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
value |
|---|---|---|---|---|---|---|---|---|---|---|
filter.tendsto.is_cobounded_under_ge [ne_bot f] (h : tendsto u f (𝓝 a)) :
f.is_cobounded_under (≥) u | h.is_bounded_under_le.is_cobounded_flip | lemma | filter.tendsto.is_cobounded_under_ge | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥) | bounded_ge_nhds_class.is_bounded_ge_nhds _ | lemma | is_bounded_ge_nhds | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.is_bounded_under_ge (h : tendsto u f (𝓝 a)) :
f.is_bounded_under (≥) u | (is_bounded_ge_nhds a).mono h | lemma | filter.tendsto.is_bounded_under_ge | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"is_bounded_ge_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.bdd_below_range_of_cofinite [is_directed α (≥)]
(h : tendsto u cofinite (𝓝 a)) : bdd_below (set.range u) | h.is_bounded_under_ge.bdd_below_range_of_cofinite | lemma | filter.tendsto.bdd_below_range_of_cofinite | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"bdd_below",
"is_directed",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.bdd_below_range [is_directed α (≥)] {u : ℕ → α} (h : tendsto u at_top (𝓝 a)) :
bdd_below (set.range u) | h.is_bounded_under_ge.bdd_below_range | lemma | filter.tendsto.bdd_below_range | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"bdd_below",
"is_directed",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded_le_nhds (a : α) : (𝓝 a).is_cobounded (≤) | (is_bounded_ge_nhds a).is_cobounded_flip | lemma | is_cobounded_le_nhds | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"is_bounded_ge_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.is_cobounded_under_le [ne_bot f] (h : tendsto u f (𝓝 a)) :
f.is_cobounded_under (≤) u | h.is_bounded_under_ge.is_cobounded_flip | lemma | filter.tendsto.is_cobounded_under_le | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_top.to_bounded_le_nhds_class [order_top α] : bounded_le_nhds_class α | ⟨λ a, is_bounded_le_of_top⟩ | instance | order_top.to_bounded_le_nhds_class | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"bounded_le_nhds_class",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_bot.to_bounded_ge_nhds_class [order_bot α] : bounded_ge_nhds_class α | ⟨λ a, is_bounded_ge_of_bot⟩ | instance | order_bot.to_bounded_ge_nhds_class | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"bounded_ge_nhds_class",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology.to_bounded_le_nhds_class [is_directed α (≤)] [order_topology α] :
bounded_le_nhds_class α | ⟨λ a, (is_top_or_exists_gt a).elim (λ h, ⟨a, eventually_of_forall h⟩) $ Exists.imp $ λ b,
ge_mem_nhds⟩ | instance | order_topology.to_bounded_le_nhds_class | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Exists.imp",
"bounded_le_nhds_class",
"is_directed",
"is_top_or_exists_gt",
"order_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_topology.to_bounded_ge_nhds_class [is_directed α (≥)] [order_topology α] :
bounded_ge_nhds_class α | ⟨λ a, (is_bot_or_exists_lt a).elim (λ h, ⟨a, eventually_of_forall h⟩) $ Exists.imp $ λ b,
le_mem_nhds⟩ | instance | order_topology.to_bounded_ge_nhds_class | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Exists.imp",
"bounded_ge_nhds_class",
"is_bot_or_exists_lt",
"is_directed",
"order_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_of_Limsup_eq_Liminf {f : filter α} {a : α}
(hl : f.is_bounded (≤)) (hg : f.is_bounded (≥)) (hs : f.Limsup = a) (hi : f.Liminf = a) :
f ≤ 𝓝 a | tendsto_order.2 $ and.intro
(assume b hb, gt_mem_sets_of_Liminf_gt hg $ hi.symm ▸ hb)
(assume b hb, lt_mem_sets_of_Limsup_lt hl $ hs.symm ▸ hb) | theorem | le_nhds_of_Limsup_eq_Liminf | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"filter"
] | If the liminf and the limsup of a filter coincide, then this filter converges to
their common value, at least if the filter is eventually bounded above and below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Limsup_nhds (a : α) : Limsup (𝓝 a) = a | cInf_eq_of_forall_ge_of_forall_gt_exists_lt (is_bounded_le_nhds a)
(assume a' (h : {n : α | n ≤ a'} ∈ 𝓝 a), show a ≤ a', from @mem_of_mem_nhds α _ a _ h)
(assume b (hba : a < b), show ∃c (h : {n : α | n ≤ c} ∈ 𝓝 a), c < b, from
match dense_or_discrete a b with
| or.inl ⟨c, hac, hcb⟩ := ⟨c, ge_mem_nhds hac... | theorem | Limsup_nhds | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"cInf_eq_of_forall_ge_of_forall_gt_exists_lt",
"dense_or_discrete",
"ge_mem_nhds",
"gt_mem_nhds",
"is_bounded_le_nhds",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Liminf_nhds : ∀ (a : α), Liminf (𝓝 a) = a | @Limsup_nhds αᵒᵈ _ _ _ | theorem | Liminf_nhds | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Limsup_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Liminf_eq_of_le_nhds {f : filter α} {a : α} [ne_bot f] (h : f ≤ 𝓝 a) : f.Liminf = a | have hb_ge : is_bounded (≥) f, from (is_bounded_ge_nhds a).mono h,
have hb_le : is_bounded (≤) f, from (is_bounded_le_nhds a).mono h,
le_antisymm
(calc f.Liminf ≤ f.Limsup : Liminf_le_Limsup hb_le hb_ge
... ≤ (𝓝 a).Limsup :
Limsup_le_Limsup_of_le h hb_ge.is_cobounded_flip (is_bounded_le_nhds a)
... = a... | theorem | Liminf_eq_of_le_nhds | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Liminf_nhds",
"Limsup_nhds",
"filter",
"is_bounded_ge_nhds",
"is_bounded_le_nhds"
] | If a filter is converging, its limsup coincides with its limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Limsup_eq_of_le_nhds : ∀ {f : filter α} {a : α} [ne_bot f], f ≤ 𝓝 a → f.Limsup = a | @Liminf_eq_of_le_nhds αᵒᵈ _ _ _ | theorem | Limsup_eq_of_le_nhds | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Liminf_eq_of_le_nhds",
"filter"
] | If a filter is converging, its liminf coincides with its limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.limsup_eq {f : filter β} {u : β → α} {a : α} [ne_bot f]
(h : tendsto u f (𝓝 a)) : limsup u f = a | Limsup_eq_of_le_nhds h | theorem | filter.tendsto.limsup_eq | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Limsup_eq_of_le_nhds",
"filter"
] | If a function has a limit, then its limsup coincides with its limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.liminf_eq {f : filter β} {u : β → α} {a : α} [ne_bot f]
(h : tendsto u f (𝓝 a)) : liminf u f = a | Liminf_eq_of_le_nhds h | theorem | filter.tendsto.liminf_eq | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Liminf_eq_of_le_nhds",
"filter"
] | If a function has a limit, then its liminf coincides with its limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_liminf_eq_limsup {f : filter β} {u : β → α} {a : α}
(hinf : liminf u f = a) (hsup : limsup u f = a)
(h : f.is_bounded_under (≤) u . is_bounded_default)
(h' : f.is_bounded_under (≥) u . is_bounded_default) :
