ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.upperCrossingTime_mono
theorem upperCrossingTime_mono (hnm : n ≤ m) :
    upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω} (hnm : n ≤ m)
Goal: MeasureTheory.upperCrossingTime a b f N n ω ≤ MeasureTheory.upperCrossingTime a b f N m ω
-- MeasureTheory.stoppedValue_lowerCrossingTime
theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
    stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (h : MeasureTheory.lowerCrossingTime a b f N n ω ≠ N)
Goal: MeasureTheory.stoppedValue f (MeasureTheory.lowerCrossingTime a b f N n) ω ≤ a
-- MeasureTheory.stoppedValue_upperCrossingTime
theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
    b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (h : MeasureTheory.upperCrossingTime a b f N (n + 1) ω ≠ N)
Goal: b ≤ MeasureTheory.stoppedValue f (MeasureTheory.upperCrossingTime a b f N (n + 1)) ω
-- MeasureTheory.upperCrossingTime_lt_lowerCrossingTime
theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
    (hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
    upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hab : a < b) (hn : MeasureTheory.lowerCrossingTime a b f N (n + 1) ω ≠ N)
Goal: MeasureTheory.upperCrossingTime a b f N (n + 1) ω < MeasureTheory.lowerCrossingTime a b f N (n + 1) ω
-- MeasureTheory.lowerCrossingTime_lt_upperCrossingTime
theorem lowerCrossingTime_lt_upperCrossingTime (hab : a < b)
    (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
    lowerCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hab : a < b) (hn : MeasureTheory.upperCrossingTime a b f N (n + 1) ω ≠ N)
Goal: MeasureTheory.lowerCrossingTime a b f N n ω < MeasureTheory.upperCrossingTime a b f N (n + 1) ω
-- MeasureTheory.upperCrossingTime_lt_succ
theorem upperCrossingTime_lt_succ (hab : a < b) (hn : upperCrossingTime a b f N (n + 1) ω ≠ N) :
    upperCrossingTime a b f N n ω < upperCrossingTime a b f N (n + 1) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hab : a < b) (hn : MeasureTheory.upperCrossingTime a b f N (n + 1) ω ≠ N)
Goal: MeasureTheory.upperCrossingTime a b f N n ω < MeasureTheory.upperCrossingTime a b f N (n + 1) ω
-- MeasureTheory.lowerCrossingTime_stabilize
theorem lowerCrossingTime_stabilize (hnm : n ≤ m) (hn : lowerCrossingTime a b f N n ω = N) :
    lowerCrossingTime a b f N m ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n m : ℕ} {ω : Ω} (hnm : n ≤ m) (hn : MeasureTheory.lowerCrossingTime a b f N n ω = N)
Goal: MeasureTheory.lowerCrossingTime a b f N m ω = N
-- MeasureTheory.upperCrossingTime_stabilize
theorem upperCrossingTime_stabilize (hnm : n ≤ m) (hn : upperCrossingTime a b f N n ω = N) :
    upperCrossingTime a b f N m ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n m : ℕ} {ω : Ω} (hnm : n ≤ m) (hn : MeasureTheory.upperCrossingTime a b f N n ω = N)
Goal: MeasureTheory.upperCrossingTime a b f N m ω = N
-- MeasureTheory.lowerCrossingTime_stabilize'
theorem lowerCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ lowerCrossingTime a b f N n ω) :
    lowerCrossingTime a b f N m ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n m : ℕ} {ω : Ω} (hnm : n ≤ m) (hn : N ≤ MeasureTheory.lowerCrossingTime a b f N n ω)
Goal: MeasureTheory.lowerCrossingTime a b f N m ω = N
-- MeasureTheory.upperCrossingTime_stabilize'
theorem upperCrossingTime_stabilize' (hnm : n ≤ m) (hn : N ≤ upperCrossingTime a b f N n ω) :
    upperCrossingTime a b f N m ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n m : ℕ} {ω : Ω} (hnm : n ≤ m) (hn : N ≤ MeasureTheory.upperCrossingTime a b f N n ω)
Goal: MeasureTheory.upperCrossingTime a b f N m ω = N
-- MeasureTheory.exists_upperCrossingTime_eq
theorem exists_upperCrossingTime_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
    ∃ n, upperCrossingTime a b f N n ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b)
Goal: ∃ n, MeasureTheory.upperCrossingTime a b f N n ω = N
-- MeasureTheory.upperCrossingTime_lt_bddAbove
theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
    BddAbove {n | upperCrossingTime a b f N n ω < N} 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hab : a < b)
Goal: BddAbove {n | MeasureTheory.upperCrossingTime a b f N n ω < N}
-- MeasureTheory.upperCrossingTime_lt_nonempty
theorem upperCrossingTime_lt_nonempty (hN : 0 < N) :
    {n | upperCrossingTime a b f N n ω < N}.Nonempty 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hN : 0 < N)
Goal: {n | MeasureTheory.upperCrossingTime a b f N n ω < N}.Nonempty
-- MeasureTheory.upperCrossingTime_bound_eq
theorem upperCrossingTime_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
    upperCrossingTime a b f N N ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b)
Goal: MeasureTheory.upperCrossingTime a b f N N ω = N
-- MeasureTheory.upperCrossingTime_eq_of_bound_le
theorem upperCrossingTime_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
    upperCrossingTime a b f N n ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hab : a < b) (hn : N ≤ n)
Goal: MeasureTheory.upperCrossingTime a b f N n ω = N
-- MeasureTheory.Adapted.isStoppingTime_crossing
theorem Adapted.isStoppingTime_crossing (hf : Adapted ℱ f) :
    IsStoppingTime ℱ (upperCrossingTime a b f N n) ∧
      IsStoppingTime ℱ (lowerCrossingTime a b f N n) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f)
Goal: MeasureTheory.IsStoppingTime ℱ (MeasureTheory.upperCrossingTime a b f N n) ∧
  MeasureTheory.IsStoppingTime ℱ (MeasureTheory.lowerCrossingTime a b f N n)
-- MeasureTheory.Adapted.isStoppingTime_upperCrossingTime
theorem Adapted.isStoppingTime_upperCrossingTime (hf : Adapted ℱ f) :
    IsStoppingTime ℱ (upperCrossingTime a b f N n) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f)
Goal: MeasureTheory.IsStoppingTime ℱ (MeasureTheory.upperCrossingTime a b f N n)
-- MeasureTheory.Adapted.isStoppingTime_lowerCrossingTime
theorem Adapted.isStoppingTime_lowerCrossingTime (hf : Adapted ℱ f) :
    IsStoppingTime ℱ (lowerCrossingTime a b f N n) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f)
Goal: MeasureTheory.IsStoppingTime ℱ (MeasureTheory.lowerCrossingTime a b f N n)
-- MeasureTheory.upcrossingStrat_nonneg
theorem upcrossingStrat_nonneg : 0 ≤ upcrossingStrat a b f N n ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω}
Goal: 0 ≤ MeasureTheory.upcrossingStrat a b f N n ω
-- MeasureTheory.upcrossingStrat_le_one
theorem upcrossingStrat_le_one : upcrossingStrat a b f N n ω ≤ 1 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω}
Goal: MeasureTheory.upcrossingStrat a b f N n ω ≤ 1
-- MeasureTheory.Adapted.upcrossingStrat_adapted
theorem Adapted.upcrossingStrat_adapted (hf : Adapted ℱ f) :
    Adapted ℱ (upcrossingStrat a b f N) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f)
Goal: MeasureTheory.Adapted ℱ (MeasureTheory.upcrossingStrat a b f N)
-- MeasureTheory.Submartingale.sum_upcrossingStrat_mul
theorem Submartingale.sum_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
      ∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) ℱ μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ℕ → Ω → ℝ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (a b : ℝ) (N : ℕ)
Goal: MeasureTheory.Submartingale (fun n => ∑ k ∈ Finset.range n, MeasureTheory.upcrossingStrat a b f N k * (f (k + 1) - f k))
  ℱ μ
-- MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul
theorem Submartingale.sum_sub_upcrossingStrat_mul [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (a b : ℝ) (N : ℕ) : Submartingale (fun n : ℕ =>
      ∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ℕ → Ω → ℝ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (a b : ℝ) (N : ℕ)
Goal: MeasureTheory.Submartingale
  (fun n => ∑ k ∈ Finset.range n, (1 - MeasureTheory.upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ
-- MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le
theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
