statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
tendsto_Ioc_at_top_at_top : tendsto_Ixx_class Ioc (at_top : filter α) at_top | tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self) | instance | filter.tendsto_Ioc_at_top_at_top | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_at_top_at_top : tendsto_Ixx_class Ioo (at_top : filter α) at_top | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Icc_self) | instance | filter.tendsto_Ioo_at_top_at_top | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_at_bot_at_bot : tendsto_Ixx_class Icc (at_bot : filter α) at_bot | (has_basis_infi_principal_finite _).tendsto_Ixx_class $ λ s hs,
set.ord_connected.out $ ord_connected_bInter $ λ i hi, ord_connected_Iic | instance | filter.tendsto_Icc_at_bot_at_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter",
"set.ord_connected.out"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_at_bot_at_bot : tendsto_Ixx_class Ico (at_bot : filter α) at_bot | tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self) | instance | filter.tendsto_Ico_at_bot_at_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_at_bot_at_bot : tendsto_Ixx_class Ioc (at_bot : filter α) at_bot | tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self) | instance | filter.tendsto_Ioc_at_bot_at_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_at_bot_at_bot : tendsto_Ixx_class Ioo (at_bot : filter α) at_bot | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Icc_self) | instance | filter.tendsto_Ioo_at_bot_at_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_connected.tendsto_Icc {s : set α} [hs : ord_connected s] :
tendsto_Ixx_class Icc (𝓟 s) (𝓟 s) | tendsto_Ixx_class_principal.2 hs.out | instance | filter.ord_connected.tendsto_Icc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_Ici_Ici {a : α} : tendsto_Ixx_class Ico (𝓟 (Ici a)) (𝓟 (Ici a)) | tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self) | instance | filter.tendsto_Ico_Ici_Ici | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_Ioi_Ioi {a : α} : tendsto_Ixx_class Ico (𝓟 (Ioi a)) (𝓟 (Ioi a)) | tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self) | instance | filter.tendsto_Ico_Ioi_Ioi | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_Iic_Iio {a : α} : tendsto_Ixx_class Ico (𝓟 (Iic a)) (𝓟 (Iio a)) | tendsto_Ixx_class_principal.2 $ λ a ha b hb x hx, lt_of_lt_of_le hx.2 hb | instance | filter.tendsto_Ico_Iic_Iio | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_Iio_Iio {a : α} : tendsto_Ixx_class Ico (𝓟 (Iio a)) (𝓟 (Iio a)) | tendsto_Ixx_class_of_subset (λ _ _, Ico_subset_Icc_self) | instance | filter.tendsto_Ico_Iio_Iio | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_Ici_Ioi {a : α} : tendsto_Ixx_class Ioc (𝓟 (Ici a)) (𝓟 (Ioi a)) | tendsto_Ixx_class_principal.2 $ λ x hx y hy t ht, lt_of_le_of_lt hx ht.1 | instance | filter.tendsto_Ioc_Ici_Ioi | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_Iic_Iic {a : α} : tendsto_Ixx_class Ioc (𝓟 (Iic a)) (𝓟 (Iic a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self) | instance | filter.tendsto_Ioc_Iic_Iic | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_Iio_Iio {a : α} : tendsto_Ixx_class Ioc (𝓟 (Iio a)) (𝓟 (Iio a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self) | instance | filter.tendsto_Ioc_Iio_Iio | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_Ioi_Ioi {a : α} : tendsto_Ixx_class Ioc (𝓟 (Ioi a)) (𝓟 (Ioi a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioc_subset_Icc_self) | instance | filter.tendsto_Ioc_Ioi_Ioi | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_Ici_Ioi {a : α} : tendsto_Ixx_class Ioo (𝓟 (Ici a)) (𝓟 (Ioi