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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|---|---|---|---|---|---|---|---|---|---|---|
tendsto_comp_coe_Iio_at_top :
tendsto (λ x : Iio a, f x) at_top l ↔ tendsto f (𝓝[<] a) l | by rw [← map_coe_Iio_at_top, tendsto_map'_iff] | lemma | tendsto_comp_coe_Iio_at_top | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"map_coe_Iio_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_at_top {f : β → Ioo a b} :
tendsto f l at_top ↔ tendsto (λ x, (f x : α)) l (𝓝[<] b) | by rw [← comap_coe_Ioo_nhds_within_Iio, tendsto_comap_iff] | lemma | tendsto_Ioo_at_top | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"comap_coe_Ioo_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioo_at_bot {f : β → Ioo a b} :
tendsto f l at_bot ↔ tendsto (λ x, (f x : α)) l (𝓝[>] a) | by rw [← comap_coe_Ioo_nhds_within_Ioi, tendsto_comap_iff] | lemma | tendsto_Ioo_at_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"comap_coe_Ioo_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Ioi_at_bot {f : β → Ioi a} :
tendsto f l at_bot ↔ tendsto (λ x, (f x : α)) l (𝓝[>] a) | by rw [← comap_coe_Ioi_nhds_within_Ioi, tendsto_comap_iff] | lemma | tendsto_Ioi_at_bot | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"comap_coe_Ioi_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_Iio_at_top {f : β → Iio a} :
tendsto f l at_top ↔ tendsto (λ x, (f x : α)) l (𝓝[<] a) | by rw [← comap_coe_Iio_nhds_within_Iio, tendsto_comap_iff] | lemma | tendsto_Iio_at_top | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"comap_coe_Iio_nhds_within_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.exists_countable_dense_subset_no_bot_top [nontrivial α]
{s : set α} [separable_space s] (hs : dense s) :
∃ t ⊆ s, t.countable ∧ dense t ∧ (∀ x, is_bot x → x ∉ t) ∧ (∀ x, is_top x → x ∉ t) | begin
rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩,
refine ⟨t \ ({x | is_bot x} ∪ {x | is_top x}), _, _, _, _, _⟩,
{ exact (diff_subset _ _).trans hts },
{ exact htc.mono (diff_subset _ _) },
{ exact htd.diff_finite ((subsingleton_is_bot α).finite.union (subsingleton_is_top α).finite) },
... | lemma | dense.exists_countable_dense_subset_no_bot_top | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"finite",
"is_bot",
"is_top",
"nontrivial"
] | Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a
separable space (e.g., if `α` has a second countable topology), then there exists a countable
dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_countable_dense_no_bot_top [separable_space α] [nontrivial α] :
∃ s : set α, s.countable ∧ dense s ∧ (∀ x, is_bot x → x ∉ s) ∧ (∀ x, is_top x → x ∉ s) | by simpa using dense_univ.exists_countable_dense_subset_no_bot_top | lemma | exists_countable_dense_no_bot_top | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"dense",
"is_bot",
"is_top",
"nontrivial"
] | If `α` is a nontrivial separable dense linear order, then there exists a
countable dense set `s : set α` that contains neither top nor bottom elements of `α`.
For a dense set containing both bot and top elements, see
`exists_countable_dense_bot_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Sup_mem_closure {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) :
Sup s ∈ closure s | (is_lub_Sup s).mem_closure hs | lemma | Sup_mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"complete_linear_order",
"is_lub_Sup",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_mem_closure {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) :
Inf s ∈ closure s | (is_glb_Inf s).mem_closure hs | lemma | Inf_mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"closure",
"complete_linear_order",
"is_glb_Inf",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.Sup_mem {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :
Sup s ∈ s | (is_lub_Sup s).mem_of_is_closed hs hc | lemma | is_closed.Sup_mem | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"complete_linear_order",
"is_closed",
"is_lub_Sup",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.Inf_mem {α : Type u} [topological_space α] [complete_linear_order α]
[order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :
Inf s ∈ s | (is_glb_Inf s).mem_of_is_closed hs hc | lemma | is_closed.Inf_mem | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"complete_linear_order",
"is_closed",
"is_glb_Inf",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.map_Sup_of_continuous_at' {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (hs : s.nonempty) :
f (Sup s) = Sup (f '' s) | --This is a particular case of the more general is_lub.is_lub_of_tendsto
((is_lub_Sup _).is_lub_of_tendsto (λ x hx y hy xy, Mf xy) hs $
Cf.mono_left inf_le_left).Sup_eq.symm | lemma | monotone.map_Sup_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"inf_le_left",
"is_lub_Sup",
"monotone"
] | A monotone function continuous at the supremum of a nonempty set sends this supremum to
the supremum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_Sup_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (fbot : f ⊥ = ⊥) :
f (Sup s) = Sup (f '' s) | begin
cases s.eq_empty_or_nonempty with h h,
{ simp [h, fbot] },
{ exact Mf.map_Sup_of_continuous_at' Cf h }