tendsto u f (𝓝 a) | le_nhds_of_Limsup_eq_Liminf h h' hsup hinf | theorem | tendsto_of_liminf_eq_limsup | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"filter",
"le_nhds_of_Limsup_eq_Liminf"
] | If the liminf and the limsup of a function coincide, then the limit of the function
exists and has the same value | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_le_liminf_of_limsup_le {f : filter β} {u : β → α} {a : α}
(hinf : a ≤ liminf u f) (hsup : limsup u f ≤ a)
(h : f.is_bounded_under (≤) u . is_bounded_default)
(h' : f.is_bounded_under (≥) u . is_bounded_default) :
tendsto u f (𝓝 a) | if hf : f = ⊥ then hf.symm ▸ tendsto_bot
else by haveI : ne_bot f := ⟨hf⟩; exact tendsto_of_liminf_eq_limsup
(le_antisymm (le_trans (liminf_le_limsup h h') hsup) hinf)
(le_antisymm hsup (le_trans hinf (liminf_le_limsup h h'))) h h' | theorem | tendsto_of_le_liminf_of_limsup_le | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"filter",
"tendsto_of_liminf_eq_limsup"
] | If a number `a` is less than or equal to the `liminf` of a function `f` at some filter
and is greater than or equal to the `limsup` of `f`, then `f` tends to `a` along this filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_no_upcrossings [densely_ordered α]
{f : filter β} {u : β → α} {s : set α} (hs : dense s)
(H : ∀ (a ∈ s) (b ∈ s), a < b → ¬((∃ᶠ n in f, u n < a) ∧ (∃ᶠ n in f, b < u n)))
(h : f.is_bounded_under (≤) u . is_bounded_default)
(h' : f.is_bounded_under (≥) u . is_bounded_default) :
∃ (c : α), tendsto u f ... | begin
by_cases hbot : f = ⊥, { rw hbot, exact ⟨Inf ∅, tendsto_bot⟩ },
haveI : ne_bot f := ⟨hbot⟩,
refine ⟨limsup u f, _⟩,
apply tendsto_of_le_liminf_of_limsup_le _ le_rfl h h',
by_contra' hlt,
obtain ⟨a, ⟨⟨la, au⟩, as⟩⟩ : ∃ a, (f.liminf u < a ∧ a < f.limsup u) ∧ a ∈ s :=
dense_iff_inter_open.1 hs (set.I... | lemma | tendsto_of_no_upcrossings | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"dense",
"densely_ordered",
"filter",
"is_open_Ioo",
"le_rfl",
"set.Ioo",
"tendsto_of_le_liminf_of_limsup_le"
] | Assume that, for any `a < b`, a sequence can not be infinitely many times below `a` and
above `b`. If it is also ultimately bounded above and below, then it has to converge. This even
works if `a` and `b` are restricted to a dense subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eventually_le_limsup (hf : is_bounded_under (≤) f u . is_bounded_default) :
∀ᶠ b in f, u b ≤ f.limsup u | begin
obtain ha | ha := is_top_or_exists_gt (f.limsup u),
{ exact eventually_of_forall (λ _, ha _) },
by_cases H : is_glb (set.Ioi (f.limsup u)) (f.limsup u),
{ obtain ⟨u, -, -, hua, hu⟩ := H.exists_seq_antitone_tendsto ha,
have := λ n, eventually_lt_of_limsup_lt (hu n) hf,
exact (eventually_countable_f... | lemma | eventually_le_limsup | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"exists_prop",
"ge_of_tendsto",
"is_glb",
"is_greatest",
"is_top_or_exists_gt",
"lower_bounds",
"not_and",
"not_forall",
"set.Ioi",
"set.mem_Ioi",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_liminf_le (hf : is_bounded_under (≥) f u . is_bounded_default) :
∀ᶠ b in f, f.liminf u ≤ u b | @eventually_le_limsup αᵒᵈ _ _ _ _ _ _ _ _ hf | lemma | eventually_liminf_le | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"eventually_le_limsup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_eq_bot : f.limsup u = ⊥ ↔ u =ᶠ[f] ⊥ | ⟨λ h, (eventually_le.trans eventually_le_limsup $ eventually_of_forall $ λ _, h.le).mono $ λ x hx,
le_antisymm hx bot_le, λ h, by { rw limsup_congr h, exact limsup_const_bot }⟩ | lemma | limsup_eq_bot | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"bot_le",
"eventually_le_limsup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
liminf_eq_top : f.liminf u = ⊤ ↔ u =ᶠ[f] ⊤ | @limsup_eq_bot αᵒᵈ _ _ _ _ _ _ _ _ | lemma | liminf_eq_top | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"limsup_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.map_Limsup_of_continuous_at {F : filter R} [ne_bot F]
{f : R → S} (f_decr : antitone f) (f_cont : continuous_at f (F.Limsup)) :
f (F.Limsup) = F.liminf f | begin
apply le_antisymm,
{ have A : {a : R | ∀ᶠ (n : R) in F, n ≤ a}.nonempty, from ⟨⊤, by simp⟩,
rw [Limsup, (f_decr.map_Inf_of_continuous_at' f_cont A)],
apply le_of_forall_lt,
assume c hc,
simp only [liminf, Liminf, lt_Sup_iff, eventually_map, set.mem_set_of_eq, exists_prop,
set.mem_image, ... | lemma | antitone.map_Limsup_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"antitone",
"bot_le",
"continuous_at",
"eq_or_lt_of_le",
"exists_Ioc_subset_of_mem_nhds",
"exists_exists_and_eq_and",
"exists_prop",
"filter",
"le_of_forall_lt",
"lt_Sup_iff",
"set.Ioc",
"set.Ioo",
"set.mem_image",
"set.not_nonempty_iff_eq_empty"
] | An antitone function between complete linear ordered spaces sends a `filter.Limsup`
to the `filter.liminf` of the image if it is continuous at the `Limsup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_limsup_of_continuous_at
{f : R → S} (f_decr : antitone f) (a : ι → R) (f_cont : continuous_at f (F.limsup a)) :
f (F.limsup a) = F.liminf (f ∘ a) | f_decr.map_Limsup_of_continuous_at f_cont | lemma | antitone.map_limsup_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"antitone",
"continuous_at"
] | A continuous antitone function between complete linear ordered spaces sends a `filter.limsup`
to the `filter.liminf` of the images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_Liminf_of_continuous_at {F : filter R} [ne_bot F]
{f : R → S} (f_decr : antitone f) (f_cont : continuous_at f (F.Liminf)) :
f (F.Liminf) = F.limsup f | @antitone.map_Limsup_of_continuous_at
(order_dual R) (order_dual S) _ _ _ _ _ _ _ _ f f_decr.dual f_cont | lemma | antitone.map_Liminf_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"antitone",
"antitone.map_Limsup_of_continuous_at",
"continuous_at",
"filter",
"order_dual"
] | An antitone function between complete linear ordered spaces sends a `filter.Liminf`
to the `filter.limsup` of the image if it is continuous at the `Liminf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.map_liminf_of_continuous_at
{f : R → S} (f_decr : antitone f) (a : ι → R) (f_cont : continuous_at f (F.liminf a)) :
f (F.liminf a) = F.limsup (f ∘ a) | f_decr.map_Liminf_of_continuous_at f_cont | lemma | antitone.map_liminf_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"antitone",
"continuous_at"
] | A continuous antitone function between complete linear ordered spaces sends a `filter.liminf`
to the `filter.limsup` of the images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_Limsup_of_continuous_at {F : filter R} [ne_bot F]
{f : R → S} (f_incr : monotone f) (f_cont : continuous_at f (F.Limsup)) :