a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Ioc_self) | instance | filter.tendsto_Ioo_Ici_Ioi | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_Iic_Iio {a : α} : tendsto_Ixx_class Ioo (𝓟 (Iic a)) (𝓟 (Iio a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Ico_self) | instance | filter.tendsto_Ioo_Iic_Iio | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_Ioi_Ioi {a : α} : tendsto_Ixx_class Ioo (𝓟 (Ioi a)) (𝓟 (Ioi a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Ioc_self) | instance | filter.tendsto_Ioo_Ioi_Ioi | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_Iio_Iio {a : α} : tendsto_Ixx_class Ioo (𝓟 (Iio a)) (𝓟 (Iio a)) | tendsto_Ixx_class_of_subset (λ _ _, Ioo_subset_Ioc_self) | instance | filter.tendsto_Ioo_Iio_Iio | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_Icc_Icc {a b : α} :
tendsto_Ixx_class Icc (𝓟 (Icc a b)) (𝓟 (Icc a b)) | tendsto_Ixx_class_principal.mpr $ λ x hx y hy, Icc_subset_Icc hx.1 hy.2 | instance | filter.tendsto_Icc_Icc_Icc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_Icc_Icc {a b : α} : tendsto_Ixx_class Ioc (𝓟 (Icc a b)) (𝓟 (Icc a b)) | tendsto_Ixx_class_of_subset $ λ _ _, Ioc_subset_Icc_self | instance | filter.tendsto_Ioc_Icc_Icc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_pure_pure {a : α} : tendsto_Ixx_class Icc (pure a) (pure a : filter α) | by { rw ← principal_singleton, exact tendsto_Ixx_class_principal.2 ord_connected_singleton.out } | instance | filter.tendsto_Icc_pure_pure | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ico_pure_bot {a : α} : tendsto_Ixx_class Ico (pure a) ⊥ | ⟨by simp⟩ | instance | filter.tendsto_Ico_pure_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_pure_bot {a : α} : tendsto_Ixx_class Ioc (pure a) ⊥ | ⟨by simp⟩ | instance | filter.tendsto_Ioc_pure_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_pure_bot {a : α} : tendsto_Ixx_class Ioo (pure a) ⊥ | ⟨by simp⟩ | instance | filter.tendsto_Ioo_pure_bot | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Icc_uIcc_uIcc {a b : α} : tendsto_Ixx_class Icc (𝓟 [a, b]) (𝓟 [a, b]) | filter.tendsto_Icc_Icc_Icc | instance | filter.tendsto_Icc_uIcc_uIcc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter.tendsto_Icc_Icc_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioc_uIcc_uIcc {a b : α} : tendsto_Ixx_class Ioc (𝓟 [a, b]) (𝓟 [a, b]) | filter.tendsto_Ioc_Icc_Icc | instance | filter.tendsto_Ioc_uIcc_uIcc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter.tendsto_Ioc_Icc_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uIcc_of_Icc {l : filter α} [tendsto_Ixx_class Icc l l] :
tendsto_Ixx_class uIcc l l | begin
refine ⟨λ s hs, mem_map.2 $ mem_prod_self_iff.2 _⟩,
obtain ⟨t, htl, hts⟩ : ∃ t ∈ l, ∀ p ∈ (t : set α) ×ˢ t, Icc (p : α × α).1 p.2 ∈ s,
from mem_prod_self_iff.1 (mem_map.1 (tendsto_fst.Icc tendsto_snd hs)),
refine ⟨t, htl, λ p hp, _⟩,
cases le_total p.1 p.2,
{ rw [mem_preimage, uIcc_of_le h], exact h... | instance | filter.tendsto_uIcc_of_Icc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.uIcc {l : filter α} [tendsto_Ixx_class Icc l l] {f g : β → α} {lb : filter β}
(hf : tendsto f lb l) (hg : tendsto g lb l) :
tendsto (λ x, [f x, g x]) lb l.small_sets | tendsto_Ixx_class.tendsto_Ixx.comp $ hf.prod_mk hg | lemma | filter.tendsto.uIcc | order.filter | src/order/filter/interval.lean | [
"data.set.intervals.ord_connected",
"order.filter.small_sets",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (f : filter α) (g : set α → filter β) | ⨅s ∈ f, g s | def | filter.lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"lift"
] | A variant on `bind` using a function `g` taking a set instead of a member of `α`.