end | lemma | monotone.map_Sup_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"monotone"
] | A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
this supremum to the supremum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_supr_of_continuous_at' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Mf : monotone f) :
f (⨆ i, g i) = ⨆ i, f (g i) | by rw [supr, Mf.map_Sup_of_continuous_at' Cf (range_nonempty g), ← range_comp, supr] | lemma | monotone.map_supr_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"monotone",
"supr"
] | A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
supremum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_supr_of_continuous_at {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Mf : monotone f) (fbot : f ⊥ = ⊥) :
f (⨆ i, g i) = ⨆ i, f (g i) | by rw [supr, Mf.map_Sup_of_continuous_at Cf fbot, ← range_comp, supr] | lemma | monotone.map_supr_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"monotone",
"supr"
] | If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_Inf_of_continuous_at' {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (hs : s.nonempty) :
f (Inf s) = Inf (f '' s) | @monotone.map_Sup_of_continuous_at' αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s Cf Mf.dual hs | lemma | monotone.map_Inf_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"monotone",
"monotone.map_Sup_of_continuous_at'"
] | A monotone function continuous at the infimum of a nonempty set sends this infimum to
the infimum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_Inf_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (ftop : f ⊤ = ⊤) :
f (Inf s) = Inf (f '' s) | @monotone.map_Sup_of_continuous_at αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s Cf Mf.dual ftop | lemma | monotone.map_Inf_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"monotone",
"monotone.map_Sup_of_continuous_at"
] | A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
this infimum to the infimum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_infi_of_continuous_at' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Mf : monotone f) :
f (⨅ i, g i) = ⨅ i, f (g i) | @monotone.map_supr_of_continuous_at' αᵒᵈ βᵒᵈ _ _ _ _ _ _ ι _ f g Cf Mf.dual | lemma | monotone.map_infi_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"infi",
"monotone",
"monotone.map_supr_of_continuous_at'"
] | A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed infimum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_infi_of_continuous_at {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Mf : monotone f) (ftop : f ⊤ = ⊤) :
f (infi g) = infi (f ∘ g) | @monotone.map_supr_of_continuous_at αᵒᵈ βᵒᵈ _ _ _ _ _ _ ι f g Cf Mf.dual ftop | lemma | monotone.map_infi_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"continuous_at",
"infi",
"monotone",
"monotone.map_supr_of_continuous_at"
] | If a monotone function sending `top` to `top` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_Sup_of_continuous_at' {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Af : antitone f) (hs : s.nonempty) :
f (Sup s) = Inf (f '' s) | monotone.map_Sup_of_continuous_at'
(show continuous_at (order_dual.to_dual ∘ f) (Sup s), from Cf) Af hs | lemma | antitone.map_Sup_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"monotone.map_Sup_of_continuous_at'",
"order_dual.to_dual"
] | An antitone function continuous at the supremum of a nonempty set sends this supremum to
the infimum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_Sup_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Af : antitone f) (fbot : f ⊥ = ⊤) :
f (Sup s) = Inf (f '' s) | monotone.map_Sup_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (Sup s), from Cf) Af fbot | lemma | antitone.map_Sup_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"monotone.map_Sup_of_continuous_at",
"order_dual.to_dual"
] | An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
this supremum to the infimum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_supr_of_continuous_at' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Af : antitone f) :
f (⨆ i, g i) = ⨅ i, f (g i) | monotone.map_supr_of_continuous_at'
(show continuous_at (order_dual.to_dual ∘ f) (supr g), from Cf) Af | lemma | antitone.map_supr_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"monotone.map_supr_of_continuous_at'",
"order_dual.to_dual",
"supr"
] | An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
indexed supremum to the indexed infimum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_supr_of_continuous_at {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (supr g)) (Af : antitone f) (fbot : f ⊥ = ⊤) :
f (⨆ i, g i) = ⨅ i, f (g i) | monotone.map_supr_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (supr g), from Cf) Af fbot | lemma | antitone.map_supr_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"monotone.map_supr_of_continuous_at",
"order_dual.to_dual",
"supr"
] | An antitone function sending `bot` to `top` is continuous at the indexed supremum over
a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_Inf_of_continuous_at' {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Af : antitone f) (hs : s.nonempty) :