f (F.Limsup) = F.limsup f | @antitone.map_Limsup_of_continuous_at R (order_dual S) _ _ _ _ _ _ _ _ f f_incr f_cont | lemma | monotone.map_Limsup_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"antitone.map_Limsup_of_continuous_at",
"continuous_at",
"filter",
"monotone",
"order_dual"
] | A monotone function between complete linear ordered spaces sends a `filter.Limsup`
to the `filter.limsup` of the image if it is continuous at the `Limsup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_limsup_of_continuous_at
{f : R → S} (f_incr : monotone f) (a : ι → R) (f_cont : continuous_at f (F.limsup a)) :
f (F.limsup a) = F.limsup (f ∘ a) | f_incr.map_Limsup_of_continuous_at f_cont | lemma | monotone.map_limsup_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"continuous_at",
"monotone"
] | A continuous monotone function between complete linear ordered spaces sends a `filter.limsup`
to the `filter.limsup` of the images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_Liminf_of_continuous_at {F : filter R} [ne_bot F]
{f : R → S} (f_incr : monotone f) (f_cont : continuous_at f (F.Liminf)) :
f (F.Liminf) = F.liminf f | @antitone.map_Liminf_of_continuous_at R (order_dual S) _ _ _ _ _ _ _ _ f f_incr f_cont | lemma | monotone.map_Liminf_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"antitone.map_Liminf_of_continuous_at",
"continuous_at",
"filter",
"monotone",
"order_dual"
] | A monotone function between complete linear ordered spaces sends a `filter.Liminf`
to the `filter.liminf` of the image if it is continuous at the `Liminf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.map_liminf_of_continuous_at
{f : R → S} (f_incr : monotone f) (a : ι → R) (f_cont : continuous_at f (F.liminf a)) :
f (F.liminf a) = F.liminf (f ∘ a) | f_incr.map_Liminf_of_continuous_at f_cont | lemma | monotone.map_liminf_of_continuous_at | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"continuous_at",
"monotone"
] | A continuous monotone function between complete linear ordered spaces sends a `filter.liminf`
to the `filter.liminf` of the images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_eq_of_forall_le_of_tendsto {x : R} {as : ι → R}
(x_le : ∀ i, x ≤ as i) {F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
(⨅ i, as i) = x | begin
refine infi_eq_of_forall_ge_of_forall_gt_exists_lt (λ i, x_le i) _,
apply λ w x_lt_w, ‹filter.ne_bot F›.nonempty_of_mem (eventually_lt_of_tendsto_lt x_lt_w as_lim),
end | lemma | infi_eq_of_forall_le_of_tendsto | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"eventually_lt_of_tendsto_lt",
"filter",
"filter.ne_bot",
"filter.tendsto",
"infi_eq_of_forall_ge_of_forall_gt_exists_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_eq_of_forall_le_of_tendsto {x : R} {as : ι → R}
(le_x : ∀ i, as i ≤ x) {F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
(⨆ i, as i) = x | @infi_eq_of_forall_le_of_tendsto ι (order_dual R) _ _ _ x as le_x F _ as_lim | lemma | supr_eq_of_forall_le_of_tendsto | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"filter",
"filter.ne_bot",
"filter.tendsto",
"infi_eq_of_forall_le_of_tendsto",
"order_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
{F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
(⋃ (i : ι), Ici (as i)) = Ioi x | begin
have obs : x ∉ range as,
{ intro maybe_x_is,
rcases mem_range.mp maybe_x_is with ⟨i, hi⟩,
simpa only [hi, lt_self_iff_false] using x_lt i, } ,
rw ← infi_eq_of_forall_le_of_tendsto (λ i, (x_lt i).le) as_lim at *,
exact Union_Ici_eq_Ioi_infi obs,
end | lemma | Union_Ici_eq_Ioi_of_lt_of_tendsto | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Union_Ici_eq_Ioi_infi",
"filter",
"filter.ne_bot",
"filter.tendsto",
"infi_eq_of_forall_le_of_tendsto",
"lt_self_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Iic_eq_Iio_of_lt_of_tendsto (x : R) {as : ι → R} (lt_x : ∀ i, as i < x)
{F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
(⋃ (i : ι), Iic (as i)) = Iio x | @Union_Ici_eq_Ioi_of_lt_of_tendsto ι Rᵒᵈ _ _ _ _ _ lt_x F _ as_lim | lemma | Union_Iic_eq_Iio_of_lt_of_tendsto | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"Union_Ici_eq_Ioi_of_lt_of_tendsto",
"filter",
"filter.ne_bot",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_eq_tendsto_sum_indicator_nat_at_top (s : ℕ → set α) :
limsup s at_top =
{ω | tendsto (λ n, ∑ k in finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω) at_top at_top} | begin
ext ω,
simp only [limsup_eq_infi_supr_of_nat, ge_iff_le, set.supr_eq_Union,
set.infi_eq_Inter, set.mem_Inter, set.mem_Union, exists_prop],
split,
{ intro hω,
refine tendsto_at_top_at_top_of_monotone' (λ n m hnm, finset.sum_mono_set_of_nonneg
(λ i, set.indicator_nonneg (λ _ _, zero_le_one) ... | lemma | limsup_eq_tendsto_sum_indicator_nat_at_top | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"by_contra",
"exists_prop",
"finset.Ico",
"finset.card_range",
"finset.range",
"finset.range_mono",
"forall_apply_eq_imp_iff'",
"forall_exists_index",
"ge_iff_le",
"imp_false",
"le_rfl",
"mem_upper_bounds",
"mul_one",
"set.infi_eq_Inter",
"set.mem_Inter",
"set.mem_Union",
"set.mem_ra... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limsup_eq_tendsto_sum_indicator_at_top
(R : Type*) [strict_ordered_semiring R] [archimedean R] (s : ℕ → set α) :
limsup s at_top =
{ω | tendsto (λ n, ∑ k in finset.range n, (s (k + 1)).indicator (1 : α → R) ω) at_top at_top} | begin
rw limsup_eq_tendsto_sum_indicator_nat_at_top s,
ext ω,
simp only [set.mem_set_of_eq],
rw (_ : (λ n, ∑ k in finset.range n, (s (k + 1)).indicator (1 : α → R) ω) =
(λ n, ↑(∑ k in finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω))),
{ exact tendsto_coe_nat_at_top_iff.symm },
{ ext n,
simp onl... | lemma | limsup_eq_tendsto_sum_indicator_at_top | topology.algebra.order | src/topology/algebra/order/liminf_limsup.lean | [
"algebra.big_operators.intervals",
"algebra.big_operators.order",
"algebra.indicator_function",
"order.liminf_limsup",
"order.filter.archimedean",
"order.filter.countable_Inter",
"topology.order.basic"
] | [
"archimedean",
"finset.range",
"finset.sum_boole",
"limsup_eq_tendsto_sum_indicator_nat_at_top",
"nat.cast_id",
"pi.one_apply",
"set.indicator",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.continuous_at_right_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≥] a)
(hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
continuous_within_at f (Ici a) a | begin
have ha : a ∈ Ici a := left_mem_Ici,
have has : a ∈ s := mem_of_mem_nhds_within ha hs,
refine tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩,
{ filter_upwards [hs, self_mem_nhds_within] with _ hxs hxa
using hb.trans_le ((h_mono.le_iff_le has hxs).2 hxa) },
{ rcases hfs b hb with ⟨c, hcs, hac, hcb⟩,
rw... | lemma | strict_mono_on.continuous_at_right_of_exists_between | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"Ico_mem_nhds_within_Ici",