This is essentially a push-forward along a function mapping each set to a filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_top (g : set α → filter β) : (⊤ : filter α).lift g = g univ | by simp [filter.lift] | lemma | filter.lift_top | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"filter.lift",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis.mem_lift_iff {ι} {p : ι → Prop} {s : ι → set α} {f : filter α}
(hf : f.has_basis p s) {β : ι → Type*} {pg : Π i, β i → Prop} {sg : Π i, β i → set γ}
{g : set α → filter γ} (hg : ∀ i, (g $ s i).has_basis (pg i) (sg i)) (gm : monotone g)
{s : set γ} :
s ∈ f.lift g ↔ ∃ (i : ι) (hi : p i) (x : β i) (hx : ... | begin
refine (mem_binfi_of_directed _ ⟨univ, univ_sets _⟩).trans _,
{ intros t₁ ht₁ t₂ ht₂,
exact ⟨t₁ ∩ t₂, inter_mem ht₁ ht₂, gm $ inter_subset_left _ _,
gm $ inter_subset_right _ _⟩ },
{ simp only [← (hg _).mem_iff],
exact hf.exists_iff (λ t₁ t₂ ht H, gm ht H) }
end | lemma | filter.has_basis.mem_lift_iff | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | If `(p : ι → Prop, s : ι → set α)` is a basis of a filter `f`, `g` is a monotone function
`set α → filter γ`, and for each `i`, `(pg : β i → Prop, sg : β i → set α)` is a basis
of the filter `g (s i)`, then `(λ (i : ι) (x : β i), p i ∧ pg i x, λ (i : ι) (x : β i), sg i x)`
is a basis of the filter `f.lift g`.
This bas... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis.lift {ι} {p : ι → Prop} {s : ι → set α} {f : filter α} (hf : f.has_basis p s)
{β : ι → Type*} {pg : Π i, β i → Prop} {sg : Π i, β i → set γ} {g : set α → filter γ}
(hg : ∀ i, (g $ s i).has_basis (pg i) (sg i)) (gm : monotone g) :
(f.lift g).has_basis (λ i : Σ i, β i, p i.1 ∧ pg i.1 i.2) (λ i : Σ i, β i,... | begin
refine ⟨λ t, (hf.mem_lift_iff hg gm).trans _⟩,
simp [sigma.exists, and_assoc, exists_and_distrib_left]
end | lemma | filter.has_basis.lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"exists_and_distrib_left",
"filter",
"monotone"
] | If `(p : ι → Prop, s : ι → set α)` is a basis of a filter `f`, `g` is a monotone function
`set α → filter γ`, and for each `i`, `(pg : β i → Prop, sg : β i → set α)` is a basis
of the filter `g (s i)`, then `(λ (i : ι) (x : β i), p i ∧ pg i x, λ (i : ι) (x : β i), sg i x)`
is a basis of the filter `f.lift g`.
This bas... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_lift_sets (hg : monotone g) {s : set β} :
s ∈ f.lift g ↔ ∃t∈f, s ∈ g t | (f.basis_sets.mem_lift_iff (λ s, (g s).basis_sets) hg).trans $
by simp only [id, exists_mem_subset_iff] | lemma | filter.mem_lift_sets | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter_lift_sets (hg : monotone g) :
⋂₀ {s | s ∈ f.lift g} = ⋂ s ∈ f, ⋂₀ {t | t ∈ g s} | by simp only [sInter_eq_bInter, mem_set_of_eq, filter.mem_sets, mem_lift_sets hg,
Inter_exists, @Inter_comm _ (set β)] | lemma | filter.sInter_lift_sets | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter.mem_sets",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_lift {s : set β} {t : set α} (ht : t ∈ f) (hs : s ∈ g t) :
s ∈ f.lift g | le_principal_iff.mp $ show f.lift g ≤ 𝓟 s,
from infi_le_of_le t $ infi_le_of_le ht $ le_principal_iff.mpr hs | lemma | filter.mem_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"infi_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_le {f : filter α} {g : set α → filter β} {h : filter β} {s : set α}
(hs : s ∈ f) (hg : g s ≤ h) : f.lift g ≤ h | infi₂_le_of_le s hs hg | lemma | filter.lift_le | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi₂_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_lift {f : filter α} {g : set α → filter β} {h : filter β} :
h ≤ f.lift g ↔ ∀ s ∈ f, h ≤ g s | le_infi₂_iff | lemma | filter.le_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"le_infi₂_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mono (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : f₁.lift g₁ ≤ f₂.lift g₂ | infi_mono $ λ s, infi_mono' $ λ hs, ⟨hf hs, hg s⟩ | lemma | filter.lift_mono | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"infi_mono",