f (Inf s) = Sup (f '' s) | monotone.map_Inf_of_continuous_at'
(show continuous_at (order_dual.to_dual ∘ f) (Inf s), from Cf) Af hs | lemma | antitone.map_Inf_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"monotone.map_Inf_of_continuous_at'",
"order_dual.to_dual"
] | An antitone function continuous at the infimum of a nonempty set sends this infimum to
the supremum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_Inf_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Af : antitone f) (ftop : f ⊤ = ⊥) :
f (Inf s) = Sup (f '' s) | monotone.map_Inf_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (Inf s), from Cf) Af ftop | lemma | antitone.map_Inf_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"monotone.map_Inf_of_continuous_at",
"order_dual.to_dual"
] | An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
this infimum to the supremum of the image of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_infi_of_continuous_at' {ι : Sort*} [nonempty ι] {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Af : antitone f) :
f (⨅ i, g i) = ⨆ i, f (g i) | monotone.map_infi_of_continuous_at'
(show continuous_at (order_dual.to_dual ∘ f) (infi g), from Cf) Af | lemma | antitone.map_infi_of_continuous_at' | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"infi",
"monotone.map_infi_of_continuous_at'",
"order_dual.to_dual"
] | An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
infimum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_infi_of_continuous_at {ι : Sort*} {f : α → β} {g : ι → α}
(Cf : continuous_at f (infi g)) (Af : antitone f) (ftop : f ⊤ = ⊥) :
f (infi g) = supr (f ∘ g) | monotone.map_infi_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (infi g), from Cf) Af ftop | lemma | antitone.map_infi_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"continuous_at",
"infi",
"monotone.map_infi_of_continuous_at",
"order_dual.to_dual",
"supr"
] | If an antitone function sending `top` to `bot` is continuous at the indexed infimum over
a `Sort`, then it sends this indexed infimum to the indexed supremum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cSup_mem_closure {s : set α} (hs : s.nonempty) (B : bdd_above s) : Sup s ∈ closure s | (is_lub_cSup hs B).mem_closure hs | lemma | cSup_mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_above",
"closure",
"is_lub_cSup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cInf_mem_closure {s : set α} (hs : s.nonempty) (B : bdd_below s) : Inf s ∈ closure s | (is_glb_cInf hs B).mem_closure hs | lemma | cInf_mem_closure | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_below",
"closure",
"is_glb_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.cSup_mem {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_above s) :
Sup s ∈ s | (is_lub_cSup hs B).mem_of_is_closed hs hc | lemma | is_closed.cSup_mem | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_above",
"is_closed",
"is_lub_cSup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.cInf_mem {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_below s) :
Inf s ∈ s | (is_glb_cInf hs B).mem_of_is_closed hs hc | lemma | is_closed.cInf_mem | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_below",
"is_closed",
"is_glb_cInf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.map_cSup_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Mf : monotone f) (ne : s.nonempty) (H : bdd_above s) :
f (Sup s) = Sup (f '' s) | begin
refine ((is_lub_cSup (ne.image f) (Mf.map_bdd_above H)).unique _).symm,
refine (is_lub_cSup ne H).is_lub_of_tendsto (λx hx y hy xy, Mf xy) ne _,
exact Cf.mono_left inf_le_left
end | lemma | monotone.map_cSup_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_above",
"continuous_at",
"inf_le_left",
"is_lub_cSup",
"monotone",
"unique"
] | If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the supremum of the image of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_csupr_of_continuous_at {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨆ i, g i)) (Mf : monotone f) (H : bdd_above (range g)) :
f (⨆ i, g i) = ⨆ i, f (g i) | by rw [supr, Mf.map_cSup_of_continuous_at Cf (range_nonempty _) H, ← range_comp, supr] | lemma | monotone.map_csupr_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_above",
"continuous_at",
"monotone",
"supr"
] | If a monotone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the supremum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_cInf_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Mf : monotone f) (ne : s.nonempty) (H : bdd_below s) :
f (Inf s) = Inf (f '' s) | @monotone.map_cSup_of_continuous_at αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s Cf Mf.dual ne H | lemma | monotone.map_cInf_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_below",
"continuous_at",
"monotone",
"monotone.map_cSup_of_continuous_at"
] | If a monotone function is continuous at the infimum of a nonempty bounded below set `s`,
then it sends this infimum to the infimum of the image of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_cinfi_of_continuous_at {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨅ i, g i)) (Mf : monotone f) (H : bdd_below (range g)) :
