statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
eventually_le.mul_nonneg [ordered_semiring β] {l : filter α} {f g : α → β}
(hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g) :
0 ≤ᶠ[l] f * g | by filter_upwards [hf, hg] with x using mul_nonneg | lemma | filter.eventually_le.mul_nonneg | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_sub_nonneg [ordered_ring β] {l : filter α} {f g : α → β} :
0 ≤ᶠ[l] g - f ↔ f ≤ᶠ[l] g | eventually_congr $ eventually_of_forall $ λ x, sub_nonneg | lemma | filter.eventually_sub_nonneg | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.sup [semilattice_sup β] {l : filter α} {f₁ f₂ g₁ g₂ : α → β}
(hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) :
f₁ ⊔ g₁ ≤ᶠ[l] f₂ ⊔ g₂ | by filter_upwards [hf, hg] with x hfx hgx using sup_le_sup hfx hgx | lemma | filter.eventually_le.sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"semilattice_sup",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.sup_le [semilattice_sup β] {l : filter α} {f g h : α → β}
(hf : f ≤ᶠ[l] h) (hg : g ≤ᶠ[l] h) :
f ⊔ g ≤ᶠ[l] h | by filter_upwards [hf, hg] with x hfx hgx using sup_le hfx hgx | lemma | filter.eventually_le.sup_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"semilattice_sup",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.le_sup_of_le_left [semilattice_sup β] {l : filter α} {f g h : α → β}
(hf : h ≤ᶠ[l] f) :
h ≤ᶠ[l] f ⊔ g | by filter_upwards [hf] with x hfx using le_sup_of_le_left hfx | lemma | filter.eventually_le.le_sup_of_le_left | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"le_sup_of_le_left",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_le.le_sup_of_le_right [semilattice_sup β] {l : filter α} {f g h : α → β}
(hg : h ≤ᶠ[l] g) :
h ≤ᶠ[l] f ⊔ g | by filter_upwards [hg] with x hgx using le_sup_of_le_right hgx | lemma | filter.eventually_le.le_sup_of_le_right | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"le_sup_of_le_right",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
join_le {f : filter (filter α)} {l : filter α} (h : ∀ᶠ m in f, m ≤ l) : join f ≤ l | λ s hs, h.mono $ λ m hm, hm hs | lemma | filter.join_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (m : α → β) (f : filter α) : filter β | { sets := preimage m ⁻¹' f.sets,
univ_sets := univ_mem,
sets_of_superset := λ s t hs st, mem_of_superset hs $ preimage_mono st,
inter_sets := λ s t hs ht, inter_mem hs ht } | def | filter.map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | The forward map of a filter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_principal {s : set α} {f : α → β} :
map f (𝓟 s) = 𝓟 (set.image f s) | filter.ext $ λ a, image_subset_iff.symm | lemma | filter.map_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.ext",
"set.image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_map {P : β → Prop} :
(∀ᶠ b in map m f, P b) ↔ ∀ᶠ a in f, P (m a) | iff.rfl | lemma | filter.eventually_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_map {P : β → Prop} :
(∃ᶠ b in map m f, P b) ↔ ∃ᶠ a in f, P (m a) | iff.rfl | lemma | filter.frequently_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map : t ∈ map m f ↔ m ⁻¹' t ∈ f | iff.rfl | lemma | filter.mem_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map' : t ∈ map m f ↔ {x | m x ∈ t} ∈ f | iff.rfl | lemma | filter.mem_map' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_map (hs : s ∈ f) : m '' s ∈ map m f | f.sets_of_superset hs $ subset_preimage_image m s | lemma | filter.image_mem_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_map_iff (hf : injective m) : m '' s ∈ map m f ↔ s ∈ f | ⟨λ h, by rwa [← preimage_image_eq s hf], image_mem_map⟩ | lemma | filter.image_mem_map_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_mem_map : range m ∈ map m f | by { rw ←image_univ, exact image_mem_map univ_mem } | lemma | filter.range_mem_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_iff_exists_image : t ∈ map m f ↔ (∃ s ∈ f, m '' s ⊆ t) | ⟨λ ht, ⟨m ⁻¹' t, ht, image_preimage_subset _ _⟩,