"infi_mono'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mono' (hg : ∀s ∈ f, g₁ s ≤ g₂ s) : f.lift g₁ ≤ f.lift g₂ | infi₂_mono hg | lemma | filter.lift_mono' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"infi₂_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lift {m : γ → β} {l : filter γ} :
tendsto m l (f.lift g) ↔ ∀ s ∈ f, tendsto m l (g s) | by simp only [filter.lift, tendsto_infi] | lemma | filter.tendsto_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"filter.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lift_eq {m : β → γ} (hg : monotone g) : map m (f.lift g) = f.lift (map m ∘ g) | have monotone (map m ∘ g),
from map_mono.comp hg,
filter.ext $ λ s,
by simp only [mem_lift_sets hg, mem_lift_sets this, exists_prop, mem_map, function.comp_app] | lemma | filter.map_lift_eq | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"exists_prop",
"filter.ext",
"mem_map",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_lift_eq {m : γ → β} : comap m (f.lift g) = f.lift (comap m ∘ g) | by simp only [filter.lift, comap_infi] | lemma | filter.comap_lift_eq | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_lift_eq2 {m : β → α} {g : set β → filter γ} (hg : monotone g) :
(comap m f).lift g = f.lift (g ∘ preimage m) | le_antisymm
(le_infi₂ $ λ s hs, infi₂_le (m ⁻¹' s) ⟨s, hs, subset.rfl⟩)
(le_infi₂ $ λ s ⟨s', hs', (h_sub : m ⁻¹' s' ⊆ s)⟩, infi₂_le_of_le s' hs' $ hg h_sub) | theorem | filter.comap_lift_eq2 | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi₂_le",
"infi₂_le_of_le",
"le_infi₂",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_map_le {g : set β → filter γ} {m : α → β} :
(map m f).lift g ≤ f.lift (g ∘ image m) | le_lift.2 $ λ s hs, lift_le (image_mem_map hs) le_rfl | lemma | filter.lift_map_le | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"le_rfl",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lift_eq2 {g : set β → filter γ} {m : α → β} (hg : monotone g) :
(map m f).lift g = f.lift (g ∘ image m) | lift_map_le.antisymm $ le_lift.2 $ λ s hs, lift_le hs $ hg $ image_preimage_subset _ _ | lemma | filter.map_lift_eq2 | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comm {g : filter β} {h : set α → set β → filter γ} :
f.lift (λs, g.lift (h s)) = g.lift (λt, f.lift (λs, h s t)) | le_antisymm
(le_infi $ assume i, le_infi $ assume hi, le_infi $ assume j, le_infi $ assume hj,
infi_le_of_le j $ infi_le_of_le hj $ infi_le_of_le i $ infi_le _ hi)
(le_infi $ assume i, le_infi $ assume hi, le_infi $ assume j, le_infi $ assume hj,
infi_le_of_le j $ infi_le_of_le hj $ infi_le_of_le i $ infi_l... | lemma | filter.lift_comm | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi_le",
"infi_le_of_le",
"le_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_assoc {h : set β → filter γ} (hg : monotone g) :
(f.lift g).lift h = f.lift (λs, (g s).lift h) | le_antisymm
(le_infi $ assume s, le_infi $ assume hs, le_infi $ assume t, le_infi $ assume ht,
infi_le_of_le t $ infi_le _ $ (mem_lift_sets hg).mpr ⟨_, hs, ht⟩)
(le_infi $ assume t, le_infi $ assume ht,
let ⟨s, hs, h'⟩ := (mem_lift_sets hg).mp ht in
infi_le_of_le s $ infi_le_of_le hs $ infi_le_of_le t $... | lemma | filter.lift_assoc | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi_le",
"infi_le_of_le",
"le_infi",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lift_same_le_lift {g : set α → set α → filter β} :
f.lift (λs, f.lift (g s)) ≤ f.lift (λs, g s s) | le_lift.2 $ λ s hs, lift_le hs $ lift_le hs le_rfl | lemma | filter.lift_lift_same_le_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lift_same_eq_lift {g : set α → set α → filter β}
(hg₁ : ∀s, monotone (λt, g s t)) (hg₂ : ∀t, monotone (λs, g s t)) :
f.lift (λs, f.lift (g s)) = f.lift (λs, g s s) | lift_lift_same_le_lift.antisymm $
le_lift.2 $ λ s hs, le_lift.2 $ λ t ht, lift_le (inter_mem hs ht) $