f (⨅ i, g i) = ⨅ i, f (g i) | @monotone.map_csupr_of_continuous_at αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ _ _ _ Cf Mf.dual H | lemma | monotone.map_cinfi_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_below",
"continuous_at",
"monotone",
"monotone.map_csupr_of_continuous_at"
] | A continuous monotone function sends indexed infimum to indexed infimum in conditionally
complete linear order, under a boundedness assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_cSup_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Sup s))
(Af : antitone f) (ne : s.nonempty) (H : bdd_above s) :
f (Sup s) = Inf (f '' s) | monotone.map_cSup_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (Sup s), from Cf) Af ne H | lemma | antitone.map_cSup_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"bdd_above",
"continuous_at",
"monotone.map_cSup_of_continuous_at",
"order_dual.to_dual"
] | If an antitone function is continuous at the supremum of a nonempty bounded above set `s`,
then it sends this supremum to the infimum of the image of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_csupr_of_continuous_at {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨆ i, g i)) (Af : antitone f) (H : bdd_above (range g)) :
f (⨆ i, g i) = ⨅ i, f (g i) | monotone.map_csupr_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (⨆ i, g i), from Cf) Af H | lemma | antitone.map_csupr_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"bdd_above",
"continuous_at",
"monotone.map_csupr_of_continuous_at",
"order_dual.to_dual"
] | If an antitone function is continuous at the indexed supremum of a bounded function on
a nonempty `Sort`, then it sends this supremum to the infimum of the composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_cInf_of_continuous_at {f : α → β} {s : set α} (Cf : continuous_at f (Inf s))
(Af : antitone f) (ne : s.nonempty) (H : bdd_below s) :
f (Inf s) = Sup (f '' s) | monotone.map_cInf_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (Inf s), from Cf) Af ne H | lemma | antitone.map_cInf_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"bdd_below",
"continuous_at",
"monotone.map_cInf_of_continuous_at",
"order_dual.to_dual"
] | If an antitone function is continuous at the infimum of a nonempty bounded below set `s`,
then it sends this infimum to the supremum of the image of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_cinfi_of_continuous_at {f : α → β} {g : γ → α}
(Cf : continuous_at f (⨅ i, g i)) (Af : antitone f) (H : bdd_below (range g)) :
f (⨅ i, g i) = ⨆ i, f (g i) | monotone.map_cinfi_of_continuous_at
(show continuous_at (order_dual.to_dual ∘ f) (⨅ i, g i), from Cf) Af H | lemma | antitone.map_cinfi_of_continuous_at | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"antitone",
"bdd_below",
"continuous_at",
"monotone.map_cinfi_of_continuous_at",
"order_dual.to_dual"
] | A continuous antitone function sends indexed infimum to indexed supremum in conditionally
complete linear order, under a boundedness assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.tendsto_nhds_within_Iio {α β : Type*}
[linear_order α] [topological_space α] [order_topology α]
[conditionally_complete_linear_order β] [topological_space β] [order_topology β]
{f : α → β} (Mf : monotone f) (x : α) :
tendsto f (𝓝[<] x) (𝓝 (Sup (f '' (Iio x)))) | begin
rcases eq_empty_or_nonempty (Iio x) with h|h, { simp [h] },
refine tendsto_order.2 ⟨λ l hl, _, λ m hm, _⟩,
{ obtain ⟨z, zx, lz⟩ : ∃ (a : α), a < x ∧ l < f a,
by simpa only [mem_image, exists_prop, exists_exists_and_eq_and]
using exists_lt_of_lt_cSup (nonempty_image_iff.2 h) hl,
exact (mem_... | lemma | monotone.tendsto_nhds_within_Iio | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"bdd_above_Iio",
"conditionally_complete_linear_order",
"exists_exists_and_eq_and",
"exists_lt_of_lt_cSup",
"exists_prop",
"le_cSup",
"mem_nhds_within_Iio_iff_exists_Ioo_subset'",
"monotone",
"order_topology",
"self_mem_nhds_within",
"topological_space"
] | A monotone map has a limit to the left of any point `x`, equal to `Sup (f '' (Iio x))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.tendsto_nhds_within_Ioi {α β : Type*}
[linear_order α] [topological_space α] [order_topology α]
[conditionally_complete_linear_order β] [topological_space β] [order_topology β]
{f : α → β} (Mf : monotone f) (x : α) :
tendsto f (𝓝[>] x) (𝓝 (Inf (f '' (Ioi x)))) | @monotone.tendsto_nhds_within_Iio αᵒᵈ βᵒᵈ _ _ _ _ _ _ f Mf.dual x | lemma | monotone.tendsto_nhds_within_Ioi | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"conditionally_complete_linear_order",
"monotone",
"monotone.tendsto_nhds_within_Iio",
"order_topology",
"topological_space"
] | A monotone map has a limit to the right of any point `x`, equal to `Inf (f '' (Ioi x))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_nhds_within_pos_mem_Ioo {ε : α} (h : 0 < ε) :
∀ᶠ x in 𝓝[>] 0, x ∈ Ioo 0 ε | Ioo_mem_nhds_within_Ioi (left_mem_Ico.2 h) | lemma | eventually_nhds_within_pos_mem_Ioo | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"Ioo_mem_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within_pos_mem_Ioc {ε : α} (h : 0 < ε) :
∀ᶠ x in 𝓝[>] 0, x ∈ Ioc 0 ε | Ioc_mem_nhds_within_Ioi (left_mem_Ico.2 h) | lemma | eventually_nhds_within_pos_mem_Ioc | topology.order | src/topology/order/basic.lean | [
