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
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mul_action [monoid α] [mul_action α β] : mul_action (filter α) (filter β) | { one_smul := λ f, map₂_pure_left.trans $ by simp_rw [one_smul, map_id'],
mul_smul := λ f g h, map₂_assoc mul_smul } | def | filter.mul_action | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"monoid",
"mul_action",
"one_smul"
] | A multiplicative action of a monoid `α` on a type `β` gives a multiplicative action of
`filter α` on `filter β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_action_filter [monoid α] [mul_action α β] : mul_action α (filter β) | { mul_smul := λ a b f, by simp only [←map_smul, map_map, function.comp, ←mul_smul],
one_smul := λ f, by simp only [←map_smul, one_smul, map_id'] } | def | filter.mul_action_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"monoid",
"mul_action",
"one_smul"
] | A multiplicative action of a monoid on a type `β` gives a multiplicative action on `filter β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distrib_mul_action_filter [monoid α] [add_monoid β] [distrib_mul_action α β] :
distrib_mul_action α (filter β) | { smul_add := λ _ _ _, map_map₂_distrib $ smul_add _,
smul_zero := λ _, (map_pure _ _).trans $ by rw [smul_zero, pure_zero] } | def | filter.distrib_mul_action_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"add_monoid",
"distrib_mul_action",
"filter",
"monoid",
"smul_add",
"smul_zero"
] | A distributive multiplicative action of a monoid on an additive monoid `β` gives a distributive
multiplicative action on `filter β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_distrib_mul_action_filter [monoid α] [monoid β] [mul_distrib_mul_action α β] :
mul_distrib_mul_action α (set β) | { smul_mul := λ _ _ _, image_image2_distrib $ smul_mul' _,
smul_one := λ _, image_singleton.trans $ by rw [smul_one, singleton_one] } | def | filter.mul_distrib_mul_action_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"monoid",
"mul_distrib_mul_action",
"smul_mul'"
] | A multiplicative action of a monoid on a monoid `β` gives a multiplicative action on `set β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_bot.smul_zero_nonneg (hf : f.ne_bot) : 0 ≤ f • (0 : filter β) | le_smul_iff.2 $ λ t₁ h₁ t₂ h₂, let ⟨a, ha⟩ := hf.nonempty_of_mem h₁ in
⟨_, _, ha, h₂, smul_zero _⟩ | lemma | filter.ne_bot.smul_zero_nonneg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.zero_smul_nonneg (hg : g.ne_bot) : 0 ≤ (0 : filter α) • g | le_smul_iff.2 $ λ t₁ h₁ t₂ h₂, let ⟨b, hb⟩ := hg.nonempty_of_mem h₂ in ⟨_, _, h₁, hb, zero_smul _ _⟩ | lemma | filter.ne_bot.zero_smul_nonneg | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_smul_filter_nonpos : (0 : α) • g ≤ 0 | begin
refine λ s hs, mem_smul_filter.2 _,
convert univ_mem,
refine eq_univ_iff_forall.2 (λ a, _),
rwa [mem_preimage, zero_smul],
end | lemma | filter.zero_smul_filter_nonpos | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_smul_filter (hg : g.ne_bot) : (0 : α) • g = 0 | zero_smul_filter_nonpos.antisymm $ le_map_iff.2 $ λ s hs, begin
simp_rw [set.image_eta, zero_smul, (hg.nonempty_of_mem hs).image_const],
exact zero_mem_zero,
end | lemma | filter.zero_smul_filter | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"set.image_eta",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (f : filter α) (g : filter β) : filter (α × β) | f.comap prod.fst ⊓ g.comap prod.snd | def | filter.prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | Product of filters. This is the filter generated by cartesian products
of elements of the component filters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mem_prod {s : set α} {t : set β} {f : filter α} {g : filter β}
(hs : s ∈ f) (ht : t ∈ g) : s ×ˢ t ∈ f ×ᶠ g | inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht) | lemma | filter.prod_mem_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod_iff {s : set (α×β)} {f : filter α} {g : filter β} :