"continuous_within_at",
"mem_of_mem_nhds_within",
"self_mem_nhds_within",
"strict_mono_on"
] | If `f` is a function strictly monotone on a right neighborhood of `a` and the
image of this neighborhood under `f` meets every interval `(f a, b]`, `b > f a`, then `f` is
continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` is required because otherwise the
function `f : ℝ → ℝ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_right_of_monotone_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : monotone_on f s) (hs : s ∈ 𝓝[≥] a)
(hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
continuous_within_at f (Ici a) a | begin
have ha : a ∈ Ici a := left_mem_Ici,
have has : a ∈ s := mem_of_mem_nhds_within ha hs,
refine tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩,
{ filter_upwards [hs, self_mem_nhds_within] with _ hxs hxa
using hb.trans_le (h_mono has hxs hxa) },
{ rcases hfs b hb with ⟨c, hcs, hac, hcb⟩,
have : a < c, fr... | lemma | continuous_at_right_of_monotone_on_of_exists_between | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"Ico_mem_nhds_within_Ici",
"continuous_within_at",
"mem_of_mem_nhds_within",
"monotone_on",
"self_mem_nhds_within"
] | If `f` is a monotone function on a right neighborhood of `a` and the image of this neighborhood
under `f` meets every interval `(f a, b)`, `b > f a`, then `f` is continuous at `a` from the right.
The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b` cannot be replaced by the weaker
assumption `hfs : ∀ b > f a, ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : monotone_on f s)
(hs : s ∈ 𝓝[≥] a) (hfs : closure (f '' s) ∈ 𝓝[≥] (f a)) :
continuous_within_at f (Ici a) a | begin
refine continuous_at_right_of_monotone_on_of_exists_between h_mono hs (λ b hb, _),
rcases (mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩,
rcases exists_between hab' with ⟨c', hc'⟩,
rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') is_open_Ioo hc'
... | lemma | continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"closure",
"continuous_at_right_of_monotone_on_of_exists_between",
"continuous_within_at",
"densely_ordered",
"exists_between",
"is_open_Ioo",
"mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset",
"monotone_on"
] | If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and
the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f`
is continuous at `a` from the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_right_of_monotone_on_of_image_mem_nhds_within [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : monotone_on f s) (hs : s ∈ 𝓝[≥] a)
(hfs : f '' s ∈ 𝓝[≥] (f a)) :
continuous_within_at f (Ici a) a | continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within h_mono hs $
mem_of_superset hfs subset_closure | lemma | continuous_at_right_of_monotone_on_of_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within",
"continuous_within_at",
"densely_ordered",
"monotone_on",
"subset_closure"
] | If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and
the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at
`a` from the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_right_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≥] a)
(hfs : closure (f '' s) ∈ 𝓝[≥] (f a)) :
continuous_within_at f (Ici a) a | continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within
(λ x hx y hy, (h_mono.le_iff_le hx hy).2) hs hfs | lemma | strict_mono_on.continuous_at_right_of_closure_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"closure",
"continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within",
"continuous_within_at",
"densely_ordered",
"strict_mono_on"
] | If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood
of `a` and the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`,
then `f` is continuous at `a` from the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_right_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≥] a)
(hfs : f '' s ∈ 𝓝[≥] (f a)) :
continuous_within_at f (Ici a) a | h_mono.continuous_at_right_of_closure_image_mem_nhds_within hs
(mem_of_superset hfs subset_closure) | lemma | strict_mono_on.continuous_at_right_of_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_within_at",
"densely_ordered",
"strict_mono_on",
"subset_closure"
] | If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood
of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is
continuous at `a` from the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_right_of_surj_on {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≥] a) (hfs : surj_on f s (Ioi (f a))) :
continuous_within_at f (Ici a) a | h_mono.continuous_at_right_of_exists_between hs $ λ b hb, let ⟨c, hcs, hcb⟩ := hfs hb in
⟨c, hcs, hcb.symm ▸ hb, hcb.le⟩ | lemma | strict_mono_on.continuous_at_right_of_surj_on | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_within_at",
"strict_mono_on"
] | If a function `f` is strictly monotone on a right neighborhood of `a` and the image of this
neighborhood under `f` includes `Ioi (f a)`, then `f` is continuous at `a` from the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_left_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≤] a)
(hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) :
continuous_within_at f (Iic a) a | h_mono.dual.continuous_at_right_of_exists_between hs $
λ b hb, let ⟨c, hcs, hcb, hca⟩ := hfs b hb in ⟨c, hcs, hca, hcb⟩ | lemma | strict_mono_on.continuous_at_left_of_exists_between | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_within_at",
"strict_mono_on"
] | If `f` is a strictly monotone function on a left neighborhood of `a` and the image of this
neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, then `f` is continuous at `a`
from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` is required because otherwise the
function `f : ℝ → ℝ` ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_left_of_monotone_on_of_exists_between {f : α → β} {s : set α} {a : α}
(hf : monotone_on f s) (hs : s ∈ 𝓝[≤] a)
(hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) :
continuous_within_at f (Iic a) a | @continuous_at_right_of_monotone_on_of_exists_between αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s a hf.dual hs $
λ b hb, let ⟨c, hcs, hcb, hca⟩ := hfs b hb in ⟨c, hcs, hca, hcb⟩ | lemma | continuous_at_left_of_monotone_on_of_exists_between | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at_right_of_monotone_on_of_exists_between",
"continuous_within_at",
"monotone_on"
] | If `f` is a monotone function on a left neighborhood of `a` and the image of this neighborhood
under `f` meets every interval `(b, f a)`, `b < f a`, then `f` is continuous at `a` from the left.