"data.set.intervals.pi",
"data.set.pointwise.interval",
"order.filter.interval",
"topology.support",
"topology.algebra.order.left_right"
] | [
"Ioc_mem_nhds_within_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_inf (L : Type*) [topological_space L] [has_inf L] : Prop | (continuous_inf : continuous (λ p : L × L, p.1 ⊓ p.2)) | class | has_continuous_inf | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous",
"continuous_inf",
"has_inf",
"topological_space"
] | Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be an infimum. Then `L` is said to have *(jointly) continuous infimum* if the map
`⊓:L×L → L` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_sup (L : Type*) [topological_space L] [has_sup L] : Prop | (continuous_sup : continuous (λ p : L × L, p.1 ⊔ p.2)) | class | has_continuous_sup | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous",
"continuous_sup",
"has_sup",
"topological_space"
] | Let `L` be a topological space and let `L×L` be equipped with the product topology and let
`⊓:L×L → L` be a supremum. Then `L` is said to have *(jointly) continuous supremum* if the map
`⊓:L×L → L` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_dual.has_continuous_sup
(L : Type*) [topological_space L] [has_inf L] [has_continuous_inf L] : has_continuous_sup Lᵒᵈ | { continuous_sup := @has_continuous_inf.continuous_inf L _ _ _ } | instance | order_dual.has_continuous_sup | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous_sup",
"has_continuous_inf",
"has_continuous_sup",
"has_inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_dual.has_continuous_inf
(L : Type*) [topological_space L] [has_sup L] [has_continuous_sup L] : has_continuous_inf Lᵒᵈ | { continuous_inf := @has_continuous_sup.continuous_sup L _ _ _ } | instance | order_dual.has_continuous_inf | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous_inf",
"has_continuous_inf",
"has_continuous_sup",
"has_sup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_lattice (L : Type*) [topological_space L] [lattice L]
extends has_continuous_inf L, has_continuous_sup L | class | topological_lattice | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"has_continuous_inf",
"has_continuous_sup",
"lattice",
"topological_space"
] | Let `L` be a lattice equipped with a topology such that `L` has continuous infimum and supremum.
Then `L` is said to be a *topological lattice*. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_dual.topological_lattice
(L : Type*) [topological_space L] [lattice L] [topological_lattice L] :
topological_lattice Lᵒᵈ | {} | instance | order_dual.topological_lattice | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"lattice",
"topological_lattice",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_order.topological_lattice {L : Type*} [topological_space L] [linear_order L]
[order_closed_topology L] : topological_lattice L | { continuous_inf := continuous_min, continuous_sup := continuous_max } | instance | linear_order.topological_lattice | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous_inf",
"continuous_max",
"continuous_min",
"continuous_sup",
"order_closed_topology",
"topological_lattice",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inf [has_inf L] [has_continuous_inf L] :
continuous (λp:L×L, p.1 ⊓ p.2) | has_continuous_inf.continuous_inf | lemma | continuous_inf | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous",
"has_continuous_inf",
"has_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.inf [has_inf L] [has_continuous_inf L]
{f g : X → L} (hf : continuous f) (hg : continuous g) :
continuous (λx, f x ⊓ g x) | continuous_inf.comp (hf.prod_mk hg : _) | lemma | continuous.inf | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous",
"has_continuous_inf",
"has_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sup [has_sup L] [has_continuous_sup L] :
continuous (λp:L×L, p.1 ⊔ p.2) | has_continuous_sup.continuous_sup | lemma | continuous_sup | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous",
"has_continuous_sup",
"has_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.sup [has_sup L] [has_continuous_sup L]
{f g : X → L} (hf : continuous f) (hg : continuous g) :
continuous (λx, f x ⊔ g x) | continuous_sup.comp (hf.prod_mk hg : _) | lemma | continuous.sup | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"continuous",
"has_continuous_sup",
"has_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.sup_right_nhds' {ι β} [topological_space β] [has_sup β] [has_continuous_sup β]
{l : filter ι} {f g : ι → β} {x y : β}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) :
tendsto (f ⊔ g) l (𝓝 (x ⊔ y)) | (continuous_sup.tendsto _).comp (tendsto.prod_mk_nhds hf hg) | lemma | filter.tendsto.sup_right_nhds' | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"filter",
"has_continuous_sup",
"has_sup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.sup_right_nhds {ι β} [topological_space β] [has_sup β] [has_continuous_sup β]
{l : filter ι} {f g : ι → β} {x y : β}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) :
tendsto (λ i, f i ⊔ g i) l (𝓝 (x ⊔ y)) | hf.sup_right_nhds' hg | lemma | filter.tendsto.sup_right_nhds | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"filter",