s ∈ f ×ᶠ g ↔ (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s) | begin
simp only [filter.prod],
split,
{ rintro ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, rfl⟩,
exact ⟨s₁, hs₁, s₂, hs₂, λ p ⟨h, h'⟩, ⟨hts₁ h, hts₂ h'⟩⟩ },
{ rintro ⟨t₁, ht₁, t₂, ht₂, h⟩,
exact mem_inf_of_inter (preimage_mem_comap ht₁) (preimage_mem_comap ht₂) h }
end | lemma | filter.mem_prod_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mem_prod_iff {s : set α} {t : set β} {f : filter α} {g : filter β}
[f.ne_bot] [g.ne_bot] :
s ×ˢ t ∈ f ×ᶠ g ↔ s ∈ f ∧ t ∈ g | ⟨λ h, let ⟨s', hs', t', ht', H⟩ := mem_prod_iff.1 h in (prod_subset_prod_iff.1 H).elim
(λ ⟨hs's, ht't⟩, ⟨mem_of_superset hs' hs's, mem_of_superset ht' ht't⟩)
(λ h, h.elim
(λ hs'e, absurd hs'e (nonempty_of_mem hs').ne_empty)
(λ ht'e, absurd ht'e (nonempty_of_mem ht').ne_empty)),
λ h, prod_mem_prod h.1 h.2⟩ | lemma | filter.prod_mem_prod_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod_principal {f : filter α} {s : set (α × β)} {t : set β}:
s ∈ f ×ᶠ 𝓟 t ↔ {a | ∀ b ∈ t, (a, b) ∈ s} ∈ f | begin
rw [← @exists_mem_subset_iff _ f, mem_prod_iff],
refine exists₂_congr (λ u u_in, ⟨_, λ h, ⟨t, mem_principal_self t, _⟩⟩),
{ rintros ⟨v, v_in, hv⟩ a a_in b b_in,
exact hv (mk_mem_prod a_in $ v_in b_in) },
{ rintro ⟨x, y⟩ ⟨hx, hy⟩,
exact h hx y hy }
end | lemma | filter.mem_prod_principal | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"exists₂_congr",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod_top {f : filter α} {s : set (α × β)} :
s ∈ f ×ᶠ (⊤ : filter β) ↔ {a | ∀ b, (a, b) ∈ s} ∈ f | begin
rw [← principal_univ, mem_prod_principal],
simp only [mem_univ, forall_true_left]
end | lemma | filter.mem_prod_top | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"forall_true_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_prod_principal_iff {p : α × β → Prop} {s : set β} :
(∀ᶠ (x : α × β) in (f ×ᶠ (𝓟 s)), p x) ↔ ∀ᶠ (x : α) in f, ∀ (y : β), y ∈ s → p (x, y) | by { rw [eventually_iff, eventually_iff, mem_prod_principal], simp only [mem_set_of_eq], } | lemma | filter.eventually_prod_principal_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_prod (f : α → β × γ) (b : filter β) (c : filter γ) :
comap f (b ×ᶠ c) = (comap (prod.fst ∘ f) b) ⊓ (comap (prod.snd ∘ f) c) | by erw [comap_inf, filter.comap_comap, filter.comap_comap] | lemma | filter.comap_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.comap_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top {f : filter α} : f ×ᶠ (⊤ : filter β) = f.comap prod.fst | by rw [filter.prod, comap_top, inf_top_eq] | lemma | filter.prod_top | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"inf_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_prod (f₁ f₂ : filter α) (g : filter β) : (f₁ ⊔ f₂) ×ᶠ g = (f₁ ×ᶠ g) ⊔ (f₂ ×ᶠ g) | by rw [filter.prod, comap_sup, inf_sup_right, ← filter.prod, ← filter.prod] | lemma | filter.sup_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"inf_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_sup (f : filter α) (g₁ g₂ : filter β) : f ×ᶠ (g₁ ⊔ g₂) = (f ×ᶠ g₁) ⊔ (f ×ᶠ g₂) | by rw [filter.prod, comap_sup, inf_sup_left, ← filter.prod, ← filter.prod] | lemma | filter.prod_sup | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"inf_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_prod_iff {p : α × β → Prop} {f : filter α} {g : filter β} :
(∀ᶠ x in f ×ᶠ g, p x) ↔ ∃ (pa : α → Prop) (ha : ∀ᶠ x in f, pa x)
(pb : β → Prop) (hb : ∀ᶠ y in g, pb y), ∀ {x}, pa x → ∀ {y}, pb y → p (x, y) | by simpa only [set.prod_subset_iff] using @mem_prod_iff α β p f g | lemma | filter.eventually_prod_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"set.prod_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_fst {f : filter α} {g : filter β} : tendsto prod.fst (f ×ᶠ g) f | tendsto_inf_left tendsto_comap | lemma | filter.tendsto_fst | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_snd {f : filter α} {g : filter β} : tendsto prod.snd (f ×ᶠ g) g | tendsto_inf_right tendsto_comap | lemma | filter.tendsto_snd | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.prod_mk {f : filter α} {g : filter β} {h : filter γ} {m₁ : α → β} {m₂ : α → γ}