calc g (s ∩ t) (s ∩ t) ≤ g s (s ∩ t) : hg₂ (s ∩ t) (inter_subset_left _ _)
... ≤ g s t : hg₁ s (inter_subset_right _ _) | lemma | filter.lift_lift_same_eq_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_principal {s : set α} (hg : monotone g) :
(𝓟 s).lift g = g s | (lift_le (mem_principal_self _) le_rfl).antisymm (le_lift.2 $ λ t ht, hg ht) | lemma | filter.lift_principal | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"le_rfl",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_lift [preorder γ] {f : γ → filter α} {g : γ → set α → filter β}
(hf : monotone f) (hg : monotone g) : monotone (λc, (f c).lift (g c)) | assume a b h, lift_mono (hf h) (hg h) | theorem | filter.monotone_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_ne_bot_iff (hm : monotone g) : (ne_bot $ f.lift g) ↔ (∀s∈f, ne_bot (g s)) | by simp only [ne_bot_iff, ne.def, ← empty_mem_iff_bot, mem_lift_sets hm, not_exists] | lemma | filter.lift_ne_bot_iff | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_const {f : filter α} {g : filter β} : f.lift (λx, g) = g | infi_subtype'.trans infi_const | lemma | filter.lift_const | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_inf {f : filter α} {g h : set α → filter β} :
f.lift (λx, g x ⊓ h x) = f.lift g ⊓ f.lift h | by simp only [filter.lift, infi_inf_eq] | lemma | filter.lift_inf | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"filter.lift",
"infi_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_principal2 {f : filter α} : f.lift 𝓟 = f | le_antisymm
(assume s hs, mem_lift hs (mem_principal_self s))
(le_infi $ assume s, le_infi $ assume hs, by simp only [hs, le_principal_iff]) | lemma | filter.lift_principal2 | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"le_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_infi_le {f : ι → filter α} {g : set α → filter β} :
(infi f).lift g ≤ ⨅ i, (f i).lift g | le_infi $ λ i, lift_mono (infi_le _ _) le_rfl | lemma | filter.lift_infi_le | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi",
"infi_le",
"le_infi",
"le_rfl",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_infi [nonempty ι] {f : ι → filter α} {g : set α → filter β}
(hg : ∀ s t, g (s ∩ t) = g s ⊓ g t) : (infi f).lift g = (⨅i, (f i).lift g) | begin
refine lift_infi_le.antisymm (λ s, _),
have H : ∀ t ∈ infi f, (⨅ i, (f i).lift g) ≤ g t,
{ intros t ht,
refine infi_sets_induct ht _ (λ i s t hs ht, _),
{ inhabit ι,
exact infi₂_le_of_le default univ (infi_le _ univ_mem) },
{ rw hg,
exact le_inf (infi₂_le_of_le i s $ infi_le _ hs) ht... | lemma | filter.lift_infi | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"exists_imp_distrib",
"filter",
"infi",
"infi_le",
"infi₂_le_of_le",
"le_inf",
"lift",
"monotone.of_map_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_infi_of_directed [nonempty ι] {f : ι → filter α} {g : set α → filter β}
(hf : directed (≥) f) (hg : monotone g) : (infi f).lift g = (⨅i, (f i).lift g) | lift_infi_le.antisymm $ λ s,
begin
simp only [mem_lift_sets hg, exists_imp_distrib, mem_infi_of_directed hf],
exact assume t i ht hs, mem_infi_of_mem i $ mem_lift ht hs
end | lemma | filter.lift_infi_of_directed | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"directed",
"exists_imp_distrib",
"filter",
"infi",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_infi_of_map_univ {f : ι → filter α} {g : set α → filter β}
(hg : ∀ s t, g (s ∩ t) = g s ⊓ g t) (hg' : g univ = ⊤) :
(infi f).lift g = (⨅i, (f i).lift g) | begin
casesI is_empty_or_nonempty ι,
{ simp [infi_of_empty, hg'] },
{ exact lift_infi hg }
end | lemma | filter.lift_infi_of_map_univ | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi",
"infi_of_empty",
"is_empty_or_nonempty",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift' (f : filter α) (h : set α → set β) | f.lift (𝓟 ∘ h) | def | filter.lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter"
] | Specialize `lift` to functions `set α → set β`. This can be viewed as a generalization of `map`.
