-- Mathlib/Probability/Process/HittingTime.lean
theorem hitting_eq_hitting_of_exists {m₁ m₂ : ι} (h : m₁ ≤ m₂)
    (h' : ∃ j ∈ Set.Icc n m₁, u j ω ∈ s) : hitting u s n m₁ ω = hitting u s n m₂ ω

Proof state:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s})

Tactic:
refine le_csInf ⟨j, Set.Icc_subset_Icc_right h hj₁, hj₂⟩ fun i hi => ?_

Result:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Proof state:
case intro.intro.hc
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ ∃ j ∈ Set.Icc n m₂, u j ω ∈ s

Tactic:
exact ⟨j, ⟨hj₁.1, hj₁.2.trans h⟩, hj₂⟩

Result:
no goals

Proof state:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Tactic:
by_cases hi' : i ≤ m₁

Result:
case pos
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi' : i ≤ m₁
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi' : ¬i ≤ m₁
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Proof state:
case pos
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi' : i ≤ m₁
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Tactic:
exact csInf_le bddBelow_Icc.inter_of_left ⟨⟨hi.1.1, hi'⟩, hi.2⟩

Result:
no goals

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
h' : ∃ j ∈ Set.Icc n m₁, u j ω ∈ s
⊢ hitting u s n m₁ ω = hitting u s n m₂ ω

Tactic:
simp only [hitting, if_pos h']

Result:
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
h' : ∃ j ∈ Set.Icc n m₁, u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) =
    if ∃ j ∈ Set.Icc n m₂, u j ω ∈ s then sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s}) else m₂

Proof state:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi' : ¬i ≤ m₁
hj₂ : j ∈ {i | u i ω ∈ s}
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Tactic:
exact ((csInf_le bddBelow_Icc.inter_of_left ⟨hj₁, hj₂⟩).trans (hj₁.2.trans le_rfl)).trans
  (le_of_lt (not_le.1 hi'))

Result:
no goals

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
h' : ∃ j ∈ Set.Icc n m₁, u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) =
    if ∃ j ∈ Set.Icc n m₂, u j ω ∈ s then sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s}) else m₂

Tactic:
obtain ⟨j, hj₁, hj₂⟩ := h'

Result:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) =
    if ∃ j ∈ Set.Icc n m₂, u j ω ∈ s then sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s}) else m₂

Proof state:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi' : ¬i ≤ m₁
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Tactic:
change j ∈ {i | u i ω ∈ s} at hj₂

Result:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
i : ι
hi : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi' : ¬i ≤ m₁
hj₂ : j ∈ {i | u i ω ∈ s}
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ i

Proof state:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) =
    if ∃ j ∈ Set.Icc n m₂, u j ω ∈ s then sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s}) else m₂

Tactic:
rw [if_pos]

Result:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) = sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s})

case intro.intro.hc
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ ∃ j ∈ Set.Icc n m₂, u j ω ∈ s

Proof state:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) = sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s})

Tactic:
refine le_antisymm ?_ (csInf_le_csInf bddBelow_Icc.inter_of_left ⟨j, hj₁, hj₂⟩
  (Set.inter_subset_inter_left _ (Set.Icc_subset_Icc_right h)))

Result:
case intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s}) ≤ sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s})