(h₁ : tendsto m₁ f g) (h₂ : tendsto m₂ f h) : tendsto (λ x, (m₁ x, m₂ x)) f (g ×ᶠ h) | tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩ | lemma | filter.tendsto.prod_mk | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_swap {α1 α2 : Type*} {a1 : filter α1} {a2 : filter α2} :
tendsto (prod.swap : α1 × α2 → α2 × α1) (a1 ×ᶠ a2) (a2 ×ᶠ a1) | tendsto_snd.prod_mk tendsto_fst | lemma | filter.tendsto_prod_swap | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.prod_inl {la : filter α} {p : α → Prop} (h : ∀ᶠ x in la, p x) (lb : filter β) :
∀ᶠ x in la ×ᶠ lb, p (x : α × β).1 | tendsto_fst.eventually h | lemma | filter.eventually.prod_inl | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.prod_inr {lb : filter β} {p : β → Prop} (h : ∀ᶠ x in lb, p x) (la : filter α) :
∀ᶠ x in la ×ᶠ lb, p (x : α × β).2 | tendsto_snd.eventually h | lemma | filter.eventually.prod_inr | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.prod_mk {la : filter α} {pa : α → Prop} (ha : ∀ᶠ x in la, pa x)
{lb : filter β} {pb : β → Prop} (hb : ∀ᶠ y in lb, pb y) :
∀ᶠ p in la ×ᶠ lb, pa (p : α × β).1 ∧ pb p.2 | (ha.prod_inl lb).and (hb.prod_inr la) | lemma | filter.eventually.prod_mk | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq.prod_map {δ} {la : filter α} {fa ga : α → γ} (ha : fa =ᶠ[la] ga)
{lb : filter β} {fb gb : β → δ} (hb : fb =ᶠ[lb] gb) :
prod.map fa fb =ᶠ[la ×ᶠ lb] prod.map ga gb | (eventually.prod_mk ha hb).mono $ λ x h, prod.ext h.1 h.2 | lemma | filter.eventually_eq.prod_map | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.prod_map {δ} [has_le γ] [has_le δ] {la : filter α} {fa ga : α → γ}
(ha : fa ≤ᶠ[la] ga) {lb : filter β} {fb gb : β → δ} (hb : fb ≤ᶠ[lb] gb) :
prod.map fa fb ≤ᶠ[la ×ᶠ lb] prod.map ga gb | eventually.prod_mk ha hb | lemma | filter.eventually_le.prod_map | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.curry {la : filter α} {lb : filter β} {p : α × β → Prop}
(h : ∀ᶠ x in la ×ᶠ lb, p x) :
∀ᶠ x in la, ∀ᶠ y in lb, p (x, y) | begin
rcases eventually_prod_iff.1 h with ⟨pa, ha, pb, hb, h⟩,
exact ha.mono (λ a ha, hb.mono $ λ b hb, h ha hb)
end | lemma | filter.eventually.curry | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.diag_of_prod {f : filter α} {p : α × α → Prop}
(h : ∀ᶠ i in f ×ᶠ f, p i) : (∀ᶠ i in f, p (i, i)) | begin
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h,
apply (ht.and hs).mono (λ x hx, hst hx.1 hx.2),
end | lemma | filter.eventually.diag_of_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | A fact that is eventually true about all pairs `l ×ᶠ l` is eventually true about
all diagonal pairs `(i, i)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually.diag_of_prod_left {f : filter α} {g : filter γ}
{p : (α × α) × γ → Prop} :
(∀ᶠ x in (f ×ᶠ f ×ᶠ g), p x) →
(∀ᶠ (x : α × γ) in (f ×ᶠ g), p ((x.1, x.1), x.2)) | begin
intros h,
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h,
refine (ht.diag_of_prod.prod_mk hs).mono (λ x hx, by simp only [hst hx.1 hx.2, prod.mk.eta]),
end | lemma | filter.eventually.diag_of_prod_left | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.diag_of_prod_right {f : filter α} {g : filter γ}
{p : α × γ × γ → Prop} :
(∀ᶠ x in (f ×ᶠ (g ×ᶠ g)), p x) →
(∀ᶠ (x : α × γ) in (f ×ᶠ g), p (x.1, x.2, x.2)) | begin
intros h,
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h,
refine (ht.prod_mk hs.diag_of_prod).mono (λ x hx, by simp only [hst hx.1 hx.2, prod.mk.eta]),