"has_continuous_sup",
"has_sup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inf_right_nhds' {ι β} [topological_space β] [has_inf β] [has_continuous_inf β]
{l : filter ι} {f g : ι → β} {x y : β}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) :
tendsto (f ⊓ g) l (𝓝 (x ⊓ y)) | (continuous_inf.tendsto _).comp (tendsto.prod_mk_nhds hf hg) | lemma | filter.tendsto.inf_right_nhds' | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"filter",
"has_continuous_inf",
"has_inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.inf_right_nhds {ι β} [topological_space β] [has_inf β] [has_continuous_inf β]
{l : filter ι} {f g : ι → β} {x y : β}
(hf : tendsto f l (𝓝 x)) (hg : tendsto g l (𝓝 y)) :
tendsto (λ i, f i ⊓ g i) l (𝓝 (x ⊓ y)) | hf.inf_right_nhds' hg | lemma | filter.tendsto.inf_right_nhds | topology.order | src/topology/order/lattice.lean | [
"topology.order.basic",
"topology.constructions"
] | [
"filter",
"has_continuous_inf",
"has_inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_lower_topology | α | def | with_lower_topology | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [] | Type synonym for a preorder equipped with the lower topology | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_lower : α ≃ with_lower_topology α | equiv.refl _ | def | with_lower_topology.to_lower | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"equiv.refl",
"with_lower_topology"
] | `to_lower` is the identity function to the `with_lower_topology` of a type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_lower : with_lower_topology α ≃ α | equiv.refl _ | def | with_lower_topology.of_lower | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"equiv.refl",
"with_lower_topology"
] | `of_lower` is the identity function from the `with_lower_topology` of a type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_with_lower_topology_symm_eq : (@to_lower α).symm = of_lower | rfl | lemma | with_lower_topology.to_with_lower_topology_symm_eq | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_with_lower_topology_symm_eq : (@of_lower α).symm = to_lower | rfl | lemma | with_lower_topology.of_with_lower_topology_symm_eq | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lower_of_lower (a : with_lower_topology α) : to_lower (of_lower a) = a | rfl | lemma | with_lower_topology.to_lower_of_lower | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"with_lower_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lower_to_lower (a : α) : of_lower (to_lower a) = a | rfl | lemma | with_lower_topology.of_lower_to_lower | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lower_inj {a b : α} : to_lower a = to_lower b ↔ a = b | iff.rfl | lemma | with_lower_topology.to_lower_inj | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lower_inj {a b : with_lower_topology α} : of_lower a = of_lower b ↔ a = b | iff.rfl | lemma | with_lower_topology.of_lower_inj | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"with_lower_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec {β : with_lower_topology α → Sort*}
(h : Π a, β (to_lower a)) : Π a, β a | λ a, h (of_lower a) | def | with_lower_topology.rec | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"with_lower_topology"
] | A recursor for `with_lower_topology`. Use as `induction x using with_lower_topology.rec`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_preimage_of_lower (S : set α) :
is_open (with_lower_topology.of_lower ⁻¹' S) ↔
(generate_from {s : set α | ∃ (a : α), (Ici a)ᶜ = s}).is_open S | iff.rfl | lemma | with_lower_topology.is_open_preimage_of_lower | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"is_open",
"with_lower_topology.of_lower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_def (T : set (with_lower_topology α)) :
is_open T ↔ (generate_from {s : set α | ∃ (a : α), (Ici a)ᶜ = s}).is_open
(with_lower_topology.to_lower ⁻¹' T) | iff.rfl | lemma | with_lower_topology.is_open_def | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"is_open",
"with_lower_topology",
"with_lower_topology.to_lower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_topology (α : Type*) [t : topological_space α] [preorder α] : Prop | (topology_eq_lower_topology [] : t = generate_from {s | ∃ a, (Ici a)ᶜ = s}) | class | lower_topology | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"topological_space"
] | The lower topology is the topology generated by the complements of the closed intervals to infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_basis (α : Type*) [preorder α] | {s : set α | ∃ t : set α, t.finite ∧ (upper_closure t : set α)ᶜ = s} | def | lower_topology.lower_basis | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"upper_closure"
] | The complements of the upper closures of finite sets are a collection of lower sets
which form a basis for the lower topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_lower_topology_homeomorph : with_lower_topology α ≃ₜ α | { continuous_to_fun := by { convert continuous_id, apply topology_eq_lower_topology },
continuous_inv_fun := by { convert ← continuous_id, apply topology_eq_lower_topology },
