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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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pi_mem_pi {I : set ι} (hI : I.finite) (h : ∀ i ∈ I, s i ∈ f i) :
I.pi s ∈ pi f | begin
rw [pi_def, bInter_eq_Inter],
refine mem_infi_of_Inter hI (λ i, _) subset.rfl,
exact preimage_mem_comap (h i i.2)
end | lemma | filter.pi_mem_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pi {s : set (Π i, α i)} : s ∈ pi f ↔
∃ (I : set ι), I.finite ∧ ∃ t : Π i, set (α i), (∀ i, t i ∈ f i) ∧ I.pi t ⊆ s | begin
split,
{ simp only [pi, mem_infi', mem_comap, pi_def],
rintro ⟨I, If, V, hVf, hVI, rfl, -⟩, choose t htf htV using hVf,
exact ⟨I, If, t, htf, Inter₂_mono (λ i _, htV i)⟩ },
{ rintro ⟨I, If, t, htf, hts⟩,
exact mem_of_superset (pi_mem_pi If $ λ i _, htf i) hts }
end | lemma | filter.mem_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pi' {s : set (Π i, α i)} : s ∈ pi f ↔
∃ (I : finset ι), ∃ t : Π i, set (α i), (∀ i, t i ∈ f i) ∧ set.pi ↑I t ⊆ s | mem_pi.trans exists_finite_iff_finset | lemma | filter.mem_pi' | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"finset",
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_pi_mem_pi [∀ i, ne_bot (f i)] {I : set ι} (h : I.pi s ∈ pi f) {i : ι} (hi : i ∈ I) :
s i ∈ f i | begin
rcases mem_pi.1 h with ⟨I', I'f, t, htf, hts⟩,
refine mem_of_superset (htf i) (λ x hx, _),
have : ∀ i, (t i).nonempty, from λ i, nonempty_of_mem (htf i),
choose g hg,
have : update g i x ∈ I'.pi t,
{ intros j hj, rcases eq_or_ne j i with (rfl|hne); simp * },
simpa using hts this i hi
end | lemma | filter.mem_of_pi_mem_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"eq_or_ne",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_mem_pi_iff [∀ i, ne_bot (f i)] {I : set ι} (hI : I.finite) :
I.pi s ∈ pi f ↔ ∀ i ∈ I, s i ∈ f i | ⟨λ h i hi, mem_of_pi_mem_pi h hi, pi_mem_pi hI⟩ | lemma | filter.pi_mem_pi_iff | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.eval_pi {i : ι} (hf : ∀ᶠ (x : α i) in f i, p i x) :
∀ᶠ (x : Π (i : ι), α i) in pi f, p i (x i) | (tendsto_eval_pi _ _).eventually hf | lemma | filter.eventually.eval_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_pi [finite ι] (hf : ∀ i, ∀ᶠ x in f i, p i x) :
∀ᶠ (x : Π i, α i) in pi f, ∀ i, p i (x i) | eventually_all.2 $ λ i, (hf _).eval_pi | lemma | filter.eventually_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_pi {ι' : ι → Type} {s : Π i, ι' i → set (α i)} {p : Π i, ι' i → Prop}
(h : ∀ i, (f i).has_basis (p i) (s i)) :
(pi f).has_basis (λ If : set ι × Π i, ι' i, If.1.finite ∧ ∀ i ∈ If.1, p i (If.2 i))
(λ If : set ι × Π i, ι' i, If.1.pi (λ i, s i $ If.2 i)) | begin
have : (pi f).has_basis _ _ := has_basis_infi' (λ i, (h i).comap (eval i : (Π j, α j) → α i)),
convert this,
ext,
simp
end | lemma | filter.has_basis_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_inf_principal_univ_pi_eq_bot :
pi f ⊓ 𝓟 (set.pi univ s) = ⊥ ↔ ∃ i, f i ⊓ 𝓟 (s i) = ⊥ | begin
split,
{ simp only [inf_principal_eq_bot, mem_pi], contrapose!,
rintros (hsf : ∀ i, ∃ᶠ x in f i, x ∈ s i) I If t htf hts,
have : ∀ i, (s i ∩ t i).nonempty, from λ i, ((hsf i).and_eventually (htf i)).exists,
choose x hxs hxt,
exact hts (λ i hi, hxt i) (mem_univ_pi.2 hxs) },
{ simp only [inf_p... | lemma | filter.pi_inf_principal_univ_pi_eq_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_inf_principal_pi_eq_bot [Π i, ne_bot (f i)] {I : set ι} :
pi f ⊓ 𝓟 (set.pi I s) = ⊥ ↔ ∃ i ∈ I, f i ⊓ 𝓟 (s i) = ⊥ | begin
rw [← univ_pi_piecewise I, pi_inf_principal_univ_pi_eq_bot],
refine exists_congr (λ i, _),
