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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prod_lift_lift
{f₁ : filter α₁} {f₂ : filter α₂} {g₁ : set α₁ → filter β₁} {g₂ : set α₂ → filter β₂}
(hg₁ : monotone g₁) (hg₂ : monotone g₂) :
(f₁.lift g₁) ×ᶠ (f₂.lift g₂) = f₁.lift (λs, f₂.lift (λt, g₁ s ×ᶠ g₂ t)) | begin
simp only [prod_def, lift_assoc hg₁],
apply congr_arg, funext x,
rw [lift_comm],
apply congr_arg, funext y,
apply lift'_lift_assoc hg₂
end | lemma | filter.prod_lift_lift | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_lift'_lift'
{f₁ : filter α₁} {f₂ : filter α₂} {g₁ : set α₁ → set β₁} {g₂ : set α₂ → set β₂}
(hg₁ : monotone g₁) (hg₂ : monotone g₂) :
f₁.lift' g₁ ×ᶠ f₂.lift' g₂ = f₁.lift (λ s, f₂.lift' (λ t, g₁ s ×ˢ g₂ t)) | calc f₁.lift' g₁ ×ᶠ f₂.lift' g₂ = f₁.lift (λ s, f₂.lift (λ t, 𝓟 (g₁ s) ×ᶠ 𝓟 (g₂ t))) :
prod_lift_lift (monotone_principal.comp hg₁) (monotone_principal.comp hg₂)
... = f₁.lift (λ s, f₂.lift (λ t, 𝓟 (g₁ s ×ˢ g₂ t))) :
by simp only [prod_principal_principal] | lemma | filter.prod_lift'_lift' | order.filter | src/order/filter/lift.lean | [
"order.filter.bases"
] | [
"filter",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_modeq {n : ℕ} (h : n ≠ 0) (d : ℕ) : ∃ᶠ m in at_top, m ≡ d [MOD n] | ((tendsto_add_at_top_nat d).comp (tendsto_id.nsmul_at_top h.bot_lt)).frequently $
frequently_of_forall $ λ m, by { simp [nat.modeq_iff_dvd, ← sub_sub] } | lemma | nat.frequently_modeq | order.filter | src/order/filter/modeq.lean | [
"data.nat.parity",
"order.filter.at_top_bot"
] | [
"nat.modeq_iff_dvd"
] | Infinitely many natural numbers are equal to `d` mod `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frequently_mod_eq {d n : ℕ} (h : d < n) : ∃ᶠ m in at_top, m % n = d | by simpa only [nat.modeq, mod_eq_of_lt h] using frequently_modeq h.ne_bot d | lemma | nat.frequently_mod_eq | order.filter | src/order/filter/modeq.lean | [
"data.nat.parity",
"order.filter.at_top_bot"
] | [
"nat.modeq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_even : ∃ᶠ m : ℕ in at_top, even m | by simpa only [even_iff] using frequently_mod_eq zero_lt_two | lemma | nat.frequently_even | order.filter | src/order/filter/modeq.lean | [
"data.nat.parity",
"order.filter.at_top_bot"
] | [
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_odd : ∃ᶠ m : ℕ in at_top, odd m | by simpa only [odd_iff] using frequently_mod_eq one_lt_two | lemma | nat.frequently_odd | order.filter | src/order/filter/modeq.lean | [
"data.nat.parity",
"order.filter.at_top_bot"
] | [
"odd",
"one_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂ (m : α → β → γ) (f : filter α) (g : filter β) : filter γ | { sets := {s | ∃ u v, u ∈ f ∧ v ∈ g ∧ image2 m u v ⊆ s},
univ_sets := ⟨univ, univ, univ_sets _, univ_sets _, subset_univ _⟩,
sets_of_superset := λ s t hs hst,
Exists₂.imp (λ u v, and.imp_right $ and.imp_right $ λ h, subset.trans h hst) hs,
inter_sets := λ s t,
begin
simp only [exists_prop, mem_set_of_eq... | def | filter.map₂ | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"Exists₂.imp",
"and.imp_right",
"exists_prop",
"filter"
] | The image of a binary function `m : α → β → γ` as a function `filter α → filter β → filter γ`.
