟶ G
x✝ : C
⊢ (a✝ + 0).app x✝ = a✝.app x✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | apply add_zero | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
⊢ ∀ (a b : F ⟶ G), a - b = a + -b | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ b✝ : F ⟶ G
⊢ a✝ - b✝ = a✝ + -b✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | ext | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
case w.h
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ b✝ : F ⟶ G
x✝ : C
⊢ (a✝ - b✝).app x✝ = (a✝ + -b✝).app x✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | apply sub_eq_add_neg | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
⊢ ∀ (a : F ⟶ G), -a + a = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ : F ⟶ G
⊢ -a✝ + a✝ = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | ext | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
case w.h
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ : F ⟶ G
x✝ : C
⊢ (-a✝ + a✝).app x✝ = 0.app x✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | apply add_left_neg | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
⊢ ∀ (a b : F ⟶ G), a + b = b + a | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ b✝ : F ⟶ G
⊢ a✝ + b✝ = b✝ + a✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | ext | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
case w.h
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
F G : C ⥤ D
a✝ b✝ : F ⟶ G
x✝ : C
⊢ (a✝ + b✝).app x✝ = (b✝ + a✝).app x✝ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | apply add_comm | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
C : Type u_1
D : Type u_2
inst✝² : Category.{?u.38, u_1} C
inst✝¹ : Category.{?u.42, u_2} D
inst✝ : Preadditive D
⊢ ∀ (P Q R : C ⥤ D) (f f' : P ⟶ Q) (g : Q ⟶ R), (f + f') ≫ g = f ≫ g + f' ≫ g | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f... | intros | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G :=
{ add := fun α β => { app := fun X => α.app X + β.app X }
zero := { app := fun X => 0 }
neg := fun α => { app := fun X => -α.app X }
sub := fun α β => { app := fun X => α.app X - β.app X }
add_assoc := by
... | Mathlib.CategoryTheory.Preadditive.FunctorCategory.27_0.Nvs9V8Hq6lv6L8Y | instance functorCategoryPreadditive : Preadditive (C ⥤ D)
where
homGroup F G | Mathlib_CategoryTheory_Preadditive_FunctorCategory |
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