by_cases hi : i ∈ I; simp [hi, (‹Π i, ne_bot (f i)› i).ne]
end | lemma | filter.pi_inf_principal_pi_eq_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_inf_principal_univ_pi_ne_bot :
ne_bot (pi f ⊓ 𝓟 (set.pi univ s)) ↔ ∀ i, ne_bot (f i ⊓ 𝓟 (s i)) | by simp [ne_bot_iff] | lemma | filter.pi_inf_principal_univ_pi_ne_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"set.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_inf_principal_pi_ne_bot [Π i, ne_bot (f i)] {I : set ι} :
ne_bot (pi f ⊓ 𝓟 (I.pi s)) ↔ ∀ i ∈ I, ne_bot (f i ⊓ 𝓟 (s i)) | by simp [ne_bot_iff] | lemma | filter.pi_inf_principal_pi_ne_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_inf_principal_pi.ne_bot [h : ∀ i, ne_bot (f i ⊓ 𝓟 (s i))] {I : set ι} :
ne_bot (pi f ⊓ 𝓟 (I.pi s)) | (pi_inf_principal_univ_pi_ne_bot.2 ‹_›).mono $ inf_le_inf_left _ $ principal_mono.2 $
λ x hx i hi, hx i trivial | instance | filter.pi_inf_principal_pi.ne_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"inf_le_inf_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_eq_bot : pi f = ⊥ ↔ ∃ i, f i = ⊥ | by simpa using @pi_inf_principal_univ_pi_eq_bot ι α f (λ _, univ) | lemma | filter.pi_eq_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_ne_bot : ne_bot (pi f) ↔ ∀ i, ne_bot (f i) | by simp [ne_bot_iff] | lemma | filter.pi_ne_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eval_pi (f : Π i, filter (α i)) [∀ i, ne_bot (f i)] (i : ι) :
map (eval i) (pi f) = f i | begin
refine le_antisymm (tendsto_eval_pi f i) (λ s hs, _),
rcases mem_pi.1 (mem_map.1 hs) with ⟨I, hIf, t, htf, hI⟩,
rw [← image_subset_iff] at hI,
refine mem_of_superset (htf i) ((subset_eval_image_pi _ _).trans hI),
exact nonempty_of_mem (pi_mem_pi hIf (λ i hi, htf i))
end | lemma | filter.map_eval_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_le_pi [∀ i, ne_bot (f₁ i)] : pi f₁ ≤ pi f₂ ↔ ∀ i, f₁ i ≤ f₂ i | ⟨λ h i, map_eval_pi f₁ i ▸ (tendsto_eval_pi _ _).mono_left h, pi_mono⟩ | lemma | filter.pi_le_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_inj [∀ i, ne_bot (f₁ i)] : pi f₁ = pi f₂ ↔ f₁ = f₂ | begin
refine ⟨λ h, _, congr_arg pi⟩,
have hle : f₁ ≤ f₂ := pi_le_pi.1 h.le,
haveI : ∀ i, ne_bot (f₂ i) := λ i, ne_bot_of_le (hle i),
exact hle.antisymm (pi_le_pi.1 h.ge)
end | lemma | filter.pi_inj | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod (f : Π i, filter (α i)) : filter (Π i, α i) | ⨆ i : ι, comap (eval i) (f i) | def | filter.Coprod | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter"
] | Coproduct of filters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_Coprod_iff {s : set (Π i, α i)} :
(s ∈ filter.Coprod f) ↔ (∀ i : ι, (∃ t₁ ∈ f i, eval i ⁻¹' t₁ ⊆ s)) | by simp [filter.Coprod] | lemma | filter.mem_Coprod_iff | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mem_Coprod {s : set (Π i, α i)} :
sᶜ ∈ filter.Coprod f ↔ ∀ i, (eval i '' s)ᶜ ∈ f i | by simp only [filter.Coprod, mem_supr, compl_mem_comap] | lemma | filter.compl_mem_Coprod | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_ne_bot_iff' :
ne_bot (filter.Coprod f) ↔ (∀ i, nonempty (α i)) ∧ ∃ d, ne_bot (f d) | by simp only [filter.Coprod, supr_ne_bot, ← exists_and_distrib_left, ← comap_eval_ne_bot_iff'] | lemma | filter.Coprod_ne_bot_iff' | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"exists_and_distrib_left",
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_ne_bot_iff [∀ i, nonempty (α i)] :