Mathematically this should be thought of as the image of the corresponding function `α × β → γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map₂_iff : u ∈ map₂ m f g ↔ ∃ s t, s ∈ f ∧ t ∈ g ∧ image2 m s t ⊆ u | iff.rfl | lemma | filter.mem_map₂_iff | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image2_mem_map₂ (hs : s ∈ f) (ht : t ∈ g) : image2 m s t ∈ map₂ m f g | ⟨_, _, hs, ht, subset.rfl⟩ | lemma | filter.image2_mem_map₂ | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod_eq_map₂ (m : α → β → γ) (f : filter α) (g : filter β) :
filter.map (λ p : α × β, m p.1 p.2) (f ×ᶠ g) = map₂ m f g | begin
ext s,
simp [mem_prod_iff, prod_subset_iff]
end | lemma | filter.map_prod_eq_map₂ | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter",
"filter.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod_eq_map₂' (m : α × β → γ) (f : filter α) (g : filter β) :
filter.map m (f ×ᶠ g) = map₂ (λ a b, m (a, b)) f g | (congr_arg2 _ (uncurry_curry m).symm rfl).trans (map_prod_eq_map₂ _ _ _) | lemma | filter.map_prod_eq_map₂' | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"congr_arg2",
"filter",
"filter.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_mk_eq_prod (f : filter α) (g : filter β) : map₂ prod.mk f g = f ×ᶠ g | by simp only [← map_prod_eq_map₂, prod.mk.eta, map_id'] | lemma | filter.map₂_mk_eq_prod | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_mono (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : map₂ m f₁ g₁ ≤ map₂ m f₂ g₂ | λ _ ⟨s, t, hs, ht, hst⟩, ⟨s, t, hf hs, hg ht, hst⟩ | lemma | filter.map₂_mono | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_mono_left (h : g₁ ≤ g₂) : map₂ m f g₁ ≤ map₂ m f g₂ | map₂_mono subset.rfl h | lemma | filter.map₂_mono_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_mono_right (h : f₁ ≤ f₂) : map₂ m f₁ g ≤ map₂ m f₂ g | map₂_mono h subset.rfl | lemma | filter.map₂_mono_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map₂_iff {h : filter γ} :
h ≤ map₂ m f g ↔ ∀ ⦃s⦄, s ∈ f → ∀ ⦃t⦄, t ∈ g → image2 m s t ∈ h | ⟨λ H s hs t ht, H $ image2_mem_map₂ hs ht, λ H u ⟨s, t, hs, ht, hu⟩, mem_of_superset (H hs ht) hu⟩ | lemma | filter.le_map₂_iff | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_bot_left : map₂ m ⊥ g = ⊥ | empty_mem_iff_bot.1 ⟨∅, univ, trivial, univ_mem, (image2_empty_left).subset⟩ | lemma | filter.map₂_bot_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_bot_right : map₂ m f ⊥ = ⊥ | empty_mem_iff_bot.1 ⟨univ, ∅, univ_mem, trivial, (image2_empty_right).subset⟩ | lemma | filter.map₂_bot_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_eq_bot_iff : map₂ m f g = ⊥ ↔ f = ⊥ ∨ g = ⊥ | begin
simp only [←empty_mem_iff_bot, mem_map₂_iff, subset_empty_iff, image2_eq_empty_iff],
split,
{ rintro ⟨s, t, hs, ht, rfl | rfl⟩,
{ exact or.inl hs },
{ exact or.inr ht } },
{ rintro (h | h),
{ exact ⟨_, _, h, univ_mem, or.inl rfl⟩ },
{ exact ⟨_, _, univ_mem, h, or.inr rfl⟩ } }
end | lemma | filter.map₂_eq_bot_iff | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_ne_bot_iff : (map₂ m f g).ne_bot ↔ f.ne_bot ∧ g.ne_bot | by { simp_rw ne_bot_iff, exact map₂_eq_bot_iff.not.trans not_or_distrib } | lemma | filter.map₂_ne_bot_iff | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.map₂ (hf : f.ne_bot) (hg : g.ne_bot) : (map₂ m f g).ne_bot | map₂_ne_bot_iff.2 ⟨hf, hg⟩ | lemma | filter.ne_bot.map₂ | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_map₂_left (h : (map₂ m f g).ne_bot) : f.ne_bot | (map₂_ne_bot_iff.1 h).1 | lemma | filter.ne_bot.of_map₂_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot.of_map₂_right (h : (map₂ m f g).ne_bot) : g.ne_bot | (map₂_ne_bot_iff.1 h).2 | lemma | filter.ne_bot.of_map₂_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_sup_left : map₂ m (f₁ ⊔ f₂) g = map₂ m f₁ g ⊔ map₂ m f₂ g | begin
ext u,
split,
{ rintro ⟨s, t, ⟨h₁, h₂⟩, ht, hu⟩,
exact ⟨mem_of_superset (image2_mem_map₂ h₁ ht) hu,
mem_of_superset (image2_mem_map₂ h₂ ht) hu⟩ },
{ rintro ⟨⟨s₁, t₁, hs₁, ht₁, hu₁⟩, s₂, t₂, hs₂, ht₂, hu₂⟩,
refine ⟨s₁ ∪ s₂, t₁ ∩ t₂, union_mem_sup hs₁ hs₂, inter_mem ht₁ ht₂, _⟩,
rw image2_... | lemma | filter.map₂_sup_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_sup_right : map₂ m f (g₁ ⊔ g₂) = map₂ m f g₁ ⊔ map₂ m f g₂ | begin
ext u,
split,
{ rintro ⟨s, t, hs, ⟨h₁, h₂⟩, hu⟩,
exact ⟨mem_of_superset (image2_mem_map₂ hs h₁) hu,