end | lemma | filter.eventually.diag_of_prod_right | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_diag : tendsto (λ i, (i, i)) f (f ×ᶠ f) | tendsto_iff_eventually.mpr (λ _ hpr, hpr.diag_of_prod) | lemma | filter.tendsto_diag | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_infi_left [nonempty ι] {f : ι → filter α} {g : filter β}:
(⨅ i, f i) ×ᶠ g = (⨅ i, (f i) ×ᶠ g) | by { rw [filter.prod, comap_infi, infi_inf], simp only [filter.prod, eq_self_iff_true] } | lemma | filter.prod_infi_left | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"infi_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_infi_right [nonempty ι] {f : filter α} {g : ι → filter β} :
f ×ᶠ (⨅ i, g i) = (⨅ i, f ×ᶠ (g i)) | by { rw [filter.prod, comap_infi, inf_infi], simp only [filter.prod, eq_self_iff_true] } | lemma | filter.prod_infi_right | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"inf_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) :
f₁ ×ᶠ g₁ ≤ f₂ ×ᶠ g₂ | inf_le_inf (comap_mono hf) (comap_mono hg) | lemma | filter.prod_mono | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"inf_le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_left (g : filter β) {f₁ f₂ : filter α} (hf : f₁ ≤ f₂) :
f₁ ×ᶠ g ≤ f₂ ×ᶠ g | filter.prod_mono hf rfl.le | lemma | filter.prod_mono_left | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_right (f : filter α) {g₁ g₂ : filter β} (hf : g₁ ≤ g₂) :
f ×ᶠ g₁ ≤ f ×ᶠ g₂ | filter.prod_mono rfl.le hf | lemma | filter.prod_mono_right | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u v w x} prod_comap_comap_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} :
(comap m₁ f₁) ×ᶠ (comap m₂ f₂) = comap (λ p : β₁×β₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) | by simp only [filter.prod, comap_comap, eq_self_iff_true, comap_inf] | lemma | filter.prod_comap_comap_eq | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm' : f ×ᶠ g = comap (prod.swap) (g ×ᶠ f) | by simp only [filter.prod, comap_comap, (∘), inf_comm, prod.fst_swap,
eq_self_iff_true, prod.snd_swap, comap_inf] | lemma | filter.prod_comm' | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter.prod",
"inf_comm",
"prod.fst_swap",
"prod.snd_swap",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm : f ×ᶠ g = map (λ p : β×α, (p.2, p.1)) (g ×ᶠ f) | by { rw [prod_comm', ← map_swap_eq_comap_swap], refl } | lemma | filter.prod_comm | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_fst_prod (f : filter α) (g : filter β) [ne_bot g] : map prod.fst (f ×ᶠ g) = f | begin
refine le_antisymm tendsto_fst (λ s hs, _),
rw [mem_map, mem_prod_iff] at hs,
rcases hs with ⟨t₁, h₁, t₂, h₂, hs⟩,
rw [← image_subset_iff, fst_image_prod] at hs,
exacts [mem_of_superset h₁ hs, nonempty_of_mem h₂]
end | lemma | filter.map_fst_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_snd_prod (f : filter α) (g : filter β) [ne_bot f] : map prod.snd (f ×ᶠ g) = g | by rw [prod_comm, map_map, (∘), map_fst_prod] | lemma | filter.map_snd_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod {f₁ f₂ : filter α} {g₁ g₂ : filter β} [ne_bot f₁] [ne_bot g₁] :
f₁ ×ᶠ g₁ ≤ f₂ ×ᶠ g₂ ↔ f₁ ≤ f₂ ∧ g₁ ≤ g₂ | ⟨λ h, ⟨map_fst_prod f₁ g₁ ▸ tendsto_fst.mono_left h, map_snd_prod f₁ g₁ ▸ tendsto_snd.mono_left h⟩,
λ h, prod_mono h.1 h.2⟩ | lemma | filter.prod_le_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inj {f₁ f₂ : filter α} {g₁ g₂ : filter β} [ne_bot f₁] [ne_bot g₁] :
f₁ ×ᶠ g₁ = f₂ ×ᶠ g₂ ↔ f₁ = f₂ ∧ g₁ = g₂ | begin
refine ⟨λ h, _, λ h, h.1 ▸ h.2 ▸ rfl⟩,
have hle : f₁ ≤ f₂ ∧ g₁ ≤ g₂ := prod_le_prod.1 h.le,
haveI := ne_bot_of_le hle.1, haveI := ne_bot_of_le hle.2,
exact ⟨hle.1.antisymm $ (prod_le_prod.1 h.ge).1, hle.2.antisymm $ (prod_le_prod.1 h.ge).2⟩
end | lemma | filter.prod_inj | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_swap_iff {p : (α × β) → Prop} : (∀ᶠ (x : α × β) in (f ×ᶠ g), p x) ↔
∀ᶠ (y : β × α) in (g ×ᶠ f), p y.swap | by { rw [prod_comm, eventually_map], simpa, } | lemma | filter.eventually_swap_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_assoc (f : filter α) (g : filter β) (h : filter γ) :
