-- Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
    (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
    (bdd_above : F.IsBounded (· ≤ ·)

Proof state:
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
h' : ∀ c < F.limsSup, (Set.Ioo c F.limsSup).Nonempty
H : f F.limsSup < liminf f F
not_bot : ¬IsBot F.limsSup
l : R
l_lt : l < F.limsSup
h'l : Set.Ioc l F.limsSup ⊆ {x | f x < liminf f F}
m : R
l_m : l < m
m_lt : m < F.limsSup
⊢ ∃ᶠ (x : R) in F, f x ≤ f m

Tactic:
apply (frequently_lt_of_lt_limsSup cobdd m_lt).mono

Result:
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
h' : ∀ c < F.limsSup, (Set.Ioo c F.limsSup).Nonempty
H : f F.limsSup < liminf f F
not_bot : ¬IsBot F.limsSup
l : R
l_lt : l < F.limsSup
h'l : Set.Ioc l F.limsSup ⊆ {x | f x < liminf f F}
m : R
l_m : l < m
m_lt : m < F.limsSup
⊢ ∀ (x : R), m < x → f x ≤ f m

Proof state:
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
c : S
hc : c < sSup (f '' {a | ∀ᶠ (n : R) in F, n ≤ a})
d : S
hd : d ∈ f '' {a | ∀ᶠ (n : R) in F, n ≤ a}
h'd : c < d
⊢ d ∈ {a | ∀ᶠ (a_1 : R) in F, a ≤ f a_1}

Tactic:
rcases hd with ⟨e, ⟨he, fe_eq_d⟩⟩

Result:
case intro.intro
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
c : S
hc : c < sSup (f '' {a | ∀ᶠ (n : R) in F, n ≤ a})
d : S
h'd : c < d
e : R
he : e ∈ {a | ∀ᶠ (n : R) in F, n ≤ a}
fe_eq_d : f e = d
⊢ d ∈ {a | ∀ᶠ (a_1 : R) in F, a ≤ f a_1}

Proof state:
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
h' : ∀ c < F.limsSup, (Set.Ioo c F.limsSup).Nonempty
H : f F.limsSup < liminf f F
not_bot : ¬IsBot F.limsSup
l : R
l_lt : l < F.limsSup
h'l : Set.Ioc l F.limsSup ⊆ {x | f x < liminf f F}
⊢ (Set.Ioo l F.limsSup).Nonempty

Tactic:
contrapose! h'

Result:
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
H : f F.limsSup < liminf f F
not_bot : ¬IsBot F.limsSup
l : R
l_lt : l < F.limsSup
h'l : Set.Ioc l F.limsSup ⊆ {x | f x < liminf f F}
h' : Set.Ioo l F.limsSup = ∅
⊢ ∃ c < F.limsSup, Set.Ioo c F.limsSup = ∅

Proof state:
case a
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
⊢ liminf f F ≤ f F.limsSup

Tactic:
by_cases h' : ∃ c, c < F.limsSup ∧ Set.Ioo c F.limsSup = ∅

Result:
case pos
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
h' : ∃ c < F.limsSup, Set.Ioo c F.limsSup = ∅
⊢ liminf f F ≤ f F.limsSup

case neg
R : Type u_4
S : Type u_5
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : ConditionallyCompleteLinearOrder S
inst✝² : TopologicalSpace S
inst✝¹ : OrderTopology S
F : Filter R
inst✝ : F.NeBot
f : R → S
f_decr : Antitone f
f_cont : ContinuousAt f F.limsSup
bdd_above : autoParam (IsBounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
cobdd : autoParam (IsCobounded (fun x1 x2 => x1 ≤ x2) F) _auto✝
h' : ¬∃ c < F.limsSup, Set.Ioo c F.limsSup = ∅
⊢ liminf f F ≤ f F.limsSup