mem_of_superset (image2_mem_map₂ hs h₂) hu⟩ },
{ rintro ⟨⟨s₁, t₁, hs₁, ht₁, hu₁⟩, s₂, t₂, hs₂, ht₂, hu₂⟩,
refine ⟨s₁ ∩ s₂, t₁ ∪ t₂, inter_mem hs₁ hs₂, union_mem_sup ht₁ ht₂, _⟩,
rw image2_... | lemma | filter.map₂_sup_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_inf_subset_left : map₂ m (f₁ ⊓ f₂) g ≤ map₂ m f₁ g ⊓ map₂ m f₂ g | le_inf (map₂_mono_right inf_le_left) (map₂_mono_right inf_le_right) | lemma | filter.map₂_inf_subset_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"inf_le_left",
"inf_le_right",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_inf_subset_right : map₂ m f (g₁ ⊓ g₂) ≤ map₂ m f g₁ ⊓ map₂ m f g₂ | le_inf (map₂_mono_left inf_le_left) (map₂_mono_left inf_le_right) | lemma | filter.map₂_inf_subset_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"inf_le_left",
"inf_le_right",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_pure_left : map₂ m (pure a) g = g.map (λ b, m a b) | filter.ext $ λ u, ⟨λ ⟨s, t, hs, ht, hu⟩,
mem_of_superset (image_mem_map ht) ((image_subset_image2_right $ mem_pure.1 hs).trans hu),
λ h, ⟨{a}, _, singleton_mem_pure, h, by rw [image2_singleton_left, image_subset_iff]⟩⟩ | lemma | filter.map₂_pure_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_pure_right : map₂ m f (pure b) = f.map (λ a, m a b) | filter.ext $ λ u, ⟨λ ⟨s, t, hs, ht, hu⟩,
mem_of_superset (image_mem_map hs) ((image_subset_image2_left $ mem_pure.1 ht).trans hu),
λ h, ⟨_, {b}, h, singleton_mem_pure, by rw [image2_singleton_right, image_subset_iff]⟩⟩ | lemma | filter.map₂_pure_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_pure : map₂ m (pure a) (pure b) = pure (m a b) | by rw [map₂_pure_right, map_pure] | lemma | filter.map₂_pure | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_swap (m : α → β → γ) (f : filter α) (g : filter β) :
map₂ m f g = map₂ (λ a b, m b a) g f | by { ext u, split; rintro ⟨s, t, hs, ht, hu⟩; refine ⟨t, s, ht, hs, by rwa image2_swap⟩ } | lemma | filter.map₂_swap | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_left (h : g.ne_bot) : map₂ (λ x y, x) f g = f | begin
ext u,
refine ⟨_, λ hu, ⟨_, _, hu, univ_mem, (image2_left $ h.nonempty_of_mem univ_mem).subset⟩⟩,
rintro ⟨s, t, hs, ht, hu⟩,
rw image2_left (h.nonempty_of_mem ht) at hu,
exact mem_of_superset hs hu,
end | lemma | filter.map₂_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_right (h : f.ne_bot) : map₂ (λ x y, y) f g = g | by rw [map₂_swap, map₂_left h] | lemma | filter.map₂_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₃ (m : α → β → γ → δ) (f : filter α) (g : filter β) (h : filter γ) : filter δ | { sets := {s | ∃ u v w, u ∈ f ∧ v ∈ g ∧ w ∈ h ∧ image3 m u v w ⊆ s},
univ_sets := ⟨univ, univ, univ, univ_sets _, univ_sets _, univ_sets _, subset_univ _⟩,
sets_of_superset := λ s t hs hst, Exists₃.imp
(λ u v w, and.imp_right $ and.imp_right $ and.imp_right $ λ h, subset.trans h hst) hs,
inter_sets := λ s t,
... | def | filter.map₃ | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"Exists₃.imp",
"and.imp_right",
"exists_prop",
"filter"
] | The image of a ternary function `m : α → β → γ → δ` as a function
`filter α → filter β → filter γ → filter δ`. Mathematically this should be thought of as the image
of the corresponding function `α × β × γ → δ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_map₂_left (m : δ → γ → ε) (n : α → β → δ) :
map₂ m (map₂ n f g) h = map₃ (λ a b c, m (n a b) c) f g h | begin
ext w,
split,
{ rintro ⟨s, t, ⟨u, v, hu, hv, hs⟩, ht, hw⟩,
refine ⟨u, v, t, hu, hv, ht, _⟩,
rw ←image2_image2_left,
exact (image2_subset_right hs).trans hw },
{ rintro ⟨s, t, u, hs, ht, hu, hw⟩,
exact ⟨_, u, image2_mem_map₂ hs ht, hu, by rwa image2_image2_left⟩ }
end | lemma | filter.map₂_map₂_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_map₂_right (m : α → δ → ε) (n : β → γ → δ) :
map₂ m f (map₂ n g h) = map₃ (λ a b c, m a (n b c)) f g h | begin
ext w,
split,
{ rintro ⟨s, t, hs, ⟨u, v, hu, hv, ht⟩, hw⟩,
refine ⟨s, u, v, hs, hu, hv, _⟩,
rw ←image2_image2_right,
exact (image2_subset_left ht).trans hw },
{ rintro ⟨s, t, u, hs, ht, hu, hw⟩,
exact ⟨s, _, hs, image2_mem_map₂ ht hu, by rwa image2_image2_right⟩ }
end | lemma | filter.map₂_map₂_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map₂ (m : α → β → γ) (n : γ → δ) : (map₂ m f g).map n = map₂ (λ a b, n (m a b)) f g | by rw [← map_prod_eq_map₂, ← map_prod_eq_map₂, map_map] | lemma | filter.map_map₂ | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_map_left (m : γ → β → δ) (n : α → γ) :