The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)` cannot be replaced by the weaker
assumption `hfs : ∀ b < f a, ∃ ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_left_of_monotone_on_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (hf : monotone_on f s)
(hs : s ∈ 𝓝[≤] a) (hfs : closure (f '' s) ∈ 𝓝[≤] (f a)) :
continuous_within_at f (Iic a) a | @continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within αᵒᵈ βᵒᵈ _ _ _ _ _ _ _
f s a hf.dual hs hfs | lemma | continuous_at_left_of_monotone_on_of_closure_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"closure",
"continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within",
"continuous_within_at",
"densely_ordered",
"monotone_on"
] | If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and
the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is
continuous at `a` from the left | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_left_of_monotone_on_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : monotone_on f s)
(hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] (f a)) :
continuous_within_at f (Iic a) a | continuous_at_left_of_monotone_on_of_closure_image_mem_nhds_within h_mono hs
(mem_of_superset hfs subset_closure) | lemma | continuous_at_left_of_monotone_on_of_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at_left_of_monotone_on_of_closure_image_mem_nhds_within",
"continuous_within_at",
"densely_ordered",
"monotone_on",
"subset_closure"
] | If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and
the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at
`a` from the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_left_of_closure_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≤] a)
(hfs : closure (f '' s) ∈ 𝓝[≤] (f a)) :
continuous_within_at f (Iic a) a | h_mono.dual.continuous_at_right_of_closure_image_mem_nhds_within hs hfs | lemma | strict_mono_on.continuous_at_left_of_closure_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"closure",
"continuous_within_at",
"densely_ordered",
"strict_mono_on"
] | If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of
`a` and the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`,
then `f` is continuous at `a` from the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_left_of_image_mem_nhds_within [densely_ordered β]
{f : α → β} {s : set α} {a : α} (h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≤] a)
(hfs : f '' s ∈ 𝓝[≤] (f a)) :
continuous_within_at f (Iic a) a | h_mono.dual.continuous_at_right_of_image_mem_nhds_within hs hfs | lemma | strict_mono_on.continuous_at_left_of_image_mem_nhds_within | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_within_at",
"densely_ordered",
"strict_mono_on"
] | If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of
`a` and the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is
continuous at `a` from the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_left_of_surj_on {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_on f s) (hs : s ∈ 𝓝[≤] a) (hfs : surj_on f s (Iio (f a))) :
continuous_within_at f (Iic a) a | h_mono.dual.continuous_at_right_of_surj_on hs hfs | lemma | strict_mono_on.continuous_at_left_of_surj_on | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_within_at",
"strict_mono_on"
] | If a function `f` is strictly monotone on a left neighborhood of `a` and the image of this
neighborhood under `f` includes `Iio (f a)`, then `f` is continuous at `a` from the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : strict_mono_on f s) (hs : s ∈ 𝓝 a)
(hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
continuous_at f a | continuous_at_iff_continuous_left_right.2
⟨h_mono.continuous_at_left_of_exists_between (mem_nhds_within_of_mem_nhds hs) hfs_l,
h_mono.continuous_at_right_of_exists_between (mem_nhds_within_of_mem_nhds hs) hfs_r⟩ | lemma | strict_mono_on.continuous_at_of_exists_between | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at",
"mem_nhds_within_of_mem_nhds",
"strict_mono_on"
] | If a function `f` is strictly monotone on a neighborhood of `a` and the image of this
neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, and every interval
`(f a, b]`, `b > f a`, then `f` is continuous at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_of_closure_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : strict_mono_on f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a | continuous_at_iff_continuous_left_right.2
⟨h_mono.continuous_at_left_of_closure_image_mem_nhds_within (mem_nhds_within_of_mem_nhds hs)
(mem_nhds_within_of_mem_nhds hfs),
h_mono.continuous_at_right_of_closure_image_mem_nhds_within (mem_nhds_within_of_mem_nhds hs)
(mem_nhds_within_of_mem_nhds hfs)⟩ | lemma | strict_mono_on.continuous_at_of_closure_image_mem_nhds | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"closure",
"continuous_at",
"densely_ordered",
"mem_nhds_within_of_mem_nhds",
"strict_mono_on"
] | If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a`
and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.continuous_at_of_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : strict_mono_on f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a | h_mono.continuous_at_of_closure_image_mem_nhds hs (mem_of_superset hfs subset_closure) | lemma | strict_mono_on.continuous_at_of_image_mem_nhds | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at",
"densely_ordered",
"strict_mono_on",
"subset_closure"
] | If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a`
and the image of this set under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_monotone_on_of_exists_between {f : α → β} {s : set α} {a : α}
(h_mono : monotone_on f s) (hs : s ∈ 𝓝 a)
(hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
continuous_at f a | continuous_at_iff_continuous_left_right.2
⟨continuous_at_left_of_monotone_on_of_exists_between h_mono
(mem_nhds_within_of_mem_nhds hs) hfs_l,
continuous_at_right_of_monotone_on_of_exists_between h_mono
(mem_nhds_within_of_mem_nhds hs) hfs_r⟩ | lemma | continuous_at_of_monotone_on_of_exists_between | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at",
"continuous_at_right_of_monotone_on_of_exists_between",
"mem_nhds_within_of_mem_nhds",
"monotone_on"
] | If `f` is a monotone function on a neighborhood of `a` and the image of this neighborhood under
`f` meets every interval `(b, f a)`, `b < f a`, and every interval `(f a, b)`, `b > f a`, then `f`