λ ⟨s, hs, ht⟩, mem_of_superset (image_mem_map hs) ht⟩ | lemma | filter.mem_map_iff_exists_image | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : filter.map id f = f | filter_eq $ rfl | lemma | filter.map_id | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.map",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id' : filter.map (λ x, x) f = f | map_id | lemma | filter.map_id' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.map",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_compose : filter.map m' ∘ filter.map m = filter.map (m' ∘ m) | funext $ λ _, filter_eq $ rfl | lemma | filter.map_compose | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map : filter.map m' (filter.map m f) = filter.map (m' ∘ m) f | congr_fun (@@filter.map_compose m m') f | lemma | filter.map_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.map",
"filter.map_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_congr {m₁ m₂ : α → β} {f : filter α} (h : m₁ =ᶠ[f] m₂) :
map m₁ f = map m₂ f | filter.ext' $ λ p,
by { simp only [eventually_map], exact eventually_congr (h.mono $ λ x hx, hx ▸ iff.rfl) } | lemma | filter.map_congr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.ext'",
"map_congr"
] | If functions `m₁` and `m₂` are eventually equal at a filter `f`, then
they map this filter to the same filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap (m : α → β) (f : filter β) : filter α | { sets := { s | ∃ t ∈ f, m ⁻¹' t ⊆ s },
univ_sets := ⟨univ, univ_mem, by simp only [subset_univ, preimage_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, inter_subset_inter ... | def | filter.comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | The inverse map of a filter. A set `s` belongs to `filter.comap m f` if either of the following
equivalent conditions hold.
1. There exists a set `t ∈ f` such that `m ⁻¹' t ⊆ s`. This is used as a definition.
2. The set `{y | ∀ x, m x = y → x ∈ s}` belongs to `f`, see `filter.mem_comap'`.
3. The set `(m '' sᶜ)ᶜ` belon... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_comap' : s ∈ comap f l ↔ {y | ∀ ⦃x⦄, f x = y → x ∈ s} ∈ l | ⟨λ ⟨t, ht, hts⟩, mem_of_superset ht $ λ y hy x hx, hts $ mem_preimage.2 $ by rwa hx,
λ h, ⟨_, h, λ x hx, hx rfl⟩⟩ | lemma | filter.mem_comap' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap_prod_mk {x : α} {s : set β} {F : filter (α × β)} :
s ∈ comap (prod.mk x) F ↔ {p : α × β | p.fst = x → p.snd ∈ s} ∈ F | by simp_rw [mem_comap', prod.ext_iff, and_imp, @forall_swap β (_ = _), forall_eq, eq_comm] | lemma | filter.mem_comap_prod_mk | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"and_imp",
"filter",
"forall_eq",
"forall_swap",
"prod.ext_iff"
] | RHS form is used, e.g., in the definition of `uniform_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_comap : (∀ᶠ a in comap f l, p a) ↔ ∀ᶠ b in l, ∀ a, f a = b → p a | mem_comap' | lemma | filter.eventually_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frequently_comap : (∃ᶠ a in comap f l, p a) ↔ ∃ᶠ b in l, ∃ a, f a = b ∧ p a | by simp only [filter.frequently, eventually_comap, not_exists, not_and] | lemma | filter.frequently_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.frequently",
"not_and",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap_iff_compl : s ∈ comap f l ↔ (f '' sᶜ)ᶜ ∈ l | by simp only [mem_comap', compl_def, mem_image, mem_set_of_eq, not_exists, not_and', not_not] | lemma | filter.mem_comap_iff_compl | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"not_and'",
"not_exists",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mem_comap : sᶜ ∈ comap f l ↔ (f '' s)ᶜ ∈ l | by rw [mem_comap_iff_compl, compl_compl] | lemma | filter.compl_mem_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind (f : filter α) (m : α → filter β) : filter β | join (map m f) | def | filter.bind | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | The monadic bind operation on filter is defined the usual way in terms of `map` and `join`.