map₂ m (f.map n) g = map₂ (λ a b, m (n a) b) f g | begin
rw [← map_prod_eq_map₂, ← map_prod_eq_map₂, ← @map_id _ g, prod_map_map_eq, map_map, map_id],
refl
end | lemma | filter.map₂_map_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_map_right (m : α → γ → δ) (n : β → γ) :
map₂ m f (g.map n) = map₂ (λ a b, m a (n b)) f g | by rw [map₂_swap, map₂_map_left, map₂_swap] | lemma | filter.map₂_map_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_curry (m : α × β → γ) (f : filter α) (g : filter β) :
map₂ (curry m) f g = (f ×ᶠ g).map m | (map_prod_eq_map₂' _ _ _).symm | lemma | filter.map₂_curry | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_uncurry_prod (m : α → β → γ) (f : filter α) (g : filter β) :
(f ×ᶠ g).map (uncurry m) = map₂ m f g | by rw [←map₂_curry, curry_uncurry] | lemma | filter.map_uncurry_prod | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_assoc {m : δ → γ → ε} {n : α → β → δ} {m' : α → ε' → ε} {n' : β → γ → ε'}
{h : filter γ} (h_assoc : ∀ a b c, m (n a b) c = m' a (n' b c)) :
map₂ m (map₂ n f g) h = map₂ m' f (map₂ n' g h) | by simp only [map₂_map₂_left, map₂_map₂_right, h_assoc] | lemma | filter.map₂_assoc | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_comm {n : β → α → γ} (h_comm : ∀ a b, m a b = n b a) : map₂ m f g = map₂ n g f | (map₂_swap _ _ _).trans $ by simp_rw h_comm | lemma | filter.map₂_comm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_left_comm {m : α → δ → ε} {n : β → γ → δ} {m' : α → γ → δ'} {n' : β → δ' → ε}
(h_left_comm : ∀ a b c, m a (n b c) = n' b (m' a c)) :
map₂ m f (map₂ n g h) = map₂ n' g (map₂ m' f h) | by { rw [map₂_swap m', map₂_swap m], exact map₂_assoc (λ _ _ _, h_left_comm _ _ _) } | lemma | filter.map₂_left_comm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_right_comm {m : δ → γ → ε} {n : α → β → δ} {m' : α → γ → δ'} {n' : δ' → β → ε}
(h_right_comm : ∀ a b c, m (n a b) c = n' (m' a c) b) :
map₂ m (map₂ n f g) h = map₂ n' (map₂ m' f h) g | by { rw [map₂_swap n, map₂_swap n'], exact map₂_assoc (λ _ _ _, h_right_comm _ _ _) } | lemma | filter.map₂_right_comm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map₂_distrib {n : γ → δ} {m' : α' → β' → δ} {n₁ : α → α'} {n₂ : β → β'}
(h_distrib : ∀ a b, n (m a b) = m' (n₁ a) (n₂ b)) :
(map₂ m f g).map n = map₂ m' (f.map n₁) (g.map n₂) | by simp_rw [map_map₂, map₂_map_left, map₂_map_right, h_distrib] | lemma | filter.map_map₂_distrib | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map₂_distrib_left {n : γ → δ} {m' : α' → β → δ} {n' : α → α'}
(h_distrib : ∀ a b, n (m a b) = m' (n' a) b) :
(map₂ m f g).map n = map₂ m' (f.map n') g | map_map₂_distrib h_distrib | lemma | filter.map_map₂_distrib_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map₂_map_left_comm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_map₂_distrib_right {n : γ → δ} {m' : α → β' → δ} {n' : β → β'}
(h_distrib : ∀ a b, n (m a b) = m' a (n' b)) :
(map₂ m f g).map n = map₂ m' f (g.map n') | map_map₂_distrib h_distrib | lemma | filter.map_map₂_distrib_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map_map₂_right_comm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_map_left_comm {m : α' → β → γ} {n : α → α'} {m' : α → β → δ} {n' : δ → γ}
(h_left_comm : ∀ a b, m (n a) b = n' (m' a b)) :
map₂ m (f.map n) g = (map₂ m' f g).map n' | (map_map₂_distrib_left $ λ a b, (h_left_comm a b).symm).symm | lemma | filter.map₂_map_left_comm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map_map₂_distrib_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_map₂_right_comm {m : α → β' → γ} {n : β → β'} {m' : α → β → δ} {n' : δ → γ}
(h_right_comm : ∀ a b, m a (n b) = n' (m' a b)) :
map₂ m f (g.map n) = (map₂ m' f g).map n' | (map_map₂_distrib_right $ λ a b, (h_right_comm a b).symm).symm | lemma | filter.map_map₂_right_comm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map_map₂_distrib_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_distrib_le_left {m : α → δ → ε} {n : β → γ → δ} {m₁ : α → β → β'} {m₂ : α → γ → γ'}
{n' : β' → γ' → ε} (h_distrib : ∀ a b c, m a (n b c) = n' (m₁ a b) (m₂ a c)) :
map₂ m f (map₂ n g h) ≤ map₂ n' (map₂ m₁ f g) (map₂ m₂ f h) | begin
rintro s ⟨t₁, t₂, ⟨u₁, v, hu₁, hv, ht₁⟩, ⟨u₂, w, hu₂, hw, ht₂⟩, hs⟩,