map (equiv.prod_assoc α β γ) ((f ×ᶠ g) ×ᶠ h) = f ×ᶠ (g ×ᶠ h) | by simp_rw [← comap_equiv_symm, filter.prod, comap_inf, comap_comap, inf_assoc, function.comp,
equiv.prod_assoc_symm_apply] | lemma | filter.prod_assoc | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"equiv.prod_assoc",
"filter",
"filter.prod",
"inf_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_assoc_symm (f : filter α) (g : filter β) (h : filter γ) :
map (equiv.prod_assoc α β γ).symm (f ×ᶠ (g ×ᶠ h)) = (f ×ᶠ g) ×ᶠ h | by simp_rw [map_equiv_symm, filter.prod, comap_inf, comap_comap, inf_assoc, function.comp,
equiv.prod_assoc_apply] | theorem | filter.prod_assoc_symm | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"equiv.prod_assoc",
"filter",
"filter.prod",
"inf_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_assoc {f : filter α} {g : filter β} {h : filter γ} :
tendsto (equiv.prod_assoc α β γ) (f ×ᶠ g ×ᶠ h) (f ×ᶠ (g ×ᶠ h)) | (prod_assoc f g h).le | lemma | filter.tendsto_prod_assoc | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"equiv.prod_assoc",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_assoc_symm {f : filter α} {g : filter β} {h : filter γ} :
tendsto (equiv.prod_assoc α β γ).symm (f ×ᶠ (g ×ᶠ h)) (f ×ᶠ g ×ᶠ h) | (prod_assoc_symm f g h).le | lemma | filter.tendsto_prod_assoc_symm | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"equiv.prod_assoc",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_swap4_prod {f : filter α} {g : filter β} {h : filter γ} {k : filter δ} :
map (λ p : (α × β) × (γ × δ), ((p.1.1, p.2.1), (p.1.2, p.2.2))) ((f ×ᶠ g) ×ᶠ (h ×ᶠ k)) =
(f ×ᶠ h) ×ᶠ (g ×ᶠ k) | by simp_rw [map_swap4_eq_comap, filter.prod, comap_inf, comap_comap, inf_assoc, inf_left_comm] | lemma | filter.map_swap4_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"inf_assoc",
"inf_left_comm"
] | A useful lemma when dealing with uniformities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_swap4_prod {f : filter α} {g : filter β} {h : filter γ} {k : filter δ} :
tendsto (λ p : (α × β) × (γ × δ), ((p.1.1, p.2.1), (p.1.2, p.2.2)))
((f ×ᶠ g) ×ᶠ (h ×ᶠ k)) ((f ×ᶠ h) ×ᶠ (g ×ᶠ k)) | map_swap4_prod.le | lemma | filter.tendsto_swap4_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u v w x} prod_map_map_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} :
(map m₁ f₁) ×ᶠ (map m₂ f₂) = map (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) | le_antisymm
(λ s hs,
let ⟨s₁, hs₁, s₂, hs₂, h⟩ := mem_prod_iff.mp hs in
filter.sets_of_superset _ (prod_mem_prod (image_mem_map hs₁) (image_mem_map hs₂)) $
calc (m₁ '' s₁) ×ˢ (m₂ '' s₂) = (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) '' s₁ ×ˢ s₂ :
set.prod_image_image_eq
... ⊆ _ : by rwa [image_subs... | lemma | filter.prod_map_map_eq | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"le_rfl",
"set.prod_image_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map_map_eq' {α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*}
(f : α₁ → α₂) (g : β₁ → β₂) (F : filter α₁) (G : filter β₁) :
(map f F) ×ᶠ (map g G) = map (prod.map f g) (F ×ᶠ G) | prod_map_map_eq | lemma | filter.prod_map_map_eq' | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_prod_map_fst_snd {f : filter (α × β)} : f ≤ map prod.fst f ×ᶠ map prod.snd f | le_inf le_comap_map le_comap_map | lemma | filter.le_prod_map_fst_snd | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.prod_map {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β}
{c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) :
tendsto (prod.map f g) (a ×ᶠ b) (c ×ᶠ d) | begin
erw [tendsto, ← prod_map_map_eq],
exact filter.prod_mono hf hg,
end | lemma | filter.tendsto.prod_map | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod (m : α × β → γ) (f : filter α) (g : filter β) :
map m (f ×ᶠ g) = (f.map (λ a b, m (a, b))).seq g | begin