Unfortunately, this `bind` does not result in the expected applicative. See `filter.seq` for the
applicative instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq (f : filter (α → β)) (g : filter α) : filter β | ⟨{ s | ∃ u ∈ f, ∃ t ∈ g, (∀ m ∈ u, ∀ x ∈ t, (m : α → β) x ∈ s) },
⟨univ, univ_mem, univ, univ_mem,
by simp only [forall_prop_of_true, mem_univ, forall_true_iff]⟩,
λ s₀ s₁ ⟨t₀, t₁, h₀, h₁, h⟩ hst, ⟨t₀, t₁, h₀, h₁, λ x hx y hy, hst $ h _ hx _ hy⟩,
λ s₀ s₁ ⟨t₀, ht₀, t₁, ht₁, ht⟩ ⟨u₀, hu₀, u₁, hu₁, hu⟩,
⟨t₀ ∩... | def | filter.seq | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"forall_prop_of_true",
"forall_true_iff"
] | The applicative sequentiation operation. This is not induced by the bind operation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pure_sets (a : α) : (pure a : filter α).sets = {s | a ∈ s} | rfl | lemma | filter.pure_sets | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_pure {a : α} {s : set α} : s ∈ (pure a : filter α) ↔ a ∈ s | iff.rfl | lemma | filter.mem_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_pure {a : α} {p : α → Prop} :
(∀ᶠ x in pure a, p x) ↔ p a | iff.rfl | lemma | filter.eventually_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_singleton (a : α) : 𝓟 {a} = pure a | filter.ext $ λ s, by simp only [mem_pure, mem_principal, singleton_subset_iff] | lemma | filter.principal_singleton | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pure (f : α → β) (a : α) : map f (pure a) = pure (f a) | rfl | lemma | filter.map_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
join_pure (f : filter α) : join (pure f) = f | filter.ext $ λ s, iff.rfl | lemma | filter.join_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_bind (a : α) (m : α → filter β) :
bind (pure a) m = m a | by simp only [has_bind.bind, bind, map_pure, join_pure] | lemma | filter.pure_bind | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monad : monad filter | { map := @filter.map } | def | filter.monad | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.map"
] | The monad structure on filters. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lawful_monad : is_lawful_monad filter | { id_map := λ α f, filter_eq rfl,
pure_bind := λ α β, pure_bind,
bind_assoc := λ α β γ f m₁ m₂, filter_eq rfl,
bind_pure_comp_eq_map := λ α β f x, filter.ext $ λ s,
by simp only [has_bind.bind, bind, functor.map, mem_map', mem_join, mem_set_of_eq,
comp, mem_pure] } | lemma | filter.is_lawful_monad | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"bind_assoc",
"filter",
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_def {α β} (m : α → β) (f : filter α) : m <$> f = map m f | rfl | lemma | filter.map_def | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_def {α β} (f : filter α) (m : α → filter β) : f >>= m = bind f m | rfl | lemma | filter.bind_def | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap : s ∈ comap m g ↔ ∃ t ∈ g, m ⁻¹' t ⊆ s | iff.rfl | theorem | filter.mem_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mem_comap (ht : t ∈ g) : m ⁻¹' t ∈ comap m g | ⟨t, ht, subset.rfl⟩ | theorem | filter.preimage_mem_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually.comap {p : β → Prop} (hf : ∀ᶠ b in g, p b) (f : α → β) :
∀ᶠ a in comap f g, p (f a) | preimage_mem_comap hf | lemma | filter.eventually.comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id : comap id f = f | le_antisymm (λ s, preimage_mem_comap) (λ s ⟨t, ht, hst⟩, mem_of_superset ht hst) | lemma | filter.comap_id | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id' : comap (λ x, x) f = f | comap_id | lemma | filter.comap_id' | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_const_of_not_mem {x : β} (ht : t ∈ g) (hx : x ∉ t) :