..with_lower_topology.of_lower } | def | lower_topology.with_lower_topology_homeomorph | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"continuous_id",
"with_lower_topology",
"with_lower_topology.of_lower"
] | If `α` is equipped with the lower topology, then it is homeomorphic to `with_lower_topology α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_iff_generate_Ici_compl : is_open s ↔ generate_open {t | ∃ a, (Ici a)ᶜ = t} s | by rw topology_eq_lower_topology α; refl | lemma | lower_topology.is_open_iff_generate_Ici_compl | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Ici (a : α) : is_closed (Ici a) | is_open_compl_iff.1 $ is_open_iff_generate_Ici_compl.2 $ generate_open.basic _ ⟨a, rfl⟩ | lemma | lower_topology.is_closed_Ici | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"is_closed",
"is_closed_Ici"
] | Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_upper_closure (h : s.finite) : is_closed (upper_closure s : set α) | begin
simp only [← upper_set.infi_Ici, upper_set.coe_infi],
exact is_closed_bUnion h (λ a h₁, is_closed_Ici a),
end | lemma | lower_topology.is_closed_upper_closure | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"is_closed",
"is_closed_Ici",
"is_closed_bUnion",
"upper_closure",
"upper_set.coe_infi",
"upper_set.infi_Ici"
] | The upper closure of a finite set is closed in the lower topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lower_set_of_is_open (h : is_open s) : is_lower_set s | begin
rw is_open_iff_generate_Ici_compl at h,
induction h,
case generate_open.basic : u h { obtain ⟨a, rfl⟩ := h, exact (is_upper_set_Ici a).compl },
case univ : { exact is_lower_set_univ },
case inter : u v hu1 hv1 hu2 hv2 { exact hu2.inter hv2 },
case sUnion : _ _ ih { exact is_lower_set_sUnion ih },
end | lemma | lower_topology.is_lower_set_of_is_open | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"ih",
"is_lower_set",
"is_lower_set_sUnion",
"is_lower_set_univ",
"is_open",
"is_upper_set_Ici"
] | Every set open in the lower topology is a lower set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_upper_set_of_is_closed (h : is_closed s) : is_upper_set s | is_lower_set_compl.1 $ is_lower_set_of_is_open h.is_open_compl | lemma | lower_topology.is_upper_set_of_is_closed | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"is_closed",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_singleton (a : α) : closure {a} = Ici a | subset_antisymm (closure_minimal (λ b h, h.ge) $ is_closed_Ici a) $
(is_upper_set_of_is_closed is_closed_closure).Ici_subset $ subset_closure rfl | lemma | lower_topology.closure_singleton | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"closure",
"closure_minimal",
"closure_singleton",
"is_closed_Ici",
"is_closed_closure",
"subset_antisymm",
"subset_closure"
] | The closure of a singleton `{a}` in the lower topology is the left-closed right-infinite interval
[a, ∞). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis :
is_topological_basis (lower_basis α) | begin
convert is_topological_basis_of_subbasis (topology_eq_lower_topology α),
simp_rw [lower_basis, coe_upper_closure, compl_Union],
ext s,
split,
{ rintro ⟨F, hF, rfl⟩,
refine ⟨(λ a, (Ici a)ᶜ) '' F, ⟨hF.image _, image_subset_iff.2 $ λ _ _, ⟨_, rfl⟩⟩, _⟩,
rw sInter_image },
{ rintro ⟨F, ⟨hF, hs⟩, r... | lemma | lower_topology.is_topological_basis | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"coe_upper_closure",
"subtype.forall'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_hom.continuous (f : Inf_hom α β) : continuous f | begin
convert continuous_generated_from _,
{ exact lower_topology.topology_eq_lower_topology β },
rintro _ ⟨b, rfl⟩,
rw [preimage_compl, is_open_compl_iff],
convert lower_topology.is_closed_Ici (Inf $ f ⁻¹' Ici b),
refine subset_antisymm (λ a, Inf_le) (λ a ha, le_trans _ $ order_hom_class.mono f ha),
simp... | lemma | Inf_hom.continuous | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"Inf_hom",
"Inf_le",
"continuous",
"continuous_generated_from",
"is_open_compl_iff",
"lower_topology.is_closed_Ici",
"order_hom_class.mono",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_topology.to_has_continuous_inf : has_continuous_inf α | ⟨(inf_Inf_hom : Inf_hom (α × α) α).continuous⟩ | instance | lower_topology.to_has_continuous_inf | topology.order | src/topology/order/lower_topology.lean | [
"topology.homeomorph",
"topology.order.lattice",
"order.hom.complete_lattice"
] | [
"Inf_hom",
"has_continuous_inf",
"inf_Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
priestley_space (α : Type*) [preorder α] [topological_space α] | (priestley {x y : α} : ¬ x ≤ y → ∃ U : set α, is_clopen U ∧ is_upper_set U ∧ x ∈ U ∧ y ∉ U) | class | priestley_space | topology.order | src/topology/order/priestley.lean | [
"order.upper_lower.basic",
"topology.separation"
] | [
"is_clopen",
"is_upper_set",
"topological_space"
] | A Priestley space is an ordered topological space such that any two distinct points can be
separated by a clopen upper set. Compactness is often assumed, but we do not include it here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_clopen_upper_of_not_le :
¬ x ≤ y → ∃ U : set α, is_clopen U ∧ is_upper_set U ∧ x ∈ U ∧ y ∉ U | priestley_space.priestley | lemma | exists_clopen_upper_of_not_le | topology.order | src/topology/order/priestley.lean | [