simp [filter.ext_iff, mem_prod_iff, mem_map_seq_iff],
intro s,
split,
exact λ ⟨t, ht, s, hs, h⟩, ⟨s, hs, t, ht, λ x hx y hy, @h ⟨x, y⟩ ⟨hx, hy⟩⟩,
exact λ ⟨s, hs, t, ht, h⟩, ⟨t, ht, s, hs, λ ⟨x, y⟩ ⟨hx, hy⟩, h x hx y hy⟩
end | lemma | filter.map_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.ext_iff",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq {f : filter α} {g : filter β} : f ×ᶠ g = (f.map prod.mk).seq g | have h : _ := f.map_prod id g, by rwa [map_id] at h | lemma | filter.prod_eq | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inf_prod {f₁ f₂ : filter α} {g₁ g₂ : filter β} :
(f₁ ×ᶠ g₁) ⊓ (f₂ ×ᶠ g₂) = (f₁ ⊓ f₂) ×ᶠ (g₁ ⊓ g₂) | by simp only [filter.prod, comap_inf, inf_comm, inf_assoc, inf_left_comm] | lemma | filter.prod_inf_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod",
"inf_assoc",
"inf_comm",
"inf_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_bot {f : filter α} : f ×ᶠ (⊥ : filter β) = ⊥ | by simp [filter.prod] | lemma | filter.prod_bot | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_prod {g : filter β} : (⊥ : filter α) ×ᶠ g = ⊥ | by simp [filter.prod] | lemma | filter.bot_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_principal_principal {s : set α} {t : set β} :
(𝓟 s) ×ᶠ (𝓟 t) = 𝓟 (s ×ˢ t) | by simp only [filter.prod, comap_principal, principal_eq_iff_eq, comap_principal, inf_principal];
refl | lemma | filter.prod_principal_principal | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_prod {a : α} {f : filter β} : pure a ×ᶠ f = map (prod.mk a) f | by rw [prod_eq, map_pure, pure_seq_eq_map] | lemma | filter.pure_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pure_prod (f : α → β → γ) (a : α) (B : filter β) :
filter.map (function.uncurry f) (pure a ×ᶠ B) = filter.map (f a) B | by { rw filter.pure_prod, refl } | lemma | filter.map_pure_prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.map",
"filter.pure_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_pure {f : filter α} {b : β} : f ×ᶠ pure b = map (λ a, (a, b)) f | by rw [prod_eq, seq_pure, map_map] | lemma | filter.prod_pure | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_pure_pure {a : α} {b : β} : (pure a) ×ᶠ (pure b) = pure (a, b) | by simp | lemma | filter.prod_pure_pure | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_bot {f : filter α} {g : filter β} : f ×ᶠ g = ⊥ ↔ (f = ⊥ ∨ g = ⊥) | begin
split,
{ intro h,
rcases mem_prod_iff.1 (empty_mem_iff_bot.2 h) with ⟨s, hs, t, ht, hst⟩,
rw [subset_empty_iff, set.prod_eq_empty_iff] at hst,
cases hst with s_eq t_eq,
{ left, exact empty_mem_iff_bot.1 (s_eq ▸ hs) },
{ right, exact empty_mem_iff_bot.1 (t_eq ▸ ht) } },
{ rintro (rfl | rf... | lemma | filter.prod_eq_bot | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"set.prod_eq_empty_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ne_bot {f : filter α} {g : filter β} : ne_bot (f ×ᶠ g) ↔ (ne_bot f ∧ ne_bot g) | by simp only [ne_bot_iff, ne, prod_eq_bot, not_or_distrib] | lemma | filter.prod_ne_bot | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.prod {f : filter α} {g : filter β} (hf : ne_bot f) (hg : ne_bot g) :
ne_bot (f ×ᶠ g) | prod_ne_bot.2 ⟨hf, hg⟩ | lemma | filter.ne_bot.prod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_ne_bot' {f : filter α} {g : filter β} [hf : ne_bot f] [hg : ne_bot g] :
ne_bot (f ×ᶠ g) | hf.prod hg | instance | filter.prod_ne_bot' | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_iff {f : α × β → γ} {x : filter α} {y : filter β} {z : filter γ} :
filter.tendsto f (x ×ᶠ y) z ↔
∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W | by simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self] | lemma | filter.tendsto_prod_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"exists_prop",
"filter",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_prod_iff' {f : filter α} {g : filter β} {g' : filter γ}
{s : α → β × γ} :