comap (λ y : α, x) g = ⊥ | empty_mem_iff_bot.1 $ mem_comap'.2 $ mem_of_superset ht $ λ x' hx' y h, hx $ h.symm ▸ hx' | lemma | filter.comap_const_of_not_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_const_of_mem {x : β} (h : ∀ t ∈ g, x ∈ t) : comap (λ y : α, x) g = ⊤ | top_unique $ λ s hs, univ_mem' $ λ y, h _ (mem_comap'.1 hs) rfl | lemma | filter.comap_const_of_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"top_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_const [ne_bot f] {c : β} : f.map (λ x, c) = pure c | by { ext s, by_cases h : c ∈ s; simp [h] } | lemma | filter.map_const | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap {m : γ → β} {n : β → α} : comap m (comap n f) = comap (n ∘ m) f | filter.coext $ λ s, by simp only [compl_mem_comap, image_image] | lemma | filter.comap_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.coext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comm (F : filter α) : map ψ (map φ F) = map ρ (map θ F) | by rw [filter.map_map, H, ← filter.map_map] | lemma | filter.map_comm | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comm (G : filter δ) : comap φ (comap ψ G) = comap θ (comap ρ G) | by rw [filter.comap_comap, H, ← filter.comap_comap] | lemma | filter.comap_comm | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"filter.comap_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.semiconj.filter_map {f : α → β} {ga : α → α} {gb : β → β}
(h : function.semiconj f ga gb) : function.semiconj (map f) (map ga) (map gb) | map_comm h.comp_eq | lemma | function.semiconj.filter_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"function.semiconj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.commute.filter_map {f g : α → α} (h : function.commute f g) :
function.commute (map f) (map g) | h.filter_map | lemma | function.commute.filter_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"function.commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.semiconj.filter_comap {f : α → β} {ga : α → α} {gb : β → β}
(h : function.semiconj f ga gb) : function.semiconj (comap f) (comap gb) (comap ga) | comap_comm h.comp_eq.symm | lemma | function.semiconj.filter_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"function.semiconj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.commute.filter_comap {f g : α → α} (h : function.commute f g) :
function.commute (comap f) (comap g) | h.filter_comap | lemma | function.commute.filter_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"function.commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_principal {t : set β} : comap m (𝓟 t) = 𝓟 (m ⁻¹' t) | filter.ext $ λ s,
⟨λ ⟨u, (hu : t ⊆ u), (b : preimage m u ⊆ s)⟩, (preimage_mono hu).trans b,
λ h, ⟨t, subset.refl t, h⟩⟩ | theorem | filter.comap_principal | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_pure {b : β} : comap m (pure b) = 𝓟 (m ⁻¹' {b}) | by rw [← principal_singleton, comap_principal] | theorem | filter.comap_pure | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap : map m f ≤ g ↔ f ≤ comap m g | ⟨λ h s ⟨t, ht, hts⟩, mem_of_superset (h ht) hts, λ h s ht, h ⟨_, ht, subset.rfl⟩⟩ | lemma | filter.map_le_iff_le_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap (m : α → β) : galois_connection (map m) (comap m) | λ f g, map_le_iff_le_comap | lemma | filter.gc_map_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mono : monotone (map m) | (gc_map_comap m).monotone_l | lemma | filter.map_mono | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mono : monotone (comap m) | (gc_map_comap m).monotone_u | lemma | filter.comap_mono | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bot : map m ⊥ = ⊥ | (gc_map_comap m).l_bot | lemma | filter.map_bot | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup : map m (f₁ ⊔ f₂) = map m f₁ ⊔ map m f₂ | (gc_map_comap m).l_sup | lemma | filter.map_sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr {f : ι → filter α} : map m (⨆ i, f i) = (⨆ i, map m (f i)) | (gc_map_comap m).l_supr | lemma | filter.map_supr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"map_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_top (f : α → β) : map f ⊤ = 𝓟 (range f) | by rw [← principal_univ, map_principal, image_univ] | lemma | filter.map_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_top : comap m ⊤ = ⊤ | (gc_map_comap m).u_top | lemma | filter.comap_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf : comap m (g₁ ⊓ g₂) = comap m g₁ ⊓ comap m g₂ | (gc_map_comap m).u_inf | lemma | filter.comap_inf | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_infi {f : ι → filter β} : comap m (⨅ i, f i) = (⨅ i, comap m (f i)) | (gc_map_comap m).u_infi | lemma | filter.comap_infi | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_top (f : α → β) (l : filter α) : l ≤ comap f ⊤ | by { rw [comap_top], exact le_top } | lemma | filter.le_comap_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_le : map m (comap m g) ≤ g | (gc_map_comap m).l_u_le _ | lemma | filter.map_comap_le | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_map : f ≤ comap m (map m f) | (gc_map_comap m).le_u_l _ | lemma | filter.le_comap_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bot : comap m ⊥ = ⊥ | bot_unique $ λ s _, ⟨∅, mem_bot, by simp only [empty_subset, preimage_empty]⟩ | lemma | filter.comap_bot | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"bot_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot_of_comap (h : (comap m g).ne_bot) : g.ne_bot | begin
rw ne_bot_iff at *,
contrapose! h,
rw h,
exact comap_bot
end | lemma | filter.ne_bot_of_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf_principal_range : comap m (g ⊓ 𝓟 (range m)) = comap m g | by simp | lemma | filter.comap_inf_principal_range | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_comap (h : disjoint g₁ g₂) : disjoint (comap m g₁) (comap m g₂) | by simp only [disjoint_iff, ← comap_inf, h.eq_bot, comap_bot] | lemma | filter.disjoint_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"disjoint",
"disjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_supr {ι} {f : ι → filter β} {m : α → β} :
comap m (supr f) = (⨆ i, comap m (f i)) | le_antisymm
(λ s hs,
have ∀ i, ∃ t, t ∈ f i ∧ m ⁻¹' t ⊆ s,
by simpa only [mem_comap, exists_prop, mem_supr] using mem_supr.1 hs,
let ⟨t, ht⟩ := classical.axiom_of_choice this in
⟨⋃ i, t i, mem_supr.2 $ λ i, (f i).sets_of_superset (ht i).1 (subset_Union _ _),
begin
rw [preimage_Union, U... | lemma | filter.comap_supr | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"exists_prop",
"filter",
"le_supr",
"supr",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_Sup {s : set (filter β)} {m : α → β} : comap m (Sup s) = (⨆ f ∈ s, comap m f) | by simp only [Sup_eq_supr, comap_supr, eq_self_iff_true] | lemma | filter.comap_Sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"Sup_eq_supr",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_sup : comap m (g₁ ⊔ g₂) = comap m g₁ ⊔ comap m g₂ | by rw [sup_eq_supr, comap_supr, supr_bool_eq, bool.cond_tt, bool.cond_ff] | lemma | filter.comap_sup | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"bool.cond_ff",
"bool.cond_tt",
"sup_eq_supr",
"supr_bool_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap (f : filter β) (m : α → β) : (f.comap m).map m = f ⊓ 𝓟 (range m) | begin
refine le_antisymm (le_inf map_comap_le $ le_principal_iff.2 range_mem_map) _,
rintro t' ⟨t, ht, sub⟩,
refine mem_inf_principal.2 (mem_of_superset ht _),
rintro _ hxt ⟨x, rfl⟩,
exact sub hxt
end | lemma | filter.map_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_of_mem {f : filter β} {m : α → β} (hf : range m ∈ f) : (f.comap m).map m = f | by rw [map_comap, inf_eq_left.2 (le_principal_iff.2 hf)] | lemma | filter.map_comap_of_mem | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
can_lift (c) (p) [can_lift α β c p] :
can_lift (filter α) (filter β) (map c) (λ f, ∀ᶠ x : α in f, p x) | { prf := λ f hf, ⟨comap c f, map_comap_of_mem $ hf.mono can_lift.prf⟩ } | instance | filter.can_lift | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"can_lift",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_le_comap_iff {f g : filter β} {m : α → β} (hf : range m ∈ f) :
comap m f ≤ comap m g ↔ f ≤ g | ⟨λ h, map_comap_of_mem hf ▸ (map_mono h).trans map_comap_le, λ h, comap_mono h⟩ | lemma | filter.comap_le_comap_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_of_surjective {f : α → β} (hf : surjective f) (l : filter β) :
map f (comap f l) = l | map_comap_of_mem $ by simp only [hf.range_eq, univ_mem] | theorem | filter.map_comap_of_surjective | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.surjective.filter_map_top {f : α → β} (hf : surjective f) : map f ⊤ = ⊤ | (congr_arg _ comap_top).symm.trans $ map_comap_of_surjective hf ⊤ | lemma | function.surjective.filter_map_top | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_coe_map_comap (s : set α) (f : filter α) :