This is essentially a push-forward along a function mapping each set to a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift'_top (h : set α → set β) : (⊤ : filter α).lift' h = 𝓟 (h univ) | lift_top _ | lemma | filter.lift'_top | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_lift' {t : set α} (ht : t ∈ f) : h t ∈ (f.lift' h) | le_principal_iff.mp $ show f.lift' h ≤ 𝓟 (h t),
from infi_le_of_le t $ infi_le_of_le ht $ le_rfl | lemma | filter.mem_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"infi_le_of_le",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lift' {m : γ → β} {l : filter γ} :
tendsto m l (f.lift' h) ↔ ∀ s ∈ f, ∀ᶠ a in l, m a ∈ h s | by simp only [filter.lift', tendsto_lift, tendsto_principal] | lemma | filter.tendsto_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"filter.lift'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis.lift' {ι} {p : ι → Prop} {s} (hf : f.has_basis p s) (hh : monotone h) :
(f.lift' h).has_basis p (h ∘ s) | begin
refine ⟨λ t, (hf.mem_lift_iff _ (monotone_principal.comp hh)).trans _⟩,
show ∀ i, (𝓟 (h (s i))).has_basis (λ j : unit, true) (λ (j : unit), h (s i)),
from λ i, has_basis_principal _,
simp only [exists_const]
end | lemma | filter.has_basis.lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"exists_const",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_lift'_sets (hh : monotone h) {s : set β} : s ∈ f.lift' h ↔ ∃ t ∈ f, h t ⊆ s | mem_lift_sets $ monotone_principal.comp hh | lemma | filter.mem_lift'_sets | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_lift'_iff (hh : monotone h) {p : β → Prop} :
(∀ᶠ y in f.lift' h, p y) ↔ (∃ t ∈ f, ∀ y ∈ h t, p y) | mem_lift'_sets hh | lemma | filter.eventually_lift'_iff | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter_lift'_sets (hh : monotone h) :
⋂₀ {s | s ∈ f.lift' h} = ⋂ s ∈ f, h s | (sInter_lift_sets (monotone_principal.comp hh)).trans $ Inter₂_congr $ λ s hs, cInf_Ici | lemma | filter.sInter_lift'_sets | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"cInf_Ici",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_le {f : filter α} {g : set α → set β} {h : filter β} {s : set α}
(hs : s ∈ f) (hg : 𝓟 (g s) ≤ h) : f.lift' g ≤ h | lift_le hs hg | lemma | filter.lift'_le | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_mono (hf : f₁ ≤ f₂) (hh : h₁ ≤ h₂) : f₁.lift' h₁ ≤ f₂.lift' h₂ | lift_mono hf $ assume s, principal_mono.mpr $ hh s | lemma | filter.lift'_mono | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_mono' (hh : ∀s∈f, h₁ s ⊆ h₂ s) : f.lift' h₁ ≤ f.lift' h₂ | infi₂_mono $ λ s hs, principal_mono.mpr $ hh s hs | lemma | filter.lift'_mono' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"infi₂_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_cong (hh : ∀s∈f, h₁ s = h₂ s) : f.lift' h₁ = f.lift' h₂ | le_antisymm (lift'_mono' $ assume s hs, le_of_eq $ hh s hs)
(lift'_mono' $ assume s hs, le_of_eq $ (hh s hs).symm) | lemma | filter.lift'_cong | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lift'_eq {m : β → γ} (hh : monotone h) : map m (f.lift' h) = f.lift' (image m ∘ h) | calc map m (f.lift' h) = f.lift (map m ∘ 𝓟 ∘ h) :
map_lift_eq $ monotone_principal.comp hh
... = f.lift' (image m ∘ h) : by simp only [(∘), filter.lift', map_principal, eq_self_iff_true] | lemma | filter.map_lift'_eq | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter.lift'",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_map_le {g : set β → set γ} {m : α → β} : (map m f).lift' g ≤ f.lift' (g ∘ image m) | lift_map_le | lemma | filter.lift'_map_le | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lift'_eq2 {g : set β → set γ} {m : α → β} (hg : monotone g) :