tendsto s f (g ×ᶠ g') ↔ tendsto (λ n, (s n).1) f g ∧ tendsto (λ n, (s n).2) f g' | by { unfold filter.prod, simp only [tendsto_inf, tendsto_comap_iff, iff_self] } | lemma | filter.tendsto_prod_iff' | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod (f : filter α) (g : filter β) : filter (α × β) | f.comap prod.fst ⊔ g.comap prod.snd | def | filter.coprod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | Coproduct of filters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_coprod_iff {s : set (α×β)} {f : filter α} {g : filter β} :
s ∈ f.coprod g ↔ ((∃ t₁ ∈ f, prod.fst ⁻¹' t₁ ⊆ s) ∧ (∃ t₂ ∈ g, prod.snd ⁻¹' t₂ ⊆ s)) | by simp [filter.coprod] | lemma | filter.mem_coprod_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coprod (l : filter β) : (⊥ : filter α).coprod l = comap prod.snd l | by simp [filter.coprod] | lemma | filter.bot_coprod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_bot (l : filter α) : l.coprod (⊥ : filter β) = comap prod.fst l | by simp [filter.coprod] | lemma | filter.coprod_bot | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_coprod_bot : (⊥ : filter α).coprod (⊥ : filter β) = ⊥ | by simp | lemma | filter.bot_coprod_bot | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mem_coprod {s : set (α × β)} {la : filter α} {lb : filter β} :
sᶜ ∈ la.coprod lb ↔ (prod.fst '' s)ᶜ ∈ la ∧ (prod.snd '' s)ᶜ ∈ lb | by simp only [filter.coprod, mem_sup, compl_mem_comap] | lemma | filter.compl_mem_coprod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"filter.coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) :
f₁.coprod g₁ ≤ f₂.coprod g₂ | sup_le_sup (comap_mono hf) (comap_mono hg) | lemma | filter.coprod_mono | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_ne_bot_iff : (f.coprod g).ne_bot ↔ f.ne_bot ∧ nonempty β ∨ nonempty α ∧ g.ne_bot | by simp [filter.coprod] | lemma | filter.coprod_ne_bot_iff | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter.coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_ne_bot_left [ne_bot f] [nonempty β] : (f.coprod g).ne_bot | coprod_ne_bot_iff.2 (or.inl ⟨‹_›, ‹_›⟩) | lemma | filter.coprod_ne_bot_left | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_ne_bot_right [ne_bot g] [nonempty α] : (f.coprod g).ne_bot | coprod_ne_bot_iff.2 (or.inr ⟨‹_›, ‹_›⟩) | lemma | filter.coprod_ne_bot_right | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_coprod_principal (s : set α) (t : set β) :
(𝓟 s).coprod (𝓟 t) = 𝓟 (sᶜ ×ˢ tᶜ)ᶜ | by rw [filter.coprod, comap_principal, comap_principal, sup_principal, set.prod_eq, compl_inter,
preimage_compl, preimage_compl, compl_compl, compl_compl] | lemma | filter.principal_coprod_principal | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"compl_compl",
"filter.coprod",
"set.prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u v w x} map_prod_map_coprod_le {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} :
map (prod.map m₁ m₂) (f₁.coprod f₂) ≤ (map m₁ f₁).coprod (map m₂ f₂) | begin
intros s,
simp only [mem_map, mem_coprod_iff],
rintro ⟨⟨u₁, hu₁, h₁⟩, u₂, hu₂, h₂⟩,
refine ⟨⟨m₁ ⁻¹' u₁, hu₁, λ _ hx, h₁ _⟩, ⟨m₂ ⁻¹' u₂, hu₂, λ _ hx, h₂ _⟩⟩; convert hx
end | lemma | filter.map_prod_map_coprod_le | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_const_principal_coprod_map_id_principal {α β ι : Type*} (a : α) (b : β) (i : ι) :
(map (λ _ : α, b) (𝓟 {a})).coprod (map id (𝓟 {i}))
= 𝓟 (({b} : set β) ×ˢ univ ∪ univ ×ˢ ({i} : set ι)) | by simp only [map_principal, filter.coprod, comap_principal, sup_principal, image_singleton,
image_id, prod_univ, univ_prod] | lemma | filter.map_const_principal_coprod_map_id_principal | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter.coprod"
] | Characterization of the coproduct of the `filter.map`s of two principal filters `𝓟 {a}` and
`𝓟 {i}`, the first under the constant function `λ a, b` and the second under the identity function.