is continuous at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_monotone_on_of_closure_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : monotone_on f s) (hs : s ∈ 𝓝 a)
(hfs : closure (f '' s) ∈ 𝓝 (f a)) :
continuous_at f a | continuous_at_iff_continuous_left_right.2
⟨continuous_at_left_of_monotone_on_of_closure_image_mem_nhds_within h_mono
(mem_nhds_within_of_mem_nhds hs) (mem_nhds_within_of_mem_nhds hfs),
continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within h_mono
(mem_nhds_within_of_mem_nhds hs) (mem_nhds_wi... | lemma | continuous_at_of_monotone_on_of_closure_image_mem_nhds | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"closure",
"continuous_at",
"continuous_at_right_of_monotone_on_of_closure_image_mem_nhds_within",
"densely_ordered",
"mem_nhds_within_of_mem_nhds",
"monotone_on"
] | If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the
closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is
continuous at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_monotone_on_of_image_mem_nhds [densely_ordered β] {f : α → β}
{s : set α} {a : α} (h_mono : monotone_on f s) (hs : s ∈ 𝓝 a)
(hfs : f '' s ∈ 𝓝 (f a)) :
continuous_at f a | continuous_at_of_monotone_on_of_closure_image_mem_nhds h_mono hs
(mem_of_superset hfs subset_closure) | lemma | continuous_at_of_monotone_on_of_image_mem_nhds | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous_at",
"continuous_at_of_monotone_on_of_closure_image_mem_nhds",
"densely_ordered",
"monotone_on",
"subset_closure"
] | If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the
image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.continuous_of_dense_range [densely_ordered β] {f : α → β}
(h_mono : monotone f) (h_dense : dense_range f) :
continuous f | continuous_iff_continuous_at.mpr $ λ a,
continuous_at_of_monotone_on_of_closure_image_mem_nhds (λ x hx y hy hxy, h_mono hxy)
univ_mem $ by simp only [image_univ, h_dense.closure_eq, univ_mem] | lemma | monotone.continuous_of_dense_range | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous",
"continuous_at_of_monotone_on_of_closure_image_mem_nhds",
"dense_range",
"densely_ordered",
"monotone"
] | A monotone function with densely ordered codomain and a dense range is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.continuous_of_surjective [densely_ordered β] {f : α → β} (h_mono : monotone f)
(h_surj : function.surjective f) :
continuous f | h_mono.continuous_of_dense_range h_surj.dense_range | lemma | monotone.continuous_of_surjective | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous",
"densely_ordered",
"monotone"
] | A monotone surjective function with a densely ordered codomain is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (e : α ≃o β) : continuous e | begin
rw [‹order_topology β›.topology_eq_generate_intervals],
refine continuous_generated_from (λ s hs, _),
rcases hs with ⟨a, rfl|rfl⟩,
{ rw e.preimage_Ioi, apply is_open_lt' },
{ rw e.preimage_Iio, apply is_open_gt' }
end | lemma | order_iso.continuous | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [
"continuous",
"continuous_generated_from",
"is_open_gt'",
"is_open_lt'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph (e : α ≃o β) : α ≃ₜ β | { continuous_to_fun := e.continuous,
continuous_inv_fun := e.symm.continuous,
.. e } | def | order_iso.to_homeomorph | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [] | An order isomorphism between two linear order `order_topology` spaces is a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_homeomorph (e : α ≃o β) : ⇑e.to_homeomorph = e | rfl | lemma | order_iso.coe_to_homeomorph | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_homeomorph_symm (e : α ≃o β) : ⇑e.to_homeomorph.symm = e.symm | rfl | lemma | order_iso.coe_to_homeomorph_symm | topology.algebra.order | src/topology/algebra/order/monotone_continuity.lean | [
"topology.order.basic",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_convergence_class (α : Type*) [preorder α] [topological_space α] : Prop | (tendsto_coe_at_top_is_lub : ∀ (a : α) (s : set α), is_lub s a → tendsto (coe : s → α) at_top (𝓝 a)) | class | Sup_convergence_class | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"is_lub",
"topological_space"
] | We say that `α` is a `Sup_convergence_class` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a least upper bound of `set.range f`. Then `f x` tends to `𝓝 a` as
`x → ∞` (formally, at the filter `filter.at_top`). We require this for `ι = (s : set α)`, `f = coe`
in the definition, then prov... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Inf_convergence_class (α : Type*) [preorder α] [topological_space α] : Prop | (tendsto_coe_at_bot_is_glb : ∀ (a : α) (s : set α), is_glb s a → tendsto (coe : s → α) at_bot (𝓝 a)) | class | Inf_convergence_class | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"is_glb",
"topological_space"
] | We say that `α` is an `Inf_convergence_class` if the following holds. Let `f : ι → α` be a
monotone function, let `a : α` be a greatest lower bound of `set.range f`. Then `f x` tends to `𝓝 a`
as `x → -∞` (formally, at the filter `filter.at_bot`). We require this for `ι = (s : set α)`,
`f = coe` in the definition, then... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_dual.Sup_convergence_class [preorder α] [topological_space α]
[Inf_convergence_class α] : Sup_convergence_class αᵒᵈ | ⟨‹Inf_convergence_class α›.1⟩ | instance | order_dual.Sup_convergence_class | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"Inf_convergence_class",
"Sup_convergence_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_dual.Inf_convergence_class [preorder α] [topological_space α]
[Sup_convergence_class α] : Inf_convergence_class αᵒᵈ | ⟨‹Sup_convergence_class α›.1⟩ | instance | order_dual.Inf_convergence_class | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"Inf_convergence_class",
"Sup_convergence_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_order.Sup_convergence_class [topological_space α] [linear_order α]
[order_topology α] : Sup_convergence_class α | begin
refine ⟨λ a s ha, tendsto_order.2 ⟨λ b hb, _, λ b hb, _⟩⟩,
{ rcases ha.exists_between hb with ⟨c, hcs, bc, bca⟩,
lift c to s using hcs,
refine (eventually_ge_at_top c).mono (λ x hx, bc.trans_le hx) },
{ exact eventually_of_forall (λ x, (ha.1 x.2).trans_lt hb) }
end | instance | linear_order.Sup_convergence_class | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"Sup_convergence_class",
"lift",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_order.Inf_convergence_class [topological_space α] [linear_order α]