refine ⟨u₁ ∩ u₂, _, inter_mem hu₁ hu₂, image2_mem_map₂ hv hw, _⟩,
refine (image2_distrib_subset_left h_distrib).trans ((image2_subset _ _).trans hs),
{ exact (image2_subset_right $ inter_subset_left _ _).trans ht₁ },
{ exact (image2_su... | lemma | filter.map₂_distrib_le_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | The other direction does not hold because of the `f`-`f` cross terms on the RHS. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_distrib_le_right {m : δ → γ → ε} {n : α → β → δ} {m₁ : α → γ → α'}
{m₂ : β → γ → β'} {n' : α' → β' → ε} (h_distrib : ∀ a b c, m (n a b) c = n' (m₁ a c) (m₂ b c)) :
map₂ m (map₂ n f g) h ≤ map₂ n' (map₂ m₁ f h) (map₂ m₂ g h) | begin
rintro s ⟨t₁, t₂, ⟨u, w₁, hu, hw₁, ht₁⟩, ⟨v, w₂, hv, hw₂, ht₂⟩, hs⟩,
refine ⟨_, w₁ ∩ w₂, image2_mem_map₂ hu hv, inter_mem hw₁ hw₂, _⟩,
refine (image2_distrib_subset_right h_distrib).trans ((image2_subset _ _).trans hs),
{ exact (image2_subset_left $ inter_subset_left _ _).trans ht₁ },
{ exact (image2_su... | lemma | filter.map₂_distrib_le_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | The other direction does not hold because of the `h`-`h` cross terms on the RHS. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_map₂_antidistrib {n : γ → δ} {m' : β' → α' → δ} {n₁ : β → β'} {n₂ : α → α'}
(h_antidistrib : ∀ a b, n (m a b) = m' (n₁ b) (n₂ a)) :
(map₂ m f g).map n = map₂ m' (g.map n₁) (f.map n₂) | by { rw map₂_swap m, exact map_map₂_distrib (λ _ _, h_antidistrib _ _) } | lemma | filter.map_map₂_antidistrib | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map₂_antidistrib_left {n : γ → δ} {m' : β' → α → δ} {n' : β → β'}
(h_antidistrib : ∀ a b, n (m a b) = m' (n' b) a) :
(map₂ m f g).map n = map₂ m' (g.map n') f | map_map₂_antidistrib h_antidistrib | lemma | filter.map_map₂_antidistrib_left | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map₂_map_left_anticomm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_map₂_antidistrib_right {n : γ → δ} {m' : β → α' → δ} {n' : α → α'}
(h_antidistrib : ∀ a b, n (m a b) = m' b (n' a)) :
(map₂ m f g).map n = map₂ m' g (f.map n') | map_map₂_antidistrib h_antidistrib | lemma | filter.map_map₂_antidistrib_right | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map_map₂_right_anticomm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_map_left_anticomm {m : α' → β → γ} {n : α → α'} {m' : β → α → δ} {n' : δ → γ}
(h_left_anticomm : ∀ a b, m (n a) b = n' (m' b a)) :
map₂ m (f.map n) g = (map₂ m' g f).map n' | (map_map₂_antidistrib_left $ λ a b, (h_left_anticomm b a).symm).symm | lemma | filter.map₂_map_left_anticomm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map_map₂_antidistrib_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_map₂_right_anticomm {m : α → β' → γ} {n : β → β'} {m' : β → α → δ} {n' : δ → γ}
(h_right_anticomm : ∀ a b, m a (n b) = n' (m' b a)) :
map₂ m f (g.map n) = (map₂ m' g f).map n' | (map_map₂_antidistrib_right $ λ a b, (h_right_anticomm b a).symm).symm | lemma | filter.map_map₂_right_anticomm | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [] | Symmetric statement to `filter.map_map₂_antidistrib_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_left_identity {f : α → β → β} {a : α} (h : ∀ b, f a b = b) (l : filter β) :
map₂ f (pure a) l = l | by rw [map₂_pure_left, show f a = id, from funext h, map_id] | lemma | filter.map₂_left_identity | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter",
"map_id"
] | If `a` is a left identity for `f : α → β → β`, then `pure a` is a left identity for
`filter.map₂ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map₂_right_identity {f : α → β → α} {b : β} (h : ∀ a, f a b = a) (l : filter α) :
map₂ f l (pure b) = l | by rw [map₂_pure_right, funext h, map_id'] | lemma | filter.map₂_right_identity | order.filter | src/order/filter/n_ary.lean | [
"order.filter.prod"
] | [
"filter"
] | If `b` is a right identity for `f : α → β → α`, then `pure b` is a right identity for
`filter.map₂ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rmap (r : rel α β) (l : filter α) : filter β | { sets := {s | r.core s ∈ l},
univ_sets := by simp,
sets_of_superset := λ s t hs st, mem_of_superset hs $ rel.core_mono _ st,
inter_sets := λ s t hs ht, by simp [rel.core_inter, inter_mem hs ht] } | def | filter.rmap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.core_inter",