Together with the next lemma, `map_prod_map_const_id_principal_coprod_principal`, this provides an
example showing that the in... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_prod_map_const_id_principal_coprod_principal {α β ι : Type*} (a : α) (b : β) (i : ι) :
map (prod.map (λ _ : α, b) id) ((𝓟 {a}).coprod (𝓟 {i}))
= 𝓟 (({b} : set β) ×ˢ (univ : set ι)) | begin
rw [principal_coprod_principal, map_principal],
congr,
ext ⟨b', i'⟩,
split,
{ rintro ⟨⟨a'', i''⟩, h₁, h₂, h₃⟩,
simp },
{ rintro ⟨h₁, h₂⟩,
use (a, i'),
simpa using h₁.symm }
end | lemma | filter.map_prod_map_const_id_principal_coprod_principal | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [] | Characterization of the `filter.map` of the coproduct of two principal filters `𝓟 {a}` and
`𝓟 {i}`, under the `prod.map` of two functions, respectively the constant function `λ a, b` and the
identity function. Together with the previous lemma,
`map_const_principal_coprod_map_id_principal`, this provides an example s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto.prod_map_coprod {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β}
{c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) :
tendsto (prod.map f g) (a.coprod b) (c.coprod d) | map_prod_map_coprod_le.trans (coprod_mono hf hg) | lemma | filter.tendsto.prod_map_coprod | order.filter | src/order/filter/prod.lean | [
"order.filter.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets (l : filter α) : filter (set α) | l.lift' powerset | def | filter.small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | The filter `l.small_sets` is the largest filter containing all powersets of members of `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_sets_eq_generate {f : filter α} : f.small_sets = generate (powerset '' f.sets) | by { simp_rw [generate_eq_binfi, small_sets, infi_image], refl } | lemma | filter.small_sets_eq_generate | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter",
"infi_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis.small_sets {p : ι → Prop} {s : ι → set α}
(h : has_basis l p s) : has_basis l.small_sets p (λ i, 𝒫 (s i)) | h.lift' monotone_powerset | lemma | filter.has_basis.small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_small_sets (l : filter α) :
has_basis l.small_sets (λ t : set α, t ∈ l) powerset | l.basis_sets.small_sets | lemma | filter.has_basis_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_small_sets_iff {f : α → set β} :
tendsto f la lb.small_sets ↔ ∀ t ∈ lb, ∀ᶠ x in la, f x ⊆ t | (has_basis_small_sets lb).tendsto_right_iff | lemma | filter.tendsto_small_sets_iff | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | `g` converges to `f.small_sets` if for all `s ∈ f`, eventually we have `g x ⊆ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_small_sets {p : set α → Prop} :
(∀ᶠ s in l.small_sets, p s) ↔ ∃ s ∈ l, ∀ t ⊆ s, p t | eventually_lift'_iff monotone_powerset | lemma | filter.eventually_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_small_sets' {p : set α → Prop} (hp : ∀ ⦃s t⦄, s ⊆ t → p t → p s) :
(∀ᶠ s in l.small_sets, p s) ↔ ∃ s ∈ l, p s | eventually_small_sets.trans $ exists₂_congr $ λ s hsf,
⟨λ H, H s subset.rfl, λ hs t ht, hp ht hs⟩ | lemma | filter.eventually_small_sets' | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"exists₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_small_sets {p : set α → Prop} :
(∃ᶠ s in l.small_sets, p s) ↔ ∀ t ∈ l, ∃ s ⊆ t, p s | l.has_basis_small_sets.frequently_iff | lemma | filter.frequently_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_small_sets_mem (l : filter α) : ∃ᶠ s in l.small_sets, s ∈ l | frequently_small_sets.2 $ λ t ht, ⟨t, subset.rfl, ht⟩ | lemma | filter.frequently_small_sets_mem | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_antitone_basis.tendsto_small_sets {ι} [preorder ι] {s : ι → set α}
(hl : l.has_antitone_basis s) : tendsto s at_top l.small_sets | tendsto_small_sets_iff.2 $ λ t ht, hl.eventually_subset ht | lemma | filter.has_antitone_basis.tendsto_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_small_sets : monotone (@small_sets α) | monotone_lift' monotone_id monotone_const | lemma | filter.monotone_small_sets | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"monotone",
"monotone_const",
"monotone_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_sets_bot : (⊥ : filter α).small_sets = pure ∅ | by rw [small_sets, lift'_bot monotone_powerset, powerset_empty, principal_singleton] | lemma | filter.small_sets_bot | order.filter | src/order/filter/small_sets.lean | [
"order.filter.lift",
"order.filter.at_top_bot"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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