"order.upper_lower.basic",
"topology.separation"
] | [
"is_clopen",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_clopen_lower_of_not_le (h : ¬ x ≤ y) :
∃ U : set α, is_clopen U ∧ is_lower_set U ∧ x ∉ U ∧ y ∈ U | let ⟨U, hU, hU', hx, hy⟩ := exists_clopen_upper_of_not_le h in
⟨Uᶜ, hU.compl, hU'.compl, not_not.2 hx, hy⟩ | lemma | exists_clopen_lower_of_not_le | topology.order | src/topology/order/priestley.lean | [
"order.upper_lower.basic",
"topology.separation"
] | [
"exists_clopen_upper_of_not_le",
"is_clopen",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_clopen_upper_or_lower_of_ne (h : x ≠ y) :
∃ U : set α, is_clopen U ∧ (is_upper_set U ∨ is_lower_set U) ∧ x ∈ U ∧ y ∉ U | begin
obtain (h | h) := h.not_le_or_not_le,
{ exact (exists_clopen_upper_of_not_le h).imp (λ U, and.imp_right $ and.imp_left or.inl) },
{ obtain ⟨U, hU, hU', hy, hx⟩ := exists_clopen_lower_of_not_le h,
exact ⟨U, hU, or.inr hU', hx, hy⟩ }
end | lemma | exists_clopen_upper_or_lower_of_ne | topology.order | src/topology/order/priestley.lean | [
"order.upper_lower.basic",
"topology.separation"
] | [
"and.imp_left",
"and.imp_right",
"exists_clopen_lower_of_not_le",
"exists_clopen_upper_of_not_le",
"is_clopen",
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
priestley_space.to_t2_space : t2_space α | ⟨λ x y h, let ⟨U, hU, _, hx, hy⟩ := exists_clopen_upper_or_lower_of_ne h in
⟨U, Uᶜ, hU.is_open, hU.compl.is_open, hx, hy, disjoint_compl_right⟩⟩ | instance | priestley_space.to_t2_space | topology.order | src/topology/order/priestley.lean | [
"order.upper_lower.basic",
"topology.separation"
] | [
"exists_clopen_upper_or_lower_of_ne",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_order_hom (α β : Type*) [preorder α] [preorder β] [topological_space α]
[topological_space β]
extends order_hom α β | (continuous_to_fun : continuous to_fun) | structure | continuous_order_hom | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous",
"order_hom",
"topological_space"
] | The type of continuous monotone maps from `α` to `β`, aka Priestley homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_order_hom_class (F : Type*) (α β : out_param $ Type*) [preorder α] [preorder β]
[topological_space α] [topological_space β]
extends rel_hom_class F ((≤) : α → α → Prop) ((≤) : β → β → Prop) | (map_continuous (f : F) : continuous f) | class | continuous_order_hom_class | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous",
"rel_hom_class",
"topological_space"
] | `continuous_order_hom_class F α β` states that `F` is a type of continuous monotone maps.
You should extend this class when you extend `continuous_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_order_hom_class.to_continuous_map_class [preorder α] [preorder β]
[topological_space α] [topological_space β] [continuous_order_hom_class F α β] :
continuous_map_class F α β | { ..‹continuous_order_hom_class F α β› } | instance | continuous_order_hom_class.to_continuous_map_class | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous_map_class",
"continuous_order_hom_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (f : α →Co β) : C(α, β) | { ..f } | def | continuous_order_hom.to_continuous_map | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | Reinterpret a `continuous_order_hom` as a `continuous_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : α →Co β} : f.to_fun = (f : α → β) | rfl | lemma | continuous_order_hom.to_fun_eq_coe | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : α →Co β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | continuous_order_hom.ext | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →Co β) (f' : α → β) (h : f' = f) : α →Co β | ⟨f.to_order_hom.copy f' $ by exact h, h.symm.subst f.continuous_to_fun⟩ | def | continuous_order_hom.copy | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | Copy of a `continuous_order_hom` with a new `continuous_map` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : α →Co β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | continuous_order_hom.coe_copy | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →Co β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | continuous_order_hom.copy_eq | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : α →Co α | ⟨order_hom.id, continuous_id⟩ | def | continuous_order_hom.id | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | `id` as a `continuous_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(continuous_order_hom.id α) = id | rfl | lemma | continuous_order_hom.coe_id | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : continuous_order_hom.id α a = a | rfl | lemma | continuous_order_hom.id_apply | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : β →Co γ) (g : α →Co β) : continuous_order_hom α γ | ⟨f.to_order_hom.comp g.to_order_hom, f.continuous_to_fun.comp g.continuous_to_fun⟩ | def | continuous_order_hom.comp | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous_order_hom"
] | Composition of `continuous_order_hom`s as a `continuous_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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