(map m f).lift' g = f.lift' (g ∘ image m) | map_lift_eq2 $ monotone_principal.comp hg | lemma | filter.map_lift'_eq2 | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_lift'_eq {m : γ → β} : comap m (f.lift' h) = f.lift' (preimage m ∘ h) | by simp only [filter.lift', comap_lift_eq, (∘), comap_principal] | theorem | filter.comap_lift'_eq | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter.lift'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_lift'_eq2 {m : β → α} {g : set β → set γ} (hg : monotone g) :
(comap m f).lift' g = f.lift' (g ∘ preimage m) | comap_lift_eq2 $ monotone_principal.comp hg | theorem | filter.comap_lift'_eq2 | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_principal {s : set α} (hh : monotone h) :
(𝓟 s).lift' h = 𝓟 (h s) | lift_principal $ monotone_principal.comp hh | lemma | filter.lift'_principal | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_pure {a : α} (hh : monotone h) :
(pure a : filter α).lift' h = 𝓟 (h {a}) | by rw [← principal_singleton, lift'_principal hh] | lemma | filter.lift'_pure | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_bot (hh : monotone h) : (⊥ : filter α).lift' h = 𝓟 (h ∅) | by rw [← principal_empty, lift'_principal hh] | lemma | filter.lift'_bot | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_lift' {f : filter α} {h : set α → set β} {g : filter β} :
g ≤ f.lift' h ↔ ∀ s ∈ f, h s ∈ g | le_lift.trans $ forall₂_congr $ λ s hs, le_principal_iff | lemma | filter.le_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_le_lift' {t : set β} : 𝓟 t ≤ f.lift' h ↔ ∀ s ∈ f, t ⊆ h s | le_lift' | lemma | filter.principal_le_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_lift' [preorder γ] {f : γ → filter α} {g : γ → set α → set β}
(hf : monotone f) (hg : monotone g) : monotone (λc, (f c).lift' (g c)) | assume a b h, lift'_mono (hf h) (hg h) | theorem | filter.monotone_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lift'_assoc {g : set α → set β} {h : set β → filter γ}
(hg : monotone g) (hh : monotone h) :
(f.lift' g).lift h = f.lift (λs, h (g s)) | calc (f.lift' g).lift h = f.lift (λs, (𝓟 (g s)).lift h) :
lift_assoc (monotone_principal.comp hg)
... = f.lift (λs, h (g s)) : by simp only [lift_principal, hh, eq_self_iff_true] | lemma | filter.lift_lift'_assoc | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"lift",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_lift'_assoc {g : set α → set β} {h : set β → set γ}
(hg : monotone g) (hh : monotone h) :
(f.lift' g).lift' h = f.lift' (λs, h (g s)) | lift_lift'_assoc hg (monotone_principal.comp hh) | lemma | filter.lift'_lift'_assoc | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_lift_assoc {g : set α → filter β} {h : set β → set γ}
(hg : monotone g) : (f.lift g).lift' h = f.lift (λs, (g s).lift' h) | lift_assoc hg | lemma | filter.lift'_lift_assoc | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lift'_same_le_lift' {g : set α → set α → set β} :
f.lift (λs, f.lift' (g s)) ≤ f.lift' (λs, g s s) | lift_lift_same_le_lift | lemma | filter.lift_lift'_same_le_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lift'_same_eq_lift' {g : set α → set α → set β}
(hg₁ : ∀s, monotone (λt, g s t)) (hg₂ : ∀t, monotone (λs, g s t)) :
f.lift (λs, f.lift' (g s)) = f.lift' (λs, g s s) | lift_lift_same_eq_lift
(assume s, monotone_principal.comp (hg₁ s))
(assume t, monotone_principal.comp (hg₂ t)) | lemma | filter.lift_lift'_same_eq_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_inf_principal_eq {h : set α → set β} {s : set β} :
f.lift' h ⊓ 𝓟 s = f.lift' (λt, h t ∩ s) | by simp only [filter.lift', filter.lift, (∘), ← inf_principal, infi_subtype', ← infi_inf] | lemma | filter.lift'_inf_principal_eq | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter.lift",
"filter.lift'",
"infi_inf",