[order_topology α] : Inf_convergence_class α | show Inf_convergence_class αᵒᵈᵒᵈ, from order_dual.Inf_convergence_class | instance | linear_order.Inf_convergence_class | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"Inf_convergence_class",
"order_dual.Inf_convergence_class",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_is_lub (h_mono : monotone f) (ha : is_lub (set.range f) a) :
tendsto f at_top (𝓝 a) | begin
suffices : tendsto (range_factorization f) at_top at_top,
from (Sup_convergence_class.tendsto_coe_at_top_is_lub _ _ ha).comp this,
exact h_mono.range_factorization.tendsto_at_top_at_top (λ b, b.2.imp $ λ a ha, ha.ge)
end | lemma | tendsto_at_top_is_lub | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"is_lub",
"monotone",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_is_lub (h_anti : antitone f) (ha : is_lub (set.range f) a) :
tendsto f at_bot (𝓝 a) | by convert tendsto_at_top_is_lub h_anti.dual_left ha | lemma | tendsto_at_bot_is_lub | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"is_lub",
"set.range",
"tendsto_at_top_is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_is_glb (h_mono : monotone f) (ha : is_glb (set.range f) a) :
tendsto f at_bot (𝓝 a) | by convert tendsto_at_top_is_lub h_mono.dual ha.dual | lemma | tendsto_at_bot_is_glb | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"is_glb",
"monotone",
"set.range",
"tendsto_at_top_is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_is_glb (h_anti : antitone f) (ha : is_glb (set.range f) a) :
tendsto f at_top (𝓝 a) | by convert tendsto_at_bot_is_lub h_anti.dual ha.dual | lemma | tendsto_at_top_is_glb | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"is_glb",
"set.range",
"tendsto_at_bot_is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_csupr (h_mono : monotone f) (hbdd : bdd_above $ range f) :
tendsto f at_top (𝓝 (⨆i, f i)) | begin
casesI is_empty_or_nonempty ι,
exacts [tendsto_of_is_empty, tendsto_at_top_is_lub h_mono (is_lub_csupr hbdd)]
end | lemma | tendsto_at_top_csupr | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"bdd_above",
"is_empty_or_nonempty",
"is_lub_csupr",
"monotone",
"tendsto_at_top_is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_csupr (h_anti : antitone f) (hbdd : bdd_above $ range f) :
tendsto f at_bot (𝓝 (⨆ i, f i)) | by convert tendsto_at_top_csupr h_anti.dual hbdd.dual | lemma | tendsto_at_bot_csupr | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"bdd_above",
"tendsto_at_top_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_cinfi (h_mono : monotone f) (hbdd : bdd_below $ range f) :
tendsto f at_bot (𝓝 (⨅ i, f i)) | by convert tendsto_at_top_csupr h_mono.dual hbdd.dual | lemma | tendsto_at_bot_cinfi | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"bdd_below",
"monotone",
"tendsto_at_top_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_cinfi (h_anti : antitone f) (hbdd : bdd_below $ range f) :
tendsto f at_top (𝓝 (⨅ i, f i)) | by convert tendsto_at_bot_csupr h_anti.dual hbdd.dual | lemma | tendsto_at_top_cinfi | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"bdd_below",
"tendsto_at_bot_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_supr (h_mono : monotone f) : tendsto f at_top (𝓝 (⨆i, f i)) | tendsto_at_top_csupr h_mono (order_top.bdd_above _) | lemma | tendsto_at_top_supr | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"monotone",
"order_top.bdd_above",
"tendsto_at_top_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_supr (h_anti : antitone f) :
tendsto f at_bot (𝓝 (⨆i, f i)) | tendsto_at_bot_csupr h_anti (order_top.bdd_above _) | lemma | tendsto_at_bot_supr | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"order_top.bdd_above",
"tendsto_at_bot_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_infi (h_mono : monotone f) : tendsto f at_bot (𝓝 (⨅i, f i)) | tendsto_at_bot_cinfi h_mono (order_bot.bdd_below _) | lemma | tendsto_at_bot_infi | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"monotone",
"order_bot.bdd_below",
"tendsto_at_bot_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_infi (h_anti : antitone f) :
tendsto f at_top (𝓝 (⨅i, f i)) | tendsto_at_top_cinfi h_anti (order_bot.bdd_below _) | lemma | tendsto_at_top_infi | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"order_bot.bdd_below",
"tendsto_at_top_cinfi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.Sup_convergence_class' {ι : Type*} [preorder α] [topological_space α]
[Sup_convergence_class α] : Sup_convergence_class (ι → α) | pi.Sup_convergence_class | instance | pi.Sup_convergence_class' | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"Sup_convergence_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.Inf_convergence_class' {ι : Type*} [preorder α] [topological_space α]
[Inf_convergence_class α] : Inf_convergence_class (ι → α) | pi.Inf_convergence_class | instance | pi.Inf_convergence_class' | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"Inf_convergence_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_of_monotone {ι α : Type*} [preorder ι] [topological_space α]
[conditionally_complete_linear_order α] [order_topology α] {f : ι → α} (h_mono : monotone f) :
tendsto f at_top at_top ∨ (∃ l, tendsto f at_top (𝓝 l)) | if H : bdd_above (range f) then or.inr ⟨_, tendsto_at_top_csupr h_mono H⟩
else or.inl $ tendsto_at_top_at_top_of_monotone' h_mono H | lemma | tendsto_of_monotone | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"bdd_above",
"conditionally_complete_linear_order",
"monotone",
"order_topology",
"tendsto_at_top_csupr",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_tendsto_subseq_of_monotone {ι₁ ι₂ α : Type*} [semilattice_sup ι₁] [preorder ι₂]
[nonempty ι₁] [topological_space α] [conditionally_complete_linear_order α] [order_topology α]
[no_max_order α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : monotone f)
(hg : tendsto φ at_top at_top) :
tendsto f at_top (𝓝 l... | begin
split; intro h,
{ exact h.comp hg },
{ rcases tendsto_of_monotone hf with h' | ⟨l', hl'⟩,
{ exact (not_tendsto_at_top_of_tendsto_nhds h (h'.comp hg)).elim },
{ rwa tendsto_nhds_unique h (hl'.comp hg) } }
end | lemma | tendsto_iff_tendsto_subseq_of_monotone | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"conditionally_complete_linear_order",
"monotone",
"no_max_order",
"not_tendsto_at_top_of_tendsto_nhds",
"order_topology",
"semilattice_sup",
"tendsto_nhds_unique",
"tendsto_of_monotone",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.ge_of_tendsto [topological_space α] [preorder α] [order_closed_topology α]