ne_bot (filter.Coprod f) ↔ ∃ d, ne_bot (f d) | by simp [Coprod_ne_bot_iff', *] | lemma | filter.Coprod_ne_bot_iff | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_eq_bot_iff' : filter.Coprod f = ⊥ ↔ (∃ i, is_empty (α i)) ∨ f = ⊥ | by simpa [not_and_distrib, funext_iff] using not_congr Coprod_ne_bot_iff' | lemma | filter.Coprod_eq_bot_iff' | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod",
"is_empty",
"not_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_eq_bot_iff [∀ i, nonempty (α i)] : filter.Coprod f = ⊥ ↔ f = ⊥ | by simpa [funext_iff] using not_congr Coprod_ne_bot_iff | lemma | filter.Coprod_eq_bot_iff | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_bot' : filter.Coprod (⊥ : Π i, filter (α i)) = ⊥ | Coprod_eq_bot_iff'.2 (or.inr rfl) | lemma | filter.Coprod_bot' | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter",
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_bot : filter.Coprod (λ _, ⊥ : Π i, filter (α i)) = ⊥ | Coprod_bot' | lemma | filter.Coprod_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter",
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.Coprod [∀ i, nonempty (α i)] {i : ι} (h : ne_bot (f i)) :
ne_bot (filter.Coprod f) | Coprod_ne_bot_iff.2 ⟨i, h⟩ | lemma | filter.ne_bot.Coprod | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_ne_bot [∀ i, nonempty (α i)] [nonempty ι] (f : Π i, filter (α i))
[H : ∀ i, ne_bot (f i)] : ne_bot (filter.Coprod f) | (H (classical.arbitrary ι)).Coprod | lemma | filter.Coprod_ne_bot | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"classical.arbitrary",
"filter",
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Coprod_mono (hf : ∀ i, f₁ i ≤ f₂ i) : filter.Coprod f₁ ≤ filter.Coprod f₂ | supr_mono $ λ i, comap_mono (hf i) | lemma | filter.Coprod_mono | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod",
"supr_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pi_map_Coprod_le :
map (λ (k : Π i, α i), λ i, m i (k i)) (filter.Coprod f) ≤ filter.Coprod (λ i, map (m i) (f i)) | begin
simp only [le_def, mem_map, mem_Coprod_iff],
intros s h i,
obtain ⟨t, H, hH⟩ := h i,
exact ⟨{x : α i | m i x ∈ t}, H, λ x hx, hH hx⟩
end | lemma | filter.map_pi_map_Coprod_le | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter.Coprod",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.pi_map_Coprod {g : Π i, filter (β i)} (h : ∀ i, tendsto (m i) (f i) (g i)) :
tendsto (λ (k : Π i, α i), λ i, m i (k i)) (filter.Coprod f) (filter.Coprod g) | map_pi_map_Coprod_le.trans (Coprod_mono h) | lemma | filter.tendsto.pi_map_Coprod | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter",
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_one : has_one (filter α) | ⟨pure 1⟩ | def | filter.has_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | `1 : filter α` is defined as the filter of sets containing `1 : α` in locale `pointwise`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_one : s ∈ (1 : filter α) ↔ (1 : α) ∈ s | mem_pure | lemma | filter.mem_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem_one : (1 : set α) ∈ (1 : filter α) | mem_pure.2 one_mem_one | lemma | filter.one_mem_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_one : pure 1 = (1 : filter α) | rfl | lemma | filter.pure_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_one : 𝓟 1 = (1 : filter α) | principal_singleton _ | lemma | filter.principal_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_ne_bot : (1 : filter α).ne_bot | filter.pure_ne_bot | lemma | filter.one_ne_bot | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"filter.pure_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one' (f : α → β) : (1 : filter α).map f = pure (f 1) | rfl | lemma | filter.map_one' | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_iff : f ≤ 1 ↔ (1 : set α) ∈ f | le_pure_iff | lemma | filter.le_one_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.le_one_iff (h : f.ne_bot) : f ≤ 1 ↔ f = 1 | h.le_pure_iff | lemma | filter.ne_bot.le_one_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_one {p : α → Prop} : (∀ᶠ x in 1, p x) ↔ p 1 | eventually_pure | lemma | filter.eventually_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_one {a : filter β} {f : β → α} :