"infi_subtype'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_ne_bot_iff (hh : monotone h) : (ne_bot (f.lift' h)) ↔ (∀s∈f, (h s).nonempty) | calc (ne_bot (f.lift' h)) ↔ (∀s∈f, ne_bot (𝓟 (h s))) :
lift_ne_bot_iff (monotone_principal.comp hh)
... ↔ (∀s∈f, (h s).nonempty) : by simp only [principal_ne_bot_iff] | lemma | filter.lift'_ne_bot_iff | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_id {f : filter α} : f.lift' id = f | lift_principal2 | lemma | filter.lift'_id | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_infi [nonempty ι] {f : ι → filter α} {g : set α → set β}
(hg : ∀ s t, g (s ∩ t) = g s ∩ g t) : (infi f).lift' g = (⨅ i, (f i).lift' g) | lift_infi $ λ s t, by rw [inf_principal, (∘), ← hg] | lemma | filter.lift'_infi | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_infi_of_map_univ {f : ι → filter α} {g : set α → set β}
(hg : ∀{s t}, g (s ∩ t) = g s ∩ g t) (hg' : g univ = univ) :
(infi f).lift' g = (⨅ i, (f i).lift' g) | lift_infi_of_map_univ (λ s t, by rw [inf_principal, (∘), ← hg])
(by rw [function.comp_app, hg', principal_univ]) | lemma | filter.lift'_infi_of_map_univ | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_inf (f g : filter α) {s : set α → set β} (hs : ∀ t₁ t₂, s (t₁ ∩ t₂) = s t₁ ∩ s t₂) :
(f ⊓ g).lift' s = f.lift' s ⊓ g.lift' s | have (⨅ b : bool, cond b f g).lift' s = ⨅ b : bool, (cond b f g).lift' s :=
lift'_infi @hs,
by simpa only [infi_bool_eq] | lemma | filter.lift'_inf | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"infi_bool_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_inf_le (f g : filter α) (s : set α → set β) :
(f ⊓ g).lift' s ≤ f.lift' s ⊓ g.lift' s | le_inf (lift'_mono inf_le_left le_rfl) (lift'_mono inf_le_right le_rfl) | lemma | filter.lift'_inf_le | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"inf_le_left",
"inf_le_right",
"le_inf",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_eq_lift' {f : filter β} {m : α → β} :
comap m f = f.lift' (preimage m) | filter.ext $ λ s, (mem_lift'_sets monotone_preimage).symm | theorem | filter.comap_eq_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_def {f : filter α} {g : filter β} : f ×ᶠ g = (f.lift $ λ s, g.lift' $ λ t, s ×ˢ t) | have ∀(s:set α) (t : set β),
𝓟 (s ×ˢ t) = (𝓟 s).comap prod.fst ⊓ (𝓟 t).comap prod.snd,
by simp only [principal_eq_iff_eq, comap_principal, inf_principal]; intros; refl,
begin
simp only [filter.lift', function.comp, this, lift_inf, lift_const, lift_inf],
rw [← comap_lift_eq, ← comap_lift_eq],
simp only [f... | lemma | filter.prod_def | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"filter.lift'",
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_same_eq : f ×ᶠ f = f.lift' (λ t : set α, t ×ˢ t) | prod_def.trans $ lift_lift'_same_eq_lift'
(λ s, monotone_const.set_prod monotone_id)
(λ t, monotone_id.set_prod monotone_const) | lemma | filter.prod_same_eq | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone_const",
"monotone_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod_same_iff {s : set (α×α)} :
s ∈ f ×ᶠ f ↔ (∃t∈f, t ×ˢ t ⊆ s) | by { rw [prod_same_eq, mem_lift'_sets], exact monotone_id.set_prod monotone_id } | lemma | filter.mem_prod_same_iff | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"monotone_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_self_iff {f : α × α → β} {x : filter α} {y : filter β} :
filter.tendsto f (x ×ᶠ x) y ↔
∀ W ∈ y, ∃ U ∈ x, ∀ (x x' : α), x ∈ U → x' ∈ U → f (x, x') ∈ W | by simp only [tendsto_def, mem_prod_same_iff, prod_sub_preimage_iff, exists_prop, iff_self] | lemma | filter.tendsto_prod_self_iff | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"exists_prop",
"filter",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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