"rel.core_mono"
] | The forward map of a filter under a relation. Generalization of `filter.map` to relations. Note
that `rel.core` generalizes `set.preimage`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rmap_sets (r : rel α β) (l : filter α) : (l.rmap r).sets = r.core ⁻¹' l.sets | rfl | theorem | filter.rmap_sets | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_rmap (r : rel α β) (l : filter α) (s : set β) :
s ∈ l.rmap r ↔ r.core s ∈ l | iff.rfl | theorem | filter.mem_rmap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rmap_rmap (r : rel α β) (s : rel β γ) (l : filter α) :
rmap s (rmap r l) = rmap (r.comp s) l | filter_eq $
by simp [rmap_sets, set.preimage, rel.core_comp] | theorem | filter.rmap_rmap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.core_comp",
"set.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rmap_compose (r : rel α β) (s : rel β γ) : rmap s ∘ rmap r = rmap (r.comp s) | funext $ rmap_rmap _ _ | lemma | filter.rmap_compose | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rtendsto (r : rel α β) (l₁ : filter α) (l₂ : filter β) | l₁.rmap r ≤ l₂ | def | filter.rtendsto | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel"
] | Generic "limit of a relation" predicate. `rtendsto r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `r`-core of `a` is an `l₁`-neighborhood. One generalization of
`filter.tendsto` to relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rtendsto_def (r : rel α β) (l₁ : filter α) (l₂ : filter β) :
rtendsto r l₁ l₂ ↔ ∀ s ∈ l₂, r.core s ∈ l₁ | iff.rfl | theorem | filter.rtendsto_def | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap (r : rel α β) (f : filter β) : filter α | { sets := rel.image (λ s t, r.core s ⊆ t) f.sets,
univ_sets := ⟨set.univ, univ_mem, set.subset_univ _⟩,
sets_of_superset := λ a b ⟨a', ha', ma'a⟩ ab, ⟨a', ha', ma'a.trans ab⟩,
inter_sets := λ a b ⟨a', ha₁, ha₂⟩ ⟨b', hb₁, hb₂⟩,
⟨a' ∩ b', inter_mem ha₁ hb₁,
... | def | filter.rcomap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.image",
"set.inter_subset_inter",
"set.subset_univ"
] | One way of taking the inverse map of a filter under a relation. One generalization of
`filter.comap` to relations. Note that `rel.core` generalizes `set.preimage`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rcomap_sets (r : rel α β) (f : filter β) :
(rcomap r f).sets = rel.image (λ s t, r.core s ⊆ t) f.sets | rfl | theorem | filter.rcomap_sets | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap_rcomap (r : rel α β) (s : rel β γ) (l : filter γ) :
rcomap r (rcomap s l) = rcomap (r.comp s) l | filter_eq $
begin
ext t, simp [rcomap_sets, rel.image, rel.core_comp], split,
{ rintros ⟨u, ⟨v, vsets, hv⟩, h⟩,
exact ⟨v, vsets, set.subset.trans (rel.core_mono _ hv) h⟩ },
rintros ⟨t, tsets, ht⟩,
exact ⟨rel.core s t, ⟨t, tsets, set.subset.rfl⟩, ht⟩
end | theorem | filter.rcomap_rcomap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.core_comp",
"rel.core_mono",
"rel.image",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap_compose (r : rel α β) (s : rel β γ) : rcomap r ∘ rcomap s = rcomap (r.comp s) | funext $ rcomap_rcomap _ _ | lemma | filter.rcomap_compose | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rtendsto_iff_le_rcomap (r : rel α β) (l₁ : filter α) (l₂ : filter β) :
rtendsto r l₁ l₂ ↔ l₁ ≤ l₂.rcomap r | begin
rw rtendsto_def,
change (∀ (s : set β), s ∈ l₂.sets → r.core s ∈ l₁) ↔ l₁ ≤ rcomap r l₂,
simp [filter.le_def, rcomap, rel.mem_image], split,
{ exact λ h s t tl₂, mem_of_superset (h t tl₂) },
{ exact λ h t tl₂, h _ t tl₂ set.subset.rfl }
end | theorem | filter.rtendsto_iff_le_rcomap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"filter.le_def",
"rel",
"rel.mem_image",
"set.subset.rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap' (r : rel α β) (f : filter β) : filter α | { sets := rel.image (λ s t, r.preimage s ⊆ t) f.sets,
univ_sets := ⟨set.univ, univ_mem, set.subset_univ _⟩,
sets_of_superset := λ a b ⟨a', ha', ma'a⟩ ab, ⟨a', ha', ma'a.trans ab⟩,
inter_sets := λ a b ⟨a', ha₁, ha₂⟩ ⟨b', hb₁, hb₂⟩,
⟨a' ∩ b', inter_mem ha₁ hb₁,
... | def | filter.rcomap' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.image",
"rel.preimage_inter",
"set.inter_subset_inter",
"set.subset_univ"
] | One way of taking the inverse map of a filter under a relation. Generalization of `filter.comap`
to relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_rcomap' (r : rel α β) (l : filter β) (s : set α) :
s ∈ l.rcomap' r ↔ ∃ t ∈ l, r.preimage t ⊆ s | iff.rfl | lemma | filter.mem_rcomap' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap'_sets (r : rel α β) (f : filter β) :
(rcomap' r f).sets = rel.image (λ s t, r.preimage s ⊆ t) f.sets | rfl | theorem | filter.rcomap'_sets | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap'_rcomap' (r : rel α β) (s : rel β γ) (l : filter γ) :
rcomap' r (rcomap' s l) = rcomap' (r.comp s) l | filter.ext $ λ t,
begin
simp [rcomap'_sets, rel.image, rel.preimage_comp], split,
{ rintro ⟨u, ⟨v, vsets, hv⟩, h⟩,
exact ⟨v, vsets, (rel.preimage_mono _ hv).trans h⟩ },
rintro ⟨t, tsets, ht⟩,
exact ⟨s.preimage t, ⟨t, tsets, set.subset.rfl⟩, ht⟩
end | theorem | filter.rcomap'_rcomap' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"filter.ext",
"rel",
"rel.image",
"rel.preimage_comp",
"rel.preimage_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rcomap'_compose (r : rel α β) (s : rel β γ) : rcomap' r ∘ rcomap' s = rcomap' (r.comp s) | funext $ rcomap'_rcomap' _ _ | lemma | filter.rcomap'_compose | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rtendsto' (r : rel α β) (l₁ : filter α) (l₂ : filter β) | l₁ ≤ l₂.rcomap' r | def | filter.rtendsto' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel"
] | Generic "limit of a relation" predicate. `rtendsto' r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `r`-preimage of `a` is an `l₁`-neighborhood. One generalization of
`filter.tendsto` to relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rtendsto'_def (r : rel α β) (l₁ : filter α) (l₂ : filter β) :
rtendsto' r l₁ l₂ ↔ ∀ s ∈ l₂, r.preimage s ∈ l₁ | begin
unfold rtendsto' rcomap', simp [le_def, rel.mem_image], split,
{ exact λ h s hs, h _ _ hs set.subset.rfl },
{ exact λ h s t ht, mem_of_superset (h t ht) }
end | theorem | filter.rtendsto'_def | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"rel",
"rel.mem_image",
"set.subset.rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_rtendsto (l₁ : filter α) (l₂ : filter β) (f : α → β) :
tendsto f l₁ l₂ ↔ rtendsto (function.graph f) l₁ l₂ | by { simp [tendsto_def, function.graph, rtendsto_def, rel.core, set.preimage] } | theorem | filter.tendsto_iff_rtendsto | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"function.graph",
"rel.core",
"set.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_rtendsto' (l₁ : filter α) (l₂ : filter β) (f : α → β) :
tendsto f l₁ l₂ ↔ rtendsto' (function.graph f) l₁ l₂ | by { simp [tendsto_def, function.graph, rtendsto'_def, rel.preimage_def, set.preimage] } | theorem | filter.tendsto_iff_rtendsto' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"function.graph",
"rel.preimage_def",
"set.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pmap (f : α →. β) (l : filter α) : filter β | filter.rmap f.graph' l | def | filter.pmap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"filter.rmap"
] | The forward map of a filter under a partial function. Generalization of `filter.map` to partial
functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_pmap (f : α →. β) (l : filter α) (s : set β) : s ∈ l.pmap f ↔ f.core s ∈ l | iff.rfl | lemma | filter.mem_pmap | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ptendsto (f : α →. β) (l₁ : filter α) (l₂ : filter β) | l₁.pmap f ≤ l₂ | def | filter.ptendsto | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter"
] | Generic "limit of a partial function" predicate. `ptendsto r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `p`-core of `a` is an `l₁`-neighborhood. One generalization of
`filter.tendsto` to partial function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ptendsto_def (f : α →. β) (l₁ : filter α) (l₂ : filter β) :
ptendsto f l₁ l₂ ↔ ∀ s ∈ l₂, f.core s ∈ l₁ | iff.rfl | theorem | filter.ptendsto_def | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ptendsto_iff_rtendsto (l₁ : filter α) (l₂ : filter β) (f : α →. β) :
ptendsto f l₁ l₂ ↔ rtendsto f.graph' l₁ l₂ | iff.rfl | theorem | filter.ptendsto_iff_rtendsto | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pmap_res (l : filter α) (s : set α) (f : α → β) :
pmap (pfun.res f s) l = map f (l ⊓ 𝓟 s) | begin
ext t,
simp only [pfun.core_res, mem_pmap, mem_map, mem_inf_principal, imp_iff_not_or],