[semilattice_sup β] {f : β → α} {a : α} (hf : monotone f)
(ha : tendsto f at_top (𝓝 a)) (b : β) :
f b ≤ a | begin
haveI : nonempty β := nonempty.intro b,
exact ge_of_tendsto ha ((eventually_ge_at_top b).mono (λ _ hxy, hf hxy))
end | lemma | monotone.ge_of_tendsto | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"ge_of_tendsto",
"monotone",
"order_closed_topology",
"semilattice_sup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.le_of_tendsto [topological_space α] [preorder α] [order_closed_topology α]
[semilattice_inf β] {f : β → α} {a : α} (hf : monotone f)
(ha : tendsto f at_bot (𝓝 a)) (b : β) :
a ≤ f b | hf.dual.ge_of_tendsto ha b | lemma | monotone.le_of_tendsto | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"monotone",
"order_closed_topology",
"semilattice_inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.le_of_tendsto [topological_space α] [preorder α] [order_closed_topology α]
[semilattice_sup β] {f : β → α} {a : α} (hf : antitone f)
(ha : tendsto f at_top (𝓝 a)) (b : β) :
a ≤ f b | hf.dual_right.ge_of_tendsto ha b | lemma | antitone.le_of_tendsto | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"order_closed_topology",
"semilattice_sup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone.ge_of_tendsto [topological_space α] [preorder α] [order_closed_topology α]
[semilattice_inf β] {f : β → α} {a : α} (hf : antitone f)
(ha : tendsto f at_bot (𝓝 a)) (b : β) :
f b ≤ a | hf.dual_right.le_of_tendsto ha b | lemma | antitone.ge_of_tendsto | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"order_closed_topology",
"semilattice_inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_of_tendsto_at_top [topological_space α] [preorder α] [order_closed_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : monotone f)
(ha : tendsto f at_top (𝓝 a)) :
is_lub (set.range f) a | begin
split,
{ rintros _ ⟨b, rfl⟩,
exact hf.ge_of_tendsto ha b },
{ exact λ _ hb, le_of_tendsto' ha (λ x, hb (set.mem_range_self x)) }
end | lemma | is_lub_of_tendsto_at_top | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"is_lub",
"le_of_tendsto'",
"monotone",
"order_closed_topology",
"semilattice_sup",
"set.mem_range_self",
"set.range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb_of_tendsto_at_bot [topological_space α] [preorder α] [order_closed_topology α]
[nonempty β] [semilattice_inf β] {f : β → α} {a : α} (hf : monotone f)
(ha : tendsto f at_bot (𝓝 a)) :
is_glb (set.range f) a | @is_lub_of_tendsto_at_top αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ hf.dual ha | lemma | is_glb_of_tendsto_at_bot | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"is_glb",
"is_lub_of_tendsto_at_top",
"monotone",
"order_closed_topology",
"semilattice_inf",
"set.range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_of_tendsto_at_bot [topological_space α] [preorder α] [order_closed_topology α]
[nonempty β] [semilattice_inf β] {f : β → α} {a : α} (hf : antitone f)
(ha : tendsto f at_bot (𝓝 a)) :
is_lub (set.range f) a | @is_lub_of_tendsto_at_top α βᵒᵈ _ _ _ _ _ _ _ hf.dual_left ha | lemma | is_lub_of_tendsto_at_bot | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"is_lub",
"is_lub_of_tendsto_at_top",
"order_closed_topology",
"semilattice_inf",
"set.range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb_of_tendsto_at_top [topological_space α] [preorder α] [order_closed_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : antitone f)
(ha : tendsto f at_top (𝓝 a)) :
is_glb (set.range f) a | @is_glb_of_tendsto_at_bot α βᵒᵈ _ _ _ _ _ _ _ hf.dual_left ha | lemma | is_glb_of_tendsto_at_top | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"is_glb",
"is_glb_of_tendsto_at_bot",
"order_closed_topology",
"semilattice_sup",
"set.range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_eq_of_tendsto {α β} [topological_space α] [complete_linear_order α] [order_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : monotone f) :
tendsto f at_top (𝓝 a) → supr f = a | tendsto_nhds_unique (tendsto_at_top_supr hf) | lemma | supr_eq_of_tendsto | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"complete_linear_order",
"monotone",
"order_topology",
"semilattice_sup",
"supr",
"tendsto_at_top_supr",
"tendsto_nhds_unique",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_eq_of_tendsto {α} [topological_space α] [complete_linear_order α] [order_topology α]
[nonempty β] [semilattice_sup β] {f : β → α} {a : α} (hf : antitone f) :
tendsto f at_top (𝓝 a) → infi f = a | tendsto_nhds_unique (tendsto_at_top_infi hf) | lemma | infi_eq_of_tendsto | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"antitone",
"complete_linear_order",
"infi",
"order_topology",
"semilattice_sup",
"tendsto_at_top_infi",
"tendsto_nhds_unique",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_eq_supr_subseq_of_monotone {ι₁ ι₂ α : Type*} [preorder ι₂] [complete_lattice α]
{l : filter ι₁} [l.ne_bot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : monotone f)
(hφ : tendsto φ l at_top) :
(⨆ i, f i) = (⨆ i, f (φ i)) | le_antisymm
(supr_mono' $ λ i, exists_imp_exists (λ j (hj : i ≤ φ j), hf hj)
(hφ.eventually $ eventually_ge_at_top i).exists)
(supr_mono' $ λ i, ⟨φ i, le_rfl⟩) | lemma | supr_eq_supr_subseq_of_monotone | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"complete_lattice",
"filter",
"monotone",
"supr_mono'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_eq_infi_subseq_of_monotone {ι₁ ι₂ α : Type*} [preorder ι₂] [complete_lattice α]
{l : filter ι₁} [l.ne_bot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : monotone f)
(hφ : tendsto φ l at_bot) :
(⨅ i, f i) = (⨅ i, f (φ i)) | supr_eq_supr_subseq_of_monotone hf.dual hφ | lemma | infi_eq_infi_subseq_of_monotone | topology.algebra.order | src/topology/algebra/order/monotone_convergence.lean | [
"topology.order.basic"
] | [
"complete_lattice",
"filter",
"monotone",
"supr_eq_supr_subseq_of_monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.Icc_extend (f : γ → Icc a b → β) {z : γ} {l : filter α} {l' : filter β}
(hf : tendsto ↿f (𝓝 z ×ᶠ l.map (proj_Icc a b h)) l') :
tendsto ↿(Icc_extend h ∘ f) (𝓝 z ×ᶠ l) l' | show tendsto (↿f ∘ prod.map id (proj_Icc a b h)) (𝓝 z ×ᶠ l) l', from
hf.comp $ tendsto_id.prod_map tendsto_map | lemma | filter.tendsto.Icc_extend | topology.algebra.order | src/topology/algebra/order/proj_Icc.lean | [
"data.set.intervals.proj_Icc",
"topology.order.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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