tendsto f a 1 ↔ ∀ᶠ x in a, f x = 1 | tendsto_pure | lemma | filter.tendsto_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_prod_one [has_one β] : (1 : filter α) ×ᶠ (1 : filter β) = 1 | prod_pure_pure | lemma | filter.one_prod_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_one_hom : one_hom α (filter α) | ⟨pure, pure_one⟩ | def | filter.pure_one_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"one_hom"
] | `pure` as a `one_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pure_one_hom : (pure_one_hom : α → filter α) = pure | rfl | lemma | filter.coe_pure_one_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_one_hom_apply (a : α) : pure_one_hom a = pure a | rfl | lemma | filter.pure_one_hom_apply | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one [one_hom_class F α β] (φ : F) : map φ 1 = 1 | by rw [filter.map_one', map_one, pure_one] | lemma | filter.map_one | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter.map_one'",
"map_one",
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv : f.map has_inv.inv = f⁻¹ | rfl | lemma | filter.map_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"map_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv : s ∈ f⁻¹ ↔ has_inv.inv ⁻¹' s ∈ f | iff.rfl | lemma | filter.mem_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_inv (hf : f ≤ g) : f⁻¹ ≤ g⁻¹ | map_mono hf | lemma | filter.inv_le_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"inv_le_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pure : (pure a : filter α)⁻¹ = pure a⁻¹ | rfl | lemma | filter.inv_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_bot_iff : f⁻¹ = ⊥ ↔ f = ⊥ | map_eq_bot_iff | lemma | filter.inv_eq_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"map_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot_inv_iff : f⁻¹.ne_bot ↔ ne_bot f | map_ne_bot_iff _ | lemma | filter.ne_bot_inv_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.inv : f.ne_bot → f⁻¹.ne_bot | λ h, h.map _ | lemma | filter.ne_bot.inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mem_inv (hs : s ∈ f) : s⁻¹ ∈ f⁻¹ | by rwa [mem_inv, inv_preimage, inv_inv] | lemma | filter.inv_mem_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_involutive_inv : has_involutive_inv (filter α) | { inv_inv := λ f, map_map.trans $ by rw [inv_involutive.comp_self, map_id],
..filter.has_inv } | def | filter.has_involutive_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter",
"has_involutive_inv",
"inv_inv",
"map_id"
] | Inversion is involutive on `filter α` if it is on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_le_inv_iff : f⁻¹ ≤ g⁻¹ ↔ f ≤ g | ⟨λ h, inv_inv f ▸ inv_inv g ▸ filter.inv_le_inv h, filter.inv_le_inv⟩ | lemma | filter.inv_le_inv_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter.inv_le_inv",
"inv_inv",
"inv_le_inv_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_iff_le_inv : f⁻¹ ≤ g ↔ f ≤ g⁻¹ | by rw [← filter.inv_le_inv_iff, inv_inv] | lemma | filter.inv_le_iff_le_inv | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter.inv_le_inv_iff",