map (coe : s → α) (comap (coe : s → α) f) = f ⊓ 𝓟 s | by rw [map_comap, subtype.range_coe] | lemma | filter.subtype_coe_map_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_of_mem_comap {f : filter α} {c : β → α} (h : range c ∈ f) {W : set β}
(W_in : W ∈ comap c f) : c '' W ∈ f | begin
rw ← map_comap_of_mem h,
exact image_mem_map W_in
end | lemma | filter.image_mem_of_mem_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_coe_mem_of_mem_comap {f : filter α} {U : set α} (h : U ∈ f) {W : set U}
(W_in : W ∈ comap (coe : U → α) f) : coe '' W ∈ f | image_mem_of_mem_comap (by simp [h]) W_in | lemma | filter.image_coe_mem_of_mem_comap | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map {f : filter α} {m : α → β} (h : injective m) :
comap m (map m f) = f | le_antisymm
(λ s hs, mem_of_superset (preimage_mem_comap $ image_mem_map hs) $
by simp only [preimage_image_eq s h])
le_comap_map | lemma | filter.comap_map | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap_iff {f : filter β} {m : α → β} (inj : injective m)
(large : set.range m ∈ f) {S : set α} : S ∈ comap m f ↔ m '' S ∈ f | by rw [← image_mem_map_iff inj, map_comap_of_mem large] | lemma | filter.mem_comap_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_map_iff_of_inj_on {l₁ l₂ : filter α} {f : α → β} {s : set α}
(h₁ : s ∈ l₁) (h₂ : s ∈ l₂) (hinj : inj_on f s) :
map f l₁ ≤ map f l₂ ↔ l₁ ≤ l₂ | ⟨λ h t ht, mp_mem h₁ $ mem_of_superset (h $ image_mem_map (inter_mem h₂ ht)) $
λ y ⟨x, ⟨hxs, hxt⟩, hxy⟩ hys, hinj hxs hys hxy ▸ hxt, λ h, map_mono h⟩ | lemma | filter.map_le_map_iff_of_inj_on | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_map_iff {f g : filter α} {m : α → β} (hm : injective m) : map m f ≤ map m g ↔ f ≤ g | by rw [map_le_iff_le_comap, comap_map hm] | lemma | filter.map_le_map_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_map_iff_of_inj_on {f g : filter α} {m : α → β} {s : set α}
(hsf : s ∈ f) (hsg : s ∈ g) (hm : inj_on m s) :
map m f = map m g ↔ f = g | by simp only [le_antisymm_iff, map_le_map_iff_of_inj_on hsf hsg hm,
map_le_map_iff_of_inj_on hsg hsf hm] | lemma | filter.map_eq_map_iff_of_inj_on | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inj {f g : filter α} {m : α → β} (hm : injective m) :
map m f = map m g ↔ f = g | map_eq_map_iff_of_inj_on univ_mem univ_mem (hm.inj_on _) | lemma | filter.map_inj | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective {m : α → β} (hm : injective m) : injective (map m) | λ f g, (map_inj hm).1 | lemma | filter.map_injective | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot_iff {f : filter β} {m : α → β} : ne_bot (comap m f) ↔ ∀ t ∈ f, ∃ a, m a ∈ t | begin
simp only [← forall_mem_nonempty_iff_ne_bot, mem_comap, forall_exists_index],
exact ⟨λ h t t_in, h (m ⁻¹' t) t t_in subset.rfl, λ h s t ht hst, (h t ht).imp hst⟩,
end | lemma | filter.comap_ne_bot_iff | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter",
"forall_exists_index"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot {f : filter β} {m : α → β} (hm : ∀ t ∈ f, ∃ a, m a ∈ t) : ne_bot (comap m f) | comap_ne_bot_iff.mpr hm | lemma | filter.comap_ne_bot | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot_iff_frequently {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ ∃ᶠ y in f, y ∈ range m | by simp [comap_ne_bot_iff, frequently_iff, ← exists_and_distrib_left, and.comm] | lemma | filter.comap_ne_bot_iff_frequently | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"exists_and_distrib_left",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot_iff_compl_range {f : filter β} {m : α → β} :
ne_bot (comap m f) ↔ (range m)ᶜ ∉ f | comap_ne_bot_iff_frequently | lemma | filter.comap_ne_bot_iff_compl_range | order.filter | src/order/filter/basic.lean | [
"control.traversable.instances",
"data.set.finite",
"order.copy",
"tactic.monotonicity"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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