refl
end | theorem | filter.pmap_res | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"imp_iff_not_or",
"mem_map",
"pfun.core_res",
"pfun.res"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_ptendsto (l₁ : filter α) (l₂ : filter β) (s : set α) (f : α → β) :
tendsto f (l₁ ⊓ 𝓟 s) l₂ ↔ ptendsto (pfun.res f s) l₁ l₂ | by simp only [tendsto, ptendsto, pmap_res] | theorem | filter.tendsto_iff_ptendsto | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"pfun.res"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_ptendsto_univ (l₁ : filter α) (l₂ : filter β) (f : α → β) :
tendsto f l₁ l₂ ↔ ptendsto (pfun.res f set.univ) l₁ l₂ | by { rw ← tendsto_iff_ptendsto, simp [principal_univ] } | theorem | filter.tendsto_iff_ptendsto_univ | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"pfun.res"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pcomap' (f : α →. β) (l : filter β) : filter α | filter.rcomap' f.graph' l | def | filter.pcomap' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"filter.rcomap'"
] | Inverse map of a filter under a partial function. One generalization of `filter.comap` to
partial functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ptendsto' (f : α →. β) (l₁ : filter α) (l₂ : filter β) | l₁ ≤ l₂.rcomap' f.graph' | def | filter.ptendsto' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter"
] | Generic "limit of a partial function" predicate. `ptendsto' r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `p`-preimage of `a` is an `l₁`-neighborhood. One generalization of
`filter.tendsto` to partial functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ptendsto'_def (f : α →. β) (l₁ : filter α) (l₂ : filter β) :
ptendsto' f l₁ l₂ ↔ ∀ s ∈ l₂, f.preimage s ∈ l₁ | rtendsto'_def _ _ _ | theorem | filter.ptendsto'_def | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ptendsto_of_ptendsto' {f : α →. β} {l₁ : filter α} {l₂ : filter β} :
ptendsto' f l₁ l₂ → ptendsto f l₁ l₂ | begin
rw [ptendsto_def, ptendsto'_def],
exact λ h s sl₂, mem_of_superset (h s sl₂) (pfun.preimage_subset_core _ _),
end | theorem | filter.ptendsto_of_ptendsto' | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"pfun.preimage_subset_core"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ptendsto'_of_ptendsto {f : α →. β} {l₁ : filter α} {l₂ : filter β} (h : f.dom ∈ l₁) :
ptendsto f l₁ l₂ → ptendsto' f l₁ l₂ | begin
rw [ptendsto_def, ptendsto'_def],
intros h' s sl₂,
rw pfun.preimage_eq,
exact inter_mem (h' s sl₂) h
end | theorem | filter.ptendsto'_of_ptendsto | order.filter | src/order/filter/partial.lean | [
"order.filter.basic",
"data.pfun"
] | [
"filter",
"pfun.preimage_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi (f : Π i, filter (α i)) : filter (Π i, α i) | ⨅ i, comap (eval i) (f i) | def | filter.pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter"
] | The product of an indexed family of filters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.is_countably_generated [countable ι] [∀ i, is_countably_generated (f i)] :
is_countably_generated (pi f) | infi.is_countably_generated _ | instance | filter.pi.is_countably_generated | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_eval_pi (f : Π i, filter (α i)) (i : ι) :
tendsto (eval i) (pi f) (f i) | tendsto_infi' i tendsto_comap | lemma | filter.tendsto_eval_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pi {β : Type*} {m : β → Π i, α i} {l : filter β} :
tendsto m l (pi f) ↔ ∀ i, tendsto (λ x, m x i) l (f i) | by simp only [pi, tendsto_infi, tendsto_comap_iff] | lemma | filter.tendsto_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_pi {g : filter (Π i, α i)} : g ≤ pi f ↔ ∀ i, tendsto (eval i) g (f i) | tendsto_pi | lemma | filter.le_pi | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_mono (h : ∀ i, f₁ i ≤ f₂ i) : pi f₁ ≤ pi f₂ | infi_mono $ λ i, comap_mono $ h i | lemma | filter.pi_mono | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [
"infi_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pi_of_mem (i : ι) {s : set (α i)} (hs : s ∈ f i) :
eval i ⁻¹' s ∈ pi f | mem_infi_of_mem i $ preimage_mem_comap hs | lemma | filter.mem_pi_of_mem | order.filter | src/order/filter/pi.lean | [
"order.filter.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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