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_le_self : f⁻¹ ≤ f ↔ f⁻¹ = f | ⟨λ h, h.antisymm $ inv_le_iff_le_inv.1 h, eq.le⟩ | lemma | filter.inv_le_self | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul : has_mul (filter α) | /- This is defeq to `map₂ (*) f g`, but the hypothesis unfolds to `t₁ * t₂ ⊆ s` rather than all the
way to `set.image2 (*) t₁ t₂ ⊆ s`. -/
⟨λ f g, { sets := {s | ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ * t₂ ⊆ s}, ..map₂ (*) f g }⟩ | def | filter.has_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | The filter `f * g` is generated by `{s * t | s ∈ f, t ∈ g}` in locale `pointwise`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_mul : map₂ (*) f g = f * g | rfl | lemma | filter.map₂_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mul : s ∈ f * g ↔ ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ * t₂ ⊆ s | iff.rfl | lemma | filter.mem_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_mul : s ∈ f → t ∈ g → s * t ∈ f * g | image2_mem_map₂ | lemma | filter.mul_mem_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_mul : ⊥ * g = ⊥ | map₂_bot_left | lemma | filter.bot_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bot : f * ⊥ = ⊥ | map₂_bot_right | lemma | filter.mul_bot | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_bot_iff : f * g = ⊥ ↔ f = ⊥ ∨ g = ⊥ | map₂_eq_bot_iff | lemma | filter.mul_eq_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ne_bot_iff : (f * g).ne_bot ↔ f.ne_bot ∧ g.ne_bot | map₂_ne_bot_iff | lemma | filter.mul_ne_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.mul : ne_bot f → ne_bot g → ne_bot (f * g) | ne_bot.map₂ | lemma | filter.ne_bot.mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_mul_left : (f * g).ne_bot → f.ne_bot | ne_bot.of_map₂_left | lemma | filter.ne_bot.of_mul_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_mul_right : (f * g).ne_bot → g.ne_bot | ne_bot.of_map₂_right | lemma | filter.ne_bot.of_mul_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_mul : pure a * g = g.map ((*) a) | map₂_pure_left | lemma | filter.pure_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pure : f * pure b = f.map (* b) | map₂_pure_right | lemma | filter.mul_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_mul_pure : (pure a : filter α) * pure b = pure (a * b) | map₂_pure | lemma | filter.pure_mul_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_iff : h ≤ f * g ↔ ∀ ⦃s⦄, s ∈ f → ∀ ⦃t⦄, t ∈ g → s * t ∈ h | le_map₂_iff | lemma | filter.le_mul_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_mul : covariant_class (filter α) (filter α) (*) (≤) | ⟨λ f g h, map₂_mono_left⟩ | instance | filter.covariant_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"covariant_class",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_swap_mul : covariant_class (filter α) (filter α) (swap (*)) (≤) | ⟨λ f g h, map₂_mono_right⟩ | instance | filter.covariant_swap_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"covariant_class",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul [mul_hom_class F α β] (m : F) : (f₁ * f₂).map m = f₁.map m * f₂.map m | map_map₂_distrib $ map_mul m | lemma | filter.map_mul | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"map_mul",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_mul_hom : α →ₙ* filter α | ⟨pure, λ a b, pure_mul_pure.symm⟩ | def | filter.pure_mul_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | `pure` operation as a `mul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pure_mul_hom : (pure_mul_hom : α → filter α) = pure | rfl | lemma | filter.coe_pure_mul_hom | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_mul_hom_apply (a : α) : pure_mul_hom a = pure a | rfl | lemma | filter.pure_mul_hom_apply | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_div : has_div (filter α) | /- This is defeq to `map₂ (/) f g`, but the hypothesis unfolds to `t₁ / t₂ ⊆ s` rather than all the
