Theorem: MeasureTheory.hitting_of_lt
File: Mathlib/Probability/Process/HittingTime.lean

tactic: simp_rw [hitting]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
⊢ hitting u s n m ω = m
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) = m

tactic: have h_not : ¬∃ (j : ι) (_ : j ∈ Set.Icc n m), u j ω ∈ s := by
  push_neg
  intro j
  rw [Set.Icc_eq_empty_of_lt h]
  simp only [Set.mem_empty_iff_false, IsEmpty.forall_iff]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) = m
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
h_not : ¬∃ j, ∃ (_ : j ∈ Set.Icc n m), u j ω ∈ s
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) = m

tactic: simp only [exists_prop] at h_not
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
h_not : ¬∃ j, ∃ (_ : j ∈ Set.Icc n m), u j ω ∈ s
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) = m
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
h_not : ¬∃ j ∈ Set.Icc n m, u j ω ∈ s
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) = m

tactic: simp only [h_not, if_false]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
h_not : ¬∃ j ∈ Set.Icc n m, u j ω ∈ s
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) = m
state_after: no goals

tactic: push_neg
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
⊢ ¬∃ j, ∃ (_ : j ∈ Set.Icc n m), u j ω ∈ s
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
⊢ ∀ j ∈ Set.Icc n m, u j ω ∉ s

tactic: intro j
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
⊢ ∀ j ∈ Set.Icc n m, u j ω ∉ s
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
j : ι
⊢ j ∈ Set.Icc n m → u j ω ∉ s

tactic: rw [Set.Icc_eq_empty_of_lt h]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
j : ι
⊢ j ∈ Set.Icc n m → u j ω ∉ s
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
j : ι
⊢ j ∈ ∅ → u j ω ∉ s

tactic: simp only [Set.mem_empty_iff_false, IsEmpty.forall_iff]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m✝ : MeasurableSpace Ω
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
m : ι
h : m < n
j : ι
⊢ j ∈ ∅ → u j ω ∉ s
state_after: no goals