    μ[∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: ∫ (x : Ω), (∑ k ∈ Finset.range n, MeasureTheory.upcrossingStrat a b f N k * (f (k + 1) - f k)) x ∂μ ≤
  ∫ (x : Ω), f n x ∂μ - ∫ (x : Ω), f 0 x ∂μ
-- MeasureTheory.upcrossingsBefore_bot
@[simp]
theorem upcrossingsBefore_bot [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ}
    {ω : Ω} : upcrossingsBefore a b f ⊥ ω = ⊥ 
Arguments: {Ω : Type u_1} {ι : Type u_2} [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ} {ω : Ω}
Goal: MeasureTheory.upcrossingsBefore a b f ⊥ ω = ⊥
-- MeasureTheory.upcrossingsBefore_zero
theorem upcrossingsBefore_zero : upcrossingsBefore a b f 0 ω = 0 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {ω : Ω}
Goal: MeasureTheory.upcrossingsBefore a b f 0 ω = 0
-- MeasureTheory.upcrossingsBefore_zero'
@[simp]
theorem upcrossingsBefore_zero' : upcrossingsBefore a b f 0 = 0 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ}
Goal: MeasureTheory.upcrossingsBefore a b f 0 = 0
-- MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore
theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
    (hn : n ≤ upcrossingsBefore a b f N ω) : upperCrossingTime a b f N n ω < N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hN : 0 < N) (hab : a < b) (hn : n ≤ MeasureTheory.upcrossingsBefore a b f N ω)
Goal: MeasureTheory.upperCrossingTime a b f N n ω < N
-- MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt
theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
    (hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hab : a < b) (hn : MeasureTheory.upcrossingsBefore a b f N ω < n)
Goal: MeasureTheory.upperCrossingTime a b f N n ω = N
-- MeasureTheory.upcrossingsBefore_le
theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
    upcrossingsBefore a b f N ω ≤ N 
Arguments: {Ω : Type u_1} {a b : ℝ} {N : ℕ} (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b)
Goal: MeasureTheory.upcrossingsBefore a b f N ω ≤ N
-- MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt
theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
    (h : lowerCrossingTime a b f N n ω < N) :
    upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
      lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} {M : ℕ} (hNM : N ≤ M) (h : MeasureTheory.lowerCrossingTime a b f N n ω < N)
Goal: MeasureTheory.upperCrossingTime a b f M n ω = MeasureTheory.upperCrossingTime a b f N n ω ∧
  MeasureTheory.lowerCrossingTime a b f M n ω = MeasureTheory.lowerCrossingTime a b f N n ω
-- MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
    (h : upperCrossingTime a b f N (n + 1) ω < N) :
    upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
      lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} {M : ℕ} (hNM : N ≤ M) (h : MeasureTheory.upperCrossingTime a b f N (n + 1) ω < N)
Goal: MeasureTheory.upperCrossingTime a b f M (n + 1) ω = MeasureTheory.upperCrossingTime a b f N (n + 1) ω ∧
  MeasureTheory.lowerCrossingTime a b f M n ω = MeasureTheory.lowerCrossingTime a b f N n ω
-- MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt
theorem upperCrossingTime_eq_upperCrossingTime_of_lt {M : ℕ} (hNM : N ≤ M)
    (h : upperCrossingTime a b f N n ω < N) :
    upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} {M : ℕ} (hNM : N ≤ M) (h : MeasureTheory.upperCrossingTime a b f N n ω < N)
Goal: MeasureTheory.upperCrossingTime a b f M n ω = MeasureTheory.upperCrossingTime a b f N n ω
-- MeasureTheory.upcrossingsBefore_mono
theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} (hab : a < b)
Goal: Monotone fun N ω => MeasureTheory.upcrossingsBefore a b f N ω
-- MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing
theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
    (hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
    upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁) (hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω)
Goal: MeasureTheory.upcrossingsBefore a b f N ω < MeasureTheory.upcrossingsBefore a b f (N₂ + 1) ω
-- MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore
theorem lowerCrossingTime_lt_of_lt_upcrossingsBefore (hN : 0 < N) (hab : a < b)
    (hn : n < upcrossingsBefore a b f N ω) : lowerCrossingTime a b f N n ω < N 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hN : 0 < N) (hab : a < b) (hn : n < MeasureTheory.upcrossingsBefore a b f N ω)
Goal: MeasureTheory.lowerCrossingTime a b f N n ω < N
-- MeasureTheory.le_sub_of_le_upcrossingsBefore
theorem le_sub_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
    (hn : n < upcrossingsBefore a b f N ω) :
    b - a ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
      stoppedValue f (lowerCrossingTime a b f N n) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hN : 0 < N) (hab : a < b) (hn : n < MeasureTheory.upcrossingsBefore a b f N ω)
Goal: b - a ≤
  MeasureTheory.stoppedValue f (MeasureTheory.upperCrossingTime a b f N (n + 1)) ω -
    MeasureTheory.stoppedValue f (MeasureTheory.lowerCrossingTime a b f N n) ω
-- MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt
theorem sub_eq_zero_of_upcrossingsBefore_lt (hab : a < b) (hn : upcrossingsBefore a b f N ω < n) :
    stoppedValue f (upperCrossingTime a b f N (n + 1)) ω -
      stoppedValue f (lowerCrossingTime a b f N n) ω = 0 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} {ω : Ω} (hab : a < b) (hn : MeasureTheory.upcrossingsBefore a b f N ω < n)
Goal: MeasureTheory.stoppedValue f (MeasureTheory.upperCrossingTime a b f N (n + 1)) ω -
    MeasureTheory.stoppedValue f (MeasureTheory.lowerCrossingTime a b f N n) ω =
  0
-- MeasureTheory.mul_upcrossingsBefore_le
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
    (b - a) * upcrossingsBefore a b f N ω ≤
    ∑ k ∈ Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hf : a ≤ f N ω) (hab : a < b)
Goal: (b - a) * ↑(MeasureTheory.upcrossingsBefore a b f N ω) ≤
  ∑ k ∈ Finset.range N, MeasureTheory.upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
-- MeasureTheory.integral_mul_upcrossingsBefore_le_integral
theorem integral_mul_upcrossingsBefore_le_integral [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
    (b - a) * μ[upcrossingsBefore a b f N] ≤ μ[f N] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hfN : ∀ (ω : Ω), a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b)
Goal: (b - a) * ∫ (x : Ω), ↑(MeasureTheory.upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), f N x ∂μ
-- MeasureTheory.crossing_pos_eq
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 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N n : ℕ} (hab : a < b)
Goal: MeasureTheory.upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = MeasureTheory.upperCrossingTime a b f N n ∧
  MeasureTheory.lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = MeasureTheory.lowerCrossingTime a b f N n
-- MeasureTheory.upcrossingsBefore_pos_eq
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 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hab : a < b)
Goal: MeasureTheory.upcrossingsBefore 0 (b - a) (fun n ω => (f n ω - a)⁺) N ω = MeasureTheory.upcrossingsBefore a b f N ω
-- MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux
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)⁺] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hab : a < b)
Goal: (b - a) * ∫ (x : Ω), ↑(MeasureTheory.upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ
-- MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part
theorem Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part [IsFiniteMeasure μ]
    (a b : ℝ) (hf : Submartingale f ℱ μ) (N : ℕ) :
    (b - a) * μ[upcrossingsBefore a b f N] ≤ μ[fun ω => (f N ω - a)⁺] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ℕ → Ω → ℝ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (a b : ℝ) (hf : MeasureTheory.Submartingale f ℱ μ) (N : ℕ)
Goal: (b - a) * ∫ (x : Ω), ↑(MeasureTheory.upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω => (f N ω - a)⁺) x ∂μ
-- MeasureTheory.upcrossingsBefore_eq_sum
theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
    ∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ω : Ω} (hab : a < b)
Goal: MeasureTheory.upcrossingsBefore a b f N ω =
  ∑ i ∈ Finset.Ico 1 (N + 1), {n | MeasureTheory.upperCrossingTime a b f N n ω < N}.indicator 1 i
-- MeasureTheory.Adapted.measurable_upcrossingsBefore
theorem Adapted.measurable_upcrossingsBefore (hf : Adapted ℱ f) (hab : a < b) :
    Measurable (upcrossingsBefore a b f N) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f) (hab : a < b)
Goal: Measurable (MeasureTheory.upcrossingsBefore a b f N)
-- MeasureTheory.Adapted.integrable_upcrossingsBefore
theorem Adapted.integrable_upcrossingsBefore [IsFiniteMeasure μ] (hf : Adapted ℱ f) (hab : a < b) :
    Integrable (fun ω => (upcrossingsBefore a b f N ω : ℝ)) μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Adapted ℱ f) (hab : a < b)
Goal: MeasureTheory.Integrable (fun ω => ↑(MeasureTheory.upcrossingsBefore a b f N ω)) μ
-- MeasureTheory.Adapted.measurable_upcrossings
theorem Adapted.measurable_upcrossings (hf : Adapted ℱ f) (hab : a < b) :
    Measurable (upcrossings a b f) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {a b : ℝ} {f : ℕ → Ω → ℝ} {ℱ : MeasureTheory.Filtration ℕ m0} (hf : MeasureTheory.Adapted ℱ f) (hab : a < b)
Goal: Measurable (MeasureTheory.upcrossings a b f)
-- MeasureTheory.upcrossings_lt_top_iff
theorem upcrossings_lt_top_iff :
    upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {ω : Ω}
Goal: MeasureTheory.upcrossings a b f ω < ⊤ ↔ ∃ k, ∀ (N : ℕ), MeasureTheory.upcrossingsBefore a b f N ω ≤ k
-- MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part
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 ω - a)⁺) ∂μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ℕ → Ω → ℝ} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (a b : ℝ) (hf : MeasureTheory.Submartingale f ℱ μ)
Goal: ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), MeasureTheory.upcrossings a b f ω ∂μ ≤
  ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ
-- AdicCompletion.map_surjective
theorem map_surjective : Function.Surjective (map I f) 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) {M : Type v} [AddCommGroup M] [Module R M] {N : Type w} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} (hf : Function.Surjective ⇑f)
Goal: Function.Surjective ⇑(AdicCompletion.map I f)
-- AdicCompletion.map_injective
theorem map_injective {f : M →ₗ[R] N} (hf : Function.Injective f) :
    Function.Injective (map I f) 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) {M : Type u} [AddCommGroup M] [Module R M] {N : Type u} [AddCommGroup N] [Module R N] [IsNoetherianRing R] [Module.Finite R N] {f : M →ₗ[R] N} (hf : Function.Injective ⇑f)
Goal: Function.Injective ⇑(AdicCompletion.map I f)
-- _private.Mathlib.RingTheory.AdicCompletion.Exactness.0.AdicCompletion.mapExactAux.h1
private noncomputable def mapExactAux :
    (n : ℕ) → { a : M | f a - x (k + n) ∈ (I ^ (k + n) • ⊤ : Submodule R N) }
  
Arguments: {R : Type u} [CommRing R] {I : Ideal R} {N : Type u} [AddCommGroup N] [Module R N] {P : Type u} [AddCommGroup P] [Module R P] {g : N →ₗ[R] P} (hg : Function.Surjective ⇑g) {k : ℕ} (x : AdicCompletion.AdicCauchySequence I N) (hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤) (n : ℕ)
Goal: g (↑x (k + n)) ∈ Submodule.map g (I ^ (k + n) • ⊤)
-- _private.Mathlib.RingTheory.AdicCompletion.Exactness.0.AdicCompletion.mapExactAux.h2
private noncomputable def mapExactAux :
    (n : ℕ) → { a : M | f a - x (k + n) ∈ (I ^ (k + n) • ⊤ : Submodule R N) }
  
Arguments: {R : Type u} [CommRing R] {I : Ideal R} {M : Type u} [AddCommGroup M] [Module R M] {N : Type u} [AddCommGroup N] [Module R N] {P : Type u} [AddCommGroup P] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) {k : ℕ} (x : AdicCompletion.AdicCauchySequence I N) (hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤) (n : ℕ)
Goal: ∃ d y, d ∈ I ^ (k + n) • ⊤ ∧ f y = ↑x (k + n) - d