Le liminf of eventually monotone of tendsto on dense
le_liminf_of_eventually_monotone_of_tendsto_on_dense
Plain-language statement
This is the dual of limsup_le_of_eventually_monotone_of_tendsto_on_dense.
Exact Lean statement
lemma le_liminf_of_eventually_monotone_of_tendsto_on_dense {ι α β : Type*} [LinearOrder α]
[BoundedOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α]
[ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {l : Filter ι}
[l.NeBot] {D : Set α} {F : ι → α → β} {f : α → β} (hF : ∀ᶠ i in l, Monotone (F i))
(hD : Dense D) (htop : ⊤ ∈ D) (hbot : ⊥ ∈ D) {a : α} (hfa : ContinuousWithinAt f (Set.Iio a) a)
(hlim : ∀ t ∈ D, Tendsto (F · t) l (𝓝 (f t))) :
f a ≤ liminf (F · a) lFormal artifact
Lean source
lemma le_liminf_of_eventually_monotone_of_tendsto_on_dense {ι α β : Type*} [LinearOrder α] [BoundedOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {l : Filter ι} [l.NeBot] {D : Set α} {F : ι → α → β} {f : α → β} (hF : ∀ᶠ i in l, Monotone (F i)) (hD : Dense D) (htop : ⊤ ∈ D) (hbot : ⊥ ∈ D) {a : α} (hfa : ContinuousWithinAt f (Set.Iio a) a) (hlim : ∀ t ∈ D, Tendsto (F · t) l (𝓝 (f t))) : f a ≤ liminf (F · a) l := by by_cases! ha : a = ⊥ · rw [ha, (hlim ⊥ hbot).liminf_eq] · have : (comap ((↑) : D → α) (𝓝[<] a)).NeBot := hD.comap_val_nhdsWithin_Iio_neBot ha.bot_lt refine (isClosed_Iic (a := liminf (F · a) l)).mem_of_tendsto (Tendsto.comp hfa (tendsto_comap (f := ((↑) : D → α)))) ?_ rw [eventually_comap, eventually_nhdsWithin_iff] filter_upwards with z hz d rfl simp only [Function.comp_apply, Set.mem_Iic, ← (hlim d d.2).liminf_eq] refine liminf_le_liminf ?_ (hlim d d.2).isBoundedUnder_ge ?_ · filter_upwards [hF] with i hi using hi hz.le · refine (hlim ⊤ htop).isCoboundedUnder_ge.trans ?_ filter_upwards [hF] with i hi using hi le_top- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:992-1010
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.