way to `set.image2 (/) t₁ t₂ ⊆ s`. -/
⟨λ f g, { sets := {s | ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ / t₂ ⊆ s}, ..map₂ (/) f g }⟩ | def | filter.has_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | The filter `f / g` is generated by `{s / t | s ∈ f, t ∈ g}` in locale `pointwise`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_div : map₂ (/) f g = f / g | rfl | lemma | filter.map₂_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_div : s ∈ f / g ↔ ∃ t₁ t₂, t₁ ∈ f ∧ t₂ ∈ g ∧ t₁ / t₂ ⊆ s | iff.rfl | lemma | filter.mem_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mem_div : s ∈ f → t ∈ g → s / t ∈ f / g | image2_mem_map₂ | lemma | filter.div_mem_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_div : ⊥ / g = ⊥ | map₂_bot_left | lemma | filter.bot_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_bot : f / ⊥ = ⊥ | map₂_bot_right | lemma | filter.div_bot | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_bot_iff : f / g = ⊥ ↔ f = ⊥ ∨ g = ⊥ | map₂_eq_bot_iff | lemma | filter.div_eq_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_ne_bot_iff : (f / g).ne_bot ↔ f.ne_bot ∧ g.ne_bot | map₂_ne_bot_iff | lemma | filter.div_ne_bot_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.div : ne_bot f → ne_bot g → ne_bot (f / g) | ne_bot.map₂ | lemma | filter.ne_bot.div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_div_left : (f / g).ne_bot → f.ne_bot | ne_bot.of_map₂_left | lemma | filter.ne_bot.of_div_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_div_right : (f / g).ne_bot → g.ne_bot | ne_bot.of_map₂_right | lemma | filter.ne_bot.of_div_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_div : pure a / g = g.map ((/) a) | map₂_pure_left | lemma | filter.pure_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_pure : f / pure b = f.map (/ b) | map₂_pure_right | lemma | filter.div_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_div_pure : (pure a : filter α) / pure b = pure (a / b) | map₂_pure | lemma | filter.pure_div_pure | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div : f₁ ≤ f₂ → g₁ ≤ g₂ → f₁ / g₁ ≤ f₂ / g₂ | map₂_mono | lemma | filter.div_le_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"div_le_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_left : g₁ ≤ g₂ → f / g₁ ≤ f / g₂ | map₂_mono_left | lemma | filter.div_le_div_left | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"div_le_div_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_div_right : f₁ ≤ f₂ → f₁ / g ≤ f₂ / g | map₂_mono_right | lemma | filter.div_le_div_right | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"div_le_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff :
h ≤ f / g ↔ ∀ ⦃s⦄, s ∈ f → ∀ ⦃t⦄, t ∈ g → s / t ∈ h | le_map₂_iff | lemma | filter.le_div_iff | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"le_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_div : covariant_class (filter α) (filter α) (/) (≤) | ⟨λ f g h, map₂_mono_left⟩ | instance | filter.covariant_div | order.filter | src/order/filter/pointwise.lean | [
"data.set.pointwise.smul",
"order.filter.n_ary",
"order.filter.ultrafilter"
] | [
"covariant_class",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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